Surfaces and Extended Defects in Wurtzite GaN
zur Erlangung des akademischen Grades
do ctor rerum naturalium (Dr. rer. nat.)
vom
Fachbereich Physik
der
Universit
at Gesamthochschule Paderborn
genehmigte
Dissertation
von Dipl. Math. Joachim Elsner
Eingereicht am 24. September 1998
Contents
1 Intro duction 3
1.1 Into the Blue: GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Density Functional Theory 7
2.1 The Kohn{Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The lo cal density approximation (LDA) . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 AIMPRO Metho dology 13
3.1 Pseudop otentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Gaussian orbitals as basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Application of AIMPROtoGaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4DFTB Metho dology 17
4.1 Density-functional basis of TB-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Standard{DFTB formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 SCC{DFTB formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 Performance of standard{DFTB and SCC{DFTB . . . . . . . . . . . . . . . . . . . . 27
4.5 Application of standard{DFTB and SCC{DFTB to GaN . . . . . . . . . . . . . . . . 27
5 Formation Energies of Surfaces and Defects 29
5.1 Formation energies of charge neutral GaN structures . . . . . . . . . . . . . . . . . . 29
5.2 Formation energies of structures with impurities . . . . . . . . . . . . . . . . . . . . 31
5.2.1 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2.2 Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 Formation energies of charged structures . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.4 Applications to surfaces and defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6 Nonp olar GaN Surfaces 37
6.1 The electron counting rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Stoichiometric (10
10) and (11
20) surfaces . . . . . . . . . . . . . . . . . . . . . . . . 38
6.3 Non{stoichiometric (10
10) and (11
20) surfaces . . . . . . . . . . . . . . . . . . . . . 42
6.4 Oxygen at (10
10) and (11
20) surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1
2
CONTENTS
7 Threading Dislo cations and Nanopip es 47
7.1 Screw dislo cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.1.1 Full core screw dislo cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.1.2 Screw dislo cations with a narrow op ening . . . . . . . . . . . . . . . . . . . . 51
7.2 The formation of nanopip es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 Threading edge dislo cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.4 Deep acceptors trapp ed at edge dislo cations . . . . . . . . . . . . . . . . . . . . . . . 58
7.4.1 Benchmark calculations for O in GaN . . . . . . . . . . . . . . . . . . . . . . 59
7.4.2 O related defects in the dislo cation stress eld . . . . . . . . . . . . . . . . . 60
8 Domain Boundaries 63
8.1 Brief review of domain b oundaries on
f
10
10
g
planes . . . . . . . . . . . . . . . . . . 64
8.2 Domain b oundaries on
f
11
20
g
planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9 Reconstructions of (0001)/(000
1) Surfaces 69
9.1 Ga, N and H terminated surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.2 The chemisorption of oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10 Conclusions 83
10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.2 Outlo ok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A Expressions for SCC{DFTB 85
A.1 Analytical evaluation of
IJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.2 Numerical evaluation of
IJ
in p erio dic systems . . . . . . . . . . . . . . . . . . . . . 88
B (SCC)-DFTB Parameters 91
C Range of the Chemical Potentials within SCC{DFTB 93
C.1 The elemental chemical p otentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
C.2 The electro-chemical p otential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
D Structural Mo delling of Surfaces and Defects 95
D.1 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
D.2 Point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
D.3 Line defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
D.4 Domain b oundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Bibliography 99
Danksagung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter 1
Intro duction
1.1 Into the Blue: GaN
Semiconducting lasers on the basis of gallium arsenide (GaAs) and indium phosphide (InP) emit
light in the red{infrared sp ectrum. These lasers have found many applications ranging from data
storage on compact discs (CD) and data transmission via optical bres to medical diagnostics and
surgery.
Also blue and green lasers and laser dio des (LDs) are highly desirable. With their shorter wave{
lengths they would allow to reduce the storage space on CDs b ecause the data can b e "written" in
a more compact form. Moreover, blue and green lasers are also exp ected to b e employed in medical
diagnostics: the red background colour of blo o d oated tissue makes the use of the available red
laser light rather dicult whereas green or blue lightwould b e much easier to recognise. In terms
of market value, a very imp ortantby{pro duct related to the development of semiconducting lasers
are light emitting dio des (LEDs). Due to their high luminescence eciency,quickresp onse time
and long lifetime LEDs are an attractive alternativetoconventional sources of light: LEDs consume
10% of the energy of conventional light bulbs and have
10
3
times longer lifetimes. As blue and
green b elong to the three primary colours (red, green, blue), blue and green LEDs are required to
repro duce the full colour sp ectrum and achieve white light.
Having these applications in mind, many semiconductor companies started research for materials
which can provide blue and green light. For a review see the textb o ok by Nakamura and Fasol [1].
Light emission in the blue sp ectrum requires band gaps of
3 eV. Only materials with direct band
gaps are suitable to pro duce bright light b ecause indirect band gaps require also phonons for optical
transitions which reduce the luminescence eciency. During the 80's, it was thought that lattice
matching substrates were essential to grow materials in a stress{free manner. Fig. 1.1 shows the
band gaps of comp ound semiconductors in dep endence on the lattice constant.
With direct band gaps b etween
2
:
0 and
4
:
5 eV and lattice constants similar to that of GaAs, I I{
VI comp ounds such as ZnSSe were the favourite materials for a long time. However, although high
quality I I{VI materials were grown with densities of crystal defects below 10
4
/cm
2
,these state-
of-the-art materials still show severe stability problems and degrade within hours thus making
3
4
CHAPTER 1. INTRODUCTION
GaP
GaAs
Bandgap Energy (eV)
AlN
InN
SiC AlP
InP
CdSe
ZnSe
MgSe
MgS
: Indirect Bandgap
: Direct Bandgap
7.0
6.0
5.0
4.0
3.0
2.0
1.0
3.0 4.0 5.0 6.0
AlAs
GaN ZnS
Figure 1.1: Band gap energy versus lattice constantofvarious materials for visible emission [1]. The
lines indicate linear scaling of band gap energy and lattice constant for ternary comp ounds.
commercial applications imp ossible. It is generally thoughtthat the rapid degradation is due to
crystal defects b ecause I I{VI materials are very weakly b onded so that one defect can cause the
propagation of other defects leading to failure of the devices even if the densityof defects was
lowat the b eginning. Another wide band material is SiC. However, SiC has an indirect band gap
leading to very little brightness. Despite of their p o or p erformance, 6H{SiC blue LEDs have been
commercialised for a long time b ecause no alternative existed [2 , 3].
The remaining material group in Fig. 1.1 are group I I I{nitrides, in particular GaN which under
standard growth conditions crystallises in the wurtzite phase (
{GaN). Since group I I I{nitrides
p ossess direct band gaps ranging from
2
:
0
;
6
:
3eV, band gap engineering could lead to devices
emitting brightlightcovering the entire sp ectrum from green to UV light.
However, until recently
p
{typ e doping of GaN could not be achieved so that eciently working
devices which require
p
;
n
junctions could not be pro duced. This doping problem was overcome
byAmano
et al.
[4] and Nakamura
et al.
[5] so that the rst highly ecient light emitting devices
based on GaN could b e built and are commercially available now.
p
{typ e GaN lms are obtained
by Mg doping and annealing in an N
2
atmosphere [5].
The next ma jor problem in the developmentofhigh{quality GaN based devices concerns the mate-
rial quality. Sapphire is the most suitable substrate on which growth of
{GaN can b e p erformed.
However, sapphire has a lattice mismatch of 13% with resp ect to GaN. Therefore it is no surprise
that attempts to grow GaN directly on sapphire resulted in a huge numb er of defects which p ene-
trate the GaN epilayer (threading defects). The fabrication of light emitting devices based on these
poor quality epilayers were obviously imp ossible. In part, this problem was overcome by Akasaki
et al.
[6] in 1989 who suggested that GaN growth on top of an AlN buer layer (see Fig. 1.2)
reduces signicantly the defect density. This growth technique has been successfully adopted by
Nakamura
et al.
[1] who found the epilayer quality to b e still improved by replacing the AlN buer
layer byaGaN buer layer. In addition, Nakamura
et al.
used a mo died metal organic chemical
vap our dep osition (MOCVD) reactor where a subowof N
2
and H
2
molecules provides a uniform
distribution of the main ow gases to the substrate.
1.1. INTO THE BLUE: GAN
5
2
Sapphire (α- Al O )
buffer layer
(50 nm GaN
or AlN)
faulted zone
sound zone
GaN, threading
dislocation
3
Figure 1.2: Using a GaN (or AlN) buer layer, GaN can b e grown with a suciently high qualityon
a sapphire substrate despite of the very large lattice mismatch. The buer layer is normally grown
at lower temp eratures than the GaN device layers [6].
In spite of these improvements the density of threading defects (see Fig. 1.2) remains considerably
high (
10
9
/cm
2
in typical MOCVD grown epilayers) and gives rise to one of the most puzzling
questions ab out GaN based devices [1]:
Why in general do GaN baseddevices work in spite of the large number of defects incorporated?
The answer is not yet known, instead it seems that most of the typ es of threading defects are
harmless whereas the existence of some of them can seriously inuence the material quality by
inducing deep electronic states in the band gap. Deep states lead to parasitic comp onents in the
emission sp ectrum and, if o ccurring at high densities, render the sample useless for sophisticated
optical applications, in particular for lasers [1 ].
The most commonly observed parasitic comp onent in the GaN sp ectrum is the defect related broad
band yellow luminescence (YL). The YL is generally asso ciated with
n
{typ e material where it can
have a strong intensity varying all over the material as shown in Fig. 1.3. The precise origin of
the YL is not known and even the class of defect, e.g. pointdefect, line defect or planar defect,
resp onsible for the YL remains unclear. A characterisation of the prop erties and p ossibly the origin
of the most commonly observed defects in GaN is therefore a very imp ortant matter in order
to get an idea ab out their inuence on the material qualities and give growers hints on how to
avoid "dangerous sp ecies". A variety of p oint defects and related defect complexes in GaN have
already b een investigated by theoretical groups [8, 9 ] and the results can b e related to exp erimental
work [10, 11]. On the other hand, most of the frequently o ccurring extended threading defects, in
particular dislo cations, have not b een studied yet. This is mainly due to the fact that a theoretical
investigation of extended defects is computationally very exp ensive since they require mo dels which
are much larger than those employed for studying p oint defects. Recently, numerically ecient
theoretical metho ds have b een implemented on parallel machines so that larger structures can now
be investigated. The present thesis will try to ll the lack of information concerning the prop erties
of the most commonly observed extended defects in wurtzite GaN. Also the interaction of these
extended defects with stable p oint defects, impurities and related defect complexes will b e explored.
Another imp ortant topic concerns the investigation of surfaces. A detailed knowledge of their prop-
erties is vital for high quality device fabrication, where in general electrically and chemically inactive
surfaces are desired to form junctions and contacts. Therefore the stabilities of intrinsic GaN sur-
faces and their p ossible passivation by means of oxygen will be studied. These investigations are
6
CHAPTER 1. INTRODUCTION
Figure 1.3: CL sp ectra from several dierent regions of a GaN lm grown on a sapphire substrate.
There are striking variations from p oint to p oint on the sample.
C. Traeger{Cowan et al.
[7].
related to the study of extended defects b ecause some of the extended defects contain internal
surfaces. Moreover, knowledge of the surface prop erties might help to determine growth conditions
under which almost defect{ and impurity{free material can b e pro duced.
1.2 Outline of this thesis
In chapter 2, a brief summary of density functional theory which is a theoretical approach for deter-
mining total energies, structures, and to some extend electrical prop erties is given. The
AIMPRO
and
DFTB
metho ds used within this work are based on density functional theory and explained
in the following chapters. Particular emphasis is given to the extension to p erio dic systems of the
self-consistent charge
DFTB
metho d (
SCC{DFTB
)develop ed by M. Elstner for clusters. To this
end in co op eration with D. Porezag and M. Haugk, the author develop ed a new functional expres-
sion suitable to account for the energy arising from charge uctuations in extended and p erio dic
systems. A b enchmark will b e given for a parallel co de develop ed and implemented together with
M. Haugk. Chapter 5 describ es how total energies can b e transformed into formation energies which
will b e used throughout the application chapters to comment on the stabilities of the structures.
The theory is applied in chapter 6tocharacterise nonp olar wurtzite surfaces in view of the extended
defects explored in the following. Chapter 7 and 8 constitute the main part of this thesis: threading
line and planar defects are investigated in their pure form and with segregated impurities. Their
p ossible implication for the yellow luminescence is discussed.
In chapter 9 p olar surfaces corresp onding to the main growth directions are investigated. A mecha-
nism is suggested to identify the p olarity during MBE growth in dep endence on the surface recon-
structions observed. Also the adsorption of oxygen will b e discussed.
Finally,chapter 10 gives a summary of and an outlo ok on related topics in GaN.
Chapter 2
Density Functional Theory
Knowledge of the total energy of a structure is the key p oint in a theoretical investigation. Electronic
structure calculations attempt to determine the total energy of a system of nuclei and their electrons.
The structures considered in this work corresp ond to complex systems with a large numb er of degrees
of freedom. It is therefore necessary to makeanumb er of simplications.
The motion of electrons and nuclei are describ ed by the many{b o dy Schrodinger equation. Within
the Born{Opp enheimer approximation one separates the motion of the electrons from the motion
of the nuclei so that for given nuclear p ositions in a rst step one only needs to solve the many{
electron Schrodinger equation. With this solution one can in a second step calculate the forces on
the nuclei, optimise the nuclear p ositions with resp ect to the total energy and hence derive the
equilibrium geometry.
The many{electron Schrodinger equation is usually solved using either wavefunction based metho ds
such as the Hartree{Fock formalism or electron density based schemes such as density{functional
theory (DFT). The latter approach is pursued within this work. It circumvents the computationally
exp ensive asp ects of theory by treating the
electron density
with only
three degrees of freedom
as
the fundamental variable instead of the full wavefunction with 3
N
el
degrees of freedom
.
All expressions used in the theory chapters are in atomic units, i.e.
h; e; m
e
and 4
0
are set to
unity. Therefore energies are given in Hartree (1 H = 27.21 eV), lengths are given in Bohr radii (1
a
0
= 0.5292
A) and masses in multiples of the electron mass (m
e
= 9.109
10
;
31
kg). Furthermore,
R
d
r
and
R
d
r
0
are denoted by
R
and
R
0
,respectively.
2.1 Reduction of the many{electron equation into eective one{
electron (Kohn{Sham) equations
In 1964 Hohenberg and Kohn [12 ] established the basis of density{functional theory. DFT allows
to determine the ground{state energy
E
0
of an interacting electron gas in an external p otential.
Consider the many{electron problem:
H
j
'
0
(
f
r
i
g
)
i
=
E
0
j
'
0
(
f
r
i
g
)
i
:
(2.1)
7
8
CHAPTER 2. DENSITY FUNCTIONAL THEORY
H is the Hamilton op erator for N
el
electrons in the eld of N
nuc
nuclei consisting of the following
contributions: the kinetic energies of the electrons
T
e
,the Coulomb interaction between electrons
V
ee
and the Coulombinteraction b etween electrons and nuclei, denoted by a more general external
potential
V
ext
:
H
=
T
e
+
V
ee
+
V
ext
;
where:
T
e
=
N
el
X
i
;
r
2
i
2
; V
ee
=
1
2
N
el
X
i;j
i
6
=
j
1
j
r
i
;
r
j
j
and
V
ext
=
N
el
;N
nuc
X
i;I
Z
I
j
r
i
;
R
I
j
:
The many{electron problem leads to a charge density
n
0
for the ground{state:
n
0
(
r
)=
h
'
0
(
f
r
i
g
)
j
N
el
X
i
=1
(
r
;
r
i
)
j
'
0
(
f
r
i
g
)
i
:
Let us now dene the wide class of densities
A
N
=
f
n
(
r
)
j
n comes from an N{particle ground{state
g
:
Note that
A
N
contains all p ossible N{particle densities and do es not refer to any sp ecic external
potential
V
ext
. The most imp ortant features of density{functional{theory can then b e summarised
within the following two theorems:
Theorem 1 :
The ground{state energy
E
0
of an electron gas is a functional of the
ground{state charge density
n
0
(
r
):
E
0
:
A
N
!
IR
; n
!
E
0
[
n
]
:
In words: for every p ossible ground{state charge density n, there is one and only one
ground{state energy
E
0
[
n
].
Based on this theorem the energy
E
0
can b e written as a functional of the charge{density
n
0
:
E
0
[
V
ext
]=
E
0
[
V
ext
[
n
0
]] =
Z
n
0
(
r
)
V
ext
(
r
)+
F
[
n
0
]
:
Here
F
[
n
] is dened on
A
N
by Eq. (2.1) as:
F
[
n
]=
E
0
[
n
]
;
Z
n
(
r
)
V
ext
(
r
)
:
Theorem 2 :
The functional
E
0
[
V
ext
;n
] attains its minimum at
n
=
n
0
:
E
0
[
V
ext
;n
0
]
E
0
[
V
ext
;n
]
:
2.1. THE KOHN{SHAM EQUATIONS
9
In order to evaluate
E
0
[
V
ext
;n
] explicitly Kohn and Sham [13 ] prop osed to write the functional
F
[
n
]
as:
F
[
n
]=
T
s
[
n
]+
E
H
[
n
]+
E
XC
[
n
]
:
(2.2)
Here
T
s
[
n
] is the true kinetic energy of a non{interacting electron gas with density
n
.
E
H
[
n
]isthe
so{called Hartree{energy:
E
H
[
n
]=
1
2
Z Z
0
n
(
r
0
)
n
(
r
)
j
r
0
;
r
j
:
The last term in Eq. (2.2) is called exchange and correlation energy (XC{energy). In addition
to the actual exchange and correlation energy it contains contributions of the kinetic energy and
corrections arising from the self{interaction of particles:
E
XC
[
n
]=
T
e
[
n
]
;
T
s
[
n
]+
V
ee
[
n
]
;
E
H
[
n
]
:
With the decomp osition (2.2) the energy functional
E
0
[
V
ext
;n
] reads:
E
0
[
V
ext
(
n
)] =
T
s
[
n
]+
1
2
Z Z
0
n
(
r
0
)
n
(
r
)
j
r
0
;
r
j
+
Z
V
ext
(
r
)
n
(
r
) +
E
XC
[
n
]
:
(2.3)
Applying the variational{principle (Theorem 2 guarantees aminimum in
A
N
) yields a condition
for the ground{state charge density
n
0
:
T
s
n
+
V
ext
+
Z
0
n
(
r
0
)
j
r
0
;
r
j
+
V
XC
[
n
]
;
n
=
n
0
=0
:
(2.4)
is a Lagrange{multiplier, expressing particle conservation:
N
el
=
Z
n
(
r
)
:
(2.5)
V
XC
is the functional derivativeof
E
XC
and is called exchange{ and correlation p otential:
V
XC
=
E
XC
[
n
]
n
:
Eq. (2.3) and (2.4) determine the ground{state energy and the ground{state charge density.However
the functionals
T
s
[
n
],
E
XC
[
n
]and
V
XC
[
n
]in these equations are still unknown. The functionals
E
XC
[
n
]and
V
XC
[
n
] can only b e approximated. The most commonly used approximation is describ ed
in section 2.2. On the other hand,
T
s
[
n
]can be determined exactly assuming a p otential
V
e
[
n
],
for which
n
is the ground{state charge densityofasystemof non{interacting electrons. Although
this p otential
V
e
do es not always exist [14 ], it can b e assumed for most applications. Let now
j
i
i
i
=1
;::: ;N
el
denote the one{particle wavefunctions of the non{interacting electron gas describ ed
with the eective p otential
V
e
.We then have:
(
;
r
2
2
+
V
e
[
n
])
j
i
i
=
"
i
j
i
i
(2.6)
and
n
(
r
)=
occ
X
i
n
i
j
i
(
r
)
ih
i
(
r
)
j
:
(2.7)
10
CHAPTER 2. DENSITY FUNCTIONAL THEORY
The variables
n
i
are the o ccupation numb ers of the one{particle states. Assuming vanishing tem-
p erature and the system to b e in the ground{state the o ccupation numb ers are given by
n
i
=2(
"
i
;
)
:
(2.8)
Here is the Heaviside step function, i.e. (
x
)=0
; x
0and(
x
)=1
;x>
0,
is the Lagrange{
multiplier from Eq. (2.4) and corresp onds to the electron chemical p otential.
is determined by
equations (2.7) and (2.5).
Using the expansion of
n
in the one{particle states
j
i
i
the kinetic energy
T
s
[
n
] can b e written in
the simple form:
T
s
[
n
]=
occ
X
i
n
i
h
i
j;
r
2
2
j
i
i
:
(2.9)
Thus the energy functional
E
0
[
n; V
ext
] nally reads:
E
0
[
n; V
ext
] =
occ
X
i
n
i
h
i
j;
r
2
2
j
i
i
+
1
2
Z Z
0
n
(
r
0
)
n
(
r
)
j
r
0
;
r
j
(2.10)
+
Z
V
ext
(
r
)
n
(
r
) +
E
XC
[
n
]
:
Of course, the one{particle states
j
i
i
are related to
n
via (2.7).
The ground{state density
n
0
for which the functional (2.10) attains the minimum can b e calculated
using Hohenb erg and Kohn's second theorem. To this end we consider a system of non{interacting
electrons which in an external p otential
V
e
[
n
0
] has the same ground{state density
n
0
as the system
of interacting electrons. Following Hohenb erg{Kohn's second theorem the functional of the non{
interacting electron gas attains its minimum at
n
0
, to o, i.e. the functional is stationary at
n
0
:
E
s;
0
[
n; V
e
[
n
0
]] =
T
s
[
n
]+
Z
V
e
[
n
0
](
r
)
n
(
r
) (2.11)
E
s;
0
[
n; V
e
[
n
0
]]
n
;
0
n
=
n
0
=
T
s
[
n
]
n
+
V
e
[
n
0
](
r
)
;
0
n
=
n
0
=0
:
(2.12)
Comparing (2.12) and (2.4) we derive the following form for the eective one{particle p otential
V
e
[
n
0
]:
V
e
[
n
0
]=
V
ext
+
Z
0
n
0
(
r
0
)
j
r
0
;
r
j
+
V
XC
[
n
0
]
:
(2.13)
Inserting this expression into Eq. (2.6), we obtain the eigenvalue-problem:
(
;
r
2
2
+
V
ext
+
Z
0
n
0
(
r
0
)
j
r
0
;
r
j
+
V
XC
[
n
0
])
|{z }
V
e
j
i
i
=
"
i
j
i
i
:
(2.14)
n
0
(
r
)=
occ
X
i
n
i
j
i
(
r
)
ih
i
(
r
)
j
(2.15)
These two equations are coupled via the construction of V
e
and together constitute the
Kohn{Sham
equations
. They havetobesolved self{consistently, i.e. the eectivepotential V
e
constructed from
the charge densityvia (2.13) put into (2.14) must lead to the one{particle states out of whichthe
2.2. THE LOCAL DENSITY APPROXIMATION (LDA)
11
charge density has b een constructed via 2.15. The self{consistent solution gives then the ground{
state charge density of the system. Apart from the exchange and correlation p otential
E
XC
all terms
in (2.10) are known explicitly.We can thus write for the ground{state energy
E
0
of an interacting
electron gas in an external p otential:
E
0
=
occ
X
i
n
i
h
i
j;
r
2
2
j
i
i
+
1
2
Z Z
0
n
0
(
r
0
)
n
0
(
r
)
j
r
0
;
r
j
+
Z
V
ext
(
r
)
n
0
(
r
)+
E
XC
[
n
0
]
:
(2.16)
Multiplying (2.14) with
h
i
j
, summing over the o ccupied states and using (2.7) and (2.9) we derive
the identity:
T
s
[
n
0
]=
occ
X
i
n
i
"
i
;
Z
V
ext
(
r
)
n
0
(
r
)
;
Z Z
0
n
0
(
r
)
n
0
(
r
0
)
j
r
0
;
r
j
;
Z
V
XC
[
n
0
](
r
)
n
0
(
r
)
:
Inserting it in (2.3), we obtain an alternative form for the total energy:
E
0
=
occ
X
i
n
i
"
i
;
1
2
Z Z
0
n
0
(
r
0
)
n
0
(
r
)
j
r
0
;
r
j
;
Z
V
XC
[
n
0
](
r
)
n
0
(
r
)+
E
XC
[
n
0
]
:
(2.17)
The total energy is therefore the sum of the one{particle eigenvalues frequently called
band structure
energy
corrected by terms usually referred to as
double counting
contributions plus the exchange{
correlation energy
E
XC
.
2.2 Approximating the exchange and correlation energy: the lo cal
density approximation (LDA)
The transformation of the many{electron problem (2.1) to a system of eective one{electron equa-
tions (2.14) has b een exact so far, i.e. no approximations have b een made up to this p oint. However,
the one particle Hamilton op erator (2.14) and the expressions for the total energies (2.16) and
(2.17) contain the functionals
V
XC
[
n
]and
E
XC
[
n
]. Although the existence of these functionals can
be mathematically justied in most cases their explicit form is unknown. Using quantum Monte{
Carlo calculations one could in principle determine numerically the exchange and correlation energy
of a given system up to any required accuracy. However, quantum Monte{Carlo calculations are
computationally to o exp ensivetotreat any but the simplest systems such as the uniform electron
gas [15 ].
The most common approximation for the exchange and correlation energy and potential is the
local density approximation
(LDA). Within the LDA approximation one assumes that for any small
region in the system, the exchange-correlation is the same as that for the uniform electron gas
with the same electron density. This approximation applies to a
non{spinpolarized
system, and the
exchange-correlation energy is approximated by:
E
xc
=
Z
n
(
r
)
xc
(
n
)
;
where
xc
(
n
) is the exchange-correlation density for the homogeneous electron gas. For a spin p o-
larised system, one simply applies the same assumptions using the exchange-correlation energy
12
CHAPTER 2. DENSITY FUNCTIONAL THEORY
density of the spin-p olarised uniform electron-gas,
xc
(
n
"
;n
#
). This is termed the lo cal spin den-
sity approximation (LSDA) and implementing this within DFT is often called lo cal spin density
functional theory (LSDFT).
It is p ossible to go beyond such a
local
approximation and to consider further terms in a series
representation of the charge density, termed the
generalised gradient approximation
(GGA) [16].
However, the merits of such an approach are not accepted universally. Therefore in this work the
exchange and correlation energies and p otentials are exclusively approximated byLDA.
Summary
It has been shown that DFT results in a simple yet powerful way of solving the many electron
Schrodinger equation: the whole problem reduces to nd the solutions of one{particle equations, the
Kohn{Sham equations (2.14), in an eective p otential. It should however b e noted that in contrast
to Hartree{Fock theory, which in principle gives meaningful one{particle states and eigenvalues that
corresp ond to the true ionisation energies (Ko opman's theorem) of the system the wavefunctions
i
derived in density{functional theory do not satisfy this condition. Instead they are related to
o ccupation numb ers. It is p ossible to augment the density functional theory with GW theory
1
which
predicts quasi-particle energies with reasonable accuracy [18 ].
1
A full discussion of GW theory is beyond the scop e of this thesis, but the framework within which the GW
approximation is formulated is that of a p erturbation expansion of one{particle Greens functions
G
(
p; w
). See for
example reference [17 ]
Chapter 3
AIMPRO Metho dology
In the previous chapter eective one{particle equations, the Kohn{Sham equations, have been
deduced from the many{b o dy Schrodinger equation. In this chapter the main features of a practical
approach to the solution of the Kohn{Sham equations implemented in the
ab initio
mo delling
pro
gram (AIMPRO) is describ ed. The AIMPRO metho d uses pseudop otentials and a Gaussian
basis set for the expansion of the one{particle wavefunctions. A full review of the metho dology
b ehind AIMPRO is given in Reference [19]. Currently on parallel machines AIMPRO can b e applied
routinely to clusters of sizes up to
350 atoms. In this thesis AIMPRO will b e used to determine
the electronic prop erties of extended defects in GaN.
