Ultra thin ZnO on metal substrates [original]

Ultra thin ZnO on metal substrates

An ab initio study

vorgelegt von

Dipl.-Phys.

Björn Bieniek

Köthen/Anhalt

Von der Fakultät II – Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. rer. nat. Mario Dähne

1. Gutachter: Prof. Dr. rer. nat. Andreas Knorr

2. Gutachter: Prof. Dr. rer. nat. Patrick Rinke

Tag der wissenschaftlichen Aussprache: 12. Januar 2016

Berlin– 2016

ABSTRACT

In the context of catalysis and hybrid inorganic/organic systems, metal sup-

ported ultra-thin ZnO can be used as model systems within the surface science

approach. However, it is not clear to what degree the ZnO films resemble the

surfaces of ZnO or whether they exhibit significantly different properties.

To this purpose we investigate the structure of ultra-thin ZnO films (1to 4layers)

on the (111) surfaces of Ag, Cu, Ni, Rh, Pd, and Pt by means of density-functional

theory. The free-standing ZnO mono-layer adopts an α-BN structure. This struc-

ture prevails on the metal substrates, and we obtain coincidence structures in

good agreement with experiment. Thicker ZnO layers, preferentially grow in the

c-direction and adopt a wurtzite structure. The films exhibit a large random cor-

rugation. The electronic structure is modified by the intrinsic polar nature of ZnO.

The induced field leads to an upwards shift of the surface electronic states rela-

tive to the Fermi level with increasing film thickness. The states are eventually

pinned at the Fermi level provided by the metal substrate, leading to an effective

p-type doping of the thin film surface. Metal supported ultra-thin films might

thus be a way to achieve the much coveted p-type doping of ZnO.

To investigate the thermodynamic stability of ZnO on the metal substrates we

take the chemical potentials of Zn, O2, H2, and H2O environments into consider-

ation by means of ab initio thermodynamics. Up to three layers of unreconstructed

ZnO are predicted to be stable on the metal substrates at experimentally accessi-

ble pressure ranges. A hydrogen over-layer with 50% coverage is formed at chem-

ical potentials that range from low vacuum to ultra-high vacuum H2pressures.

The surface structure and the density of states of these hydrogen passivated ZnO

thin films agree well with those of the ZnO (000¯

1)–(2×1)–H surface.

ZnO ultra-thin films without hydrogen exhibit interesting and unique properties,

whereas, the (2×1)–H reconstructed films can be regarded as good ZnO surface

models. The hydrogen chemical potential provides a handle to select and poten-

tially switch between those two surface terminations.

We also include OH-reconstructions, Zn/O defects and sparse ring structures in

our approach to draw up an extensive surface phase diagram that will guide

future work on ultra-thin ZnO on metal substrates. For example, our calcula-

tions allow us to identify the ultra-thin films grown by Shiotari et al. on Ag as

hydrogen-free ZnO double-layers.

In a joined effort with the group of A. Knorr at TU Berlin, “hybrid Bloch equa-

tions” based on a density-matrix formalism are parametrized by ab initio results.

The parameterizations are based on dipole matrix elements and partial charges

fitted to the electrostatic potential. Both methods were implemented as part of

this Phd into the FHI-aims code to investigate exitonic process at ZnO/organic

interfaces. The interaction of substrate and molecule was found to be strongest at

low coverages and for a parallel alignment between the molecule and the surface

dipole.

iii

ZUSAMMENFASSUNG

In der heterogenen Katalyse und für hybride in-organische/organische Systeme

können ultra-dünne ZnO Filme auf Metallen als Modell-System genutzt werden.

Allerdings ist nicht geklärt ob diese Filme ZnO Oberflächen repräsentieren oder

ob sie abweichende und damit für sie einzigartige Eigenschaften aufweisen.

Um diesen Sachverhalt zu klären untersuchen wir ultra-dünne ZnO Filme (1bis

4Lagen) auf den (111) Oberflächen von Ag, Cu, Ni, Rh, Pd, und Pt mit Hilfe von

Dichtefunktionaltheorie. Der freistehende, einlagige ZnO Film weist eine α-BN

Struktur auf. Diese Struktur bleibt auf den Metallsubstraten zunächst erhalten

und wir können Koinzidenz-Strukturen in guter Übereinstimmung mit experi-

mentellen Resultaten bestimmen. Dickere Filme nehmen eine Wurtzit-Struktur

an. Die Filme weisen eine mit der Dicke zunehmende Korrugation auf. Die elek-

tronische Struktur wird durch die intrinsische, polare Natur von ZnO modifiziert.

Das induzierte elektrische Feld verschiebt die elektronischen Zustände an der

Oberfläche mit zunehmender Filmdicke zu höheren Energien. Die Zustände wer-

den schließlich am Fermi-Niveau des Metalls verankert. Dies führt zu einer ef-

fektiven p-Dotierung der Oberfläche des Dünnfilms.

Während ultra-dünne ZnO-Filme ohne Wasserstoff interessante und einzigar-

tige Eigenschaften aufweisen, können (2×1)–H rekonstruierte Filme zur Ober-

flächenmodellierung genutzt werden. Das chemische Potential des Wasserstoffes

fungiert hierbei als Schalter zwischen beiden Oberflächenterminierungen.

Die Stabilität der ZnO-Filme auf Metallsubstraten wurde unter Berücksichtigung

des chemischen Potentials von Zn, O2, H2und H2O mit Hilfe von ab initio

Thermodynamik analysiert. Bis zu drei Lagen ZnO können auf den Metallsub-

straten stabilisiert werden, bei experimentell realisierbaren Umgebungsbedin-

gungen. Eine zusätzliche Absättigung der ZnO Oberfläche mit einer 50%igen

Wasserstoffbedeckung ist bei chemischem Potentialen stabil die sich von niedri-

gen bis zu ultra-hoch-Vakuumbedingungen erstrecken. Die Oberflächenstruktur

und die Zustandsdichte der passivierten ZnO Filme zeigen eine große Überein-

stimmung mit der ZnO (000¯

1)–(2×1)–H Oberfläche.

Die Hinzunahme von OH-Rekonstruktionen, Zn/O Defekten und Ringstrukturen

führt zu einem umfangreichen Phasendiagramm, dass zukünftige Untersuchun-

gen von ZnO auf Metalloberflächen erleichtern wird. Zum Beispiel erlauben es

unsere Resultate die von Shiotari et al. gewachsenen Dünnfilme auf Ag als H-

freie ZnO Doppellagen zu identifizieren.

In enger Kooperation mit der AG Knorr an der TU Berlin konnten die „Hybrid

Bloch Gleichungen“ mit Hilfe von ab initio Rechnungen parametrisiert werden.

Die Parametrisierung basiert auf Dipolmatrixelementen und Partialladungen, die

an das elektrostatische Potential angepasst werden. Beide Methoden wurden im

Rahmen dieser Arbeit in FHI-aims implementiert um exitonische Prozesse zwis-

chen ZnO und organischen Molekülen zu untersuchen. Die Wechselwirkung

zwischen Substrat und Molekül ist stärker bei geringen Bedeckungen und für

parallel ausgerichtete Molekül- und Oberflächendipole.

CONTENTS

iintroduction 1

ii theoretical concepts to characterize surfaces and de-

fects 7

1 theoretical framework 9

1.1The many-body problem 9

1.2The Born-Oppenheimer approximation 10

1.3Density Functional Theory 12

1.4Approximate xc-functionals 14

1.4.1Local density approximation (LDA) 14

1.4.2Semi-Local density approximation 16

1.4.3Hybrid functionals 17

1.5Long range dispersion 18

1.6Ab initio thermodynamics 20

1.7DFT as implemented in FHI-aims 26

1.8Calculating surface vibrations 28

1.9Simulating STM apparent height maps 30

iii zno on metal substrates 33

2 introduction 35

3 bulk zno 37

4 surfaces of zno 41

5 freestanding zno thin films 47

5.1Mechanical stability 48

5.2Transition mono-layer to bulk 48

5.2.1α-BN bulk ZnO 50

6 metal supported mono-layer films 53

6.1Coincidence structures 54

6.2Position in the 1x1surface unit cell 59

6.3Corrugation 59

7 metal supported multi-layer zno films 63

7.1Atomic structure of multiple layers of ZnO 63

7.2Electronic structure of multiple layers of ZnO 65

8 thermodynamic stability of zno on metal substrates 73

8.1Hydrogen adsorption 73

8.1.1Electronic structure of the (2x1)–H reconstruction. 78

8.2Vacancies and ad-atoms 82

8.2.1O vacancies and ad-atoms 83

8.2.2Zinc defects and ad-atoms 85

8.2.3Oxygen and zinc defects 87

8.3Ring structures 89

8.4Hydroxyl absorption 91

vi contents

9 zno on ag in experiment and theory 103

9.1Apparent heights in experiment 104

9.2Results from DFT simulations 105

9.3Conclusion 108

10 special surfaces 109

10.1ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦109

10.2Rh (100)114

11 parameterization of density matrix formalism by dft 119

11.1Linking to Density-Matrix Theory 120

11.1.1Dipole Approximation 123

11.1.2Partial charge approximation 124

11.2Results 129

iv conclusion 137

vappendix 143

a constants 145

b reference energies 147

c convergence tests 149

d coincidence structures 153

e convergence with number of metal layers 157

f corrugation maps 159

g structure of multiple zno layers 161

h layer resolved corrugation maps 163

i electronic structure of multi-layer systems 169

j electronic structure of (2x1)–h systems 171

k layer heights for zno on ag (111)175

l current dependency of apparent heights 177

m defects in ideal α-bn zno mono layer 179

n charge density differences 183

publications i

colophon iii

bibliography v

Part I

INTRODUCTION

The advances of semi conductor physics and surface science have changed

the world in the last 100 years and lead to many technological applications

that are integral part of our daily lives e. g. computers, flat and touch screens.

One material, that is present in these developments since the beginning is ZnO

[125]. It is not as prominent as Si or Ge, but as widely abundant, transparent,

wide band gap semi-conductor has many applications e. g. in photo-voltaics

[287,103,141,289] or for laser diodes [264]. In this work it is of particular inter-

est in the context of hybrid inorganic/organic systems (HIOS) for optoelectronic

devices [137,281,233,139]. Organic molecules and related compounds have al-

ready shown their potential e. g. for the application as OLEDs (organic light

emitting diodes) [236,43,235,90,212], that even found their way into consumer

electronics. Because of the low charge carrier mobilities in organic semi conduc-

tors [135,80,38,256], these applications are mostly in areas that only require

low device performances. A second issue are chemical reactions with metal elec-

trodes [107,37] and the general stability under realistic environmental conditions

[115]. To realize devices, within the concepts established for inorganic semicon-

ductors, the injection of electrons from one side and holes from the other side,

transport of the charges and capture on the same molecule (as recombination

center for radiative decay) are required [205]. Hybrid devices offer these features

by combining the stability and the high charge carrier mobility of the inorganic,

semi-conducting component with the chemical flexibility and strong light-matter

coupling of the organic component. Especially resonant interactions such as effec-

tive non-radiative Förster energy transfer [82,46,34,3] between both components

are of interest. We can combine the best of two worlds: organic compounds and

inorganic semi conductors.

Another important application of ZnO lies in the field of heterogeneous cataly-

sis [184]. ZnO is required for the conversion of syn-gas in methanol production

as catalyst [19]. This process captures CO2and produces methanol as a sustain-

able, environmentally friendly energy source. Here, ZnO supported metal nano-

particles are used in industry as high performance catalysts [25], but the reaction

mechanism on the atomic scale is still under debate [95,283].

For hybrid inorganic/organic systems and heterogeneous catalysis alike, metal

supported ultra-thin oxide films have been proposed as model systems to un-

derstand the interface structure and its chemistry, because they facilitate the

application of the standard tool set of surface science, such as photo-electron

spectroscopy, and scanning tunneling microscopy, and prevent charging effects

[87,86]. However, some ultra-thin films exhibit their own interesting properties

[88,189,234,181], that differ from bulk materials. For example, the formation

of a graphitic ZnOxspecies was suggested and experimentally observed in the

vicinity of metal nano-particles [175,283]. It is thus not clear to what degree ultra-

thin metal-supported ZnO films resemble the surfaces of ZnO or whether they

exhibit significantly different properties. For ZnO ultra thin films a multitude of

different polymorphs has been theoretically predicted and some even experimen-

tally observed [83,169,170,65].

Under ambient conditions ZnO adopts a wurtzite structure [1]. Other bulk poly-

morphs show equally interesting properties [228,118]. In this work we will focus

on the surfaces of wurtzite ZnO. The most common low index surfaces are the

(10¯

10), (1120), (0001) and (000¯

1) surfaces. The discussion will mostly be restricted

to the polar (0001) and (000¯

1) surfaces. Only in the final chapter the (10¯

10) surface

is used. The advances in epitaxial thin film growth [188] and surface preparation

[93] make it possible to produce ZnO surfaces under highly controlled conditions

[276]. However, the structure of the polar (000¯

1) and (0001) surfaces, that are often

used in hybrid systems [40,5], is not fully understood [132,161,276,262,136,267]

hampering further quantitative interface studies.

To answer these questions and to characterize ultra-thin metal supported ZnO

films as model systems for HIOS and heterogeneous catalysis, we performed

density-functional theory (DFT) calculations for 1to 4layers ZnO films on vari-

ous transition metal substrates. The theoretical insight gained in this work leads

to predictions concerning the structure, thermodynamical stability and electronic

structure of ZnO thin films on metal substrates. These predictions can guide ex-

perimentalists in preparing and understanding their samples to accurately model

surfaces, interfaces and catalysts. The results of this work may help to create bet-

ter devices in opto-electronics and to optimize chemical process on an industrial

scale. We will focus on geometrical properties, e. g. the corrugation of the surface,

and electronic properties, e. g. the density of states. Similarities and differences

between ZnO thin films and bulk surfaces will be pointed out. Furthermore, the

differences in the atomic structure and thermodynamic stability of pristine and

reconstructed ultra-thin ZnO films on the transition metals: Ag, Cu, Pd, Pt, Ni,

and Rh are investigated. The choice of the metal offers an additional degree of

freedom in the selection of the model system.

In Chap 1of this thesis we introduce the theoretical concepts and techniques re-

quired to achieve the broad spectrum of goals we have set out to achieve. We will

shortly lay out the fundamentals of density-functional theory (DFT). For the prac-

tical application we used the Fritz Haber Institute ab initio simulation package

(FHI-aims) [33]. The basics of the code are explained in Sec. 1.7, while the imple-

mentations for the code we performed for the description of Föerster processes

in HIOS (collaboration with the group of A. Knorr, TU Berlin) are deferred to

Chap. 11. Based on DFT total energy calculations, we explain the construction of

reliable surface phase diagrams in Sec. 1.6and 1.8. The theoretical methods part

concludes with the description of simulating STM graphs, to directly compare

theory and experiment (Sec. 1.9).

To understand ultra-thin ZnO on metal substrates, the properties of bulk ZnO

and its most prominent surfaces known from literature are discussed at the be-

ginning of part II in Sec. 3and 4. We first investigate freestanding ZnO films

without a metal substrate in Sec. 5. Previous results [51,83,258] and calcula-

tions in this work demonstrate that it adopts an α-BN structure in analogy to

graphene. Thicker films change to the wurtzite structure after 3-4layers [255]

(see Sec. 5.2). To obtain a stable combination of metal and ZnO mono-layer we

address the lattice mismatch between the two constituents in Sec. 6and obtain

stable coincidence structures in good agreement with experimental results.

The observed differences in the atomic structure cannot only be explained by the

strain that is introduced when ZnO is brought into contact with transition met-

als of different lattice constant. The role of the electron reservoir provided by the

metal is therefore investigated in Sec. 7.2. We show that the potential difference

between the top and the bottom layer of the film, resulting from the intrinsic

dipole of ZnO (000¯

1), shifts the electronic states of ZnO upwards in energy until

they are pinned at the Fermi level of the metal. This can be understood as effec-

tive p-type doping of the ultra-thin films.

In the context of previous results on the stability of the polar ZnO bulk sur-

faces [132,267] the stability of the mono- and multi-layer ZnO films is addressed

in Chap. 8. The films can be stabilized by the mechanisms valid for both the

Zn and O-terminated bulk surface. The exact environmental conditions deter-

mine if the films are O- or Zn- terminated. For increased H2partial pressures

the surface bonds are saturated by hydrogen, and the geometric and electronic

structure resembles that of bulk terminated ZnO (000¯

1) with a (2×1) hydrogen

over layer (ZnO (000¯

1)–(2×1)-H). At low H2partial pressures the clean ultra-

thin film, exhibiting a graphitic (α-BN) character, is retained. The choice of metal

and the partial pressures of H2and H2O are additional degrees of freedom to

switch between ultra-thin ZnO films, that differ from bulk ZnO, and films that

resemble wurtzite ZnO and could serve as important models for the study of

the ZnO (000¯

1)–(2×1)-H surface. In Sec. 8.2.1to Sec. 8.4we successively develop

and discuss an extensive surface phase diagram, considering the reconstruction

mechanisms proposed for both polar ZnO surfaces. Furthermore, we discuss the

accuracy, the influence of including vdW-corrections and the reliability of the

PBE xc-functional.

In Chap. 9the experimental results from the Department of Physical Chemistry

(PC) of the FHI [237] are discussed and analyzed with the help of ab inito calcula-

tions. Special emphasis is put on the apparent heights measurements of ZnO on

Ag [237]. The experimentally observed 5x5reconstruction is discussed in Sec. 10.1

and included in the full surface phase diagram. In Sec. 10.2we analyze the stabil-

ity of ZnO on Rh (0001) [120] and compare it to the (111) surfaces, which offers

the same three-fold symmetry as the ZnO thin films.

Finally we address non-radiative Föester process in the HIOsystem obtained by

putting the ladder type quarter-phenyl molecule L4P [127] on ZnO (10¯

10) in

Sec. 11 (in collaboration with AG Knorr, TU Berlin). The "hybrid bloch equations"

derived by Verdanhalven, Richter, and Knorr are presented (see Ref. [265]). This

density matrix formalism is parametrized by the atomic and electronic structure

we obtain from DFT calculations. To model Förster processes we implemented

the calculation and the output of dipole/momentum matrix elements and elec-

trostatic potential partial charges in the FHI-aims code, that make it possible to

approximate the coupling arising from the Coulomb matrix element between the

organic and inorganic component.

The dissertation is summarized with a comprehensive discussion and an outlook

on future work in Chap. iv.

Part II

THEORETICAL CONCEPTS TO CHARACTERIZE

SURFACES AND DEFECTS

THEORETICAL FRAMEWORK

The theoretical results and predictions obtained in this thesis are based on the

developments in quantum physics in the last century. The research in solid state

physics has ultimately lead to the development of high performance computers,

that are one key tool for achieving the goals outlined in the previous section. In

the following sections the fundamental theoretical concepts used in this disserta-

tion will be presented. They are themselves based on the axioms and theorems

postulated and proven by many authors. For a rigorous derivation of the theory

the reader is referred to standard textbooks of quantum and solid state physics.

A short selection, that is entirely based on the preferences of the author are refer-

ences [11,247,223,124,231,182,81,76,180,42].

1.1 the many-body problem

At the heart of all our efforts to understand molecular systems, solids and com-

binations of the former is the Many-body problem. For a system of electrons and

nuclei the non-relativistic time-independent Schödinger equation [230] is:

HΨ =EΨ (1)

with the Hamiltonian:

H= −

∑

i=1

2∇2

∑

i=1

∑

j>i

⏐⏐ri−rj⏐⏐

−

∑

i=1

∑

a=1

|ri−Ra|

  

Helec

(2)

−

∑

a=1

2Ma∇2

∑

a=1

∑

b>a

ZaZb

|Ra−Rb|(3)

ri:position of the electrons

Ra:position of the nuclei

Za:charge of the nuclei

Throughout we are using atomic units:

me=e=

h=1

4πϵ0

=1(4)

The energy is given in Hartree:

[Ha] = 

mea0

=27.21138505(60)eV. (5)

The unit of length is the Bohr radius:

a0=4πϵ0

mee2=0.529177208592(17)Å. (6)

10 theoretical framework

The other relevant elementary constants are: me- electron mass, e- electron

charge, 

h=h/2π - Planck constant, ϵ0- permittivity. Their values can be found

in appendix A. The Schrödinger equation contains the mutual interaction of all

electrons in the system, whose number can be exceedingly large (≈1023 in solids).

To demonstrate the futility of attempting an exact solution of eq. 2let us consider

a Silicon atom. Simply wanting to store the wave function on a grid with only

10 points in each dimensions produces a grid with 103N=1042 points. Assuming

128-bit double precision complex numbers, each sample point requires 16 bytes

of storage, which gives approx 1043 bytes of data. A regular blue-ray disc holds

about 50 GB and we would therefore need ∼1032 disks. This example illustrates

that we have to use approximations to tackle the many-body problem in solid

state physics.

1.2 the born-oppenheimer approximation

For each configuration of the nuclei {Ra}the electrons are in an eigenstate of

the electronic hamiltonian Helec (eq. 2). If we consider their response to nuclear

motion as instantaneous, no transitions between the eigenstates of Helec are in-

duced by the nuclei. Nuclear and electronic degrees of freedom decouple. This is

known as the adiabatic or Born-Oppenheimer [36] approximation.

H=Helec +Hnucl (7)

Ψ({ri},{Ra})=Ψelec ({ri},{Ra})Ψnucl ({Ra})(8)

The electronic Hamiltonian can be solved by:

HelecΨelec =EelecΨelec (9)

with the electronic wave function:

Ψelec =Ψelec ({ri},{Ra})(10)

which depends on:

{ri}explicitly

{Ra}parametrically

Also the total energy depends parametrically on the positions of the nuclei.

Eelec =Eelec ({Ra})(11)

Ψnucl ({Ra})describes the vibrations, rotations and translations of a system.

We can asses the validity of this approximation with the help of first order pertur-

bation theory, where the first order coupling between different electronic eigen-

states of the system is zero. The second order scales with the ratio of electron

and nuclear mass. The nuclei are orders of magnitude heavier. Therefore, the

second order term is considered small and can be safely neglected together with

higher order terms. The potential felt by the electrons, caused by the nuclei, can

1.2 the born-oppenheimer approximation 11

be approximated by a field of fixed nuclei. The kinetic energy of the nuclei is

neglected and the repulsion between nuclei becomes a constant. The remaining

Hamiltonian is Helec (eq. 2), which describes the motion of N electrons in the

field of M point charges.

Now we can reverse the argument and solve the problem for nuclei moving in an

effective potential of the electrons. Since the electrons move much faster we can

replace the electronic coordinates by values, which are averaged over electronic

wave-functions.

H= −

∑

a=1

2Ma∇2

∑

a=1

∑

b>a

ZaZb

|Ra−Rb|

+⟨−

∑

i=1

2∇2

∑

i=1

∑

j>i

⏐⏐ri−rj⏐⏐

−

∑

i=1

∑

a=1

|ri−Ra|⟩

H= −

∑

a=1

2Ma∇2

∑

a=1

∑

b>a

ZaZb

|Ra−Rb|+Eelec ({Ra})(12)

By using the Born-Oppenheimer wave function (eq. 8) and neglecting diagonal

terms we obtain the total energy for a set of nuclear coordinates:

Etot ({Ra})=⟨Ψ({ri},{Ra})|H|Ψ({ri},{Ra})⟩=Eelec ({Ra})+

⟨Ψnucl ⏐⏐⏐⏐⏐

−

∑

a=1

2Ma∇2

∑

a=1

∑

b>a

ZaZb

|Ra−Rb|⏐⏐⏐⏐⏐

Ψnucl⟩(13)

In general, the nuclear wave function is peaked around the equilibrium position

of the nuclei and can be approximated by point charges:

Etot ({Ra})=Eelec ({Ra})+

∑

a=1

∑

b>a

ZaZb

|Ra−Rb|

  

classical electrostatic energy

+⟨Ψnucl ⏐⏐⏐⏐⏐

−

∑

a=1

2Ma∇2

a⏐⏐⏐⏐⏐

Ψnucl⟩

  

quantum corrections

(14)

The total energy of the electrons presents a potential energy surface for the mo-

tion of the nuclei. Forces can be calculated by the derivative of the total energy

with respect to the coordinates of the nuclei.

F= −dEtot

dRi

(15)

The adiabatic approximation fails for systems where non-adiabatic effects, e. g.

electron-phonon or electron-lattice interactions, are import. A variety of inter-

esting phenomena can not be described, e. g. Jahn-Teller and Peiers distortions,

superconductivity or thermal conductivity. The Born-Oppenheimer approxima-

tion also fails in cases where more than one energy landscape is present (e. g.,

when a molecule approaches a surface).

12 theoretical framework

1.3 density functional theory

Density functional theory is based on the relation between the particle density

n(r)and the many-body wave function of the ground state Ψ(r1...rN). The ad-

vantage is apparent. The many body wave function depends on the coordinates

of all N electrons (spin is omitted), while the particle density is a scalar object

that depends only on r. In DFT we try to find the system energy E([n]) as a

functional expression of the density n(r)by a variational principle. The first and

simplest theory in this spirit is the Thomas-Fermi-Theory [252,79]. It is still of

conceptual importance, but will not be discussed.

The goal is to recast the Schrödinger equation for the many body wave-function

(eq. 3) in terms of the density with the electronic Hamiltonian

Helec = −

∑

∇2

  

∑

vext (ri)

  

∑

i=j

⏐⏐ri−rj⏐⏐

  

(16)

=T+Vext +Vee. (17)

Tis the kinetic energy operator of the N-electron system, Vext - the external

potential operator and Vee the electron-electron interaction (Coulomb operator).

The foundation of DFT is formed by the Hohenberg-Kohn Theorems [102,140]:

Hohenberg-Kohn Theorem I

The ground-state density n(r)uniquely determines the potential (vext) up to an

arbitrary constant.

vext (r)⇐⇒ n(r)(18)

Hohenberg-Kohn Theorem II

A universal functional for the energy E[n]in terms of the density n(r)can be

defined, valid for any external potential vext (r). For any given vext (r), the exact

ground state energy of the system is the global minimum of this functional, and

the density that minimizes the functional is the exact ground state density.

EHK[n] = T[n] + Vee[n]

  +∫drvext (r)n(r)

=F[n] + ∫drvext (r)n(r)

By definition the Hohenberg-Kohn functional gives the minimum energy of the

system for the ground state density and if the the functional is known the ground

state density can be obtained from the variational principle. We can only obtain

the ground state density and therefore only ground state properties. A funda-

mental problem is the missing prescription to construct the kinetic energy as a

functional of the density. This was solved by a trick by Kohn and Sham [129],

which lead to a breakthrough in modern DFT. Their solution was to consider an

auxiliary system of N non-interacting electrons that has the same ground state

density as the fully interacting system.

haux = −∇2

2+V(r)(19)

1.3 density functional theory 13

The non-interacting electrons move in an effective potential. The full Hamiltonian

is a sum over the auxiliary single particle Hamiltonians.

H({xi})=∑

haux (xi)=Haux ({xi}). (20)

This gives a set of single particle orbitals.

hauxφi=ϵiφi(21)

This leads to the density

n(r)=

N/2

∑

i∫dr|φi(r,σ)|2, (22)

Hartree energy

EH[n] = 1

2∑

ij ∫drdr′|φi(r)|2⏐⏐φj(r′)⏐⏐

|r−r′|=1

2∫drdr′n(r)n(r′)

|r−r′|(23)

and kinetic energy:

TS[n]=−1

N/2

∑

i∫dr⟨φi|∇2|φi⟩=1

N/2

∑

i∫dr|∇φi|2. (24)

This leads finally to the Kohn-Sham energy functional:

EKS[n] = TS[n] + ∫drVext (r)n(r)

  

=Eext

+EH[n] + EXC[n](25)

With this energy functional we can define a Lagrange functional:

L[n] := EKS[n] − ∑

ϵi∫dxφ∗

i(x)φi(x)

and minimize it with respect to the single electron orbitals

δL

δφ∗

=0∀i(26)

This leads to the key result in our derivation:

[−1

2∇2+VKS (r)]φi(r)=ϵiφi(r)(27)

The Kohn-Sham equations [129]

with the (effective) Kohn-Sham potential:

VKS (r)=Veff (r)=vext (r)+vH(r)+vXC (r)(28)

vXC (r):= δEXC[n]

δn (r)(29)

14 theoretical framework

The Kohn-Sham equation has to be solved self-consistently because vKS depends

on the solution of the system of differential equations. Now we have to find

good approximations for Exc[n]. It is the only missing ingredient to solve the

many-body problem. We have introduced Exc[n]in order to map our fully inter-

acting system on a system of non-interacting electrons. Exc[n]is in general small,

because TSand EHcapture a large part of Tand Vee.

1.4 approximate xc-functionals

As we have seen at the end of the last chapter, by introducing the effective non-

interacting system we have obtained a well defined set of equations. The Kohn-

Sham equations can be transformed into an eigenvalue problem, which can be

solved with high (computational) efficiency [92,30,277,31,156]. The cost for this

transformation is the loss of the knowledge about the exact form of the energy

functional and the quantities derived from it. The non-interacting (model) system

can be modified to continuously switch back to the fully interacting (physical)

system by "adiabatic continuation" [225]. The solution would be exact, but the

computational effort to solve the problem is unfeasible. To take advantage of

the Kohn-Sham equations many different approximations for the notorious xc-

energy functional have been proposed. The development of new functionals, ad-

opted to special classes of systems or problem sets as well as the generalization

of functionals based on basic quantum mechanical principles is an active field

of scientific research. In this chapter only the functionals used for obtaining the

results in this work will be presented. The reader is referred to the literature for

other functionals and their applications [85,114].

1.4.1Local density approximation (LDA)

Figure 1: Schematic illustration of the local behavior of the density.

Inhomogeneous systems with a slowly varying density look locally like the

homogeneous electron gas (see Fig. 1). Making a local approximation, the xc-

energy Exc[n]can be expressed as an integral over the density and the xc-energy

density ϵHEG

xc ([n],r)at rof the homogeneous electron gas. It only depends on

1.4 approximate xc-functionals 15

the density at r.

=⇒Exc[n] = ∫drn(r)ϵHEG

xc ([n],r)(30)

The exchange-correlation potential vxc is obtained by a functional derivative:

vxc (r)=δExc[n]

δn (r)=ϵ([n],r)+n(r)δϵ ([n],r)

δn (r)(31)

The exchange part of the exchange-correlation energy for the homogeneous elec-

tron gas is known exactly [260]:

ϵHEG

x[n] = 3kF

4π , (32)

with the Fermi momentum kF=(3π2n)1/3. By inserting eq. 32 in eq. 30 we

obtain

Ex[n] = ∫drn(r)ϵx([n],r)=3

4(3

π)1/3 ∫drn(r)4/3 (33)

for the exchange energy and

vx[n] = (3

π)1/3

n1/3 (34)

for the exchange potential by inserting eq. 32 into eq. 31.

The correlation energy density for the homogeneous electron gas is not known

analytically. An approximate analytic expression was first given by Wigner in

1938 [273] (see Fig. 2):

ϵc[n]=− 0.44

rs+7.8, (35)

with rs=(3

4πn )1/3 =1.919

kF. A better parameterization was computed with high

precision Quantum Monte Carlo techniques by Perdew and Zunger [195]:

ϵc[n] = ⎧

⎪

⎨

⎪

⎩

Gell-Mann/Brückner

  

Alnrs+B+Crslnrs+Drsrs⩽1

1+β1√rs+β2rs

rs> 1

(36)

The numerical values of A,B,C,D,γ,β1and β2are given in Ref. [195].

The local density approximation is exact for the homogeneous electron gas. It

is expected to perform well for systems with slowly-varying densities such as

simple metals. Typical dissociation energies and cohesive properties are with

10% to 20% experimental values [57]. Bond lengths and lattice parameters are

typically underestimated by 1% to 2% [241,55]. LDA does not perform well for

systems with a rapidly changing density e. g. atoms.

16 theoretical framework

Figure 2:ϵc[n]as parameterized by Wigner and Perdew/Zunger.

1.4.2Semi-Local density approximation

To improve on the approximation of a homogeneous density, gradients were in-

troduced into Exc [24]

EGGA

xc [n]=∫drn(r)ϵxc (n,|∇n|,...)

=∫drn(r)ϵHEG

xFxc (n,|∇n|,...). (37)

We can define a scaled gradient, that measures on the scale of the density itself:

S(r)=|∇n(r)|

2kFn(r)(38)

leading to the generalized gradient expansion (GGA) [196].

EGGA

xc [n]=∫drn(r)ϵHEG

x[n]Fxc (n,S). (39)

There is no unique form of Fxc. Many different parameterizations exist in lit-

erature , that are often named by the initials of their creators: Perdew-Burke-

Ernzerhof (PBE) functional [196], Perdew-Burke-Ernzerhof functional for solids

(PBEsol) [200], Perdew-Burke-Ernzerhof functional for interfaces (PBEint) [78],

revised Perdew-Burke-Ernzerhof functional (revPBE) [94], Armiento-Mattsson

functional from 2005 (AM05) [9], Becke-Lee-Yang-Parr functional (BLYP) [23,138].

They are usually designed to describe a predefined class of materials (e.g. organic

molecules, surfaces). There are two major strategies to construct Fxc:

• Construct Fxc to fulfill certain exact conditions, e. g. asymptotic behavior

of the homogeneous electron gas.

• Fit the parameters in Fxc to reproduce certain properties of test sets, that

have been obtained from very accurate high level methods.

Functionals based on the general gradient approximation improve in many as-

pects the accuracy of the local density approximation [197], e. g. the errors in

1.4 approximate xc-functionals 17

total energies and the over-binding of LDA are reduced. In general GGAs per-

form best in the regime for which they were designed. The next logical step to

further improve our approximations are meta-GGA functionals, that include the

second and higher order derivatives in the exchange correlation functional [201].

We will not further discuss meta-GGAs and directly progress to functionals with

the exchange part expressed by the Kohn-Sham orbitals.

1.4.3Hybrid functionals

The energy expression obtained from DFT is eq. 25 with the Hartree energy in eq.

23. For i=jthe electron interacts with itself. This is the so called self interaction.

In Hartree-Fock theory this term is canceled exactly by the exchange energy:

Ex= −1

2∑

ij ∫drdr′ψ∗

i(r)ψj(r)ψ∗

j(r′)ψi(r′)

|r−r′|

for i=j,EH+Exis 0. For the local and semi-local functionals in DFT, this

cancellation does not occur. Exis part of Exc in eq. 39. The self interaction error

in such a functional was defined by Perdew and Zunger [195] as:

δi=1

2∫drdr′|ψi(r)|2|ψi(r′)|2

|r−r′|+EXC [|ψi(r)|2](40)

For local and semi-local functionals this value is in general not zero. The self in-

teraction error can lead to a over-delocalization of states. To improve the thermo-

chemical accuracy of LDA and GGA based functionals for molecular systems the

inclusion of exact exchange (Hartree-Fock exchange) information was proposed

by Becke [24]. An additional advantage is the remediation of the self-interaction

error.

Ehyb

xc =EDFT

xc +α(EHF

x−EDFT

x). (41)

In this simple form a portion of the exchange contribution of DFT is replaced

with exact exchange (Hartree-Fock). The correlation contribution is purely DFT.

Other hybrid functionals with more complex parameterizations exist (e. g. B3LYP

[24,138,266,244,232]). A separation of long (LR) and short range (SR) contribu-

tions in Exand Eccan be introduced

EωPBEh

xc =aEHF,SR

x(ω) + (1−a)EPBE,SR

x(ω) + EPBE,LR

x(ω) + EPBE

c, (42)

where ais the mixing parameter and ωis an adjustable parameter controlling

the short-ranginess of the interaction. The range separation is typically achieved

by an error function. With a=1

4and ω=0.2the HSE06 [99,134,100] xc-functional

is obtained. Setting α=0.25 in eq. 41 and using exchange from the GGA xc-

functional PBE [196], yields the functional known as PBE0[2]. The exchange

correlation potential is calculated as in Hartree-Fock (functional derivative with

respect to orbitals, δEXC

δφ∗

i, not the density). This leads to a non-local potential:

vhyb

XC (r,r′)=[vDFT

XC (r)−αvDFT

X(r)]δ(r−r′)+αΣHF

X(r,r′)(43)

18 theoretical framework

with

ΣHF

X(r,r′)=

∑

ψi(r)ψ∗

j(r′)

|r−r′|. (44)

The formalism introduces new parameters, that can be chosen on the basis of

the satisfaction of exact conditions, a best fit to test set properties obtained with

higher level methods or a best match for the system dependent properties under

investigations [14,158,4,243,213].

1.5 long range dispersion

In this work we follow the convention of Ref. [245] and refer to the van-der-Waals

(vdW) energy as the attractive interaction between fluctuating multi-poles. This

description is based on concepts first introduced by London [75,145,146]. The

basic idea is to add the long range dispersion contributions, which are missing in

DFT calculations employing LDA, GGA and hybrid xc-functionals. Many meth-

ods have been developed to calculate the vdW-energy and coupling them to the

underlying functional e.g. DFT-vdW[254], vdW-DF[70]. The reader is deferred to

recent reviews for discussion and developments [84,253,71,116]

In this work we apply the method developed by Tkatchenko and Scheffler [254]

with additional parameterizations from Ruiz et al. [220] and Zhang et al. [286].

The vdW energy is expressed as:

EvdW = −1

2∑

A∑

fdamp(RAB,R0

AB)CAB

RAB

, (45)

with RAB the inter atomic distance between atoms A and B, R0

AB =R0

A+R0

the vdW radius, CAB

6the corresponding vdW coefficients and fdamp a damping

function. The Tkatchenko and Scheffler method (TS) [254] obtains R0

AB and CAB

from the electronic ground-state density by Hirshfeld-volume partitioning [101].

Empirical parameters are only introduced in the damping function f(eq. 52). The

starting point of the formalism is a set of accurate vdW parameters (Ci

6,Ri

6and

αi) for the free atoms that are determined from the ground state mean field den-

sity [254] or from high level time dependent density functional theory (TD-DFT)

[157,191] calculations in improved/specialized versions [286,220]. The change

in the chemical environment of the free atoms and the system in a molecule or a

solid is captured by calculating the volume of these atoms [101].

Vfree

i=∫drnfree

i(r)|r−Ri|3(46)

Veff

i=∫drnH

i(r)|r−Ri|3(47)

with Rithe position of atom iand the density of the free atom nfree

i(r)or the

Hirshfeld-volume partitioned density nH

i(r):

i(r) = n(r)ni(r)free

∑jnj(r)free (48)

1.5 long range dispersion 19

Species C6α R6

Ag[220]122 15.4 2.57

Cu[220]59 10.9 2.40

Pd[220]110.6 14.4 3.10

Pt[220]129.5 15.0 2.83

Zn[286]46.02 13.774 2.818

O [286]4.45 4.285 2.953

Table 1: Free atom parameters from references [286,220] used as input for the TS-method

[254].

The free atom values for Ci

6,free,Ri

6,free and αi

free are rescaled by the volume

ratio to represent the chemical environment in a molecule or solid.

αi=αi

free ×Veff

Vfree

(49)

6=Ci

6,free ×(Veff

Vfree

i)2

(50)

6=Ri

6,free ×(Veff

Vfree

i)1/3

(51)

Only the damping function in eq. 45, which is required to remove the R−6

AB singu-

larity at short distances, has to be chosen. A Fermi-type function is used in the

TS-method [254].

fdamp(RAB,R0

A,R0

B) = 1

1+exp[−d(RAB

sRR0

−1)] (52)

Only dand sRare free parameters. ddetermines the steepness of the function.

d=20 has been chosen. sRreflects the damping behavior of the xc-functional.

It determines the electron correlation covered by a given functional. sRwas ob-

tained by fitting to the S22 [117] test set [254] for different xc-functionals.

The free atom parameters Ci

6,free,Ri

6,free work well for molecular systems. To

include vdW effects for ionic solids such as ZnO and polarisable surfaces these

parameters have to be adjusted. For ionic and semi-conducting crystals a combi-

nation of again TD-DFT and the Clausius-Mossotti equation [172,53,221]

α(ω) = Vϵ(ω) − 1

4π

3ϵ(ω) − 4π

3+4π (53)

is used for calculating accurate free atom parameters [220]. From TD-DFT cluster

calculations the total static polarizabilities are obtained from the microscopic di-

electric function ϵ(ω).C6,R6and αare calculated from those results in the same

manner as in the original TS-method [254].

For an inorganic (metallic) surface the many-body collective response of the bulk

20 theoretical framework

Species Veff C6,eff αeff R6,eff

Ag 1.054 109.963 14.614 2.526

Cu 1.084 50.214 10.056 2.336

Pd 1.044 101.538 13.797 3.056

Pt 0.990 132.367 15.165 2.840

Zn 0.915 54.975 15.055 2.903

O0.978 4.651 4.381 2.975

Table 2: Effective vdW-parameters.

electrons inside the surface has to be included in the description of the vdW-

interaction to calculate accurate C6,R6and αparameters. This was achieved

by applying Lifshitz-Zaremba-Kohn theory for the nonlocal Coulomb screening

within the bulk [285]. The Cab

6for atom sort aand bcan be obtained from CaB

for atom aat a solid surface B.

CaB

3=ns(π

6)Cab

6(54)

CaB

3is calculated from:

CaB

3=

4π ∫∞

dωα(iω)ϵB(iω) − 1

ϵB(iω) + 1(55)

with α(iω)the polarizability of atom aand ϵB(iω)the dielectric function of

solid B.ϵB(iω)is calculated from its imaginary part obtained from experimen-

tal data by the Kramers-Kronig relation [60,130]. By combination with the same

scheme as in the TS-method (Pade-approximant model) [254,16]C6,eff,R6,eff

and αeff are calculated.

The values acquired with the methods described above are listed in tab. 1. The

effective volume is taken from FHI-aims [33] calculations for each system employ-

ing the PBE xc-functional [196] with the tight basis set provided by the software

package [33]. The effective vdW parameters are listed in tab. 2for the systems

considered in this work.

1.6 ab initio thermodynamics

Density-functional theory provides a versatile and flexible framework to accu-

rately describe microscopic properties of many different systems. The electronic

structure is provided by the calculated density. The atomic structure can be de-

termined by minimizing the forces on the nuclei of the constituents. Many other

properties can directly be calculated or estimated from these quantities. By apply-

ing methods known from thermodynamics (and statistical mechanics) the micro-

scopic information obtained from DFT can be extended to the meso- and macro-

scopic regime.

In practice many structures of different size and atomic composition lie in a very

narrow energy range. The combination of DFT and concepts from thermodynam-

ics facilitates a comparison of these systems to access their relative stability. DFT

1.6 ab initio thermodynamics 21

is in this context referred to as ab initio or from first principles because it is fully

derived from quantum mechanics without introducing parameters, that have to

be fitted to empirical data. In this section the concepts of ab initio thermodynam-

ics [226,214] will be presented for the example of the ZnO (000¯

1) surface.

The solution of the KS-equation in DFT is the ground state of the system and

is only valid for T= 0K and p= 0mbar, but we can extend DFT to finite tem-

peratures and pressures with the help of well established concepts from thermo-

dynamics [214]. The key quantity for all further considerations is the Gibbs free

energy G for the (T, p)-ensemble:

G(T,p) = Einternal +Fvib −TSconf +pV =Nµ(T,p)(56)

The leading term is the internal energy Einternal, which can be obtained directly

from DFT calculations. The contributions from the other three terms are highly

system dependent and have to be accessed for every problem. The second term

is the vibrational free energy Fvib =EZPE −TSvib with the zero point energy

EZPE and the vibrational entropy Svib. It accounts for contributions from the

vibrational degrees of freedom of the system. The third term in Gaccounts for

the configurational entropy Sconf. The last term includes the volume Vand the

external pressure p. It is import to note that this approach is only applicable for

systems that are in thermodynamic equilibrium. Such systems can be divided

into smaller subsystems, which are again in thermodynamic equilibrium with

each other. These smaller, computational more feasible systems can be treated

more easily with DFT (e. g. a semi-infinite bulk surface in a gas atmosphere,

that both act as reservoirs). Dynamical or kinetically enabled processes cannot be

described with eq. 56.

The internal energy of a one component system (e. g. a bulk metal) fully describes

that system in equilibrium.

Einternal =TS −pV +Nµ. (57)

It is the sum of the products of entropy and temperature minus the pressure

times volume and the number of particles in the system times the chemical po-

tential. By cleaving the bulk crystal and creating a surface we have to pay a

penalty proportional to the created surface area. This energy per surface area is

defined as the surface energy γ[284]. The internal energy for a surface is

Einternal

surf =TS −pV +Nµ +γA. (58)

With the help of the Gibbs free energy (eq. 56) we can rewrite eq. 57 for the

surface energy.