3.1 Pseudop otentials
DFT as describ ed ab ove would still prove computationally to o dicult for system sizes useful for
the mo delling of surfaces, p oint and line defects in semiconductors. This is due to the fact that
these systems, esp ecially if they contain
non rst row elements
usually have a large number of
electrons
N
el
and according to equation (2.8) this numb er is correlated with the numb er of Kohn{
Sham equations (2.14)
N
KS
to be solved by
N
KS
N
el
=
2. However, not all the electrons need be
considered. They are divided into two groups: core and valence electrons. The former are b ound
close to the ions and do not play a part in b onding. It is highly advantageous to remove these
from the theory. This can b e done through the use of pseudop otentials. These rely on the fact that
only the valence electrons are involved in chemical b onding. Therefore it is p ossible to incorp orate
the core states into a bulk nuclear potential, or pseudo-p otential, and only deal with the valence
electrons separately.
For example, in Ga the 1
s;
2
s;
2
p;
3
s;
3
p
(and sometimes also the 3
d
) electrons are regarded as
part of the core while the 4
s
and 4
p
(and in the cases where they are not included in the core also
the 3
d
) electrons are part of the valence shell. In the same way, only N 2
s
and 2
p
electrons are
considered as valence electrons whereas the N 1
s
electrons are regarded as part of the core. The
four b onds asso ciated with a chemical unit of GaN in the tetrahedrally b onded GaN bulk arise from
these eightvalence electrons.
13
14
CHAPTER 3. AIMPRO METHODOLOGY
By reducing the problem in a way that without mo difying the result only the Kohn{Sham equa-
tions for these valence electrons need to be solved, a considerable simplication is achieved. The
pseudop otential is constructed by insisting that it has exactly the same valence energy levels as the
true atomic p otential, e.g. the 4
s
and 4
p
pseudop otential levels in Ga are the same as the 4
s
and 4
p
levels in an all{electron calculation. Moreover, the corresp onding pseudo-wavefunctions are exactly
equal to the true wavefunctions outside a core whose size dep ends on the typ e of the atom. Inside
the core, the pseudo-wavefunctions are not equal to the true valence ones but are very smo oth
no deless functions which are easy to represent mathematically. On the other hand, the true 4
s
and
4
p
wavefunctions of Ga oscillate inside the core and are rather dicult to represent mathemati-
cally. As the pseudo-wavefunctions are no deless inside the core the pseudop otentials have no core
states at all. Indeed, the lowest b ound state solutions are the valence energy eigenvalues and, by
construction of the pseudop otential, these are the same as the true valence energy levels.
In addition to the advantages achieved by reducing the number of one{particle equations and by
simplifying the evaluation of integrals in the core region pseudop otentials allow for the treatment
of heavier atoms in which relativistic eects are imp ortant. Whereas for a description of the core
electrons the Dirac equation is required, the valence electrons can be treated non-relativistically.
Therefore, removal of the core electrons allows a non-relativistic approachtobemaintained (some
corrections must b e included in the core electron pseudop otentials to account for relativistic eects).
The pseudop otentials used in
AIMPRO
come from work done by Bachelet, Hamann and Schl uter
[20] who pro duced pseudop otentials for all elements up to Pu.
3.2 Numerical solution via a basis set expansion into Gaussian
orbitals
Despite the great simplications gained by the use of pseudop otentials the numerical solution of the
interacting one{particle equations (2.14) in the eective potential
V
e
remains a challenging task.
For a practical solution the one{particle wavefunctions
j
i
(
r
)
i
have to be expanded in terms of a
basis
j
(
r
)
i
:
j
i
(
r
)
i
=
M
X
c
i
j
(
r
)
i
:
(3.1)
Two choices for the basis functions
j
i
are often made: plane waves and Cartesian Gaussian
orbitals. Using the rst is equivalent to making a Fourier transform of the wavefunction. This
has problems with certain elements, e.g. rst row, transition and rare-earth elements where the
wavefunction varies rapidly and a great numb er of plane waves are required.
Within the
AIMPRO
metho dology, Cartesian Gaussian orbitals are used as basis functions. Centred
at the p oint
R
they have the form:
(
r
)=(
x
;
R
x
)
n
1
(
y
;
R
y
)
n
2
(
z
;
R
z
)
n
3
e
;
a
(
r
;
R
)
2
; n
i
2
IN:
In practice, they are centred at the nuclei and sometimes at b ond centred sites b etween the nuclei.
From suitable combinations of
n
i
, one can construct the
s
,
p
,
d
, ... orbitals. For example,
n
i
=0,
i
=1
;
2
;
3 generates an
s
-Gaussian orbital.
3.3. APPLICATION OF AIMPROTOGAN
15
The use of Gaussian orbitals as a basis set has considerable advantages:
1. In contrast to a plane wave basis, Gaussian orbitals are lo calised and therefore a signicantly
smaller basis set is needed to describ e lo calised wavefunctions. In GaN this is particularly
imp ortant since Ga 3
d
and N 2
s
wavefunctions are very lo calised.
2. In contrast to Slater typ e orbitals, which are also lo calised, matrix elements in Gaussian
orbitals can b e evaluated analytically leading to a reduction in the computational task.
Inserting the expansions (3.1) for the one{particle wavefunctions into (2.14) gives:
(
;
r
2
2
+
V
e
)
j
M
X
c
i
i
=
"
i
j
M
X
c
i
i
:
(3.2)
Multiplying this equation by
h
for 1
M
reduces the dierential equation to a set of
algebraic equations:
M
X
c
i
(
H
;
"
i
S
)=0
;
1
; i
M ;
which in matrix notation read as a generalised eigenvalue problem.
H
and
S
are matrix elements
of the Hamiltonian and the overlap matrix, resp ectively:
H
=
h
j;
r
2
2
+
V
e
j
i
S
=
h
j
i
:
Once this equation has b een solved we can generate the
output
charge density:
n
out
(
r
)=
occ
X
i
n
i
j
i
(
r
)
ih
i
(
r
)
j
:
In general, this is not equal to the
input
charge density used to generate the eective p otential
V
e
according to (2.13). Of course, this is an inconsistency and as indicated in the previous chapter
we need to devise a scheme whereby the input charge density, which is used to construct
V
e
, is
equal to the output density. This is usually carried out by an iterative pro cedure and the pro cess
of achieving equality in the densities is called the self-consistent cycle. Metho ds which use a self{
consistent cycle in this way are also referred to as self{consistent eld (SCF) metho ds. With this
self{consistent density the total energy can be calculated via (2.17). The atomic forces needed
to p erform a structural relaxation follow from the derivatives of the total energy with resp ect
to the atomic co ordinates via the application of the Hellman{Feynman theorem. Further details
of the
AIMPRO
metho dology, and in particular a description of the practical realisation of the
implementation can b e found in the review written by Jones and Briddon [19].
3.3 Application of AIMPRO to GaN
In this work
AIMPRO
is used to determine the structural and electrical prop erties of p oint defects
and extended defects in GaN. All the structures investigated with
AIMPRO
are mo delled in clusters
16
CHAPTER 3. AIMPRO METHODOLOGY
(see app endix D). In this work no absolute energies are calculated with the
AIMPRO
metho d. An
explicit inclusion of the Ga 3
d
electrons as valence electrons, whichhybridise with the N 2
s
electrons
and thus might inuence the absolute energy of a structure, is therefore not necessary. In this work
AIMPRO
is rather used to investigate the electrical prop erties which are well describ ed within a
converged basis of the Ga 4
s
, 4
p
andN2
s
,2
p
electrons. In the current application we use 5 (4)
s
and
p
-Gaussian orbitals with dierent exp onents for eachGa(N) atom yielding a large real{space
basis set of 20 (16) Gaussian orbitals on every Ga (N) atom. Applied to a72 atom H-terminated
stoichiometric cluster arranged as in wurtzite GaN, gave b ond lengths within 4% of exp eriment and
a maximum vibrational frequency of 729 cm
;
1
compared with 741 cm
;
1
exp erimentally found for
the
E
1
(
LO
) mo de [21 ].
Summary
The
AIMPRO
metho d has b een presented which solves the Kohn{Sham equations within a Gaussian
basis. Since Gaussian basis functions are lo calised
AIMPRO
is a suitable program to describ e the
N2
s
wavefunctions in GaN.
Chapter 4
The density functional based tight
binding metho dology
Density{functional theory realized with pseudop otentials and a Gaussian basis set as implemented in
the
AIMPRO
metho dology describ ed in the previous chapter allows to treat accurately non{p erio dic
structures within clusters up to
350 atoms. However, some of the structures considered in this
work require mo dels which considerably exceed this size. Because of limited computer resources
these structures cannot be investigated by any metho d based on a rigorous implementation of
density functional theory, like
AIMPRO
but require more approximate schemes.
Empirically constructed potentials, derived from tting parameters to equilibrium structures are
very ecientandthus capable of dealing with extended systems. However, they suer from a trans-
ferability problem: they generally only work well within the regime in whichtheywere tted and
thus are not predictive in structural simulations. Therefore, more approximate schemes combining
the advantage of the eciency of the empirical p otentials with the transferability and accuracy of the
SCF
metho ds are highly desirable. In this context, b esides numerous traditional quantum chemical
metho ds, tight-binding (TB) mo dels have recently b ecome very p opular [22 , 23, 24 ], providing one
of the most accurate alternatives in the determination of the total energy and equilibrium geometry
of various systems. In particular, two{centre{oriented schemes considering only two{centre integrals
in the Hamiltonian give results that deviate only slightly from those of more sophisticated metho ds.
In the standard TB-metho d eigenstates of a Hamiltonian are expanded in an orthogonalised basis
of atomic-like orbitals. The exact many-b o dy Hamilton op erator is represented by a parameterised
Hamiltonian matrix, where the matrix elements are tted to the band structure of suitable reference
systems. However, the choice of reference systems to which the matrix elements are tted is rather
arbitrary. The tting is thus often complicated and do es not guarantee a general transferabilityto
all scale systems.
To circumvent this problem a density{functional based tight binding scheme
DFTB
has b een devel-
op ed [25, 26]. Using a minimal basis linear combination of atomic orbitals (LCAO) - framework [25 ]
this scheme avoids diculties arising from an empirical parametrisation. Instead, the two{centre
Hamiltonian and overlap matrix elements are calculated from atom{centred valence{electron or-
bitals and atomic p otentials are derived from
SCF
single{atom calculations within the lo cal{density
approximation. This scheme can b e seen as an approximate DFT scheme. In contrast with the usual
17
18
CHAPTER 4. DFTB METHODOLOGY
parametrised TB schemes it has a well dened pro cedure for determining the desired matrix ele-
ments. Moreover, interactions extending b eyond the rst shell of neighb ours are taken into account
which is not the case in empirical TB metho ds. This is of crucial imp ortance for GaN, b ecause the
second neighb our Ga-Ga distances (3.18
A) are comparable to the distances b etween neighb ouring
atoms in bulk Ga which range from 2.45
A to 2.71
A.
In GaN systems charge transfer may play an imp ortant role, esp ecially if bonds are broken as at
surfaces and extended defects, since although GaN is acovalently b onded material, it has partial
ionic character. Recently, we have extended the
DFTB
scheme by incorp orating a self-consistency
cycle for the atomic charges [27 ]. This self-consistentcharge density{functional based tight binding
(
SCC{DFTB
) scheme can be used to investigate systems where interatomic charge transfer plays
an imp ortant role.
In the following, the
DFTB
scheme and its charge self-consistent extension
SCC{DFTB
will be
deduced following Elstner
et al.
[27] from general density{functional theory describ ed in chapter 2.
Particular emphasis will b e given to the extension of
SCC{DFTB
to p erio dic systems where within
this thesis a well b ehaved functional is derived to describ e the energy arising from charge transfer
for extended systems [27]. The numerical eciency will be outlined. It will be shown that as we
have implemented the co de on parallel machines [28] a high degree of parallelisation can b e reached
making
SCC{DFTB
avery useful scheme for treating the extended GaN systems examined within
this thesis.
4.1 Density-functional basis of TB-theory
According to DFT describ ed in chapter 2 the total energy of asystemof
N
el
electrons in the eld
of
N
nuc
nuclei at p ositions
R
I
may b e written as a functional of a charge density
n
(
r
) (see (2.10)):
E
=
occ
X
i
n
i
h
i
j;
r
2
2
+
V
ext
+
1
2
Z
0
n
(
r
0
)
j
r
;
r
0
j
j
i
i
+
E
XC
[
n
(
r
)] +
1
2
N
nuc
X
I;J
Z
I
Z
J
j
R
I
;
R
J
j
:
(4.1)
The rst sum is over o ccupied
Kohn-Sham
eigenstates
i
with o ccupation numbers
n
i
, the second
term is the exchange-correlation (XC) contribution and the last term whichwas not considered in
(2.10) accounts for the ion-ion core repulsion,
E
ii
.
We now follow the approach of Foulkes and Haydo ck[29 ] and substitute the "true", usually self{
consistently determined charge density
n
=
n
(
r
) in (4.1) by a sup erp osition of a reference or input
density
n
0
=
n
0
(
r
) and a small uctuation
n
=
n
(
r
):
n
=
n
0
+
n :
Inserting this sup erp osition into (4.1) gives for the total energy:
E
=
occ
X
i
n
i
h
i
j;
r
2
2
+
V
ext
+
Z
0
n
0
0
j
r
;
r
0
j
+
V
XC
[
n
0
]
j
i
i;
1
2
ZZ
0
n
0
0
(
n
0
+
n
)
j
r
;
r
0
j
;
Z
V
XC
[
n
0
](
n
0
+
n
)+
1
2
ZZ
0
n
0
(
n
0
+
n
)
j
r
;
r
0
j
+
E
XC
[
n
0
+
n
]+
E
ii
;
4.2. STANDARD{DFTB FORMALISM
19
where
E
ii
has been intro duced as a shorthand for the ion{ion repulsion. The second term in this
equation corrects for the double-counting of the new
Hartree
, the third term for the new XC
contribution in the leading matrix element and the fourth term comes from dividing the full
Hartree
energy in (4.1) into a part related to
n
0
and to
n
.
Expanding
E
XC
at the reference density into a Taylors series, whichwe truncate after the second
order terms, gives the total energy correct to second order in the density uctuations. After a simple
transformation this reads:
E
=
occ
X
i
n
i
h
i
j
^
H
0
j
i
i ;
1
2
ZZ
0
n
0
0
n
0
j
r
;
r
0
j
+
E
XC
[
n
0
]
;
Z
V
XC
[
n
0
]
n
0
+
E
ii
+
1
2
ZZ
0
1
j
r
;
r
0
j
+
2
E
XC
n n
0
n
0
!
n n
0
:
(4.2)
Note, that the terms linear in
n
cancel each other at any arbitrary input density
n
0
.
^
H
0
denotes
the Hamiltonian op erator resulting from the input density
n
0
.
4.2 Zero-th order approximation: standard{DFTB
Within the
standard{DFTB
metho d one supp oses that the initial charge density
n
0
is very close
to the real self{consistent density
n
.In this case
n
0
is suciently small so that all higher order
terms, in particular the last term in the nal equation for the total energy (4.2) can b e neglected.
A frozen-core approximation is applied to reduce the computational eorts by only considering the
valence orbitals. A frozen core approximation diers from the pseudop otential approach describ ed
in chapter 3 mainly by the fact that since the core wave functions are not mo died also the valence
wavefunctions oscillate within the core region requiring a ne integration mesh. However, as will
be seen later, within the
SCC-DFTB
scheme the evaluation of integrals plays only a minor role in
the computational eort so that a frozen{core approach is suitable. The
Kohn-Sham
equations are
then solved non-self-consistently and the second-order correction is neglected. The contributions
in (4.2) that dep end on the input density
n
0
only and the core-core repulsion are taken to b e a sum
of one- and two-b o dy potentials [29 ]. The latter, denoted by
E
rep
, are strictly pairwise, repulsive
and short-ranged. The total energy then reads:
E
TB
0
=
occ
X
i
n
i
h
i
j
^
H
0
j
i
i
+
E
rep
:
(4.3)
Making a linear combination of atomic orbital (LCAO)-
ansatz
the single-particle wavefunctions
i
are expanded into a suitable set of lo calised atomic orbitals
:
j
i
(
r
)
i
=
M
X
c
i
j
(
r
;
R
I
)
i
:
(4.4)
We employ conned atomic orbitals
in a Slater-typ e representation. These are determined by
solving a mo died Schrodinger equation for afree neutral pseudoatom within SCF-LDA calcula-
tions [26]. For further details of the construction of these basis functions see also [30].
20
CHAPTER 4. DFTB METHODOLOGY
Within this expansion (4.3) transforms to:
E
TB
0
=
occ
X
i
n
i
M
X
M
X
c
i
H
0
c
i
+
E
rep
;
(4.5)
where
H
0
=
h
j
^
H
0
j
i
are the Hamilton matrix elements in the LCAO basis.
Denoting the overlap matrix elements with
S
=
h
j
i
and applying the Lagrange formalism with the condition that the numb er of electrons in the system
remains constant, i.e.
P
occ
i
n
i
=
N
el
, to this zero-th order energy functional (4.5) gives:
@
@c
i
h
E
TB
0
;
i
occ
X
n
M
X
M
X
c
S
c
;
N
el
!
i
=0
:
(4.6)
The derivatives are:
@E
TB
0
@c
i
=
n
i
M
X
H
0
c
i
and
@
@c
i
"
i
occ
X
n
M
X
M
X
c
S
c
;
N
el
!
=
i
n
i
M
X
S
c
i
:
Inserting these derivatives into (4.6) gives the
Kohn-Sham
equations of the zero-th order
DFTB
metho d as a set of algebraic equations:
M
X
c
i
(
H
0
;
"
i
S
)=0
;
1
; i
M :
(4.7)
Within the
DFTB
metho d the eective one-electron p otential of the many-atom structure
V
e
in
^
H
0
=
^
T
+
V
e
is approximated as a sum of spherical
Kohn-Sham
p otentials of neutral pseudoatoms due to their
conned
electron density.Furthermore, several terms in the Hamilton matrix elements, in particular
multicentre contributions are neglected. This is consistent with the construction of the eective one-
electron p otential [25] and gives:
H
0
=
8
>
<
>
:
"
neutral free atom
if
=
h
I
j
^
T
+
V
I
0
+
V
J
0
j
J
i
if
I
6
=
J
0 otherwise
:
(4.8)
Since indices
I
and
J
indicate the atoms on whichthewavefunctions and p otentials are centred, only
two-centre Hamiltonian matrix elements are treated and explicitly evaluated in combination with
the two-centre overlap matrix elements. As follows from (4.8), the eigenvalues of the free atom serve
as diagonal elements of the Hamiltonian, thus guaranteeing the correct limit for isolated atoms.
4.2. STANDARD{DFTB FORMALISM
21
By solving the general eigenvalue problem (4.7), the rst term in (4.5) b ecomes a simple summation
over the eigenvalues
"
i
of all o ccupied Kohn-Sham orbitals:
E
TB
0
=
occ
X
i
n
i
"
i
+
E
rep
:
(4.9)
A transferable parametrised short range p otential
E
rep
=
1
2
P
I
6
=
J
I;J
(
I; J
) can easily b e determined
as a function of distance by taking the dierence of the SCF-LDA cohesive and the corresp onding
TB band structure energy for a suitable reference system:
E
rep
(
R
)=
1
2
I
6
=
J
X
I;J
(
I; J
)=
(
E
SCF
LD A
(
R
)
;
occ
X
i
n
i
"
i
(
R
)
)
reference structure
:
(4.10)
An analytic expression for the interatomic forces follows by taking the derivative of the nal
DFTB
energy (4.9) with resp ect to the nuclear co ordinates:
M
I
R
I
=
;
@E
TB
0
@
R
I
=
;
occ
X
i
n
i
M
X
M
X
c
i
c
i
"
@H
0
@
R
I
;
"
i
@S
@
R
I
#
;
@E
rep
@
R
I
:
Atomic charges, which will b e of particular interest in the charge self-consistent second order ex-
tension of
DFTB
(see next section) can b e approximately evaluated as Mulliken charges:
q
I
=
1
2
occ
X
i
n
i
X
2
I
M
X
c
i
c
i
S
+
c
i
c
i
S
:
(4.11)
For many problems, like the calculation of surface energies or defect formation energies, it is very
useful to know the energy contribution of single atoms, which for atom
I
is derived from (4.5) as:
E
I
=
occ
X
i
n
i
X
2
I
M
X
c
i
H
0
c
i
+
J
6
=
I
X
J
(
I; J
)
:
(4.12)
Summary
In this section the
non-SCC DFTB
-approach has been describ ed, which in the following will also
be called
standard{DFTB
. Provided a clever guess of the initial or input charge density of the
system, the energies and forces are correct to second order of charge density uctuations. Further-
more, the short-range two-particle repulsion (determined once using aprop er reference system)
op erates transferably in very dierent b onding situations considering various scale systems. Indeed
standard{DFTB
has been successfully applied to systems of dierent materials with sizes ranging
from small clusters [31 , 32 , 33] over fullerenes [34, 35 ] to surfaces and interfaces [36, 37] of a variety
of semiconductors.
22
CHAPTER 4. DFTB METHODOLOGY
4.3 Second-order self-consistent charge extension, SCC-DFTB
The
standard{DFTB
scheme discussed ab ove is suitable when the electron density of the many-
atom structure may b e represented as a sum of atomic-like densities in go o d approximation. How-
ever, since the
standard{DFTB
-variant neglects eects of charge redistribution due to Coulomb{
interactions it cannot accurately describ e systems with considerable charge transfer. This means
that
standard{DFTB
will normally fail if the chemical b onding is controlled byadelicate balance
of the interatomic charge transfer. In systems containing atoms having dierent electro-negativity,
esp ecially in p olar semiconductors and in heteronuclear molecules this is often the case. Therefore,
wehave extended the approach in order to improve total energies, forces, and transferabilityinthe
presence of long-range Coulombinteractions.
We start from equation (4.2) and now explicitly consider the second order term in the density
uctuations.
In order to include asso ciated eects in a simple and ecient TB concept, we rst decomp ose
n
(
r
)
into atom-centred contributions which decay quickly with increasing distance from the corresp ond-
ing centre. The second-order term then reads:
E
2
nd
=
1
2
N
nuc
X
I;J
ZZ
0
;[
r
;
r
0
;n
0
]
n
I
(
r
)
n
J
(
r
0
)
;
(4.13)
where wehave used the functional ; to denote the
Hartree
and XC co ecients. Secondly, the
n
I
may b e expanded in a series of radial and angular functions:
n
I
(
r
)=
X
l;m
K
ml
F
I
ml
(
j
r
;
R
I
j
)
Y
lm
r
;
R
I
j
r
;
R
I
j
q
I
F
I
00
(
j
r
;
R
I
j
)
Y
00
;
(4.14)
where
F
I
ml
denotes the normalised radial dep endence of the density uctuation on atom
I
for
the corresp onding angular-momentum. While the angular deformation of the charge density, e.g.
in covalently b onded systems, is usually describ ed very well within the non-SCC approach, charge
transfers b etween dierent atoms are not prop erly handled in manycases.Truncating the multip ole
expansion (4.14) after the monop ole term accounts for the most imp ortantcontributions of this
kind while avoiding a substantial increase in the numerical complexity of the scheme. Also, it
should b e noted that higher-order interactions decaymuch more rapidly with increasing interatomic
distance. Finally, expression (4.14) preserves the total charge in the system, i.e.
P
I
q
I
=
R
n
(
r
).
Substitution of (4.14) into (4.13) yields the simple nal expression for the second-order energy term:
E
2
nd
=
1
2
N
nuc
X
I;J
q
I
q
J
IJ
;
where
IJ
=
ZZ
0
;[
r
;
r
0
;n
0
]
F
I
00
(
j
r
;
R
I
j
)
F
J
00
(
j
r
0
;
R
J
j
)
4
:
(4.15)
In the limit of large interatomic distances, the XCcontribution vanishes within LDA and
E
2
nd
may
b e viewed as a pure Coulombinteraction b etween two p ointcharges
q
I
and
q
J
. In the opp osite
case, where the charges are lo cated at one and the same atom, a rigorous evaluation of
II
would
require the knowledge of the actual charge distribution. This could b e calculated by expanding the
4.3. SCC{DFTB FORMALISM
23
charge densityinto an appropriate basis set of lo calised orbitals. In order to avoid this numerically
exp ensive expansion wemake an attempt to evaluate
IJ
analytically.To this end in a rst step the
exchange and correlation contribution is neglected (the second order contribution of
E
XC
will be
included a p osteriori for short distances) and
IJ
in (4.15) is evaluated analytically for the Coulomb
contribution only:
IJ
=
ZZ
0
1
j
r
;
r
0
j
F
I
00
(
j
r
;
R
I
j
)
F
J
00
(
j
r
0
;
R
J
j
)
4
:
(4.16)
In accordance with the Slater-typ e orbitals used as a basis set to solvethe Kohn-Sham equations
[31], we assume an exp onential decay of the monop ole term of the density uctuations (as will b e
seen, the values for the exp onentials
I
will follow from the evaluation of
II
at
R
=0):
F
I
00
(
r
;
R
I
)
2
p
=
3
I
8
e
;
I
j
r
;
R
I
j
:
(4.17)
Inserting (4.17) into (4.16) gives:
IJ
=
Z Z
0
1
j
r
;
r
0
j
3
I
8
e
;
I
j
r
;
R
I
j
3
J
8
e
;
J
j
r
0
;
R
J
j
=
Z
0
(
r
0
)
3
J
8
e
;
J
j
r
0
;
R
J
j
;
where
(
r
0
)=
Z
1
j
r
;
r
0
j
3
I
8
e
;
I
j
r
;
R
I
j
:
Via Poisson's equation we obtain for :
(
r
0
)=
1
j
r
0
;
R
I
j
;
I
2
+
1
j
r
0
;
R
I
j
e
;
I
j
r
0
;
R
I
j
:
Hence
IJ
b ecomes:
IJ
=
Z
0
1
j
r
0
;
R
I
j
;
I
2
+
1
j
r
0
;
R
I
j
e
;
I
j
r
0
;
R
I
j
3
J
8
e
;
J
j
r
0
;
R
J
j
:
(4.18)
Setting
R
=
j
R
I
;
R
J
j
, after some co ordinate transformations one gets for
R
6
= 0 (see app endix A.1):
IJ
=
1
R
;
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
e
;
I
R
4
J
I
2(
2
I
;
2
J
)
2
;
6
J
;
3
4
J
2
I
(
2
I
;
2
J
)
3
R
;
e
;
J
R
4
I
J
2(
2
J
;
2
I
)
2
;
6
I
;
3
4
I
2
J
(
2
J
;
2
I
)
3
R
if
I
6
=
J
and
;
e
;
I
R
1
R
+
11
I
16
+
3
2
I
R
16
+
3
I
R
2
48
if
I
=
J
(4.19)
In the limit of large interatomic distances,
IJ
!
1
=R
and thus represents the Coulombinteraction
between two p oint charges
q
I
and
q
J
.This accounts for the fact that at large interatomic
distances the exchange-correlation contribution vanishes within the lo cal density approximation
and only the Coulombcontribution remains imp ortant, as stated ab ove.
Now we need to determine the values for
I
. This is done by examining
IJ
in the limit
R
!
0
which corresp onds to the "on{site" contribution of the second order energy in (4.13). Expanding
the exp onentials in
IJ
we nd (see App endix A.1):
IJ
R
!
0
=
II
=
5
16
I
:
(4.20)
24
CHAPTER 4. DFTB METHODOLOGY
Parisers [38] suggested that
II
can be approximated by the dierence of the atomic ionisation
potential
I
I
and the electron anity
A
I
. This dierence is related to the chemical hardness
I
, [39 ]
or the Hubbard parameter
U
I
:
II
I
I
;
A
I
2
I
U
I
:
Inserting this approximation which is widely used in semi-empirical quantum chemistry metho ds
into (4.20) one gets:
I
=
16
5
U
I
:
This result can b e interpreted by noting that elements with a high chemical hardness tend to have
lo calised wavefunctions which in turn implies a "lo calised uctuation" of their charge densities. The
Hubbard parameters
U
I
are constants which can b e calculated for anyatomtyp e within LDA-DFT
as the second derivative of the total energy of a single atom with resp ect to the o ccupation number
of the highest o ccupied atomic orbital. This includes the inuence of the second order contribution
of
E
XC
in
IJ
for small distances where it is imp ortant.
Since
IJ
is explicitly known, the DFT total energy (4.2) can be transformed into a transparent
TB-form:
E
TB
2
=
occ
X
i
n
i
h
i
j
^
H
0
j
i
i
+
1
2
N
nuc
X
I;J
IJ
q
I
q
J
+
E
rep
;
(4.21)
where
IJ
=
IJ
(
U
I
;U
J
;
j
R
I
;
R
J
j
). As discussed earlier, the contribution due to
^
H
0
dep ends
only on
n
0
and is therefore exactly the same as in the previous non-SCC studies [31]. However,
since the atomic charges dep end on the one-particle wave functions
i
,an iterative pro cedure is
required to nd the minimum energy in expression (4.21). To solve (4.21) explicitly we use again
the pseudoatomic basis set expansion (4.4) for the wavefunctions
i
and hence obtain for (4.21):
E
TB
2
=
occ
X
i
n
i
M
X
c
H
0
c
i
+
1
2
N
nuc
X
I;J
IJ
q
I
q
J
+
E
rep
:
(4.22)
The charge uctuations
q
I
=
q
I
;
q
0
I
are estimated by means of Mulliken charges (4.11). Applica-
tion of the Lagrange formalism to expression (4.22) gives:
@
@c
i
h
E
TB
2
;
i
occ
X
n
M
X
M
X
c
S
c
;
N
el
!
i
=0
:
(4.23)
Tothisendwehavetoevaluate the derivatives
@E
TB
2
@c
i
:
@E
TB
2
@c
i
=
n
i
M
X
c
i
H
0
+
N
nuc
X
I
N
nuc
X
J
@q
I
@c
i
q
J
IJ
:
Using (4.11) we get:
@q
I
c
i
=
1
2
n
i
h
X
2
I
S
c
i
+
I
M
X
S
c
i
i
;
I
=
(
1if
2
I
0 otherwise
4.3. SCC{DFTB FORMALISM
25
and hence:
@E
TB
2
@c
i
=
n
i
M
X
c
i
H
0
+
1
2
N
nuc
X
I
N
nuc
X
J
(
X
2
I
S
c
i
+
I
M
X
S
c
i
)
q
J
IJ
=
n
i
M
X
c
i
h
H
0
+
1
2
S
N
nuc
X
I
q
I
(
JI
+
KI
)
i
;
2
J;
2
K :
Inserting these derivatives into (4.23) gives the Kohn{Sham equations (2.14) in matrix notation:
M
X
c
i
(
H
;
"
i
S
)=0
;
1
; i
M;
(4.24)
with Hamilton and overlap matrix elements:
H
=
h
j
^
H
0
j
i
+
1
2
S
N
nuc
X
K
(
IK
+
JK
)
q
K
(4.25)
=
H
0
+
H
1
;
S
=
h
j
i
; I ; J:
As in the
standard{DFTB
approach, wemake use of the two-centre approximation for the integrals
of
H
0
(4.8). It should however be noted that, since the overlap matrix elements
S
generally
extend over a few nearest neighbour distances, also multiparticle interactions are incorp orated in
the
SCC{DFTB
scheme. The second-order correction due to charge uctuations is now represented
by the non-diagonal Mulliken charge dep endentcontribution
H
1
to the matrix elements
H
.
In consistency with equation (4.10), we determine the short-range repulsive pair p otential
E
rep
as a
function of distance by taking the dierence of the SCF-LDA cohesive energy and the corresp onding
SCC-DFTB
electronic energy (rst two terms in equation (4.21)) for a suitable reference structure.
Since charge transfer eects are now considered explicitly, the transferability of
E
rep
is improved
compared to the non-SCC approach.
Equation (4.25) has b een derived for nite systems. A gamma pointapproximation to periodic
systems is found in a straightforward manner by replacing the sum over the atoms
K
by the sum
over all atoms
K
in the cell summed over all cells:
H
=
H
0
+
1
2
S
N
nuc
X
K
X
R
q
K
(
IK
(
R
)+
JK
(
R
))
=
H
0
+
1
2
S
N
nuc
X
K
q
K
X
R
IK
(
R
)+
X
R
JK
(
R
)
!
;
8
2
I;
2
J :
Here
IK
(
R
) means that the function
IK
in (4.19) is evaluated at
R
I
+
R
and
R
K
. We can now
take the Fourier transform to obtain the formula for the Hamilton matrix elements in k{space:
H
(
k
)=
H
0
(
k
)+
1
2
S
(
k
)
N
nuc
X
K
q
K
X
R
IK
(
R
)+
X
R
JK
(
R
)
!
:
where
H
(
k
) and
S
(
k
) are the Fourier transforms of
H
and
S
, resp ectively. The same cor-
rection for
H
0
(
k
)can be rigorously derived by starting from the energy expression in a periodic
26
CHAPTER 4. DFTB METHODOLOGY
system given by:
E
TB
2
=
occ
X
i
X
k
W
ik
M
X
M
X
c
(
k
)
i
c
i
(
k
)
H
0
(
k
)
+
1
2
N
nuc
X
I
N
nuc
X
J
q
I
q
J
X
R
IJ
(
R
)
:
In p erio dic systems, the evaluation of
IJ
=
P
R
IJ
(
R
) demands the evaluation of an innite sum
which is not absolutely convergent, i.e. the value dep ends on the order of summation. This is byno
means a trivial task. As can b e seen from (4.19)
IJ
consists of a long range part
P
R
1
j
R
j
and a short
range part which is the sum over the terms following the curly bracket in (4.19). The long range
part can be evaluated using the standard Ewald technique, whereas the short range part decays
exp onentially and can therefore b e summed over a small numb er of unit cells (see App endix A.2 for
a detailed description of the numerical evaluation of
IJ
for p erio dic systems). Hence the functional
form for
IJ
(4.19) derived in this work yields a well dened expression for extended and p erio dic
systems. This is in contrast to common functional forms for
IJ
frequently employed in semi{
empirical schemes for molecules. Those functional forms are based on empirical studies and may
cause severe numerical problems when applied to p erio dic systems since Coulomb-like b ehaviour is
only accomplished for large interatomic distances. For example, for p erio dic systems the expression
IJ
=
1
r
(
R
I
;
R
J
)
2
+
1
4
1
U
I
+
1
U
J
2
used in MINDO [40 ] yields ill{conditioned energies with resp ect to the Hubbard parameters. This
means that small changes in the Hubbard parameters may result in considerable variations for the
value of
IJ
=
P
R
IJ
(
R
), i.e. the derivativeof
IJ
with resp ect to the Hubbard parameters has a
large norm. Therefore expressions like this can not b e used for systems where long range Coulomb
interactions o ccur and thus limit considerably the applicabilityof the scheme.
Finally an analytic expression for the interatomic forces follows by taking the derivative of the nal
SCC-DFTB
energy (4.21) with resp ect to the nuclear co ordinates:
F
I
=
;
occ
X
i
n
i
M
X
;
c
i
c
i
"
@H
0
@
R
I
+
1
2
@S
@
R
I
N
nuc
X
L
(
KL
+
JL
)
q
L
#
;
"
i
@S
@
R
I
!
;
q
I
N
nuc
X
L
@
IL
@
R
I
q
L
;
@E
rep
@
R
I
;
2
K ;
2
J :
(4.26)
For p erio dic systems the derivativeof
IJ
is again evaluated by means of the Ewald technique (see
App endix A.2).
In analogy to (4.12) the atomic energies for the
SCC{DFTB
metho d are given by:
E
I
=
X
2
I
M
X
X
i
n
i
c
i
c
i
H
0
+
1
2
N
X
J
(
q
I
;
q
0
I
)(
q
J
;
q
0
J
)
IJ
+
J
6
=
I
X
J
(
I; J
)
:
(4.27)
4.4. PERFORMANCE OF STANDARD{DFTB AND SCC{DFTB
27
Summary
In this section the
SCC{DFTB
metho d has been describ ed. It has b een derived from DFT by
considering second order uctuations in the charge density.To this end wehave deduced an analytic
functional to include the energy arising from charge transfer. This resulted in a charge self{consistent
extension to
standard{DFTB
.Incontrast to
standard{DFTB
the
SCC-DFTB
metho d also allows to
treat systems where a correct description of the chemical b onding requires an accurate distribution
of the electronic charges. Furthermore, the
SCC-DFTB
metho d can be applied to determine the
energetic ordering of systems where electrostatic interactions play a crucial role. The b est known
example for the latter case are the well studied GaAs
(2
4) and
2(2
4) surfaces. Whereas
standard{DFTB
as well as other TB metho ds [43 ] which neglect electrostatic interactions found
(2
4) and
2(2
4) to have the same surface energies [36 ],
SCC{DFTB
favours the
2(2
4)
surface [27 ] in agreement with SCF{LDA calculations [44] and high resolution scanning tunnelling
microscopy exp eriments [45].
4.4 Performance of standard{DFTB and SCC{DFTB
The computationally most exp ensive step in
standard{DFTB
and
SCC{DFTB
calculations is the
solution of the generalised eigenvalue problem (4.7) and (4.24). While the
standard{DFTB
metho d
requires this task to be solved only once for every step in the ionic relaxation pro cess, the
SCC{
DFTB
scheme is converged after
5 iterations in the charge self{consistency cycle, provided the
structure is semiconducting. Therefore,
SCC{DFTB
is only slightly less ecient than
standard{
DFTB
.
Moreover it can b e shown that also all the other computationally exp ensive steps in
(SCC){DFTB
,
i.e. evaluation of Mulliken charges, atomic energies and calculation of forces, whichall scale with
M
3
, where
M
is the total numb er of basis functions of the system, can b e transformed into linear
algebra op erations and can thus easily and eciently b e parallelised [28 ].
As an illustrative example Fig. 4.1 shows the time scaling dep ending on the system size, i.e. the
numb er of basis functions, for one iteration in the charge self-consistency cycle. The b enchmarking
has b een p erformed on a typical workstation (HP 735/125) and on a parallel machine (
Cray T3E
)
employing dierentnumb ers of pro cessors. The discrete data p oints were tted to cubic p olynomials.
In this work 8 (13) basis functions p er GaN unit are used in the
standard
{ (
SCC){DFTB
metho d
(see section b elow). As can b e seen in Fig. 4.1 run on a parallel machine the
(SCC){DFTB
metho d
allows to treat large systems within a reasonable time making it a suitable to ol for the investigation
of surfaces and extended defects in GaN.
4.5 Application of standard{DFTB and SCC{DFTB to GaN
In this work
standard{DFTB
and
SCC{DFTB
are used to determine the structural prop erties and
energetics of surfaces, point defects, extended defects and domain b oundaries in GaN. According
to the top ology of the structures they were mo delled either in clusters or in sup ercells whichever
typ e of mo delling is more appropriate for the problem under consideration (see app endix D).
28
CHAPTER 4. DFTB METHODOLOGY
0
50
100
150
200
250
300
350
0 500 1000 1500 2000 2500 3000 3500 4000
HP 735/125
CRAY T3E
4PE’s
(1 scc-cycle ~ 5 iterations)
CRAY T3E
16 PE’s
CRAY T3E
64 PE’s
number of basis functions
time/scc-iteration in s
Figure 4.1: Time scaling of the
SCC{DFTB
metho d for a sup ercell with a GaAs (100) surface.
In most of the cases the geometries are already found to be well describ ed within the
standard{
DFTB
metho d. In order to provide geometry optimisations in a numerically ecient way in this
work the
standard{DFTB
metho d uses a minimal basis set consisting of Ga 4
s
, 4
p
and N 2
s
, 2
p
wavefunctions. The Ga 3
d
electrons are not treated as valence electrons, an approximation whichas
in the case of
AIMPRO
(see previous chapter) is p ermitted if no energetics are considered. Indeed,
test calculations show that an inclusion of the Ga 3
d
electrons into the valence band has almost no
eect on the geometries.
In order to determine the sometimes complex energetic ordering of structures the use of the more
accurate
SCC{DFTB
metho d is more appropriate. As mentioned ab ove charge self{consistency is
particularly imp ortantfortheenergetics of systems where electrostatic interactions play a crucial
role. In the current applications we use the same minimal basis set of Ga 4
s
, 4
p
and N 2
s
, 2
p
wavefunctions as in the
standard{DFTB
metho d but with the Ga 3
d
electrons included in the
valence band. As indicated by Northrup
et al.
[46 ], this is imp ortant to obtain accurate formation
energies since the Ga 3
d
and N 2
s
levels hybridise.
A detailed description of the construction of the wavefunctions used for GaN in
standard{
and
SCC{DFTB
along with the resulting electronic band structures can b e found in app endix B. In this
app endix also the structures employed to generate the repulsivepotentials
E
rep
for Ga{Ga, Ga{N
and N{N interactions are given.
Chapter 5
Formation Energies of Surfaces and
Defects in Thermo dynamic
Equilibrium
This chapter deals with the stability of surfaces and defects in a thermo dynamical context. The
energy necessary to create a structure (here and in the following structure stands for b oth, surfaces
and defects) is called the formation energy. This energy is not constant but dep ends on sp ecic
growth conditions. In GaN the relative abundance of Ga, N and impurity atoms during the crystal
growth determines the formation energy of the structure. In a thermo dynamical context, these
abundances are describ ed bythechemical p otentials asso ciated with the reservoirs from which Ga,
N and impurityatoms enter the growth pro cess.
A simple expression for the formation energy will be given where sp ecic growth conditions are
describ ed via chemical potentials. The derivation of this expression will be given for clean GaN
structures. The result will then b e generalised to formation energies of GaN structures with impu-
rities.
Finally, from these calculated formation energies some exp erimentally interesting quantities will b e
deduced.
5.1 Formation energies of charge neutral GaN structures
The equilibrium structure is determined by minimising the grand canonical p otential for the for-
mation energy. Given a sp ecic temp erature and pressure the grand canonical p otential for the
formation energy
E
as a function of comp osition reads:
E
=
G
(
p; T ; n
Ga
;n
N
)
structure
;
Ga
(
p; T
)
n
Ga
;
N
(
p; T
)
n
N
:
(5.1)
Here
G
(
p; T ; n
Ga
;n
N
)
structure
=
E
structure
+
PV
;
TS
29
30
CHAPTER 5. FORMATION ENERGIES OF SURFACES AND DEFECTS
is the Gibbs free energy of the structure.
E
structure
is the total energy of the structure which is
obtained from a calculation at
T
=0.
Ga
and
N
denote the Ga and N chemical potentials and
n
Ga
and
n
N
the number of Ga and N atoms in the structure. In the following we assume the
structure to be in equilibrium with its surroundings, i.e. with the p erfect GaN bulk and with the
gas phase Ga and N. The chemical p otential
i
=
dG=dn
i
of a given atomic sp ecies (
i
= Ga or
N) is the same in each of the phases that are in contact in equilibrium. Therefore, each
i
can b e
considered as the free energy p er particle in eachreservoir for particle
i
.
Equation (5.1) gives the formation energy of a charge neutral structure dep ending on the pres-
sure and the temp erature. As shown in [47 ] the pressure dep endence can usually be neglected in
condensed{state systems which are not easily compressible. Therefore the free energy may b e ap-
proximated at
P
= 0. On the other hand, the free energy has a temp erature dep endent contribution
arising from the vibrational entropy
;
TS
. The energy
;
TS
is not necessarily small compared with
the dierences in formation energies for dierent structures. It comes from the fact that atoms at
surfaces and defects are less strongly b ound than in bulk material and thus can vibrate more eas-
ily. A quantitativeevaluation would b e computationally very demanding. However, for comparable
structures, as e.g. dierent typ es of p oint defects, the vibrational entropy is usually very similar.
If wearenotinterested in the precise values of the absolute formation energies but rather wantto
compare the stabilities of structures we can neglect the contribution coming from the vibrational
entropyand may replace the free energy
G
structure
of the structure by the total energy
E
structure
of the structure. Note that this approximation means that
G
structure
=
E
structure
which is exact at
T
=0.
For the gaseous phase, the eect of
T
and
P
up on the chemical potential cannot be ignored, and
any value of
Ga
and
N
maybe exp erimentally attained. Qian, Martin and Chadi [48 ] showed,
however, that there are limits on the allowable range of
Ga
and
N
under equilibrium conditions
with all p ossible phases. Thermo dynamic equilibrium implies that the chemical potential of the
elementary comp ounds cannot b e larger than the chemical p otential of the lowest energy phase in
which the elementary comp ound is involved. Otherwise this phase would b e adopted.
1
Itisagoodapproximation to assume that Ga in its bulk structure and nitrogen in the form of N
2
molecules are the corresp onding phases of lowest energy. This means for the upp er limits of the
elemental chemical p otentials:
Ga
Ga(bulk)
(
p; T
)and
N
N
2
(
p; T
)
:
(5.2)
For
T
= 0 (corresp onding to an upp er limit)
Ga(bulk)
and
N
2
can b e obtained from total energy
calculations as
E
Ga(bulk)
and
1
2
E
N
2
, resp ectively. Thermo dynamic equilibrium conditions also imply
that the crystalline GaN structure can exchange pairs of Ga and N atoms with the reservoir. As
stated ab ove for a pair of Ga and N in the crystal the free energy
GaN
is the total bulk energy p er
pair
E
GaN(bulk)
. An exchange of pairs of GaN units between structure and reservoirs implies that
the sum of the chemical p otentials
Ga
and
N
in the reservoirs equals the chemical p otential of a
crystal unit:
Ga
+
N
=
GaN(bulk)
(
E
GaN(bulk)
)
=
Ga(bulk)
+
N
2
;
H
f
;
(5.3)
where
H
f
is the heat of formation of GaN. The heat of formation can b e determined from enthalpy
measurements and has b een found to b e 1.14 eV for GaN [49 ]. Of course,
H
f
can also b e derived
1
This would then result in a disso ciation of the GaN crystal, in contradiction to thermodynamic equilibrium.
5.2. FORMATION ENERGIES OF STRUCTURES WITH IMPURITIES
31
from total energy calculations for GaN in the wurtzite phase
E
GaN(bulk)
, Ga in the orthorhombic
bulk phase
E
Ga(bulk)
and the N
2
molecule
1
2
E
N
2
. Combined with Eq. (5.2), Eq. (5.3) sets a lower
limit on the elemental chemical p otentials
Ga
and
N
so that the structure can b e in equilibrium
with its surroundings only if the chemical p otentials satisfy the relations:
Ga(bulk)
;
H
f
Ga
and
N
2
;
H
f
N
:
(5.4)
For a more detailed theoretical discussion ab out the validity of the limits for the chemical p otentials
see Kley [47 ].
Finally,we can use (5.3) to eliminate one of the elemental chemical p otentials in (5.1). Here and in
the following we cho ose to eliminate
N
.With this choice the formation energies dep end only on
the gallium chemical p otential.
2
This gives the following simple expression for the formation energy
of a charge neutral structure:
E
=
E
structure
;
E
GaN(bulk)
n
N
;
Ga
(
n
Ga
;
n
N
)
;
(5.5)
where according to (5.2) and (5.4) the allowed range of the gallium chemical p otential is given by:
Ga(bulk)
;
H
f
Ga
Ga(bulk)
:
(5.6)
The way in which the limits for
Ga
used in this work are determined for calculations with the
SCC{DFTB
metho d is describ ed in App endix C.
5.2 Formation energies of structures with impurities
Impurities, in particular H, C, O, Si and Mg playan imp ortant role in GaN. In this work the
stabilities and electrical prop erties of hydrogen and oxygen as impurities and surface adsorbates
will b e examined.
The formalism for formation energies describ ed ab ove can b e generalised in a straightforward man-
ner. Denoting the impurities byIwehave:
E
=
E
structure
;
E
GaN(bulk)
n
N
;
Ga
(
n
Ga
;
n
N
)
;
X
I
n
I
I
:
(5.7)
Assuming that the elemental chemical potentials of the impurities can be chosen indep endently
from the gallium chemical p otential, one would need to represent (5.7) as a multi{dimensional plot.
However, as will b e shown belowfor the hydrogen and oxygen impurities considered in this work
this task can often b e simplied by assigning the chemical p otential of the impurities with a xed
value which corresp onds to the growth conditions at which the pro cess should b e examined.
5.2.1 Hydrogen
From statistical mechanics [50 ] the chemical p otential of hydrogen
H
intheformofH
2
gas is given
by:
2
H
=
E
H
2
+
ln
(
PV
Q
=
)
;
lnZ
rot
;
lnZ
vib
;
2
Of course, expressing the formation energies in terms of the nitrogen chemical p otential
N
would describ e the
same physics.
32
CHAPTER 5. FORMATION ENERGIES OF SURFACES AND DEFECTS
where
E
H
2
is the total energy of an H
2
molecule, P is the pressure,
V
Q
= (
h
2
=
2
mkT
)
3
=
2
is the
quantum volume and
=
kT
is the temp erature.
Z
rot
and
Z
vib
are the rotational and vibrational
contributions to the partition function describing the internal degrees of freedom of an H
2
molecule.
AtanMOCVDgrowth temp erature of
1300
o
K and with a nominal H
2
pressure of
1atm,
H
is
1 eV lower than
E
H
2
=
2. For further discussion and a temp erature dep endent plot of
H
see
reference [51 ].
5.2.2 Oxygen
Unfortunately,incontrast to hydrogen, the oxygen pressure during MOCVD or MBE growth is not
really known. It is therefore not p ossible to deriveavalue for the oxygen chemical p otential in the
same explicit way as for the H chemical p otential.
In manycases, itishowever satisfactory to get a rough idea ab out the stability of a defect. To this
end one commonly assumes [52 , 8, 9 ] that oxygen and gallium are in equilibrium with gallium oxide
(Ga
2
O
3
):
2
Ga
+3
O
=
Ga
2
O
3
:
(5.8)
This corresp onds to the upp er oxygen concentration. If
O
were larger than
Ga
2
O
3
gallium oxide
would form during growth under thermo dynamic equilibrium. In order to get an absolute formation
energy which is useful to determine the concentration of a defect (see b elow) one often furthermore
assumes Ga rich growth conditions (
Ga
=
Ga(bulk)
). The value of
O
for the upp er limit of the
oxygen concentration is then xed via (5.8):
O
(upp er)=
Ga
2
O
3
;
2
Ga(bulk)
3
:
3
(5.9)
The ideas outlined in the last paragraph are very plausible if eects of oxygen incorp oration during
GaN growth are examined. Here the growth of GaN is still the dominating pro cess while the
incorp oration of oxygen plays a secondary role. This allows to set the oxygen chemical p otential to
axedvalue. On the other hand, if the chemisorption of oxygen at surfaces is considered it is more
meaningful to express the surface formation energies via the oxygen chemical p otential. Therefore
in the following
O
will be chosen as the indep endent variable in (5.7). We assume that during
the chemisorption of O the surface is in an environment where O is in the gas phase (O
2
). Hence
the upp er limit of
mu
O
is given by equilibrium with O
2
molecules. The lower limit of
O
can be
chosen somehow arbitrary.We assume that it should b e not muchlessthan
O
in equilibrium with
gallium oxide under Ga{rich growth conditions. Otherwise the resulting oxygen conguration at
the surface would b e signicantly more stable than bulk gallium oxide which is not very likely.The
energy equation for gallium oxide is:
Ga
2
O
3
=2
Ga(bulk)
+3
O
2
;
H
f
(Ga
2
O
3
)
;
(5.10)
3
It should b e noted that the absolute formation energy obtained by inserting
O(upp er)
and
Ga(bulk)
into equation
(5.7) is also a go o d quantity for the upp er limit of the oxygen concentration under N{richgrowth conditions: oxygen
adopts a nitrogen site in GaN and oxygen defects often haveanequivalentstoichiometry to Ga
2
O
3
,ase.g.V
Ga
{(O
N
)
3
.
This means that bymoving towards a N{richenvironment the formation energy gained by increasing the upp er limit
O(upp er)
in (5.9) (now
Ga
<
Ga(bulk)
) is comp ensated by the formation energy lost due to the fact that the structure
has an N decient stoichiometry.
5.3. FORMATION ENERGIES OF CHARGED STRUCTURES
33
where
H
f
(Ga
2
O
3
)
is the heat of formation for Ga
2
O
3
, which has b een calculated from bulk Ga
2
O
3
,
the orthorhombic Ga bulk phase and the O
2
molecule to b e 12
:
05 eV (see C) in go o d agreement with
the heat of formation of 11
:
3 eV determined from enthalpy measurements [49]. Hence, equilibrium
with gallium oxide under Ga{rich growth conditions as a lower limit and with O
2
molecules as an
upp er limit implies:
O
2
;
1
3
H
f
(Ga
2
O
3
)
O
O
2
;
(5.11)
extending from the Ga
2
O
3
like(
O
=
O
2
;
1
3
H
f
(Ga
2
O
3
)
) to the O{rich(
O
=
O
2
)environment.
5.3 Formation energies of charged structures
In the previous section expressions for the formation energy of charge neutral structures were de-
duced. However, point defects frequently o ccur in a charged state, as e.g. in I I I{V semiconductors
where the energetically low (high) lying anion (cation) derived electronic states b ecome lled (emp-
tied). A charged defect is stable if its formation energy is lower than the formation energy of the
charge neutral defect. To obtain the formation energy of charged defects we need to include the
energy of the reservoir from which the electrons come from (in the case of a negatively charged
defect) or are transfered to (in the case of a p ositively charged defect). The energy of the electronic
reservoir can b e expressed via the p osition of the Fermi level
E
F
. Therefore if
q
denotes the charge
of the defect, we can extend (5.7):
E
=
E
(
q
)=
E
structure
(
q
)
;
E
GaN(bulk)
n
N
;
Ga
(
n
Ga
;
n
N
)
;
X
I
n
I
I
;
qE
F
:
(5.12)
This formula allows an easy interpretation: often negatively charged defects are particularly stable
in
n
{typ e GaN where electrons are abundant whereas in
p
{typ e material, where there is a lackof
electrons, p ositively charged defects tend to havelow energies.
5.4 Applications to surfaces and defects
In the following it will be shown how some exp erimentally interesting quantities can be derived
from the formation energy of a structure.
Absolute Surface Energies
If the surfaces are mo delled by slabs (see section D.1) the surface energy is obtained by dividing
the formation energy of the slab by the area
A
of the slab surface:
=
E=A:
4
(5.13)
Frequently, the energy needed to create a surface from p erfect bulk material, called the
absolute
surface energy
,is of particular interest. Examples for pro cesses whichcan be explained by the
4
Of course, due to the approximations in the previous section
do es not include the entropycontribution and
thus maycontain a certain error.
34
CHAPTER 5. FORMATION ENERGIES OF SURFACES AND DEFECTS
knowledge of absolute surface energies are the adaption of an equilibrium crystal shap e (ECS) [53],
the dierence in the growth rate of crystals in sp ecic directions [54] and the formation of nanopip es
in certain materials as SiC [55]. However, the surface energy derived by inserting the formation
energy of the entire slab into (5.13) do es not corresp ond to the absolute surface energy, but always
contains the energy of another, often inequivalent surface which terminates the b ottom of the slab.
For the (110), (100), (111) and (111 ) surfaces of comp ound semiconductors in the zinc{blende phase
it is p ossible to obtain absolute surface energies by dividing the slab into top and b ottom (slab
top
and a slab
b ottom
)by a b oundary consisting of crystal symmetry planes [56, 57]. Here slab
top
is a well
dened region b elow the surface which includes the surface and underlying layers that are aected
by a relaxation due to the surface reconstruction. On the other hand, slab
bottom
is the remaining
region b elow slab
top
containing only bulk{like GaN and the terminating pseudo{hydrogen layer
(see section D.1).
5
Of course, only the energy contained in slab
top
is related to the absolute surface
energy, whereas the energy contained in slab
b ottom
includes terminating hydrogens and has no
physical meaning. Therefore, if the formation energy
E
is evaluated for slab
top
via equations (4.27)
and (5.5) then
represents the desired absolute surface energy.