γ(T,pi) = 1

A[Gsurf −∑

Niµi(T,pi)](59)

Gsurf is the Gibbs free energy of surface and solid and µi(T,pi)are the chemical

potentials of the different species (i) in the system. For a ZnO surface eq. 59

includes the three species in the system: Zn, O and H. The ZnO surface is in

equilibrium with the bulk ZnO, which acts as a reservoir. The chemical potentials

22 theoretical framework

of O and Zn are not independent. They are connected by the Gibbs free energy g

(per formula unit).

gbulk

ZnO (T,p) = µZn +µO(60)

By putting µZn,µOand µHinto eq. 59 and using eq. 60 we obtain:

γZnO(T,pO,pH) = 1

A[Gsurf(T,pO,NZn,NO) − NZngbulk

ZnO (61)

−(NO−NZn)µO(T,pO) − NHµH(T,pH)]

The limits of the chemical potential can be derived from thermodynamical consid-

erations. The upper limit for the gas phase components, H and O, is determined

by the point when they are so highly concentrated, that condensation at the ox-

ide surface starts. For a ZnO surface in equilibrium with a oxygen and hydrogen

atmosphere this is determined by:

µO⩽1

2Etot

O2,µH⩽1

2Etot

H2(62)

The oxide starts to decompose at low oxygen chemical potentials when the chem-

ical potential of Zn becomes greater than its bulk Gibbs free energy.

µZn ⩽gbulk

Zn (63)

With eq. 62 and eq. 63 the limits for the oxygen chemical potential are:

gbulk

ZnO (T,p) − gbulk

Zn −1

2Etot

O2⩽∆µO(T,p) − 1

2Etot

O2⩽0(64)

The upper boundary for µOis set to zero by defining ∆µO(T,p) = µO(T,p) −

2Etot

O2and ∆µH(T,p) = µH(T,p) − 1

2Etot

H2for µH. The lower limit for µHcannot

be derived from the equilibrium conditions. In practice the limit is chosen to

give a physically reasonable pressure limit at a given temperature. The relation-

ship between the chemical potential of a gas phase component and the pressure

and temperature is calculated using statistical thermodynamics [214] or from the

ideal gas law [133,52] and tabulated values for entropy and Helmholtz energy

terms [178].

µO(T,p) = µO(T,p0) + 1

2kBTln(p

p0). (65)

The temperature and pressure dependence can be determined if µ(T, p0) is known

at one pressure. The reference value of the chemical potential of the gas phase

component at T=0K is set to the total energy of an isolated molecule.

µ(0K,p0) = 1

2Etot

O2=0(66)

With respect to this reference and using the relation between Gibbs free energy

Gand enthalpy H,G=H−TS, the chemical potential at p0is given by

µO(T,p0) = 1

2(H(T,p0) − H(0K,p0))−1

2TS(T,p0)(67)

1.6 ab initio thermodynamics 23

0 200 400 600 800 1000

T[K]

0.00

0.01

0.02

0.03

0.04

0.05

0.06

TSconf

N[eV ·˚

A2]

N=16

n= 1

n= 3

n= 5

n= 7

Figure 3: Confrontational entropy fracTSconfNAsite as a function of temperature T.

For the standard pressure p=0.1MPa, H(T,p0) − H(0K,p0)and S(T, p0) are tabu-

lated and can be obtained from the NIST data base [178] for many substances.

To obtain the surface energy for a given system we have to evaluated the three

missing terms in the Gibbs free energy (eq. 56). The pV term is discussed by

a dimension analysis. The unit of [pV/A] is [atm Å3/Å], 1atm times 1Å3is

10−3meV. For a pressure of 100 atm it is than of the order pV/A ∼0.1mev/Å2

and can be safely neglected for the surfaces in this work.

The configurational entropy is strongly system dependent. It can be calculated

very accurately with Monte Carlo techniques from statistical mechanics [29,202,

203,217]. This approach is out of scope for the present work. An estimation

of the contribution can be achieved for the relation for non-interacting particles

from statistical mechanics.

Sconf =kBln N!

N!(N−n)!(68)

This relation can be derived from combinatorics for calculating the possible con-

figurations by selecting N times from a set of n samples and putting them back.

Here we are considering a surface with N surface sites and n defects or adsor-

bate atoms to adsorb on the N sites. Eq. 68 can be approximated with the Sterling

formula for n,N≫1.

ln(n!)≈nln(n) − n(69)

For the contribution to the Gibbs free energy of the configurational entropy per

surface Area, A=NAsite, this yields.

TSconf

NAsite =kBT

Asite [ln(N

N−n)+n

Nln(N

n−1)] (70)

To further estimate the configurational entropy we consider a 4x4ZnO (000¯

surface. A total number of N=16 adsorption/defect sites are available. In Fig. 3

TSconf

N(eq. 70) is plotted as a function of the temperature Tfor N=16 and dif-

ferent numbers nof adsorbates/defects. The maximum number nis n=N/2,

24 theoretical framework

here the interaction between the adsorbates would reduce the configurational en-

tropy. These interactions were neglected in the previous considerations. They can

reduce the configurational entropy even at lower, highly symmetric coverages, as

shown by Monte Carlo calculations [217]. The line for n=N/2 in Fig. 3repre-

sents an upper limit for the contribution to the Gibbs free energy. The area of a

ZnO (000¯

1) surface unit cell is AZnO =7.5Å2. The upper limit for the contribu-

tion to the Gibbs free energy (nis n=N/2) is:

TSconf

(n/2)Asite

ZnO

⩽8meV. (71)

The contribution is small, but should be discussed or accounted for, for all con-

sidered systems.

The final contribution to the Gibbs free energy of our surface is the free energy

of vibration Fvib. From statistical mechanics and thermodynamics we know that

Fvib(T,V) = Evib −TSvib = −kBTlnZvib (72)

with Zvib the partition function of the vibrational degrees of freedom of a N-

atomic system. It is defined as

Zvib =

∑

i∫dk

(π)3

exp(−1

2β

hωi(k))

[1−exp(β

hωi(k)](73)

with β=1/kBTand ωi(k)the 3N vibrational modes of the Natom system. By

introducing the phonon density of states σ(ω)(DOS) and inserting eq. 73 into eq.

72 we get

Fvib(T,V) = ∫dω[1

2

hω +kBTln(1−e−β

hω)]σ(ω). (74)

Fvib can thus be calculated from the phonon density of states σ(ω)for all bulk

and surface phases in the system. Fvib enters the surface free energy γ(T,p)(eq.

59) as a difference between these bulk and surface phases (Gsurf and gbulk).

Calculating the phonons for the surfaces considered in this work is not feasible.

To approximate the contributions from vibrating surface adsorbates we can se-

lect one characteristic frequency in eq. 74 by using the Einstein model [74]. The

phonon density of states σ(ω)is replaced by a delta distribution at the character-

istic frequency ˆω.

Fvib(T,V) = 1

2

hˆω+kBTln(1−e−β

hˆω)(75)

As a first approximation for Fvib at moderate temperatures (T < 1000) only the

first term in eq. 75 is evaluated (zero point energy). The vibrational energy con-

tribution of a hydrogen atom adsorbed at an oxygen surface site can be approxi-

mated by the difference between the experimental H2gas phase stretching mode

at ωH2=4138cm−1[207] and the the frequency obtained by a finite difference

1.6 ab initio thermodynamics 25

DFT calculation ωsurf

O−H=3657cm−1. The vibrational contribution to the surface

free energy for nhydrogen on Npossible sites is:

γvib =1

A[1

2

hˆωsurf

O−H−1

2

hˆωH2]=n

NAsite

0.03eV (76)

The contribution is small, but has to be discussed for the system under investiga-

tion. For high coverages the contribution could be non-negligible.

To calculate the surface free energies and to determine the energetically most

stable surface phases at a given pressure and temperature, the vibrational energy

and configurational entropy have to be considered for the systems of interest.

The total energy obtained from DFT is indeed the leading term, but disorder and

vibrations could stabilize otherwise unstable surface phases.

For the systems of ZnO on metal substrates eq. 61 has to be extended to in-

clude a water atmosphere as reservoir. Under typical UHV conditions the partial

pressures of H2, O2, and H2O are of the same order of magnitude. Previously

only H2and O2were considered. The chemical potential µH2Oof water can only

be included in eq. 61 together with µH2or µO2to satisfy element conservation

for the surfaces considered. In this approach one of the surface species is al-

ways neglected. The resulting surface phase diagrams can differ significantly as

this method assumes unrealistic partial pressures in equilibrium for the missing

species. Under experimental conditions these three species are not in thermo-

dynamic equilibrium because of the negligible rate of spontaneous gas-phase

reactions. The most common approach is to only use H2and O2as gaseous

reservoirs and constrain them by the chemical potential of water:

µH2+1

2µO2> µH2O. (77)

As water is not explicitly included in the calculation, its partial pressure is ill-

defined. The chemical potential of water is therefore approximated by the DFT

total energy in the gas phase EH2O. The argument here is that outside the pro-

posed boundary in eq. 77 water would macroscopically condense on the surface

[73,132,161,261,267]. This portion of the phase diagram is considered unreli-

able because no prediction about the structure can be made. Unfortunately, the

pressure range of experiments often lies in this area.

Herrmann and Heimel recently [98] proposed a solution to this problem of the

thermodynamic theory by explicitly including the chemical potentials of all three

species µH2,µO2, and µH2O. Eq. 61 now reads:

γZnO =1

A[Gsurf −NZngbulk

ZnO −NOµO−NHµH−NH2OµH2O]. (78)

Element conservation requires

ZZn =NZnO (79)

ZH=2NH2+2NH2O(80)

ZO=2NO2+NH2O+NZnO (81)

This is a set of three equations with four unknowns and thus under-determined.

An additional condition has to be introduced. Only energy contributions which

are a multiple of the water gas-phase reaction

2H2+O2⇄2H2O(82)

26 theoretical framework

are permitted or in the terminology of the set of linear equations, to the solution

vector N=(NZnO,NH2,NO2,NH2O) a multiple of the vector v=(0,-2,-1,2) can be

added. These are the stoichiometric coefficients of the water gas-phase reaction.

Herrmann and Heimel eliminate this under-determination by demanding, that

the virtual reaction of putting all species back into their reservoirs is orthogonal

to the water gas-phase reaction

v·N=0. (83)

The surface energy now depends on three variables and has to be analyzed in

three dimensions. No constraints for µH2,µO2, and µH2Oare required and there

is no condensation of water on the surface until the vapor pressure of water is

reached.

1.7 dft as implemented in fhi-aims

In this section the practical aspects of carrying out the DFT calculations in this

work are presented. As the mayor tool, we will focus on the implementation in

the all-electron code FHI-aims [33], developed at the Fritz Haber Institute. All

electron here refers to the fact, that the wave functions of all electrons with the

full potential are treated explicitly. Valence and core electrons are treated on equal

footing. We will introduce the basis set employed and give an overview over the

algorithm(s) for constructing and solving the Kohn-Sham equations.

The particular choice in FHI-aims for the basis set are numeric atom-centered

orbitals (NAO):

φi(r) = ui(r)

rYlm(θ)(84)

The Ylm(θ)in eq. 84 are the complex spherical harmonics and ui(r)are the

numerically tabulated radial functions. They are chosen to solve the radial Schrö-

dinger equation [230] for the potential vtot

i(r) = vi(r) + vcut(r).vi(r)defines

the main behavior of ui(r)and is chosen as the self-consistent free atom po-

tential (hydrogen-like, cation-like or atom-like). vcut(r)is a steeply increasing

(exponential) confining potential, to cut off any slowly decaying tails. The radial

Schrödinger equation is solved on a dense logarithmic grid [91] and the result-

ing radial functions are orthonormalized by a Gram Schmitt process [8]. These

procedure provides a pool of basis functions that was optimized to yield a basis

set that can be gradually increased in accuracy and computational cost.

They are well tested with local (e.g. [129]), semi-local (e.g. [128]) functionals as

well as Hartree-Fock based functionals (e.g. [248]). The first advantage of these

NAOs is the seamless descriptions of all electrons and an accurate description of

the orbitals near the nuclei. The second advantage is the localized nature of the

orbitals. After a certain distance the radial functions can be cut off and set to zero

for larger distances. This is especially advantageous for large systems. Here the

overlap of orbitals localized on spatially separated atoms is zero, increasing the

sparsity of important quantities such as the Hamiltonian matrix.

The evaluation of the KS equations (eq. 27) requires a numerical integration on a

1.7 dft as implemented in fhi-aims 27

real-space grid [63,17,246,23] for the setup of the overlap and the Hamiltonian

matrices. These extended integrands are divided into atom-centered pieces

∫d3rφi(r)ˆ

HKSφj(r) = ∑

at,s,t

w(r)φi(r)[ˆ

HKSφj(r)](85)

with r=r(at,s,t),ˆ

HKS the Kohn-Sham-Hamilton operator and w(r) = pat(r)·

wrad(s)·wang(t), the total partition function pat as product of the atom cen-

tered partition functions pat(r)[63], the radial wrad(s)and angular wang(t)

partition function [17]. In practice the integration gird is divided into batches re-

cursively [96]. Taking advantage of sparsity, the non-zero basis functions in each

batch are tabulated. For those basis functions φi(r)and ˆ

HKSφj(r)are evaluated

and summed up by matrix multiplication.

For periodic boundary conditions Bloch-like generalized basis functions are de-

fined from the real-space basis functions φi(r), that are centered in unit cells.

The integration is formally carried out by the previously described integration

over one unit cell and the sums over the basis function φi(r)that are shifted by

T(N)[(N= (N1,N2,N3))] and have a contribution in the first unit cell

hij(k) = ∑

M,N

exp(ik[T(N) − T(M)]) ∫unit cell d3rφi,M(r)ˆ

HKSφj,N(r), (86)

with the k-point vectors k.

Now, the single particle eigenvalue problem can be solved. For this task highly ef-

ficient linear algebra libraries are employed. For large numbers of k-points (more

or equal number of k-points as computing tasks) the Lapack library [7] is used.

For larger and cluster systems, ScaLapack [30] and the ELPA [156] libraries are

employed to make use of the highly parallelized structure of modern super com-

puters.

From the eigenstates cij, that are the solution to the KS-eigenvalue problem, a

new density is calculated. First the density matrix is set up

nij =

Nocc

∑

flcilcjl (87)

by summing the outer product between eigenstate pairs and multiplying with

the occupation number flof state l. The density on the grid is obtained by a

matrix operation

n(r) = ∑

φi(r)nijφj(r). (88)

From this density the potential can be calculated and the KS-eigenstate problem

re-evaluated. This is the self-consistent field (s.c.f.) cycle (see Fig. 4). The pro-

cedure is repeated until the change (as compared to the previous step) of the

integrated density, energy or sum of eigenvalues falls below a certain thresh-

old. In practice only a fraction of the new density is mixed with the original

density. This results in a more stable convergence behavior. A well established

mixing scheme is Pulay’s direct inversion of the iterative subspace or Pulay mix-

ing [209,131]. It is based on the idea to find the best linear combination of the

28 theoretical framework

Figure 4: Self consistent field cycle.

residuals (difference between two consecutive densities) of all previous densities.

Another useful technique is the preconditioning of the density to improve con-

vergence and reduce long-range oscillations in the residual of the densities. This

problem is especially notorious in slab calculations. Charge-density can be trans-

fered over the vacuum region in one s.c.f. step and back in the next. This leads

to an oscillation of the density, ultimately preventing convergence. This problem

can be tackled by applying a pre-conditioner. A robust and well tested choice is

the Kerker pre-conditioner [121,177,154].

After convergence of the s.c.f. cycle the force on each atom can be calculated by

Fat = − ∂

∂Rat

Etot. (89)

By minimizing the forces on the atoms the (local) minimum on the Born-Oppen-

heimer-energy surface can be obtained. To optimize the atomic positions the trust-

radius method [179] or the standard Broyden-Fletcher-Shanno-Goldfarb (BFGS)

algorithm [206,33] are used, which try to minimize the Born-Oppenheimer-

energy surface in the direction of the steepest decent. After an optimization step

a new s.c.f. cycle is started with the new atom coordinates. This procedure is re-

peated until the residual force on the atoms is below a certain threshold. From the

density and eigenvalues of such an optimized structure many important proper-

ties can be obtained/calculated, e.g. cohesive energy, surface energy, the band

structure En(k), density of states or transition matrix elements.

1.8 calculating surface vibrations

After calculating and minimizing the energy on the Born-Oppenheimer potential

energy surface we want to calculate molecular vibrations and phonons. They are

essential to accurately account for the surface free energy of vibration (eq. 72).

1.9 simulating stm apparent height maps 29

This contribution to the total surface free energy, although typically small, can

determine the stability of surface reconstructions that are close in energy. We re-

quire the phonon density of states (eq. 74) for the surface free energy of vibration.

Its full calculation, although possible, is not feasible for the system investigated

in this work. We will restrict the discussion to the description of vibrations of sur-

face adsorbates within the harmonic approximation. The change in vibrational

modes between surface adsorbate and free molecule is expected to be the leading

term in the calculation of the surface free energy. We will omit changes in pure

surface and pure bulk modes. A comprehensive description of the phonons and

molecular vibrations can be found in standard text books [124,11,85].

The nuclei of the system move in the effective potential of the electrons (eq. 12

and 15). The forces are the derivatives of this potential with respect to the coor-

dinates of the nuclei:

ma¨

Ra=Fa= −∂Etot

∂Ra

= − ∂

∂Ra[∑

a=1∑

b>a

ZaZb

|Ra−Rb|+Eelec ({Ra})](90)

If we define R0

ias the equilibrium positions of the atoms in the system we

can Taylor expand the potential in eq. 90 around the minimum of the Born-

Oppenheimer potential energy surface:

Etot({R−R0}) = Etot({R0}) + ∑

a=1(∂Etot

∂Ra)R0

+(91)

∑

a=1∑

b=1(∂2Etot

∂Ra∂Rb)R0

a,R0

(Ra−R0

a)(Rb−R0

b) + ....

The first order term is zero for a minimum. Only the second order has to be

considered. The matrix of second derivatives is called Hessian matrix and is in

practice calculated by displacing the atoms by ±δin x, y and z direction (finite

differences):

∂2Etot

∂xi∂xj

=Fi(xj+δ) − Fi(xj−δ)

2δ . (92)

By omitting higher order terms in the potential and with the Ansatz for the

harmonic oscillator va=Aexp(iωt +ϕ)the dynamical matrix is constructed

Dab(q) = ∑

b′

ei(q·Rbb′)

√mamb

∂E

∂xa∂xb⏐⏐⏐⏐Rbb′

(93)

with the atomic masses maand the connecting vector between atoms band b′,

Rbb′. This dynamical matrix determines the equation of motion for a periodic

array of harmonic atoms for each reciprocal vector q

D(q) [v(q)] = ω2(q) [v(q)] . (94)

The eigenvalues ω2of the dynamical matrix D(q)completely describe the dy-

namics of the system (in the harmonic approximation).

30 theoretical framework

Tip

e

Figure 5: Schematic illustration of a Scanning tunneling microscope.

1.9 simulating stm apparent height maps

One of the best established experimental techniques to image surfaces is scanning

tunneling microscopy (STM). It is capable of locally imaging the electron density

of a conducting surface. The basic principle is shown in Fig. 5. A conducting tip

(e.g. tungsten) is brought in the vicinity of the surface and a voltage (bias voltage)

is applied between tip and surface. If the distance (d) between tip and surface

is small enough electrons can tunnel between tip and sample depending on the

applied voltage. This tunneling current is measured at different lateral positions,

resulting in a spatially resolved map of the density of states. The position above

the surface is kept fixed. This is the constant height operation mode. Another

operating mode is the constant current mode. The tunneling current between

sample and tip is kept constant by a feedback loop that automatically adjusts the

sample tip distance. The actuation is typically achieved by very precise pietzo-

crystals.

The theoretical description of STM measurements was given by Tersoff and Har-

man [250]. The quantity we want to calculate from theoretical considerations is

the tunneling current J(r,VB). It is a function of the tip position and the applied

bias or potential. To describe the tunneling between surface and tip the interac-

tion between tip and surface is assumed to be weak. This assumption is valid for

typical experimental tunneling conditions. It can be treated in first order pertur-

bation theory (Fermi’s Golden Rule)

J=2πe



h∑

i,j(f(Ei)[1−f(Ej)]−f(Ej)[1−f(Ei)])|Mij|2δ(Ei+V−Ej)(95)

with the Fermi function [11]f(E), V the bias voltage and Mij the tunneling matrix

element between the states φiand φjof the electrodes (sample and tip). Eiis the

energy of the state φi. By taking the limit T→0the Fermi functions become step

functions and eq. 95 simplifies to

J=2π



he2V∑

i,j

|Mij|2δ(Ei−EF)δ(Ej−EF). (96)

1.9 simulating stm apparent height maps 31

The task is to calculate the tunneling matrix element in eq. 96. Bardeen showed

that it can be expressed as [20]

Mij =

2m ∫dS(φ∗

i∇φj−φj∇φ∗

i). (97)

The integral is over any surface lying in the barrier region. This expression is

valid for a free-electron approximation for the normal state ( Wentzel, Kramers,

Brillouin (WKB) -approximation). To further simplify, a model for the tip is usu-

ally considered. The simplest approximation was put forward by Tersoff and

Hamann [250]. They idealized the STM tip as a mathematical point source of

current, that has maximum resolution and no interaction between surface and

tip. Eq. 96 is reduced to

J∼V∑

|φi(rt)|2δ(Ei+V−EF). (98)

The authors also showed, that this approximation is even valid for more realistic

models. The tip can be considered as an s-wave function and rtis the center of

the wave function. The model breaks down for considerable tunneling to several

tip atoms and if the tip and sample wave function start to overlap for small dis-

tances.

We can use eq. 98 to simulate experimental results from the density obtained

by our DFT calculations. We integrate the local density ρ(r,ϵ)from the Fermi

energy EFto EF+V. The result is a map of the current (local density of states)

at a constant distance rtabove the surface. To compare to experimental results

obtained in the constant current mode, the current has to be calculated for differ-

ent rtand be rearranged to represent the height for a constant current. Absolute

values can not be compared between theory and experiment, because electronic

and geometric properties of the tip, that are difficult to determine experimentally,

enter into the matrix elements in eq. 96. Another general issue is the necessity to

calibrate height measurements for the STM. For this task accurately known step

edge heights are measured. For systems composed of substrate and ad-layer the

STM is typically calibrated on the step edges of the substrate. This introduces

systematic errors if the interaction between tip and substrate differs from the

interaction between tip and ad-layer. Here only height differences between ad-

layers or substrate step edges can be compared to theoretical results calculated

by the above method. Another approach is to calculate heights by subtracting

the height obtained from simulating the bare substrate from the ad-layer system.

This theoretical calibration is in analogy to experiment and allows to qualitatively

and quantitatively compare theoretical and experimental results.

Part III

ZNO ON METAL SUBSTRATES

INTRODUCTION

Zinc oxide (ZnO) is a group II-VI semiconductor with a 3.4eV large, direct band

gap [1]. In nature it is found as the mineral zincite. Its color is yellow to red

because of the naturally occurring manganese content [125]. The ZnO used in

industry and science is a clear and colorless synthetic material [208]. It is not poi-

sonous and Zn as its main production source is easily available [185]. Zn forms

0.0076 % of the earth’s crust [160] and is therefore one of the more abundant

materials.

ZnO has many applications in industry and science. It is produced on industrial

scales in the order of 105tons per year [125,165]. It is used as an additive for the

production of ceramics. By adding ZnO to ceramics the thermal conductivity is

increased and the elasticity improved [165]. Another major application lies in tire

production. The admixture of ZnO to rubber helps to dissipate the heat during

vulcanization [125,165]. Other fields of application include medicine [190] and

cosmetics [105]. Here ZnO is used as UV-blocker in suntan solutions [164]. In

photo-voltaics it is used as front contact for solar cells or liquid crystal displays

[126,18].

A first period of strong interest in ZnO peaked in the middle of the 20th century.

The focus was on possible application in opto-electronics [125]. These efforts sub-

sided because is was deemed impossible to p-type or n-type dope ZnO. A second

period of intensified ZnO research started in the middle of the 1990 when the ad-

vances in thin-film technology made it possible to grow epitaxial films, quantum

dots and other nano structures [187]. The goal is to replace GaN as material

for light emitting diodes in opto-electronics (Ga is expensive and poisonous) by

ZnO [18,148]. Other applications might include spin-tronics for ZnO doped with

atoms of high magnetic moment[204]. A large effort goes into replacing tin oxide

by ZnO doped with Al, Ga, etc. to achieve high conductivity while sustaining

opaqueness.[126]

Another major field of research is heterogeneous catalysis. ZnO is used as cata-

lyst in industrial scale synthesis of methyl [25]. The active components are ZnO

supported metal nano particles. However, the actual catalytic mechanism as well

as the atomic structure of the catalytically active ZnO are still under investigation

[136].

We will start with a short description of the bulk phases of ZnO and their sta-

bility [1]. Than we will shortly discuss the low index surfaces of ZnO. From

the polar surfaces ZnO (0001) and (000¯

1) ultra-thin films will be derived and

sub-sequentially fitted to metal substrates. We will investigate the atomic and

electronic structure of these ultra-thin ZnO films on metal substrates and access

their stability by means of ab initio thermodynamics (see Sec. 1.6) with respect to

their chemical environment.

BULK ZNO

(a) Zinc-blend (b) Rock-salt (c) Wurtzite

Figure 6: Crystal structures of ZnO. The unitcell is marked in black. Zn - purple, O - red

The most stable bulk phase of ZnO under ambient conditions is the wurtzite

structure which is shown in Fig. 6c. The c-axis of the crystal is chosen parallel to

the z-direction. Wurtzite exhibits a hexagonal symmetry along the c-axis as well

as strong polarity along this axis. Zn atoms are 4-fold coordinated with O and

wise versa. The tetrahedra created by this coordinations are indicated in Fig. 6c

for one Zn and O atom. The ideal value for the c/a ratio is c/a=√8/3=1.633 [1].

Another ZnO polymorph is zinc-blend. Zinc-blend can be stabilized by epitaxial

growth on suitable cubic substrates [12]. The zinc-blend structure is shown in

Fig. 6a. The four-fold coordination between Zn and O are indicated by yellow

tetrahedra. By application of high pressures ZnO can be stabilized in the rock-

salt structure [68]. This structure is shown in fig. 6b. ZnO in rock-salt structure

has a much closer packing and a 6-fold co-ordination which is indicated by the

yellow polyhedra in fig. 6b. Other structures have been proposed to grow at even

higher pressures [109,282].

To determine the lattice parameters a fit to the Birch-Murnighan equation of

state (eq. 99) [174]

E(V) = E0+B0V

B′

0((V0/V)B′

B′

0−1+1)−B0V0

B′

0−1(99)

was performed for the three most common crystal structures of ZnO: wurtzite,

rock-salt and zinc-blend. These structures represent only a small portion of a

much larger phase space [109,282]. The theoretical investigation of all these

phases is not in the scope of this work and the reader is referred to the litera-

ture.

The Energy per atom E(V)curves for the three crystal structures is shown in Fig. 7

for the local PW-LDA [199], semi-local PBE [196] and the hybrid xc-functional

HSE06 [100]. By fitting to eq. 99 the equilibrium cohesive energy E0, the bulk

modulus B0, its derivative B′

0and the equilibrium volume V0are determined.

38 bulk zno

8 9 10 11 12 13 14

V[˚

A3]

−4.5

−4.4

−4.3

−4.2

−4.1

−4.0

Ecoh [eV per Atom]

Rocksalt

Zinc Blend

Wurtzite

Common Tangent

(a) LDA.

8 9 10 11 12 13 14

V[˚

A3]

−3.5

−3.4

−3.3

−3.2

−3.1

−3.0

Ecoh [eV per Atom]

Rocksalt

Zinc Blend

Wurtzite

Common Tangent

(b) PBE.

8 9 10 11 12 13 14

V[˚

A3]

−3.5

−3.4

−3.3

−3.2

−3.1

−3.0

Ecoh [eV per Atom]

Rocksalt

Zinc Blend

Wurtzite

Common Tangent

Figure 7: Cohesive energy versus crystal volume for different bulk phases of ZnO for

different xc-functionals.

The fitted values for the three xc-functionals of Fig. 7are presented in Tab. 3to-

gether with experimental reference data. From the values for the cohesive prop-

erties, the general remarks from sections 1.4.1,1.4.2and 1.4.3are confirmed. PBE

improves the over-binding of LDA, leading to lattice parameters in good agree-

ment with experimental values in Tab. 3. By using HSE06, the overall agreement

between theory and experimental values is slightly increased for wurtzite and

rock salt. For the wurtzite structure the lattice parameter in c-direction is larger

than observed in experiment, while the lattice parameter in the a/b-plane is now

below the experimental value. However, the comparability of experiment is lim-

ited, because experimental values were measured at room temperature and no

finite temperature effects were included in the calculations (T=0K). By means of

the Maxwell construction [54] a common tangent is fitted to the E(V)curves in

Fig. 7for wurtzite and rock salt. The slope of the tangent determines the transi-

tion pressure between both structures. The slope of the common tangent in Fig. 7

a is 9.58 GPa, Fig. 7b is 11.6GPa and Fig. 7c is 10.8GPa.

In Fig. 8the electronic band structures and (projected) density of state (DOS)

for rock-salt (a and b), zinc-blend (c and d), and wurtzite (e and f) are shown

for the PBE [196] and HSE06 xc-functional. Calculating the electronic structure

with local or semi-local DFT leads to an underestimation of the band gap and

inter-band transition energies. The effect of the notorious self interaction error is

greatly reduced by applying the hybrid xc-functional HSE06 [100]. The band gaps

calculated with the PBE (HSE06) xc-functional are 0.73 eV (2.57 eV) for wurtzite,

0.64 eV (2.51 eV) for zinc-blend and 1.97 eV (3.71 eV) for rock-salt at the Γ-point.

For wurtzite and zinc blend this is the direct band gap. For rock-salt a indirect

band gap between the Γ- and L-point of 0.75 eV (2.69 eV) is observed. The large

bulk zno 39

PW-LDA PBE HSE06 Exp.

Wurtzite

V0[Å3]22.700 24.619 23.96 23.99 [61]

E0[eV] 9.13 7.425 6.77 7.52[61]

B0[GPa] 177 125 140 181[61]

B′

05.09 4.13 3.9 4.0[61]

c [Å] 5.150 5.291 5.239 5.220[61]

a [Å] 3.190 3.279 3.250 3.258[61]

u0.379 0.379 0.379 0.382[61]

Zinc-blend

V0[Å3]22.694 24.656 23.434

E0[eV] 11.316 7.040 3.463

B0[GPa] 158.76 129.655 155.460

B′

04.244 3.950 4.005

a [Å] 4.083 4.620 4.543 4.463[13]

Rock-salt

V0[Å3]18.669 20.283 19.540 19.484[119]

E0[eV] 11.114 6.744 6.650

B0[GPa] 205.549 163.469 190.659 228[119]

B′

04.781 4.343 4.185

a [Å] 4.211 4.329 4.276 4.271[119]

Table 3: Lattice parameters for zinc blend, rock salt and wurtzite bulk ZnO. Experimental

data was measured at room temperature.

peaks in the DOS in the valence band originate from Zn 3d states. The valence

band highest in energy is dominated by states of O 2p character. The conduc-

tion band lowest in energy around the Γ-point is dominated by Zn 4s-like states.

Qimin et al. showed, that by a quasi particle correction within the GW approxi-

mation [280] the agreement of theoretical and experimental electronic structure

can be further increased.

The semi-local PBE xc-functional will be the basis for the surface calculations in

the following sections. The obtained cohesive parameters are in good agreement

with experimental data and calculations with the HSE06 hybrid xc-functional.

The band gap is significantly underestimated, but the electronic structure is oth-

erwise in good agreement with HSE06 results. If feasible the accuracy of the

results will be tested with HSE06.

In the following sections, due to its prevalence, the wurtzite structure is the major

structure for our investigation.

40 bulk zno

(a) Rock-salt (PBE). (b) Rock-salt (HSE06).

(e) Wurtzite (PBE). (f) Wurtzite (HSE06).

Figure 8: Electronic band structure and density of states (DOS) for the three prevalent

bulk crystal structures of ZnO calculated with the PBE [196] (a, c, e) and HSE06

[100] (b, d, f) xc-functional. The occupied states in the DOS are colored in light

red. The Fermi energy is set as zero for all graphs.

SURFACES OF ZNO

Figure 9: ZnO planes

ZnO has a wide range of applications, especially in opto-electronics and het-

erogeneous catalysis. Surfaces and interfaces are of crucial importance for these

fields. Most chemical processes occur at surfaces and devices can have interfaces

(e. g. hetero junctions). The experimental techniques available are manifold. The

standard technique for the experimental investigation of ZnO surfaces is low

energy electron diffraction (LEED) [284]. It is very sensitive to periodically or-

dered structures on the surfaces. The LEED diffraction patterns of all low index

surfaces have been reported [72]. Another method is the scattering of helium

atoms [284]. Its particular advantage is its sensitivity for hydrogen. Hydrogen

over-layers on ZnO could be detected with this techniques [72,275]. Other meth-

ods for obtaining information about the chemical composition of ZnO surfaces

are: X-ray photo electron spectroscopy (XPS) [284], photo electron spectroscopy

(UPS) citeZangwill88, X-ray emissions spectroscopy (XES) and near edge X-ray

absorption fine structure spectroscopy (NEXAFS) [284]. For measuring binding

energies thermal desorption spectroscopy (TDS) can be used. To investigate and

identify adsorbates at surfaces, vibrational spectroscopy is used. Infrared spec-

troscopy (IR) [284] is a widely applied technique for this purpose. For direct

structural characterization and local spectroscopy scanning tunneling microscopy

and spectroscopy are important methods (see Sec. 1.9). The four scientifically

most important low index surfaces are indicated in Fig. 9. A general problem

is the requirement of a conducting sample for many experimental techniques.

42 surfaces of zno

(a) ZnO 10¯

10 (b) ZnO 11¯

Figure 10: ZnO non-polar surfaces

Low conductance can lead to charging effects and ultimately the destruction of

the sample. Methods capable of atomic resolution e. g. scanning tunneling mi-

croscopy are therefore difficult to apply. A comprehensive presentation of the

results obtained with the experimental methods mentioned previously can be

found in the review by Wöll [276].

The scientifically interesting surfaces of ZnO are divided into two categories:

the non-polar surfaces ZnO (10¯

10) and ZnO (11¯

20) (Fig. 10) and the polar sur-

faces ZnO-Zn (0001) and ZnO-O (000¯

1) (Fig. 11). For ionic crystals, such as ZnO,

without a mirror plane in the z-direction and no inversion symmetry, both types

of surfaces exist. They can be defined by the projection of the bulk unit cell

dipole moment on the surface normal. A polar surface is observed if this pro-

jection is non-zero. For ZnO this is the case for the {0001} surfaces, because the

bulk dipole moment is directed along the {0001} direction. The non-vanishing

dipole moment in this direction creates an electric field. Within a simple ionic

model 1/4of the Zn+2-ions on the Zn-terminated and 3/4of the Zn- and O-ions

at the O-terminated surfaces are missing in every building block of a slab (see

Fig. 11). This results in an excess charge of +1/2electrons and -1/2electrons at

the (0001) and (000¯

1) surface, respectively. This ultimately leads to a polarization

catastrophe for the ZnO (0001) surfaces. These stability problems were first ad-

dressed by Tasker [249] (Tasker-Type-II surfaces). The infinite potential difference

(a) ZnO-Zn 0001 (b) ZnO-O 000¯

Figure 11: ZnO polar surfaces

surfaces of zno 43

(a) Zn vacancy in a 4x4surface super cell. (b) O ad-atom in a 2x2surface super cell.

Figure 12: ZnO-Zn (0001) stabilization mechanisms.

is unphysical and basically a consequence of the simple ionic model. There are

three dominant mechanisms to stabilizes these polar ZnO (0001) surfaces. Charge

redistribution of 1/2electron form the Zn to the O terminated surface or a ge-

ometry reconstruction would quench the dipole moment. The third possibility is

the adsorption of atoms or molecules, that provide a compensating charge. For

the non-polar (10¯

10) and (11¯

20) surfaces a structure close to a bulk truncation is

expected [162,242].

The main topics of this work are the atomic and electronic structure of ZnO

thin films on metal substrates. Before we start investigating thin films, previous

results on reconstruction mechanisms and stability for the ZnO (0001) and (000¯

surface will be presented. These results are a valuable guideline for our discus-

sions of ultra-thin ZnO on metal substrates.

The Zn terminated ZnO (0001) surface is experimentally and theoretically better

understood than the O terminated surface ZnO (000¯

1). In joined experimentally

and theoretically effort different stabilizing mechanisms for the ZnO (0001) sur-

face were investigated [132,261]. Both works applied DFT based on the PW91

xc-functional [199] and the projector augmented wave method [32]. The recon-

structions investigated by Kresse et al. and Valtiner et. al for the Zn terminated

ZnO (0001) surface are:

• Zn vacancies or O ad-atoms (Fig. 12 a and b)

• Adsorption of hydrogen or OH groups (Fig. 12 c and d)

• Triangular pits (see Fig. 13 a to c), experimentally observed by Dulub et al. [72]

44 surfaces of zno

(a) Triangular pit with side length n=4. (b) Triangular pit of side length 6with upside-

down triangular pit with side length n=3in

the second layer and oxygen ad-atom.

down triangular pit with side length n=3in

the second layer.

(d) (√3×√3)R30◦) − (2×1)Hreconstruction.

Figure 13: ZnO-Zn (0001) stabilization mechanisms.

• A (√3×√3)R30◦)reconstruction with hydrogen arranged in a (2×1) struc-

ture [261] (see Fig. 13 d)

By taking into account these structures the authors constructed surface phase

diagrams. Valtiner et. al introduced the explicit temperature dependence in the

phase diagram by including the vibrational free energy (see. chapter 1.6). The

authors found that at low hydrogen and oxygen chemical potentials the trian-

gular pits are favored. At high hydrogen chemical potential OH-reconstructed

surfaces are stable. The structure with 50% OH coverage is most stable for a

wide range of chemical potentials. At intermediate oxygen and hydrogen poten-

tials the (√3×√3)R30◦–(2×1)–H and 2×2O-ad atom reconstructions are most

stable.

For the oxygen terminated surface ZnO-O (000¯

1) an extensive theoretical inves-

tigation was conducted by Wahl et al. [267]. The PW91 xc-functional [199] was

applied within the projector augmented wave method [32]. Vibrational free en-

ergy and configurational entropy were neglected. The considered structures were:

• O vacancies or Zn ad-atoms (Fig. 12 a and b)

• Adsorption of hydrogen (Fig. 14 a)

• Triangular pits (see Fig. 13 a to c)

surfaces of zno 45

(a) (2×1)–H reconstruction. (b) (2×2) hexagonal reconstruction.

ditional hydrogen atoms adsorbed.

Figure 14: Reconstructions for ZnO-O (000¯

1), that are not observed for ZnO-Zn (0001).

For triangular pits see Fig. 13.

• A hexagonal honeycomb phase (see Fig. 14 c and d)

• A (2x2) reconstruction by removing two oxygen and one zinc atom from

the surface (see Fig. 14 b)

A full surface phase diagram was constructed by Wahl et al. [267] with these

structures. The authors found, that the (2x2) reconstruction is preferred at inter-

mediate to low oxygen chemical potentials. At very low oxygen chemical poten-

tials the 25% oxygen vacancy reconstruction prevails. At high oxygen chemical

potential and low hydrogen chemical potential the honeycomb structure is most

stable. For intermediate hydrogen chemical potential honeycomb structures with

additional hydrogen atoms adsorbed at the surface are stable for most oxygen

chemical potentials. At higher hydrogen chemical potentials the 50% hydrogen

covered surface is most stable.

Apart from the geometric structure, it is interesting to look at the electronic struc-

ture of ZnO surfaces and compare it to bulk ZnO. In Fig. 15 the band structure

for ZnO (000¯

1)-(2×1)–H and (10¯

10) is projected on the bands of bulk ZnO. The

band structure and DOS was calculated with the HSE xc-functional. We chose

the fraction of exact exchange in the HSE xc-functional to be α=0.4, yielding the

calculated band width [211] and band gap within 0.1eV of error (Moll et al. [167]:

α=0.4, Oba et al. [183]: α=0.375). The bulk like bands are shaded in gray. For ZnO

(000¯

1) the density of states of the surface is quite similar as compared to bulk.

There are no pronounced surface bands emerging. The situation is very different

46 surfaces of zno

(a) ZnO (000¯

1) (b) ZnO (10¯

10)

Figure 15: Projected band structures for two ZnO surfaces (HSE, α=0.4). The surface

band structure is projected on the bulk bands. The bulk-like bands are shaded

in gray, surface bands are blue. The DOS is plotted to the right of the band

structure for bulk ZnO in gray and in blue for the surfaces.

for ZnO (10¯

10). Here two surface bands close to the conduction band and close

to the valence band are observed. These bands are formed by the states that origi-

nated from the dumbbells at the surface, breaking the bulk symmetry (see Fig. 10

a) in analogy to the Si (0001) surface. These surface bands result in a reduction

of the band gap and a pronounced increase of the density of states close to the

Fermi level.

FREESTANDING ZNO THIN FILMS

(a) Top view. (b) Sied view.

Figure 16: Atomic structure of an ideal ZnO mono-layer in α-BN structure.

Before we proceed to putting ZnO films on metal substrates, we will briefly

investigate the structure and stability of a hypothetical free-standing ZnO mono-

layer. Our calculations predict the ZnO mono-layer to be stable in the α-BN struc-

ture in agreement with previous results [258], see Fig. 16 a. The equilibrium cor-

rugation for the mono-layer is zero. Therefore, only the in-plane lattice parameter

has to be varied for the Murnaghan-EOS fit. The same values for the k-point grid

(kx-/ky-plane) and basis set settings as for bulk wurtzite ZnO were used (see

Appendix C). In Tab. 4the equilibrium lattice parameter of the freestanding ZnO

mono-layer is listed for the local PW-LDA [199], the semi-local PBE [196] and

the hybrid xc-functional HSE06 [100]. The general remarks of Sec. 1.4.1,1.4.2

and 1.4.3are confirmed by the comparison. The over-binding of LDA is reduced

when using the PBE xc-functional. The lattice parameters obtained with PBE

and HSE06 are similar, following the same trend as observed for bulk wurtzite

(a) Band structure and DOS (PBE). (b) Band structure and DOS (HSE06).

Figure 17: Electronic structure of an ideal ZnO mono-layer in α-BN structure.

48 freestanding zno thin films

xc PW-LDA PBE HSE06

a [Å] 3.198 3.277 3.252

Gap [eV] 1.78 1.77 3.49

Table 4: In-plane lattice parameter aof the ZnO mono-layer for different functionals.

ZnO (see Tab. 3). Topsakal et al. [255] confirmed the stability of the freestanding

ZnO mono-layer by frozen phonon-calculations and ab initio molecular dynamics

simulations. No negative phonon modes were observed, which would lead to a

dissociation of the mono-layer. It is a direct band-gap semi-conductor, the band

gaps for the LDA, PBE and HSE06 xc-functional are listed in Tab. 4. The band

structure and the DOS are shown in Fig. 17 a for PBE and in Fig. 16 b for HSE06

[100]. The bonding is of sp2hybrid orbital nature. The upper part of the valence

band is derived from O-2p orbitals, and the lower part of the conduction band

from the Zn-3d orbitals.

5.1 mechanical stability

In agreement with the results of Peng et al. [194] we found that the freestand-

ing ZnO mono-layer is much softer than graphene-like hexagonal boron nitride

[269]. Smaller strains could be applied to the mono-layer before destruction. Peng

et al. [194] performed a least square fit of the stress-strain responses and con-

cluded that the ultimate strength of the hypothetical freestanding ZnO mono-

layer is about half of graphene and hexagonal boron nitride. The elastic constants

were obtained by straining the film in the direction of an Zn-O bond and perpen-

dicular to this direction. The strain ηwas defined as the relative change of the

lattice vector in the direction of straining.

5.2 transition mono-layer to bulk

The transition from the α-BN (graphene-like) ZnO structure is of particular inter-

est in the context of layer by layer film growth e. g. by molecular beam epitaxy.

This transition was reported to occur at 18 single layers ZnO [51]. The authors

compared the cleavage energy of the ZnO film in α-BN structure with an inter-

layer spacing of 2.4Å to the binding energy of the ZnO thin films in wurtzite

structure. The fixed inter-layer spacing between the ZnO layers has been pointed

out as possible source for errors by other authors [258]. By considering the in-

terlayer attraction coming from the spontaneous polarization of the layers, when

switching to wurtzite, Tu et. al. [258] determined the critical layer number as 3or

greater. To confirm these previous results we calculated the cohesive energy of

ZnO thin-film slabs of different thickness in wurtzite and α-BN. For both struc-

tures the lattice parameter cis kept at a constant value of 5.24 AA. This translates

into an inter-layer spacing of 2.62 Å for the films in α-BN structure. The result

is shown in Fig. 18 a for the PBE xc-functional. The transition is observed at

4double layers of ZnO, 8layers of planar ZnO. The effect of including contri-

5.2 transition mono-layer to bulk 49

012 3 4 5 678

Slab thickness in double layers

3.8

3.7

3.6

3.5

3.4

3.3

3.2

3.1

Ecoh [eV / Zn-O-pair]

α-BN

Wurtzite

(a) PBE xc-functional.

012 3 4 5 678

Slab thickness in double layers

3.8

3.7

3.6

3.5

3.4

3.3

3.2

Ecoh [eV / Zn-O-pair]

α-BN

Wurtzite

(b) PBE+vdWT S xc-functional.