The wurtzite structure has a lower crystalline symmetry than the zinc{blende structure. However,
for the nonp olar (10
10) and (11
20) surfaces and for the p olar (10
11) and (10
1
1) surfaces an accurate
splitting into slab
top
and slab
b ottom
can easily b e found (see thesis of M. Haugk [59]). The splitting
of slabs with zinc{blende (111) and (
1
1
1) surfaces into slab
top
and slab
b ottom
implicitly makes use
of the equivalence of the four tetrahedral b onds in the zinc{blende phase. It cannot therefore be
strictly applied to (0001) and (000
1) surfaces in the wurtzite phase, where the non{ideal c/a ratio
and u parameter lead to only three equivalent b onds. In GaN
c=a
=1
:
626 and
u
=0
:
377 are very
close to the ideal values, however, thus yielding four nearly equivalent b onds in bulk material. We
may therefore apply the same slab division as in zinc{blende material and exp ect it to give very
reasonable absolute surface energies. For a rigorous mathematical justication see reference [58].
Concentration of Point Defects
For p oint defects it is useful to evaluate the absolute concentrations in thermo dynamic equilibrium.
To this end we also need to consider the congurational entropy and the energy contribution
;
TS
arising from the vibrational entropy. The latter has b een neglected in all equations following (5.1).
Estimates based on an Einstein mo del givevalues b etween 3 and 5
k
B
for the vibrational entropy
S
of
nativepoint defects in GaN [8 ]. The congurational entropy for p oint defects is simply given bythe
number of sites
N
sites
at which the defect can b e created.
6
We can then write for the concentration
in thermo dynamic equilibrium:
c
=
N
sites
e
S=k
B
e
;
E=k
B
T
:
(5.14)
Whether thermo dynamic equilibrium conditions are reached dep ends on the mobility of the defect at
the sp ecic temp erature. A high mobility implies that the assumption of thermo dynamic equilibrium
conditions should b e valid so that the defect concentration will b e given via (5.14). On the other
hand, at a lowmobility the defect concentration is rather likely to b e controlled by surface kinetics.
5
For a detailed description of the splitting of zinc{blende surface slabs into top and bottom see [36 , 47, 58 ].
6
N
sites
is the concentration of p ossible sites. As an example a gallium vacancy can replace any gallium atom giving
N
sites
=2
:
2
10
22
=cm
3
.
5.4. APPLICATIONS TO SURFACES AND DEFECTS
35
Line energy of one{dimensional defects
The energy p er unit length of a one{dimensional defect is dened as:
E
line
=
E=L ;
(5.15)
where
L
is the length of the line defect and
E
the formation energy in the structural mo del. Line
energies can sometimes b e used as a guide line to determine how frequently a sp ecic typ e of line
defect o ccurs.
In the case of dislo cations
E
line
is mainly the core energy of the dislo cation along with the elastic
energy stored in the bulk{like region of the mo del. If the dislo cations are mo delled in a sup ercell
by a dislo cation dip ole the elastic energy p er dislo cation is stored in a cylinder of radius
R
roughly
equal to half the distance b etween the cores of the dislo cation dip ole.
7
On the other hand, following linear elasticity theory, the energy p er unit length within a cylinder
of radius
R
, of a general straight dislo cation with Burgers vector
b
is given by:
E
line
(
R
)=
b
2
4
cos
2
+
sin
2
1
;
!
ln
R
j
b
j
:
(5.16)
Here
is the Poisson's ratio and
is the shear mo dulus of the medium.
is the angle between
the dislo cation line and the Burgers vector. The parameter
represents the core energy of the
dislo cation. Due to the heavily distorted b onds at the dislo cation core the core energy cannot be
derived from linear elasticity theory. Instead
could b e obtained from a comparison of equations
(5.15) and (5.16). To achieve a reasonable accuracy for the parameter
it would b e necessary to
evaluate
E
line
via (5.15) for dierent sup ercell sizes to check whether the stress led in the region
of the dislo cation core and thus
is converged. This is a computationally very exp ensive task and
not within the scop e of this work.
Wall Energies of Domain Boundaries
The wall energy for a domain b oundary is dened as:
E
wall
=
E=A;
(5.17)
where
E
and
A
are formation energy and area of the wall contained in the mo del. Often it is useful
to compare domain wall energies with the energy of two surfaces which brought together form the
domain b oundary.
Summary
Assuming thermo dynamic equilibrium it has b een shown that the formation energy of a structure
can b e derived from total energy calculations. Moreover, it dep ends on the growth conditions and
7
Note that according to 5.16 the elastic energy of a dislo cation in an innite crystal diverges, so that it makes
more sense to dene
E
line
(
R
).
36
CHAPTER 5. FORMATION ENERGIES OF SURFACES AND DEFECTS
the position of the Fermi level describ ed by the elemental chemical p otentials and the electro-
chemical p otential, resp ectively. Some examples of how exp erimental interesting quantities can b e
derived from the formation energy are given.
Chapter 6
Nonp olar GaN Surfaces
Nonp olar planes in comp ound semiconductors are characterised by an equal numb er of cations and
anions. In wurtzite material these are the
f
10
10
g
and
f
11
20
g
planes. A nonp olar surface is a surface
which lies in a nonp olar plane. Nonp olar surfaces have been investigated for along time in I I I-V
semiconductors [60, 61 ]. They can b e obtained by cleavage and are imp ortant for the fabrication of
lasers where they are employed as resonators. It might also b e p ossible to use them as alternative
growth directions for MBE or MOCVD. Finally in wurtzite GaN many of the extended defects
threading in the [0001] direction exhibit internal nonp olar surfaces or contain atomic arrangements
which are similar to them. Studies of the nonp olar surfaces may therefore provide information on the
electronic prop erties of these defects. Oxygen is an impuritywhich is built in at high concentrations
during MBE and MOCVD growth and may segregate to nonp olar surfaces and related extended
defects. It is therefore of interest to explore how these impurities adsorb at the surfaces, inuence
the electrical prop erties and change the absolute surface energies.
In this chapter an extensive study of GaN (101 0) and (112 0) surfaces is presented. Atomic geome-
tries, electrical prop erties and absolute surface energies are determined for stoichiometric as well as
for Ga and N terminated surfaces. In addition the adsorption of O on (101 0) and (1120) surfaces is
investigated. We identify the gallium vacancy surrounded by three oxygen impurities (V
Ga
{(O
N
)
3
)
to be a particularly stable and electrically inert complex whichmightbe involved during the for-
mation of nanopip es (see chapter 7.2).
All stable surfaces ob ey a simple electron counting rule which has b een derived from the nonp olar
stoichiometric GaAs (110) surface [62 ] and is a useful to ol for predicting the stabilities of surfaces
and defects of I I I{V semiconductors.
37
38
CHAPTER 6. NONPOLAR GAN SURFACES
6.1 Common relaxation mechanism for I I I{V semiconductor sur-
faces: the electron counting rule
Nonp olar surfaces have b een investigated intensively for a variety of I I I{V semiconductors, in partic-
ular the (110) surface in zinc{blende GaAs has b een sub ject to several detailed analysis [63, 64 , 65].
A relaxation mechanism within the (1
1) surface unit cell was prop osed by low{energy electron
diraction (LEED) studies [64 ]. The conclusion of this analysis was that the As atoms rotate out
of the (110) surface plane and the Ga atoms inward from this plane by an angle 27
o
!
31
o
.
The b ond lengths are nearly conserved.
Theoretical investigations conrmed these results and interpreted them by suggesting that the Ga
derived dangling b ond, which has a high energy in the band gap, transfers its charge to ll the lower
lying As derived dangling bond. The resulting conguration has thus alower energy. The charge
transfer can also explain the surface geometry: the surface cation which has lost an electron favours
an
sp
2
likehybridisation. This makes the Ga atom at the stoichiometric GaAs (110) surface relax
inward and form a more planar conguration. On the other hand the dangling b ond of the anion
is completely lled and
s
like. Therefore the anion forms b onds with the remaining three
p
orbitals
explaining why the As atom relaxes outwards into a
p
3
conguration. It should b e noted that for
many III{V semiconductors the empty
p
like cation orbital is pushed ab ove the conduction band
edge, whereas the full
s
like anion orbital is just b elow the valence band edge.
This phenomenon has since b een observed at a variety of I I I{V semiconductor surfaces and can b e
summarised in the following
electron counting rule
:
A stable III-V semiconductor surface has al l cation dangling bonds emptied and al l anion dangling
bonds l led resulting in a semiconducting surface.
The stabilities of a variety of III{V semiconductor surfaces could be explained by the electron
counting mo del. In particular, all reconstructions o ccurring at GaAs (100) and (111) surfaces match
this simple rule. Only at a few surfaces, growth under Ga{rich conditions sometimes results in
metallic surfaces which do not ob ey the electron counting rule. Examples for this failure are the
(
p
19
p
19 reconstruction of the GaAs (1 11) surface [66 ] and the (1
1) reconstructions at the
GaN (0001) and (0001 ) surfaces (see chapter 9).
6.2 Stoichiometric (10
10) and (11
20) surfaces
Stoichiometric nonp olar comp ound semiconductor surfaces are terminated by an equal number of
cation and anion atoms on the top surface layer.
The (10
10) surface
The (101 0) surface o ccurs at the walls of nanopip es (see chapter 7.2) and a similar atomic arrange-
ment is found at the core of the threading edge dislo cation (see chapter 7.3). Also domain b oundaries
of typ e DB{I I terminate in (101 0) planes (see chapter 8).
6.2. STOICHIOMETRIC (10
10) AND (11
20) SURFACES
39
[0001]
[10-10]
Type I
Type II
X
Y
12
34
Figure 6.1: Side view along [1120] (
left
)and top view (
right
) of the GaN (1010) surface. Surfaces
of typ e I have one dangling b ond p er atom and are therefore more stable than surfaces of typ e II
which have two dangling b onds per atom (see Fig. 6.4). Ga (N) atoms rehybridise into
sp
2
(
p
3
)
resulting in an electrically inactive surface (see Fig. 6.2).
The (101 0) surface exhibits two inequivalent surface typ es (typ e I and typ e I I in Fig. 6.1). Surfaces
of typ e I have Ga{N surface dimers and one dangling b ond per surface atom. Surfaces of typ e II
have two dangling b onds p er atom.
Typ e I has b een investigated theoretically byPandey
et al.
[67 ] with a multicongurational Hartree-
Fockscheme, by Northrup
et al.
[46] using SCF{LDA within a plane wave expansion and byus[68]
within the
standard{DFTB
approximation. All these works found atomic geometries with the Ga
atoms rehybridized into
sp
2
and the N atoms into
p
3
as predicted by the electron counting rule.
However, as a surprising result the rotation angle was found to be
6
o
which is signicantly
smaller than those known from other stoichiometric nonp olar I I I{V semiconductor surfaces. Also a
contraction of the b ond{lengths of the surface dimers of
6% has been calculated in contrast to
the b ond{length conserving rotations rep orted for other I I I-V's.
Within
SCC{DFTB
the (101 0) surface is mo delled in a (3
2), i.e. 9.5
A
10.1
A sup ercell. Table
6.2 gives details of the calculated geometrical structure along with the results of the rst{principles
calculations by Northrup and Neugebauer [46].
Table 6.1: Atomic displacements in
Afor the top two layers of atoms at the GaN(1010) surface.
Atom numb ers refer to Fig. 6.1. Values in brackets are results of reference [46 ].
Atom
x
y
z
1(Ga
3
coor d:
) -0.10 (-0.11) 0.00 -0.23 (-0.20)
2(N
3
coor d:
)0.03 (0.01) 0.00 -0.01 (0.02)
3(Ga
4
coor d:
)0.01 (0.05) 0.00 0.08 (0.05)
4(N
4
coor d:
)0.04 (0.05) 0.00 0.07 (0.05)
The
SCC{DFTB
metho d gives an absolute surface energy of 121 meV/
A
2
which is in go o d agree-
ment with the plane wave calculations (118 meV/
A
2
). The stoichiometric surface of typ e I I is found
40
CHAPTER 6. NONPOLAR GAN SURFACES
ΧΜ ΧΓΓΜ
-14.0
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0 N
eV
Figure 6.2: Valence band structure of the relaxed GaN (1010) surface calculated within
SCC-
DFTB
. The full line represents the N derived surface band. The shaded region corresp onds to the
bulk pro jected band structure. The Ga derived dangling b onds (not shown) are
4
:
1 eV (i.e.
clearly more than 3.4 eV) ab ove the valence band maximum. Therefore the surface is found to b e
electrically inactive.
to have a high surface energy of 152 meV/
A
2
and will therefore transform into a typ e Isurface
under equilibrium growth conditions.
Finally we calculated the electrical prop erties of the stoichiometric typ e I surface. The band struc-
ture is shown in Fig. 6.2. In agreement with Northrup and Neugebauer [46 ] we nd that the relaxed
structure has no deep gap states ab ove the valence band maximum (the Ga derived dangling b onds
lie
4
:
1 eV ab ove VBM). Northrup and Neugebauer [46] conclude that the relaxed structure is
electrically inactive.
1
The (11
20) surface
The second non{p olar surface in wurtzite GaN lies in
f
11
20
g
planes which are also the terminating
planes of domain b oundaries of typ e DB{I (see chapter 8).
As can be seen in Fig. 6.3 there is only one typ e of (1120) surface with three{fold co ordinated
Ga and N surface atoms arranged in zickzag chains. It has been investigated by Northrup and
Neugebauer [46] with SCF{LDA plane wave and in our previous work [68 ] with
standard{DFTB
.
In analogy to the (1010) surface the b ond{length contraction of
5% is found to be larger than
1
Due to the minimal basis set employed in
SCC{DFTB
we cannot describ e the conduction band appropriately.
Therefore here and in the following we supp ose that any lo calised state which is clearly more than 3.4 eV ab ove the
valence band maximum is not a deep gap state. We will call a structure with no deep gap states electrically inactive.
Within this approach also
SCC{DFTB
nds (101 0) to b e electrically inactive.
6.2. STOICHIOMETRIC (10
10) AND (11
20) SURFACES
41
[10-10]
Ga
N
[11-20] Y
1
23
4
56
7
8
X
Figure 6.3: Side view along [0001] (
left
)and top view of the GaN (1120) surface. In analogy to
(101 0) surfaces Ga (N) atoms rehybridise into
sp
2
(
p
3
) making the surface electrically inactive.
usually observed at nonp olar surfaces of I I I-V semiconductors, whereas the tilt angle of 6.5
o
is again
signicantly smaller.
Within
SCC{DFTB
we mo del the (112 0) surface in a (2
2), i.e. 11.0
A
10.3
A sup ercell. Table 6.2
shows details of the calculated geometrical structure compared with the values of Northrup and
Neugebauer [46 ].
Table 6.2: Atomic displacements in
Afor the top two layers of atoms at the GaN(1120) surface.
Atom numb ers refer to Fig. 6.3. Values in brackets are results of reference [46 ].
Atom
x
y
z
1 (Ga) -0.08 (-0.10) -0.13 (-0.16) -0.16 (-0.17)
2 (N) 0.04 (0.02) -0.02 (-0.02) 0.05 (0.05)
3 (Ga) -0.08 (-0.10) 0.13 (0.16) -0.16 (-0.17)
4 (N) 0.04 (0.02) -0.02 (0.02) 0.05 (0.05)
5 (Ga) 0.01 (0.02) 0.00 (0.00) 0.05 (0.05)
6 (N) 0.03 (0.01) 0.02 (0.01) 0.03 (0.02)
7 (Ga) 0.01 (0.02) 0.00 (0.00) 0.05 (0.05)
8 (N) -0.01 (0.01) -0.02 (-0.01) 0.03 (0.02)
The
SCC{DFTB
metho d gives an absolute surface energy of 127 meV/
A
2
(123 meV/
A
2
in refer-
ence [46 ]). We note that the energy dierence b etween (101 0) and (1120) surfaces, although small,
might be the reason that GaN crystallites grow in hegaxons, i.e. they terminate in (1010) and
not in the prismatic (1120) planes. Also the (1120) surface fulls the electron counting rule. The
SCC-DFTB
metho d nds again that the N derived lone pairs are slightly b elowthe valence band
maximum and the Ga derived dangling b onds lie 4.0 eV ab ove the valence band maximum. We
thus claim that the surface is electrically inactive.
42
CHAPTER 6. NONPOLAR GAN SURFACES
50
100
150
200
250
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Type I
Type I (Ga dimers)
Type II (Ga dimers)
(GaN dimers)
(GaN dimers)
Surface energy in meV/A
Type II
2
Ga-rich
µ
Ga µGa(bulk) in eV-
N-rich
Figure 6.4: Absolute surface energies of dierent reconstructions of the GaN (1010) surface dep end-
ing on the Ga chemical p otential
Ga
. Only typ e I surfaces are stable.
6.3 Non{stoichiometric (10
10) and (11
20) surfaces
Under As-rich growth conditions the GaAs (110) surface is stable with As surface dimers [65 ]. It
is therefore useful to consider also GaN (10
10) and (11
20) surfaces with mo died terminations, i.e.
with terminations which are dierent from the stoichiometric surface. In analogy to the GaAs (110)
surface the most promising mo dels are those where the top layer cations (anions) are substituted by
the other sp ecies resulting in a complete anion (cation) coverage. Both surfaces ob ey the electron
counting rule: the cation terminated surface has emptied cation dangling b onds, whereas the anion
terminated surface has lled lone pairs.
The (10
10) surface
For the typ e I surface the full cation coverage is obtained if the N atom (No. 2 in Fig. 6.1) is replaced
byGa giving Ga{Ga dimers. The full anion coverage follows if the Ga atom (No. 1 in Fig. 6.1) is
replaced by N yielding N{N dimers. The same can be done for the typ e II surface. The resulting
surface energies are shown in Fig. 6.4.
Over a wide range of the gallium chemical p otential
Ga
the stoichiometric surface of typ e I is the
most stable conguration. Ga dimers at the typ e I surface could be stable under Ga{rich growth
conditions. Such a Ga{rich surface should have a higher sticking co ecient for N, and so it has b een
suggested [46] that it maybe advantageous to employ such a surface as an intermediate stage in
atomic layer epitaxy. The nitrogen terminated surfaces of typ e I and typ e I I havevery high surface
6.4. OXYGEN AT(10
10) AND (11
20) SURFACES
43
energies and are not drawn in the diagram. This comes from the fact that the N{N b onds in a
nitrogen dimer are signicantly shorter (
1
:
5
A) than the Ga{N b onds (1.95
A) resulting in a very
strained conguration with a high surface energy. Surfaces of typ e I I are never stable.
The (11
20) surface
In analogy to the Ga{N surface dimers at the (1010) surface, the Ga{N zickzag chains at the (1120)
surface can b e replaced by Ga{Ga or N{N chains. Ga{Ga (N{N) chains are achieved by substituting
atoms No. 2 and 4 (No. 1 and 3) in Fig. 6.3 by Ga (N) atoms. In agreement with the plane wave
calculations [46 ]
SCC{DFTB
shows that again the N{N chains are very strained and thus p ossess
very high surface energies and also the Ga{Ga zickzag chains are unfavourable, even under Ga{rich
growth conditions. The (1120) surface should therefore only o ccur in its stoichiometric form.
6.4 Oxygen at (10
10) and (11
20) surfaces
Oxygen is a very common impurity in GaN whichunintentionally enters the material during growth.
O has a size which is very similar to N and thus favours a nitrogen substitutional site O
N
[8].
Therefore O is a single donor which mightattract acceptors to reduce the Coulomb energy.
The stability of extended defects, in particular nanopip es which are surrounded by(10
10) surfaces,
has recently b een linked to the O concentration in wurtzite GaN [69 ].
In the following weshow that oxygen segregates to the (1010) and (1120) surfaces and discuss the
stabilityat dierent adsorption sites.
O at (
1010
)
Placing O
N
into a bulk{like p osition (six layers belowthe surface) gives an energy which is by
1
:
5 eV larger than the energy found for O
N
at the surface where it can sit three{fold co ordinated.
This shows that there is a tendency for O to segregate to the surface. To some extend this tendency
is o{set by the congurational entropy encouraging the defect to remain in the bulk.
We have then investigated O in a variety of p ositions at the (1010) surface including O
N
, neigh-
bouring O
N
{O
Ga
and O as an adatom where it sits as a bridge b etween the Ga dangling b ond and
the N lone pair, but nd that if equilibrium with Ga
2
O
3
is assumed all of them have higher energies
than the ideal surface (see Fig. 6.6).
We next consider the V
Ga
{(O
N
)
3
defect (see Fig. 6.5) whichis obtained by removing a surface
gallium atom (No. 1in Fig. 6.1) and replacing the surrounding nitrogen atoms by oxygen. A
calculation showed that also this defect complex is more stable at the surface than in bulk material
(2.2 eV). Furthermore, the defect is electrically inactive with the O atoms passivating the vacancy
in the same way as the fully hydrogenated vacancy, VH
4
,in Si. Two O neighb ours of the surface
vacancy are sub-surface and each b onded to three Ga neighbours (Ga{O bond length 1.78
A to
surface Ga, 1.81
Ato second layer Ga and 1.98
A to third layer Ga atom), but the surface O is
44
CHAPTER 6. NONPOLAR GAN SURFACES
1
3
2
[1-210]
[0001]
[000-1]
1
2
3
GaN
O
[10-10]
Figure 6.5: Schematic top view of the V
Ga
{(O
N
)
3
defect complex at the (1010) surface (
left
) and
at the (11
20) surface (
right
). White (black) circles represent Ga (N) atoms and large (small) circles
top (second) layer atoms. At (101 0) atoms 1 and 2 are three{fold co ordinated second layer O atoms
each with one lone pair, atom 3 is a two{fold co ordinated rst layer O with two lone pairs. At
(11
20) atoms 1 and 2 are two{fold co ordinated surface layer O atoms each with two lone pairs,
atom 3 is a three{fold co ordinated second layer O with one lone pairs.
b onded to two subsurface Ga atoms at a normal oxygen bridge site (1.77
A). With a formation
energy of 1.7 eV p er V
Ga
{O
3N
site (calculated for Ga{rich growth conditions and O in equilibrium
with Ga
2
O
3
)below the defect free (101 0) surface, V
Ga
{(O
N
)
3
at the (1010) surface is a very stable
arrangement. Furthermore V
Ga
{(O
N
)
3
do es not encourage overgrowth. Growth must pro ceed by
adding a Ga atom to the vacant site but this leaves three electrons in shallow levels near the
conduction band probably resulting in an unstable defect (O
N
)
3
whicheven under Ga{richgrowth
conditions is by 1.5 eV higher than V
Ga
{(O
N
)
3
. At the growth temp erature, these O atoms will
drift away diusing to the new surface.
O at (11
20)
The energetic order of the investigated oxygen congurations was exactly the same as at the (1010)
surface with V
Ga
{(O
N
)
3
b eing a very stable and electrically inert defect complex. Two oxygen
atoms of V
Ga
{(O
N
)
3
at the (11
20) surface sit two{fold co ordinated and only one oxygen is three{
fold co ordinated (see Fig. 6.5).
Comparison between the oxygen adsorption at (10
10) and (11
20) surfaces
Although the V
Ga
{(O
N
)
3
defect complex was found to be very stable at both surfaces, there is a
fundamental dierence for the formation of this complex during growth.
V
Ga
{(O
N
)
3
can form very easily on (1010) surfaces. Supp ose, one oxygen atom has emerged to the
6.4. OXYGEN AT(10
10) AND (11
20) SURFACES
45
100
120
140
160
180
200
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
O surface layer
V
ideal
Surface energy in meV/A
O adatom
2
3
µµ in eV Ga-rich
N-rich Ga(bulk)Ga -
N
- O Ga
O
N
Ga -(O
N)
Figure 6.6: Absolute surface energies for the oxygen adsorption at the GaN (101 0) surface dep ending
on the Ga chemical p otential
Ga
. Oxygen is assumed to b e in equilibrium with Ga
2
O
3
.
surface where it sits at a nitrogen site, i.e. atom No. 3 in Fig. 6.5 is the only oxygen. A gallium
vacancy is very likely to b e attracted reducing the Coulomb energy. Already this gallium vacancy
oxygen complex is stable: the oxygen sits two fold co ordinated in a bridge p osition whereas the
two nitrogen atoms have three b onds. This V
Ga
{O
N
defect complex will then attract the other two
oxygens to form a charge neutral V
Ga
{(O
N
)
3
complex.
This mechanism however do es not exist at the (1120) surface. Assuming one oxygen at a surface
nitrogen site, i.e. atom No. 1 in Fig. 6.5 is the only oxygen. A gallium vacancy would lead to a high
energy conguration since there would b e a two{fold co ordinated surface nitrogen at p os. 2. A stable
gallium vacancy at the surface would only be created if two surface oxygens sit on neighb ouring
nitrogen sites, i.e. at pos. 1and pos. 2. This is not very likely to happ en, since the oxygens are
positively charged donors and thus rep el each other.
As an illustrative example for the energy dierence during growth we calculated the energy for a
gallium vacancy surrounded by only one surface O. We found that due to the two{fold co ordinated N
at the (112 0) surface the energy of V
Ga
{O is indeed by 1.5 eV higher than at the (101 0) surface. We
therefore conclude that although V
Ga
{(O
N
)
3
defect complexes are stable at b oth typ es of nonp olar
GaN surfaces they are only likely to o ccur at the (101 0) surfaces.
Summary
Ga, N and O terminated nonp olar surfaces have been investigated. In agreement with SCF{LDA
calculations the stoichiometric surfaces were found to be electrically inactive and to be stable
46
CHAPTER 6. NONPOLAR GAN SURFACES
over a wide range of growth conditions. Only in a Ga{rich environment a Ga terminated (10
10)
surface is energetically favourable. N terminated surfaces havevery high formation energies in any
environment. O segregation to the stoichiometric surfaces is likely to o ccur. In particular at the
(10
10) surface this can lead to the formation of the energetically favourable V
Ga
{(O
N
)
3
defect
complex which is electrically and chemically inactive.
Chapter 7
Line Defects: Threading Dislo cations
and Nanopip es
Frequently sapphire substrates are used to grow device quality wurtzite{(
)GaN with the metal-
organic chemical vap our phase dep osition (MOCVD) technique. In this case, growth pro ceeds along
the
c
{axis. Figure 7.1 shows a cross{sectional TEM weak beam image of a typical sample. The
Figure 7.1: Dislo cation arrangement in aGaN sample grown on sapphire by MOCVD: (a) cross-
sectional TEM (
g
=
3
g
) weak b eam image,
g
=(011 0). Screw dislo cations with
b
=
[0001] are out
of contrast. Dislo cations with a
b
-comp onent in the interface, i.e.
b
=
1
3
[1210] are visible. (b)
cross-sectional TEM (
g
=
3
g
) weak beam image,
g
= (0002). Screw dislo cations with
b
=
[0001]
are visible. Dislo cations with a
b
-comp onentintheinterface are out of contrast. S. Christiansen
et
al.
[70].
large lattice mist between GaN and the sapphire substrate of 13% results in dislo cation tangles
47
48
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
near the interface. In addition to these geometric mist dislo cations whichhave dislo cation lines in
the basal plane, also isolated threading dislo cations with dislo cation lines parallel to
c
and Burgers
vectors
c, a
and
c+a
p ersist b eyond the interface [71, 72 , 73]. Since many of them p enetrate the
entire epilayer from the substrate to the surface they are called threading dislo cations.
An unexp ected nding [74, 75] is that inspite of their high density(typically
10
9
cm
;
2
) and the
fact that they cross the active region of the devices (starting typically
0
:
5
mab ove interface)
these threading dislo cations in GaN do not lead to a pronounced reduction in the device-lifetime of
the light-emitting dio des [76 ] or blue lasers [77]. This can b e contrasted with GaAs where radiation
enhanced dislo cation motion [78] readily o ccurs and leads to an increase in non-radiative pro cesses.
It is therefore of considerable interest to understand the structural and electrical prop erties of
threading dislo cations in GaN and to compare them with those of dislo cations in more traditional
semiconductors.
Threading dislo cations are often asso ciated with the app earance of long nanopip es which are parallel
to
c
and have hexagonal cross sections with constant diameters ranging from 20{250
A [79 , 80, 81 ,
69]. Nanopip es degrade the material quality. In particular, they can get lled by metal during
the formation metal of contacts with GaN and have already caused short circuits in laser devices.
Frank [82] predicted that a dislo cation whose Burgers vector exceeds a critical value should havea
hollowtubeat the core. The equilibrium radius is achieved by balancing the elastic strain energy
released by the formation of a hollow core against the energy of the resulting free surfaces. Liliental{
Web er
et al.
[54] suggested another p ossible mechanism for the formation of nanopip es. They
found the density of nanopip es to b e increased with the impurityconcentration and prop osed that
impurities p oison the walls of the nanopip es whichprevents the nanopip es from growing out.
Finally, the origin of defect{induced electronic states, which lie deep in the GaN band gap and
can thus signicantly alter the optical p erformance, is p ossibly related to threading dislo cations.
Esp ecially in laser devices deep gap states are of concern since parasitic comp onents in the emission
sp ectrum are highly undesirable. The most commonly observed emission in unintentionally dop ed
n
{
typ e GaN, the yellow luminescence (YL), is centred at 2.2-2.3 eV with a line width of
1 eV. Several
mo dels for the origin of the YL in GaN have b een prop osed. Most of them assume the transition to b e
between a shallow donor and a deep acceptor [83] or a deep donor and a shallow acceptor [84]. Recent
work has however found evidence for the deep acceptor mo del [85]. Catho doluminescence (CL)
studies of the yellow luminescence haveshown that the YL is spatially non-uniform (see Figure 1.3 in
the intro duction of this thesis). A p ossible reason for this non{uniform distribution of the YL could
b e related to threading dislo cations, whichare non{uniformly distributed throughout the epilayer
and might b e electrically active. Indeed, also atomic force microscopy (AFM) in combination with
CL has led to the conclusion that threading dislo cations act as non-radiative recombination centres
and degrade the luminescence eciency in the blue light sp ectrum of the epilayers [86 ]. However,
the typ e of dislo cations involved in the YL is not clear: Christiansen
et al.
[70] suggest that the YL
arises from threading dislo cations with a screw comp onent whereas Ponce
et al.
[87 ] lo calise the YL
at low angle grain b oundaries which predominantly contain threading edge dislo cations. Moreover,
it is not clear whether dislo cations in the pure, i.e. impurity free form or defects trapp ed in the
stress eld of dislo cations are resp onsible for the YL.
In this chapter, the atomic geometries, electrical prop erties and line energies of threading screw
and edge dislo cations with full and op en cores are investigated [88 ]. The results are interpreted by
comparing elements of the dislo cation cores with nonp olar (101 0) surfaces (see chapter 6). Possible
7.1. SCREW DISLOCATIONS
49
mechanisms for the formation of nanopip es are then examined [89]. Finally, we also explore the
segregation of gallium vacancies and oxygen as well as related defect complexes to threading edge
dislo cations and discuss their implication for the yellow luminescence [91].
7.1 Screw dislo cations
Threading screw dislo cations in wurtzite material have a Burgers vector parallel to the dislo cation
line [0001]. The smallest screw dislo cations have thus elementary Burgers vector
c
. Screw dis-
lo cations o ccur at a density
10
6
cm
;
2
in
{GaN grown by MOCVD on (0001) sapphire. Since
they nucleate in the early stages of growth at the sapphire interface and thread to the surface of
the crystallites (see Fig. 7.1.b), screw dislo cations are b elieved to arise from the collisions of islands
during growth [72 ]. At a screw dislo cation the surface is rough and has a high energy whichfavours
the nucleation of islands. They are thus vital for the growth pro cess. In GaN screw dislo cations
are unusual in often b eing asso ciated with nanopip es [92 ]. However, full core screw dislo cations [93]
and screw dislo cations with a very narrow op ening of
8
A [94 ] are also rep orted.
7.1.1 Full core screw dislo cations
Screw dislo cations with a full core have been observed by Xin
et al.
[93] using high resolution
Z{contrast imaging (see Fig. 7.2).
Within the
SCC{DFTB
metho d the dislo cations are mo delled in 210 atom clusters p erio dic along
the dislo cation line with p erio dicity
c
and in 576 atom (12
12
1) sup ercells. Because of the large
lateral extension of the sup ercell (12
12), only
k
{p oints for the sampling along the
c
{direction are
necessary. Two
k
-p oints parallel to this direction were found sucient to carry out the sum over
the Brillouin zone: using four
k
{p oints gave only a dierence of
0
:
02 eV/
A in the dislo cation line
energy.Inthe
AIMPRO
case, relaxations were carried out in 392 atom stoichiometric clusters. For
further details concerning the mo delling of dislo cations see app endix D.3.
Both metho ds found heavily distorted b ond lengths for the full core screw dislo cation (see Fig. 7.3
and Table 7.1) yielding deep gap states ranging from 0.9{1.6 eV ab ove the valence band maximum,
VBM, and shallow gap states at
0.2 eV b elow the conduction band minimum, CBM. An analysis
of these gap states revealed that the states ab ove the VBM are lo calised on N core atoms, whereas
the states b elow CBM are lo calised on core atoms but have mixed Ga and N character. Therefore the
full core screw dislo cation is electrically active and could act as a non{radiativecentre [88 ]. Similarly
one could exp ect that dislo cations of mixed typ e would also have deep states in the gap as a result of
the distortion arising from their screw comp onent. Indeed, CL exp eriments haverelatedtheyellow
luminescence centred at 2.2 eV to screw dislo cations [70 ]. In addition, atomic force microscopy
in combination with CL imaging has shown that threading dislo cations with ascrew comp onent
act as nonradiative combination sites [86 ]. A calculation in a sup ercell containing a screw dip ole
consisting of two dislo cations with
b
= [0001] and
;
[0001], which are symmetrically equivalent,
conrmed these results and gave a high line energy of 4.88 eV/
A. This is mainly the core energy
of each screw dislo cation together with the elastic energy stored in a cylinder of diameter roughly
equal to the distance b etween the cores, 19.1
A. See chapter 5 for a more detailed interpretation of
dislo cation line energies.
50
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
Figure 7.2:
Left:
Low magnication angular dark eld image along [0001]. Threading dislo cations
show as bright dots due to their strain eld.
Right:
High resolution Z{contrast image of an end{on
pure screw dislo cation showing a full core. Y. Xin
et al.
[93].
core
[0001]
[10-10]
Figure 7.3: Side view (in [11
20]) of the relaxed core of the full{core screw dislo cation (
b
= [0001]).
The atoms at the dislo cation core adopt heavily distorted congurations (see Table 7.1) yielding
deep gap states.
7.1. SCREW DISLOCATIONS
51
Table 7.1: Bond lengths, min-max (average), in
Aand bond angles (min-max) in
o
for the most
distorted atoms at the core centre of the full{core screw dislo cation (
b
= [0001]).
Atom b ond lengths b ond angles
1(Ga
4
coor d:
) 1.85-2.28 (2.14) 68-137
2(N
4
coor d:
) 1.89-2.28 (2.13) 71-136
7.1.2 Screw dislo cations with a narrow op ening
Wenowinvestigate whether the line energy of the full core screw dislo cation is reduced if material
is taken from the core. Accordingly, calculations were then carried out using the same sup ercell as
for the full core screw dislo cations, but with the hexagonal core of each screw dislo cation removed
leading to a pair of op en{core dislo cations with diameters
d
7
:
2
A. The relaxed structure (Fig. 7.4)
preserved the hexagonal core character, demonstrating that the internal surfaces of the dislo cation
cores shown in Fig. 7.5 are similar to
f
10
10
g
typ e facets except for the top ological singularity
required by a Burgers circuit.
[10-10]
[1-210]
12
3/4
Figure 7.4:
Left:
Top view (in [0001]) of the relaxed core of the op en{core screw dislo cation (
b
=
[0001]). The three fold co ordinated atoms 1 (Ga) and 2(N) adopt a hybridisation similar to the
(1010) surface atoms.
Right:
TEM image of a nanopip e containing a dislo cation with ascrew
comp onent. During growth the nanopip e closes leaving the dislo cation with an op ening of three
rows (
8
A) wide (see black arrangement within the nanopip e). Z. Liliental{Web er [94 ]
.
It is instructive to compare the distortions of the atoms situated at the wall of the op en{core
(Table 7.2) with the corresp onding atoms at the (1010) surface (Table 7.3). In both cases, the
three fold co ordinated Ga (N) atoms adopt an
sp
2
-(
p
3
)- likehybridisation whichlowers the surface
energy and cleans the band gap [46 ]. Indeed, we nd that unlike the full{core screw dislo cation, the
gap is free from deep states [88].
52
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
Y
[0001]
1
2
3
4
Figure 7.5: Pro jection of the wall of the op en{core (d=7.2
A) screw dislo cation (
b
= [0001]). The
three fold co ordinated atoms 1 (Ga) and 2 (N) adopt a hybridisation similar to the (1010) surface
atoms.
Table 7.2: Bond lengths, min-max (average) in
A and b ond angles, min-max (average) in
o
for the
most distorted atoms at the wall of the op en core screw dislo cation (
b
= [0001]). Atom numb ers
refer to Fig. 7.4 and 7.5.
Atom b ond lengths b ond angles
1 (Ga
3
coor d:
) 1.86-1.89 (1.88) 107-123 (117)
2(N
3
coor d:
) 1.88-2.05 (1.96) 102-111 (108)
3 (Ga
4
coor d:
) 1.89-2.07 (1.96) 100-122
4(N
4
coor d:
) 1.93-2.03 (1.97) 98 -120
There are, however, in contrast to the (1010) surface, energetically shallow gap states. Calculations
were carried out for a distorted (101 0) surface, i.e. a (1010) surface in a unit cell where the unit cell
vectors were mo died to give a distorted surface corresp onding to that of the wall of the op en{core
screw dislo cation with diameter
d
=7
:
2
A. We nd that the distorted (101 0) surface has a sp ectrum
with shallowstates very similar to those of the op en{core screw dislo cation with
d
=7
:
2
A. Wealso
calculated the sp ectrum for a nanopip e with
d
=7
:
2
Abut without a dislo cation core. This, like
the undistorted (1010) surface, p ossesses a gap free from deep states, although there are N (Ga)
derived surface states lying slightly b elow(above) the VBM (CBM). These results indicate that the
shallow states in the op en{core screw dislo cation with diameter
d
=7
:
2
A can b e attributed to the
distortion arising from the dislo cation Burgers vector. Calculations for a series of dierent distortions
of the (1010) surface corresp onding to op en{core screw dislo cations with dierent diameters also
suggest that op en{core screw dislo cations with diameters greater than
20
A should have no
gap states at all. As can be seen in Table 7.2 the distortion in the op en{core screw dislo cation
is signicantly less than that in the full{core screw dislo cation (see Table 7.1). It is therefore not
surprising that the calculated line energy of 4.55 eV/
Aislower than the line energy of the full{core
screw dislo cation. The energy required to form the surface at the wall is comp ensated by the energy
gained by reducing the strain. However, a further op ening gave a higher line energy and we conclude
that the equilibrium diameter is
7
:
2
A. This op ening has also b een rep orted by Liliental{Web er
et al.
[94] who found some of the screw dislo cations to have holes which are three atomic rows wide
(see Fig. 7.4).
7.2. THE FORMATION OF NANOPIPES
53
Table 7.3: Bond lengths, min-max (average) in
A and b ond angles, min-max (average) in
o
for the
top twolayers of atoms at the GaN(101 0) surface. Atom numb ers refer to Fig. 6.1.
Atom b ond lengths b ond angles
1 (Ga
3
coor d:
) 1.83-1.88 (1.86) 116-117 (117)
2(N
3
coor d:
) 1.83-1.92 (1.89) 107-111 (108)
3 (Ga
4
coor d:
) 1.91-2.02 (1.94) 107-112
4(N
4
coor d:
) 1.88-2.03 (1.93) 99-115
A theoretical approach to predict the op ening of a screw dislo cation was deduced byFrank [82 ]. By
balancing the elastic dislo cation strain energy released by the formation of a hollow core against the
energy of the resulting free surfaces, he showed that, for isotropic linear elasticity and a cylindrical
core, the equilibrium core radius
r
eq
is
r
eq
=
b
2
8
2
;
(7.1)
where
is the surface energy,
is the shear mo dulus and
b
is the Burgers vector. For a rough
estimate of
r
eq
,we use the theoretical value for the surface energy of
f
1010
g
facets whichwe found
to be
= 121 meV/
A
2
=1
:
9Jm
;
2
(see chapter 6). Taking
=8
10
10
Nm
2
as an upp er limit
and
b
=0
:
5 nm for the Burgers vector of an elementary screw dislo cation yields
r
eq
0
:
2nm. It
is unlikely, that isotropic elasticity theory can describ e the severely distorted full core dislo cation
which limits the usefulness of Frank's expression (7.1) concerning the precise quantitativevalue of
the equilibrium diameter. Our calculated value of
7
:
2
Aand Frank's value are reasonably close
since the relatively small line energy dierence found b etween full core and op en core (
d
7
:
2
A)
screw dislo cations suggests a shallow minimum which probably allows all intermediate structures
to exist. In our calculations only structures constructed by removing entire hexagons, but not those
obtained by removing single rows were considered. Calculating the latter ones, may lead to slightly
lower energies.
In summary, it can be concluded from our calculations and from Frank's theorem that in GaN
screw dislo cations with an elementary Burgers vector
c
can exist with a full core and with a narrow
op ening up to
7
:
2
A. The full core screw dislo cation is electrically active whereas the screw
dislo cation with a hexagonal op ening has only shallow gap states. These states are induced bythe
distortion arising from the Burgers vector.
7.2 The formation of nanopip es
Nanopip es in
{GaN thread along the
c
{axis and have hexagonal cross sections, i.e. they are inclosed
by
f
10
10
g
typ e walls (see Fig. 7.6). Nanopip es are commonly observed in MOCVD grown epilayers
on sapphire [79, 80 , 81 ]. However, they have also been rep orted in samples grown by MBE on
SiC [69]. Nanopip es o ccur at a densityupto
10
8
cm
2
and have constant diameters ranging from
20{250
A. The rst suggestion was that they were the manifestation of screw dislo cations with empty
cores as discussed by Frank a long time ago [82]. However, as shown ab ove
ab initio
calculations
54
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
1
0
2
6
5
4
3
[10-10]
[1-210]
Figure 7.6:
Left:
High resolution Z{contrast image along [0001] of a nanopip e. Y. Xin unpublished.
Right:
Suggested mechanism for the formation of a nanopip e (area No. 0). Three hexagons (No.
1,2,3) are growing together. As the surface to bulk ratio at ledges (No. 4,5,6) is very large, they
grow out quickly leaving a nanopip e (area No. 0) with
f
10
10
g
typ e facets.
as well as Frank's theorem do not supp ort the idea that in GaN the core of a screw dislo cation
with Burgers vector equal to
c
is op en with such a large diameter. Pirouz [55 ] has therefore argued
that sup erscrew dislo cations with Burgers vectors
n
c
where
n>
1 are formed during growth bythe
collision of islands. Clearly,if
n
were big enough, then the cores would b e op en. However, there is
at present no microscopic evidence for such dislo cations [95 ]. We also note that in a typical sample
some nanopip es (
<
10%) were observed which could not b e asso ciated with a screw dislo cation [95 ].
Another p ossibilityis that the
f
10
10
g
typ e surface walls of the nanopip es are coated by hydrogen
which is always present during MOCVD growth. This might result in a very low surface energy
explaining the large diameter of the nanopip es via formula (7.1). The adsorption of H on (1010)
surfaces has b een investigated by Northrup and Neugebauer [51 ]. They found however, that at a
usual MOCVD growth temp erature of
1000
o
C the energy of the (10
10) surface is not lowered by
the adsorption of H and concluded that hydrogen is not resp onsible for the formation of nanopip es.
This has b een conrmed by the recent work of Liliental-Web er
et al.
[69] who detected nanopip es
also in MBE grown material where the concentration of hydrogen is negligible. On the other hand,
Liliental-Web er
et al.
[69] found the diameters and densities of nanotub es to be increased in the
presence of impurities, e.g. O, Mg, In and Si, and argued that these impurities decorate the
f
10
10
g
walls of the nanotub es inhibiting overgrowth. O b eing the main source of unintentional doping in
GaN, wewillnow discuss how O can cause the formation of nanopip es.
From Fig. 7.7 it can be seen that the surface walls of nanopip es are (10
10) surfaces whichare
predominantly of typ e I and at, i.e. they usually have one dangling b ond p er atom and only little
irregularities caused by surface steps. GaN samples usually contain a considerable concentration
of gallium vacancies and oxygen which as our calculations show, have both a tendency to diuse
to (10
10) surfaces. It is therefore very likely that many gallium vacancies and oxygen atoms have
segregated to the nanopip e walls where they can form V
Ga
{(O
N
)
3
defect complexes. In chapter 6 it
7.2. THE FORMATION OF NANOPIPES
55
Figure 7.7:
Bottom:
Straight edge (
left)
and corner (
right
)atthe
f
10
10
g
typ e wall. Y. Xin
et al.
[96 ].
was shown that V
Ga
{(O
N
)
3
are very stable defect complexes on (10
10) surfaces of typ e I. Moreover,
overgrowth was determined to be dicult as oxygen atoms would drift away diusing to the new
surface. Since V
Ga
{(O
N
)
3
defect complexes do not lead to any noticeable change of the atomic
p ositions at the surface they are consistent with the HRTEM image in Fig. 7.7. Unfortunately at
present there seems to b e no direct way for detecting V
Ga
{(O
N
)
n
at nanopip e walls by exp eriments.
To explain the formation of nanopip es, we supp ose [89, 90] that oxygen atoms constantly diuse
to the
f
10
10
g
typ e surfaces. Within the framework of Stranski-Krastanow growth, the internal
f
10
10
g
typ e surfaces between GaN islands are shrinking along with the spaces between colliding
GaN islands (see Fig. 7.6). Therefore, the O coverage and densityof V
Ga
{(O
N
)
3
defects is exp ected
to increase. The maximum concentration of this defect would be reached if 50% (100%) of the
rst (second) layer N atoms were replaced by O (see Fig. 6.5). It is, however, likely that far lower
concentrations are necessary to stabilise the surface and make further shrinkage of the inter-island
spaces imp ossible thus leaving a nanopip e. Provided oxygen could diuse to the surface fast enough,
the diameter and density of the holes would b e related to the densityofoxygen atoms in the bulk.
This has indeed b een observed by Liliental{Web er
et al.
[54] who found that as the concentration of
oxygen in the material changed by ab out an order of magnitude the numb er of nanopip es increased
by a factor of
3 and the diameter of the nanopip es changed from (3{10) nm to (6{12) nm. A more
detailed prediction of the radii and density of nanopip es dep ending on the oxygen partial pressure
would require thermo dynamic equilibrium for the formation of the nanopip es. However, as can b e
seen in the large distribution of nanopip e radii, this is obviously not reached.
It is also necessary to explain why the tub es have
f
10
10
g
typ e surface walls. The other low index
surface p erp endicular to the growth direction which could b ecome p oisoned by O impurities and
thus b e resp onsible for the formation of nanopip es is the (11
20) surface.
f
1120
g
typ e surfaces are
not observed presumably b ecause of their higher absolute surface energy (see chapter 6). Moreover
we suggested that b ecause of the dierent surface top ologies V
Ga
{(O
N
)
3
is likely to form on (10
10)
surfaces but not on (11
20) surfaces during growth.
Finally, we pointout that our arguments are still valid if eachnanopip e is asso ciated with a
screw dislo cation since the walls of the tub e with a dislo cation are lo cally equivalent to a (10
10)
surface which is distorted to form a helix (see 7.1.2). We therefore conclude that rather than b eing
56
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
resp onsible for the formation of nanopip es screw dislo cation are attracted to nanopip es in order to
reduce the elastic energy.
7.3 Threading edge dislo cations
Pure edge dislo cations lie on
f
1010
g
planes and have a Burgers vector
b
=
a
=[1210]
=
3. They are
a dominant sp ecies of dislo cation, o ccurring at extremely high densities of
10
8
;
10
11
cm
;
2
in
{GaN grown by MOCVD on (0001) sapphire (Fig. 7.1.a) and in analogy to screw dislo cations are
thought to arise from the collisions of islands during growth [72 ].
Within the
SCC{DFTB
metho d threading edge dislo cations are mo delled in 210 atom clusters
p erio dic along the dislo cation line with periodicity
c
and in 576 atom (12
12
1) sup ercells
containing a dislo cation dip ole. In analogy to the mo dels for the screw dislo cations, two
k
-p oints
parallel to
c
were used to carry out the sum over the Brillouin zone. In the
AIMPRO
case, relaxations
were carried out in 286 atom stoichiometric clusters.
The relaxed core of the threading edge dislo cation is shown in Fig. 7.8. The corresp onding b ond{
lengths and b ond angles of the most distorted atoms are given in Table 7.4. With resp ect to the
p erfect lattice the distance b etween columns (1/2) and (3/4) [and the equivalentonthe right] are
9 % contracted while the distance between columns (9/10) and (7/8) [and the equivalent on the
right] are 13 % stretched. This atomic geometry for the threading edge dislo cation has recently b een
conrmed by Xin
et al.
[93] using atomic resolution Z{contrast imaging (see Fig. 7.9). Consistent
with our calculation they determined a contraction (stretching) of 15
10 % of the distances b etween
the columns at the dislo cation core. Our calculations show that in a manner identical to the (10
10)
surface, the three{fold co ordinated Ga (N) atoms (no. 1 and 2in Fig. 7.8) relax towards
sp
2
(
p
3
)
leading to empty Ga dangling bonds pushed towards the CBM, and lled lone pairs on N atoms
lying near the VBM. Thus we nd threading edge dislo cations to b e electrically inactive[88].
From a sup ercell calculation, we obtain a line energy of 2.19 eV/
A for the threading edge dislo cation.
We note that this line energy is considerably lower than the one found for the screw dislo cation
with a narrow op ening. This can be interpreted by noting that the edge dislo cation has a smaller
Table 7.4: Bond lengths, min-max (average) in
A and b ond angles, min-max (average) in
o
for the
most distorted atoms at the core of the threading edge dislo cation (
b
=
1
3
[12 10]). Atom numb ers
refer to Fig. 7.8.
Atom b ond lengths b ond angles
1(Ga
3
coor d:
) 1.85-1.86 (1.85) 112-118 (116)
2(N
3
coor d:
) 1.88-1.89 (1.86) 106-107 (106)
3/4 (Ga/N
4
coor d:
) 1.86-1.95 (1.91) 97-119
5/6 (Ga/N
4
coor d:
) 1.92-2.04 (1.97) 100-129
7/8 (Ga/N
4
coor d:
) 1.94-2.21 (2.06) 94-125
9/10 (Ga/N
4
coor d:
) 1.95-2.21 (2.11) 100-122
7.3. THREADING EDGE DISLOCATIONS
57
[1-210]
[10-10]
1
2
3/4
5/6
7/8 9/10
Figure 7.8: Top view (in [0001]) of the relaxed core of the threading edge dislo cation (
b
=
1
3
[1210]).
The three fold co ordinated atoms 1 (Ga) and 2 (N) adopt a hybridisation similar to the (1010)
surface atoms. The distance between columns (1/2) and (3/4) are by 9% contracted while the
distance b etween columns (7/8) and (9/10) is by 13% stretched.
Figure 7.9:
Left
: High{resolution Z{contrast image of a threading edge dislo cation lo oking down
[0001]. The bright dots are atomic columns of alternating Ga and N atoms. The dislo cation core is
shown in the b oxed region.
Right
: Maximum entropy image showing most probable column p ositions.
The distance b etween the column of three{fold co ordinated atoms and the columns on the left [and
right] is found to be by15
10 % contracted. The distance between the column b elow the three{
fold co ordinated atoms and the neighb ouring columns is found to b e by15
10 % stretched. These
results are consistent with our calculations. Y. Xin
et al.
[93].
58
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
number of three fold co ordinated atoms than the op en core screw dislo cation as well as a smaller
elastic strain energy arising from the smaller Burgers vector. This last energy is prop ortional to
k
b
2
. Here,
b
is the magnitude of the Burgers vector and the constant
k
is equal to 1 for the screw
dislo cation, and
1
1
;
for the edge dislo cations, where
is Poisson's ratio (0.37 for GaN [97]). Thus
the ratio of the elastic energies is
E
scr ew
=E
edg e
whichisapproximately 1.66. Our calculations give
the ratio of the line energies, which includes the core energies, to b e 2.08. This could explain why
threading edge dislo cations o ccur at a higher density than threading screw dislo cations.
In analogy to the op en{core screw dislo cations we have investigated whether the energy of the
threading edge dislo cation could b e lowered by removing the most distorted core atoms (see Fig. 7.8).
However, removal of either the columns of atoms (9,10), or the columns (1,2), (3,4), (5,6), (7,8)
and their equivalents on the right, leads to considerably higher line energies. This implies that, in
contrast with screw dislo cations, which as discussed ab ove can exist with a variety of cores, the
threading edge dislo cations should exist with a full core.
7.4 Deep acceptors trapp ed at threading edge dislo cations: V
Ga
and V
Ga
{(O
N
)
n
In the previous section we showed that in the defect free form the threading edge dislo cation has a
band gap free from deep lying states, hence implying that the pure dislo cation cannot b e resp onsible
for the yellow luminescence detected in
n
{typ e GaN. However, as can be seen from Fig. 7.8 and
table 7.4 the core atoms adopt a very particular geometry with atoms 1 and 2being three{fold
co ordinated and atoms 9 and 10 having very stretched b onds with b ond{lengths ranging from 2.0
to 2.2
A. This geometry diers considerably from a p osition in bulk{like material and thus gives
rise to a stress eld which could act as a trap for intrinsic defects and impurities. Gallium vacancies
(V
Ga
)have b een detected by p ositron annihilation studies in bulk GaN and their concentration was
found to b e related to the intensity of the YL [10]. The relevant transition level in
n
{typ e GaN is at
the centre of the YL sp ectrum (
E
2
;
=
3
;
1.1 eV referenced to the top of the valence band [52]). As a
triple acceptor the gallium vacancy is three{fold negatively charged in
n
{typ e GaN and can attract
up to three p ositively charged donors. Recent exp erimental [11 , 98, 99 ] and theoretical [100] works
suggest that oxygen at a nitrogen site (O
N
) is the main cause of unintentional
n
-typ e conductivity
in GaN. V
Ga
forms defect complexes with O
N
which sits as a next neighb our of V
Ga
to reduces
the Coulomb energy [52 , 9]. V
Ga
related defect complexes in GaN were found to have electrical
prop erties dominated by the Ga vacancy [52 ], i.e. they are acceptors and exhibit gap states ab ove
the top of the valence band arising from the N dangling b onds surrounding V
Ga
. Furthermore,
Youngman and Harris [101 ] studied the violet luminescence (VL) in AlN, which is b elieved to have
essentially the same origin as the YL in GaN [9]. They found the VL in AlN to b e correlated with
the oxygen incorp oration and extended defects which are also known to contain substantial amounts
of oxygen [102 ]. Hence, in analogy to the VL in AlN it has b een suggested that the YL in
n
{typ e
GaN is caused by O related defect complexes.
7.4. DEEP ACCEPTORS TRAPPED AT EDGE DISLOCATIONS
59
7.4.1 Benchmark calculations for V
Ga
, O
N
and V
Ga
{(O
N
)
in bulk material
In bulk material the (V
Ga
{O
N
)
2
;
defect complex (see Fig. 7.10) as well as its constituents, V
3
;
Ga
and O
+
N
have previously b een investigated by Neugebauer
et al.
[52, 8] and Mattila
et al.
[9] using
plane wave SCF{LDA metho ds. As abenchmark they are now investigated by the
SCC{DFTB
[0001]
Ga Ga Ga
O
V
NN
N
Ga
Figure 7.10: Schematic view of the V
Ga
{O defect complex. Substituting further three{fold co ordi-
nated N by O leads to V
Ga
{(O
N
)
2
and V
Ga
{(O
N
)
3
.
metho d where the defects are mo delled in 128 atom wurtzite sup ercells using two k-p oints to
sample the Brillouin zone (see app endix D). As in references [52 , 8, 9], formation energies are
evaluated assuming Ga{rich growth conditions, whichare common in many growth techniques, O
in equilibrium with Ga
2
O
3
, corresp onding to an upp er limit for the O concentration [8], and
n
{
typ e material, i.e. the Fermi level is pinned close to the conduction band minimum. The atomic
geometry of the triply charged Ga vacancy is characterised by a strong outward relaxation of the
surrounding N atoms. The three equivalentN atoms relax by 10.2% (11.8% in ref. [52]) outwards
and the remaining N atom moves 9.5% in [0001] (10.6% in ref. [52]). The formation energy is low
(1.6 eV in this work,
1
:
3 eV in ref. [52] and
1
:
5 eV in ref. [9]). Oxygen on a nitrogen site has
slightly larger Ga{O b onds than the Ga{N b ond length in bulk GaN (1.95
A). We obtain again a
low formation energy of 1.7 eV (
1.7 eV in ref. [52 ] and
1.6 eV in ref. [9 ]). Bringing V
3
;
Ga
and
O
+
N
together, one gets (V
Ga
{O
N
)
2
;
.We nd the distance b etween the vacancy core and the O (N)
increased by 13.5% (8.9%) which is close to the values of 14.9% (9.8%) given by Neugebauer
et
al.