Figure 18: Cohesive energy per ZnO-pair for ZnO in α-BN and wurtzite structure against

slab thickness for semi-local functionals.

butions from van-der-Waals interactions is shown in Fig. 18 b). The TS-scheme

[254] with parameters provided by Zhang et al. [286] was used (also see Sec. 1.5).

The cohesive energy of both structures is lower. For wurtzite ZnO the effect is

more pronounced. The transition from α-BN is observed one planar layer ear-

lier as compared to the plain PBE xc-functional. If we use the hybrid functional

HSE06 [100] including contributions from van-der-Waals interactions with the TS-

scheme (Fig. 19 a) the transition occurs at 4and a half double layers (Fig. 19 a).

With the xc-functional HSE06* [167,211] (HSE06 [100] with 40% exact exchange,

α=0.4) the transition is predicted above 5double layers.

We can conclude, after taking into account van-der-Waals contributions and re-

sults obtained with the PBE and HSE06 xc-functional, that the α-BN to wurtzite

transition will occur above 4double layers or 8planar mono-layers.

12 3 4

Slab thickness in double layers

4.2

4.1

4.0

3.9

3.8

3.7

3.6

3.5

Ecoh [eV / Zn-O-pair]

α-BN

Wurtzite

(a) HSE06+vdWT S xc-functional.

12 3 4 5

Slab thickness in double layers

4.2

4.1

4.0

3.9

3.8

3.7

3.6

3.5

3.4

Ecoh [eV / Zn-O-pair]

α-BN

Wurtzite

(b) HSE06*+vdWT S xc-functional.

Figure 19: Cohesive energy per ZnO-pair for ZnO in α-BN and wurtzite structure against

slab thickness for hybrid functionals.

50 freestanding zno thin films

Figure 20:α-BN bulk ZnO.

5.2.1α-BN bulk ZnO

Metastable α-BN bulk ZnO was synthesized experimentally through a molecu-

lar precursor route at low temperatures (2◦C) [144]. The unit cell of the material

is shown in Fig. 20 a. The major difference to wurtzite ZnO is that Zn and O

form layers with Zn and O having the same position in the lateral direction

for each layer. The experimental lattice parameters obtained by powder x-ray

diffraction are a=3.099Å and c=3.858Å [144], resulting in a crystal volume of

V= (√3a2)c/4 =16.044Å. The material was found to be stable up to 200◦C,

when it transforms into the thermodynamically most stable wurtzite structure.

We obtained the PBE and HSE06 lattice parameters from an fit to the Mur-

naghan equation of state [174]. They are shown in Tab. 5. The results are in

agreement with previous results [210,268]. The experimentally observed struc-

ture is meta-stable. By comparing the energy versus volume curves of ZnO in

α-BN and wurtzite structure (see Fig. 21), we find the α-BN phase to be energet-

xc PBE HSE06

V0[Å3]23.70 22.87

E0[eV] 6.89 6.63

B0[GPa] 238.24 296.40

B′

07.77 7.90

a [Å] 3.45 3.40

c [Å] 4.598 4.57

Table 5: Equilibrium lattice parameters for α-BN ZnO.

5.2 transition mono-layer to bulk 51

16 18 20 22 24 26 28

V[◦

A3]

8.0

7.5

7.0

6.5

6.0

5.5

5.0

4.5

4.0

E(V) [eV]

Wurtzite

α-BN

Exp. conditions

Murnaghan EOS for bulk ZnO - PBE+vdw

(a) PBE+vdWT S xc-functional.

16 18 20 22 24 26 28

V[◦

A3]

E(V) [eV]

Wurtzite

α-BN

Exp. conditions

Murnaghan EOS for ZnO - HSE06

(b) HSE06 xc-functional.

Figure 21: Comparison of E(V) for bulk wurtzite and α-BN ZnO.

ically most stable at the experimentally observed volume. The lattice parameters

for the PBE (HSE06) xc-functional a=3.08 Å (3.1Å) and c=3.90 Å (3.84 Å) are in

good agreement with experimental results [144]. However, we find in agreement

with Rakshit et al. [210] and Wang et al. [268], that the experimental structure has

negative phonon modes for the PW-LDA and PBE xc-functional and is therefore

unstable. This disagreement between experiment and theory, which works well

for wurtzite ZnO, is not fully understood. Rakshit et al. [210] showed that by ap-

plying PBE+U the α-BN crystal can be stabilized and the negative phonons are

removed. They attributed this effect to the wrong energetic position of the Zn-d

orbitals, which is corrected by the +U formalism.

METAL SUPPORTED MONO-LAYER FILMS

mm1

(m1)

metal

ZnO Zn

Figure 22: Exemplary coincidence structure of 6×6surface unit cells of Ag (111) and

5×5unit cells of mono-layer α-BN ZnO.

Ultra-thin ZnO films have been experimentally observed on the (111) surface

of the metal substrates Ag [259,237], Pd [270], Pt [159], Cu (brass) [229], Au [67],

and also on Ru (0001) [120]. For Ag, Au, Pd, and Pt, coincidence structures have

been deduced from the observation of Moiré patterns and crystallographic data.

The observed coincidence structures are combinations of m×mmany metal (111)

surface unit cells and m−1×m−1many mono- or multi-layers of ZnO (see Fig.

21). Sub-mono-layer coverages yield flat graphitic (α-BN) islands of ZnO. For

ZnO on Ag, the transition to the wurtzite structure occurs between 3and 4layers

[259], but already at 2layers the onset of corrugation has been observed on Ag

and Pd [259,270]. Theoretical investigations of freestanding and supported ZnO

films show that lateral strain influences the corrugation of the films [278,66] and

may decrease the number of ZnO layers necessary for the transition to wurtzite.

Weirum et al. [270] found that oxygen promotes the growth of bilayer structures

for low coverages, leading to the transformation of mono-layer islands to bilayer

islands on Pd. The authors assigned the film thickness based on STM apparent

height measurements. For the bulk ZnO surfaces, hydrogen has a significant ef-

fect on the surface structures [132,261,267]. However, at present it is unclear if

hydroxyl groups form on the ultra-thin films and how they effect the atomic and

electronic structure. The influence of the different metal substrates has also not

yet been investigated systematically.

In this chapter we will investigate the lattice mismatch between an ideal mono-

layer of ZnO and the metal substrates Ag, Cu, Pd, Ni and Rh. Furthermore we

54 metal supported mono-layer films

will address the differences in the atomic structure and thermodynamic stability

of pristine ultra-thin ZnO and ZnO on transition metals. The starting point for

our investigation will be hypothetical freestanding ZnO films we introduced in

Chap. 5.

6.1 coincidence structures

(a) Top view of an fcc (111) surface. (b) ZnO mono-layer in an α-BN structure.

Figure 23: Hexagonal structure of fcc (111) and ZnO (0001/000¯

Fcc (111) surfaces as well as mono-layer ZnO in the α-BN structure have a

hexagonal symmetry (see Fig. 23). Ideally the "honeycombs" of the ZnO sheet

would fit on the "honeycombs" of the fcc (111) surface. The in-plane lattice param-

eter of these commensurable ZnO mono-layers is easily calculated by dividing

the metal (fcc) lattice parameter by √2. In Fig. 24 the resulting ZnO lattice pa-

rameters are indicated by black vertical lines, labeled with the chemical symbol

of the metal. Together with these values we show in Fig. 24 the cohesive energy

of a freestanding ZnO mono-layer as a function of the in-plane lattice parameter

in the α-BN structure (dz=0Å, see Fig. 24 for definition), with wurtzite-like spac-

ing between Zn and O along the surface normal (dz=0.63Å) and with the atomic

positions (z) along the surface normal relaxed. We find that commensurable struc-

tures would favor wurtzite-like corrugation because of the strong compression

of the in-plane lattice parameter of ZnO for most transition metal (111) surfaces.

The only exception is the (111) surface of Pb, which has a larger in-plane lattice

parameter than ZnO (see Fig. 24). Too much strain will force the film out of its

preferred planar structure towards a wurtzite-like structure [278]. According to

the results shown in Fig. 24 the switch from planar to wurtzite occurs at a ZnO

in-plane lattice parameter of 3Å (∼9% strain), indicated by the dashed line. The

strain in the films will be minimized when the in-plane lattice parameter of the

metal-supported ultra-thin films matches the lattice parameter of freestanding

mono-layer ZnO. The most likely coincidence structure can be estimated by min-

6.1 coincidence structures 55

2.4 2.6 2.8 3.0 3.2 3.4 3.6

a[◦

Ecoh [eV]

abulk

dz=0.63 ◦

A(bulk)

dz=0.0 ◦

A(ultrathin-film)

dz relaxed

Figure 24: Cohesive energy Ecoh of an ideal α-BN (dz=0Å), a wurtzite/zincblend

(dz=0.63Å) and a relaxed ZnO mono-layer as a function of the in-plane lat-

tice parameter a. The in-plane lattice parameters (a/√2) of the (111) surface

of selected fcc transition metals are indicated by vertical lines.

imizing the strain on the in-plane lattice parameter aof the ZnO mono-layer as

compared to the ideal freestanding film:

min

m(aZnO −m

m−1

√2). (100)

In Tab. 6the most likely coincidence structures for different transition metal

fcc (111) surfaces are calculated from the equilibrium lattice parameters obtained

with the PBE xc-functional. To confirm the predicted coincidence structure we

consider the formation energy ∆H of the combined system (ZnO mono-layer

on metal) as a function of the number of unit cells malong the in-plane lattice

vectors of the metal

∆H =(ETot −m2EM− (m−1)2EZnO)/A. (101)

Here ETot is the total energy of the coincidence slab, EMis the energy of the 1x1

metal surface, EZnO is the energy of freestanding 1x1ZnO mono-layer and A

the surface area of the structure. Reference energies and lattice parameters for

metals and ZnO are reported in Appendix B. For m×mmetal unit cells with

a fixed lattice parameter aMthis results in (m−1)×(m−1) unit cells of ZnO

with a ZnO lattice parameter of aZnO =m−1

√2. The formation energy (∆H)

curves for one layer of ZnO on different metals are shown in Fig. 25 and the coin-

cidence structures with the lowest formation energy ∆H are listed in Tab. 7. The

agreement with experimental values for sub-mono-layer ZnO islands is good (Ag

[259,237], Pd [270], Pt [159], Au [67], Cu (brass) [229], Ru (0001) [120]). The coin-

cidence structures for which the ZnO in-plane lattice parameter is closest to that

of the freestanding structure gives the lowest formation energy. These theoreti-

cally predicted coincidence structures depend on the relation between the metal

56 metal supported mono-layer films

Table 6: Coincidence structures calculated from the lattice parameters obtained with the

PBE xc-functional for fcc metals by eq. 100.

Metal aExp [Å] a [Å] a

√2[Å] coincidence Exp.

Ag 4.08 4.15 2.934 9 /8 8 /7[259,237]

Pt 3.92 3.97 2.807 7 /6 6 /5[159]

Pd 3.89 3.942 2.787 7 /6 6 /5[270]

Au 4.07 4.158 2.94 10 /9 8 /7[67]

Pb 4.95 5.041 3.565 15 /14 -

Al 4.04 4.041 2.857 8 /7-

Cu 3.62 3.632 2.568 5 /4-

Ni 3.52 3.51 2.482 4 /3-

and ZnO lattice parameters obtained with a specific xc-functional. For larger co-

incidence structures (Ag, Au) small changes in the lattice parameters can lead to

different predicted coincidence structures, while systems with small coincidence

structures (Pt, Pd, Ni, Cu) are less sensitive (see Appendix Dfor a list of predicted

coincidence structures for different xc-functionals). For the metals with lattice pa-

rameters larger than 4Å (Ag, Au) the predicted coincidence structures can differ

by up to two digits for the BLYP xc-functional, which was optimized for soft

matter. Overall, the differences between xc-functionals are small and we attribute

them to small differences in the equilibrium lattice parameters. For metals with

lattice parameters smaller than 4Å (Pt, Pd, Cu, Ni) the predicted coincidences do

not depend on the employed xc-functional. In the following sections we will use

the PBE xc-functional because of its good agreement with experimental results

246 8 10 12

0.04

0.03

0.02

0.01

0.00

0.01

0.02

∆ H [eV/ ◦

A2]

Figure 25: Formation energy of differently sized coincidence structures for selected tran-

sition metals.

6.1 coincidence structures 57

Metal strain coincidence experimental ∆Φ [eV] dz [Å]

Ag 0.7%9/8 8 /7[259,237]0.14 0.127

Pd 0.8%7/6 6 /5[270]0.06 0.237

Pt 0.2%7/6 6 /5[159] -0.04 0.246

Ni 1.0%4/3- -0.05 0.266

Cu -2.0%5/4-0.43 0.292

Rh 0.8%6/5-0.07 0.337

Table 7: Experimental and calculated (PBE) coincidence structures (m ×m-1) for ZnO

mono-layers on the (111) surface of different transition metals. dz is the corru-

gation of the ZnO mono-layer and the strain is the lattice mismatch between

adsorbate film and freestanding mono-layer. For ideal bulk wurtzite ZnO dz is

0.63 Å and 0Å for an ideal freestanding ZnO α-BN mono-layer. ∆Φ is the work

function change between the bare metal surface and the ZnO mono-layer on the

metal substrate.

and its generality [196].

Another aspect, that could influence the coincidence structure observed in exper-

iment is the chemical environment. Different concentrations of Zn on the surface

or the O2-partial pressure could lead to larger or smaller coincidence structures.

We will use the framework of ab initio thermodynamics, which we introduced in

Figure 26: Surface phase diagram for differently sized coincidence structures of one layer

of ZnO on Cu (1ZnO/Cu). The size of the coincidence structures is indicated

by the color-bar (m/n).The dashed line represents the condition of bulk ZnO

as reservoir for Zn/O atoms (eq. 60).

58 metal supported mono-layer films

Figure 27: Surface phase diagram for differently sized coincidence structures of one layer

of ZnO on Ag (1ZnO/Ag). The dashed line represents the condition of bulk

ZnO as reservoir for Zn/O atoms (eq. 60).

Chap. 1.6, to analyze the stability of differently sized coincidence structures. In

Fig. 26 a the surface free energy of differently sized coincidence structures for

one layer of ZnO on Cu (1ZnO/Cu) is shown as a function of the oxygen and

zinc chemical potentials. In Fig. 27 the same surface phase diagram is shown

for one layer of ZnO on Ag (1ZnO/Ag). We lifted the condition of bulk ZnO as

reservoir for Zn, that is imposed by eq. 60.∆µZn and ∆µOare therefore treated

as independent variables. The oxygen chemical potential is translated to partial

pressures with the help of thermodynamical tables [178]. As temperature for the

partial pressure we chose 800 K, which lies in the vicinity of the temperature

used in experiments during the growth of ZnO films by molecular beam epitaxy

(MBE) [188,270,159,237] (see Sec. 9for a more detailed discussion of experi-

mental results). If we would impose the condition of a Zn reservoir via bulk ZnO

(eq. 60) we would only find one stable structure (see Tab. 7). The path through the

surface phase diagram corresponding to eq. 60 is marked in Fig. 26 and Fig. 27

by dashed lines. Along this path only the coincidence structure found by mini-

mizing the strain in the ZnO film is stable (Ag: 9/8, Cu: 5/4). This coincidence

is also the most stable structure in the full phase diagrams of Fig. 26 and Fig. 27

for oxygen partial pressures typically encountered in experiment (1·10−10 mbar)

and for a wide range of Zn chemical potentials (Zn surface concentration). This

situation is encountered for all metals discussed previously and we will focus on

the coincidence structures presented in Tab. 7for the following discussions.

6.2 position in the 1x1 surface unit cell 59

6.2 position in the 1x1 surface unit cell

(a) Zn at top site (O at hcp hollow) in a 1x1fcc

(111) metal surface cell.

(b) Binding energy between (111) Ag surface and

commensurable ZnO layer as function of the

position of the ZnO within the surface unit

cell.

Figure 28: ZnO mono-layer on Ag in a 1×1geometry.

In addition to the lattice match between ZnO and a given metal discussed in

the previous section we must also consider the relative position of the Zn and

O atoms with respect to the surface. The most favorable position of Zn and O

is best investigated within a 1by 1coincidence structure in which ZnO has the

same lattice constant as the in-plane lattice constant of the metal. An illustration

is shown in Fig. 28 a. First Zn is positioned above the topmost metal atom and

O above the metal atom in the third layer (hcp hollow site). Then ZnO is moved

within the 1×1surface unit cell of the metal and the binding energy between

metal and ZnO is determined. In Fig. 28 b the binding energy between ZnO and

the metal surface is plotted with respect to the positioning of Zn in the Ag 1×1

cell. The data is extended to the hexagonal cell for better assessment of the results.

The most favorable geometry is Zn at the hcp hollow site and O at the top site.

The fcc bridge site is unfavorable.

6.3 corrugation

We will now return to the discussion of the full coincidence structures found in

Sec. 6.1. The ZnO is placed relative to the metal substrate as discussed in the

previous section. To distinguish between the α-BN and the wurtzite structure

we define the corrugation dz as the mean distance of the O/Zn atoms from

the plane spanned by its three surrounding Zn/O atoms. This definition has

as limiting cases ideal α-BN, where dz would be 0Å and wurtzite ZnO with

dz=0.63 Å. The average dz values for each film are listed in Tab. 7of the previous

section. The corrugation dz does not correlate with the lattice mismatch between

metal and film. The freestanding mono-layer would not exhibit any corrugation

within the range of residual strains observed for ZnO on the metal substrates

(see Fig. 24). We attribute the larger corrugation of ZnO on Cu, Rh, Pd and Pt

60 metal supported mono-layer films

(a) Atomic structure of one layer of ZnO on Cu. (b) Atomic structure of one layer of ZnO on Ag.

Figure 29: Atomic structure of ZnO on Cu and Ag.

to chemical effects such as a larger affinity for oxygen and thus a prevalence for

oxide formation and the different distances between surface metal atoms and

ZnO due to the varying size of the coincidence structures. In Fig. 29,30 and 31

the corrugation within the super-cell is visualized for Cu and Ag by assigning a

color scale to the height of the Zn and O atoms. The corrugation maps for Ni, Rh,

Pd, and Pt are shown in Appendix F. For smaller coincidence structures such

as ZnO on Cu (m=5) (Fig. 29), the structure is rather disordered as compared to

ZnO on Ag (Fig. 30 b). The overall corrugation follows the mismatch between

the surface metal atoms and ZnO. If Zn is in registry with the metal fcc top site

the ZnO-metal distance is smallest. If in registry with the hcp hollow site the

ZnO metal distance is largest. This results in the also experimentally observed

Moiré pattern [237]. Closely connected to the corrugation is the mean distance

between Zn and O atoms. In the case of ZnO on Cu, the lateral distance between

Zn and O reaches the wurtzite limit of 0.63 Å if the O is located at the fcc top site.

For ZnO on Ag the lateral distance between O and the surrounding Zn atoms

is about 0.1Å and basically follows the mismatch between ZnO and the Ag fcc

Cu A layer

Cu B layer

Cu C layer −0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

∆zfrom ¯

zin [ ˚

1ZnO/Cu, 1. Layer

Figure 30: Corrugation map (unit cell tripled) of ZnO on Cu. Each Zn and O atom is

assigned a color according to their lateral position.

6.3 corrugation 61

(111) surface. The ZnO mono-layer on Pd and Pt behaves similar as on Ag, while

the behavior on ZnO on Ni is comparable to the Cu case. For ZnO on Rh the

corrugation is not as pronounced as for Cu and Ni, but the film is not as flat as

in the case of Ag. For ZnO on Pd, Pt, and Au the same behavior is observed.

In summary we find, that the mono-layer ZnO films are characterized by a α-BN

structure. For larger coincidence structure, such as ZnO/Ag, ZnO/Au, ZnO/Pd,

ZnO/Pd and ZnO/Rh the α-BN structure is retained almost ideally. For smaller

coincidences we find a strong corrugation. In both cases the corrugation follows

the mismatch between metal surface atoms and ZnO atoms. Therefore, the films

exhibit significantly different geometric properties in comparison to the polar

ZnO surfaces ((0001)-Zn and (000¯

1)-O). They may not be suited as model system

for these surfaces. We will proceed to analyze the atomic structure of thicker

films and the electronic structure to gain further insight.

Ag A layer

Ag B layer

Ag C layer −0.16

−0.12

−0.08

−0.04

0.00

0.04

0.08

0.12

0.16

∆zfrom ¯

zin [ ˚

1ZnO/Ag, 1. Layer

Figure 31: Corrugation map (unit cell tripled) of ZnO on Ag. Each Zn and O atom is

assigned a color according to their lateral position.

METAL SUPPORTED MULTI-LAYER ZNO FILMS

After investigating mono-layer films of ZnO on metal substrates we will proceed

to look at films with thicknesses of up to 4layers. In this section we will focus

on defect-free, bare ZnO films. The discussion regarding defects and ad-atoms

is presented in Sec. 8.1. All calculations, unless stated otherwise, were carried

out with the PBE+vdWTS [196,254] xc-functional with additional parameters

by Zhang et al. and Ruiz et al.. All ZnO films were initialized in an ideal α-BN

structure and the forces on all atoms in the films and the top most metal layers

were relaxed below 5·10−3eV/Å. We will first discuss the atomic structure and

than proceed to the electronic structure.

7.1 atomic structure of multiple layers of zno

dz [Å] / Metal Ag Cu Pd Pt Rh Ni

1Layer

1. Layer 0.159 0.314 0.228 0.282 0.338 0.334

2Layers

1. Layer 0.267 0.471 0.335 0.430 0.430 0.426

2. Layer 0.219 0.348 0.278 0.269 0.305 0.276

3Layers

1. Layer 0.426 0.544 0.455 0.553 0.514 0.520

2. Layer 0.388 0.502 0.394 0.459 0.453 0.481

3. Layer 0.285 0.366 0.290 0.290 0.333 0.341

4Layers

1. Layer 0.436 0.631 0.589 0.654 0.549 0.577

2. Layer 0.415 0.551 0.493 0.559 0.509 0.425

3. Layer 0.387 0.513 0.451 0.509 0.487 0.444

4. Layer 0.226 0.340 0.292 0.312 0.324 0.267

Table 8: Layer resolved corrugation (see text for definition) for 1to 4layers of ZnO on

Ag, Cu, Pd, Pt, Ni, and Rh. The corrugation dz for ideal α-BN ZnO would be

0Å and 0.63 Å for wurtzite ZnO.

With growing number of ZnO layers on the metal substrates the corrugation of

the ZnO ultra-thin films increases. The corrugation is defined, as in the previous

section, as the mean distance of the O/Zn atoms from the plane spanned by its

three surrounding Zn/O atoms. Strain in the ZnO films (Cu, Ni, Pd, Pt) facilitates

the formation of a wurtzite-like structure. To illustrate the transition from the α-

64 metal supported multi-layer zno films

(a) (b) (c) (d)

Figure 32: The relaxed structures for hydrogen free ZnO films on Cu (111); a) mono-

layer, b) bilayer, c) 3layers, d) 4layers of ZnO.

BN structure to a more wurtzite-like structure, the relaxed geometries for ZnO

on Cu are shown in Fig. 32, for Ni in Fig. 33, for Ag in Fig. 34, for Pd, Pt, and

Rh in Appendix G. The films gradually progress from the α-BN to the wurtzite

structure. O-terminated structures (ZnO (000¯

1)) always emerge as the dominant

surface termination once the ZnO films reach a thickness of 4layers on any metal

substrate we investigated. However, the full transformation to ZnO (000¯

1) is not

accomplished for all systems within this 4layers limit.

To investigate the stability of the O-terminated ZnO films with respect to their

Zn-terminated form we initialized four layers of ZnO on Cu in the Zn-terminated

structure before relaxation. The Zn-termination prevails after the force relaxation.

However, the O-terminated structure has a lower total energy. The difference

amounts to ∼20 meV per 1x1Cu surface unit cell. In Sec. 8.1we will return to the

question of film termination in the context of surface defects and ad-atoms. For

now we will focus on the defect-free, bare films.

The corrugation of the multi-layer ZnO films for all six investigated metal sub-

strates is listed in Tab. 8. We calculated the corrugation for each layer of the 1to

4layer systems separately. The trend to form wurtzite ZnO is observed for all

the investigated metal substrates. For Cu, Pd, and Pt the structure has already

transformed to wurtzite at a film thickness of 4layers. For Ni this is achieved at

six layers. For Ag and Rh more than 5layers are required for the transformation

to wurtzite. In Tab. 8we observe a decrease of the corrugation with increasing

distance from the metal substrates. At the surface the O retracts into the surface,

reducing the corrugation. This is a well known behavior from the investigation

of the bulk ZnO-O (000¯

1) surface [162].

While the thicker films (4and more layers) nearly completely transform to

wurtzite, the thinner films (less than 4layers) exhibit patches of α-BN and simul-

taneously patches of wurtzite-structure. Morgan et al. [171,169,170] predicted a

transition from the layered α-BN (h-MgO) to d-BCT (body-centered tetragonal)

structure, to wurtzite (B4) for free-standing ZnO films (no substrate). They pre-

dict the transition to d-BCT to occur at 8ZnO layers. The transition to wurtzite

(a) (b) (c) (d)

Figure 33: The relaxed structures for hydrogen free ZnO films on Ni (111); a) mono-layer,

b) bilayer, c) 3layers, d) 4layers of ZnO.

7.2 electronic structure of multiple layers of zno 65

(a) (b) (c) (d)

Figure 34: The relaxed structures for hydrogen free ZnO films on Ag (111); a) mono-

layer, b) bilayer, c) 3layers, d) 4layers of ZnO.

would only occur at rather thick films of 56 layers, although the wurtzite film

was found to be meta-stable starting from 16 layers. Morgan et al. further report

that for up to 6layers of free-standing ZnO the α-BN and d-BCT structures are

indistinguishable because the α-BN film slightly buckles during relaxation while

the d-BCT film contracts. However, the lack of the trigonal basal plane symmetry

in the d-BCT structure leads to an incompatibility with epitaxial growth on ( 111)

metal surfaces.

By applying network theory Demiroglu et al. [65] suggest further stable struc-

tures, that are similar to the α-BN and d-BCT structure. We could not identify

any of the structures proposed by Morgan et al. [171,169,170] or Demiroglu

et al. [65] in our calculations. The transition to wurtzite (B4) leads to a competi-

tion between O- and Zn-terminated patches at the interface, that are determined

by the distance between Zn/O and metal atoms due to the differently sized co-

incidence structures. Only for Ni at 3and 4layers (Fig. 33) and Rh at 4layers

(Fig. 109) we find a d-BCT structure. For Ni with four layers of ZnO the structure

is almost perfectly ordered. The observation of the d-BCT structure is in contrast

to the results by Morgan et al. [171,169,170] for free-standing ultra-thin films

where the films are stretched (positive strain) with respect to their free-standing

equilibrium lattice parameter. They predict positive strain to facilitate the lay-

ered α-BN structure. In our calculations six layers of ZnO on Ni then relax to the

wurtzite structure.

In summary the prevalence for oxide formation and the different distances be-

tween surface metal atoms and ZnO due to the varying size of the coincidence

structures influence the final structure and the transition to wurtzite. The evolu-

tion of the electronic structure is of particular interest and will be investigated in

the following section.

7.2 electronic structure of multiple layers of zno

We have seen in the previous section that the combination of metal surfaces

and ZnO yields interesting effects on the atomic structure of multi-layer films.

The electronic structure is affected in a similarly interesting fashion. The over-

all alignment of the electronic structure of metals and ZnO thin-films has been

investigated within the context of Schottky barrier heights in electronic devices

with DFT [58]. The authors analyzed the interface electronic structure for metals

adsorbed on ZnO in a 1x1and (√3x2)R30-geometry. The nature of the metal-

semiconductor interface was found to be very sensitive to the chemical bonding.

Ohmic contacts of Schottky barriers could be observed depending on the ZnO ter-

mination and the metal. The authors identified two types of interface states, metal

66 metal supported multi-layer zno films

8 6 42 0 2

E−F[eV]

100

150

(pro)DOS [states/eV]

1st layer

2nd layer

(a) 2ZnO/Ag.

8 6 42 0 2

E−F[eV]

100

120

140

160

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) 4ZnO/Ag.

Figure 35: Layer resolved projected density of states for two and four layers of ZnO on

Ag obtained with the PBE+vdWTS [196,254] xc-functional.

induced gap states and energetically well defined states due to chemical bonding.

The overall behavior can be understood as an interplay of surface charge and im-

age charge effects [26,257]. The electric field due to the polar nature of the bulk

truncated ZnO (0001) or (000¯

1) surface is screened by the metal. An image charge

of opposite sign forms in the metal, which results in the creation of an interface

dipole. The electric field is mainly due to the negatively charged O ions at the O-

terminated side and due to the positively charged Zn ions at the Zn terminated

side. The interface dipole between metal and ZnO shifts the oxygen bands up or

down with respect to the metal and the Fermi energy.

For our systems of mono and multi-layers of ZnO on fcc (111) transition metals

the situation is different; the thickness of the ZnO films and their structure are

additional degrees of freedom that have to be considered. We will focus on the

influence of film thickness and the choice of the metal substrate. Possible surface

reconstructions of the ZnO will be discussed later.

In the previous section we observed the emergence of the wurtzite structure

in the ZnO films on different metal substrates for increasing film thickness. We

can further investigate this behavior by looking at the density of states (DOS) of

these systems. In Fig. 35, Fig. 36, and Fig. 37 the projected DOS obtained with

the PBE+vdWTS [196,254] xc-functional for each layer of a slab with 2and 3(4)

layers of ZnO on Ag and Cu is shown. The influence of exact-exchange admix-

ture in the DFT-functional, which is quantitative rather than qualitative will be

discussed throughout the chapter for the HSE06 xc-functional.

The electronic states of each successive layer are pushed further up in energy.

In Fig. 35,36 and 37 the increase in the projected DOS with respect to the pre-

vious layer is indicated by the colored areas. The arrows show the direction of

the shift for the Zn 3d bands. The situation is shown schematically in Fig 38.

Figure 38 a depicts the isolated system, and Fig. 38 b the combined system. The

intrinsic dipole of ZnO leads to a potential difference across the ultra-thin film.

7.2 electronic structure of multiple layers of zno 67

10 8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

(a) 2ZnO/Cu (PBE+vdWT S).

10 8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

(b) 3ZnO/Cu (PBE+vdWT S).

10 8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

10 8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

(d) 3ZnO/Cu (HSE06).

Figure 36: Layer resolved projected density of states for two layers and three layers of

ZnO on Cu obtained with the PBE+vdWTS [196,254] and HSE06 [100] xc-

functional.

This mechanism can be understood within the ionic picture of Tasker [249] for

the ZnO part of the system. Taskers model yields 1/2electrons (per unit cell) of

excess charge at the oxygen side and (-1/2) electrons at the Zn terminated side of

a finite ZnO slab. This gives rise to a potential difference (in analogy to a capac-

itor) between the Zn and the O terminated site of the ZnO slab. The unrealistic

divergence with increasing thickness can be removed from the model by trans-

ferring half an electron from the O- to the Zn-side. This means that for the ideal

bulk-truncated ZnO on the O-terminated side the valence band is not entirely

filled and electrons enter the conduction band. It was found by Kresse et al. [132]

that this model still lacks some rather important details. The states arising from

the Zn-surface atoms penetrate deep into the bulk, leading to a charge transfer to

these states. The resulting counter-field shifts the O 2p and Zn 4s states to higher

68 metal supported multi-layer zno films

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

Figure 37: Layer resolved projected density of states for four layers of ZnO on Cu

(4ZnO/Cu) obtained with the PBE+vdWTS [196,254] xc-functional.

energies. This effect is referred to as band bending. Due to the finite nature of

the slab approach electrons are transferred (back) from the Zn-terminated site

to holes at the O-terminated site. In calculations for the bulk truncated surfaces,

this effect is considered an artifact of the finite model [132].

However, the situation for metal supported ZnO thin-films is different. The ZnO

mono-layer does not have a wurtzite, but an α-BN structure on metal supports.

The bonding is thus different from wurtzite and the ionic model cannot be ap-

plied. With increasing thickness the polar character of wurtzite ZnO emerges

as the films structurally become more and more wurtzite-like. Charge is again

transfered from the Zn-terminated interface to adjacent layers. The resulting field

shifts the electronic states upwards until the Fermi energy, provided by the metal,

is reached and the film becomes effectively p-type doped at its surface by pin-

ning the electronic states of the top most layer(s) at the Fermi level. Here, p-type

doping refers to a position of the Fermi level in the semi-conductor below the

center of the band gap or close to the valence band maximum.

This mechanism can be observed e. g. in the layer-projected DOS of 2to 4layers

of ZnO on Cu and Ag in Fig. 35, Fig. 36, and Fig. 37. For 4layers of ZnO on

Ag in Fig. 35 the electronic states of the 2nd to fourth layer are shifted upwards

in energy, relative to the Fermi level and with respect to the electronic states of

the previous layer. The portion of the DOS, that has higher density than the DOS

of the previous layer, is filled with the color of the respective curve. The shift

is observed best at the Zn d-band between −4and −6eV. The electronic states

of the fourth and topmost layer have reached the Fermi energy, leading to an

increased DOS at E=ϵF. The states are pinned at the Fermi level, which is pro-

vided by the metal. For 4ZnO/Cu in Fig. 37 this effect is even more pronounced.

The electronic states of the third layer are already pinned at the Fermi level and

for the electronic states of the fourth layer no further upwards shift is observed,

7.2 electronic structure of multiple layers of zno 69

123

(a) Isolated metal surface and

isolated ZnO slab with three

layers. The bend bending

in the ZnO has already oc-

curred.

123

(b) Metal surface with

three layers of ZnO.

Pinning of the occu-

pied ZnO states is al-

most reached.

1234

layers of ZnO. The elec-

tronic states of the fourth

layer are pinned at the

Fermi energy provided

by the metal.

Figure 38: (a) Illustration of the level alignment for the isolated systems, (b) for the com-

bined system, (c) for the combined system with a sufficient number of ZnO

layers to achieve pinning.

thus making the surface of these films effectively p-type doped. The effect is

qualitatively the same regardless if we apply the PBE+vdWTS or the HSE06 hy-

brid xc-functional. In Fig. 36 the projected DOS for 2ZnO/Cu and 3ZnO/Cu are

shown for both functionals. The band gap of the ZnO is increased by admixture

of exact exchange and the onset of the electronic states of the ZnO is about 1eV

lower in energy. However, the upwards shift per layer is quantitatively the same

(∼0.5eV).

P-type doping is notoriously difficult to achieve in bulk ZnO samples, which are

usually intrinsically n-type doped [276]. The origin of both types of doping in

bulk ZnO is still under debate [111,215]. For p-type doping in ZnO only very

few candidate shallow acceptors are available. Nitrogen is the most promising

candidate for the formation of shallow acceptor levels. The other column V ele-

ments are deep acceptors [193]. On the other hand there are quite a few defects

that can act as compensating centers for p-type doping [112,110,113]. Another

fundamental reason for the difficulties to achieve p-type doping in ZnO is that

the valence band edge lies too deep below the Fermi level. Attempts to p-type

doping will be compensated by other defects before the Fermi energy is shifted

far enough [215,149].

In the mechanism for achieving p-type doping, we describe for the surface of

a ZnO thin film, the Fermi level is fixed by the metal substrate. The valence

band is moved upwards in energy, towards the Fermi level. This mechanism for

achieving p-type doping at the surface of ZnO films could explain experimental

reports of p-type doping for ZnO, that are difficult to reproduce [163,222,147,

122,279,49] or provide a novel way to achieve p-type doping in ZnO. The (ther-

modynamic) stability of the films under conditions realized by experiment will

be investigate in the next section.

Preliminary experimental evidence for the behavior described above is the effect

on the measured dI/dV spectra (STS measurement, see Sec. 1.9) as a function

of bias voltage for ZnO films of different height reported by Shiotari et al. [237].

The authors observe the onset of the signal at lower bias voltages for the thick

70 metal supported multi-layer zno films

(a) Upwards shift. (b) Flat band conditions. (c) Downwards shift.

Figure 39: Possible effect of intrinsic dipolar field and surface reconstruction in ZnO

thin films on metal substrates. See text for further explanations.

film ("high" ZnO) than for the thinner film ("low" ZnO). The dI/dV spectra can

be considered approximately proportional to the (local) DOS of the sample [250].

This is an indication for a (downwards) shift of the unoccupied electronic states

of the thicker ZnO film with respect to its thinner version. In our model, derived

from our DFT calculations, we had, however, found an upwards shift of the oc-

cupied electronic states with increasing film thickness. For the results reported

by Shiotari et al. [237] two scenarios are plausible, which are depicted in Fig 39.

The first scenario is that the electronic states of the " low" ZnO are indeed shifted

upwards in energy (Fig 39 a), but the "high" ZnO is (2×1)–H reconstructed (see

Sec. 8.1) and no shift occurs (Fig 39 b). The electronic states of the "high" ZnO

appear to be shifted downwards in energy with respect to the "low" ZnO. The

second scenario is based on a different orientation of the ZnO film. If the ZnO

in experiment is (0001)-Zn terminated and not (000¯

1)-O terminated, the intrinsic

field would be of opposite sign. This would shift the electronic states downwards

in energy with increasing film thickness. Both scenarios are possible, because a

(2×1)–H reconstruction is difficult to determine experimentally and the differ-

ence in formation energies for a (0001)-Zn and (000¯

1)-O terminated films are

small (see Sec. 7.1). However, it has to be noted that Shiotari et al. [237] mainly

focused on other properties of the film, that we will discuss is Sec. 9and further

careful experimental and theoretical analysis is required. Especially the role of

the substrate requires further consideration.

The number of layers necessary to reach pinning and effective p-type doping

depends on the metallic substrate. More layers are required to reach pinning if

the initial valence band maximum of the ZnO ultra-thin film is positioned further

below the Fermi level of the metal. In the PBE+vdWTS calculations for the Ag

substrate 4layers are required to reach pinning while for Cu the upwards shift

of the ZnO states already stops after three layers (Fig. 35, Fig. 36 a and b, and

Fig. 37). The electronic structure of ZnO on Pd shows a similar behavior as on Cu

with pinning after three layers (Fig. 117). For Pt and Rh the onset of the pinning

occurs already after 2layers (Fig. 116 and Fig. 118). In the case of Ni five layers

are required (Fig. 119).

The potential difference over the extent of the films eventually shifts the valence

band minimum on the top layer into resonance with the conduction band mini-

mum of the layer closest to the metal, thus closing the band gap of the ZnO film.

For the higher level HSE06 hybrid xc-functional the band gap is larger and hence

the band gap closes at a higher number of ZnO layers. This is shown for ZnO

on Cu in Fig. 36 c and d (for a direct comparison of PBE and HSE06 see Fig. 40).

7.2 electronic structure of multiple layers of zno 71

(a) 2ZnO/Cu. (b) 3ZnO/Cu.

Figure 40: Layer resolved projected density of states for two layers and three layers of

ZnO on Cu, direct comparison between the PBE+vdWTS [196,254] and HSE06

[100] xc-functionals. The Fermi energy is set to zero.

Analogously the pinning and effective p-type doping occurs at higher layer num-

bers because the initial valence band maximum is further away from the Fermi

energy while the upwards shift is similar to the situation with the PBE+vdWTS

xc-functional.

In conclusion, we describe a mechanism for achieving effective p-type doping in

ZnO thin films. The Fermi level, which is typically moved towards the valence

band edge by group V dopants, is kept fixed. The position of the Fermi level is

determined by the metal substrate. The valence band is moved toward the Fermi

energy, because of the intrinsic field that forms over the extent of the film. The

number of layers required for achieving effective p-type doping is determined

by the band gap and the initial alignment of the electronic levels of ZnO and

metal substrate. The mechanism might help to understand experimental reports

of p-type doped ZnO and construct new devices.

In the next section we will analyze the stability of mono- and multi-layer ZnO

films on metal substrates with respect to their chemical environment. We will

investigate vacancies, ad-atoms and adsorbates. In Sec. 8.1.1we will return to the

topic of effective p-type doping at the surface when we compare the electronic

structure of bare ZnO films, which we discussed in this section, with hydrogen

terminated ZnO films.

THERMODYNAMIC STABILITY OF ZNO ON METAL

SUBSTRATES

The experimental and theoretical work on the polar surfaces, (0001) and (000¯

1),

of ZnO revealed a manifold of reconstructions and terminations [73,132,161,

261,267,275,276]. For the two surface terminations (0001)-Zn and (000¯

1)-O dif-

ferent reconstructions were observed. For the ultra-thin films all of them have

to be considered, effectively doubling the amount of structures, that have to be

investigated. Initially the ultra-thin films have a α-BN structure (see Sec. 5), by

choosing a surface preferred reconstruction (e. g. H- or OH-adsorbates) either the

(0001) or (000¯

1) side is selected. The competition of these structures as a function

of the chemical potentials of H2, O2and H2O has to be investigated.

The most stable and simplest surface reconstruction of the ZnO (000¯

1) surface is

a (2×1) hydrogen covered surface (50% hydrogen concentration) [161,267]. The

hydrogen saturates the surface bonds and cancels the charge accumulation at the

surface. The equivalent structure on the ZnO (0001) surface is a 2×1hydroxyl

(OH) reconstruction. At low oxygen chemical potentials oxygen vacancies could

be stabilized. In intermediate chemical potential ranges triangular pits are pre-

dicted to form for both surface terminations, although with different stoichiome-

try. Experimentally they were only observed for the ZnO (0001)-Zn surface. Ad-

ditionally ring structures and oxygen ad-atoms were predicted and observed

[270,267]. In this section the possible reconstructions will be discussed for 1to 4

layers of ZnO on Ag, Cu, Pd, Pt, Rh and Ni. The focus will be on the ZnO/Ag and

ZnO/Cu systems. We will begin the discussion with H-reconstructions, followed

by Zn/O-vacancies and ad-atoms. Ring structures will be briefly addressed af-

terwards. Finally, we will address OH-reconstructions and present a full surface

phase diagram for ZnO on ultra-thin films. In addition we will discuss the ex-

perimentally observed 5×5coincidence structure on Ag (111) [237] and ZnO

ultra-thin films on Rh (0001) [120].

8.1 hydrogen adsorption

We will now analyze the phase diagrams for different coverages of H on ZnO

ultra-thin films on different metals as function of the number of ZnO layers with

ab initio atomistic thermodynamics (see [226,214] and Sec. 1.6). We consider the

surface free energy γin eq. 61 (see Sec. 7.1) as a function of the change of the

H2chemical potential, ∆µ =µH−EH2/2. In all structures we will discuss in this

section the number of Zn and O atoms is equal and we can therefore drop the oxy-

gen (and zinc) chemical potential from eq. 61. Reference energies for bulk ZnO

and H2are tabulated in App. B. First, we will construct the (one-dimensional)

phase diagrams by neglecting the vibrational free energy (Fvib) and the configu-

rational entropy (Sconf). These will be included later in this section. In Fig. 42 a

74 thermodynamic stability of zno on metal substrates

(a) No hydrogen. (b) 25% hydrogen coverage. (c) 50% hydrogen coverage -

(2×1)–H reconstruction.

Figure 41: Atomic structure of three different amounts of hydrogen coverage for 2layers

of ZnO on Ag. Only parts of the unit cell are shown.

the result is shown for two layers of ZnO on Ag. The considered surface struc-

tures are the bare metal-supported ZnO films discussed in the previous chapters

and reconstructions with different amounts of hydrogen adsorbed at the sur-

face oxygen sites. The chemical potential is translated into partial pressures for

a given, exemplary temperature at 400K with the help of the ideal gas law and

thermodynamic tables [178]. In Fig. 41 a two layers of ZnO on Ag without hy-

drogen are shown. Fig. 41 b shows 25% of hydrogen coverage on two layers of

ZnO on Ag and Fig. 41 c shows the (2×1)–H reconstruction (50% coverage of the

surface oxygen sites). The different coverages result in lines with different incli-

nation in these one-dimensional surface phase diagrams. The unreconstructed,

bare ZnO on Ag is only stable (lowest line at the considered chemical potential)

at low chemical potentials, that are not experimentally accessible at the chosen

temperature of 400 K. The reconstruction with the lowest amount of hydrogen is

stable over a wide range of low chemical potentials. This region is in the range of

typical UHV conditions. The transition to the (2×1)–H reconstruction progresses

quickly, lower hydrogen concentrations are only stable in a very narrow window

of hydrogen chemical potentials. The (2×1)–H reconstruction is most stable for

a wide range of chemical potentials, because the oxygen surface bonds are fully

saturated. Higher coverages of hydrogen can only be realized at very high hydro-

gen chemical potentials, that are not shown in Fig. 41 a.

Now we will discuss the missing terms in the Gibbs free energy (eq. 56). Follow-

ing the arguments in Sec. 1.6the pV term can be safely neglected. The approxima-

tion of the configurational entropy follows the arguments in Sec. 1.6and is given

by eq. 70. Fig. 42 b shows the previous phase diagram now also including the

configurational entropy. It is only included up to 50% coverage. For higher cov-

erages the configurational freedom is further decreased by interaction between

the adsorbed H-atoms. This might already be the case for lower coverages and

highly symmetric arrangements of the H-atoms [217]. The contribution to the

surface free energy of a single hydrogen atom adsorbed at the surface is smallest,

but the effect in the phase diagram is the strongest. Due to the low inclination of

8.1 hydrogen adsorption 75

(a) Total energy only. (b) Including configurational entropy.

brational free energy.