[52]. Furthermore, we determined the energy
E
for the reaction,
(
V
Ga
;
O
N
)
2
;
!
V
3
;
Ga
+
O
+
N
;
to be 2.2 eV in good agreement with the plane wave metho ds (1.8 eV in ref. [52 ] and
2
:
1 eV
in ref. [9]). We thus get an absolute formation energy of
1
:
1eV which is again very close to
the plane wave values (
1
:
1 eV in ref. [52 ] and
0
:
9 eV in ref. [9]) implying a high equilibrium
concentration of
10
18
=
cm
3
[52, 9] at a usual MOCVD growth temp erature of
1300 K.
The go o d agreement of the
SCC{DFTB
metho d for V
3
;
Ga
, O
+
N
and (V
Ga
{O
N
)
2
;
with SCF{LDA
plane wave calculations suggests that
SCC{DFTB
allows a valid description of oxygen in GaN.
60
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
Table 7.5: Formation energiesineVofV
3
;
Ga
,O
+
N
,(V
Ga
{O
N
)
2
;
,(V
Ga
{(O
N
)
2
)
1
;
and V
Ga
{(O
N
)
3
in
a 128 atom bulk cell and at the threading edge dislo cation (see Fig. 7.8 and D.2). Ga{richgrowth
conditions, O in equilibrium with Ga
2
O
3
and
n
{typ e material are assumed.
p osition E(V
3
;
Ga
) E(O
+
N
)E(V
Ga
{O
N
)
2
;
E(V
Ga
{O
2N
)
;
E(V
Ga
{O
3N
)
bulk cell 1.8 1.7 1.1 0.7 0.8
p os. L (bulk{like) 1.7 1.5 1.0 0.9 0.7
p os. (1,2) (core) -0.2 0.2 -2.3 -2.5 -3.0
p os. (5,6) 0.3 1.0 -1.0 -1.0 -0.8
p os. (9,10) -0.3 1.3 -0.6 -0.3 -0.3
7.4.2 Prop erties of V
Ga
, O
N
and V
Ga
{(O
N
)
n
(
n
=1
;
2
;
3
) in the stress eld of the
threading edge dislo cation
In the following, the geometries, electrical prop erties and formation energies of V
Ga
,O
N
and V
Ga
{
(O
N
)
n
(
n
= 1
;
2
;
3) are investigated in the stress eld of threading edge dislo cations [91 ]. In the
SCC{DFTB
case the dislo cations are mo delled in a 312 atom sup ercell containing a dislo cation
dip ole (see Fig. D.2). In order to reduce the interaction b etween the p oint defects we doubled the
312 atom sup ercell along the dislo cation line, i.e. in [0001], to obtain a624atom sup ercell. In the
AIMPRO case, we used 286 atom stoichiometric clusters with one dislo cation.
Firstly we place the p oint defects into a bulk{like p osition, i.e. a p osition with a very small stress
eld, far away from the dislo cation core in the sup ercell (p osition Lin Fig. D.2). At p osition L
in this cell we nd the atomic geometries and formation energies of the point defects to be in
go o d agreement with the values obtained in the 128 atom p erfect lattice sup ercell (see rst two
lines in Table 7.5). We now put the defects at dierent p ositions (column (1/2), (5/6), (9/10)) in
Fig. 7.8) in the dislo cation stress eld and evaluate the formation energies and electrical prop erties.
As will be seen, some of the formation energies are negative suggesting that under equilibrium
conditions the corresp onding p osition would certainly be adopted by the defect. However, since
gallium vacancies and oxygen are not necessarily in equilibrium with the dislo cation stress eld,
the precise concentration of defect complexes in the dislo cation stress eld dep ends on the history
of the sample.
V
3
;
Ga
is trapp ed in the dislo cation stress eld, in particular, at the dislo cation core (p os. 1 in
Fig. 7.8) and at p os. 9. Ga atoms in these p ositions would have high energies, caused by the
under{co ordination in pos. 1or by the strongly strained b onds in p os. 9 (2.11
A average b ond
length). This makes the formation of vacancies at these p ositions energetically favourable (see Ta-
ble 7.5). It should b e noted that at p os. 1 a Ga vacancy creates a two{fold co ordinated N atom at
pos. 2which would result in a high energy. However, since Ga atoms at pos. 7 and its equivalent
at the right are quite close to the N atom at p os. 2, this N atom forms a b ond (2.00
A) with one of
these Ga atoms and thus achieves three{fold co ordination. The new conguration has a distorted
core and lo oks like a rst step of a kink formation. This suggests that V
Ga
play an imp ortantrole
in the dislo cation motion.
7.4. DEEP ACCEPTORS TRAPPED AT EDGE DISLOCATIONS
61
Oxygen atoms sit preferentially two or three{fold co ordinated. This explains whyO
+
N
is by 1.3 eV
more stable at the dislo cation core (p os. 2) where it replaces a three{fold co ordinated N atom than
in a bulk{like region (p os. 0) where it is four{fold co ordinated (see Table 7.5).
The high stabilities of V
3
;
Ga
and O
+
N
at the dislo cation core imply also a very low formation energy for
(V
Ga
{O
N
)
2
;
(-3.3 eV b elow the energy for p os. L) and hence a high concentration. Here O sits two{
fold co ordinated in a bridge p osition with very strong Ga{O b onds (1.72
A). Due to these strong
b onds and the high complex binding energy of 2.3 eV at the dislo cation core we exp ect (V
Ga
{O
N
)
2
;
to b e immobile. Finally,weinvestigated (V
Ga
{(O
N
)
2
)
;
and V
Ga
{(O
N
)
3
,which in analogy to (V
Ga
{
O
N
)
2
;
are found to b e particularly stable at the core of the threading edge dislo cations where they
are likely to b e immobile. See tables 7.5 for the detailed formation energies. All these results suggest
that (V
Ga
{(O
N
)
n
)
(3
;
n
)
;
(
n
= 1
;
2
;
3) defect complexes increase the oxygen concentration near to
threading edge dislo cations and in particular at the dislo cation core. Threading edge dislo cations
may therefore b e used as a trap for undesired impurities. This has b een suggested byNakamura
et
al.
[103] who prop osed that during the initial stages of GaN growth threading edge dislo cations
should be p ermitted to clean the sample from impurities which emerge from the substrate. In a
following step, a very thin SiO mask is then used to reduce the numb er of threading edge dislo cations
in the following region of the epilayer which will b e used as the active region of the devices.
Concerning the electrical prop erties the
SCC{DFTB
calculations reveal that at bulk p ositions
V
3
;
Ga
, (V
Ga
{O
N
)
2
;
, (V
Ga
{(O
N
)
2
)
;
and V
Ga
{(O
N
)
3
defects are deep acceptors with gap states
1
:
0
;
1
:
2eVabove VBM. In order to obtain information ab out the contribution of these defects to
the YL we then calculated the dierence of the formation energies dep ending on the charge states
relevant to the transition in
n
{typ e material. The results referenced to VBM are given in Table 7.6.
Subtracting them from the band gap (
3
:
4 eV) gives an estimate for the transition energies in
n
{typ e material. Since the energies for the dierent charge states are derived from total energies
asso ciated with fully relaxed atomic conguration, the calculated energy dierences corresp ond to
zero{phonon transitions. As can b e seen, at a variety of p ositions the defects could contribute to the
yellow luminescence. It is interesting to note that in a bulk{like p osition V
Ga
{(O
N
)
3
has a deep gap
state (
1
:
0 eV ab ove VBM) which comes from a three{fold co ordinated nitrogen atom in a bulk
p osition. At the dislo cation core (col. (1/2)), however, for V
Ga
{(O
N
)
3
all three{fold co ordinated
nitrogens surrounding the Ga vacancy are replaced byoxygen. At this p osition V
Ga
{(O
N
)
3
adopts
the same conguration as at the (101 0) surface (see chapter 6) and do es not induce deep states in
the band gap.
Table 7.6: Transition energies of V
Ga
,V
Ga
{O
N
,V
Ga
{(O
N
)
2
and V
Ga
{(O
N
)
3
at the threading edge
dislo cation (see Fig. 7.8 and D.2) referenced to VBM.
p osition (V
Ga
)
2
;
=
3
;
(V
Ga
{O
N
)
1
;
=
2
;
(V
Ga
{O
2N
)
0
=
1
;
(V
Ga
{O
3N
)
1+
=
0
p os. L (bulk{like) 1.4 1.0 0.7 0.9
p os. (1/2) (core) 0.8 1.0 0.7 0.4
p os. (5/6) 0.8 1.4 1.0 0.9
p os. (9/10) 0.4 0.3 0.6 0.8
62
CHAPTER 7. THREADING DISLOCATIONS AND NANOPIPES
Summary
Line defects threading along the
c
{axis have b een explored. We found full core screw dislo cations
to have a large distortion of the b onds at the dislo cation core resulting in deep states in the band
gap. Op en core screw dislo cations and in particular threading edge dislo cations, which o ccur at
very high densities, have a core structure similar to (10
10) surfaces and are therefore electrically
inactive in their pure, i.e. impurity{free form. This and the fact that threading dislo cations do not
lie on the basal glide plane makes movement and the generation of large numbers of point defects
dicult. In contrast, dislo cations in GaAs glide and climb easily through recombination{enhanced
mechanisms. This motion generates ecient radiative recombination centres which degrade optical
emissions.
Oxygen{related defect complexes, some of which are electrically active, are found to p ossess very
low formation energies at the core of threading edge dislo cations. One sp ecic oxygen related defect
complex, V
Ga
{(O
N
)
3
is b elieved to b e resp onsible for the formation of nanopip es.
Chapter 8
Domain Boundaries
In addition to the threading line defects discussed in the previous chapter, also planar defects thread
along the
c
{axis in GaN and may inuence the electrical prop erties of the devices. Fig. 8.1 shows a
transmission electron microscopy image along [0001] of a typical sample grown by MBE on a GaP
(111) substrate. Two kinds of threading planar defects, called "domain b oundaries" (DB) can be
distinguished [104 , 105 , 106, 107, 108 , 109 ]. They layon
f
11
20
g
and
f
10
10
g
planes and following
Xin
et al.
[109] are denoted by DB{I and DB{I I resp ectively. Domain b oundaries are either describ ed
in terms of a double p osition b oundary (DPB) [otherwise termed a stacking mismatch b oundary
(SMB)] consisting of a dierent stacking sequence across the b oundary, or an inversion domain
b oundary (IDB) whichischaracterised by a p olarityinversion across the b oundary.
Figure 8.1: Plan view TEM bright eld image of epitaxial wurtzite GaN of the region close to the
growth surface. Domain b oundaries on
f
1
210
g
planes (DB{I) and on
f
10
10
g
planes (DB{I I) are
visible. In the faceted DB{I b oundary, the faceted segments are on
f
1
210
g
. Y. Xin
et al.
[109].
Domain b oundaries on
f
10
10
g
planes have been extensively explored exp erimentally and theoret-
63
64
CHAPTER 8. DOMAIN BOUNDARIES
ically.The results are summarised in the section below. For domain b oundaries of the DB{I typ e
structural mo dels have b een prop osed based on high resolution transmission electron microscopy
(HRTEM) studies by Xin
et al.
[109] and Rouviere
et al.
[108 ]. However, no theoretical investiga-
tions for the energetics and electrical prop erties of these mo dels have been rep orted, presumably
b ecause of the larger sup ercells required to mo del domain b oundaries terminating in
f
11
20
g
planes.
To ll this gap weinvestigate DB{I domain b oundaries with the
SCC{DFTB
metho d in this work.
8.1 Brief review of domain b oundaries on
f
10
10
g
planes
Domain b oundaries of typ e DB{I I have b een explored extensively using transmission electron mi-
croscopy (TEM) [104 , 105 , 106, 107].
Total energy calculations by Northrup
et al.
[110 ] show that an inversion domain b oundary involving
a
c=
2 translation along the
h
0001
i
direction has a very low domain wall energy and is thus a
suitable candidate for many of the vertical defects observed on
f
10
10
g
planes. At this shifted
inversion domain b oundary denoted by IDB
{I I all atoms remain fourfold co ordinated with Ga{N
b onds across the b oundary and therefore do not induce electronic states in the band gap. Domain
b oundaries of the IDB
{typ e originate at the substrate interface and thread along the whole epilayer
since it would b e energetically very exp ensive to terminate them byovergrowth.
Furthermore Northrup
et al.
[110] investigated a double p osition b oundary (DPB{I I). DPB{I I could
account for those domain b oundaries on
f
10
10
g
planes for whichnoinversion of p olarity across the
boundary is observed [109 ]. Across the b oundary DPB{I I would have three{fold co ordinated Ga
and N atoms both in
sp
2
hybridisations whichgives rise to a deep acceptor state lo calised at the
lone pair of the
sp
2
hybridised N atoms.
8.2 Domain b oundaries on
f
11
20
g
planes
In contrast to many of the DB{I I typ e b oundaries which originate at the epilayer substrate interface
the DB{I typ e b oundaries found in a GaN sample grown by MBE on GaP (see Fig. 8.1) extend only
a short distance along the
c
{axis [109 ]. A high resolution Z{contrast image down [0001] rep orted
by Xin
et al.
[96] shows clearly that DB{I has a
horizontal
displacement of
R
h
= 1
=
2
h
1010
i
(see
Fig. 8.2). This conguration which is called prismatic stacking fault is comp osed of four{ and eight{
fold rings along the fault.
In this work [111 ] DB{I domain b oundaries are mo delled within 256 atom sup ercells containing two
b oundaries and eightlayers of atoms b etween the b oundaries (see app endix D.4).
Assuming no additional displacementinthe
vertical
, i.e.
h
0001
i
direction gives a mo del for a double
p osition b oundary denoted by DPB{I. As can b e seen in the side view in Fig. 8.3 DPB{I contains
wrong, i.e. Ga{Ga and N{N b onds. Due to the very dierent b ond lengths of b oth sp ecies (
2
:
7
A
in Ga bulk and
1
:
5
A in the N
2
molecule) wrong b onds give rise to a high energy and thus reduce
the stability of the system. The lowest energy conguration is achieved for a spacing of 2.8
A
between the b oundary planes (in the ideal lattice the corresp onding distance would be
1
:
6
A)
8.2. DOMAIN BOUNDARIES ON
f
11
20
g
PLANES
65
2
4
1
3
[11-20]
N
Ga
[10-10]
Figure 8.2: Top view along [0001] of a domain b oundary of DB{I typ e, i.e. on
f
11
20
g
planes. From
TEM exp eriments (
left
)the
horizontal
shift across the b oundary is found to be 1
=
2
h
1010
i
.
Right:
Structural mo del. All mo dels discussed below (DPB{I, DPB
{I and IDB{I) agree with this top
view. (Of course, in this gure the b onding across the b oundary is arbitrary and varies with the
dierent mo dels). Here and in the following gures atom numbers 1(2) refer to Ga (N) atoms
in eight{fold rings close to the b oundary, whereas atom numbers 3 (4) refer to Ga (N) atoms in
four{fold rings with b onds across the b oundary.Y. Xin
et al.
[96].
which is comparable with the bond length in bulk Ga. Our calculations nd a high domain wall
energy (see Eq. (5.17)) of
wall
= 246 meV/
A
2
which is only slightly less than the energy of twofree
surfaces (256 meV/
A
2
). This suggests, that DPB{I should not o ccur frequently and if it o ccurs it
should exist with dierent spacings. Indeed, we nd that varying the spacing b etween the b oundaries
changes the wall energy only slightly since the wrong bonds across the b oundary are very weak.
We note that at the equilibrium distance of 2.8
Athe structure has shallow o ccupied N-derived
states at ca. 0.2 eV ab ove the valence band maximum (VBM) and uno ccupied states at ca. 0.4
eV below the conduction band minimum (CBM). At larger distances the inuence of the Ga{Ga
b onds across the b oundary should vanish so that the electrical prop erties corresp ond to free (11
20)
surfaces whichwe found to b e electrically inactive.
Wenow examine the structure with an additional vertical displacementof 1
=
2
h
0001
i
giving a total
displacementof1
=
2
h
1011
i
as derived from TEM by Xin
et al.
[109 ]. In this double p osition b oundary
denoted byDPB
{I all atoms along the b oundary are four{fold co ordinated and form Ga{N b onds
across the b oundary (see Fig. 8.3). Since Ga{N b onds are very strong DPB
{I has a clearly dened
spacing of
1
:
90
Abetween the
f
11
20
g
planes at the b oundary. The calculated domain wall energy
of 99 meV/
A
2
is signicantly lower than the energy of the unshifted DPB{I mo del suggesting that
DPB
{I is a promising candidate for domain b oundaries in
f
11
20
g
planes for which no p olarity
inversion across the b oundary has b een observed [109 ]. DPB
{I are thought to b e asso ciated with
single growth faults in the basal plane [112 , 109 ]: DPB
{I starts and ends with a basal plane stacking
fault. Since these basal plane stacking faults have a low energy and thus are easily formed during
growth, there are many p ossibilities for DPB
{I to nucleate but also to b e overgrown. This explains
why DPB
{I are observed throughout the whole epilayer but extend only over a short distance
along the
c
{axis [109 ].
66
CHAPTER 8. DOMAIN BOUNDARIES
[11-20]
Ga
N
[0001]
[11-20]
42
13
N
Ga
[0001]
[11-20]
34
12
[0001]
N
Ga
Figure 8.3: Side view along [10
10] of the DPB{I (
left
), DPB
{I (
midd le
) and IDB{I (
right
) structures
whichhave total displacements of 1
=
2
h
1010
i
, 1
=
2
h
1011
i
and 1
=
2
h
1010
i
resp ectively.IntheDPB{I
structure wrong b onds yield a high energy which is only slightly less than that of two free (11
20)
surfaces. In the DPB
{I and IDB{I structures all atoms are four{fold co ordinated and exhibit strong
Ga{N b onds across the b oundary. DPB
{I has the lowest wall energy among all domain b oundaries
of typ e DB{I, the energy of IDB{I is slightly higher.
Table 8.1: Bond lengths in
A and b ond angles in degree at the DPB
{I domain b oundary. Atom
numb ers refer to Fig. 8.2 and 8.3.
Atom Bond Lengths (min, max) Bond Angles (min, max)
1 (Ga) 1.86, 1.95 107.0, 112.6
2 (N) 1.88, 1.96 106.1, 111.5
3 (Ga) 1.86, 2.11 80.6, 130.2
4 (N) 1.88, 2.11 86.3, 127.8
Details of the geometry of DPB
{I can be foundinTable 8.1. As can b e seen, some of the b onds
are quite distorted which makes that DPB
{I induces shallow electronic states
0
:
35 eV ab ove
VBM in the band gap. However, these states are not deep enough to b e resp onsible for the yellow
luminescence whichiscentred at
2
:
2 eV and observed in
n
{typ e GaN. On the other hand p oint
defects may segregate to the DPB
{I b oundary and change the electrical prop erties. As stated
in section 7.4 gallium vacancies were exp erimentally found to be related to the intensity of the
YL [10], a fact which is also supp orted by theoretical calculations [52, 9]. We therefore evaluated
the formation energy of V
3
;
Ga
at the domain b oundary and found it to be lower by 1.1 eV at pos.
3with resp ect to a p osition in a bulk{like environment. The electronic prop erties of
V
Ga
at the
DPB
{I were found to be similar to
V
Ga
at a p erfect lattice p osition with deep acceptor states
1
:
1eVabove VBM and
E
2
;
=
3
;
1
:
6 eV with resp ect to VBM (in a bulk{like p osition we found
E
2
;
=
3
;
1
:
4 eV). This suggests that if Ga vacancies diuse easily in GaN alot of them will b e
trapp ed at DPB
{I where they would intro duce deep acceptor states and can act as electron traps,
in agreement with recent electron energy loss sp ectroscopy (EELS) measurements by Natusch
et
al.
[113].
8.2. DOMAIN BOUNDARIES ON
f
11
20
g
PLANES
67
Table 8.2: Bond lengths in
A and b ond angles in degree at the IDB{I domain b oundary. Atom
numb ers refer to Fig. 8.2 and 8.3.
Atom Bond Lengths (min, max) Bond Angles (min, max)
1(Ga) 1.88, 1.95 105.4, 112.4
2(N) 1.87, 1.95 103.9, 111.7
3(Ga) 1.87, 2.04 87.6, 142.3
4(N) 1.87, 2.04 91.6, 141.0
A mo del for an inversion domain b oundary on
f
11
20
g
planes (IDB{I) has been suggested by
Rouviere
et al.
[108]. It has a total displacement of 1
=
2
h
1010
i
(see Fig. 8.3) and again four fold
co ordinated atoms with Ga{N bonds across the b oundary yielding a spacing of
2
:
0
Abetween
the b oundary planes. Features of the geometry are listed in Table 8.2. The domain wall energy
for IDB{I of 122 meV/
A
2
is slightly ab ove the wall energy for DPB
{I. This can be understo o d
by analysing the structural prop erties. At DPB
{I each of the b oundary atoms (No. 3 and 4 in
Fig. 8.2) has four b ond angles near to the ideal
sp
3
value of 109
:
3
o
.Onlytwo angles at each atom
deviate considerably (
80
o
and
130
o
). At IDB{I only three angles at each b oundary atom are
near to the ideal value whereas each atom has two angles of
90
o
and one angle as large as
140
o
.
The b ond angles are signicantly more distorted at IDB{I compared to DPB
{I. This explains the
higher domain wall energy found for IDB{I. In spite of the considerable distortion also IDB{I has
only shallow gap states
0
:
3eVaboveVBM.Itisworth noting that in contrast to DPB
{I which
can be terminated by a low energy basal plane stacking fault, a mechanism to end IDB{I will b e
energetically much more costly. Therefore, domain b oundaries of typ e IDB{I should thread over a
long distance along the
c
{axis.
Summary
In summary, our calculations for structural mo dels of domain b oundaries in
f
11
20
g
planes reveal
that in analogy to domain b oundaries on
f
10
10
g
planes only structures which have Ga{N b onds
across the b oundary have low formation energies. The mo del with the lowest domain wall energy
has a total displacement of 1
=
2
h
1011
i
which is in agreement with recent transmission electron
exp eriments [109 ]. This b oundary do es not induce deep states in the band gap. However, gallium
vacancies which are a common point defect in GaN could segregate to the domain b oundary and
adversely inuence the electrical prop erties.
68
CHAPTER 8. DOMAIN BOUNDARIES
Chapter 9
Reconstructions of Ga, N, H and O
terminated (0001)/(000
1) surfaces
The knowledge of the surface prop erties, in particular the typ e of reconstruction observed, can b e
imp ortant to pro duce high quality material. Indeed, the growth of material with sp ecic prop erties
is often related to the observation of a sp ecic surface reconstruction pattern during the growth
pro cess. Also for an understanding of the growth mechanism it is essential to know the prop erties
of the energetically favourable surface reconstructions. Up on these one can then simulate the dif-
fusion of atoms (in particular Ga atoms which in GaN are the rate limiting sp ecies) and suggest
improvements in the growth technique. Finally,many semiconducting devices dep end crucially up on
phenomena that o ccur at a surface or interface. Often an electrically and chemically inert surface
is desired prior to device fabrication.
In this chapter the reconstructions of the main growth surfaces in wurtzite GaN, the (0001) and
(000
1) surfaces, are explored. These surfaces are p olar, i.e. they lie in planes which are characterised
by an unequal number of cations and anions. The polar direction in which a crystal is grown is
also called the p olarity. We consider intrinsic, i.e. Ga and N terminated structures which account
for surfaces observed during MBE growth. The stabilities of the reconstructions dep ending on the
growth conditions expressed via the chemical potentials are discussed. From these results it is
p ossible to determine the p olarityof the material during growth. We also investigate the p ossible
adsorption of H which could o ccur during the MOCVD growth pro cess. Finally, we study the
adsorption of oxygen, which with a size similar to that of nitrogen is a promising candidate for
surface passivation, on top of the most common reconstructions.
9.1 Reconstructions of Ga, N and H terminated surfaces
The most common techniques for growing device quality wurtzite GaN are molecular b eam epi-
taxy (MBE) or metal-organic chemical vap our phase dep osition (MOCVD). In MBE, the growth
temp erature ranges from 600-800
o
Cand very little hydrogen is present during growth. MOCVD
growth requires a temp erature of
1000
o
Candasubstantial amount of H is present in the pre-
cursors. Thus the surface characteristics of GaN epilayers during growth may strongly dep end on
69
70
CHAPTER 9. RECONSTRUCTIONS OF (0001)/(000
1) SURFACES
the employed technique. The common growth direction is normal to the hexagonal
f
0001
g
basal
plane which exhibits a p olar conguration with two atomic sub{planes each consisting of either the
cationic or the anionic element of the binary comp ound. Hence, the ideal GaN basal plane surface is
either Ga{ or N{ terminated. Such p olar surfaces are exp ected to havevery dierentcharacteristics.
Recently Ponce
et al.
[73 ] studied the p olarity of MOCVD grown GaN and found that lms grown
in the (0001), Ga terminated, direction exhibit smo oth facets, whereas rough facets also found cor-
resp ond to (0001 ), N terminated, planes. Also Sung
et al.
[114 ] rep ort that MOCVD growth in the
(0001 ) direction results in rather rough surfaces with (1
1) p erio dicity.
Real{time monitoring has been very useful for the study of the crystal growth pro cess. Usually,
reection high{energy electron diraction (RHEED) is used in MBE for real{time monitoring of
the growth pro cess. Unfortunately in MOCVD more sophisticated techniques have to b e develop ed
since RHEED requires ultra high vacuum. RHEED exp eriments [115, 116 , 117 ] observed transitions
between (1
1), (2
2) and (4
4) reconstructions during MBE growth and co oling of GaN on
sapphire. In these works the growth direction and thus surface p olarity is not rep orted. Knowledge
of the atomic p ositions and the corresp onding absolute surface energies of the most stable surface
reconstructions may help to establish a relation between the observed RHEED pattern and the
growth direction and furthermore give a guidance for controlling the epitaxial growth pro cess. Ga
and N terminated surfaces havebeeninvestigated by Smith
et al.
exp erimentally using a varietyof
techniques, in particular STM [118] and theoretically with the SCF{LDA plane wave metho d [118].
In a recent work Smith
et al.
[119] related the surface p erio dicities observed by RHEED during
MBE growth to the lattice p olarity. As shown b elow the conclusions of this exp erimental work are
in agreement with our
SCC{DFTB
calculations [120 ].
Schematic illustration of the surfaces
We examine the geometries and stabilities of ideal surfaces, mo dels containing of a Ga mono{ and
aGabilayer and hydrogen terminated surfaces (see Fig. 9.1) to provide p ossible candidates for the
observed (1
1) RHEED pattern.
[10-10]
[000+/-1]
Ga(N)
N(Ga)
Ga/H atop
ideal
Figure 9.1: Side view of the Ga/H monolayer structure. Empty (lled) circles represent Ga (N)
atoms for the (0001) surface and N (Ga) atoms for the (0001 ). The (0001) Ga bilayer structure is
obtained by removing the Ga adlayer and changing the top layer N atoms into Ga atoms. In H
terminated structures some of the adlayer Ga atoms are replaced byH.
9.1. GA, N AND H TERMINATED SURFACES
71
[11-20]
Ga (N)
N (Ga)
[10-10]
N(Ga)
H3
[10-10]
[11-20]
T4
Ga(N)
Figure 9.2: Top view of the (2
2) vacancy (
left
) and adatom (
right
) structures at the (0001) and
(0001) surfaces. Empty (lled) circles represent Ga (N) atoms for the (0001) surface and N (Ga)
atoms for the (0001) surface. The triangle structure can be obtained by replacing the adatom at
the adatom structure by a trimer arranged in a unilateral triangle.
Following previous theoretical studies for (2
2) p erio dicities at AlN (0001)/(0001 ) [121 ] and GaAs
(111)/(
1
1
1) surfaces [53, 36 ] we have investigated the geometries and stabilities of the vacancy,
adatom and trimer induced reconstructions (see Fig. 9.2) which satisfy the electron counting rule.
The (0001) Surface
MBE growth: Ga and N surface terminations
The absolute surface energies dep ending on the gallium chemical p otential (see equation (5.5))
for the examined Ga and N covered reconstructions are shown in Fig. 9.3. Under Ga rich growth
conditions the (1
1) reconstruction consisting of a Ga monolayer added in the atop conguration
(Ga atop) has the lowest surface energy. This is in apparent contradiction with the recent works
of Smith [118] who found the Ga atop structure to be unstable in anyenvironment. We b elieve,
however, that this dierence can b e explained by the dierent symmetry constraints employed in the
calculations. By restricting the Ga atop structure to a small unit cell and therefore symmetry Smith
et al.
did not allow a Peierls distortion which often occurs at metallic surfaces. Such a distortion
would reduce the symmetry (denoted
n
m
), makes the surface semiconducting and thus lowers the
energy. Since each Ga atom in the adlayer contributes 1.25 electrons to ll the b ond with the Ga
atom in the layer b elow, to achieve a semiconducting band structure the (
n
m
) reconstruction must
satisfy the condition
nm
=(8
;
16
;
24
; :::
). In our work weevaluated the surface energy within a (4
4)
unitcell thus allowing a Peierls distortion which indeed reduces signicantly the surface energy (0.31
eV/(1
1) cell
40 meV/
A
2
) and therefore gives a stable surface under a Ga{richenvironment[120].
In Fig. 9.3 the energy is therefore drawn for the distorted (1
1) Ga atop structure. Although the
distortion breaks the (1
1) symmetry, the Peierls distortion do es not exhibit any regularity within
the (4
4) cell and is probably only weakly correlated between neighb ouring (4
4) cells, i.e. it
is likely to vary all over the surface. The Ga atop structure might therefore be acandidate for
the (1
1) RHEED pattern observed in a Ga{rich environment[116, 117]. The disorder arising
from the Peierls distortion would furthermore explain the fact that the RHEED images in Smith
72
CHAPTER 9. RECONSTRUCTIONS OF (0001)/(000
1) SURFACES
et al
[119 ] indicate that the reconstruction is not a clear (1
1) (for this reason Smith
et al.
denote
it with "(1
1)"). Going further towards N{rich growth conditions we nd that the Ga adatom
mo del in aT4 p osition b ecomes favoured. In an extreme N{rich environment the stoichiometric
(2
2) mo dels, i.e. the Ga vacancy mo del and the N adatom mo del, could b e stable having nearly
degenerate energies. In the N adatom mo del due to its negativecharge state the N adatom resides
in the H3 p osition in order to reduce the electrostatic interaction with the third layer N atom. All
other examined structures are unstable in anyenvironment. Some of them havevery high energies
and are not drawn in Fig. 9.3. In particular, the N trimer structure has an extremely high surface
energy, since the short N{N b onds in the trimer can only b e achieved by a strong distortion of the
underlying GaN surface layer which is energetically very exp ensive. This is in striking contrast to
the corresp onding GaAs (111) surface where the anion (As) trimer is very stable under As{rich
growth conditions.
These results for the Ga and N covered (0001) surface suggest that if MBE growth pro ceeds in the
(0001) direction, varying growth conditions should yield changes in the p erio dicity from (1
1)
to (2
2) in the observed RHEED patterns. Indeed the diagram in Fig. 9.3 could explain the
transitions observed during MBE growth [116 , 117] where a (1
1) p erio dicitychanges to a (2
2)
p erio dicity if the Ga ux is lowered and/or the temp erature is increased. The results are also in
very go o d agreement with the recent results of Smith
et al.
[119] who nd a "(1
1)" RHEED
pattern in Ga{richenvironment and a (2
2) pattern under an N{richenvironment. Moreover they
rep ort (6
4) and (5
5) pattern in the intermediate range.
MOCVD growth: H adsorption at the surface
During MOCVD growth a large amount of H is presentand may b e adsorb ed at the surface. Fig.
9.4 shows the energies of the ideal surface covered with 50% and 75% H together with the most
stable Ga and N terminated surfaces from Fig. 9.3. We see that under typical MOCVD growth
conditions the 75% and the 50% H{terminated surfaces have similar energies and are stable under
N{richenvironment. The 100% H coverage has a very high surface energy.We can therefore suggest
that a 50-75% H coverage is b ound to the surface under N{rich growth conditions. This agrees
with the work of Rap cewicz
et al.
[122] who state that a considerable energy gain is achieved by
H adsorption. However, no quantitativevalue is given in that work and furthermore the hydrogen
chemical p otential is xed at
H
=
H
2
which is not realistic for MOCVD growth temp eratures [51].
The 75% H covered surface has one empty Ga dangling b ond p er (2
2) surface cell and is therefore
semiconducting. If the 75% H conguration is ordered, it should b e arranged in a (2
2) p erio dicity.
If the sample is then transferred into vacuum this (2
2) p erio dicity could b e distinguished by LEED
and RHEED since all Ga atoms b ound to H are
sp
3
hybridised whereas the Ga atoms with the empty
dangling bond adopt a planar
sp
2
conguration. However, calculations within a (4
4) sup ercell
showvery little correlation b etween the p ositions of the H atoms giving an energy dierence of less
than 0.4 meV/
A
2
between the ordered and disordered p ositions. This suggests that "on average" a
(1
1) periodicity should b e observed.
From the energies shown in Fig. 9.4 wemay supp ose that MOCVD growth in the (0001) direction
pro duces only surfaces with (1
1) p erio dicity no matter whether Ga or N{richgrowth conditions
are pursued. It should however be emphasised that, although the ideal surfaces terminated with
hydrogen seem to be the most plausible H terminated surfaces, dierent mo dels for H terminated
surfaces, e.g. H on the Ga atop structure, exist whichhave not yet b een examined.
9.1. GA, N AND H TERMINATED SURFACES
73
50
100
150
200
250
300
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
N-rich Ga-rich
Ga vacancy/N adatom
Ga trimer
Ga atop
ideal
Ga adatom
surface energy in meV/A
2
µGa - µGa(bulk) in eV
Figure 9.3: Surface energies in meV/
A
2
of Ga and Ncovered GaN (0001) surfaces plotted versus
Ga
;
Ga(bulk)
. The part on the left (right) of the diagram corresp onds to N(Ga) rich growth
conditions. This diagram might explain phase transitions observed during MBE growth.
50
100
150
200
250
300
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
N-rich Ga-rich
75% H 50% H
surface energy in meV/A
µ
Ga atop
N adatom/Ga vacancy
2
Ga µGa(bulk) in eV
−
Figure 9.4: Surface energies in meV/
A
2
of the most stable Ga, N and H covered GaN (0001) surfaces
plotted versus
Ga
;
Ga(bulk)
.Thepart on the left (right) of the diagram corresp onds to N (Ga)
rich growth conditions. This diagram might explain why MOCVD growth pro duces only surfaces
with (1
1) p erio dicity.
74
CHAPTER 9. RECONSTRUCTIONS OF (0001)/(000
1) SURFACES
Finally, the b onding congurations and electrical prop erties of the stable reconstructions at the
(0001) surface are listed in Table 9.1. The eigenvalues of the highest (lowest) o ccupied (uno ccupied)
molecular orbitals HOMO (LUMO) are to b e taken as a guidance for further exp erimental studies.
The calculated
SCC{DFTB
band gap is 6.6 eV. The exp erimental band gap for bulk GaN is
3
:
4 eV. The values in brackets are "interpreted values", i.e. values the author believes to be
the true ones if a larger basis set is used so that
SCC{DFTB
pro duces the correct band gap.
1
Table 9.1: Bonding conguration and electrical prop erties of the most stable GaN (0001) surface
reconstructions. In an
sp
3
conguration one of the four bonds is highly distorted. HOMO and
LUMO are given in eV ab ove the VBM. The calculated
SCC{DFTB
band gap is 6.6 eV. The
exp erimental band gap for bulk GaN is
3
:
4 eV. In brackets are "interpreted values" (see text).
surface 1./2. layer Bonding 1./2. layer HOMO LUMO
Ga monolayer Ga/Ga non{directional 1.0 1.1
Ga adatom Ga/Ga
p
3
/
sp
3
1.3 2.2
N adatom N/Ga
p
3
/
sp
3
1.0 3.9 (3.4)
Ga vacancy Ga/N
sp
2
=p
3
0.8 4.0 (3.4)
75% H H/Ga
s=sp
3
0.0 4.0 (3.4)
The (000
1) Surface
MBE growth: Ga and N surface terminations
Fig. 9.5 shows the absolute surface energies of p ossible mo dels for (1
1) and (2
2) reconstructions
which are Ga and N covered. We see that except for extremely Ga{rich growth conditions where
an accumulation of Ga resulting in a bilayer coverage app ears to be stable, the Ga monolayer
again arranged in the atop conguration is by far the most probable conguration. In analogy to
the Ga atop reconstruction at the (0001) surface a calculation within a(4
4) cell shows that a
p ossible distortion is unlikely to b e ordered and will therefore result in a (1
1) RHEED pattern.
It is worth noting that in the recent exp erimental work of Smith
et al.
[119 ] the (1
1) Ga atop
structure is observed under N{rich growth conditions and cedes to (3
3), (6
6) and c(6
12)
p erio dicities with even more Ga atoms at the surface in a Ga{rich environment. The fact that a
(2
2) reconstruction is never observed agrees very well with our theoretical investigations but
contradicts the theoretical studies by Smith
et al.
[118]. In analogy to the (0001) surface in the
calculations of Smith
et al.
aPeierls distortion has not b een allowed.
SCC{DFTB
calculations show
that the energy gained byaPeierls distortion is considerable (0.10 eV/(1
1) cell
12 meV/
A
2
).
This might explain why the range where the Ga atop structure is stable was found to be larger
in our work [120 ] than in the work of Smith
et al.
who suggest that reconstructions with (2
2)
p erio dicity should be stable under N{rich growth conditions. In our work the ideal surface (not
drawn in the diagram) and the examined (2
2) structures are unstable under typical growth
1
The author b elieves that
sp
2
{typ e Ga derived states do not mix strongly with the conduction band states so that
a basis extension will not change their positions signicantly. Studies with an extended basis set would, however, b e
needed to justify this assumption.
9.1. GA, N AND H TERMINATED SURFACES
75
conditions. Esp ecially, the N terminated structures have very high surface energies. This is again
in contrast to the corresp onding GaAs (
1
1
1) surface where As terminated surfaces are stable over
a wide range of the growth conditions and only in a Ga{richenvironment cede to metallic surfaces
which are terminated by b oth Ga and As atoms [66].
We can therefore suggest that if GaN is grown in the (0001) direction by MBE then a (1
1)
p erio dicity and following the exp erimental work by Smith
et al.
[119 ] (3
3), (6
6) and c(6
12)
p erio dicities should b e observed. Also from our calculations we exclude (2
2) p erio dicities at the
(000
1) surface.
MOCVD growth: H adsorption at the surface
We nally examine some H terminated (0001) surfaces. Fig. 9.6 shows the energies of the ideal
surface covered with 50% and 75% H together with the most stable surfaces from Fig. 9.5. Under
typical MOCVD growth conditions we see that in an N{rich environment a semiconducting 75%
Hcoverage passivating three of the four N dangling b onds p er (2
2) cell and leaving one N lone
pair should b e stable. This mo del has already b een prop osed to account for the (1
1) p erio dicity
observed on MOCVD grown GaN (0001 ) surfaces [114 ]. Indeed, a calculation in a (4
4) cell showed
very little correlation between the H atoms (
<
0
:
2meV/
A
2
). It is also worth noting that even if
the arrangement were ordered the (2
2) pattern would probably not been seen b ecause of the
fact that nitrogen atoms which b ound to hydrogen sit in
sp
3
hybridised congurations and have a
geometry very similar to nitrogens with a lled lone pair in a
p
3
conguration. This is in contrast
to the (0001) surface where a (2
2) pattern arising from 75% hydrogen coverage could in principle
b e distinguished with RHEED and LEED due to the
sp
2
hybridised Ga atoms at the surface.
These results suggest that samples grown by MOCVD in the (000
1) direction eectively show a
(1
1) p erio dicity. The stabilities of more complicated mo dels, in particular with H on the Ga atop
reconstruction, havenotyet b een examined though.
Finally,the b onding congurations and electrical prop erties of the stable structures at the (000
1)
surface are summarised in Table 9.2.
Table 9.2: Bonding conguration and electrical prop erties of the stable reconstructions at the GaN
(0001 ) surface (see Table 9.1 and corresp onding text).
surface 1./2. layer Bonding 1./2. layer HOMO LUMO
Ga bilayer Ga/Ga non{directional 1.6 1.7
Ga monolayer Ga/Ga non{directional 0.6 1.2
75% H H/N
s=sp
3
0.0 6.6 (3.4)
76
CHAPTER 9. RECONSTRUCTIONS OF (0001)/(000
1) SURFACES
50
100
150
200
250
300
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
N-rich Ga-rich
200% Ga
Ga trimer
N vacancy
Ga adatom
Ga atop
surface energy in meV/A
2
µ
Ga µGa(bulk) in eV
-
Figure 9.5: Surface energies in meV/
A
2
of Ga and Ncovered GaN (0001) surfaces plotted versus
Ga
;
Ga(bulk)
. The part on the left (right) of the diagram corresp onds to N (Ga){rich growth
conditions. In contrast to MBE growth in the (0001) direction no (2
2) p erio dicity seems to be
stable.
50
100
150
200
250
300
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
N-rich Ga-rich
50% H
75% H
200% Ga
Ga atop
surface energy in meV/A
2
µGa − µGa(bulk) in eV
Figure 9.6: Surface energies in meV/
A
2
of the stable Ga, Nand Hcovered GaN (0001) surfaces
plotted versus
Ga
;
Ga(bulk)
. The part on the left (right) of the diagram corresp onds to N (Ga)
rich growth conditions. In MOCVD growth in both the (0001) and (000
1) direction app ears to
exclusively result in surfaces with a (1
1) periodicity.
9.2. THE CHEMISORPTION OF OXYGEN
77
Summary: Polarity determination during MBE growth
In this section we have presented a theoretical study of p ossible mo dels for the Ga, Nand H
terminated GaN (0001) and (0001) surfaces. In particular, absolute surface energies and relative
stabilities dep ending on the growth conditions have been examined. We suggest that GaN grown
by MBE in the (0001) direction exhibits transitions between "(1
1)" and (2
2) p erio dicities.
On the other hand, a (1
1) periodicity should be observed during MBE growth in the (0001)
direction over a wide range of growth conditions including the N{richenvironment. In an extremely
Ga{richenvironmentmore complicated p erio dicities can o ccur whichhave not yet b een examined
theoretically.
9.2 The chemisorption of oxygen
A detailed knowledge of surface comp osition and electronic structure is imp ortant for understanding
the contact formation which in turn is necessary for device fabrication. Often electrically and
chemically inactive surfaces are desired. Since the intrinsic surface reconstructions determined in
the previous section are Ga terminated and thus not chemically inert, it is of interest to knowhow
common adsorbates react with the surface. In particular group VI elements have b een successfully
employed to passivate I I I{V surfaces, e.g. S at GaAs. For surfaces of group I I I{nitrides, O seems a
promising candidate since its size is comparable to nitrogen. Exp erimentally it has b een found that
surface oxides are predominantly in the Ga
2
O
3
form [123 ]. Cleaning mechanisms for GaN surfaces
from surface oxides are describ ed in [106].
A detailed exp erimental study of oxygen chemisorption on the GaN (0001) surface has b een carried
out by Bermudez [124 ]. Exp osing the clean (1
1) surface to an excited O
2
ux he rep orts that
chemisorption of O at the surface takes place. Although low energy electron diraction (LEED)
exp eriments indicate an ordered adsorbate layer, and x{ray photo emission sp ectroscopy (XPS)
suggests a single chemically distinct adsorption site, a mo del for the chemisorb ed layer could not
be dened. In particular, the typ e of the observed b onding, i.e. Ga{O or N{O b onding could not
b e determined, and also the p olarity of the underlying GaN surface, i.e. (0001) or (0001 ) remained
unclear.
In this work [125 ] we use
SCC{DFTB
to examine a variety of p ossible adsorption places for oxygen
on top of the most stable reconstructions at the (0001) and (0001 ) surfaces.
The (0001) Surface
The surface energies according to (5.7) of some of the examined mo dels are shown in Fig. 9.7. Here
we cho ose the oxygen chemical p otential as the free variable in (5.7) and x the gallium chemical
p otential at
Ga
=
Ga(bulk)
.Therangeof
O
is then given by (5.11).
78
CHAPTER 9. RECONSTRUCTIONS OF (0001)/(000
1) SURFACES
-10
-7.5
-5
-2.5
0
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
100%
Ga atop
O on Ga atop
32
Ga O
ideal
(eV)µ
O O2
µ -
eV/1x1
O-rich
100% O on ideal
37.5% O on ideal
Figure 9.7: The relative energies calculated for p ossible mo dels of O at the GaN (0001) surface
are shown as a function of the O chemical potential (see (5.7)). All energies are evaluated for
Ga
=
Ga(bulk)
. The part on the left of the diagram corresp onds to equilibrium with Ga
2
O
3
,the
part on the righttoanO{richenvironment. The zero of energy in this gure is not related to the
zero of energy in Fig. 9.9.
O on top of the ideal surface
We start with placing O on top of the ideal surface. The ideal surface (see Fig. 9.1) is never observed
during GaN growth, no matter whatever growth conditions are chosen (see Fig. 9.3). However, with
one free orbital p er atom p ointing along the surface normal the chemical b ehaviour of this surface
with resp ect to the adsorption of O should b e similar to the chemical b ehaviour of the Ga adatom,
N adatom and Ga vacancy structures. As mentioned b efore these structures are b elieved to give
rise to the exp erimentally observed (2
2) reconstructions [116 , 118 , 119, 120 ]. They have b onding
congurations dominated by free Ga orbitals p ointing along the surface normal. Therefore, the ideal
surface is a go o d starting p oint for the investigation of O adsorption on Ga adatom, N adatom and
Ga vacancy structures. Moreover, during the adsorption pro cess, some diusion of Ga and N atoms
could o ccur changing the top ology of Ga adatom, N adatom and Ga vacancy structures more and
more to that of the ideal surface which clearly oers more free orbitals suitable for the oxygen
adsorption than any of the (2
2) reconstructions.
Over a wide range of the oxygen chemical potential the 37.5% mo del (see Fig. 9.8) is stable with
resp ect to the ideal surface and with resp ect to all of the (2
2) reconstructions which are not
shown in the diagram. The mo del consists of the ideal surface covered by six oxygen adatoms per
(4
4) cell. Four of the oxygens (No. 1{4 in Fig. 9.8) are three{fold co ordinated in H3 p ositions
and two oxygens (No. 5,6 in Fig. 9.8) sit two{fold co ordinated in an asymmetric bridge p osition.
The three{fold co ordinated O have Ga{O bond lengths of 2.14
Aand the two{fold co ordinated
9.2. THE CHEMISORPTION OF OXYGEN
79
Ga
O
N
1
2
[11-20]
3
4
5
6
[10-10]
Figure 9.8: Top view of the 37.5% O mo del at the ideal (0001) surface. Empty (lled) circles
represent Ga (N) atoms, grey circles O atoms. In this gure all three{fold co ordinated O atoms sit
in H3 p ositions. However, some of them might adopt T4 p ositions, i.e. they sit in top of the third
layer N, yielding a disordered structure and therefore a (1
1) LEED pattern.
O in the asymmetrical bridges have b ond lengths 1.92
Aand 2.07
A. These lengths compare well
with the b ond lengths found in bulk Ga
2
O
3
ranging from
1
:
8
;
2
:
1
A. Each of the underlying Ga
atoms contributes 0.75 electrons per bond. Therefore eachof the three{fold co ordinated oxygens
has a lled lone pair and 0.25 extra electrons. On the other hand, a two{fold co ordinated O atom
in a bridge p osition needs 0.5 electrons to ll its two lone pairs. Therefore, at a charge neutral
surface charge transfer occurs from the three{fold to the two{fold co ordinated O atoms yielding
an emptied conduction band and lled O lone pairs. Hence, this mo del is semiconducting which
explains the low surface energy. Placing some of the three{fold co ordinated O from the H3 into
a T4 p osition where they sit three{fold co ordinated ab ove the third{layer Natoms gives nearly
degenerate energies whichcan b e understo o d by the fact that the three{fold co ordinated oxygens
are nearly uncharged and thus have very little Coulombinteraction with the third layer N atoms.
The structure is therefore exp ected to b e disordered with some of the three{fold co ordinated O in
H3 and some in T4 p ositions and could give rise to a (1
1) LEED pattern.
Increasing the oxygen coverage to eight O atoms per (4
4) cell arranged in bridges results in a
metallic surface and a signicantly higher surface energy.Removing twoOper(4
4) cell from the
37.5% mo del gives a structure with a 25% oxygen coverage which is only stable in a small range of
the oxygen chemical p otential near the Ga
2
O
3
{like environment. Another p ossibilitywould be to
place 100% oxygen as an adlayer on top of the ideal surface, i.e. one oxygen p er (1
1) cell in top of
the gallium atom. The O form dimers so that each OhasoneGa{O and one O{O b ond. This full
monolayer coverage is stable in a very O{richenvironment (see Fig. 9.7). Here the dierentchemical
behaviour between N and O b ecomes clear. N
2
has a very high binding energy and is chemically
inert. Therefore N{N b onds in GaN are not formed. On the other hand, the O
2
molecule has not
such a high binding energy so that O{O b onds can b e formed at GaN surfaces.
80
CHAPTER 9. RECONSTRUCTIONS OF (0001)/(000
1) SURFACES
We therefore conclude that over a wide range of the O chemical potential an oxygen coverage of
O
=
6
16
ML = 0
:
375 ML is energetically favourable at the ideal surface. Under O{rich growth
conditions also a fully oxidised surface can b e achieved.
O on top of the Ga atop reconstruction
As discussed in the previous section, the Ga atop surface (see Fig. 9.1) could be apromising
candidate to explain the (1
1){like RHEED pattern rep orted during and after growth in the
(0001) direction [124, 115, 116, 117, 126, 119]. The exp erimentally observed surface might b e slightly
dierent from the ideal Ga atop surface in terms of p ossible Peierls distortions and irregularities in
the arrangement of the top layer Ga atoms [119 ]. However, since the atoms in the top surface layer
have exclusively Ga{Ga b onds the ideal Ga atop structure is very likely to have b onding prop erties
similar to the Ga{rich (1
1){like structures rep orted exp erimentally and serves thus asagood
starting p oint.
As shown in Fig. 9.7 a 100% O coverage on top of the Ga adlayer is very stable over the entire range
of the oxygen chemical p otential. Inthismodel alloxygens sit in three fold co ordinated p ositions.
Since the typical b ond lengths in Ga
2
O
3
are quite close to the b ond length in GaN bulk (1.95
A),
no stress is induced in the surface layers. In addition, the ideal structure ob eys nearly the electron
counting rule: if one starts to count from the ideal GaN (0001) surface the 100% Oon Ga atop
surface has four b onds (one Ga{Ga and three Ga{O b onds) and one lone pair (situated on the O
atom) per 1
1 unit cell. On the other hand, the surface contains 9.75 electrons per (1
1) unit
cell (0.75e
;
from the Ga atom at the ideal surface, 3e
;
from the Ga atom in top of it and 6e
;
from the O atom). Therefore, to match the electron counting rule completely, 0.25 electrons per
(1
1) unit cell are required. Our calculation within a (4
4) cell shows that two Ga{Ga b onds
are broken thus giving the four electrons needed. Moreover, calculation shows that these three{
fold co ordinated Ga atoms adopt an
sp
2
conguration thus lowering the energy and making the
structure semiconducting.
It is also worth noting that even if O is assumed to be in equilibrium with Ga
2
O
3
(left part
in Fig. 9.7) the 100% O coverage has approximately the same energy as the Ga atop structure,
showing that the oxygens adopt very low energy p ositions which are similar to those adopted in
the lowest energy phase Ga
2
O
3
. This explains why our calculations nd any other oxygen coverage
on the Ga atop reconstruction to b e unstable with resp ect to the 100% coverage mo del.
The (
0001
) Surface
O on top of the Ga atop reconstruction
Fig. 9.9 shows the surface energies dep ending on the O chemical potential. According to section
(9.1) the Ga atop structure seems a suitable reconstruction for the investigation of O on GaN (000
1)
surfaces. We rst consider a mo del which fulls the electron counting rule. The 75% O + 25% N
on Ga atop mo del (see Fig. 9.10) is energetically favourable over awide range of
O
. 12 O atoms
per (4
4) cell have adsorb ed on the Ga atop structure and the remaining four sites are o ccupied
with N which could have diused to the surface.
9.2. THE CHEMISORPTION OF OXYGEN
81
-18
-17
-16
-15
-14
-13
-12
-11
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
2
Ο
µ
ΟO rich
µ-
Ga atop
ideal
100% O on Ga atop
eV/1x1
eV
75% O + 25% N on Ga atop
Ga O
23
Figure 9.9: The relative energies calculated for p ossible mo dels for O at the GaN (0001) surface
are shown as a function of the O chemical p otential (see (5.7)). All energies are evaluated for
Ga
=
Ga(bulk)
.The part on the left of the diagram corresp onds to equilibrium with Ga
2
O
3
,the
part on the rightto an O{richenvironment. The zero of energy in this gure is not related to the
zero of energy in Fig. 9.7.
In this mo del each O atom has one lled lone pair and 0.25 extra electrons. The N atoms haveeach
1.25 electrons in dangling b onds. Therefore, the extra electrons of three oxygens ll one N dangling
b ond. This results in an empty conduction band, one lled lone pair at all N and O atoms and thus
in a semiconducting surface. The Ga{O b ond lengths at the surface are very similar to the Ga{N
b ond lengths yielding very little stress at the surface which explains the very low surface energy.
Any related structure with a lower oxygen concentration was found to have a higher energy. We
have also calculated structures where we replaced some of the Ga atoms in the adlayer by O so
that N{O b onds could b e formed. However, these structures turn out to haveavery high formation
energy suggesting that in any stable conguration only Ga{O b onds should exist.
Another p ossibilitywould b e to place 100% O on top of the Ga atop reconstruction. This structure
has four electrons p er 4
4 cell to o much, which in the case of the ideal structure, where all O sit
threefold on N sites, would have to be placed in the conduction band. However, GaN has a wide
band gap (
3
:
4 eV) which usually results in high energies for structures which p ossess electrons in
the conduction band. Our calculations show that it is energetically more favourable to break two
of the Ga{O b onds within the (4
4) cell giving rise to two additional O lone pairs and two Ga
derived orbitals into which the four extra electrons can be placed. Since the Ga derived orbitals
are o ccupied the Ga atoms
do not
rehybridise into
sp
2
but remain in
sp
3
p ositions. Of course, they
intro duce deep gap states which lie
1
:
4eV ab ove VBM. This explains why the 100% O on Ga
atop structure is less stable than the 75% O + 25% N on Ga atop structure with resp ect to Ga
2
O
3
and b ecomes only stable under a more O{richenvironment (see Fig. 9.9).
82
CHAPTER 9. RECONSTRUCTIONS OF (0001)/(000
1) SURFACES
O
Ga
[11-20]
N
[10-10]
Figure 9.10: Top view of the 75% O + 25% N mo del at the (0001) surface. Empty (lled) circles
represent Ga (N) atoms, grey circles O. Although the structure can be arranged within a (2
2)
p erio dicity (as shown in the gure) it is not likely to b e ordered (cf. text). Replacing the N atoms
byOgives the 100% O on Ga atop mo del.