Figure 42: One dimensional surface phase diagrams for two layers of ZnO on Ag, includ-

ing/excluding the different contributions to the surface free energy.

the line and the fact that the bare surface is not stabilized by the configurational

entropy term, the stability range of the bare surface is significantly decreased.

The final contribution to the Gibbs free energy of our surface is the free en-

ergy of vibration Fvib. Only free energy differences between bulk and surface

phases (Gsurf and gbulk) enter into the surface free energy γ(T,p)(eq. 61). As a

first approximation for Fvib at moderate temperatures (T < 1000) only the first

term in eq. 75 is evaluated (zero point energy). The vibrational energy contri-

bution of a hydrogen atom adsorbed at a oxygen surface site is approximated

by the difference between the experimental H2gas phase stretching mode at

ωH2=4138cm−1[207] and the the frequency obtained by a finite difference

DFT calculation. The frequencies we calculated for the O-H stretching mode of

hydrogen adsorbed on 1to 3layers of ZnO are shown in Tab. 9. The change as a

function of layer numbers or metal substrate is small (only the difference to the

H2gas phase stretching mode enters eq. 75).

The phase diagram including the vibrational contribution is shown in Fig. 42 c.

The additional contributions are small. The H reconstructed structures are fur-

ther stabilized but no qualitative changes are observed. The final phase diagram

including all discussed contributions is shown in Fig. 42 d. They are included in

the following discussions of this section.

Other H distributions than the highly ordered structures considered for the

phase diagrams are close in surface free energy, but less stable. The situation is

the same as discussed for ZnO (000¯

1) by Kresse et al. [132]. The surface phase

diagram for two layers of ZnO on Cu is shown in Fig. 43 a. The disordered H

terminations (only 50% coverage) are represented by parallel black lines. The con-

76 thermodynamic stability of zno on metal substrates

Metal Layers H-coverage ω[cm−1]1

2

hω[eV]

Ag 1 1.5% (1H) 3632.974 0.2252

Ag 1 50% ((2×1)-H) 3643.378 0.2259

Ag 2 1.5% (1H) 3690.411 0.2288

Ag 2 50% ((2×1)-H) 3693.310 0.2290

Cu 1 6.25% (1H) 3658.597 0.2268

Cu 1 50% ((2×1)-H) 3657.129 0.2267

Cu 2 6.25% (1H) 3688.506 0.2287

Cu 2 50% ((2×1)-H) 3701.007 0.2294

Cu 3 6.25% (1H) 3700.087 0.2294

Pt 1 50% ((2×1)-H) 3673.911 0.2278

Pt 2 50% ((2×1)-H) 3698.991 0.2293

Table 9: Vibration frequencies for O-H stretching mode of H adsorbed on ZnO on metal

substrates.

figuration lowest in energy is the "striped" (2×1)–H reconstruction. In Fig. 43 b

the surface phase diagram for two layers of ZnO on Cu is shown with up to 50%

of the adsorbed hydrogen at the interface between ZnO and Cu. The atomic struc-

ture is shown in the inset of Fig. 43 b. The total amount of H is the same as for the

(2×1)–H reconstruction, which is the structure lowest in energy. Every H atom,

that is moved from the surface to the interface increase the surface free energy

by about 40 meV per 1×1metal surface unit cell. From this we conclude that

hydrogen at the metal/ZnO interface is not a stable structure. Another possible

adsorption geometry for hydrogen is the adsorption at the Zn sites. Adsorbing H

at the Zn forces the ZnO film into a Zn-terminated (0001) surface structure (Fig.

44 b) as compared to the O-terminated structure for H-adsorption at the O-sites.

The structure in Fig. 42 b (2layers of ZnO on Cu - (2×1)–H) is approximately

0.5eV per 1×1metal surface unit cell higher in energy as compared to adsorp-

tion at the O-sites.

Another considered geometry we show in Fig. 44 a is zinc-blend stacking. The

ZnO mono-layer is simultaneously in the zinc-blend and wurtzite structure, be-

cause both structures can only be distinguished from two layers on. The zinc-

blend stacking is analogous to the stacking of graphite. A top view for two layers

of ZnO with zinc-blend stacking on Cu is shown in Fig. 44 a. This geometry is

approximately 150 meV to 200 meV higher in energy than wurtzite stacking, i. e.

thermodynamically not stable.

Finally we discuss the evolution of the surface phase diagram with the number of

ZnO layers. The surface phase diagrams for 1to 4layers of ZnO on Ag are shown

in Fig. 45. These results show that for Ag the clean surface as well as the (2×1)–

H reconstruction is thermodynamically stable for layer numbers greater than 2at

400K. Thus the H2partial pressure (chemical potential) can be used to select one

of the two phases. The mono-layer is very stable in its α-BN structure. Only at

intermediate to high chemical potentials first the (2×1)–H reconstructions and

8.1 hydrogen adsorption 77

(a) Surface phase diagram including disor-

dered surface adsorption geometries, that

are higher in energy than the (2×1)–H

reconstruction. The inset shows one exem-

plary structure of randomly distributed hy-

drogen (50% coverage).

(b) Surface phase diagram including up-to 50%

of the hydrogen present in the (2×1)–H re-

constructed unit cell adsorbed at the inter-

face between ZnO and Cu. The inset shows

the atomic structure with half of the H

atoms in the (2×1)–H reconstruction at the

interface.

Figure 43: Surface phase diagrams for 2layers of ZnO on Cu for different adsorption

geometries.

at high hydrogen chemical potential even the full coverage will become stable.

With increasing number of ZnO layers, the (2×1)–H reconstructed surface is fur-

ther and further stabilized. Already for three layers the unreconstructed surface

cannot be realized under typical experimental hydrogen partial pressures. This

behavior is even more pronounced for the other metals.

For all calculated systems, the partial pressure region for the transition from

the clean (α-BN) film to the (2×1)–H reconstruction is shown in Fig. 46. Below

the colored bars (dark gray regions, lower pressures), the graphite-like films are

stable, above (light gray region, higher pressures) (2×1)–H is stabilized. In the

region marked by the bars H-coverages larger than 0% and smaller than 50%

are stable. H coverages corresponding to one H per unit cell (determined by the

coincidence structure) dominate this transition regime. For Cu and Ni the struc-

ture without H and the intermediate H-coverages are reachable only at elevated

temperatures. For Pd and Rh only the (2×1)–H reconstruction is within experi-

mentally accessible pressure ranges (also see Ref. [28]).

(a) Zinc-blend (111) stacking. (b) Hydrogen adsorption at the Zn sites.

Figure 44: Atomic structure for the a) zinc-blend stacking and b) hydrogen adsorption

at the Zn sites.

78 thermodynamic stability of zno on metal substrates

(a) 1Layer (b) 2Layers

Figure 45: Evolution of the surface phase diagram for 1to 4layers of ZnO on Ag.

For increased H pressures, the difference in formation energies between systems

with different numbers of ZnO layers is significantly reduced. For (2×1)–H re-

constructed structures this difference is very small and thicker layers (3-4layers)

can exhibit a lower surface free energy than 1or 2layers of ZnO. The situation

is shown for Ag in Fig. 47. The separation in surface free energy for the un-

reconstructed structure at low hydrogen chemical potentials is quite large. The

(2×1)–H reconstruction is stabilized at lower chemical potentials with increas-

ing thickness of the ZnO. Except for the mono-layer it is the most stable struc-

ture throughout most of the chemical potential range considered. The energetic

separation for the (2×1)–H reconstructed ZnO films is low. The 3and 4layer

ZnO films are lower in surface free energy than 1and 2layers. Under experi-

mental conditions, some of these structures could be kinetically stabilized and

further growth be hindered [159]. The formation of the (2×1)–H reconstruction,

although thermodynamically most stable, could be blocked by an energy barrier

for the dissociation of H2at the surface.

8.1.1Electronic structure of the (2x1)–H reconstruction.

In Sec. 7.2we discussed the pronounced upwards shift for the electronic states of

each successive layer in the unreconstructed ZnO films on the metal substrates.

The situation is shown for 4layers of ZnO on Ag in Fig. 48 a and schematically

in the inset of the same figure. If we now look at the projected density of states

of the ZnO ultra-thin films saturated with H we can analyze their fundamentally

different behavior as compared to the unreconstructed films. Figure 48 b shows

the projected DOS of 4layers of ZnO on Ag with 50% H-coverage ((2×1)–H).

8.1 hydrogen adsorption 79

Ag Cu Pd Ni Rh Pt

−2.5

−2.0

−1.5

−1.0

−0.5

∆µH2[eV]

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

log(pH2) [mbar] at 400K

2x1-H stable

1 L

2 L

3 L

4 L

1 L

2 L

3 L

4 L

1 L

2 L

3 L

4 L

2 L

3 L

4 L

1 L

2 L

3 L

4 L

1 L

2 L

3 L

1 L

bare surface stable

Figure 46: Surface free energy γper (1x1) surface unit cell area as function of the change

in chemical potential ∆µ (see eq. 61) for 1to 4layers of ZnO (L) on Ag, Cu,

Pd, Ni, Rh and Pt (111) with different coverages of H. The partial pressure,

as calculated from thermodynamical tables, of H2at 400 K is plotted on the

right axis. The colored area indicates the transition region between the ultra-

thin film without H (dark gray) and the (2×1)–H with 50% H coverage (light

gray).

The onset of the Zn d levels exhibits no shift between successive layers. The first

layer being an exception due to its proximity to the metal. In the inset of Fig. 48

b the situation is shown schematically. The bonds at the surface are saturated

by H, providing electrons. The intrinsic dipole moment of ZnO is quenched, no

polarization of the film occurs and the position of the ZnO electronic states stays

constant throughout the ultra-thin film. The position of the electronic states of

the ZnO ultra-thin films with respect to the Fermi level, provided by the metal,

−1.4−1.2−1.0−0.8−0.6−0.4−0.2 0.0

∆µH2[eV ]

0.5

0.6

0.7

0.8

0.9

1.0

1.1

surface free energy γper A1x1[eV ]

1ZnO/Ag

2ZnO/Ag

3ZnO/Ag

4ZnO/Ag

-25 -20 -15 -10 -5 0 5

log(pH2) [mbar] at 400K

Figure 47: Comparison of the surface free energy of 1to 4layers of ZnO on Ag covered

with different amounts of H. The colored lines correspond to the black lines

in Fig. 45 a–d.

80 thermodynamic stability of zno on metal substrates

8 6 42 0 2

E−F[eV]

100

120

140

160

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(a) Projected density for 4layers ZnO on Ag

without H (4ZnO/Ag).

8 6 42 0 2

E−F[eV]

100

150

200

250

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Projected density of states for 4layers ZnO

on Ag with H (4ZnO/Ag–(2×1)–H)

Figure 48: Projected density of states for 4layers (8x8) ZnO on (9x9x4) Ag with and

without H. The densities are projected on each successive layer of ZnO. The

metal is shown in gray. The behavior of the electronic states in each layer is

shown schematically in the insets.

is highly system dependent. The discussion of the mechanism governing the

alignment of the electronic structure of ZnO with respect to the metal was al-

ready discussed in Sec. 7.2and further information can be found in the literature

[26,257,58]. The projected density of states for 2and 4layers of ZnO on all the

previously discussed transition metals can be found in Appendix J.

In order to test the accuracy of our calculations we calculated the atomic and

electronic structure for three layers of (2×1)–H reconstructed ZnO on Cu with

the HSE06 [100] hybrid xc-functional. The ZnO film on Cu was initialized in an

α-BN structure and as a first step we relaxed the geometry using the PBE xc-

functional. Finally, we relaxed the geometry with the HSE06 xc-functional and

calculated the projected density of states. The differences in the atomic structure

are small. The projected DOS for three layers of ZnO on Cu calculated with PBE

and HSE06 are compared in Fig. 49 and Fig. 50. The overall agreement between

both calculations is rather good. The structure of the projected DOS for both xc-

functionals shows no qualitative differences. The Zn d-band is shifted by -2eV

to lower energies for HSE06, while higher lying states are only shifted by -1eV.

We can conclude, that for our systems the accuracy of PBE+vdWT S is sufficient

to describe the atomic and electronic structure.

Finally, we want to compare the electronic structure of the (2×1)–H recon-

structed ZnO to unreconstructed ZnO (see Sec. 7.1) on metal substrates and the

(2×1)–H reconstructed ZnO surface without substrate. The comparison of the

density of states for the (2×1)–H reconstructed ZnO (000¯

1) surface and the ZnO

films on the metals in Fig. 51 b shows that systems with 4and more layers already

resemble the (2×1)–H reconstructed ZnO (000¯

1) surface very well. However, the

films without H (Fig. 51 b) retain a unique character. The electronic structure dif-

fers from the ZnO (000¯

1)–(2×1)–H surface and the geometry combines aspects

8.1 hydrogen adsorption 81

10 8 6 42 0 2

E−F[eV]

100

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

(a) PBE.

10 8 6 42 0 2

E−F[eV]

100

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

(b) HSE06.

Figure 49: Projected density of states for three layers of ZnO on Cu with H, 3ZnO/Cu–

(2×1)–H. The DOS is projected on the individual layers.

of wurtzite and α-BN (see Sec. 7.1).

We conclude that the choice of metal and the H2partial pressure are two ad-

ditional degrees of freedom to select between unreconstructed ultra-thin ZnO

films, that differ from bulk ZnO, and films ((2×1)–H reconstructed) that resem-

Figure 50: Projected density of states for three layers of ZnO on Cu with H, 3ZnO/Cu–

(2×1)–H. Direct comparison between the PBE and HSE06 xc-functionals.

82 thermodynamic stability of zno on metal substrates

(a) 4ZnO/Ag (b) 4ZnO/Ag–(2×1)-H

Figure 51: (a) Comparison of projected DOS of 4Layers ZnO on Ag (4ZnO/Ag) without

H and with 50% H coverage with the DOS of a ZnO (2×1)–H surface slab.

The projected DOS of the ZnO (2×1)–H surface is shifted by -1eV in (a) and

(b) with respect to their Fermi level.

ble wurtzite ZnO. This dual character make metal-supported thin films particu-

larly interesting. Under certain conditions these films resemble novel materials

with unique properties whereas under different conditions they could serve as

important models for the study of the ZnO (000¯

1)–(2×1)–H surface.

8.2 vacancies and ad-atoms

In Sec. 8.1we have seen that the structure of the films is typically neither the

structure of the Zn-terminated nor the O-terminated ZnO surface. It is rather a

mixture of both. If we offer hydrogen to the films, the O-terminated (2×1)–H

reconstructed surface forms. A similar behavior is expected for O/Zn vacancies

and ad-atoms. Within the ionic model of Tasker [249] O vacancies and Zn ad-

atoms would stabilize the (000¯

1)-O surface [161,267], while O ad-atoms and

Zn vacancies would achieve the same for the (0001)-Zn surface [132,261]. We

will first investigate the thermodynamic stability of O vacancies and ad-atoms

before progressing to their Zn counterparts. Afterwards we will discuss more

extended surface reconstructions and hydroxyl adsorption. Due to the number

of surface reconstructions and their fundamentally different structures, we will

neglect contributions from the vibrational free energy and entropic contributions

in the following discussion, unless stated otherwise.

8.2 vacancies and ad-atoms 83

(a) 25% oxygen vacancies. (b) 60% additional oxygen atoms. Inter-facial

oxygen is colored in magenta.

Figure 52: Atomic structure for oxygen defects in one layer of ZnO on Cu.

8.2.1O vacancies and ad-atoms

By removing oxygen atoms from the surface, excess electrons are removed and

the surface energy is reduced. For the distribution of the vacancies for a given

vacancy concentration we will use the structures found by Wahl et al. for the ZnO

(000¯

1)-O surface [267]. Wahl et al. calculated the formation energy of all possible

surface vacancy distributions for a 4×4ZnO (000¯

1)-O surface slab. The struc-

tures lowest in energy are the input for our calculations. For a detailed analysis a

structure search based on a cluster expansion would be required for ZnO on the

different metal substrates. This is in general possible, but not in the scope of this

dissertation. We will therefore restrict the pool of investigated structures to those

of Ref. [267]. The structure of mono-layer ZnO on Cu with 1/4of the oxygen

atoms removed is shown in Fig. 52 a. When more than 1/4of the oxygen atoms

are removed the remaining oxygen atoms start to form hexagonal ring structures,

that eventually start to break up for even higher oxygen deficiencies.

The equivalent O induced mechanism for the ZnO (0001)-Zn surface are oxygen

ad-atoms. One layer of ZnO with 60% additional oxygen is shown in Fig. 52 b.

The oxygen ad-atoms were initially distributed randomly on top of the ZnO layer.

For all systems the oxygen relaxes to the interface between metal and ZnO. For

100% of additional oxygen the film "floats" on top of these oxygen atoms. The

only exception of this behavior is for the case of a gold substrate. Here the oxy-

gen remains on top of the ZnO film. The structures discussed in this section all

contain the same amount of Zn in their super cell. Structures with increased or

decreased Zn- and O-concentration are discussed in Sec. 8.2.3and 8.3.

The surface free energy is given in eq. 61 as a function of the change in chemical

potential of oxygen ∆µO(T,pO)and Zn ∆µZn(T,pZn). Because all structures con-

tain the same amount of Zn (NZn=const.), we can chose a value for ∆µZn(T,pZn).

We set ∆µZn(T,pZn)=-1eV. The formally two-dimensional equation is reduced to

one dimension.

The surface phase diagram for one layer of ZnO on Cu is shown in Fig. 53

and for one layer of ZnO on Ag in Fig. 54. At low oxygen chemical potentials

vacancies are the prevalent structure for both systems. They are more stable for

ZnO on Cu. Oxygen vacancies in ZnO on Cu are stable until -2.4eV of oxygen

84 thermodynamic stability of zno on metal substrates

Figure 53: Surface phase diagram (eq. 61) for one layer of ZnO on Cu considering oxy-

gen defects only. The oxygen amount in the system increases with the oxygen

chemical potential. The lines of the stable structures at a certain chemical po-

tential are colored. Thermodynamically unstable structures are represented

by gray lines. The section of the line of the energetically lowest structure at

a given oxygen chemical potential is colored in black. The stability region of

the structures lowest in energy are marked in the same color as the line. The

structures are labeled by their Zn and O content within the super cell. Below

the phase diagram top views of the thermodynamically stable structures are

shown, ordered by oxygen chemical potential. The chemical potential of oxy-

gen is translated to partial pressures at the exemplary temperature of 800 K

with the help of thermodynamical tables [178]. The Zn chemical potential is

set to ∆µ Zn(T,pZn)=-1eV.

chemical potential when the defect free mono-layer is stabilized. On Ag the ideal

mono-layer is very stable. Only below -3.0eV oxygen defects are stabilized. A

similar situation is observed for oxygen ad-atoms. Pristine ZnO films on Ag are

stable up to -0.25 eV, on Cu incorporation of oxygen starts at -1.5eV of oxygen

chemical potential. The oxygen concentration at the interface is increased quickly

with the chemical potential. Additional 50% of oxygen are stable for a wide range

of oxygen chemical potentials for ZnO on Cu.

We conclude that the stability of oxygen defects in mono-layer ZnO depends on

the oxygen affinity of the metal substrate. For Cu it is high, for Au it is very low.

For the other investigated metals an intermediate situation is observed. In the

next section we will extend our investigations to Zn defects and ad-atoms.

8.2 vacancies and ad-atoms 85

Figure 54: Surface phase diagram for one layer of ZnO on Ag considering oxygen defects

only. See caption of Fig. 53 for further explanations.

8.2.2Zinc defects and ad-atoms

Alternate reconstruction mechanisms for the ZnO (000¯

1)-O and ZnO (000¯

1)-Zn

surface are Zn ad-atoms and Zn vacancies. For the ZnO (000¯

1)-Zn surface the

removal of 25% of the surface Zn atoms should compensate for the excess elec-

trons in the surface state. For the ZnO (000¯

1)-O surface the surface state can

be saturated by additional Zn ad-atoms. For the determination of the surface

phase diagrams we consider the surface free energy as a function of the change

in Zn chemical potential ∆µZn. The change in oxygen chemical potential is (ar-

bitrarily) fixed to ∆µO2(T,pO2)=-1eV, because the amount of oxygen atoms in

the super cells of all considered structures is constant. Again, the formally two-

dimensional equation is reduced to one dimension. The chemical potential of Zn

can be understood as a measure for the concentration of Zn at the surface.

The surface phase diagram considering Zn defects for one layer of ZnO on Cu

(a) 31% zinc vacancies. (b) 13% additional zinc atoms. Zinc ad-atoms

are colored in blue.

Figure 55: Atomic structure for zinc defects in one layer of ZnO on Cu.

86 thermodynamic stability of zno on metal substrates

Figure 56: Surface phase diagram for one layer of ZnO on Cu considering zinc defects

only. See caption of Fig. 53 for further explanations.

is shown in Fig. 56 and for ZnO on Ag in Fig. 57. It is evident, that Zn defects

are less stable than O defects. For Cu, Zn vacancies are observed until -1.4eV of

Zn chemical potential. For Ag they do not show up in the selected window of

chemical potentials. Analogously, Zn ad-atoms are less stable than O ad-atoms.

They adsorb on top of Zn surface sites and are only observed at very high Zn

chemical potentials greater than zero. At this point Zn starts to wet the surface.

A very high concentration of Zn is required to form Zn ad-atoms during film

growth. The observation, that Zn ad-atoms and vacancies are much higher in

energy than defect free ZnO is in agreement with previous work on the (000¯

1)-O

Figure 57: Surface phase diagram for one layer of ZnO on Ag considering zinc defects

only. See caption of Fig. 53 for further explanations.

8.2 vacancies and ad-atoms 87

4.500

50.0% Zn, 50.0% O

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

O vacancies

Zn vacancies

Zn and O vacancies

Defect free

O ad-atoms

Zn ad-atoms

100.0% Zn, 62.5% O

100.0% Zn, 56.3% O

100.0% Zn, 50.0% O

50.0% Zn, 100.0% O

87.5% Zn, 100.0% O

100.0% Zn, 118.8% O

100.0% Zn, 125.0% O

100.0% Zn, 150.0% O

100.0% Zn, 175.0% O

112.5% Zn, 100.0% O

100.0% Zn, 187.5% O

56.3% Zn, 56.3% O

100.0% Zn, 100.0% O

1ZnO/Cu

Figure 58: Two-dimensional surface phase diagram for ZnO on Cu including Zn-/O-

defects, ad-atoms, and random combinations of the former. The projection of

the lowest energy planes on the planes spanned by the changes in chemical

potential of Zn and O are shown. The plane of the defect free mono-layer

is colored black. The defect structures are indicated by a rainbow color map.

The labels give the Zn and O content in percent with respect to the defect free

mono-layer on the metal. The O partial pressure is determined from thermo-

dynamical tables [178].

bulk surface [267]. We will extent our surface phase diagrams further, to ana-

lyze the stability of the simultaneous occurrence of Zn and O defects in the next

section.

8.2.3Oxygen and zinc defects

The previous sections showed, that oxygen defects in the ZnO ultra-thin films

are favored considering the chemical potentials of the species. To compare the

stability of the defective structures directly, we have to consider the surface free

energy as a function of the change in chemical potential of Zn ∆µZn and O ∆µO.

Because of the simultaneously varying amount of Zn and O atoms none of the

two chemical potential can be fixed and the surface phase diagrams become two-

dimensional. The oxygen chemical potential is, again, translated to partial pres-

sures at 800 K. This is a typical temperature for the annealing step of film growth

by molecular beam epitaxy [188]. The surface phase diagrams as a function of

∆µZn and ∆µO2for one layer of ZnO on Cu and Ag are shown in Fig. 58 and

Fig. 59. The surface phase diagram for ZnO on Cu includes randomly distributed

88 thermodynamic stability of zno on metal substrates

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

100.0% Zn, 95.9% O

100.0% Zn, 98.0% O

100.0% Zn, 75.0% O

100.0% Zn, 100.0% O

100.0% Zn, 104.1% O

100.0% Zn, 102.0% O

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

O vacancies

Defect free

O ad-atoms

Zn ad-atoms

102.0% Zn, 100.0% O

1ZnO/Ag

Figure 59: Two-dimensional surface phase diagram for ZnO on Ag including Zn-/O-

defects and ad-atoms.

combinations of Zn and O vacancies. They are not expected to be the structures

lowest in energy for a fixed vacancy concentration. For this purpose a more rig-

orous analysis (e. g. a cluster expansion) is required, which is not in the scope

of this work. We understand our candidate structure as exemplary and expect

only a small error compared to other defect arrangements. This was previously

shown for defects at ZnO surfaces by Wahl et al. [267] and Kresse et al. [132].

The lines in the previous phase diagrams representing different phases are now

intersecting planes. Their orientation is determined by the Zn and O content. The

defect free ZnO mono-layer on the metals in Fig. 58 is colored in black. The defect

structures are marked by a rainbow color map and are labeled by their Zn and O

content with respect to the defect free mono-layer. By comparing the two phase

diagrams for ZnO on Cu and Ag in Fig. 58 and Fig. 59 we observe an increased

stability of the defect free ZnO mono-layer. On Cu, defects (Zn and O) are stabi-

lized at lower chemical potentials. Random distributions of Zn and O vacancies

are only stabilized at even lower chemical potentials. On Ag the mono-layer is

more stable with respect to these reconstructions. At low chemical potentials we

expect even sparser structures to be stabilized. The remaining oxygen at high

oxygen vacancy concentration prefers hexagonal ring-like arrangements. We will

investigate these types of structures in the next section.

For Zn and O defects, we conclude that O vacancies and O ad-atoms are in gen-

eral more stable than their Zn counter parts. Similar trends were previously re-

8.3 ring structures 89

ported for the (0001) and (000¯

1) ZnO surface by Wahl et al. [267] and Kresse et al.

[132]. The ZnO films behave similarly as the two polar ZnO surfaces with respect

to the relative stability of defects, but the overall stability of the defects is system

dependent. For ZnO on Ag a defect free ideal mono-layer is predicted to be stable

over a wide range of Zn and O chemical potentials. In the case of a Cu substrate

this stability range is reduced. In particular, O ad-atoms are stabilized, which we

attribute to the higher oxygen affinity of the substrate.

8.3 ring structures

(a) 3Zn and 2O atoms in a 2×2Cu surface cell. (b) 2Zn and 3O atoms in a 2×2Cu surface cell.

cell.

Figure 60: Atomic structure of hexagonal rings on Cu. The hexagons are constructed

by putting Zn or O at the corners of the ideal α-BN ZnO mono-layer and

inserting units of Zn and/or O in between. For even numbers of inserted

atoms the corner atoms of the hexagons alternate. For odd numbers the Zn

and O content is not equivalent. The species with the higher content is selected

by the atom type placed at the corner of the hexagons.

Sparse ZnO surface reconstructions have been observed in experiment in Ref.

[227,272,261,45,263,50] and their stability was analyzed by theory in Ref. [261,

272,267]. Weirum et al. observed ring structures at low coverages and specifically

chosen annealing conditions [270]. Other structures include a (5×5) reconstruc-

tion and a (√3×√3)R30◦(on ZnO) rotated ring structure.

The structures we will use in the discussion of the surface phase diagrams of

ZnO on metals are shown in Fig. 60 and 61 for Cu as substrate. The structures in

Fig. 60 are constructed by choosing Zn or O as the corner elements of a hexagon

90 thermodynamic stability of zno on metal substrates

(a) 12 Zn and 8O atoms in a 5×5Cu surface

cell.

(b) 18 Zn and 14 O atoms in a 6×6Cu surface

cell.

Figure 61: Atomic structure of ring structures from Valtiner et al. [261] and Wahl

et al. [267].

and inserting units of Zn and O in between (1and 3element units in Fig. 60

a, b and d. For even numbers of inserted atoms the corner elements have to be

alternated to ensure bonds only between Zn and O atoms. With increasing size

of the inserted Zn-O units the sparsity of the structure is increased and greater

areas of metal are exposed. For odd numbers of inserted atoms, the Zn and O

content is not equivalent. The species with the higher content is selected by the

atom type placed at the corner of the hexagons. In Fig. 61 a the (√3×√3)R30◦

reconstruction, investigated by Valtiner et al. [261], is shown. The basic structure

is equivalent to the ring structure in Fig. 60 b. The coincidence is fitted to the

structure proposed by Valtiner et al. Fig 61 b shows the (5×5) reconstruction dis-

cussed by Wahl et al. [267] and measured by Lauritsen et al. [136]. The hexagons

typical for the mono-layer form larger bands creating large pores of exposed

metal surface. In the theoretical studies for the ZnO surfaces in Ref. [261,267]

the (√3×√3)R30◦and (5×5) reconstructions are predicted to be stable at low

to intermediate O chemical potentials.

The surface phase diagrams for ZnO on Cu and Ag, including ring structures,

are shown in Fig. 62 and Fig. 63. We have used the same form of the surface

free energy as in the previous section for their construction. The defects, Zn and

O ad-atoms and vacancies, are marked by the same colors as in Fig. 58 of the

previous section (Sec. 8.2). The defect free, ideal coincidence structure is, again,

shown in black and the different ring structures are colored in orange. The labels

of the ring structures refer to the amount of Zn and O in the chosen metal sur-

face unit cell (see Fig. 60 and 61). The ring structures are predominantly stable in

regions of low Zn and O chemical potential. The lower Zn and O content of the

structures is one obvious reason for this behavior. The other reason is their well

ordered hexagonal structure. If ZnO cannot sustain the 3-fold bonding required

for the ideal mono-layer because of the chemical environment the 3-fold bond-

ing can be conserved for some of the constituents by forming sparser hexagonal

ring structures. The Zn and O vacancies we have previously observed in Sec. 8.2

have been completely replaced by ring structures. The behavior is similar to our

observation for high contents of oxygen vacancies in Sec. 8.2.1. For ZnO on Cu,

oxygen vacancies are pushed from the phase diagram at low oxygen chemical

8.4 hydroxyl absorption 91

4.500

50.0% Zn, 50.0% O

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

Rings (2x2), 3 Zn, 2 O

Rings (5x5), 6 Zn, 5 O

Rings (5x5), 5 Zn, 6 O

Rings (5x5), 12 Zn, 8 O

Rings (6x6), 18 Zn, 14 O

O vacancies

Zn vacancies

Defect free

O ad-atoms

Zn ad-atoms

Rings

1ZnO/Cu

100.0% Zn, 62.5% O

100.0% Zn, 56.3% O

100.0% Zn, 50.0% O

50.0% Zn, 100.0% O

100.0% Zn, 118.8% O

100.0% Zn, 125.0% O

100.0% Zn, 150.0% O

100.0% Zn, 175.0% O

112.5% Zn, 100.0% O

100.0% Zn, 187.5% O

100.0% Zn, 100.0% O

Figure 62: Surface phase diagram for Zn and O defects and ring structures on Cu. The

defects are colored in green, purple, red or blue. The color of the ring struc-

tures is organge. The unreconstructed ideal mono-layer coincidence structure

is shown in black. The labels for defects are as in the previous surface phase

diagrams. For the ring structures the labels refer to the Zn and O content on

the chosen metal surface unit cell.

potentials and high to low Zn chemical potentials. For ZnO on Ag, no oxygen va-

cancies remain in the surface phase diagram, only ring structures are present at

low oxygen chemical potentials. At high oxygen chemical potentials O ad-atoms

prevail for both metals.

We conclude, that sparse, ordered structures are more stable than disordered

vacancy distributions even at intermediate chemical potential. So far we have dis-

cussed H adsorption, spare ring structures, Zn and O defects. In the next section

we will extend our surface phase diagram further by combining Zn/O defects

with structures containing hydrogen.

8.4 hydroxyl absorption

In section 8.1we discussed the adsorption of hydrogen on ZnO ultra-thin films

on different metal substrates. The phase diagrams were constructed from the one

dimensional surface free energy as function of the change in hydrogen chemical

potential. This was possible because the content of Zn and O remained constant

for all investigated structures. In this section we will combine hydrogen adsorp-

tion with Zn and O defects. The surface free energy will, therefore, be a function

of up-to 4variables (∆µH2,∆µO2,∆µZn,∆µH2O).

92 thermodynamic stability of zno on metal substrates

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

Defect free

O ad-atoms

Zn ad-atoms

1ZnO/Ag

100.0% Zn, 100.0% O

100.0% Zn, 104.1% O

100.0% Zn, 102.0% O

102.0% Zn, 100.0% O

Rings (4x4), 5 Zn, 6 O

Rings (4x4), 6 Zn, 5 O

Rings (2x2), 3 Zn, 2 O

Rings

Figure 63: Surface phase diagram for Zn and O defects and ring structures on Ag. See

caption of Fig. 62.

The hydrogen adsorption of Sec. 8.1is one of the stabilization mechanisms for

the O-terminated ZnO (0001) surface. The corresponding stabilization mecha-

nism for the Zn-terminated (000¯

1) surface is the adsorption of hydroxyl groups.

The reservoir for the hydroxyl adsorbate has to be provided by at least two of the

three gaseous phases of molecular oxygen, hydrogen or water. Typically, gaseous

H2O, H2and oxygen O2are considered to be in equilibrium with each other:

2H2+O2⇄2H2O. (102)

In the description by chemical potentials this amounts to the condition:

µO2+2µH2=2gH2O. (103)

In this formalism the partial pressure of the gaseous reservoir, not included ex-

plicitly in the calculation, is ill-defined leading to unrealistic partial pressures.

This problem is overcome by explicitly including all three chemical potentials

in the surface free energy by the approach described by Hermann and Heimel

[98] (also see Sec. 1.6). The surface free energy is a function of the four chemical

potentials:

γ(T,pH) = 1

A[Etot −NMEbulk

Metal −NZn∆µZn −NH2∆µH2(T,pH2)

−NO2∆µO2(T,pO2) − NH2O∆µH2O(T,pH2O)]. (104)

Ebulk

Metal is the bulk (fcc) reference energy of the metal (see Appendix B). The

8.4 hydroxyl absorption 93

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

=-0.0eV, =-1.5eV (10−6mbar, 800K)

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

log(pH2)[mbar] at 800K

100.0% Zn, 93.8% O, 50.0% H

clean surface

O vacancies

Defect free

O ad-atoms

Rings

1ZnO/Cu

100.0% Zn, 62.5% O

100.0% Zn, 56.3% O

100.0% Zn, 50.0% O

100.0% Zn, 118.8% O

100.0% Zn, 125.0% O

100.0% Zn, 150.0% O

100.0% Zn, 175.0% O

6.3% H

50.0% H

100.0% H

12.5% OH

25.0% OH

50.0% OH

100.0% OH

Rings (5x5), 6 Zn, 5 , 5 H

Rings (4x4), 6 Zn, 5 , 5 H

Rings (5x5), 12 Zn, 8 , 3 H

Figure 64: Surface phase diagram for one layer of ZnO on Cu. The surface free energy

is plotted as a function of ∆µH2and ∆µO2, for ∆µZn=0eV and for ∆µH2O=-

1.5eV (see eq. 107). H-adsorption shown in light blue, hydroxyl adsorption in

turquoise. Other structures are colored as in Fig. 62. The new structures are

labeled by their OH-content with respect to the number of available surface

sites. The chemical potential for H2, O2and H2O are translated to partial

pressures at 800 K with the help of thermodynamical tables.

change of the chemical potentials of Zn and H2O will be treated as parameters

to visualize a two-dimensional surface phase diagram. In order to correctly de-

termine NZn,NH2,NO2and NH2Owe have to consider element conservation

ZZn =NZn

ZH=2NH2+2NH2O

ZO=2NO2+NH2O

0= −2NH2−NO2+2NH2O. (105)

Here the ZXare the actual numbers of atoms in the considered structures. These

equations ensure, for example, that if one hydrogen atom coming from a water

molecule is adsorbed, another hydrogen atom and a oxygen atom are put in

to the reservoirs. The fourth equation enforces the satisfaction of the water gas-

phase reaction (eq. 103). The system of equations in eq. 105 has as many equations

94 thermodynamic stability of zno on metal substrates

=-0.0eV,

log(pH2)[mbar] at 800K

O vacancies

Defect free

O ad-atoms

Rings

1ZnO/Cu

=-2.0eV (

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

100.0% Zn, 93.8% O, 50.0% H

clean surface

100.0% Zn, 62.5% O

100.0% Zn, 56.3% O

100.0% Zn, 50.0% O

100.0% Zn, 118.8% O

100.0% Zn, 125.0% O

100.0% Zn, 150.0% O

100.0% Zn, 175.0% O

6.3% H

50.0% H

12.5% OH

25.0% OH

50.0% OH

100.0% OH

Rings (5x5), 6 Zn, 5 , 5 H

Rings (4x4), 6 Zn, 5 , 5 H

Rings (5x5), 12 Zn, 8 , 3 H

10−12 mbar, 800K)

Figure 65: Surface phase diagram for one layer of ZnO on Cu. The surface free energy is

plotted as a function of ∆µH2and ∆µO2, for ∆µZn=0eV and for ∆µH2O=-2.0eV

(see eq. 107). See caption of Fig. 64.

as unknowns and can therefore be inverted to yield solutions for the number of

gaseous species entering into the surface free energy

NZn =ZZn

NH2O=1

9(ZO+2ZH)

NH2=1

9(5

2ZH−ZO)

NO2=1

9(4ZO−ZH). (106)

Putting eqs. 106 into eq. 104 yields

γ(T,pH) = 1

A[Etot −ZMEbulk

M−ZZn∆µZn

−1

9(5

2ZH−ZO)∆µH2(T,pH2)

−1

9(4ZO−ZH)∆µO2(T,pO2)

−1

9(ZO+2ZH)∆µH2O(T,pH2O)]. (107)

This is a function of 4independent variables. The plotting style of the previ-

ous section will be continued and the surface free energy treated as a function

of ∆µH2and ∆µO2.∆µZn and ∆µH2Owill be treated as parameters, that will be

8.4 hydroxyl absorption 95

1ZnO/Cu

=-2.0eV (

=-0.5eV,

Defect free

Oad-atoms

Rings

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

log(pH2)[mbar] at 800K

clean surface

100.0% Zn, 118.8% O

100.0% Zn, 125.0% O

100.0% Zn, 150.0% O

100.0% Zn, 175.0% O

6.3% H

50.0% H

12.5% OH

25.0% OH

50.0% OH

100.0% OH

Rings (5x5), 6 Zn, 5 , 0 H

Rings (4x4), 4 Zn, 4 , 4 H

Rings (5x5), 6 Zn, 5 , 5 H

Rings (5x5), 12 Zn, 8 , 1 H

Rings (5x5), 12 Zn, 8 , 2 H

Rings (6x6), 18 Zn, 14 , 0 H

10−12 mbar, 800K)

Figure 66: Surface phase diagram for one layer of ZnO on Cu. The surface free energy

is plotted as a function of ∆µH2and ∆µO2, for ∆µZn=-0.5eV and for ∆µH2O=-

2.0eV (see eq. 107). See caption of Fig. 64.

chosen to investigate the full phase space. In Fig. 64 the surface phase diagram is

shown for one layer of ZnO on Cu for ∆µH2O=-1.5eV and in Fig. 65 for ∆µH2O=-

2.0eV. The general appearance of the surface phase diagram is similar to the

graphs presented in the previous section. The areas representing the different

most stable phases (lowest in surface free energy) for a given set of chemical po-

tentials are slightly deformed in comparison to the example in Fig. 62. In Fig. 64

the change in chemical potential of the gaseous water reservoir (∆µH2O=-1.5eV)

corresponds to a partial pressure pH2O=1·10−6mbar at 800 K. By explicitly in-

cluding ∆µH2Owe can make predictions for ∆µH2+∆µO2>EH2O. The condensation

of water at the surface is predicted to occur when the critical vapor pressure of

H2O is reached. The highest impact of including ∆µH2Oin Fig. 64 is on structures

containing hydrogen. By decreasing the water chemical potential (Fig. 65) we re-

duce the available hydrogen in the system (cost to take H from the reservoir)

and an increased hydrogen chemical potential is required to stabilize structures

containing H, e. g. H- or OH-adsorbates. By adjusting the water partial pressure

to typical experimental values (<1·10−12 mbar) the unreconstructed ZnO mono-

layer is stabilized even for the previously unstable case of a Cu substrate for O2

and H2partial pressure, that are of the same order as the water partial pressure

(at 800 K). The affinity of the mono-layer on Cu for the formation of inter-facial

oxygen (high O chemical potential) and O vacancies (low O chemical potential)

prevails.

To analyze the influence of the Zn chemical potential on the composition of the

96 thermodynamic stability of zno on metal substrates

=-2.0eV (

=-0.5eV, 10−12 mbar, 800K)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

log(pH2)[mbar] at 800K

clean surface

Rings (5x5), 28 Zn, 24 , 0 H

50.0% H

56.3% H

6.3% OH

25.0% OH

50.0% OH

100.0% OH

2ZnO/Cu

Defect free

Rings

Figure 67: Surface phase diagram for 2layers of ZnO on Cu at 800 K. The surface free

energy is plotted as a function of ∆µH2and ∆µO2, for ∆µZn=-0.5eV and for

∆µH2O=-2.0eV (see eq. 107). See caption of Fig. 64.

=-2.0eV (

=-0.5eV, 10−12 mbar, 800K)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

log(pH2)[mbar] at 800K

6.3% H

50.0% H

50.0% OH

3ZnO/Cu

Figure 68: Surface phase diagram for 3layers of ZnO on Cu at 800 K. The surface free

energy is plotted as a function of ∆µH2and ∆µO2, for ∆µZn=-0.5eV and for

∆µH2O=-2.0eV (see eq. 107). See caption of Fig. 64.

8.4 hydroxyl absorption 97

=-2.0eV (

=-0.5eV, 10−12 mbar, 800K)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

log(pH2)[mbar] at 800K

clean surface

100.0% Zn, 102.0% O

100.0% Zn, 103.1% O

Rings (4x4), 6 Zn, 5 , 0 H Rings (4x4), 5 Zn, 6 , 6 H

Rings (4x4), 6 Zn, 5 , 5 H

6.1% OH

24.5% OH

Defect free

O ad-atoms

Rings

1ZnO/Ag

Figure 69: Surface phase diagram for 1layer of ZnO on Ag at 800 K. The surface free

energy is plotted as a function of ∆µH2and ∆µO2, for ∆µZn=-0.5eV and for

∆µH2O=-2.0eV (see eq. 107). See caption of Fig. 64.

surface phase diagrams we keep the water chemical potential fixed, while ∆µZn

is varied. In Fig. 66 the Zn chemical potential is decreased from ∆µZn=0.0eV in

Fig. 65 to ∆µZn=-0.5eV. ∆µH2Ois in both cases -1.85 eV. For ∆µZn=-0.5eV sparse

ring structures dominate the regime of low oxygen chemical potential. For in-

creased hydrogen chemical potentials variants of these ring structures with hy-

drogen adsorbed at the O-site are energetically most favorable. For ∆µZn=0.0eV

in Fig. 65 the sparse ring structures have almost entirely disappeared from the

surface phase diagram, only at low oxygen chemical potential and high hydrogen

chemical potential they prevail against oxygen vacancies. The other structures in

the surface phase diagram, H- and OH-adsorbates and O ad-atoms, are not influ-

enced by the change in Zn chemical potential because their Zn content is constant.

We will restrict the Zn chemical potential to ∆µZn=-0.5eV and the water chemical

potential to ∆µH2O=-2.0eV (pH2O=10−12 mbar at 800 K) for the following discus-

sion of thicker ZnO films on the metal substrates.

Fig. 66 to Fig. 68 show the surface phase diagrams for one to three layers of

ZnO on Cu. The most important observation is that with increasing film thick-

ness the unreconstructed surface is destabilized, while structures with H- and

OH-adsorbates are stabilized. Already for two layers, the largest part of the sur-

face phase diagram is occupied by structures containing H- and OH-adsorbates.

At high hydrogen chemical potentials H-adsorbates, most prominently (2×1)–H

reconstructed structures, dominate. For high oxygen (and hydrogen) chemical

potential hydroxyl adsorbates are most stable. Only at very low chemical poten-

98 thermodynamic stability of zno on metal substrates

=-2.0eV (

=-0.5eV, 10−12 mbar, 800K)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

log(pH2)[mbar] at 800K

clean surface

Rings (9x9), 112 Zn, 96 , 0 H

1.6% H

12.5% H

50.0% H

1.6% OH

50.0% OH

Defect free

Rings

2ZnO/Ag

Figure 70: Surface phase diagram for 2layers of ZnO on Ag at 800 K. The surface free

energy is plotted as a function of ∆µH2and ∆µO2, for ∆µZn=-0.5eV and for

∆µH2O=-2.0eV (see eq. 107). See caption of Fig. 64.

=-2.0eV (

=-0.5eV, 10−12 mbar, 800K)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−30 −20 −10 0 10

log(pO2)[mbar] at 800K

−10

−5

log(pH2)[mbar] at 800K

clean surface

6.3% H

12.5% H

25.0% H

50.0% H

50.0% OH

Defect free

3ZnO/Ag

Figure 71: Surface phase diagram for 3layers of ZnO on Ag at 800 K. The surface free

energy is plotted as a function of ∆µH2and ∆µO2, for ∆µZn=-0.5eV and for

∆µH2O=-2.0eV (see eq. 107). See caption of Fig. 64.

8.4 hydroxyl absorption 99

tials O-vacancies and ring structures prevail for two layers of ZnO. However, for

two layers of ZnO on Cu the predicted stability range for the unreconstructed

surface is within partial pressure ranges typically realized in experiments. Only

for the three layers of ZnO in Fig. 68 the unreconstructed surface is pushed

from the surface phase diagram by H- reconstructions. While the stability of

H-reconstructed surfaces is increased with layer thickness the stability of OH-

adorbates is unchanged. The portion of the surface phase diagram occupied by

those reconstructions stays approximately constant with increasing film thick-

ness (Fig. 66 to Fig. 68). This again demonstrates the previously observed high

stability of H-reconstructions on Cu (see Sec. 8.1).

For Ag as substrate, we observe the unreconstructed surface in the surface phase

diagrams for one to three layers in Fig. 69 to Fig. 71. The mono-layer in Fig. 69

is especially stable for a large range of O- and H- chemical potentials. For two

layers of ZnO in Fig. 70 the unreconstructed system is substantially destabilized,

especially with respect to H-adsorption. Hydroxyl reconstructions are stabilized

for high O-chemical potentials. For three layers of ZnO on Ag in Fig. 71, the

region of stability for the unreconstructed surface is hardly decreased in com-

parison to two layers of ZnO on Ag. Only the (2×1)–H reconstructed surface

is further stabilized. Similar to the case of ZnO on Cu the portion of the phase

diagram occupied by OH-reconstructions stays approximately constant for two

and three layers of ZnO on Ag, while the stability of H-reconstructions increases.

It is important to note that at 800 K the unreconstructed surface is stable under

experimentally accessible conditions for 1,2, and 3layers of ZnO on Ag. For Cu

this is only true for one and two layers. The stability of the unreconstructed sur-

face is further increased by going to even lower water chemical potentials. The

explicit inclusion of the water chemical potential leads to a destabilization of H-

and OH-reconstructed structures at partial pressures (chemical potentials) that

are relevant in experiment. The prediction of H and OH-free ultra-thin films is in

agreement with previously observed experimental results [259,237,270,159]. In

order to systematically compare the behavior of mono- and multi-layer systems

of ZnO on different metal substrates the dimensionality of the surface phase dia-

grams has to be reduced.

We can reduce the number of variables in our surface phase diagrams further

by defining paths through the phase space represented by the previous surface

phase diagrams. To analyze the stability with respect to H adsorption, ∆µZn,

∆µH2Oand ∆µO2are set to fixed values (∆µZn=-0.5eV, ∆µH2O=-2.0eV, ∆µO2=-

2.0eV). For the OH-adsorption the change in chemical potential of H2and

O2is coupled by the relation ∆µO2=∆µH2and ∆µZn=-0.5eV, ∆µH2O=-2.0eV. In

both cases we can now treat the surface free energy as one dimensional objects.

The results are shown in Fig. 72 for H-reconstructions and in Fig. 73 for OH-

reconstructions. The structures with coverages of up-to 50% of H and OH are

represented by differently shaded colored bars. The color indicates the substrate,

while the shading stands for the H-/OH-content. Below the bars the unrecon-

structed surface is stable (low H chemical potential, dark gray), above the bars

the (2×1)–H reconstruction or OH-coverages equal or larger than 50% are stable.

The chemical potentials of H2, O2and H2O are translated to partial pressure

at the selected temperatures. The structure of the graphs is similar to those we

100 thermodynamic stability of zno on metal substrates

presented in Sec. 8.1, but here we have explicitly included the chemical poten-

tials of oxygen, hydrogen, zinc and water. This treatment not only allows us to

describe OH-reconstructions, it also changes the predicted stability range of H-

reconstructions.

As we have already observed in the discussions of the full surface phase dia-

grams for 1to 3layers of ZnO on Cu and Ag the stability of the unreconstructed

surface with respect to H-reconstructions is increased for Ag, Cu, Pd, Ni, Rh and

Pt by explicitly including the chemical potential of water in the surface free en-

ergy. For Ag, Cu, Ni and Rh as substrate up-to 3layers of ZnO are predicted to

be stable for partial pressure that are realized in experiment. For Pd and Pt only

the first layer is predicted to be stable under such conditions. The spread with

H2chemical potential of the H-reconstructed structures in Fig. 72 is much larger

than for the OH-reconstructed structures in Fig. 73. The OH-reconstructed struc-

tures become stable in a narrow range of H2chemical potential around -1.0eV,

almost independent of the metal substrate and the film thickness. The stability of

H-adsorbates is increased with each additional layer and becomes highly system

dependent. This result is in agreement with our observations from the full surface

phase diagrams for Cu and Ag, where the portion of the phase diagram occupied

by OH-reconstructions hardly changes as a function of film thickness (and metal

substrate). For 800 K in Fig. 73 up to three layers of unreconstructed ZnO can be

stabilized with respect to OH-reconstructions on most metal substrates for exper-

imentally accessible partial pressures. The stability of H-reconstructed structures

is typically smaller than for OH-reconstructions for one and two layers. For film

thicknesses greater than 2layers of ZnO, H-reconstructions become more stable

than OH-reconstructions.

The comparison to theoretical surface phase diagrams for the polar ZnO surfaces

Ag Cu Pd Ni Rh Pt

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

∆µH2[eV]

-20

-15

-10

-5

log(pH2) [mbar] at 800K

2x1-H stable

µO2=-2eV (10−11.5mbar, 800K)

µH2O=-2eV (10−12.1mbar, 800K)

1 L

2 L

3 L

4 L

1 L

2 L

3 L

4 L

1 L

2 L

3 L

4 L

2 L

3 L

4 L

1 L

2 L

3 L

4 L

1 L

2 L

3 L

1 L

bare surface stable

Figure 72: Stability of H-reconstruction for different metal substrates and ultra-thin films

of different thicknesses at 800 K. The shade of the colored bars represents H-

coverages up tp 50%. The base color indicates the metal substrate. Above the

bars 50% and higher coverages are stable, below the unreconstructed surface

is stable. The set values for ∆µZn,∆µH2O,∆µO2are given in the plot.

8.4 hydroxyl absorption 101

Ag Cu Ni Rh

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

∆µH2,∆µO2[eV]

-20

-15

-10

-5

log(pH2) [mbar] at 800K

OH reconstruction

µZn=0.0eV

µH2O=-1.85eV

(10−10 mbar, 800K)

1 L

2 L

3 L 1 L 2 L

3 L 4 L

2 L

1 L 2 L

1 L

bare surface stable -15

-10

-5

log(pO2) [mbar] at 800K

Figure 73: Stability of OH-reconstruction for different metal substrates and ultra-thin

films of different thicknesses at 800 K. The shade of the colored bars represents

OH-coverages up tp 50%. The base color indicates the metal substrate. Above

the bars 50% and higher coverages are stable, below the unreconstructed sur-

face is stable. The set values for ∆µZn,∆µH2O,∆µO2are given in the plot.

by Wahl et al. [267] and Valtiner et al. [261] is difficult as we include aspects

of the ZnO (0001)-Zn and (000¯

1)-O surface in the phase diagrams for the ZnO

thin films on metal substrates. Additionally, Wahl et al. and Valtiner et al. did

not explicitly include the H2O chemical potential and could couple the chemi-

cal potential of O2and Zn via eq. 60 (ZnO bulk reservoir). The high stability

of OH- and H-reconstructions we predict for ZnO films was qualitatively also

observed by Wahl et al. and Valtiner et al.. Large scale defective reconstructions,

as observed/predicted for the polar ZnO surfaces, are less stable than OH- and

H-reconstructions for ZnO thin films. We expect them to appear in the surface

phase diagrams of thicker films, because their construction is based on the re-

moval of atoms in the 2nd and 3rd layer below the surface.

In conclusion we find an increased stability of the unreconstructed ZnO thin

films on all metal substrates by explicitly including the chemical potentials of

Zn, O2, H2and H2O. H- or OH-reconstructions are the most stable reconstruc-

tions in the surface phase diagrams. Zn- and O-defects are only present at very

large/small chemical potentials. Only at low O2chemical potentials sparse ring

structure prevail, that are stabilized for increased H2chemical potentials by ab-

sorbing hydrogen at their oxygen sites. Up-to three unreconstructed layers of

ZnO are predicted to be stable on Ag, which is in agreement with experimental

results of Shiotari et al. [237]. These experimental findings and their comparison

to our theoretical predictions are the topic of the next section.

ZNO ON AG IN EXPERIMENT AND THEORY

(a) STM image of an Ag (111) sample covered

with 60% ZnO (VB=1V, It=0.1nA).

(b) Small ZnO island on a Ag (111) terrace

(VB=1V, It=1nA).

Figure 74: STM images of Ag (111) covered with 60% ZnO, measured at 5K, courtesy

of T. Kumagai [237]. The sample was annealed at 670 K after Zn deposition.

The dark red area in a) and the black area and the island center in b) are the

bare Ag (111) substrate. Lighter colors correspond to ZnO with different layer

heights.

Apart from the question of thermodynamic stability and termination, which

was addressed in the previous Chapters, the number of ZnO layers observed in

experiment is a key issue. In Fig 74 a a STM graph of ZnO on Ag, annealed

at 670 K, is shown. The ZnO coverage is approximately 60% and the data was

measured at 5K (courtesy of T. Kumagai) [237]. The dark red patches in Fig 74 a

correspond to the Ag (111) substrate. The other areas are covered with ZnO lay-

ers of different heights. Fig 74 b shows an enlargement of a small ZnO island on

an Ag (111) terrace. The black area and the central spot of the island are consid-

ered to be part of the Ag (111) substrate. Two different areas are visible in Fig 74

b. They are labeled “low ZnO” and “high ZnO”. They were initially attributed to

be 1and 2layers or 2and 3layers of ZnO, respectively [237]. However, the deter-

mination of absolute heights in STM measurements is very difficult. Electronic

states of the surface and the substrate typically contribute to the signal. Shiotari

et al. [237] tried to improve their measurement by using a set-up that allows for

simultaneous STM and AFM measurements. While STM, strictly speaking, does

not allow for a direct measurement of atomic heights, the AFM mode records the

forces between tip and surface. The AFM measurements are thus not dominated

by the electronic states of the surface but by weak chemical interaction between

tip and surface and are therefore more amenable to height measurements. To

103

104 zno on ag in experiment and theory

(a) STM graph of Ag (111) sample covered with

60% ZnO (VB=1V, It=0.1nA) measured at

5K.

(b) STM and AFM apparent height line scans

along the dotted line in a).

Figure 75: STM image of Ag (111) covered with 60% ZnO, measured at 5K a) and

apparent height line scans in STM and AFM mode b), courtesy of T. Kumagai

[237]. The path of the line scans are indicated by the white dotted line in a).

The area at the right and left ends of the scan are attributed to the Ag (111)

substrate. The area to the right of the island is considered to be a single step

Ag (111) higher than the left side. The high/low ZnO areas are marked in

both plots.

reliably assign the number of layers, the apparent heights obtained from STM or

AFM measurements have to be compared to the results of theoretical simulations.

9.1 apparent heights in experiment

The resulting measured apparent height maps need to be calibrated by adjusting

to a well known sample height. Shiotari et al. [237] calibrate their instrument to

a single step on Ag (111). This necessary calibration is a major source of system-

atic errors in (any) STM/AFM measurement. As already mentioned, the STM

measurements are dominated by the electronic states of the substrate, that dif-

fer significantly for a metal substrate and a semi-conductor thin film. The same

is true for the chemical interactions between the tip and the surface in the AFM

mode. In the AFM mode the condition of the tip has to be controlled and checked

very carefully as contaminations such as adsorbed molecules can change the in-

teraction between tip and surface significantly.

Figure 75 a shows the STM image of an Ag (111) sample covered with 60% ZnO

(courtesy of T. Kumagai) [237]. The Moiré pattern of the (8x8) Ag (7x7) ZnO co-

incidence structure is visible. This is absent in the AFM image (not shown, see

Shiotari et al. [237]), indicating the electronic structure as source of the Moiré pat-

tern.

In Fig. 75 b the results of line scans along the white dashed line in Fig. 75 a in

AFM and STM mode are shown [237]. Both measurement modes were calibrated

to the Ag (111) single step height (2.36 Å). The right and left end of the line scans

9.2 results from dft simulations 105

are attributed to the Ag substrate. The right side is one Ag single step higher. The

results of both measurements are shown in Tab. 10. AFM results are listed first,

STM results are in brackets. The height of “low ZnO” is 4.0±0.3Å. This is high

compared to results obtained for Pd (111) (2.0Å [271]) and Pt (111) (1.8Å [159]).

In combination with the high oxygen partial pressure during deposition, which

lead to ZnO double layers on Pd (111) [271], Shiotari et al. [237] concluded that

“low ZnO” is two layers thick and “high ZnO” three layers thick. The height dif-

ference between low and high ZnO agrees well with step height measurements

obtained by SXRD of 2.1Å [259]. Further evidence has to be collected and the

error due to the calibration procedure requires quantification to solidify this con-

clusion.

System Apparent height AFM (STM) [Å]

Low ZnO 3.8(4.0)±0.3

High ZnO 5.8(6.1)±0.3

Ag(111) step 2.36

Table 10: Experimental heights obtained from the measurement in Fig. 75 a. The AFM

results are given first, STM results are in brackets.

9.2 results from dft simulations

Experimental results do not give a clear picture of the thickness of ultra-thin ZnO

films on Ag. Simulations based on DFT provide absolute values for all atomic po-

sitions, that can be compared to experimental results. In Tab. 11 the thickness of

ZnO ultra-thin films on Ag (111) is presented, more details are shown in Fig. 126

in Appendix K. The heights presented in Tab. 11 are the difference in the mean

value of the z-components of the atoms in the top Ag layer and the top ZnO

layer in the super cell. The atomic structure of these films was already discussed

in detail in Sec. 7.1. Here we will focus on the film heights and the comparison

of theory and experiment.

The values for the layer heights of unreconstructed ZnO on Ag (111) obtained

from the calculation do not fit the measured values. One layer of ZnO is not

thick enough to compare to the “low ZnO” and two layers are too thick. Exper-

iment and theory apparently do not match. In the previous sections we have

learned that the ZnO on Ag (111) reconstructs by absorbing hydrogen on 50% of

the available surface sites. This (2×1)–H reconstruction is predicted to be stable

under thermodynamic conditions (partial pressure of the residual gases in the

measurement chamber) that are met during the experimental data acquisition.

The layer heights obtained from DFT simulations employing the PBE+vdWTS

xc-functional for the (2×1)–H reconstructed ZnO on Ag (111) are presented in

Tab. 11. Further details can be found in Fig. 127 in Appendix Kand Sec. 7.1. The

film thickness for the (2×1)–H reconstructed ZnO on Ag (111) agree well with

the values reported from experiments. The mono-layer film is 4.02 Å in theory

and “low ZnO” 3.8±0.3Å in experiment. The double layer is 6.48Å in theory

106 zno on ag in experiment and theory

System h DFT [Å] σDFT [Å] App. h DFT [Å]

1layer ZnO 2.61 0.095 1.74

2layers ZnO 6.48 0.317 4.16

3layers ZnO 9.16 0.315 6.82

1layer ZnO–(2×1)–H 4.02 0.354 2.01

2layers ZnO–(2×1)–H 6.48 0.317 4.62

3layers ZnO–(2×1)–H 9.16 0.315 7.33

Ag(111) step 2.36

Table 11: Height of ZnO layers on Ag (111) obtained by DFT calculations employing the

PBE+vdWTS xc-functional [196,254]. The heights are measured as difference

between the mean value of the lateral position of the atoms in the top Ag layers

and the mean value of the lateral position of the atoms in the top ZnO layers.

The deviation from the calculated mean height, due to corrugation, is quanti-

fied by the standard deviation σ. See text for a description of the procedure to

calculate apparent heights from DFT calculations.

whereas “high ZnO” is 5.8±0.3Å in experiment. This indicates that the surface

could be covered with hydrogen, but a careful analysis of the surface vibrations

by infrared spectroscopy (courtesy of Bo-Hong Liu and Shamil Shaikhutdinov,

FHI, Berlin) did not reveal any characteristic O-H vibration modes. The exper-

imental conclusion is that no hydrogen is adsorbed at the surface. Indeed the

analysis of the thermodynamic stability of the (2×1)–H reconstruction in Sec. 8.1

revealed, that the unreconstructed surface is stable under annealing conditions

(800 K) for up to three layers. Additionally, the formation of the (2×1)–H over-

layer could be blocked under measurement conditions (5K) by a barrier for the

dissociation and/or diffusion of hydrogen.

Further theoretical evidence for a hydrogen free surface can be gained by a

comparison of formation energies for a clean Ag (111) surface and the unrecon-

structed and reconstructed ZnO mono- and double-layer films. By adding the

total energy from the DFT simulation of a clean Ag (111) surface and 2layers of

ZnO on Ag (111) and subtracting twice the total energy of a mono-layer film on

Ag (111) we obtain:

The energy balance is positive. Thus the system prefers to form ZnO double lay-

ers and leave patches of Ag (111) surface uncovered rather than to form a closed

mono-layer. This is in agreement with experimental observations for ZnO on Ag

(111) [237,192] and Pt(111) [159] of island formation after catalytic experiments.

For the (2×1)–H reconstructed films adding two layers of (2×1)–H reconstructed

ZnO, a layer of hydrogen and the clean Ag (111) surface and subtracting twice

the (2×1)–H reconstructed ZnO mono-layer yield a negative value:

9.2 results from dft simulations 107

This implies that a closed (2×1)–H reconstructed ZnO mono-layer should be

more stable than two layers and patches of Ag. This is in disagreement with

experimental observations. To further investigate the difference between experi-

mental apparent heights and the atomic structure obtained from DFT calculations

theoretical STM graphs were calculated within the Tersoff-Hamann approach

(Ref. [250] and Sec. 1.9). The density was summed up from the Fermi level to the

bias voltage at Cartesian grid points. This yields a theoretical current map for

different (constant) heights. To compare to experimental constant current STM

measurements, the maps are sorted by their current values at each grid point.

For these (constant) current values the corresponding heights are plotted. These

simulated constant current STM graphs for one and two layers of ZnO and ZnO–

(a) One layer of ZnO on Ag (111). (b) Two layers of ZnO on Ag (111).

(111).

Figure 76: Simulated constant current STM graphs of 1and 2layers of ZnO and ZnO–

(2×1)–H on Ag (111). The super cell used in the DFT calculation is indicated

in the graphs.

108 zno on ag in experiment and theory

(2×1)–H on Ag (111) are shown in Fig. 76 for a constant current of 1·10−4eV/Å.

The (2 times1)–H reconstructed films show stripes, that correspond to the rows

of adsorbed hydrogen. In experiment no stripes were observed (see Fig. 75 and

Ref. [237]), indicating the lack of hydrogen.

From the simulated constant current maps the film height cannot be obtained

directly. In analogy to experiment, where the instrument is calibrated at Ag sin-

gle steps, the film thickness has to be calculated as height difference between a

simulated Ag (111) STM graph and ZnO on Ag (111). The required values are

calculated by averaging the absolute heights at the same constant current for

the bare Ag and ZnO on Ag over the super cell. A further discussion concern-

ing the choice of the current for the image creation of the constant current STM

graphs can be found in Appendix L. The calculated apparent heights are listed

in Tab. 11. With 1.74 Å the height of the mono-layer is too small if compared

to experimental results. The apparent height of the double layer (4.16 Å) and of

three ZnO layers (6.82 Å) are in good agreement with the experimental results.

The same procedure can be carried out for (2×1)–H reconstructed ZnO on Ag

(111). The calculated apparent heights are listed in Tab. 11. The obtained values

are too high compared to experiment. We can confirm the assessment of Shiotari

et al. [237] that their experimentally observed films are indeed two and three lay-

ers of hydrogen free ZnO. The agreement between theory and experiment can be

explained by the improved theoretical description of the measurement process.

9.3 conclusion

In conclusion, the combination of experiment and theory gives further evidence

for the proposition that the ZnO ultra-thin films observed by Shiotari et al. are

indeed at least two layers in thickness. The low ZnO is two layers of unrecon-

structed ZnO on Ag and the high ZnO is three layers of unreconstructed ZnO

on Ag (111). Thermodynamic and total energy results lead to this conclusion.

The disagreement between experimental apparent height and the atomic struc-

ture can be remedied by calculating the apparent height from simulated constant

current STM graphs based on DFT calculations.

SPECIAL SURFACES

In this chapter we will focus on coincidence structures and reconstructions of

ultra-thin ZnO films on Ag (111) and Rh (100). For Ag we will discuss a (5×5)

coincidence structure of ZnO on Ag that was experimentally observed by Shio-

tari et al. [237] as minority phase together with the more abundant (8×8) coin-

cidence structure. We will analyze the stability of the (5×5) reconstruction and

include it in the surface phase diagrams of Sec. 8.4. For Rh experimental results

are available for the (100) surface. This substrate provides a completely different

symmetry than the previously discussed (111) surface. We will analyze and com-

pare the stability of ZnO on both surfaces of Rh. This will lead us to a broader

understanding of strain in the ZnO films and how it influences their stability

with respect to H-reconstructions.

10.1 zno (0001)-(5×5)/ag (111)-(3√3×3√3)R30◦

(a) One layer of ZnO. (b) Two layers of ZnO.

Figure 77: Side view of one a) and two b) layers of ZnO on Ag in the ZnO (0001)-(5×

5)/Ag (111)-(3√3×3√3)R30◦reconstruction proposed by Shiotari et al. [237].

A highly corrugated, domed structure is formed for one and two layers.

Several authors reported coincidence structures from direct (STM) or indirect

observation (LEED) that differ from the structure deduced in Sec. 6[237,159]. The

focus in this section will be on the ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦

reconstruction observed and described by Shiotari et al. [237]. In their work ZnO

was grown on an Ag (111) substrate, that was cleaned by Ar ion sputtering and

annealing prior to Zn deposition. The deposition chamber (p< 5 ×10−10mbar)

was separated from the measurement chamber. The Zn was deposited on the sub-

strate at room temperature by heating a Zn-rod in a Knudsen cell [188] to 490 K.

During growth a high partial pressure of O2was set (5×10−5mbar). This proce-

dure followed an annealing step under UHV conditions (p< 5 ×10−10mbar) at

670 K. In the following, we will consider a temperature of 600 K.

Shiotari et al. [237] showed that the ZnO islands and films become well-ordered

109

110 special surfaces

(a) One layer of (2×1)–H reconstructed ZnO. (b) Two layers of (2×1)–H reconstructed ZnO.

Figure 78: Side view of one a) and two b) layers of ZnO on Ag in the ZnO (0001)-(5×

5)/Ag (111)-(3√3×3√3)R30◦reconstruction proposed by Shiotari et al. [237].

The ZnO is (2×1)–H reconstructed. The corrugated and domed structure is

regularized by H-adsorption and a nearly regular wurtzite structure in the

ZnO is observed.

only for annealing temperatures above 600 K. Significantly higher temperatures

lead to the sublimation of ZnO. The edges of the observed islands at low cov-

erages act as aggregation point for subsequent layers during annealing. The

films grow in preferential directions following the crystallographic plains of the

Ag (111) substrate. The predominantly observed structure is the ZnO (0001)-

(7x7)/Ag (111)-(8x8) reconstruction, also reported by other authors [259,192].

LEED patterns revealed streaky ZnO refraction spots, indicating different az-

imuthal orientations of the films with respect to the Ag substrate [237] (for ZnO

on Pt see Ref. [159]). Apart from the ZnO (0001)-(7x7)/Ag (111)-(8x8) reconstruc-

tion Shiotari et al. [237] could image another Moiré structure that is less abundant.

The structure has a periodicity of about 16Å and an azimuthal orientation of 30◦

relative to the Ag (111) high symmetry directions. Shiotari et al. [237] proposed a

ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction as a possible model

for this structure. We will assess the stability of this model structure with the

ab initio thermodynamics framework presented in the previous sections (see [214]

and Sec. 1.6).

In Fig. 77 the structure of one layer of ZnO (0001)-(5×5)/Ag (111)-(3√3×

3√3)R30◦is shown. The structure was initialized in the ideal α-BN structure

with an in-plane lattice parameter of a=3.05Å which results in 7% strain with

respect to the ideal freestanding ZnO mono-layer. The strain for the theoretically

predicted ZnO (8x8)/Ag (9x9) reconstruction is only ∼1%. The lattice parameter

of the substrate was kept at the equilibrium value obtained with the PBE xc-

functional [196]. The strain in the ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦

reconstruction leads to a highly corrugated structure for one and two layers of

ZnO (see Fig. 77). The complete film is partially lifted from the substrate and a

"domed" structure is formed. The regular but highly strained wurtzite structure

of ZnO can be retained by absorbing H at the surface. In Fig. 78 the ZnO (0001)-

(5×5) /Ag (111)-(3√3×3√3)R30◦reconstruction is shown with an additional

(2×1)–H reconstruction of the ZnO for one and two layers of ZnO. The ZnO

(0001)-O structure is not recovered as perfectly as for the regular coincidence

10.1 zno (0001)-(5×5)/ag (111)-(3√3×3√3)R30◦111

1ZnO/Ag

=-1.0eV, =-1.5eV (10−13 mbar, 600K)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 −30 −20 −10 0 10

log(pO2)[mbar] at 600K

−10

−5

log(pH2)[mbar] at 600K

clean surface

Rings (4x4), 6 Zn, 5 , 0 H

Rings (2x2), 2 Zn, 3 , 3 H

Rings (4x4), 5 Zn, 6 , 6 H

Rings (4x4), 6 Zn, 5 , 5 H

6.1% OH

24.5% OH

Defect free

Rings

- clean surface

Figure 79: Surface phase diagram for one layer of ZnO on Ag including the ZnO (0001)-

(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction [237] for µZn=-1.0eV.

=-1.5eV (10−13 mbar, 600K)

=-0.5eV,

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 −30 −20 −10 0 10

log(pO2)[mbar] at 600K

−10

−5

log(pH2)[mbar] at 600K

clean surface

Rings (4x4), 6 Zn, 5 , 0 H

Rings (2x2), 2 Zn, 3 , 3 H

Rings (4x4), 5 Zn, 6 , 6 H

Rings (4x4), 6 Zn, 5 , 5 H

6.1% OH

24.5% OH

Defect free

Rings

1ZnO/Ag

- clean surface

Figure 80: Surface phase diagram for one layer of ZnO on Ag including the ZnO (0001)-

(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction [237] for µZn=-0.5eV.

112 special surfaces

structure. A few O and Zn atoms still deviate from the wurtzite structure, but

no large scale corrugation or partial lifting of the film from the substrate was

found. The ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction has a

much denser structure as compared to the ZnO (8x8)/Ag (9x9) coincidence. We

expect the Zn and O chemical potentials to influence the stability of those two

types of structures and we therefore will investigate this dependence next.

In Fig. 79 and Fig. 80 the surface phase diagram for one layer of ZnO on Ag is

shown for two different values of µZnO. The pool of structures includes the ZnO

(0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction with and without hydro-

gen. The water chemical potential is kept constant at -1.5eV for both surface

phase diagrams. The temperature is set to 600 K to be comparable to experimen-

tal results. The previously discussed behavior of sparse ring structures being sta-

bilized at low Zn chemical potentials is observed for low O chemical potentials

(see sec. 8.3). For low Zn chemical potentials (Fig. 79) the ZnO (0001)-(5×5)/Ag

(111)-(3√3×3√3)R30◦reconstruction is only stabilized at high O chemical po-

tentials and no H-reconstruction is present for this reconstruction. At low O2

chemical potential sparse ring structures prevail. Upon increasing the Zn chem-

ical potential the ring structures start to retreat from the surface phase diagram.

The stability region of the ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦recon-

struction on the other hand is increased to O partial pressures at 600 K that

are encountered in experiment. Both coincidence structures (reconstructions) lie

in the region of experimental conditions. Further increase of the Zn chemical

potential results in a further stabilization of the ZnO (0001)-(5×5)/Ag (111)-

(3√3×3√3)R30◦reconstruction, which then dominates the surface phase dia-

=-1.0eV, =-1.5eV (10−13 mbar, 600K)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 −30 −20 −10 0 10

log(pO2)[mbar] at 600K

−10

−5

log(pH2)[mbar] at 600K

clean surface

Rings (9x9), 112 Zn, 96 , 0 H

1.6% H

12.5% H

50.0% H

1.6% OH

50.0% OH

Defect free

Rings

2ZnO/Ag

- clean surface

Figure 81: Surface phase diagram for two layers of ZnO on Ag including the ZnO (0001)-

(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction [237] for µZn=-1.0eV.

10.1 zno (0001)-(5×5)/ag (111)-(3√3×3√3)R30◦113

=-1.5eV (10−13 mbar, 600K)

Defect free

Rings

2ZnO/Ag

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 −30 −20 −10 0 10

log(pO2)[mbar] at 600K

−10

−5

log(pH2)[mbar] at 600K

clean surface

Rings (9x9), 112 Zn, 96 , 0 H

1.6% H

12.5% H

50.0% H

1.6% OH

50.0% OH

- clean surface

=-0.5eV,

Figure 82: Surface phase diagram for two layers of ZnO on Ag including the ZnO (0001)-

(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction [237] for µZn=-0.5eV.

gram. This would contradict experimental results, which report the ZnO (0001)-

(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction to be less abundant than the

ZnO (0001)-(7x7)/Ag (111)-(8x8) reconstruction.

Shiotari et al. [237] suggested that their films have a thickness of two layers,

although some uncertainty remains. In Fig. 81 and Fig. 82 the surface phase dia-

grams for two layers of ZnO on Ag for two different Zn chemical potentials are

shown including the ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruc-

tion. The behavior is similar to the one layer case. With increasing Zn chemi-

cal potential the ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦reconstruction is

stabilized. We expect that the observation of the ZnO (0001)-(5×5)/Ag (111)-

(3√3×3√3)R30◦reconstruction is highly dependent on the conditions during

film growth. The initial high surface concentration of Zn during film growth by

MBE is subsequently reduced during the annealing step, when ZnO forms by

oxidation. ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦will initially form, but

is expected to transform to ZnO (0001)-(7x7)/Ag (111)-(8x8). This (full) transfor-

mation could be kinetically hindered e. g. by a reduced diffusion length due to

the low temperature (5K) during the measurement process.

In conclusion, we find the experimentally observed ZnO (0001)-(5×5)/Ag (111)-

-(3√3×3√3)R30◦reconstruction of Ref. [237] to be stable under experimental

conditions for high Zn chemical potentials. During growth/annealing the Zn

chemical potential is expected to decrease due to the formation of ZnO and

possible sublimation of Zn. The ZnO (0001)-(5×5)/Ag (111)-(3√3×3√3)R30◦

structure is than expected to transform to the regular coincidence structure, that

is predicted to be the most stable structure at low Zn chemical potentials (also

114 special surfaces

(a) 1layer of ZnO. (b) 2layers of ZnO.

Figure 83: Atomic structure of 1to 4layers (a - c) of ZnO deposited on Rh (100). The

calculation was initialized in the α-BN structure and the forces on the atoms

relaxed by a DFT calculation employing the semi-local functional PBE [196].

The coincidence structure between ZnO and Rh is based on the data measured

by Kato et al.[120] and the model proposed by the same authors.

see discussion in Sec. 8.4). The transformation of the ZnO (0001)-(5×5)/Ag (111)-

(3√3×3√3)R30◦reconstruction could be hindered due to kinetic effects, such as

small diffusion lengths due to the very low temperatures during measurements

(5K).

10.2 rh (100)

In the previous sections we discussed the structure of ZnO (0001) on different

transition metals. The focus was entirely on the (111) surfaces of these metals.

The choice of this crystallographic surface is obvious. The three-fold symme-

try of ZnO is matched with the same symmetry on the (111) metal surface (see

Sec. 6). The ZnO films align with the substrate and the mismatch in lattice pa-

rameter is compensated by a coincidence structure. To investigate the influence

of non-uniaxial strain, other substrates without three-fold symmetry can be used.

Experimentally the same technique to deposit ZnO on the (111) surface (MBE,

see Ref. [188]) can be applied for other crystallographic surfaces. In this section,

we will investigate the atomic structure, stability and electronic structure of ZnO

on the (100) surface of Rh. This combination of transition metal surface and ZnO

thin films has been experimentally prepared and investigated by Kato et al. [120].

The analysis will make use of the surface/interface model developed by those

10.2 rh (100)115

(a) One layer of ZnO. (b) Two layers of ZnO.

Figure 84: Atomic structure of one a) and two b) layers of (2×1)—H reconstructed ZnO

on Rh (100). The geometries were initialized in α-BN structure and relaxed

employing the semi-local xc-functional PBE [196].

authors.

Kato et al.[120] deposited a Zn film on a Rh (100) crystal surface, that was cleaned

by Ar sputtering and annealing. Zn was deposited by a degassed quartz crucible

evaporator at room temperature. The sample was subsequently annealed up to

600 K at oxygen partial pressures ranging from 1×10−9mbar to 2×10−7mbar.

The base pressure of the vacuum system consisting of a preparation and mea-

surement chamber was 5×10−10 mbar (4×10−11 mbar in measurement chamber).

The long range order of the sample was determined by LEED and local high res-

olution data was acquired by STM at room temperature. Based on their high

resolution STM measurements Kato et al. [120] proposed a ZnO (13x2)/Rh (100)

(16x2) coincidence structure. In their model the ZnO film is compressed by 5%

in the [1000] direction and, 2% stretched in the [0¯

100] direction. Kato et al. [120]

measured the corrugation to be 0.04 nm.

We will use this model structure of Kato et al. [120] as input for a DFT investi-

gation based on the semi-local PBE xc-functional [196]. For our calculations we

employed the equilibrium lattice parameter for Rh, obtained with the PBE xc-

functional, to minimize strain in the substrate. The lattice parameter of ZnO we

adjusted to produce the experimentally observed ZnO–(13x2)/Rh(100)–(16x2) co-

incidence structure. In this structure, based on the theoretical lattice parameter,

ZnO is strained by -3.8% in the [1000] direction and +1.6% in the [0¯

100] direction.

The compression/stretching is slightly less than in experiment. We initialized

the ZnO film in the α-BN structure and allowed it to relax until the forces on

the atoms in the film and the two top most Rh layers were below 0.05 eV/Å. The

results for one to four layers of ZnO in the ZnO (13x2)/Rh (100) (16x2) recon-

struction are shown in Fig. 83.

In our calculations, the strain leads to the formation of a d-BCT structure pro-

posed for ZnO thin films by Morgan et al. [171,169,170] and Demiroglu et al. [65].

The actual realization of this structure depends strongly on the strain because of

the lattice mismatch between the Rh substrate and the ZnO ultra-thin film. The

inhomogeneous strain (extension/compression) seems to facilitate the formation

of the d-BCT for ZnO on Rh (100). For ZnO on the (111) surface of different tran-

sition metal substrates no such pronounced d-BCT structure could be observed

116 special surfaces

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Two layers of ZnO on Rh (100)

8 6 42 0 2

E−F[eV]

100

120

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Four layers of ZnO on Rh (100)

Figure 85: Projected density of states for two layers and four layers of ZnO on Rh (100).

The density is projected on Rh and each ZnO layer. The area underneath the

curves for layer numbers greater than one, which have shifted up in energy is

filled with the color of the curve of this layer.

(see Sec. 7.1). Only for 3and 4layers of ZnO on Ni and 4layers on Rh (111) the

d-BCT ZnO was observed. For ZnO on Rh (100) already the mono-layer shows

the regular corrugation leading to the full d-BCT structure for thicker films.

To assess the stability of the ZnO films on Rh (100) we will investigate the ef-

fect of hydrogen adsorption by considering the unreconstructed surface and the

(2×1)–H reconstruction. The focus lies on the point in the surface phase diagram,

at which the (2×1)–H reconstructed ZnO thin film becomes stable with respect

to the unreconstructed film. The atomic structures of (2×1)–H reconstructed one

and two layer thick ZnO films on Rh (100) are shown in Fig. 84. The ZnO films

have adopted the wurtzite structure. The alignment of the Zn and O atoms is not

as perfect as in the case of the (111) metal surface substrate. Some residual dis-

order remains. With the help of the modified ab initio thermodynamics approach

proposed by Hermann and Heimel [98] (also see sec. 1.6) we can determine the

transition point from the clean to the (2×1)–H reconstructed surface as a function

of H2chemical potential. For fixed Zn (∆µZn=0.0eV) , O2(∆µO2=-0.9eV) and

H2O (∆µH2O=-0.9eV) chemical potential the transition occurs at a H2chemical

potential of ∆µH2=-0.1eV for one layer of ZnO and at even higher H2chemi-

cal potential for two layers. For ZnO on Rh (111) the transition is predicted to

occur already at -1.0eV for one layer and at -1.5eV for two layers (see Fig. 72).

Expressed in partial pressure at 400 K this would amount to 1×10−20 mbar and

1×10−25 mbar for one and two layers of ZnO on Rh (111). We therefore predict

the unreconstructed ZnO films on Rh (100) for the coincidence structure by Kato

et al.[120] to be more stable than the films on Rh (111) with respect to hydrogen

absorption.

Finally, we will discuss the electronic structure of ZnO on Rh (100). The projected

density of states for one and two layers of ZnO on Rh (100) with and without

hydrogen are shown in Fig. 85 and Fig. 86. The density is projected on the single

10.2 rh (100)117

8 6 42 0 2

E−F[eV]

100

120

140

(pro)DOS [states/eV]

1st layer

2nd layer

Figure 86: Projected density of states for two layers of (2×1)–H reconstructed ZnO on

Rh (100). The density is projected on the metal and each ZnO layer. The area

underneath the curves for layer numbers greater than one, which have shifted

up in energy is filled with the color of the curve of this layer.

layers of ZnO. A shift towards the Fermi level provided by the metal is observed

for both cases. For the hydrogen free case the effect is larger. For four layers of

ZnO on Rh (100) the behavior is similar to the situation for ZnO on metal (111)

substrates. Due to the intrinsic field inside the ZnO films the electronic states

of each successive layer are pushed higher in energy, closer to the Fermi level,

where they are eventually pinned. The electronic states of four layers of ZnO on

Rh (100) are shifted slightly less up in energy than for ZnO on Rh (111) and the

states close to the Fermi energy show a smaller density. For the (2×1)–H recon-

structed surface in Fig. 86 the intrinsic field is quenched and the position of the

electronic states stays fixed throughout the extend of the film.

In conclusion we find that the uniaxial strain imposed on the ZnO films by de-

positing on the Rh (100) surface promotes the formation of a d-BCT structure.

The regular wurtzite structure can be retained by absorbing hydrogen at the

surface, leading to a (2×1)–H surface reconstruction. With the help of ab initio

thermodynamics we predict the (2×1)–H reconstruction to be much less stable

than in the case of a (111) metal substrate. The surface without hydrogen is pre-

dicted to be stable for a wide range of chemical potentials and therefore different

from the polar ZnO surfaces (0001)-Zn and (000¯

1)-O. We find the evolution of the

electronic structure with film thickness to be governed by the same effects as for

a (111) metal substrate described in Sec. 7.2.

PARAMETERIZATION OF DENSITY MATRIX FORMALISM BY

DFT

(a) Side view of L4P adsorbed on ZnO

10¯

10.

(b) Top view of 2×2surface unit cells of L4P (light blue)

on 12 ×4surface unit cells of ZnO 10¯

10 (black).

Figure 87: Ladder-type oligomers (L4P) [127] adsorbed on ZnO 10¯

10.

In the previous Chapters the focus of the theoretical work was on the descrip-

tion of the atomic and electronic structure of ZnO ultra-thin films on transition

metals. These systems are the first step to quantitatively characterize hybrid sys-

tems composed of an inorganic semiconductor and an organic light harvesting

molecule [21,176,3,142], which opens the path to new device functionality.

Recently, experimental evidence for non-radiative energy transfer processes be-

tween ZnO and organic molecule has been found [108,46,34]. However, the

exact excitation transfer mechanism is difficult to determine experimentally. A

proper description of those processes could open the path for tailoring new de-

vices for opto-electronics. While it is crucial to understand the inorganic substrate

(ZnO) fully (by modeling via ultra-thin films) a significant effort is required to

accurately describe the interaction processes between organic molecule and the

ZnO surface. As a first step towards this description, we investigated the opti-

cal absorption in such a hybrid system in collaboration with E. Verdenhalven,

A. Knorr and M. Richter from the Technical University of Berlin (TU Berlin) as

part of this work [265]. Within a Heisenberg equation of motion technique, Bloch

equations for such hybrid systems were derived by E. Verdenhalven, A. Knorr

and M. Richter. We then parameterized the model with the atomic and electronic

structure obtained from DFT calculations (this work) employing the hybrid xc-

functional HSE06 [100]. A more detailed description and derivation of the "hy-

brid Bloch equations" can be found in the next section and in Ref. [265].

119

120 parameterization of density matrix formalism by dft

Figure 88: Illustration of the operators and values entering the Hamiltonian (eq.110).

11.1 linking to density-matrix theory

Density functional theory can be used to investigate equilibrium properties of

surfaces. To gain inside into the dynamical properties of inorganic/organic hy-

brid systems a density matrix formalism based on a Heisenberg equation of

motion technique [143,218] is applied [265]. Of special interest are excitonic

processes between the organic and inorganic component of our system with-

out electronic wave-function overlap (Förster processes [82]). We here consider

energy transfer from molecular excitations to semiconductor continuum states.

The semiconductor offers many states for coupling to the molecule. If the cou-

pling proofs to be strong enough this would provide an efficient way for exciting

the molecule by external electrical pumping. The equations presented below are

based on a canonically quantized description (second quantization). We will fo-

cus on ladder-type oligomers (L4P) [127] adsorbed on a semiconducting surface

(ZnO (10¯

10)). An example of such a model systems is shown in Fig. 87. The

description of excitonic processes can in principle be achieved on the level of a

first principle calculation e.g. by Greens function based perturbation theory (GW)

[97,106,10] extended by the Bethe-Salpeter equation (BSE) [224,216]. These meth-

ods provide a powerful tool set for investigating excited states, but are not (yet)

feasible for the system of interest in this work. The reader is referred to recent

reviews form the literature for further information [10,186]. Here we follow the

approach developed by E. Verdenhalven, A. Knorr and M. Richter based on a

density matrix formalism [265].

The Hamiltonian for the systems described above can be defined as:

H=HM+HSC +He−l+Hint (108)

which is divided into four parts. The Hamiltonian for the isolated molecule

HM=∑

A,ν

EAνa†

AνaAν. (109)

and for the isolated semiconductor

HSC =∑

k,λ

λa†

λ,kaλ,k. (110)

11.1 linking to density-matrix theory 121

The electron-light interaction, described as interaction between the quantized

electrons and an external classical field [108], is treated semi-classically within

the dipole approximation [123]

He−l=∑

k,λ=λ′

λ,λ′·E(t)a†

λ,kaλ′,k+∑

ν,A=B

dAν,Bν·E(t)a†

AνaBν(111)

and the Coulomb interaction between the electrons in the semiconductor and the

molecule

Hint =∑

k,k′,ν(VkvLν

k′cHνa†

kva†

LνaHνak′c+VkcHν

k′vLνa†

kca†

HνaLνak′v). (112)

All contributions are illustrated in Fig. 88. The indexes Aνnumber the states of

the ν-th molecule in the model. Here we only take the highest occupied molecu-

lar orbital (HOMO) and lowest unoccupied molecular orbital into account. They

are indexed with Hνand Lνfor the ν-th molecule. kis the wave vector in the

semiconductor. For a surface it is a two-dimensional vector, corresponding to the

in-plane directions. The bands are indexed by λ. The conduction band is labeled

cand the valence band is labeled v. EAνis the quasi electron/hole energy of

the molecule and Ek

λthe quasi-electron/hole energy of the semiconductor. They

will be approximated by the Kohn-Shan eigenenergies obtained from DFT calcu-

lations.

We have described the molecule in a localized, non-lattice periodic basis. In or-

der to describe semiconductor and molecules in the same Bloch momentum rep-

resentation [240] we assume a periodic arrangement of equal molecules. This is

achieved by introducing two dimensional wave-vectors lfor the molecular oper-

ators in eq. 110-112

aAν=∑

√NM

e−il·RνaA,l. (113)

NMis the total number of molecules in our model. By definition the area of the

super cell containing the molecules A=NMAMmust be equal to the area of

the surface unit cell containing the semiconductor unit cells A=NSCASC to

not violate periodic boundary conditions. With eq. 110-112 and the properties of

the periodically arranged molecules the Hamiltonian can be transformed to the

Bloch basis representation

HM=∑

EA∑

a†

A,laA,l, (114)

He−l=∑

k,λ=λ′

λ,λ′·E(t)a†

λ,kaλ′,k+∑

A,B

dAB ·E(t)∑

a†

A,laB,l, (115)

Hint =1

ASCNSC ∑

k,k′,l,l′∑

δl−l′,k−k′+G(VkvL

k′cHa†

v,ka†

L,laH,l′ac,k′(116)

+VkcH

k′vLa†

cka†

H,laL,l′avk′).

122 parameterization of density matrix formalism by dft

The Coulomb matrix element Vkλ′B

k′λA is now defined without the molecular posi-

tions Rν

Vkλ,A

k′λ′,B=ASCNSC ∑

e−i(k−k′)·RνVkλ,Aν

k′λ′,Bν. (117)

The combination of the two periodic lattices of molecule and semiconductor sur-

face give the Kronecker-symbol in eq. 114-116, which yields a selection rule for

the momentum transfer.

The goal is to calculate the (macroscopic) absorption of the semiconductor PSC(t)

and the molecule PM(t). They are linked to the microscopic expectation values

of the electronic transition amplitudes σkk′

vc and σll′

HL via

PSC(t) = 2

V∑

Re[dvc

kσkk

vc](118)

PM(t) = 2

V∑

Re[dHLσll

HL](119)

where Vis the volume of the full system. The microscopic inter-band polarization

in the semiconductor σkk′

vc =⟨a†

k,vak′,c⟩and the microscopic molecular polariza-

tion σll′

HL =⟨a†

l,Hal′,L⟩can be calculated by the Heisenberg equation of motion

dtσ(t) = i



h[H,σ(t)] + ∂σ

∂t . (120)

Here the hierarchy problem resulting from the correlation expansion of the Cou-

lomb interaction is truncated at the level of Hartree-Fock [143,123,15,89]. The

resulting closed set of equations can be solved in frequency space by Fourier

transformation. Here we only reproduce the closed solution

σl,l

HL ≡σl

HL(ω) = XLH(ω)ˆ

E(ω)·dLH(ρH−ρL)[1−XLH(ω)Nm

A2×

∑

kl∑

G⏐⏐⏐VklvL

kl+GcH⏐⏐⏐

2Ykl,kl+G(ω)(ρkl

v−ρkl+G

c)(ρH−ρL)⎤

⎦

−1

(121)

with the single energy particle poles

XLH(ω) = [

hω − (EL−EH) + iγM]−1(122)

and

Yk,k′(ω) = [

hω − (Ek′

c−Ek

v) + iγSC]−1. (123)

We introduced the phenomenological line-widths γMand γSC for transitions in

the molecular layer and semiconductor surface. The missing ingredients are the

dipole-matrix-elements dLH and the Coulomb matrix elements VklvL

kl+GcH in eq.

121. The input geometry and energy (band structure) have to be obtained as well.

We will approximate those with the band structure of ZnO and the eigen-energies

11.1 linking to density-matrix theory 123

of L4P obtained from DFT calculations carried out with the hybrid-xc-functional

HSE06 [100]. The dipole-matrix-elements are taken from the same DFT calcula-

tion. For the Coulomb-matrix elements (Förster Transfer elements), facilitating

the interaction between L4P and ZnO appropriate approximations have to be

found. We will see in the next section, that this can be achieved by a parameteri-

zation derived from DFT calculations.

11.1.1Dipole Approximation

The parameterization of the density matrix formalism has previously been devel-

oped for purely crystalline semiconductor surfaces [41]. The optical excitations

and electron relaxation dynamics of Silicon surfaces were calculated by a DFT

parameterization based on the band structure, momentum-matrix elements and

phonon band structure obtained with a (semi)-local xc-functional. The first order

approximation for the Coulomb-matrix elements appearing in eq. 121 is equiva-

lent to the dipole-matrix elements between the semiconductor and the molecular

system. The Coulomb matrix element is given by

VLνkc

Hνk′v=∫d3r∫d3r′Ψ∗

Lν(r)Ψ∗

kv(r′)V(r−r′)ΨHν(r)Ψk′c(r′)(124)

with the Coulomb potential

V(r−r′) = 1

|r−r′|. (125)

ΨAνare the wave-functions of the νth molecule. Next we Taylor expand [59]

eq. 125. This leads to a factorization of the nested integrals in eq. 124. The in-

tegrals over the molecules and the semiconductor are evaluated independently.

By taking into account the two-dimensionality of our surface we obtain a closed

analytical form of the Förster transfer element:

VLνkv

Hνk+qc=1

2ϵ0

eiq·R||

νe−|q|∆z

|q|⎡

⎢

⎣⎛

⎜

⎝

−i|q|

⎞

⎟

⎠·dLH⎤

⎥

⎦×(126)

⎡

⎢

⎣⎛

⎜

⎝

−i|q|

⎞

⎟

⎠·dvc

k,k+q⎤

⎥

⎦

Here we have defined q=k−k′,R|| is the in-plane component of the center

of mass of the semiconductor/molecule unit cell and ∆z the distance between

surface and molecule.

The dipole-matrix elements entering into eq. 126 and subsequently into eq. 121

can be calculated as a post-processing step in the same DFT-calculation. The

option to calculate these matrix elements was implemented into FHI-aims as part

of this dissertation. The implementation consists of three steps. First the integral

over the unit cell is evaluated in real-space:

⟨ϕiM|r|ϕiN⟩=∫unit cell d3rϕi,M(r)rϕj,N(r)(127)

124 parameterization of density matrix formalism by dft

with the numerical atom base functions ϕi(r)centered at unit cells shifted by



T(

N),

N= (N1,N2,N3). Further information about the basis set used in FHI-

aims can be found in Sec. 1.7and Ref. [33]. From these translation vectors the

phase of the matrix elements is calculated. In the last step the matrix elements are

transformed into the Kohn-Sham basis by matrix multiplication with the Kohn-

Sham orbitals ck

iλ.

dλ′λ

k′,k=⟨ψλ′k|r|ψλk⟩=∑

c∗k

iλ′ck

jλ ∑

N,M

eik[T(N)−T(M)] ⟨ϕiM|r|ϕjN⟩(128)

11.1.2Partial charge approximation

The dipole approximation for the Coulomb matrix elements is quite drastic. To

improve on the description, a well known method from the force field community

is adapted to our problem. A similar approach was used for the description of

strongly coupled pigments in light-harvesting complexes [150].

Here we will present a simplified derivation of the Coulomb matrix element

from electro-static potential (ESP) partial charges. The starting point is again the

Coulomb matrix element

VAνkλ

Bνk′λ′=∫d3r∫d3r′Ψ∗

Aν(r)Ψ∗

kλ(r′)V(r−r′)ΨBν(r)Ψk′λ′(r′). (129)

One-particle transition densities are defined as:

ρBν

Aν(r) = Ψ∗

Aν(r)ΨBν(r). (130)

With the help of this definition we can rewrite the Coulomb matrix as:

VAνkλ

Bνk′λ′=∫d3r∫d3r′ρBν

Aν(r)ρk′λ′

kλ(r′)

|r−r′|. (131)

We can define the potential of the molecular part as

ϕBν

Aν(r)=−∫d3r′ρBν

Aν(r′)

|r−r′|≈∑

qBν

I,Aν

|r−RIν|(132)

and approximate it by atomic partial charges qBν

I,Aν, summed up over the atomic

positions RIνwithin the ν-th molecule: RIν=Rν+rI. The Coulomb matrix element

then reduces to

VAνkλ

Bνk′λ′≈−∑

qBν

I,Aν∫d3r′ρk′λ′

kλ(r′)

|RIν−r′|. (133)

We introduced the potential for the semiconductor ϕk′λ′

kλin a similar fashion

as for the molecular part of the system in eq. 132. The final expression for the

Coulomb matrix approximated by sums over the partial charges from the molec-

11.1 linking to density-matrix theory 125

ular part qBν

I,Aνat the atomic positions RIand the semiconductor part qk′λ′

J,kλat the

atomic positions RJis

VAνkλ

Bνk′λ′≈∑

qBν

I,Aν

Nunitcell

∑

i=1∫unitcell dri

ρk′λ′

kλ(ri)

|ri−RIν|

Nunitcell

∑

i=1

ei(k−k′)·Ri∥∑

I,J

qk′λ′

IkλqBν

I,Aν

|Rν+rJ−Ri−rI|. (134)

The challenge is to find reliable partial charges for molecule and semiconductor.

Such charges are widely used in the context of force fields ([168,56,239,27],

CHELP [48], CHELPG [39], RESP-charges [22], CHELP-BOW/CHELMO [238] or

ESP-charge from multi-pole-moments [104]). As part of this dissertation a sim-

ple method for cluster calculations (molecules) as well as two methods for solids

(periodic boundary conditions) [44,47] were implemented in the FHI-aims code.

The starting point for these methods is the calculation of the electrostatic po-

tential at a sufficiently high number of grid points outside the vdW radius of

the atoms. To define a spatial region for the grid, two parameters are neces-

sary: a minimal and a maximal radius around the atoms. These radii are defined

as multiples of the vdW-radius of the atoms, see Fig. 89 for details. The val-

ues for the vdW radii of most atoms in the periodic table have been taken from

Ref. [35,219,155]. For the generation of the points, the atom-centered radial grids

are used. All points lie on N spheres with radii between the minimal and maxi-

mal multiples of the vdW radius. The spacing between the radii of the N spheres

is equidistant (default) or logarithmic (optional). Alternatively, cubic (Cartesian)

grids are available. For cluster calculations, points within a cube encapsulating

the spheres with the maximal radius (multiple of the vdW radius) around all

atoms are generated. For periodic boundary conditions the provided unit cell is

used. The points within the superposition of the spheres with the minimal radius

(minimal multiple of the vdW radius) are again excluded.

For molecules and clusters (no periodic boundary conditions) the electrostatic

(a) Atom centered radial basis. (b) Cubic grid.

Figure 89: Definition of the volume used for the selection of grid points at which the

potential is evaluated.

126 parameterization of density matrix formalism by dft

potential can be expressed by a sum of Coulomb potentials with charges qi, the

ESP-charges, at the atomic position Ri:

VESP(r) =

Nat

∑

i=1

|r−Ri|(135)

The qiare calculated by a least squares fit with the additional constraint of con-

stant total charge qtot =∑Nat

i=1qi. We use the method of Lagrange multipliers to

minimize the function

Ngrid

∑

k=1

(VDFT (rk) − VESP(rk))2−λ(qtot −

Nat

∑

i=1

qi)2

. (136)

This can be translated into a system of linear equations

Aq =B. (137)

with the Nat+1×Nat+1matrix ˆ

Aij =

Ngrid

∑

k=1

|rk−Ri|

|rk−Rj|with i,j⩽Nat (138)

Ai=Nat+1,j=Ai,j=Nat+1=1;Ai=Nat+1,j=Nat+1=0

and the Nat +1vector B

Bi=

Ngrid

∑

k=1

VDFT (rk)

|rk−Ri|with i⩽Nat (139)

Bi=Nat+1=qtot. (140)

qare the Nat charges.

For solids (periodic boundary conditions) the situation is more complicated be-

cause all charges are repeated infinitely and the potential is only defined up-to an

arbitrary offset. The methods implemented as part of this work solve this prob-

lem by Ewald summation [77]. They were developed by Campana et.al. [44] and

later further improved by Chen et al. [47]. The electrostatic potential is split into

a real space and reciprocal space part by an error function (Ewald summation

[77]). The function for the potential generated by the ESP charges centered at the

atoms of the unit cell then reads

VESP(r) =

Nat

∑

i=1,T

erfc(α|r−Ri,T|)

|r−Ri,T|(141)

+4π

Vcell

Nat

∑

i=1,k

qicos(k(r−Ri))e−k2

4α2

with T=n1a1+n2a2+n3a3the real space (periodic) translation vector and ai

being the lattice vectors with ni∈Z.k=m1b1+m2b2+m3b3is the reciprocal

space translation vector and the biare the reciprocal lattice vectors with mi∈Z.

11.1 linking to density-matrix theory 127

Vcell is the volume of the unit cell. The parameter αis defined as α=√π

Rcwith

Rcthe cutoff radius of the Ewald summation [77]. Apart from Ewald summation,

the electrostatic potential can be calculated by Wolf summation [274,47] (also

implemented in the FHI-aims code as part of this work):

VESP(r) =

Nat

∑

i=1

qi[erfc(√α|r−Ri|)

|r−Ri|−erfc(√αRc)

(142)

+(erfc(√αRc)

+2√α

√π

exp(αR2

Rc)]×(|r−Ri|−Rc)

The function to minimize, derived by Campana et al. [44], then is:

FPBC

Ngrid

∑

k=1⎛

⎝VDFT (rk) − VESP(rk) −

Ngrid

∑

j=1(VDFT (rj) − VESP(rj))⎞

⎠

−λ(qtot −

Nat

∑

i=1

qi)+

Nat

∑

i=1

wi(E0

i+χiqi+1

2J00

iq2

i).

The χiis the electro-negativity and J00

ithe self-Coulomb interaction of the re-

spective elements, which can be used to constrain the desired charges. wiare

weighting factors for the constraints. The fit produces the elements of ˆ

Aand B

Aij =

Ngrid

∑

k=1⎛

⎝

∂VESP(rk)

∂qi

−1

Ngrid

∑

j=1

∂VESP(rj)

∂qi⎞

⎠×(143)

⎛

⎝

∂VESP(rk)

∂qj

−1

Ngrid

∑

m=1

∂VESP(rm)

∂qj⎞

⎠

+wi

2J00

iδij ;i,j⩽Nat

Ai=Nat+1,j=Ai,j=Nat+1=1;Ai=Nat+1,j=Nat+1=0

Bi=

Ngrid

∑

k=1⎛

⎝VDFT (rk) − 1

Ngrid

∑

j=1

VDFT (rj)⎞

⎠×

⎛

⎝

∂VESP(rk)

∂qi

−1

Ngrid

∑

m=1

∂VESP(rm)

∂qi⎞

⎠

−wm

χm

2;i⩽Nat

Bi=Nat+1=qtot

128 parameterization of density matrix formalism by dft

The function to minimize, derived by Chen et al. [47] is:

FPBC

Ngrid

∑

k=1(VDFT (rk) − (VESP(rk) + Voffset

DFT ))2(144)

−λ(qtot −

Nat

∑

i=1

qi)+β

Nat

∑

i=1

(qi−qi0)2.

The constraint charges qi0 can be determined with other methods (e. g. Mulliken

charge analysis [173]). βis the weighting factor. This gives ˆ

Aand B

Aij =

Ngrid

∑

k=1(∂VESP(rk)

∂qi

∂VESP(rk)

∂qj)+βδij ;i,j⩽Nat (145)

Ai=Nat+1,j=Ai,j=Nat+1=

Ngrid

∑

k=1

∂VESP(rk)

∂qj

j⩽Nat

Ai=Nat+2,j=Ai,j=Nat+2=1

Ai=Nat+2,j=Nat+2=Ai=Nat+1,j=Nat+2=0

Ai=Nat+2,j=Nat+1=0

Bi=

Ngrid

∑

k=1(VDFT (rk)∂VESP(rk)

∂qi)−βq0i ;i⩽Nat (146)

Bi=Nat+1=

Ngrid

∑

j=1

VDFT (rj)(147)

Bi=Nat+2=qtot

Here the arbitrary offset of the potential Voffset is an additional fitting parameter.

The matrix ˆ

Ais of dimension Nat+2×Nat+2and Bof dimension Nat+2. The

arbitrary offset of the electrostatic potential is calculated for the method derived

by Campana et al. [44] by

Voffset =1

Ngrid

∑

k=1

(VDFT (rk) − VESP(rk))(148)

from the fitted charges qi. As a measure for the quality of the fit the root-mean-

square (RRMS) is defined as

RRMS ={∑Ngrid

k=1((VESP(rk) + Voffset)−VDFT (rk))2

∑Ngrid

k=1(VDFT (rk))2}2

. (149)

The partial charges obtained by fitting to the electrostatic potential from a DFT

calculation give direct excess to the dipole-moment, calculated directly as de-

scribed in Sec. 11.1.1by summing over the charges at the atomic positions

d=∑

qiRi. (150)

11.2 results 129

(a) L4P. (b) L4P on ZnO (10¯

10)

Figure 90: Atomic structure of the ladder type quarterphenyl molecule L4P on

ZnO (10¯

10). The structure was determined by employing the semi-local xc-

functional PBE+vdwTS [196,254]

The goal of this work is to approximate the Coulomb matrix element (eq. 124).

To accomplish this task we need to calculate the partial charges for the electronic

states (orbitals) or rather products of states. The charges for those states can be

calculated analogous to the charges calculated for the full potential, once the

transition potential has been obtained. The full electrostatic potential is typically

calculated by solving the Poisson equation for the full density. In FHI-aims a very

efficient algorithm is used, that employs the multi-pole moments of the density

[63,64]. Instead of using the full density as input for this algorithm we imple-

mented the option to calculate a transition density. The ESP-charges can be fitted

to the electrostatic potential calculated on different grids, based on these transi-

tion densities.

11.2 results

As a model structure for demonstrating the combination of the density matrix

formalism and ab initio calculations, we selected a ladder type quarterphenyl

(L4P) molecule adsorbed on ZnO (10¯

10). This class of molecules provides an

electronic structure, which is well suited for opto-electronics and can be easily

processed and tailored to fit the electronic structure of quantum wells that are

used in devices and experiment [127]. In Fig. 90 a the atomic structure of L4P

relaxed with the hybrid functional HSE06 [100] is shown. In Fig. 90 b the ad-

sorption geometry of L4P on ZnO (10¯

10) calculated employing the semi-local

xc-functional PBE+vdW is shown. We only considered adsorption geometries of

flat lying molecules, these are best suited for Förster type exciton transfer [82].

The alignment between the flat lying molecule follows the description brought

forward by F. Della Sala et al. [62]. The interaction is governed by the electrostatic

coupling of the molecule to the periodic dipolar electric field originating from

the Zn-O surface dimers. The molecules either align along the field created by

the surface dimers or perpendicular to it. In Fig. 91 and Fig. 92 the situation

is depicted for the equilibrium adsorption geometry of L2P (flurorene) on ZnO

(10¯

10) obtained with the semi-local xc-functional PBE+vdWTS [196,254]. Along

the axis of the dimers a periodically oscillating field is formed by Zn-O rows

130 parameterization of density matrix formalism by dft

Figure 91: Alignment of L2P (Fluorene) perpendicular to the Zn-O dimers within the

long range potential of the ZnO (10¯

10) surface. The relaxed adsorption ge-

ometry is shown. The potential was taken from the unperturbed ZnO (10¯

10)

surface. The long range potential is calculated by Ewald summation in recip-

rocal space [77].

of opposite charge. Perpendicular to the dimers the change of the potential is

small. The molecules align in such a way, that the hydrogen atoms at the ends

of the long axis of the molecule are in the vicinity of the positive potential. The

largest overlap between the positive potential region and the hydrogen atoms of

the molecule is achieved by alignment of the long molecular axis perpendicular

to the dimers. The alignment along the axis of the dimers is achieved by bridging

one or more positive/negative potential regions while the ends of the molecule

are in the vicinity of positive potential regions.

The adsorption geometry for L4P on ZnO (10¯

10) is shown in Fig. 90 b. The

atomic structure of L4P and ZnO (10¯

10) are the first inputs for parameterizing

Figure 92: Alignment of L2P (Fluorene) parallel to the Zn-O dimers within the long

range potential of the ZnO (10¯

10) surface. See caption of Fig. 91.

11.2 results 131

Atom ESP charge in e0

Zn 1.02344470

O -1.02344470

Zn 1.02344470

O -1.02344470

RRMS 0.34405930

Table 12: ESP charges fitted to the electrostatic potential of bulk wurtzite ZnO

the model based on a density matrix formalism developed by E. Verdenhalven,

A. Knorr and M. Richter [265]. The parametrization of the electronic structure ob-

tained by ab initio calculations for the isolated molecule and ZnO (10¯

10) surface

is achieved within the dipole approximation (see Sec. 11.1.1and Ref. [265]). The

transition dipole moment matrix elements < φi|r|φj>are calculated on the real

space grid of the FHI-aims code and transformed to the KS-basis (see sec. 11.1.1).

The transition dipole matrix element for the HOMO-LUMO transition of the

molecule and the CBM-VBM transition of the ZnO (10¯

10) surface are:

|dHL|=0.35e0nm |dCV |=0.012e0nm (151)

The direction of the substrate dipole dHL is chosen and fixed to point out of the

surface plane, while the direction of the molecule transition dipole is tuned via

the angle αbetween both transition dipole moments.

Putting together the theoretical framework derived by by E. Verdenhalven, A.

Knorr and M. Richter [265] and the DFT parametrization we investigated the

interaction between molecule and substrate. At low coverages of molecules the

interaction is stronger than at high coverages because the oscillator strength of

the semiconductor substrate is distributed between fewer molecules. From vary-

ing the orientation of the molecule on the surface and thus the angle αbetween

(a) Atomic structure of

wurtzite ZnO.

(b) Electrostatic potential ob-

tained from a DFT calcu-

lation employing the hy-

brid xc-functional HSE06

[100].

constructed from partial

charges fitted to the poten-

tial of b).

Figure 93: Electrostatic potential of ZnO in wurtzite structure. The unit cell is shown in

black.

132 parameterization of density matrix formalism by dft

0 1 2 3 4 5

zin [ ˚

−0.12

−0.10

−0.08

−0.06

−0.04

Φ[a.u.]

DFT potential ESP potential

atomic positions

Figure 94: Electrostatic potential of the ZnO (10¯

10) surface from a DFT calculation

(HSE06) (green) and reconstructed from ESP-charges (blue) in Tab. 12 summed

up perpendicular to the c-axis of the crystal. The atomic positions (z) of Zn

and O are marked by red dots.

surface and molecular transition dipole we found that the interaction is strongest

for a parallel alignment of the dipoles. The decrease follows a cosine behavior

and is lowest for perpendicular moments. With increasing interaction strength

the HOMO-LUMO transition line is shifted toward lower energies. By optimiz-

ing the coverage and the alignment of surface and molecular dipole moments

the performance of hybrid inorganic/organic devices can be tuned or adjusted

to possible applications in photo voltaics and opto-electronics.

The dipole and momentum transition matrix elements implemented into the

(a) Atomic structure of

the ZnO (10¯

10) sur-

face.

(b) Electrostatic poten-

tial obtained from a

DFT calculation em-

ploying the hybrid

xc-functional HSE06

[100].

tial reconstructed

from partial charges

fitted to the potential

of b).

Figure 95: Electrostatic potential of the ZnO (10¯

10) surface. The unit cell is periodically

extended perpendicular to the surface.

11.2 results 133

Atom ESP charge in e0

O (top) -1.24697845

Zn (top) 1.09027791

Zn 1.29658012

O -1.07867024

O -1.24578893

Zn 1.15653704

O -1.15471412

Zn 1.19946914

O -1.14578008

Zn 1.14171604

O -1.17959763

Zn 1.13330173

O -0.97913785

Zn 1.20932943

O (bottom) -1.20174231

Zn (bottom) 1.00519817

Table 13: ESP charges fitted to the electrostatic potential of the ZnO (10¯

10) surface.

FHI-aims DFT code as part of this work have many further applications. They are

utilized in different theoretical approaches based on first order perturbation the-

ory (Fermi’s Golden rule). They can be used to calculate the macroscopic dielec-

tric function and absorption spectra [6], which were used to test the implemen-

tation. Other applications are the calculation of thermo-electric constants such as

the Seebeck coefficient [151,6], the simulation of NEXAFS data [69] and the pa-

rameterization of Förster interaction/energy transfer for functionalized graphene

[152,153]. Additionally we added the versatile HDF5container file format [251]

to the FHI-aims code in this work. It allows for a flexible input/output structure

to distribute data between different working groups. All data is collected within

one container file, that hosts different data structures and allows for a straight

forward annotation of all contents. A key advantage for the application in highly

parallelized scientific codes such as FHI-aims is the capability of synchronous

output from many computing processes. The data does not have to be collected

on one processor, which is typically limited by the available memory, but can be

written directly to the container file.

The modeling was so far based on a parametrization, that essentially reduces

the information contained in the electronic structure obtained by the DFT cal-

culation to two transition dipole matrix elements. This approximation is quite

severe considering the spatial extent of the molecule and the surface that clearly

will contribute differently to the interaction between the constituents. The next

step for a better modeling of the materials at hand is the partial charge technique

134 parameterization of density matrix formalism by dft

0 5 10 15 20 25

zin [ ˚

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0.00

Φ[a.u.]

DFT potential ESP potential

atomic positions

Figure 96: Electrostatic potential from a DFT calculation (HSE06) (green) and recon-

structed from ESP-charges of the ZnO (10¯

10) surface, summed up perpen-

dicular to the surface normal. Atomic positions (z) are marked by red dots.

[150], described in Sec 11.1.2.

In Fig. 93 b the electrostatic potential for ZnO in wurtzite structure is shown.

To the four atoms in the unit cell (black in Fig. 93, two Zn and two O) partial

charges are assigned by fitting with the method of Chen et al. [47] to the po-

tential in Fig. 93 b. The partial charges are listed in Tab. 12. ZnO is a strongly

ionic crystal. The partial charges slightly deviate from the values of +1and -1

for Zn and O, respectively, predicted by the simple ionic model of Tasker [249].

From the partial charges the electrostatic potential is reconstructed and shown in

Fig. 93 b as an iso-surface. In Fig. 94 the electrostatic potential obtained directly

from the DFT calculation and the reconstruction are summed up along the c-axis

of the crystal. The agreement between the DFT potential and the reconstructed

potential is good. The quality of the fit crucially depends on the real space region

selected for fitting. If the atomic structure is dense, less space is available for

fitting points. For bulk wurtzite ZnO the partial charge approach yields a good

approximation for the electrostatic potential.

As a next step the electrostatic potential for a ZnO (10¯

10) is approximated by

partial charges in Fig. 95. The electrostatic potential directly obtained from the

DFT calculation shows two different regions because of the broken symmetry at

the surface. Inside the slab, used for modeling the surface, the potential nearly

behaves as for bulk wurtzite ZnO in Fig. 93. At the surface a strong potential

is formed perpendicular to the Zn-O dimers of the surface. The fitted partial

charges are listed in Tab. 13 and the reconstructed potential is shown in Fig. 95

c. The partial charges inside the slab are close to the partial charges of bulk

wurtzite ZnO. In Fig. 96 we show the electrostatic potential obtained from the

DFT calculation and the reconstruction summed up along the surface normal.

At the surface the deviation of the reconstructed potential from the potential ob-

tained from the DFT calculation is the strongest. The few partial charges placed

at the surface atoms are not capable of fully describing the behavior of the poten-

11.2 results 135

tial, but the overall agreement is good and might be sufficient for approximating

the Coulomb matrix elements required as input for the density matrix formalism.

The accuracy of the fit could be improved in the future by introducing additional

charges at the surface.

Part IV

CONCLUSION

139

We have performed a first principles investigation of ultra-thin ZnO films

on metal supports. These films are of interest as model systems in e. g. opto-

electronics and heterogeneous catalysis. Despite the multitude of applications

the actual structure of ZnO surfaces and interfaces, which are essential for de-

vices and chemical reactions, is still under debate [132,161,276,262,136,267].

Starting from bulk ZnO in Chap. 3, we employed the tool set of surface science

to develop well defined model systems for the notoriously difficult to character-

ize polar ZnO surfaces, ZnO (0001)-Zn and (000¯

1)-O. By going from the ZnO

surface to ultra-thin films on conducting substrates experimental techniques that

rely on the conductivity of the sample can be applied. The rapid progress in

thin film growth makes it possible to prepare these films with very high preci-

sion regarding orientation and termination. The most evident question, which

arises by switching from surfaces to ultra thin films, is how well these films

resemble the surface they are supposed to model. Ultra-thin films are known

to sometimes have unique properties of their own, that yield new applications

[88,189,234,181].

In Chap. 5we found, that the hypothetical ideal ZnO mono-layer has a very

stable α-BN structure, which can be derived from the ZnO bulk wurtzite struc-

ture by bringing the Zn and O atoms into the same plains. By minimizing the

strain in the ZnO mono-layer we deduced coincidence structures between the

(111) surfaces of Ag, Cu, Pd, Pt, Ni, and Rh, which are in good agreement with

experiment (Chap. 6). The mono-layer retains its α-BN structure on the metal

substrates, however, the corrugation is highly system dependent. After assessing

the stability and atomic structure of the coincidences we progressed to thicker

films in Sec. 7.1. With increasing thickness the corrugation of the films grows

substantially and eventually the wurtzite structure is formed. The number of lay-

ers required for the formation of a full wurtzite structure depends on the metal

substrates. For Cu 6layers are necessary, while for Ni already 4layers are suffi-

cient.

In addition to the atomic structure, the electronic structure gives insight into the

mechanisms governing the transition from mono- to multi-layer systems. Due to

the intrinsic polar nature of the ZnO films a dipolar electric field forms across the

film (Sec. 7.2). This field shifts the electronic structure of each successive layer up

in energy. This upwards shift is only stopped when the Fermi level, which is pro-

vided by the metal substrate, is reached. This mechanism results in an effectively

p-type doped sample. Our model presents a fundamentally different approach

than doping with shallow acceptors and might even lead to new applications

within the rapidly developing field of epitaxial film growth.

Next, we investigated the stability of ultra-thin metal supported ZnO films in

Sec. 8.1to Sec. 8.4by developing an extensive surface phase diagram for 1to

4layers of ZnO on the different metal substrates. We found that the stability

of the well known (2×1)-H reconstruction is increased with growing film thick-

ness, but the affinity for H-adsorption is highly system dependent. While ZnO

on Cu is very susceptible to hydrogen adsorption, ZnO on Ag is very stable in its

clean, unreconstructed form. By analyzing the atomic and electronic structure of

(2×1)–H reconstructed ZnO on metal substrates we found that it resembles the

wurtzite structure of bulk ZnO very well. Both the atomic and electronic struc-

140

ture of four layer thick ZnO films are comparable to (2×1)–H reconstructed ZnO

surfaces without metal substrates. In conclusion, the (2×1)–H reconstructed films

are good models for the polar ZnO surfaces.

We also investigated Zn and O defects in Sec. 8.2and 8.2.1and confirmed the

validity of results for the ZnO surfaces [132,161,276,262,136,267] for our ultra

thin films. Oxygen defects (vacancies and ad-atoms) are more stable than their

Zn counterparts. The stability of the O defects depends on the oxygen affinity

of the metal substrate. For Cu it is strong, while it is weaker for Ag. The other

metals form an intermediate situation between those extremes. At low chemical

potentials sparse ring structures (Sec. 8.3) with or without hydrogen are stabi-

lized, which allow the Zn/O atoms to retain a three-fold bonding environment.

The ring structures are more stable than normal defects and dominate the phase

diagram at low oxygen chemical potentials.

Another reconstruction, known from the (0001)-Zn terminated ZnO surface, we

found to be dominant in the surface phase diagrams at high oxygen chemical

potentials is OH adsorption. It is the equivalent reconstruction to H adsorption

on the (000¯

1)-O terminated ZnO surface. In contrast to H adsorption we did not

find a strong dependence of the stability of OH reconstructions on the film thick-

ness or the metal substrates.

The explicit inclusion of the chemical potentials of hydrogen, oxygen, zinc and

gaseous water lead us to complete surface phase diagrams [98]. Reconstructions

containing hydrogen are destabilized with respect to the unreconstructed surface,

which are predicted to be stable up-to 3layers of thickness.

We compared our ab initio results to experimental measurements of ZnO on Ag

by Shiotari et. al. [237]. The combination of experiment and theory allows for

a conclusive determination of film thickness and surface termination. Addition-

ally, we were able to explain the appearance of the (5×5) minority coincidence

structure reported by Shiotari et al. in Chap. 9. The reduction of the Zn concen-

tration during annealing is predicted to lead to a transformation of the (5×5)

coincidence, which is only stable at high Zn chemical potentials, to the regular

coincidence structure. The complete transformation could be hindered by kinetic

effects such as reduced diffusion length because of low temperatures during the

measurement process.

In the final chapter 11 of this work we made a first effort to bridge the gap be-

tween our ab initio calculations of model structures and more realistic hybrid

inorganic organic systems for opto-electronics. In close collaboration with Ver-

denhalven, Richter and Knorr from the TU Berlin we parameterized their "hy-

brid Bloch equations". This model based on a density matrix formalism allowed

us to investigate excitonic effects in hybrid systems of organic molecules and in-

organic semi-conductors. We chose the ladder-type oligomer L4P and the ZnO

(10¯

10) surface to test our formalism. As part of this work we implemented the

calculation of dipole matrix elements into the FHI-aims [33] code. The further de-

velopment of the model let to a more detailed parametrization based on partial

charges, which are fitted to the electrostatic potential obtained from DFT calcula-

tions. We implemented the calculation of the full and the transition potential into

FHI-aims, as well as the fitting procedure for periodic and non-periodic systems.

The transition potentials are calculated from the density matrix, constructed from

141

two different eigenstates.

We have established that under certain conditions ultra-thin ZnO on metal sub-

strates can be used to model the polar ZnO surfaces. The H2partial pressure and

the choice of the metal allow to select between unreconstructed ultra-thin ZnO

films, that differ from the polar ZnO surfaces, and films ((2×1)–H reconstructed)

that resemble these surfaces. The unreconstructed ultra-thin ZnO films thereby

have a unique atomic and electronic structure, that may lead to interesting new

applications in e. g. opto-electronics. The extensive results of this work regard-

ing atomic and electronic structure and stability of ZnO ultra thin films will be

a valuable guideline for experimentalists in their efforts to prepare well defined

samples.

As an outlook, the further refinement of our models developed together with

the TU Berlin is expected to result in intriguing insights in the mechanisms gov-

erning excitonic effects in HIOS systems. This will eventually lead to a better

understanding of charge separation in realistic devices. The next step in the in-

vestigation of ZnO ultra-thin films on metal substrates will be the inclusion of

organic molecules into the calculations. In close collaboration with experimen-

talists the effect of the unique electronic structure of ZnO on metal substrates

and the Fermi level provided by the metal on the molecules will be investigated

(effective p-type doping). This has the potential to be the basis for further work

that will eventually lead to better devices and catalysts. To further improve the

quality of our theoretical results, image charge effects could be included in the

description of the ultra-thin ZnO on metal substrates.

Part V

APPENDIX

CONSTANTS

In this section the physical constants used throughout this dissertation are listed

[166].

Quantity Symbol Value

Atomic mass unit amu ≡u 1.660 538 921 (73)·10−27 kg

Avogadro constant NA6.022 141 29 (27)·1023 1

mol

Bohr magneton µB=e

2 me9.274 009 68 (20)·10−24 J

Bohr radius a0=4πε0

e2me5.291 772 109 2 (17)·10−11 m

Boltzmann constant kB1.380 648 8 (13)·10−23 J

Electric constant ϵ0(µ0c2)−1

4πε0299 792 4582·10−7Vm

Electron charge e 1.602 176 565 (35)·10−19 C

Electron mass me9.109 382 91 (40)·10−31 kg

Fine struc. constant α=µ0e2c0

2 h 7.297 352 569 8 (24)·10−3

Hartree energy Eh4.359 744 34 (19)·10−18 J

Magnetic constant µ04×10−7N A−2

Planck constant h 6.626 069 57 (29)·10−34 Js



h=h

2 π 1.054 571 726 (47)·10−34 Js

Rydberg energy R∞c h 13.605 692 53 (30)eV

Rydberg constant R∞c 3.289841960364(17)·1015 Hz

Speed of light c,c0299 792 458 m s−1

Universal gas const. R0=NAkB8.314 462 1 (75)J

K·mol

Table 14: Physical constants in SI units [166].

145

REFERENCE ENERGIES

In this section the total energies calculated with FHI-aims [33] are listed for atoms

and bulk structures used to calculate cohesive and surface free energies.

System Basis[33] xc-functional Total energy [eV]

Zn tight PBE -49117.0526

Zn tight HSE06 -49117.1589

O tight PBE -2043.2223

H tight PBE -12.4875

Ag tight PBE -146385.2978

Cu tight PBE -45242.3900

Cu tight HSE06 -45241.8872

Pd tight PBE -138878.7157

Pt tight PBE -518285.9446

Au tight PBE -535650.8830

Rh tight PBE -131611.9493

C tight PBE -1027.8640

O2tight PBE -4093.4322

H2tight PBE -31.7461

H2tight PBE+vdW -31.7480

H2O tight PBE -2080.9643

Table 15: Reference energy for different xc-functionals. For the bulk values equilibrium

lattice constants obtained by a fit to the Birch-Murnaghan equation of states

[174] were used.

147

148 reference energies

System Basis[33] xc-functional Total energy [eV]

Ag bulk (fcc) (exp) tight PBE -146388.0076

Ag bulk (fcc) tight PBE -146388.0192

Cu bulk (fcc) tight PBE -45246.1515

Pd bulk (fcc) tight PBE -138882.4397

Pt bulk (fcc) tight PBE -518291.9131

Au bulk (fcc) tight PBE -535654.1015

Rh bulk (fcc) tight PBE -131618.3056

Ni bulk (fcc) tight PBE -41565.3733

Zn bulk (bcc ) tight PBE -98235.8869

ZnO bulk wrz tight PBE -102335.4173

ZnO bulk wrz tight PBE+vdW -102335.8198

ZnO bulk zb tight PBE -51167.6922

ZnO bulk rs tight PBE -204669.5881

Table 16: Reference energy for different xc-functionals. For the bulk values equilibrium

lattice constants obtained by a fit to the Birch-Murnaghan equation of states

[174] were used.

CONVERGENCE TESTS

(a) Total energy vs. tiers

(b) Total energy vs. k-points

Figure 97: Convergence tests for bulk wurtzite ZnO

In this section the convergence tests for determining the settings of the basis

set and the k-point sampling for bulk ZnO are presented. Based on these settings

the parameters for larger super cells and surface calculations are selected. The

149

150 convergence tests

(a) Total energy vs. tiers

(b) Total energy vs. k-points

Figure 98: Convergence tests for bulk zinc-blend ZnO

number of k-points in each reciprocal space direction is reduced by the same fac-

tor the real space lattice vectors have increased. The basis sets are the numerically

tabulated atomic orbitals provided by the FHI-aims package [33]. The default set-

tings allow a successive increase of the basis set size by hydrogen-like spherical

orbitals. The grid size can be adjusted accordingly, but typically is selected in

steps, that also increase the basis set size (light,tight,really tight). In Fig. 97 the

change of the total energy of bulk wurtzite ZnO as a function of basis set size

and k-point sampling is shown. For the ZnO bulk wurtzite structure 18 ×18 ×18

k-points with off-Γk-point grid and the standard light/tight basis are sufficiently

convergence tests 151

(a) Lattice parameter vs. tiers

(b) Lattice parameter vs. k-points

Figure 99: Additional convergence tests for bulk zinc-blend ZnO

converged. All surface slab calculations are scaled to these values. The lattice pa-

rameters have been taken from a fit to the Murnaghan equation of states [174]

and the PBE xc-functional was used.

In Fig. 98 the change of the total energy of bulk zinc-blend ZnO as a function of

basis set size and k-point sampling is shown. For the ZnO bulk zinc-blend struc-

ture 12 ×12 ×12 k-points with off-Γk-point grid and the standard light/tight

basis are sufficiently converged. All surface slab calculations are scaled to these

values. The lattice parameters have been taken from a fit to the Murnaghan equa-

tion of states [174] and the PBE xc-functional was used. Additional convergence

152 convergence tests

test are shown in Fig. 99.

(a) Band gap vs. tiers

(b) Valence/Conduction band vs. tiers

Figure 100: Additional convergence tests for bulk zinc-blend ZnO

COINCIDENCE STRUCTURES

The most likely coincidence structure can be estimated by minimizing the strain

on the in-plane lattice parameter aof the ZnO mono layer.

min

m(aZnO −m

m−1

√2)(152)

The lattice parameters of ZnO as well as for the metal depend on the xc-functional.

Small difference in their ratio can lead to different coincidence structures if large

structures (m>6) are favored. In the tables below the lattice parameters obtained

by a Birch-Murnigan-fit [174] for different metals and the ZnO mono-layer are

listed. The coincidence structure leading to an in-plane lattice parameter of the

ZnO, which is closest to the free-standing case is given for various xc-functionals

(PBE [196], PW-LDA [198],PZ-LDA [195], VWN [266], AM05 [9], BLYP [23,138],

PBEint [78], PBEsol [200], rPBE [94] and revPBE [288]. The 2nd and 3rd likeliest

coincidence structures are reported, too.

xc / Ag aa/√(2)damin coincidences

pbe 4.15 2.934 0.017 10 /9 9 /8 11 /10

pw-lda 4.003 2.831 0.014 9 /8 8 /7 10 /9

pz-lda 4.004 2.831 0.015 9 /8 8 /7 10 /9

vwn 4.005 2.832 0.006 9 /8 8 /7 10 /9

am05 4.055 2.867 0.015 9 /8 8 /7 10 /9

blyp 4.262 3.014 0.008 11 /10 12 /11 10 /9

pbeint 4.077 2.883 0.009 9 /8 8 /7 10 /9

pbesol 4.054 2.867 0.01 9 /8 8 /7 10 /9

rpbe 4.215 2.98 0.002 10 /9 11 /10 9 /8

revpbe 4.196 2.967 0.005 10 /9 9 /8 11 /10

Table 17: Lattice parameter of Ag,1st, 2nd and 3rd likeliest coincidence structure leading

to an in-plane lattice parameter of the ZnO, which is closest to the free-standing

case for various xc-functionals.

153

154 coincidence structures

xc / Pt aa/√(2)damin coincidences

pbe 3.97 2.807 0.002 7 /6 8 /7 6 /5

pw-lda 3.894 2.754 0.014 7 /6 8 /7 9 /8

pz-lda 3.895 2.754 0.014 7 /6 8 /7 9 /8

vwn 3.889 2.75 0.016 7 /6 8 /7 9 /8

am05 3.908 2.764 0.016 7 /6 6 /5 8 /7

blyp 4.054 2.867 0.031 8 /7 7 /6 9 /8

pbeint 3.924 2.775 0.015 7 /6 6 /5 8 /7

pbesol 3.918 2.77 0.004 7 /6 8 /7 6 /5

rpbe 3.993 2.823 0.015 7 /6 6 /5 8 /7

revpbe 3.986 2.819 0.012 7 /6 8 /7 6 /5

Table 18: Lattice parameter of Pt,1st, 2nd and 3rd likeliest coincidence structure leading

to an in-plane lattice parameter of the ZnO, which is closest to the free-standing

case for various xc-functionals.

xc / Pd aa/√(2)damin coincidences

pbe 3.942 2.787 0.025 7 /6 6 /5 8 /7

pw-lda 3.839 2.715 0.031 7 /6 6 /5 8 /7

pz-lda 3.84 2.715 0.031 7 /6 6 /5 8 /7

vwn 3.844 2.718 0.021 7 /6 6 /5 8 /7

am05 3.869 2.736 0.042 6 /5 7 /6 8 /7

blyp 4.03 2.85 0.017 7 /6 8 /7 9 /8

pbeint 3.887 2.749 0.045 7 /6 6 /5 8 /7

pbesol 3.874 2.74 0.039 7 /6 6 /5 8 /7

rpbe 3.98 2.814 0.026 7 /6 6 /5 8 /7

revpbe 3.969 2.807 0.027 7 /6 6 /5 8 /7

Table 19: Lattice parameter of Pd,1st, 2nd and 3rd likeliest coincidence structure leading

to an in-plane lattice parameter of the ZnO, which is closest to the free-standing

case for various xc-functionals.

coincidence structures 155

xc / Au aa/√(2)damin coincidences

pbe 4.158 2.94 0.01 10 /9 9 /8 11 /10

pw-lda 4.049 2.863 0.017 10 /9 9 /8 11 /10

pz-lda 4.05 2.864 0.017 10 /9 9 /8 11 /10

vwn 4.043 2.859 0.016 10 /9 9 /8 11 /10

am05 4.077 2.883 0.003 9 /8 10 /9 8 /7

blyp 4.271 3.02 0.013 12 /11 11 /10 13 /12

pbeint 4.095 2.895 0.005 9 /8 10 /9 8 /7

pbesol 4.083 2.887 0.012 9 /8 10 /9 11 /10

rpbe 4.199 2.969 0.011 10 /9 9 /8 11 /10

revpbe 4.187 2.961 0.011 10 /9 9 /8 11 /10

Table 20: Lattice parameter of Au,1st, 2nd and 3rd likeliest coincidence structure leading

to an in-plane lattice parameter of the ZnO, which is closest to the free-standing

case for various xc-functionals.

xc / Cu aa/√(2)damin coincidences

pbe 3.632 2.568 0.067 5 /4 4 /3 6 /5

pw-lda 3.518 2.488 0.089 5 /4 4 /3 6 /5

pz-lda 3.519 2.488 0.089 5 /4 4 /3 6 /5

vwn 3.521 2.489 0.08 5 /4 4 /3 6 /5

am05 3.56 2.517 0.093 5 /4 4 /3 6 /5

blyp 3.708 2.622 0.03 5 /4 6 /5 4 /3

pbeint 3.582 2.533 0.086 5 /4 4 /3 6 /5

pbesol 3.564 2.52 0.085 5 /4 4 /3 6 /5

rpbe 3.676 2.599 0.061 5 /4 4 /3 6 /5

revpbe 3.663 2.59 0.063 5 /4 4 /3 6 /5

Table 21: Lattice parameter of Cu,1st, 2nd and 3rd likeliest coincidence structure leading

to an in-plane lattice parameter of the ZnO, which is closest to the free-standing

case for various xc-functionals.

156 coincidence structures

xc / Ni aa/√(2)damin coincidences

pbe 3.51 2.482 0.033 4 /3 5 /4 6 /5

pw-lda 3.416 2.416 0.022 4 /3 5 /4 6 /5

pz-lda 3.417 2.416 0.022 4 /3 5 /4 6 /5

vwn 3.416 2.416 0.029 4 /3 5 /4 6 /5

am05 3.45 2.439 0.012 4 /3 5 /4 6 /5

blyp 3.57 2.524 0.058 4 /3 5 /4 6 /5

pbeint 3.469 2.453 0.019 4 /3 5 /4 6 /5

pbesol 3.455 2.443 0.022 4 /3 5 /4 6 /5

rpbe 3.544 2.506 0.032 4 /3 5 /4 6 /5

revpbe 3.535 2.499 0.032 4 /3 5 /4 6 /5

Table 22: Lattice parameter of Ni,1st, 2nd and 3rd likeliest coincidence structure leading

to an in-plane lattice parameter of the ZnO, which is closest to the free-standing

case for various xc-functionals.

CONVERGENCE WITH NUMBER OF METAL LAYERS

For the found coincidence the convergence of the surface energy is tested with

respect to the number of metal layers. The following form of the surface energy

is used for the tests:

γ=1

A(Eslab(nZn,nO) − nmetal

2Ebulk

metal −nO

2EO2(153)

−nZn

2Ebulk

Zn )·A0,

with nZn,nO,nmetal the number of atoms of the species in the slab and A0

the Area of a 1x1unit-cell of metal surface. As reference energies the values

presented in Appendix Bare used. The evolution of the surface free energy γ

of a mono-layer of ZnO with the number of metal substrate layers is shown in

Fig. 101 for Ag, and in Fig. 102 for Cu. The defect free structure is plotted in red,

the behavior with one additional oxygen atom is shown in red. By taking into

account the scale of both plots it can be summarized, that four layers of metal

substrate are sufficient. The change of the surface free energy for the substrate

thicknesses shown in Fig. 101 and Fig. 102 does not exceed 5meV.

Figure 101: Convergence of the surface energy with the thickness of the metal substrate

for one layer of ZnO on Ag. The ideal mono layer is plotted in red, with an

additional oxygen atom in blue.

157

158 convergence with number of metal layers

Figure 102: Convergence of the surface energy with the thickness of the metal substrate

for one layer of ZnO on Cu. The ideal mono layer is plotted in red, with an

additional oxygen atom in blue.

CORRUGATION MAPS

In this section the corrugations maps of mono-layer ZnO on the (111) surface of

Ni, Rh, Pd, and Pt are reproduced. The corrugation maps for Cu and Ag can be

found in chapter 6.3together with the discussion of these results.

(a) Atomic structure of coincidence struc-

ture super cell.

Pd A layer

Pd B layer

Pd C layer −0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

∆zfrom ¯

zin [ ˚

1ZnO/Pd, 1. Layer

(b) Corrugation map (unit cell tripled).

Figure 103: Atomic structure and corrugation of ZnO on Pd. Each Zn and O atom is

assigned a color according to their lateral position.

(a) Atomic structure of coincidence struc-

ture super cell.

Ni A layer

Ni B layer

Ni C layer −0.24

−0.16

−0.08

0.00

0.08

0.16

0.24

0.32

∆zfrom ¯

zin [ ˚

1ZnO/Ni, 1. Layer

(b) Corrugation map (unit cell tripled).

Figure 104: Atomic structure and corrugation of ZnO on Ni. Each Zn and O atom is

assigned a color according to their lateral position.

159

160 corrugation maps

(a) Atomic structure of coincidence struc-

ture super cell.

Rh A layer

Rh B layer

Rh C layer −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆zfrom ¯

zin [ ˚

1ZnO/Rh, 1. Layer

(b) Corrugation map (unit cell tripled).

Figure 105: Atomic structure and corrugation of ZnO on Rh. Each Zn and O atom is

assigned a color according to their lateral position.

(a) Atomic structure of coincidence struc-

ture super cell.

Pt A layer

Pt B layer

Pt C layer −0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆zfrom ¯

zin [ ˚

1ZnO/Pt, 1. Layer

(b) Corrugation map (unit cell tripled).

Figure 106: Atomic structure and corrugation of ZnO on Pt. Each Zn and O atom is

assigned a color according to their lateral position.

STRUCTURE OF MULTIPLE ZNO LAYERS

The atomic structure of 1to 4layers of ZnO on Pd, Pt, Rh and Ni is presented

in this section. With growing number of ZnO layers on the metal substrates the

corrugation of the ZnO ultra-thin films increases. The films gradually progress

from an α-BN to the wurtzite structure. They exhibit patches of α-BN and si-

multaneously patches of wurtzite structure. The disorder can be understood by

a competition between different phases that are stable in a very narrow energy

range as previously demonstrated for unsupported ZnO films [65]. The strain in

the films plays a key role for the occurrence of the observed structures. Only for

the 3layers of ZnO on Ni the structure of the ZnO resembles the phases proposed

by Demiroglu et al. [65].

(a) (b) (c) (d)

Figure 107: The relaxed structures for the hydrogen free ZnO on Pd. a) Mono-layer, b)

Bilayer, c) 3Layers, d) 4Layers of ZnO.

(a) (b) (c) (d)

Figure 108: The relaxed structures for the hydrogen free ZnO on Pt. a) Mono-layer, b)

Bilayer, c) 3Layers, d) 4Layers of ZnO.

(a) (b) (c) (d)

Figure 109: The relaxed structures for the hydrogen free ZnO on Rh. a) Mono-layer, b)

Bilayer, c) 3Layers, d) 4Layers of ZnO.

161

LAYER RESOLVED CORRUGATION MAPS

In this section the evolution of the surface corrugation of up to 6layers of ZnO

on the different transition metals is shown. The results reflect the fact, that with

increasing thickness the behavior of the ZnO becomes more bulk like. The struc-

ture transforms from the flat α-BN structure to the wurtzite structure with an

intrinsic lateral distance between Zn and O atoms of 0.63 Å. This ideal value is

not reached due to the effect known from bulk ZnO surfaces, that the top surface

layers contract inwards [162].

Ag A layer

Ag B layer

Ag C layer −0.16

−0.12

−0.08

−0.04

0.00

0.04

0.08

0.12

0.16

∆zfrom ¯

zin [ ˚

1ZnO/Ag, 1. Layer

(a) 1Layer.

Ag A layer

Ag B layer

Ag C layer −0.16

−0.08

0.00

0.08

0.16

0.24

0.32

0.40

0.48

∆zfrom ¯

zin [ ˚

2 Layers of ZnO (9x9) on Ag (8x8), 2. Layer

(b) 2Layers, top layer.

Ag A layer

Ag B layer

Ag C layer −0.24

−0.16

−0.08

0.00

0.08

0.16

0.24

0.32

∆zfrom ¯

zin [ ˚

3 Layers of ZnO (9x9) on Ag (8x8), 3. Layer

Ag A layer

Ag B layer

Ag C layer

−0.16

−0.08

0.00

0.08

0.16

0.24

0.32

0.40

∆zfrom ¯

zin [ ˚

4 Layers of ZnO (9x9) on Ag (8x8), 4. Layer

(d) 4Layers, top layer.

Ag A layer

Ag B layer

Ag C layer −0.12

−0.06

0.00

0.06

0.12

0.18

0.24

0.30

∆zfrom ¯

zin [ ˚

5 Layers of ZnO (9x9) on Ag (8x8), 5. Layer

(e) 5Layers, top layer.

Figure 110: Layer resolved corrugation maps (unit cell tripled) for 1to 5layers of ZnO

on Ag.

163

164 layer resolved corrugation maps

Cu A layer

Cu B layer

Cu C layer −0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

∆zfrom ¯

zin [ ˚

1ZnO/Cu, 1. Layer

(a) 1Layer.

Cu A layer

Cu B layer

Cu C layer −0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

∆zfrom ¯

zin [ ˚

2 Layers of ZnO (5x5) on Cu (4x4), 2. Layer

(b) 2Layers, top layer.

Cu A layer

Cu B layer

Cu C layer −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

∆zfrom ¯

zin [ ˚

3 Layers of ZnO (5x5) on Cu (4x4), 3. Layer

Cu A layer

Cu B layer

Cu C layer −0.16

−0.12

−0.08

−0.04

0.00

0.04

0.08

0.12

0.16

∆zfrom ¯

zin [ ˚

4 Layers of ZnO (5x5) on Cu (4x4), 4. Layer

(d) 4Layers, top layer.

Cu A layer

Cu B layer

Cu C layer −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆zfrom ¯

zin [ ˚

5 Layers of ZnO (5x5) on Cu (4x4), 5. Layer

(e) 5Layers, top layer.

Cu A layer

Cu B layer

Cu C layer −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆zfrom ¯

zin [ ˚

6 Layers of ZnO (5x5) on Cu (4x4), 6. Layer

(f) 6Layers, top layer.

Figure 111: Layer resolved corrugation maps (unit cell tripled) for 1to 6layers of ZnO

on Cu.

layer resolved corrugation maps 165

Ni A layer

Ni B layer

Ni C layer −0.24

−0.16

−0.08

0.00

0.08

0.16

0.24

0.32

∆zfrom ¯

zin [ ˚

1ZnO/Ni, 1. Layer

(a) 1Layer.

Ni A layer

Ni B layer

Ni C layer −0.24

−0.16

−0.08

0.00

0.08

0.16

0.24

0.32

∆zfrom ¯

zin [ ˚

2 Layers of ZnO (4x4) on Ni (3x3), 2. Layer

(b) 2Layers, top layer.

Ni A layer

Ni B layer

Ni C layer −0.18

−0.12

−0.06

0.00

0.06

0.12

0.18

0.24

0.30

∆zfrom ¯

zin [ ˚

3 Layers of ZnO (4x4) on Ni (3x3), 3. Layer

Ni A layer

Ni B layer

Ni C layer −0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

∆zfrom ¯

zin [ ˚

4 Layers of ZnO (4x4) on Ni (3x3), 4. Layer

(d) 4Layers, top layer.

Ni A layer

Ni B layer

Ni C layer −0.12

−0.08

−0.04

0.00

0.04

0.08

0.12

0.16

0.20

∆zfrom ¯

zin [ ˚

5 Layers of ZnO (4x4) on Ni (3x3), 5. Layer

(e) 5Layers, top layer.

Ni A layer

Ni B layer

Ni C layer −0.12

−0.09

−0.06

−0.03

0.00

0.03

0.06

0.09

0.12

∆zfrom ¯

zin [ ˚

6 Layers of ZnO (4x4) on Ni (3x3), 6. Layer

(f) 6Layers, top layer.

Figure 112: Layer resolved corrugation maps (unit cell tripled) for 1to 6layers of ZnO

on Ni.

166 layer resolved corrugation maps

Pd A layer

Pd B layer

Pd C layer −0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

∆zfrom ¯

zin [ ˚

1ZnO/Pd, 1. Layer

(a) 1Layer.

Pd A layer

Pd B layer

Pd C layer

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

∆zfrom ¯

zin [ ˚

2 Layers of ZnO (7x7) on Pd (6x6), 2. Layer

(b) 2Layers, top layer.

Pd A layer

Pd B layer

Pd C layer −0.45

−0.30

−0.15

0.00

0.15

0.30

0.45

∆zfrom ¯

zin [ ˚

3 Layers of ZnO (7x7) on Pd (6x6), 3. Layer

Pd A layer

Pd B layer

Pd C layer −0.16

−0.12

−0.08

−0.04

0.00

0.04

0.08

0.12

0.16

∆zfrom ¯

zin [ ˚

4 Layers of ZnO (7x7) on Pd (6x6), 4. Layer

(d) 4Layers, top layer.

Figure 113: Layer resolved corrugation maps (unit cell tripled) for 1to 4layers of ZnO

on Pd.

Pt A layer

Pt B layer

Pt C layer −0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆zfrom ¯

zin [ ˚

1ZnO/Pt, 1. Layer

(a) 1Layer.

Pt A layer

Pt B layer

Pt C layer −0.32

−0.24

−0.16

−0.08

0.00

0.08

0.16

0.24

∆zfrom ¯

zin [ ˚

2 Layers of ZnO (7x7) on Pt (6x6), 2. Layer

(b) 2Layers, top layer.

Pt A layer

Pt B layer

Pt C layer −0.24

−0.18

−0.12

−0.06

0.00

0.06

0.12

0.18

∆zfrom ¯

zin [ ˚

3 Layers of ZnO (7x7) on Pt (6x6), 3. Layer

Pt A layer

Pt B layer

Pt C layer −0.16

−0.12

−0.08

−0.04

0.00

0.04

0.08

0.12

0.16

∆zfrom ¯

zin [ ˚

4 Layers of ZnO (7x7) on Pt (6x6), 4. Layer

(d) 4Layers, top layer.

Figure 114: Layer resolved corrugation maps (unit cell tripled) for 1to 4layers of ZnO

on Pt.

layer resolved corrugation maps 167

Rh A layer

Rh B layer

Rh C layer −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆zfrom ¯

zin [ ˚

1ZnO/Rh, 1. Layer

(a) 1Layer.

Rh A layer

Rh B layer

Rh C layer −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆zfrom ¯

zin [ ˚

2 Layers of ZnO (6x6) on Rh (5x5), 2. Layer

(b) 2Layers, top layer.

Rh A layer

Rh B layer

Rh C layer −0.24

−0.16

−0.08

0.00

0.08

0.16

0.24

∆zfrom ¯

zin [ ˚

3 Layers of ZnO (6x6) on Rh (5x5), 3. Layer

Rh A layer

Rh B layer

Rh C layer −0.24

−0.16

−0.08

0.00

0.08

0.16

0.24

0.32

0.40

∆zfrom ¯

zin [ ˚

4 Layers of ZnO (6x6) on Rh (5x5), 4. Layer

(d) 4Layers, top layer.

Figure 115: Layer resolved corrugation maps (unit cell tripled) for 1to 4layers of ZnO

on Rh.

ELECTRONIC STRUCTURE OF MULTI-LAYER SYSTEMS

8 6 42 0 2

E−F[eV]

100

120

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected

DOS for two layers of ZnO

on Pd.

8 6 42 0 2

E−F[eV]

100

150

200

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected

DOS for four layers of ZnO

on Pd.

8 6 42 0 2

E−F[eV]

100

150

200

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

5th layer

6th layer

DOS for six layers of ZnO on

Pd.

Figure 116: Projected density of states for a) two layers of ZnO on Pd, b) four layers

of ZnO on Pd, and c) six layers of ZnO on Pd. The DOS is projected on

the individual layers. The area that corresponds to higher density than the

projected DOS of the previous layer is filled with the color of the layer. The

direction of the shift of the Zn 3d bands is indicated by a black arrow.

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected

DOS for two layers of ZnO

on Rh.

8 6 42 0 2

E−F[eV]

100

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected

DOS for four layers of ZnO

on Rh.

8 6 42 0 2

E−F[eV]

100

120

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

5th layer

6th layer

DOS for six layers of ZnO on

Rh.

Figure 117: Projected density of states for a) two layers of ZnO on Rh, b) four layers

of ZnO on Rh, and c) six layers of ZnO on Rh. The DOS is projected on

the individual layers. The area that corresponds to higher density than the

projected DOS of the previous layer is filled with the color of the layer. The

direction of the shift of the Zn 3d bands is indicated by a black arrow.

In this section the layer resolved projected DOS for Pd, Pt, Rh and Ni are

presented. The DOS is projected separately on the metal and the single ZnO

layers in the systems. In the plots their evolution is marked by filling the area

169

170 electronic structure of multi-layer systems

underneath the curves that have shifted higher in energy, closer to the Fermi level,

as compared to the previous layer. The direction of the evolution is indicated by

an arrow. The behavior is discussed in Sec. 7.2.

8 6 42 0 2

E−F[eV]

100

120

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected DOS for two layers

of ZnO on Pt.

8 6 42 0 2

E−F[eV]

100

150

200

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected DOS for four lay-

ers of ZnO on Pt.

Figure 118: Projected density of states for a) two layers of ZnO on Pt and b) four layers

of ZnO on Pt. The DOS is projected on the individual layers. The area that

corresponds to higher density than the projected DOS of the previous layer

is filled with the color of the layer. The direction of the shift of the Zn 3d

bands is indicated by a black arrow.

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected

DOS for two layers of ZnO

on Ni.

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected

DOS for four layers of ZnO

on Ni.

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

5th layer

6th layer

DOS for six layers of ZnO on

Ni.

Figure 119: Projected density of states for a) two layers of ZnO on Ni, b) four layers

of ZnO on Ni, and c) six layers of ZnO on Ni. The DOS is projected on

the individual layers. The area that corresponds to higher density than the

projected DOS of the previous layer is filled with the color of the layer. The

direction of the shift of the Zn 3d bands is indicated by a black arrow.

ELECTRONIC STRUCTURE OF (2X1)–H SYSTEMS

In this section the layer projected DOS for (2x1)–H reconstructed ZnO on the

various transition metals is shown. The DOS is projected separately on the metal

and the single ZnO layers in the systems. In the plots their evolution is marked

by filling the area underneath the curves that have shifted higher in energy, closer

to the Fermi level, as compared to the previous layer. Typically the change is very

small and no direction is indicated as in the previous section. The behavior of the

electronic states is discussed in Sec. 8.1.1and compared to the results of Sec. 7.2.

8 6 42 0 2

E−F[eV]

100

150

200

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected DOS for two lay-

ers of ZnO on Pd, (2x1)–H.

8 6 42 0 2

E−F[eV]

100

150

200

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected DOS for four

layers of ZnO on Pd, (2x1)–H.

Figure 120: Projected density of states for a) two layers of ZnO on Pd, (2x1)–H, b) four

layers of ZnO on Pd, (2x1)–H. The DOS is projected on the individual layers.

The area that corresponds to higher density than the projected DOS of the

previous layer is filled with the color of the layer.

171

172 electronic structure of (2x1)–h systems

8 6 42 0 2

E−F[eV]

100

150

200

250

300

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected DOS for two lay-

ers of ZnO on Ag, (2x1)–H.

8 6 42 0 2

E−F[eV]

100

150

200

250

300

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected DOS for four

layers of ZnO on Ag, (2x1)–H.

Figure 121: Projected density of states for a) two layers of ZnO on Ag, (2x1)–H, b) four

layers of ZnO on Ag, (2x1)–H. The DOS is projected on the individual layers.

The area that corresponds to higher density than the projected DOS of the

previous layer is filled with the color of the layer.

8 6 42 0 2

E−F[eV]

100

120

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected DOS for two lay-

ers of ZnO on Rh, (2x1)–H.

8 6 42 0 2

E−F[eV]

100

120

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected DOS for four

layers of ZnO on Rh, (2x1)–H.

Figure 122: Projected density of states for a) two layers of ZnO on Rh, (2x1)–H, b) four

layers of ZnO on Rh, (2x1)–H. The DOS is projected on the individual layers.

The area that corresponds to higher density than the projected DOS of the

previous layer is filled with the color of the layer.

electronic structure of (2x1)–h systems 173

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected DOS for two lay-

ers of ZnO on Cu, (2x1)–H.

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

(b) Layer resolved projected DOS for four

layers of ZnO on Cu, (2x1)–H.

Figure 123: Projected density of states for a) two layers of ZnO on Cu, (2x1)–H, b) four

layers of ZnO on Cu, (2x1)–H. The DOS is projected on the individual layers.

The area that corresponds to higher density than the projected DOS of the

previous layer is filled with the color of the layer.

8 6 42 0 2

E−F[eV]

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

4th layer

Figure 124: Projected density of states for four layers of ZnO on Ni, (2x1)–H. The DOS

is projected on the individual layers. The area that corresponds to higher

density than the projected DOS of the previous layer is filled with the color

of the layer.

174 electronic structure of (2x1)–h systems

10 8 6 42 0 2

E−F[eV]

100

(pro)DOS [states/eV]

1st layer

2nd layer

(a) Layer resolved projected DOS for two lay-

ers of ZnO on Cu, (2x1)–H.

10 8 6 42 0 2

E−F[eV]

100

(pro)DOS [states/eV]

1st layer

2nd layer

3rd layer

(b) Layer resolved projected DOS for four

layers of ZnO on Cu, (2x1)–H.

Figure 125: Projected density of states for a) two layers of ZnO on Cu, (2x1)–H, b) three

layers of ZnO on Cu, (2x1)–H calculated with HSE06. The DOS is projected

on the individual layers. The area that corresponds to higher density than

the projected DOS of the previous layer is filled with the color of the layer.

LAYER HEIGHTS FOR ZNO ON AG (111)

The film thicknesses and interlayer spacing including their standard deviation

for one to four layers of ZnO (unreconstructed and (2x1)–H reconstructed) are

presented in this section. A discussion and further explanations can be found in

Sec. 9.

O Zn Ag

2.61 ˚

σ=0.095

d=2.61 ˚

1 Layer of ZnO on Ag

(a) One layer of ZnO on Ag (111)

O Zn Ag

2.45 ˚

σ=0.182

2.7 ˚

σ=0.212

d=5.15 ˚

2 Layers of ZnO on Ag

(b) Two layers of ZnO on Ag (111)

O Zn Ag

2.55 ˚

σ=0.242

2.56 ˚

σ=0.184

2.69 ˚

σ=0.258

d=7.8 ˚

3 Layers of ZnO on Ag

(111)

O Zn Ag

2.56 ˚

σ=0.257

2.59 ˚

σ=0.241

2.51 ˚

σ=0.201

2.57 ˚

σ=0.256

d=10.24 ˚

4 Layers of ZnO on Ag

(d) Four layers of ZnO on Ag

(111)

Figure 126: Film thickness and interlayer thicknesses for 1to 4layers of ZnO on Ag

(111). The distances are the differences between the mean value of the atoms

z-positions of the layer in question and the atom z-positions one layer below.

From the data for the mean values the standard deviation σis calculated.

175

176 layer heights for zno on ag (111)

O Zn H Ag

2.6 ˚

σ=0.354

4.02 ˚

Ad=4.02 ˚

1 Layer of ZnO (2x1-H) on Ag

(a) One layer of ZnO-(2x1)–H on Ag

(111)

O Zn H Ag

2.64 ˚

σ=0.317

2.56 ˚

σ=0.342

3.93 ˚

Ad=6.48 ˚

2 Layers of ZnO (2x1-H) on Ag

(b) Two layers of ZnO-(2x1)–H on Ag

(111)

O Zn H Ag

2.64 ˚

σ=0.315

2.6 ˚

σ=0.315

2.64 ˚

σ=0.337

3.89 ˚

Ad=9.16 ˚

3 Layers of ZnO (2x1-H) on Ag

(111)

O Zn H Ag

2.66 ˚

σ=0.316

2.64 ˚

σ=0.321

2.63 ˚

σ=0.311

2.73 ˚

σ=0.34

3.91 ˚

Ad=11.94 ˚

4 Layers of ZnO (2x1-H) on Ag

(d) Four layers of ZnO-(2x1)–H on Ag

(111)

Figure 127: Film thickness and interlayer thicknesses for 1to 4layers of (2x1)–H re-

constructed ZnO on Ag (111). The distances are the differences between the

mean value of the atoms z-positions of the layer in question and the atom

z-position one layer below. From the data for the mean values the standard

deviation σis calculated.

CURRENT DEPENDENCY OF APPARENT HEIGHTS

−6−4−20246

VBias in eV

∆zin ˚

1L ZnO

1L ZnO, 2x1-H

2L ZnO

2L ZnO, 2x1-H

3L ZnO

3L ZnO, 2x1-H

ZnO on Ag

(a) Ag substrate.

−6−4−20246

VBias in eV

∆zin ˚

1L ZnO, H 0

1L ZnO, H 18

2L ZnO, H 0

2L ZnO, H 18

(b) Pd substrate.

−6−4−20246

VBias in eV

∆zin ˚

1L ZnO

1L ZnO, 2x1-H

2L ZnO

2L ZnO, 2x1-H

3L ZnO

Figure 128: Bias voltage dependence of simulated STM graphs for (2x1)–H reconstructed

and unreconstructed ZnO on Ag (111), Pd (111) and Pt (111) for one to

three layers of ZnO. The underlying STM graphs were calculated within

the Tersoff-Hamann approach as constant height STM graphs. The constant

heights were sorted by current to yield constant current graphs, that were

averaged to yield absolute height values. The apparent heights are the differ-

ence between the absolute height of the ZnO on the metal and the metal at

the same constant current of 1·10−5.

In this chapter the constant current dependency of the calculated apparent

heights is presented. The apparent heights are calculated as averages over sim-

ulated constant current STM graphs. The constant height STM graphs are cal-

culated from the local density by summing from the Fermi energy to the Fermi

energy plus the selected bias voltage (VBias). The apparent heights thus yield

a bias dependence, that depends on the electronic structure of the system. In

Fig. 128 the bias dependence of reconstructed and unreconstructed ZnO on Ag,

Pd and Pt is shown. For bias voltages below 0eV only a slight decline on the

177

178 current dependency of apparent heights

calculated height is observed. For Pd a small dip exists around the Fermi level.

With increasing positive bias the calculated apparent height grows almost lin-

early until a saturation is reached. The large distance from the surface yields

a low electron density and therefore a low tunneling current. The increase for

positive bias voltages is stronger for the (2x1)–H reconstructed ZnO films.

DEFECTS IN IDEAL α-BN ZNO MONO LAYER

(a) Defect free. (b) With oxygen vacancy.

Figure 129: Atomic structure of 4x4unit-cells of an ideal α-BN ZnO mono layer without

a) and with a oxygen vacancy b). The unit-cell is periodically continued.

To asses the error resulting from employing the semi-local xc-functional PBE

[196], the formation energies of a oxygen vacancy in an ideal α-BN ZnO mono-

layer is discussed and compared to results from calculations with the higher level

hybrid xc-functional HSE06 [100] (includes 25% exact exchange). Calculations for

bulk defects of ZnO and other materials have shown substantial differences in the

formation energies obtained by PBE and HSE06 [211]. In Fig. 129 the atomic struc-

ture of a sheet of ZnO in α-BN structure is shown with and without a oxygen va-

cancy. From these two structures, relaxed with the PBE and HSE06 xc-functional

the energy for removing one oxygen atom from the lattice can be calculated. The

Formation energies are calculated as the difference between the ideal structure

and the mono-layer with one oxygen vacancy and an isolated oxygen atom. All

(a) PBE. (b) HSE06.

Figure 130: Density of states of a 4x4ZnO mono-layer with one ZnO vacancy calculated

with the PBE and HSE06 xc-functional. Both structures were relaxed with the

underlying xc-functional.

179

180 defects in ideal α-bn zno mono layer

(a) Defect free. (b) With oxygen vacancy.

Figure 131: Atomic structure of one layer of ZnO on Cu (111) without a) and with a

oxygen vacancy b). The unit-cell is periodically continued.

structures were relaxed with the xc-functional used to calculate the energies. The

reference energies are taken from Tab. 15

EForm = (EZnO-O +1

2EO) − EZnO.

(154)

The formation energies defined in such a way are 6.54 eV for HSE06 and 6.48 eV

for PBE for a 4x4sheet of ZnO and one oxygen vacancy (1/16 of defect concen-

tration). The difference per ZnO unit cell is 4meV. The error encountered in the

calculation is therefore considered small.

A further qualitative error estimation is achieved by comparing the electronic

structures of the structure with oxygen vacancy calculated by HSE06 and PBE. In

Fig. 130 density of states for PBE and HSE06 are shown. The qualitative agree-

ment between both graphs is very good. For both calculations a defect state is

observed around the Fermi energy. For HSE06 the rest of the electronic states are

further shifted up or downwards in energy as compared to the PBE calculation

resulting in an increased band gap.

The errors introduced by using PBE are expected to be negligible for the ZnO

(a) PBE. (b) HSE06.

Figure 132: Density of states of a ZnO mono-layer on Cu (111) with one ZnO vacancy cal-

culated with the PBE and HSE06 xc-functional. Both structures were relaxed

with the underlying xc-functional.

defects in ideal α-bn zno mono layer 181

mono-layer.

In this work ZnO on metal substrates is investigated. To quantify the errors for

those systems a similar analysis as for the mono-layer is conducted. In Fig. 131

the atomic structure of one layer of ZnO on Cu (111) is shown with and without

an oxygen vacancy. The full super cell is shown, being the coincidence structure

previously determined (see Sec. 6). The structures were relaxed with HSE06 and

PBE. The defect formation energies as calculated before, are 2.56 eV for the PBE

xc-functional and 2.52 eV for HSE06. This results in a difference of 3meV per

1x1ZnO surface unit cell. The error can be considered negligible for the inves-

tigation of the surface free energies. The DOS for the calculation with PBE and

HSE06 are shown in Fig 132. The qualitative differences in the DOS have further

reduced as compared to the free-standing mono-layer. The most prominent dif-

ference is due to the stronger localization of the Zn d-states because of the use of

exact exchange in HSE06. Also in this case we can conclude, that the semi-local

xc-functional PBE is sufficient for the calculation of surface phase diagrams (total

energy) and the description of the electronic structure.

CHARGE DENSITY DIFFERENCES

(a) 1Layer of ZnO (b) 2Layers of ZnO

Figure 133: Charge density differences for 1to 4layers of ZnO on Ag (111). The dif-

ferences are calculated between the full coincidence structures and the Ag

substrate and ZnO ad-layer with frozen geometries. The cutoff for all graphs

is 0.02 electrons/a3

B. Yellow refers to electron density decrease and blue to

electron density increase.

183

PUBLICATIONS

Some ideas and figures have appeared previously in the following publications:

Bjoern Bieniek, Oliver T. Hofmann, and Patrick Rinke. Influence of hydrogen on

the structure and stability of ultra-thin zno on metal substrates. Applied Physics

Letters,106(13):–, 2015

Eike Verdenhalven, Andreas Knorr, Marten Richter, Bjoern Bieniek, and Patrick

Rinke. Theory of optical excitations in dipole-coupled hybrid molecule-semi-

conductor layers: Coupling of a molecular resonance to semiconductor contin-

uum states. Phys. Rev. B,89:235314, Jun 2014

COLOPHON

This document was typeset using the typographical look-and-feel classicthesis

developed by André Miede. The style was inspired by Robert Bringhurst’s sem-

inal book on typography “The Elements of Typographic Style”. classicthesis is

available for both L

TEX and LYX:

http://code.google.com/p/classicthesis/

Postprocessing of data obtained by DFT calculations was done by scientific li-

braries numpy and scipy for python:

http://www.scipy.org http://www.numpy.org

Plots of density of states, phase diagrams, band structure, etc. were prepared

with the 2D plotting library matplotlib for python:

http://matplotlib.org

Illustrations of atomic structures were created by Jmol:

http://www.jmol.org

and rendered with povray:

www.povray.org

iii

BIBLIOGRAPHY

[1] Sadao Adachi. Zinc oxide (zno). In Handbook on Physical Properties of Semi-

conductors, pages 65–97. Springer US, 2004.

[2] C. Adamo and V. Barone. J. Chem. Phys.,110:6158,1999.

[3] V. M. Agranovich, Yu. N. Gartstein, and M. Litinskaya. Hybrid resonant

organic-inorganic nanostructures for optoelectronic applications. Chemical

Reviews,111(9):5179–5214,2011.

[4] Audrius Alkauskas, Peter Broqvist, and Alfredo Pasquarello. Defect levels

through hybrid density functionals: Insights and applications. physica status

solidi (b),248(4):775–789,2011.

[5] C. G. Allen, D. J. Baker, T. M. Brenner, C. C. Weigand, J. M. Albin, K. X.

Steirer, D. C. Olson, C. Ladam, D. S. Ginley, R. T. Collins, and T. E. Furtak.

Alkyl surface treatments of planar zinc oxide in hybrid organic/inorganic

solar cells. The Journal of Physical Chemistry C,116(16):8872–8880,2012.

[6] C. Ambrosch-Draxl and J. O. Sofo. Linear optical properties of solids

within the full-potential linearized augmented planewave method. Com-

puter Physics Communications, (175):1–14,2006.

[7] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra,

J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen.

LAPACK Users’ Guide. Society for Industrial and Applied Mathematics,

Philadelphia, PA, third edition, 1999.

[8] George B. Arfken, Hans J. Weber, and Frank E. Harris. Mathematical Meth-

ods for Physicists, Sixth Edition: A Comprehensive Guide. Academic Press, 6

edition, July 2005.

[9] R. Armiento and Ann E. Mattsson. Phys. Rev. B,72:085108,2005.

[10] F Aryasetiawan and O Gunnarsson. The gw method. Reports on Progress in

Physics,61(3):237,1998.

[11] N.W. Ashcroft and N.D. Mermin. Solid State Physics. Saunders College,

Philadelphia, 1976.

[12] A. Ashrafi and C. Jagadish. Review of zincblende zno: Stability of

metastable zno phases. Journal of Applied Physics,102(7):–, 2007.

[13] A. B. M. Almamun Ashrafi, Akio Ueta, Adrian Avramescu, Hidekazu Ku-

mano, Ikuo Suemune, Young-Woo Ok, and Tae-Yeon Seong. Growth and

characterization of hypothetical zinc-blende zno films on gaas(001) sub-

strates with zns buffer layers. Applied Physics Letters,76(5):550–552,2000.

[14] Viktor Atalla, Mina Yoon, Fabio Caruso, Patrick Rinke, and Matthias Schef-

fler. Hybrid density functional theory meets quasiparticle calculations: A

consistent electronic structure approach. Phys. Rev. B,88:165122, Oct 2013.

[15] V.M. Axt and A. Stahl. A dynamics-controlled truncation scheme for the

hierarchy of density matrices in semiconductor optics. Zeitschrift für Physik

B Condensed Matter,93(2):195–204,1994.

[16] George A. Baker and Peter Graves-Morris. Padé Approximants (Encyclopedia

of Mathematics and its Applications). Encyclopedia of mathematics and its

applications, v. 59. Cambridge University Press, 2edition, January 1996.

[17] J. Baker, J. Andzelm, A. Scheiner, and B. Delley. J. Chem. Phys.,101:8894,

1994.

[18] A. Bakin, A. El-Shaer, A. C. Mofor, M. Al-Suleiman, E. Schlenker, and

A. Waag. Znmgo-zno quantum wells embedded in zno nanopillars: To-

wards realisation of nano-leds. physica status solidi (c),4(1):158–161,2007.

[19] C. Baltes, S. Vukojevi´c, and F. Schüth. Correlations between synthesis, pre-

cursor, and catalyst structure and activity of a large set of cuo/zno/al2o3

catalysts for methanol synthesis. Journal of Catalysis,258(2):334 –344,2008.

[20] J. Bardeen. Tunnelling from a many-particle point of view. Phys. Rev. Lett.,

6:57–59, Jan 1961.

[21] Basko, D., La Rocca, G. C., Bassani, F., and Agranovich, V. M. Förster

energy transfer from a semiconductor quantum well to an organic material

overlayer. Eur. Phys. J. B,8(3):353–362,1999.

[22] C. I. Bayly, P. Cieplak, W. D. Cornell, and P. A. Kollman. A well-behaved

electrostatic potential based method using charge restraints for deriving

atomic charges: The resp model. J. Phys. Chem.,97:10269–10280,1993.

[23] A.D. Becke. J. Chem. Phys.,88:1053,1988.

[24] Axel D. Becke. A new mixing of hartree-fock and local density-functional

theories. The Journal of Chemical Physics,98(2):1372–1377,1993.

[25] Malte Behrens, Felix Studt, Igor Kasatkin, Stefanie Kühl, Michael

Hävecker, Frank Abild-Pedersen, Stefan Zander, Frank Girgsdies, Patrick

Kurr, Benjamin-Louis Kniep, Michael Tovar, Richard W. Fischer, Jens K.

NÃÿrskov, and Robert Schlögl. The active site of methanol synthesis over

cu/zno/al2o3industrial catalysts. Science,336(6083):893–897,2012.

[26] C. Berthod, N. Binggeli, and A. Baldereschi. Schottky barrier heights at

polar metal/semiconductor interfaces. Phys. Rev. B,68:085323, Aug 2003.

[27] B. H. Besler, K. M. Merz, and P. A. Kollman. Atomic charges derived from

semiempirical methods. J. Comput. Chem.,11:431–439,1990.

[28] Bjoern Bieniek, Oliver T. Hofmann, and Patrick Rinke. Influence of hy-

drogen on the structure and stability of ultra-thin zno on metal substrates.

Applied Physics Letters,106(13):–, 2015.

[29] K. Binder. Recent trends in the development and application of the monte

carlo method. In Kurt Binder, editor, Monte Carlo Methods in Statistical

Physics, volume 7of Topics in Current Physics, pages 357–392. Springer Berlin

Heidelberg, 1986.

[30] L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon,

J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker,

and R. C. Whaley. ScaLAPACK Users’ Guide.

[31] P. Blaha, H. Hofstätter, O. Koch, R. Laskowski, and K. Schwarz. Iterative di-

agonalization in augmented plane wave based methods in electronic struc-

ture calculations. Journal of Computational Physics,229(2):453 –460,2010.

[32] P. E. Blöchl. Projector augmented-wave method. Phys. Rev. B,50:17953–

17979, Dec 1994.

[33] Volker Blum, Ralf Gehrke, Felix Hanke, Paula Havu, Ville Havu, Xinguo

Ren, Karsten Reuter, and Matthias Scheffler. Ab initio molecular sim-

ulations with numeric atom-centered orbitals. Comput. Phys. Commun.,

180(11):2175–2196,2009.

[34] S Blumstengel, S Sadofev, and F Henneberger. Electronic coupling of opti-

cal excitations in organic/inorganic semiconductor hybrid structures. New

Journal of Physics,10(6):065010,2008.

[35] A. Bondi. van der waals volumes and radii. J. Phys. Chem.,68:441–451,

1964.

[36] M. Born and R. Oppenheimer. Zur quantentheorie der molekeln. Annalen

der Physik,389(20):457–484,1927.

[37] Christoph J. Brabec, Sean E. Shaheen, Christoph Winder, N. Serdar Sari-

ciftci, and Patrick Denk. Effect of lif/metal electrodes on the performance

of plastic solar cells. Applied Physics Letters,80(7):1288–1290,2002.

[38] J. L. Bredas, J. P. Calbert, D. A. da Silva Filho, and J. Cornil. Organic

semiconductors: A theoretical characterization of the basic parameters gov-

erning charge transport. Proceedings of the National Academy of Sciences,

99(9):5804–5809,2002.

[39] C. M. Breneman and K. B. Wiberg. Determining atom-centered monopoles

from molecular electrostatic potentials. the need for high sampling density

in formamide conformational analysis. J. Comput. Chem.,11:361–373,1990.

[40] Alejandro L. Briseno, Thomas W. Holcombe, Akram I. Boukai, Erik C. Gar-

nett, Steve W. Shelton, Jean J. M. Fréchet, and Peidong Yang. Oligo- and

polythiophene/zno hybrid nanowire solar cells. Nano Letters,10(1):334–340,

2010. PMID: 20000808.

[41] N. Buecking, P. Kratzer, M. Scheffler, and A. Knorr. Linking density func-

tional and density-matrix theory: Picosecond electron relaxation at the

si(100) surface. Phys. Rev. B,77:233305, Jun 2008.

[42] Kieron Burke et al. The abc of dft. Department of Chemistry, University of

California,2007.

[43] Burroughes J. H., Bradley D. D. C., Brown A. R., Marks R. N., Mackay

K., Friend R. H., Burns P. L., and Holmes A. B. Light-emitting diodes

based on conjugated polymers. Nature,347(6293):539–541, oct 1990.

10.1038/347539a0.

[44] C. Campana, B. Mussard, and T. K. Woo. Electrostatic potential derived

atomic charges for periodic systems using a modified error functional. J.

Chem. Theory Comput.,5:2866–2878,2009.

[45] Shih-Chia Chang and Peter Mark. The crystallography of the polar (0001)

zn and (000-1)o surfaces of zinc oxide. Surface Science,46(1):293 –300,1974.

[46] S. Chanyawadee, P. G. Lagoudakis, R. T. Harley, D. G. Lidzey, and

M. Henini. Nonradiative exciton energy transfer in hybrid organic-

inorganic heterostructures. Phys. Rev. B,77:193402, May 2008.

[47] D.-J. Chen, A. C. Stern, B. Space, and J. K. Johnson. Atomic charges derived

from electrostatic potential for molecular and periodic systems. J. Phys.

Chem. A,114:10225–10233,2010.

[48] L. E. Chirlian and M. M. Francl. Atomic charges derived from electrostatic

potentials: A detailed study. J. Comput. Chem.,8:894–905,1987.

[49] S. Chu, J. H. Lim, L. J. Mandalapu, Z. Yang, L. Li, and J. L. Liu. Sb-doped

p-znoâ´

L¸Tga-doped n-zno homojunction ultraviolet light emitting diodes.

Applied Physics Letters,92(15):–, 2008.

[50] M.F. Chung and H.E. Farnsworth. Investigations of surface stability of

iiâ ˘

A¸Svi wurtzite compounds by {LEED}. Surface Science,22(1):93 –110,

1970.

[51] Frederik Claeyssens, Colin L. Freeman, Neil L. Allan, Ye Sun, Michael N. R.

Ashfold, and John H. Harding. Growth of zno thin films-experiment and

theory. J. Mater. Chem.,15:139–148,2005.

[52] R. Clausius. Ueber die art der bewegung, welche wir wärme nennen. An-

nalen der Physik,176(3):353–380,1857.

[53] R. Clausius. Die mechanische Wärmetheorie, volume 2. F. Vieweg und Sohn,

1879.

[54] J. Clerk-Maxwell. Xxii.-on the dynamical evidence of the molecular consti-

tution of bodies. J. Chem. Soc.,28:493–508,1875.

[55] Aron J. Cohen, Paula Mori-Sánchez, and Weitao Yang. Challenges for den-

sity functional theory. Chemical Reviews,112(1):289–320,2012.

[56] S. R. Cox and D. E. Williams. Representation of the molecular electrostatic

potential by a net atomic charge model. J. Comput. Chem.,2:304–323,1981.

[57] Gábor I. Csonka, John P. Perdew, Adrienn Ruzsinszky, Pier H. T. Philipsen,

Sébastien Lebègue, Joachim Paier, Oleg A. Vydrov, and János G. Ángyán.

Assessing the performance of recent density functionals for bulk solids.

Phys. Rev. B,79:155107, Apr 2009.

[58] N R D’Amico, G Cantele, C A Perroni, and D Ninno. Electronic proper-

ties and schottky barriers at znoâ ˘

A¸Smetal interfaces from first principles.

Journal of Physics: Condensed Matter,27(1):015006,2015.

[59] J. Danckwerts, K. J. Ahn, J. Förstner, and A. Knorr. Theory of ultrafast

nonlinear optics of coulomb-coupled semiconductor quantum dots: Rabi

oscillations and pump-probe spectra. Phys. Rev. B,73:165318, Apr 2006.

[60] R. de L. Kronig. On the theory of dispersion of x-rays. J. Opt. Soc. Am.,

12(6):547–556, Jun 1926.

[61] F. Decremps, F. Datchi, A. M. Saitta, A. Polian, S. Pascarelli, A. Di Cicco,

J. P. Itié, and F. Baudelet. Local structure of condensed zinc oxide. Phys.

Rev. B,68:104101, Sep 2003.

[62] F. Della Sala, S. Blumstengel, and F. Henneberger. Electrostatic-field-driven

alignment of organic oligomers on zno surfaces. Phys. Rev. Lett.,107:146401,

Sep 2011.

[63] B. Delley. J. Chem. Phys.,92:508,1990.

[64] B. Delley. J. Comp Chem.,17:1152,1995.

[65] Ilker Demiroglu and Stefan T. Bromley. Nanofilm versus Bulk Polymor-

phism in Wurtzite Materials. Phys. Rev. Lett.,110(24):245501,2013.

[66] Ilker Demiroglu, Daniele Stradi, Francesc Illas, and Stefan T. Bromley. A

theoretical study of a ZnO graphene analogue: adsorption on Ag(111) and

hydrogen transport. J. Phys.: Condens. Matter,23(33):334215,2011.

[67] Xingyi Deng, Kun Yao, Keju Sun, Wei-Xue Li, Junseok Lee, and Christopher

Matranga. Growth of single- and bilayer zno on au(111) and interaction

with copper. J. Phys. Chem. C,117(21):11211–11218, MAY 30 2013.

[68] Serge Desgreniers. High-density phases of zno: structural and compressive

parameters. Phys. Rev. B,58:14102–14105, Dec 1998.

[69] Katharina Diller, Florian Klappenberger, Francesco Allegretti, Anthoula C.

Papageorgiou, Sybille Fischer, David A. Duncan, Reinhard J. Maurer, Ju-

lian A. Lloyd, Seung Cheol Oh, Karsten Reuter, and Johannes V. Barth.

Temperature-dependent templated growth of porphine thin films on the

(111) facets of copper and silver. The Journal of Chemical Physics,141(14):–,

2014.

[70] M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist.

Van der waals density functional for general geometries. Phys. Rev. Lett.,

92:246401, Jun 2004.

[71] John F Dobson and Tim Gould. Calculation of dispersion energies. Journal

of Physics: Condensed Matter,24(7):073201,2012.

[72] Olga Dulub, Lynn A Boatner, and Ulrike Diebold. {STM} study of the

geometric and electronic structure of zno(0001)-zn, (000-1)-o, (1010) and

(1120) surfaces. Surface Science,519(3):201 –217,2002.

[73] Olga Dulub, Ulrike Diebold, and G. Kresse. Novel stabilization mechanism

on polar surfaces: Zno(0001)-zn. Phys. Rev. Lett.,90:016102, Jan 2003.

[74] A. Einstein. Die plancksche theorie der strahlung und die theorie der spez-

ifischen wärme. Annalen der Physik,327(1):180–190,1906.

[75] R. Eisenschitz and F. London. Über das verhältnis der van der waalsschen

kräfte zu den homöopolaren bindungskräften. Zeitschrift für Physik,60(7-

8):491–527,1930.

[76] Helmut Eschrig. The fundamentals of density functional theory, volume 32.

Springer.

[77] P. P. Ewald. Die berechnung optischer und elektrostatischer gitterpoten-

tiale. Annalen der Physik,369(3):253–287,1921.

[78] E. Fabiano, L . A. Constantin, and F. Della Sala. Generalized gradient

approximation bridging the rapidly and slowly varying density regimes:

A pbe-like functional for hybrid interfaces. Phys. Rev. B,82:113104,2010.

[79] E. Fermi. Eine statistische methode zur bestimmung einiger eigenschaften

des atoms und ihre anwendung auf die theorie des periodischen systems

der elemente. Zeitschrift für Physik,48(1-2):73–79,1928.

[80] Denis Fichou. Structural order in conjugated oligothiophenes and its im-

plications on opto-electronic devices. J. Mater. Chem.,10:571–588,2000.

[81] T. Fließbach. Quantenmechanik. Lehrbuch zur Theoretischen Physik. BI-

Wiss.-Verlag, 1991.

[82] Th. Förster. Zwischenmolekulare energiewanderung und fluoreszenz. An-

nalen der Physik,437(1-2):55–75,1948.

[83] Colin L. Freeman, Frederik Claeyssens, Neil L. Allan, and John H. Harding.

Graphitic nanofilms as precursors to wurtzite films: Theory. Phys. Rev. Lett.,

96:066102, Feb 2006.

[84] Roger H. French, V. Adrian Parsegian, Rudolf Podgornik, Rick F. Rajter,

Anand Jagota, Jian Luo, Dilip Asthagiri, Manoj K. Chaudhury, Yet-ming

Chiang, Steve Granick, Sergei Kalinin, Mehran Kardar, Roland Kjellander,

David C. Langreth, Jennifer Lewis, Steve Lustig, David Wesolowski, John S.

Wettlaufer, Wai-Yim Ching, Mike Finnis, Frank Houlihan, O. Anatole von

Lilienfeld, Carel Jan van Oss, and Thomas Zemb. Long range interactions

in nanoscale science. Rev. Mod. Phys.,82:1887–1944, Jun 2010.

[85] Daan Frenkel and Berend Smit. Understanding Molecular Simulation, Second

Edition: From Algorithms to Applications (Computational Science). Academic

Press, 2edition, November 2001.

[86] Hans-Joachim Freund, Markus Heyde, Niklas Nilius, Swetlana Schauer-

mann, Shamil Shaikhutdinov, and Martin Sterrer. Model studies on hetero-

geneous catalysts at the atomic scale: From supported metal particles to

two-dimensional zeolites. Journal of Catalysis,308(0):154 –167,2013.50th

Anniversary Special Issue.

[87] Hans-Joachim Freund and Gianfranco Pacchioni. Oxide ultra-thin films on

metals: new materials for the design of supported metal catalysts. Chem.

Soc. Rev.,37:2224–2242,2008.

[88] Christoph Freysoldt, Patrick Rinke, and Matthias Scheffler. Ultrathin ox-

ides: Bulk-oxide-like model surfaces or unique films? Phys. Rev. Lett.,

99:086101, Aug 2007.

[89] Jens Fricke. Transport equations including many-particle correlations for

an arbitrary quantum system: A general formalism. Annals of Physics,

252(2):479 –498,1996.

[90] Friend R. H., Gymer R. W., Holmes A. B., Burroughes J. H., Marks R. N.,

Taliani C., Bradley D. D. C., Santos D. A. Dos, Bredas J. L., Logdlund M.,

and Salaneck W. R. Electroluminescence in conjugated polymers. Nature,

397(6715):121–128, jan 1999.10.1038/16393.

[91] Martin Fuchs and Matthias Scheffler. Ab initio pseudopotentials for

electronic structure calculations of poly-atomic systems using density-

functional theory. Computer Physics Communications,119(1):67–98,1999.

[92] Gene H. Golub and Henk A. van der Vorst. Eigenvalue computation

in the 20th century. Journal of Computational and Applied Mathematics,

123(1â˘

A¸S2):35 –65,2000. Numerical Analysis 2000. Vol. III: Linear Algebra.

[93] Jan Götzen and Gregor Witte. Rapid preparation of highly ordered ultraflat

zno surfaces. Applied Surface Science,258(24):10144 –10147,2012.

[94] B. Hammer, L.B. Hansen, and J.K. Nørskov. Phys. Rev. B,59:7413,1999.

[95] J. B. Hansen and P. E. Højlund Nielsen. Methanol Synthesis. Wiley-VCH,

2008.

[96] V. Havu, V. Blum, P. Havu, and M. Scheffler. Efficient integration for all-

electron electronic structure calculation using numeric basis functions. Jour-

nal of Computational Physics,228(22):8367 –8379,2009.

[97] Lars Hedin. New method for calculating the one-particle green’s function

with application to the electron-gas problem. Phys. Rev.,139:A796–A823,

Aug 1965.

[98] Philipp Herrmann and Georg Heimel. Structure and stoichiometry predic-

tion of surfaces reacting with multicomponent gases. Advanced Materials,

27(2):255–260,2015.

[99] Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof. J. Chem. Phys.,

118:8207,2003.

[100] Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof. Hybrid func-

tionals based on a screened coulomb potential. The Journal of Chemical

Physics,118(18):8207–8215,2003.

[101] F.L. Hirshfeld. Theor. Chim. Acta (Berl.),44:129,1977.

[102] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev.,

136:B864–B871, Nov 1964.

[103] Harald Hoppe and Niyazi Serdar Sariciftci. Organic solar cells: An

overview. Journal of Materials Research,19:1924–1945,2004.

[104] H. Hu, Z. Lu, and W. yang yang yang yang. Fitting molecular electrostatic

potentials from quantum mechanical calculations. J. Chem. Theory Comput.,

3:1004–1013,2007.

[105] G Hughes and N R McLean. Zinc oxide tape: a useful dressing for the

recalcitrant finger-tip and soft-tissue injury. Archives of Emergency Medicine,

5(4):223–227,1988.

[106] Mark S. Hybertsen and Steven G. Louie. Electron correlation in semicon-

ductors and insulators: Band gaps and quasiparticle energies. Phys. Rev. B,

34:5390–5413, Oct 1986.

[107] Hisao Ishii, Kiyoshi Sugiyama, Eisuke Ito, and Kazuhiko Seki. Energy level

alignment and interfacial electronic structures at organic/metal and organ-

ic/organic interfaces. Advanced Materials,11(8):605–625,1999.

[108] G. Itskos, G. Heliotis, P. G. Lagoudakis, J. Lupton, N. P. Barradas, E. Alves,

S. Pereira, I. M. Watson, M. D. Dawson, J. Feldmann, R. Murray, and

D. D. C. Bradley. Efficient dipole-dipole coupling of mott-wannier and

frenkel excitons in (ga,in)n quantum well/polyfluorene semiconductor het-

erostructures. Phys. Rev. B,76:035344, Jul 2007.

[109] John E. Jaffe, James A. Snyder, Zijing Lin, and Anthony C. Hess. Lda and

gga calculations for high-pressure phase transitions in zno and mgo. Phys.

Rev. B,62:1660–1665, Jul 2000.

[110] Anderson Janotti and Chris G. Van de Walle. New insights into the role of

native point defects in zno. Journal of Crystal Growth,287(1):58 –65,2006.

Proceedings of the International Conference on Materials for Advanced

Technologies (ICMAT 2005) Symposium N ZnO and Related Materials Pro-

ceedings of the International Conference on Materials for Advanced Tech-

nologies (ICMAT 2005) Symposium N.

[111] Anderson Janotti and Chris G Van de Walle. Fundamentals of zinc oxide

as a semiconductor. Reports on Progress in Physics,72(12):126501,2009.

[112] Anderson Janotti and Chris G. Van de Walle. Oxygen vacancies in zno.

Applied Physics Letters,87(12):–, 2005.

[113] Anderson Janotti and Chris G. Van de Walle. Native point defects in zno.

Phys. Rev. B,76:165202, Oct 2007.

[114] Frank Jensen. Introduction to Computational Chemistry. Wiley, 2edition,

November 2006.

[115] Mikkel Jørgensen, Kion Norrman, and Frederik C. Krebs. Stability/degra-

dation of polymer solar cells. Solar Energy Materials and Solar Cells,92(7):686

–714,2008. Degradation and Stability of Polymer and Organic Solar Cells.

[116] Robert A DiStasio Jr, Vivekanand V Gobre, and Alexandre Tkatchenko.

Many-body van der waals interactions in molecules and condensed mat-

ter. Journal of Physics: Condensed Matter,26(21):213202,2014.

[117] Petr Jurecka, Jiri Sponer, Jiri Cerny, and Pavel Hobza. Benchmark database

of accurate (mp2and ccsd(t) complete basis set limit) interaction energies

of small model complexes, dna base pairs, and amino acid pairs. Phys.

Chem. Chem. Phys.,8:1985–1993,2006.

[118] M. Kalay, H.H. Kart, S. Oezdemir Kart, and T. Cagin. Elastic properties

and pressure induced transitions of zno polymorphs from first-principle

calculations. Journal of Alloys and Compounds,484(1â˘

A¸S2):431 –438,2009.

[119] H. Karzel, W. Potzel, M. Köfferlein, W. Schiessl, M. Steiner, U. Hiller, G. M.

Kalvius, D. W. Mitchell, T. P. Das, P. Blaha, K. Schwarz, and M. P. Paster-

nak. Lattice dynamics and hyperfine interactions in zno and znse at high

external pressures. Phys. Rev. B,53:11425–11438, May 1996.

[120] D. Kato, T. Matsui, and J. Yuhara. Oxidation of ultra-thin zinc films on

rh(100) surface. Surface Science,604(15â˘

A¸S16):1283 –1286,2010.

[121] G.P. Kerker. Phys. Rev. B,23:3082,1981.

[122] Kyoung-Kook Kim, Hyun-Sik Kim, Dae-Kue Hwang, Jae-Hong Lim, and

Seong-Ju Park. Realization of p-type zno thin films via phosphorus doping

and thermal activation of the dopant. Applied Physics Letters,83(1):63–65,

2003.

[123] M. Kira and S.W. Koch. Many-body correlations and excitonic effects in

semiconductor spectroscopy. Progress in Quantum Electronics,30(5):155 –

296,2006.

[124] Charles Kittel. Introduction to Solid State Physics. Wiley, NY, 8edition,

November 2005.

[125] C. Klingshirn. Zno: From basics towards applications. physica status solidi

(b),244(9):3027–3073,2007.

[126] Claus Klingshirn, J. Fallert, H. Zhou, J. Sartor, C. Thiele, F. Maier-Flaig,

D. Schneider, and H. Kalt. 65 years of zno research â ˘

A¸S old and very

recent results. physica status solidi (b),247(6):1424–1447,2010.

[127] Bjorn Kobin, Lutz Grubert, Sylke Blumstengel, Fritz Henneberger, and Ste-

fan Hecht. Vacuum-processable ladder-type oligophenylenes for organic-

inorganic hybrid structures: synthesis, optical and electrochemical proper-

ties upon increasing planarization as well as thin film growth. J. Mater.

Chem.,22:4383–4390,2012.

[128] Klaus Koepernik and Helmut Eschrig. Full-potential nonorthogonal local-

orbital minimum-basis band-structure scheme. Phys. Rev. B,59:1743–1757,

Jan 1999.

[129] W. Kohn and L. J. Sham. Self-consistent equations including exchange and

correlation effects. Phys. Rev.,140:A1133–A1138, Nov 1965.

[130] H.A. Kramers. La diffusion de la lumiere par les atome. Atti Cong. Intern.

Fisici, (Transactions of Volta Centenary Congress),2:545–â ˘

A¸S557,1927.

[131] G. Kresse and J. Furthmüller. Efficiency of ab-initio total energy calcula-

tions for metals and semiconductors using a plane-wave basis set. Comp.

Mat. Sci.,6:15–50,1996.

[132] Georg Kresse, Olga Dulub, and Ulrike Diebold. Competing stabilization

mechanism for the polar zno(0001)-zn surface. Phys. Rev. B,68:245409, Dec

2003.

[133] A. Krönig. Grundzüge einer theorie der gase. Annalen der Physik,

175(10):315–322,1856.

[134] Aliaksandr V. Krukau, Oleg A. Vydrov, Artur F. Izmaylov, and Gustavo E.

Scuseria. J. Chem. Phys.,125:224106,2006.

[135] Joyce G. Laquindanum, Howard E. Katz, Andrew J. Lovinger, and Ananth

Dodabalapur. Morphological origin of high mobility in pentacene thin-film

transistors. Chemistry of Materials,8(11):2542–2544,1996.

[136] Jeppe V. Lauritsen, Soeren Porsgaard, Morten K. Rasmussen, Mona C. R.

Jensen, Ralf Bechstein, Kristoffer Meinander, Bjerne S. Clausen, Stig Helveg,

Roman Wahl, Georg Kresse, and Flemming Besenbacher. Stabilization prin-

ciples for polar surfaces of zno. ACS Nano,5(7):5987–5994,2011. PMID:

21671628.

[137] Matt Law, Lori E. Greene, Justin C. Johnson, Richard Saykally, and Peidong

Yang. Nanowire dye-sensitized solar cells. Nat. Mater.,4(6):455–459,2005.

[138] C.L. Lee, W. Yang, and R.G. Parr. Phys. Rev. B,37:785,1988.

[139] J. W. Levell, M. E. Giardini, and I. D. W. Samuel. A hybrid organic semi-

conductor/silicon photodiode for efficient ultraviolet photodetection. Opt.

Express,18(4):3219–3225, Feb 2010.

[140] Mel Levy. Universal variational functionals of electron densities, first-

order density matrices, and natural spin-orbitals and solution of the v-

representability problem. Proceedings of the National Academy of Sciences,

76(12):6062–6065,1979.

[141] Liang Li, Tianyou Zhai, Yoshio Bando, and Dmitri Golberg. Recent

progress of one-dimensional zno nanostructured solar cells. Nano Energy,

1(1):91 –106,2012.

[142] Ting Liang, Yuanjing Cui, Jiancan Yu, Wenxin Lin, Yu Yang, and Guodong

Qian. Large nonlinear optical activity from hybrid inorganicâ ˘

A¸Sorganic

films with fluorinated benzene as isolation group. Thin Solid Films,

544(0):407 –411,2013. The 6th International Conference on Technological

Advances of Thin Films and Surface Coatings.

[143] M. Lindberg and S. W. Koch. Effective bloch equations for semiconductors.

Phys. Rev. B,38:3342–3350, Aug 1988.

[144] Carlos Lizandara Pueyo, Stephan Siroky, Steve Landsmann, Maurits W. E.

van den Berg, Markus R. Wagner, Juan S. Reparaz, Axel Hoffmann, and

Sebastian Polarz. Molecular precursor route to a metastable form of zinc

oxide. Chemistry of Materials,22(14):4263–4270,2010.

[145] F. London. Zur theorie und systematik der molekularkräfte. Zeitschrift für

Physik,63(3-4):245–279,1930.

[146] F. London. The general theory of molecular forces. Trans. Faraday Soc.,

33:8b–26,1937.

[147] D. C. Look, D. C. Reynolds, C. W. Litton, R. L. Jones, D. B. Eason, and

G. Cantwell. Characterization of homoepitaxial p-type zno grown by

molecular beam epitaxy. Applied Physics Letters,81(10):1830–1832,2002.

[148] D.C. Look. Recent advances in zno materials and devices. Materials Science

and Engineering: B,80(1â˘

A¸S3):383 –387,2001.

[149] J. L. Lyons, A. Janotti, and C. G. Van de Walle. Why nitrogen cannot lead

to p-type conductivity in zno. Applied Physics Letters,95(25):–, 2009.

[150] M. E. Madjet, A. Abdurahman, and T. Renger. Intermolecular coulomb

couplings from ab initio electrostatic potentials:â ˘

AL’ application to optical

transitions of strongly coupled pigments in photosynthetic antennae and

reaction centers. The Journal of Physical Chemistry B,110(34):17268–17281,

2006.

[151] Georg K.H. Madsen and David J. Singh. Boltztrap. a code for calculat-

ing band-structure dependent quantities. Computer Physics Communications,

175(1):67 –71,2006.

[152] Ermin Malic, Heiko Appel, Oliver T. Hofmann, and Angel Rubio. Energy-

transfer in porphyrin- functionalized graphene. physica status solidi (b),

251(12):2495–2498,2014.

[153] Ermin Malic, Heiko Appel, Oliver T. Hofmann, and Angel Rubio. Förster-

induced energy transfer in functionalized graphene. The Journal of Physical

Chemistry C,118(17):9283–9289,2014.

[154] M. Manninen, R. Nieminen, and P. Hautojärvi. Phys. Rev. B,12:4012,1975.

[155] M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, and D. G. Truhlar.

Consistent van der waals radii for the whole main group. J. Phys. Chem. A,

113:5806–5812,2009.

[156] A Marek, V Blum, R Johanni, V Havu, B Lang, T Auckenthaler, A Heinecke,

H-J Bungartz, and H Lederer. The elpa library: scalable parallel eigenvalue

solutions for electronic structure theory and computational science. Journal

of Physics: Condensed Matter,26(21):213201,2014.

[157] M. A. L. Marques and E. K. U. Gross. Time-Dependent Density Functional

Theory. Annual Review of Physical Chemistry,55(1):427–455, January 2004.

[158] Miguel A. L. Marques, Julien Vidal, Micael J. T. Oliveira, Lucia Reining,

and Silvana Botti. Density-based mixing parameter for hybrid functionals.

Phys. Rev. B,83:035119, Jan 2011.

[159] Y. Martynova, B.-H. Liu, M.E. McBriarty, I.M.N. Groot, M.J. Bedzyk,

S. Shaikhutdinov, and H.-J. Freund. Co oxidation over zno films on pt(111)

at near-atmospheric pressures. Journal of Catalysis,301(0):227 –232,2013.

[160] W.F. McDonough and S. s. Sun. The composition of the earth. Chemical

Geology,120(3â˘

A¸S4):223 –253,1995. Chemical Evolution of the Mantle.

[161] B. Meyer. First-principles study of the polar o-terminated zno surface

in thermodynamic equilibrium with oxygen and hydrogen. Phys. Rev. B,

69:045416, Jan 2004.

[162] B. Meyer and Dominik Marx. Density-functional study of the structure and

stability of zno surfaces. Phys. Rev. B,67:035403, Jan 2003.

[163] Kazunori Minegishi, Yasushi Koiwai, Yukinobu Kikuchi, Koji Yano,

Masanobu Kasuga, and Azuma Shimizu. Growth of p-type zinc oxide

films by chemical vapor deposition. Japanese Journal of Applied Physics,

36(11A):L1453,1997.

[164] Mark A. Mitchnick, David Fairhurst, and Sheldon R. Pinnell. Microfine

zinc oxide (z-cote) as a photostable uva/uvb sunblock agent. Journal of the

American Academy of Dermatology,40(1):85 –90,1999.

[165] Amir Moezzi, Andrew M. McDonagh, and Michael B. Cortie. Zinc oxide

particles: Synthesis, properties and applications. Chemical Engineering Jour-

nal,185â˘

A¸S186(0):1–22,2012.

[166] Peter J. Mohr, Barry N. Taylor, and David B. Newell. Codata recom-

mended values of the fundamental physical constants: 2010*. Rev. Mod.

Phys.,84:1527–1605, Nov 2012.

[167] Nikolaj Moll, Yong Xu, Oliver T Hofmann, and Patrick Rinke. Stabilization

of semiconductor surfaces through bulk dopants. New Journal of Physics,

15(8):083009,2013.

[168] F. A. Momany. Determination of partial atomic charges from ab initio

molecular electrostatic potentials. application to formamide, methanol, and

formic acid. J. Chem. Phys.,82:592–601,1978.

[169] Benjamin J. Morgan. Preferential stability of the d-bct phase in zno thin

films. Phys. Rev. B,80:174105, Nov 2009.

[170] Benjamin J. Morgan. First-principles study of epitaxial strain as a method

of stabilization in zno, zns, and cds. Phys. Rev. B,82:153408, Oct 2010.

[171] Benjamin J. Morgan and Paul A. Madden. A molecular dynamics study of

structural relaxation in tetrahedrally coordinated nanocrystals. Phys. Chem.

Chem. Phys.,9:2355–2361,2007.

[172] O. F. Mossotti. Mem. di mathem. e fisica in Modena,24(11):49,1850.

[173] R.S. Mulliken. J. Chem. Phys.,23:1833,1955.

[174] F. D. Murnaghan. The compressibility of media under extreme pressures.

Proceedings of the National Academy of Sciences,30(9):244–247,1944.

[175] R. Naumann d’Alnoncourt, X. Xia, J. Strunk, E. Loffler, O. Hinrichsen, and

M. Muhler. The influence of strongly reducing conditions on strong metal-

support interactions in cu/zno catalysts used for methanol synthesis. Phys.

Chem. Chem. Phys.,8:1525–1538,2006.

[176] Antonio Alvaro Ranha Neves, Andrea Camposeo, Roberto Cingolani, and

Dario Pisignano. Interaction scheme and temperature behavior of energy

transfer in a light-emitting inorganic-organic composite system. Advanced

Functional Materials,18(5):751–757,2008.

[177] R. Nieminen. J. Phys. F,7:375,1977.

[178] NIST. http://kinetics.nist.gov/janaf.

[179] Jorge Nocedal and Stephen J. Wright. Numerical optimization. Springer, 2.

edition, 2006.

[180] Fernando Nogueira, Alberto Castro, and Miguel Marques. A Tutorial on

Density Functional Theory A Primer in Density Functional Theory. In

Carlos Fiolhais, Fernando Nogueira, and Miguel A. L. Marques, editors, A

Primer in Density Functional Theory, volume 620 of Lecture Notes in Physics,

chapter 6, pages 218–256. Springer Berlin / Heidelberg, Berlin, Heidelberg,

June 2003.

[181] Claudine Noguera and Jacek Goniakowski. Polarity in oxide nano-objects.

Chemical Reviews,113(6):4073–4105,2013. PMID: 23206162.

[182] Wolfgang Nolting. Grundkurs Theoretische Physik. 5/1, Quantenmechanik -

Grundlagen. Springer, 2009.

[183] Fumiyasu Oba, Atsushi Togo, Isao Tanaka, Joachim Paier, and Georg

Kresse. Defect energetics in zno: A hybrid hartree-fock density functional

study. Phys. Rev. B,77:245202, Jun 2008.

[184] George A. Olah, Alain Goeppert, and G. K. Surya Prakash. Introduction,

pages 1–10. Wiley-VCH Verlag GmbH & Co. KGaA, 2009.

[185] Inchem (International Programme on Chemical Safety). Health and safety

guides (hsg).

[186] Giovanni Onida, Lucia Reining, and Angel Rubio. Electronic excitations:

density-functional versus many-body green’s-function approaches. Rev.

Mod. Phys.,74:601–659, Jun 2002.

[187] Ü. Özgür, Ya. I. Alivov, C. Liu, A. Teke, M. A. Reshchikov, S. Dogan,

V. Avrutin, S.-J. Cho, and H. Morkoc. A comprehensive review of zno

materials and devices. Journal of Applied Physics,98(4):–, 2005.

[188] Ü. Özgür, V. Avrutin, and H. Morkoc. Chapter 16 - zinc oxide materials

and devices grown by mbe. In Mohamed Henini, editor, Molecular Beam

Epitaxy, pages 369 –416. Elsevier, Oxford, 2013.

[189] Gianfranco Pacchioni. Two-dimensional oxides: Multifunctional materials

for advanced technologies. Chemistry â ˘

A¸S A European Journal,18(33):10144–

10158,2012.

[190] Nagarajan Padmavathy and Rajagopalan Vijayaraghavan. Enhanced bioac-

tivity of zno nanoparticlesâ ˘

Aˇ

Tan antimicrobial study. Science and Technology

of Advanced Materials,9(3):035004,2008.

[191] Joachim Paier, Martijn Marsman, and Georg Kresse. Dielectric properties

and excitons for extended systems from hybrid functionals. Phys. Rev. B,

78:121201, Sep 2008.

[192] Q. Pan, B.H. Liu, M.E. McBriarty, Y. Martynova, I.M.N. Groot, S. Wang,

M.J. Bedzyk, S. Shaikhutdinov, and H.-J. Freund. Reactivity of ultra-thin

zno films supported by ag(111) and cu(111): A comparison to zno/pt(111).

Catalysis Letters,144(4):648–655,2014.

[193] C. H. Park, S. B. Zhang, and Su-Huai Wei. Origin of p-type doping diffi-

culty in zno: The impurity perspective. Phys. Rev. B,66:073202, Aug 2002.

[194] Qing Peng, AmirR. Zamiri, Wei Ji, and Suvranu De. Elastic properties of

hybrid graphene/boron nitride monolayer. Acta Mechanica,223(12):2591–

2596,2012.

[195] J. P. Perdew and A. Zunger. Self-interaction correction to density-functional

approximations for many-electron systems. Phys. Rev. B,23:5048–5079,

1981.

[196] John P. Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized gradi-

ent approximation made simple. Phys. Rev. Lett.,77:3865–3868, Oct 1996.

[197] John P. Perdew, J. A. Chevary, S. H. Vosko, Koblar A. Jackson, Mark R.

Pederson, D. J. Singh, and Carlos Fiolhais. Atoms, molecules, solids, and

surfaces: Applications of the generalized gradient approximation for ex-

change and correlation. Phys. Rev. B,46:6671–6687, Sep 1992.

[198] John P. Perdew, J. A. Chevary, S. H. Vosko, Koblar A. Jackson, Mark R.

Pederson, D. J. Singh, and Carlos Fiolhais. Atoms, molecules, solids, and

surfaces: Applications of the generalized gradient approximation for ex-

change and correlation. Phys. Rev. B,46:6671–6687, Sep 1992.

[199] John P. Perdew and Yue Wang. Accurate and simple analytic representation

of the electron-gas correlation energy. Phys. Rev. B,45:13244–13249, Jun

1992.

[200] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A.

Constantin, X. Zhou, and K. Burke. Phys. Rev. Lett.,100:136406,2008.

[201] Roberto Peverati and Donald G. Truhlar. Quest for a universal density

functional: the accuracy of density functionals across a broad spectrum of

databases in chemistry and physics. Philosophical Transactions of the Royal So-

ciety of London A: Mathematical, Physical and Engineering Sciences,372(2011),

2014.

[202] Alain J. Phares. The occupation statistics for indistinguishable dumbbells

on a rectangular lattice space. i. Journal of Mathematical Physics,25(6):1756–

1770,1984.

[203] Alain J. Phares and Francis J. Wunderlich. Monomer adsorption on a

square lattice with first- and second-neighbor interactions. Phys. Rev. E,

55:2403–2408, Mar 1997.

[204] J. Philip, A. Punnoose, B. I. Kim, K. M. Reddy, S. Layne, J. O. Holmes,

B. Satpati, P. R. LeClair, T. S. Santos, and J. S. Moodera. Carrier-controlled

ferromagnetism in transparent oxide semiconductors. Nat Mater,5(4):298–

304,2006.

[205] M. Pope, H. P. Kallmann, and P. Magnante. Electroluminescence in organic

crystals. The Journal of Chemical Physics,38(8):2042–2043,1963.

[206] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P.

Flannery. Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cam-

bridge University Press, New York, NY, USA, 3edition, 2007.

[207] Frank T. Prochaska and Lester Andrews. Vibrationâ ˘

A¸Srotational and pure

rotational laserâ ˘

A¸Sraman spectra of h2, d2, and hd in matrices at 12 k. The

Journal of Chemical Physics,67(3):1139–1143,1977.

[208] L. Pugazhenthy. Zinc Handbook: Properties, Processing, and Use In Design.

MECHANICAL ENGINEERING. Taylor & Francis, 1991.

[209] P. Pulay. Convergence acceleration of iterative sequences. the case of scf

iteration. Chem. Phys. Lett.,73:393–398,1980.

[210] Bipul Rakshit and Priya Mahadevan. Stability of the bulk phase of layered

zno. Phys. Rev. Lett.,107:085508, Aug 2011.

[211] R. Ramprasad, H. Zhu, Patrick Rinke, and Matthias Scheffler. New perspec-

tive on formation energies and energy levels of point defects in nonmetals.

Phys. Rev. Lett.,108:066404, Feb 2012.

[212] Ian D. Rees, Kay L. Robinson, Andrew B. Holmes, Carl R. Towns, and

Richard O’Dell. Recent developments in light-emitting polymers. MRS

Bulletin,27:451–455,6 2002.

[213] Sivan Refaely-Abramson, Roi Baer, and Leeor Kronik. Fundamental and

excitation gaps in molecules of relevance for organic photovoltaics from an

optimally tuned range-separated hybrid functional. Phys. Rev. B,84:075144,

Aug 2011.

[214] K. Reuter, C. Stampfl, and M. Scheffler. Ab initio atomistic thermodynamics

and statistical mechanics of surface properties and functions. Springer Berlin

Heidelberg, 2005.

[215] J. Robertson and S. J. Clark. Limits to doping in oxides. Phys. Rev. B,

83:075205, Feb 2011.

[216] Michael Rohlfing and Steven G. Louie. Electron-hole excitations and optical

spectra from first principles. Phys. Rev. B,62:4927–4944, Aug 2000.

[217] F. Romá and A.J. Ramirez-Pastor. Configurational entropy of adsorbed

particles on two-dimensional heterogeneous surfaces. Physica A: Statistical

Mechanics and its Applications,328(3â˘

A¸S4):513 –524,2003.

[218] Fausto Rossi and Tilmann Kuhn. Theory of ultrafast phenomena in pho-

toexcited semiconductors. Rev. Mod. Phys.,74:895–950, Aug 2002.

[219] R. S. Rowland and R. Taylor. Intermolecular nonbonded contact distances

in organic crystal structures: Comparison with distances expected from van

der waals radii. J. Phys. Chem.,100:7384–7391,1996.

[220] Victor G. Ruiz, Wei Liu, Egbert Zojer, Matthias Scheffler, and Alexandre

Tkatchenko. Density-Functional Theory with Screened van der Waals In-

teractions for the Modeling of Hybrid Inorganic-Organic Systems. Phys.

Rev. Lett.,108(14):146103,2012.

[221] Pierre Van Rysselberghe. Remarks concerning the clausius-mossotti law.

The Journal of Physical Chemistry,36(4):1152–1155,1931.

[222] Y.R Ryu, S Zhu, D.C Look, J.M Wrobel, H.M Jeong, and H.W White. Syn-

thesis of p-type zno films. Journal of Crystal Growth,216(1â˘

A¸S4):330 –334,

2000.

[223] J. J. Sakurai. Modern Quantum Mechanics (Revised Edition). Addison Wesley,

1edition, September 1993.

[224] E. E. Salpeter and H. A. Bethe. A relativistic equation for bound-state

problems. Phys. Rev.,84:1232–1242, Dec 1951.

[225] A. Savin, F. Colonna, and R. Pollet. Adiabatic connection approach to den-

sity functional theory of electronic systems. International Journal of Quantum

Chemistry,93(3):166–190,2003.

[226] M. Scheffler and J. Dabrowski. Philos. Mag. A,58,1988.

[227] R. Schennach, F. Weber, M. Piffl, G. Weirum, and S. Surnev. Growth and

reactivity of Zn and ZnO on Pd(111). Surface Engeneering,28(2):87–90, MAR

2012.

[228] A. Schleife, F. Fuchs, J. Furthmüller, and F. Bechstedt. First-principles

study of ground- and excited-state properties of MgO, ZnO, and CdO poly-

morphs. Phys. Rev. B,73:245212, Jun 2006.

[229] Vadim Schott, Harald Oberhofer, Alexander Birkner, Mingchun Xu,

Yuemin Wang, Martin Muhler, Karsten Reuter, and Christof Wöll. Chemi-

cal activity of thin oxide layers: Strong interactions with the support yield

a new thin-film phase of zno. Angewandte Chemie International Edition,

52(45):11925–11929,2013.

[230] E. Schrödinger. An undulatory theory of the mechanics of atoms and

molecules. Phys. Rev.,28:1049–1070, Dec 1926.

[231] Franz Schwabl. Advanced Quantum Mechanics. Springer, August 2005.

[232] G.E. Scuseria and V.N. Staroverov. Progress in the development of

exchange-correlation functionals. In C.E. Dykstra, G. Frenking, K.S. Kim,

and G.E. Scuseria, editors, Theory and Applications of Computational Chem-

istry: The First 40 Years, chapter 24. Elsevier, Amsterdam, 2005. The param-

eters for "VWN"-LDA as implemented in the Gaussian code are given in

Table 1.

[233] Michele Sessolo and Henk J. Bolink. Hybrid organicâ ˘

A¸Sinorganic light-

emitting diodes. Advanced Materials,23(16):1829–1845,2011.

[234] S. Shaikhutdinov and H. J. Freund. Ultrathin Oxide Films on Metal Sup-

ports: Structure-Reactivity Relations. 63:619–633,2012.

[235] James R. Sheats, Homer Antoniadis, Mark Hueschen, William Leonard, Jeff

Miller, Ron Moon, Daniel Roitman, and Andrew Stocking. Organic electro-

luminescent devices. Science,273(5277):884–888,1996.

[236] Joseph Shinar and Vadim Savvateev. Introduction to organic light-emitting

devices. In Joseph Shinar, editor, Organic Light-Emitting Devices, pages 1–41.

Springer New York, 2004.

[237] Akitoshi Shiotari, Bo Hong Liu, Simon Jaekel, Leonhard Grill, Shamil

Shaikhutdinov, Hans-Joachim Freund, Martin Wolf, and Takashi Kumagai.

Local characterization of ultrathin zno layers on ag(111) by scanning tun-

neling microscopy and atomic force microscopy. The Journal of Physical

Chemistry C,118(47):27428–27435,2014.

[238] E. Sigfridsson and U. Ryde. Atomic charges from the electrostatic potential

and moments. J. Comput. Chem.,19(4), 1998.

[239] U. C. Singh and P. A. Kollman. An approach to computing electrostatic

charges for molecules. J. Comput. Chem.,5:129–145,1984.

[240] J. C. Slater. The electronic structure of metals. Rev. Mod. Phys.,6:209–280,

Oct 1934.

[241] Sérgio Filipe Sousa, Pedro Alexandrino Fernandes, and Maria Joao Ramos.

General performance of density functionals. The Journal of Physical Chem-

istry A,111(42):10439–10452,2007.

[242] Michelle J.S. Spencer, Kester W.J. Wong, and Irene Yarovsky. Density func-

tional theory modelling of and surfaces: Structure, properties and adsorp-

tion of {N2O}. Materials Chemistry and Physics,119(3):505 –514,2010.

[243] Tamar Stein, Helen Eisenberg, Leeor Kronik, and Roi Baer. Fundamen-

tal gaps in finite systems from eigenvalues of a generalized kohn-sham

method. Phys. Rev. Lett.,105:266802, Dec 2010.

[244] P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch. Ab initio cal-

culation of vibrational absorption and circular dichroism spectra using den-

sity functional force fields. The Journal of Physical Chemistry,98(45):11623–

11627,1994.

[245] Anthony Stone. The Theory of Intermolecular Forces. Oxford University Press,

2013.

[246] R.E. Stratmann, G.E. Scuseria, and M.J. Frisch. Chem. Phys. Lett.,257:213,

1996.

[247] Attila Szabo and Neil S. Ostlund. Modern quantum chemistry : introduction to

advanced electronic structure theory. Dover Publications, Mineola (N.Y.), 1996.

PremiÃ´

lre édition Macmillan 1982. Edition revue chez McGraw-Hill 1989.

[248] James D. Talman. Numerical methods for multicenter integrals for numeri-

cally defined basis functions applied in molecular calculations. International

Journal of Quantum Chemistry,93(2):72–90,2003.

[249] P W Tasker. The stability of ionic crystal surfaces. Journal of Physics C: Solid

State Physics,12(22):4977,1979.

[250] J. Tersoff and D. R. Hamann. Theory of the scanning tunneling microscope.

Phys. Rev. B,31:805–813, Jan 1985.

[251] The HDF Group. Hierarchical Data Format, version 5,1997-NNNN.

http://www.hdfgroup.org/HDF5/.

[252] L. H. Thomas. The calculation of atomic fields. Mathematical Proceedings of

the Cambridge Philosophical Society,23:542–548,1 1927.

[253] Alexandre Tkatchenko, Lorenz Romaner, Oliver T. Hofmann, Egbert Zojer,

Claudia Ambrosch-Draxl, and Matthias Scheffler. Van der waals interac-

tions between organic adsorbates and at organic/inorganic interfaces. MRS

Bulletin,35:435–442,6 2010.

[254] Alexandre Tkatchenko and Matthias Scheffler. Accurate Molecular Van

Der Waals Interactions from Ground-State Electron Density and Free-Atom

Reference Data. Phys. Rev. Lett.,102(7):073005,2009.

[255] M. Topsakal, S. Cahangirov, E. Bekaroglu, and S. Ciraci. First-principles

study of zinc oxide honeycomb structures. Phys. Rev. B,80:235119, Dec

2009.

[256] Alessandro Troisi. Charge transport in high mobility molecular semicon-

ductors: classical models and new theories. Chem. Soc. Rev.,40:2347–2358,

2011.

[257] K. Y. Tse, D. Liu, and J. Robertson. Electronic and atomic structure of

metal-hfo2interfaces. Phys. Rev. B,81:035325, Jan 2010.

[258] Z. C. Tu and X. Hu. Elasticity and piezoelectricity of zinc oxide crystals,

single layers, and possible single-walled nanotubes. Phys. Rev. B,74:035434,

Jul 2006.

[259] C. Tusche, H. L. Meyerheim, and J. Kirschner. Observation of depolarized

zno(0001) monolayers: Formation of unreconstructed planar sheets. Phys.

Rev. Lett.,99:026102, Jul 2007.

[260] E. K. U. Gross und E. Runge. Vielteilchentheorie. B. G. Teubner, Stuttgart,

1986.

[261] Markus Valtiner, Mira Todorova, Guido Grundmeier, and Jörg Neugebauer.

Temperature stabilized surface reconstructions at polar zno(0001). Phys.

Rev. Lett.,103:065502, Aug 2009.

[262] Markus Valtiner, Mira Todorova, and Jörg Neugebauer. Hydrogen adsorp-

tion on polar zno(0001)-zn: Extending equilibrium surface phase diagrams

to kinetically stabilized structures. Phys. Rev. B,82:165418, Oct 2010.

[263] H. van Hove and R. Leysen. Leed study on the polar surfaces of zno.

physica status solidi (a),9(1):361–367,1972.

[264] Daniel Vanmaekelbergh and Lambert K. van Vugt. Zno nanowire lasers.

Nanoscale,3:2783–2800,2011.

[265] Eike Verdenhalven, Andreas Knorr, Marten Richter, Bjoern Bieniek, and

Patrick Rinke. Theory of optical excitations in dipole-coupled hybrid

molecule-semiconductor layers: Coupling of a molecular resonance to semi-

conductor continuum states. Phys. Rev. B,89:235314, Jun 2014.

[266] S.H. Vosko, L. Wilk, and M. Nusair. Can. J. Phys.,58:1200,1980.

[267] Roman Wahl, Jeppe V. Lauritsen, Flemming Besenbacher, and Georg

Kresse. Stabilization mechanism for the polar zno(0001)-o surface. Phys.

Rev. B,87:085313, Feb 2013.

[268] Vei Wang, Deming Ma, Wanli Jia, and Weili Ji. Structural and electronic

properties of hexagonal zno: A hybrid functional study. Solid State Commu-

nications,152(22):2045 –2048,2012.

[269] Kenji Watanabe, Takashi Taniguchi, and Hisao Kanda. Direct-bandgap

properties and evidence for ultraviolet lasing of hexagonal boron nitride

single crystal. Nat Mater,3(6):404–409, Jun 2004.

[270] G. Weirum, G. Barcaro, A. Fortunelli, F. Weber, R. Schennach, S. Surnev,

and F. P. Netzer. Growth and surface structure of zinc oxide layers on a

pd(111) surface. The Journal of Physical Chemistry C,114(36):15432–15439,

2010.

[271] G. Weirum, M. Kratzer, H. P. Koch, A. Tamtögl, J. Killmann, I. Bako, A. Win-

kler, S. Surnev, F. P. Netzer, and R. Schennach. Growth and desorption ki-

netics of ultrathin zn layers on pd(111). The Journal of Physical Chemistry C,

113(22):9788–9796,2009.

[272] Gunther Weirum. Structural investigation of zinc-palladium and zinc oxide-

palladium model catalyst surfaces . PhD thesis.

[273] E. Wigner. On the interaction of electrons in metals. Phys. Rev.,46:1002–

1011, Dec 1934.

[274] D. Wolf, P. Keblinski, S. R. Phillpot, and J. Eggebrecht. Exact method for the

simulation of coulombic systems by spherically truncated, pairwise râ´

LŠ1

summation. The Journal of Chemical Physics,110(17):8254–8282,1999.

[275] Ch Wöll. Hydrogen adsorption on metal oxide surfaces: a reinvestigation

using he-atom scattering. Journal of Physics: Condensed Matter,16(29):S2981,

2004.

[276] Christof Wöll. The chemistry and physics of zinc oxide surfaces. Progress

in Surface Science,82(2â˘

A¸S3):55 –120,2007.

[277] D M Wood and A Zunger. A new method for diagonalising large matrices.

Journal of Physics A: Mathematical and General,18(9):1343,1985.

[278] Dangxin Wu, M. G. Lagally, and Feng Liu. Stabilizing Graphitic Thin Films

of Wurtzite Materials by Epitaxial Strain. Phys. Rev. Lett.,107(23):236101,

2011.

[279] F. X. Xiu, Z. Yang, L. J. Mandalapu, D. T. Zhao, J. L. Liu, and W. P. Bey-

ermann. High-mobility sb-doped p-type zno by molecular-beam epitaxy.

Applied Physics Letters,87(15):–, 2005.

[280] Qimin Yan, Patrick Rinke, M Winkelnkemper, A Qteish, D Bimberg,

Matthias Scheffler, and Chris G Van de Walle. Band parameters and strain

effects in zno and group-iii nitrides. Semiconductor Science and Technology,

26(1):014037,2011.

[281] Y. Yang, G. A. Turnbull, and I. D. W. Samuel. Hybrid optoelectronics: A

polymer laser pumped by a nitride light-emitting diode. Applied Physics

Letters,92(16):–, 2008.

[282] D. Zagorac, J. C. Schön, J. Zagorac, and M. Jansen. Prediction of structure

candidates for zinc oxide as a function of pressure and investigation of

their electronic properties. Phys. Rev. B,89:075201, Feb 2014.

[283] Stefan Zander, Edward L. Kunkes, Manfred E. Schuster, Julia Schumann,

Gisela Weinberg, Detre Teschner, Nikolas Jacobsen, Robert Schlögl, and

Malte Behrens. The role of the oxide component in the development of

copper composite catalysts for methanol synthesis. Angewandte Chemie In-

ternational Edition,52(25):6536–6540,2013.

[284] A. Zangwill. University Press, Cambridge, 1988.

[285] E. Zaremba and W. Kohn. Van der waals interaction between an atom and

a solid surface. Phys. Rev. B,13:2270–2285, Mar 1976.

[286] Guo-Xu Zhang, A. Tkatchenko, J. Paier, H. Appel, and M. Scheffler. van

der Waals Interactions in Ionic and Semiconductor Solids. Phys. Rev. Lett.,

107:245501,2011.

[287] Qifeng Zhang, Christopher S. Dandeneau, Xiaoyuan Zhou, and Guozhong

Cao. Zno nanostructures for dye-sensitized solar cells. Advanced Materials,

21(41):4087–4108,2009.

[288] Y. Zhang and W. Yang. Phys. Rev. Lett.,80:890,1998.

[289] Huaxing Zhou, Liqiang Yang, and Wei You. Rational design of high

performance conjugated polymers for organic solar cells. Macromolecules,

45(2):607–632,2012.