Our results suggest that oxygen exp osure of the (0001 ) surface should yield a coverage of 0
:
75 ML
O
1
:
0ML dep ending on the O chemical potential and whether atomic N is available during
the chemisorption of O at the surface to allow the 75% O + 25% N reconstruction. We nally note
that both the 75% O + 25% N and the 100% O structure are not likely to exist in an ordered
conguration but are rather disordered and thus giverisetoa(1
1) RHEED pattern. Indeed, our
calculations show that within a (4
4) cell the energy dierence between ordered and disordered
congurations is negligible.
Summary
In conclusion wehave studied a variety of mo dels for the O chemisorption at GaN (0001) and (0001)
surfaces. The results suggest that in an O{richenvironment b oth surfaces can b e fully oxidised. In
all stable structures O is always b ound to Ga. It is worth noting that at the (000
1) surface adsorb ed
O sits already in a nitrogen p osition. During growth this is likely to make the incorp oration of O at
this surface easier. Therefore, if a lowoxygen concentration is desired, we suggest that the material
should b e grown in the (0001) direction.
Chapter 10
Conclusions
10.1 Summary
In this thesis the atomic structures, formation energies and electrical prop erties of surfaces and
extended defects in GaN were investigated with metho ds based on density functional theory. To
this end one of the metho ds, the
SCC{DFTB
scheme, has been extended to p erio dic systems so
that with reasonable accuracy formation energies of heterop olar systems can be evaluated within
sup ercells.
In view of the high density of threading defects in GaN the main ob jective of this thesis was to
derive structural mo dels for the most commonly observed line and planar defects and to discuss
their implications on the luminescence prop erties. In particular we explore the relation{ship b etween
extended defects and the frequently observed parasitic emission in the GaN sp ectrum, i.e. the yellow
luminescence, which is b elieved to arise from a transition between a shallow donor and adeep
acceptor state.
In a rst step we have predicted structures for the edge and screw dislo cations whichwere subse-
quently veried by exp eriment. We found full core screw dislo cations to havevery distorted b onds
in the core region giving rise to deep gap states which p ossibly contribute to the YL. On the other
hand, at the core structures of op en core screw dislo cations and pure edge dislo cations Ga{ and
N{dangling b onds o ccur in pairs. In analogy to the nonp olar (10
10) surface this yields a rehy-
bridisation of the threefold co ordinated Ga and Natoms into
sp
2
and
p
3
, resp ectively, resulting
in aband gap free from deep states. We therefore conclude that extended defects which exhibit
this reconstruction mechanism have no ma jor impact on the luminescence prop erties. Our nding
is consistent with exp eriments which show even material with a very high density of threading
edge dislo cations to luminesce in the blue sp ectrum. Subsequently, we investigated the b ehaviour
of intrinsic p oint defects and impurities in the stress eld of threading edge dislo cations and found
electrically active defect complexes consisting of gallium vacancies surrounded by oxygen to have
low energies at the dislo cation core. Since these p oint defects are b elieved to exist in high quantities
in MBE and MOCVD grown epilayers we suggest that if they are mobile they will segregate to
the dislo cation core. This could explain the non{uniformly distributed YL frequently rep orted for
unintentionally dop ed
n
{typ e GaN where O is b elieved to b e the main source of free carriers.
83
84
CHAPTER 10. CONCLUSIONS
Another topic which has often been related to threading dislo cations concerns the formation of
nanopip es. However, our calculations found their origin not to be caused by the elastic energy
arising from threading dislo cations. This is in agreementwith Frank's theorem where sup erscrew
dislo cations would be required to account for the large diameters of the nanopip es. Instead we
followed an exp erimental observation suggesting that impurities, in particular O, mightdiuse to
the inner walls of the nanopip es where they prevent further growth. Consequently weinvestigated
the stabilityof a varietyofoxygen related defect complexes at the nanopip e walls and identied a
very stable V
Ga
{(O
N
)
3
conguration whichis charge{neutral and passivates the surface imp eding
the growth pro cess.
Also domain b oundaries thread through GaN epilayers and might degrade the material prop erties.
They were therefore of sp ecial interest in this work.
Finally,we fo cused on the main p olar growth surfaces in order to help to understand the growth
pro cess aiming to improve the material quality.Weevaluated the formation energies for a varietyof
Ga, N and H terminated surface reconstructions at the (0001) and (000
1) surfaces dep ending on the
growth conditions. A surprising fact is that some stable p olar GaN surfaces are purely Ga terminated
and metallic with a (1
1) p erio dicity.Thisisin contrast to traditional I I I{V semiconductors which
exhibit semiconducting surfaces with b oth cation and anion termination. We found a relation{ship
between the surface p erio dicity observed by RHEED during MBE growth and the material p olarity.
This relation{ship is consistent with recently rep orted exp erimental results. Moreover, we explored
the adsorption of O at stable surface reconstructions and suggest that oxygen can b e adsorb ed up
to one monolayer if the sample is put into an oxygen richenvironment.
10.2 Outlo ok
Within this thesis total energy calculations for given surfaces and defects in wurtzite GaN have
been presented. In further investigations it will be interesting to see how the formation of these
structures o ccurs during the growth pro cess, esp ecially if they involve the segregation of p oint
defects at dislo cations and nanopip es. A high diusivity is necessary to achieve thermo dynamic
equilibrium which has b een assumed within this work. It would therefore b e desirable to determine
the diusion rate of common intrinsic p oint defects and impurities, in particular gallium vacancies
and oxygen, in bulk material and at surfaces [127 , 128]. This might also help growers to improve
the material quality.
A related topic is the investigation of dislo cation mobility based on the evaluation of kink pair
formation energies. This might explain why dislo cations in GaN are found to b e far less mobile than
those in GaAs and in I I{VI materials and thus do not degrade the material prop erties signicantly.
Finally,we b elieve that our work should stimulate exp erimentalists to carry out further investiga-
tions on the defect structures suggested. In particular, techniques which could determine the oxygen
concentration at and next to extended defects with a high spatial resolution would b e very desirable.
Furthermore, in go o d quality material deep level transient sp ectroscopy(DLTS) exp eriments could
analyse the electronic structure of defects and conrm the theoretical suggestions concerning the
origin of the yellow luminescence.
App endix A
Expressions for SCC{DFTB
A.1 Analytical evaluation of
IJ
The integral (4.18) consists of three parts:
3
J
8
Z
1
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
(A.1)
;
I
3
J
16
Z
e
;
I
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
(A.2)
;
3
J
8
Z
1
j
r
;
R
I
j
e
;
I
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
(A.3)
We therefore need to evaluate integrals of the form
Z
j
r
;
R
I
j
n
e
;
I
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
for
n
=
;
1
;
0 (A.4)
Setting
R
I
= 0 and transforming to spherical co ordinates with
r
a
=
j
r
j
; r
b
=
j
(
R
I
;
R
J
)+
r
j
and
a
=
cos
a
;
we get for (A.4):
Z
2
0
Z
1
;
1
Z
1
0
j
r
a
j
n
e
;
I
r
a
e
;
J
r
b
(
r
a
;
a
)
r
2
a
dr
a
d
a
d
We nowintro duce spheroidal co ordinates (
R
=
j
R
I
;
R
J
j
):
=
r
a
+
r
b
R
=
r
a
;
r
b
R
r
a
=
R
2
(
+
)
a
=
1+
+
and
r
b
=
R
2
(
;
)
85
86
APPENDIX A. EXPRESSIONS FOR SCC{DFTB
with the functional determinant
@
(
r
a
;
a
)
@
(
;
)
=
R
2
2
;
2
(
+
)
2
Setting
=
I
R
2
and
=
J
R
2
the integral then b ecomes:
4
R
3
R
2
n
Z
1
1
d
Z
1
;
1
d
(
+
)
n
+1
(
;
)
e
;
(
+
)
e
;
(
;
)
Setting
=
+
and
=
;
weobtainby expanding the pro duct (
+
)
n
+1
into binomials:
4
R
3
R
2
n
n
+1
X
i
=0
n
+1
i
Z
1
1
d e
;
i
+1
Z
1
;
1
d
n
+1
;
i
e
;
;
i
Z
1
;
1
d
n
+2
;
i
e
;
(A.5)
The remaining integrals can b e evaluated. For
6
= 0 one has:
Z
b
a
x
l
e
;
x
dx
=
;
x
l
e
;
x
b
a
+
l
Z
b
a
x
l
;
1
e
;
x
dx
=
;
x
l
e
;
x
b
a
;
lx
l
;
1
2
e
;
x
b
a
;
l
(
l
;
1)
x
l
;
2
3
e
;
x
b
a
=
;
l
X
m
=0
l
!
(
l
;
m
)!
m
+1
x
l
;
m
e
;
x
b
a
(A.6)
and for
=0:
Z
b
a
x
l
dx
=
1
l
+1
x
l
+1
b
a
(A.7)
Using (A.6) we get for (A.5) with
6
=0:
4
R
3
R
2
n
n
+1
X
i
=0
n
+1
i
Z
1
1
d e
;
"
i
+1
n
+1
;
i
X
m
=0
(
n
+1
;
i
)!
(
n
+1
;
i
;
m
)!
m
+1
(
;
1)
n
+1
;
i
;
m
e
;
e
;
;
i
n
+2
;
i
X
m
=0
(
n
+2
;
i
)!
(
n
+2
;
i
;
m
)!
m
+1
(
;
1)
n
+2
;
i
;
m
e
;
e
;
#
Using (A.6) once again we obtain the nal result for
6
=0:
4
R
3
R
2
n
n
+1
X
i
=0
n
+1
i
e
;
"
i
+1
X
m
=0
(
i
+ 1)!
(
i
+1
;
m
)!
m
+1
!
n
+1
;
i
X
m
=0
(
n
+1
;
i
)!
(
n
+1
;
i
;
m
)!
m
+1
(
;
1)
n
+1
;
i
;
m
e
;
e
;
;
i
X
m
=0
i
!
(
i
;
m
)!
m
+1
!
n
+2
;
i
X
m
=0
(
n
+2
;
i
)!
(
n
+2
;
i
;
m
)!
m
+1
(
;
1)
n
+2
;
i
;
m
e
;
e
;
#
(A.8)
A.1. ANALYTICAL EVALUATION OF
IJ
87
For
= 0 (A.5) b ecomes:
4
R
3
R
2
n
n
+1
X
i
=0
n
+1
i
e
;
"
i
+1
X
m
=0
(
i
+1)!
(
i
+1
;
m
)!
m
+1
!
1
n
+2
;
i
(1
;
(
;
1)
n
+2
;
i
)
;
i
X
m
=0
i
!
(
i
;
m
)!
m
+1
!
1
n
+3
;
i
(1
;
(
;
1)
n
+3
;
i
)
#
We can nowevaluate expressions (A.1-A.3).
In (A.1)
n
=
;
1and
I
= 0 giving
=
J
R
2
and
=
;
J
R
2
:
3
J
8
Z
1
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
=
3
J
8
2
R
2
e
;
;
2
2
;
e
;
;
e
)+2
e
;
=
1
R
1
;
e
;
J
R
(1 +
J
R
2
)
In (A.2)
n
=0,
=
I
+
J
2
R
and
=
I
;
J
2
R
.Thisgives for
6
= 0 from (A.8):
;
I
3
J
16
Z
e
;
I
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
=
;
3
J
I
16
4
R
3
2
e
;
1
2
+
1
;
1
2
)
(
e
;
e
;
)+
1
(
e
;
e
;
)
=
3
J
I
2
R
(
2
I
;
2
J
)
3
h
(
2
J
;
2
I
)(
I
e
;
J
R
;
J
e
;
I
R
)
R
+4
I
J
(
e
;
J
R
;
e
;
I
R
)
i
For
I
=
J
one has
=0.
n
= 0 and
=
I
+
J
2
R
gives then for (A.2):
;
I
3
J
16
Z
e
;
I
j
r
;
R
I
j
e
;
I
j
r
;
R
J
j
=
;
4
I
16
4
R
3
4
3
+
4
2
+
4
3
=
;
1
16
e
;
I
3
I
R
2
3
+
2
I
R
+
I
!
In (A.3)
n
=
;
1,
=
I
+
J
2
R
and
=
I
;
J
2
R
. This gives for
6
= 0 from (A.8):
;
3
J
8
Z
1
j
r
;
R
I
j
e
;
I
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
=
;
3
J
8
4
R
3
2
R
"
e
;
1
;
1
(
e
;
e
;
)+2
e
#
=
3
J
(
2
I
;
2
J
)
2
R
J
(
e
;
J
R
;
e
;
I
R
)
;
1
2
(
2
I
;
2
J
)
Re
;
J
R
For
I
=
J
one has
=0.
n
=
;
1and
=
I
+
J
2
R
gives then for (A.3):
;
3
I
8
Z
1
j
r
;
R
I
j
e
;
I
j
r
;
R
I
j
e
;
J
j
r
;
R
J
j
=
;
3
I
8
4
R
3
2
R
e
;
2
+
2
2
=
;
1
8
e
;
I
R
2
I
R
+
I
Adding these contributions nally gives for
IJ
in (4.18) for
R
6
=0:
IJ
=
1
R
;
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
e
;
I
R
4
J
I
2(
2
I
;
2
J
)
2
;
6
J
;
3
4
J
2
I
(
2
I
;
2
J
)
3
R
;
e
;
J
R
4
I
J
2(
2
J
;
2
I
)
2
;
6
I
;
3
4
I
2
J
(
2
J
;
2
I
)
3
R
if
I
6
=
J
and
;
e
;
I
R
1
R
+
11
I
16
+
3
2
I
R
16
+
3
I
R
2
48
if
I
=
J
(A.9)
88
APPENDIX A. EXPRESSIONS FOR SCC{DFTB
The expression for
IJ
in the case
I
=
J
could have also been derived by setting
J
=
I
+
in
IJ
(
I
6
=
J
), expanding the exp onentials and then taking the limit
!
0. This shows in a
mathematically rigorous way that
IJ
is continuous at
I
=
J
; a fact which is clear on physical
grounds since no discontinuity should arise if the Coulomb energy is evaluated for two identical
charge distributions.
Wenow consider the limit for
R
!
0. To this end the exp onentials of the terms after curly brackets
in (A.9) are expanded. The rst term gives:
(1 +
;
I
R
1!
+
O
(
R
2
))
4
J
I
2(
2
I
;
2
J
)
2
;
6
J
;
3
4
J
2
I
(
2
I
;
2
J
)
3
R
!
=
4
J
I
2(
2
I
;
2
J
)
2
;
6
J
;
3
4
J
2
I
(
2
I
;
2
J
)
3
R
+
I
6
J
;
3
4
J
3
I
(
2
I
;
2
J
)
3
+
O
(
R
)
The second term is obtained byinterchanging
I
and
J
. Adding b oth gives:
;
1
R
+
4
J
I
+
4
I
J
2(
2
I
;
2
J
)
2
+
I
6
J
;
3
4
J
3
I
;
J
6
I
+3
4
I
3
J
(
2
I
;
2
J
)
3
+
O
(
R
)
=
;
1
R
+
I
J
2(
I
+
J
)
3
(
I
;
J
)
3
h
;
5
I
;
5
2
I
3
J
+5
2
J
3
I
+
5
J
i
+
O
(
R
)
=
;
1
R
;
I
J
2(
I
+
J
)
3
(
I
;
J
)
3
h
(
I
+
J
)
2
+
I
J
(
I
;
J
)
3
i
+
O
(
R
)
=
;
1
R
;
I
J
2(
I
+
J
)
3
h
(
I
+
J
)
2
+
I
J
i
+
O
(
R
)
And hence:
IJ
=
I
J
2(
I
+
J
)
3
h
(
I
+
J
)
2
+
I
J
i
+
O
(
R
)
At
R
= 0 we are at the same atom (
I
=
J
) and thus have
I
=
J
from which relation (4.20)
follows:
II
=
5
16
I
Of course, the latter relation could also have been obtained by expanding the exp onential in the
expression for
I
=
J
in (A.9) and taking the limit
R
!
0. This shows that the function
IJ
is
continuous at
R
=0.
A.2 Numerical evaluation of
IJ
and its rst derivative for p erio dic
systems via the Ewald summation technique
Evaluation of
IJ
To compute the charge dep endent correction of the matrix elements, weneedtoevaluate terms of
the kind
X
I
q
I
JI
A.2. NUMERICAL EVALUATION OF
IJ
IN PERIODIC SYSTEMS
89
This expression can be implemented in a straightforward manner for nite systems. In periodic
systems the sum over the atoms
I
is replaced by the sum over all atoms
I
in the cell summed over
all cells:
X
R
X
I
q
I
JI
=
X
I
q
I
X
R
JI
=
X
I
q
I
JI
IJ
can b e split into a short range and a long range part. The short range part consists of the sum
of the terms following the curly bracket in (4.19). The terms in this sum decay exp onentially.The
short range part is therefore absolutely convergent. Numerically this short range part is evaluated
as a sum over a small numb er of unit cells.
The long range part representing the Coulomb interactions is not absolutely convergent, i.e. the
value changes dep ending on the order of summation. However, as the long range part can be
considered as the potential corresp onding to a charge distribution, there is only one physically
reasonable value to which it should converge. Ewald's metho d is used to compute this value:
R
6
=
R
J
;
R
I
X
R
1
j
R
J
;
(
R
+
R
I
)
j
=(
R
J
;
R
I
) (A.10)
Here is the Ewald p otential dened by [129, 130]:
(
r
)=
8
>
>
<
>
>
:
4
P
G
6
=0
e
;
G
2
4
2
e
i
Gr
j
G
2
j
+
P
R
1
;
er f
(
j
R
;
r
j
)
j
R
;
r
j
;
2
for
r
6
=0
4
P
G
6
=0
e
;
G
2
4
2
e
i
Gr
j
G
2
j
+
P
R
1
;
er f
(
j
R
;
r
j
)
j
R
;
r
j
;
2
;
2
p
for
r
=0
(A.11)
where: is the volume of the unit cell,
G
is a recipro cal lattice vector,
R
is a lattice vector,
is a
parameter determining the convergence (large
allows the neglect of the real space term) and
er f
is the error function dened by
er f
(
x
)=
2
Z
x
0
exp(
;
t
2
)
dt :
The p otential represents the p erio dic charge distribution of p ositive unit charge lo cated at 0 with
uniform negative background charge, so that the average p otential is zero.
As we have
P
I
q
I
= 0 for aneutral p erio dic structure, the background charges cancel leaving
exactly the p otential describing the charge distribution in (A.10).
If a charged structure is considered,
P
I
q
I
6
= 0. In this case, the background charges comp ensate
the ionic charges leaving a charge neutral sup ercell whichcan be p erio dically rep eated. This ap-
proximation of a comp ensating uniform background charge is often used to mo del charged defects
in sup ercells (see section D.2).
Evaluation of
@
@
R
I
IJ
In a nite system the derivativeof
IJ
canbeevaluated in a straightforward manner. For a p erio dic
system the derivative of
IJ
=
P
R
IJ
is again split into short range and long range part. While
90
APPENDIX A. EXPRESSIONS FOR SCC{DFTB
the short range part consisting of the derivatives of the terms after curly brackets in (A.9) can b e
dierentiated explicitly, the long range term is again evaluated by means of Ewald's technique:
@
@
R
I
R
6
=
R
I
;
R
J
X
R
1
j
R
I
;
(
R
+
R
J
)
j
=
d
d
R
I
(
R
I
;
R
J
)
where is the Ewald p otential dened in (A.11). The rst derivative reads for
r
6
=0:
@
(
r
)
@
r
=
4
X
G
6
=0
@
@
r
e
;
G
2
4
2
e
i
Gr
j
G
2
j
+
@
@
r
R
6
=
r
X
R
1
;
er f
(
j
R
;
r
j
)
j
R
;
r
j
Let us rst evaluate the recipro cal space term:
@
@
r
RE Z
=
@
@
r
4
X
G
6
=0
e
;
G
2
4
2
e
i
Gr
j
G
2
j
=
4
X
G
6
=0
e
;
G
2
4
2
(
;
G
) sin(
Gr
)
j
G
2
j
As the sum is absolutely convergent, we were allowed to change the order of dierentiation and
summation. In a similar waywe get for the real space term:
@
@
r
RE AL
=
@
@
r
R
6
=
r
X
R
1
;
er f
(
j
R
;
r
j
)
j
R
;
r
j
=
R
6
=
r
X
R
(
r
;
R
)
;
2
p
e
;
2
j
R
;
r
j
2
j
R
;
r
j;
1+
er f
(
j
R
;
r
j
)
j
R
;
r
j
3
Fortunately, no derivative needs to b e evaluated for the case
r
= 0, where is discontinuous, since
r
= 0 means
R
I
=
R
J
.
App endix B
(SCC)-DFTB Parameters
Table B.1 shows the values chosen to create the basis functions used for
(SCC){DFTB
following
Porezag
et al.
[31] and Elstner
et al.
[27]. The on{site energies are thoseofafree atom calculated
within SCF{LDA. The Hubbard parameters were derived from SCF{LDA calculations as the second
derivative of the total energy of a free atom with resp ect to the o ccupation numb er of the highest
o ccupied atomic orbital. According to Porezag
et al.
[31] conned atomic wavefunctions are used
for
(SCC){DFTB
as a minimal basis b ecause they are suitable to describ e the charge density in
molecules and solid state systems which usually is more contracted than the atomic charge density.
Conned atomic basis functions are created by adding a potential
r
r
0
2
to the eective p otential
V
e
in the Kohn{Sham equations for the single atom [31]. The conning radius
r
0
is chosen
2
r
cov
,
where
r
cov
is the covalent radius of the atom. Table B.1 shows the precise values of the
r
0
used in
this work. Inserted into the
DFTB
and
SCC{DFTB
metho ds the conned atomic basis functions
Table B.1: On{site energies
"
i
, Hubbard parameters
I
and conning radii
r
0
of the minimal basis set
used within the
(SCC){DFTB
metho d. All quantities are given in a.u. See text and references [31, 27]
for further explanations.
Ga 3d Ga 4s Ga 4p N2s N2p O2s O2p H1s
"
i
-0.7360 -0.3280 -0.1017 -0.6760 -0.2662 -0.8712 -0.3383 -0.2335
I
0.2084 0.2084 0.2084 0.4303 0.4303 0.4946 0.4946 0.4065
r
0
4.55 4.55 4.55 2.71 2.71 2.60 2.60 1.30
gave the electronic band structures shown in Fig. B.1. In order to calculate formation energies
within the
SCC{DFTB
metho d the Ga 3
d
electrons are included into the valence band since they
hybridise with the N 2
p
electrons. Note that due to the minimal basis set employed the band gap
calculated with
(SCC){DFTB
is by far to o large.
In Table B.2 we list the reference structures used to generate the repulsivepotentials
E
rep
according
to Eq. 4.10. For the Ga{N interaction bulk GaN in the zinc{blende (
) phase was chosen to be
the reference structure. The energy versus the lattice constant was evaluated via the Murnaghan
91
92
APPENDIX B. (SCC)-DFTB PARAMETERS
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
LAMAHKΓΓ
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
HL M KΓ A ΓA
Figure B.1: Electronic band structure for
{GaN calculated within the
DFTB
(
left
)and
SCC{DFTB
(
right
) approximation. In order to obtain more accurate formation energies, the Ga 3
d
orbitals are
included within
SCC{DFTB
since they hybridise with the N 2
s
levels. Note that the valence bands
in the Fig. on the rightshowamuch larger disp ersion.
equation of state [131 ] using the bulk mo dulus and its derivativewith resp ect to pressure.
1
All
other repulsiveinteractions were derived from fully saturated molecules. The SCF{LDAreference
energies were calculated with the all{electron
CLUSTER
co de [132]. The advantage of using fully
saturated molecules as reference structures is that they usually p ossess large band gaps so that
a crossing of o ccupied and uno ccupied electronic levels for varying distances ("level crossing") is
very unlikely. This results in constant o ccupation numb ers
n
i
so that Eq. 4.10 can easily b e used.
On the other hand, unsaturated molecules, in particular dimers, have o ccupation numb ers which
are discontinuous for varying distances. These level crossings pro duce kinks in the energy{versus{
distance curve making a determination of
E
rep
rather dicult.
Table B.2: Reference structures used to determine the repulsive p otential
E
rep
within the
(SCC){
DFTB
metho d. See text and references [31 , 27] for further explanations.
Ga{Ga Ga{N Ga{O Ga{H N{N N{O N{H O{O O{H H{H
Ga
2
H
4
{GaN GaH
2
{OH GaH
3
N
2
N{O NH
3
O
2
H
2
O H
2
1
For
{GaN the values used in this work are:
a
0
=4
:
50
A,
B
0
=1
:
84 Mbar,
B
0
0
=4
:
0.
App endix C
Range of the Chemical Potentials
within SCC{DFTB
C.1 The elemental chemical potentials
GaN
If SCF{LDA total energy calculations are used to investigate the formation energies of surfaces and
defects [46, 8] one usually calculates the heat of formation
H
f
of GaN to x the range of the
chemical p otential
Ga
given by Eq. (5.6). According to Eq. (5.3)
H
f
can b e determined from the
orthorhombic bulk phase of Ga, the wurtzite lattice phase of GaN and the N
2
molecule. The SCF{
LDA value for
H
f
is usually in reasonable agreementwith the exp erimentally derived value of
1.14 eV [49 ]. For example, Northrup
et al.
nd a value of 0.90 eV [46]. However, within
SCC{DFTB
wemake some approximations and, in particular, use a minimal basis set which is only suitable for
structures consisting of
sp
3
,
sp
2
and
p
3
{like b onding congurations. Therefore, we cannot exp ect
to obtain precise values for the total energies of structures containing more complicated b onding
congurations. In particular, energies of molecules which have triple bonds such as N
2
are not
converged within a minimal basis. Therefore, the
DFTB
energy of N
2
is considerably to o high
resulting in an increased value of
H
f
for GaN (see Table C.1). To circumvent this problem, we
plot the formation energy dep ending on the Ga chemical p otential within a range given by
Ga(bulk)
and the exp erimental heat of formation:
Ga(bulk)
;
1
:
14 eV
Ga
Ga(bulk)
:
The value of
Ga(bulk)
has b een lowered by 0.15 eV (see Table C.1) since also the orthorhombic bulk
phase of Ga lies slightly to o high in energy and therefore had to b e slightly corrected. This comes
from the fact that the extended wavefunctions of metals can not in general b e well describ ed within
our minimal basis of lo calised orbitals. The range of the chemical p otential
Ga
determined in this
way has b een successfully applied in chapters 6 and 9 to determine the formation energies of non{
stoichiometric nonp olar and p olar surfaces giving results very similar to those of Northrup
et al.
[46]
and Smith
et al.
[118]. Also in GaAs we dened the range of the gallium chemical p otential by taking
the calculated value of
Ga(bulk)
lowered by 0.15 eV for the upp er limit and used the exp erimental
93
94
APPENDIX C. RANGE OF THE CHEMICAL POTENTIALS WITHIN SCC{DFTB
value for
H
f
to x the lower limit of
Ga
. Within this range we determined the formation energies
of surfaces [36]. The results are in very go o d agreementwith SCF{LDA calculations [53] showing
that this pro cedure is transferable.
Table C.1:
SCC{DFTB
total energies E
tot
, correction of total energies (Corr) and resulting heat of
formations
H
f
for the elements and comp ounds used in this work. All quantities are given in eV.
Ga (bulk) N(N
2
)O(O
2
)H(H
2
) GaN (bulk) Ga
2
O
3
(bulk)
E
tot
-224.15 -67.32 -89.04 -9.70 -294.26 -729.3
Corr -0.15 not known -0.51 - - -
H
f
(SCC-DFTB)
- - - - 2.64 12.05
H
f
(exp) [49 ] - - - - 1.14 11.3
Ga
2
O
3
The heat of formation for Ga
2
O
3
has b een determined byenthalpy measurements from Ga in the
orthorhombic bulk phase and O
2
to be 11.3 eV [49]. In contrast to N
2
the O
2
molecule has no
triple bond and is therefore b etter describ ed within a minimal basis. We may thus attempt to
calculate
H
f
for Ga
2
O
3
within
SCC{DFTB
. However, the O
2
molecule adopts a paramagnetic
triplet structure in the ground{state whereas
SCC{DFTB
do es not include spin p olarisation eects,
i.e. it gives the energy for the singlet state as the ground{state energy. We therefore correct the
SCC{DFTB
energy for the singlet state bytheenergy dierence b etween singlet and triplet state
whichwe obtained from an SCF{LDA calculation to b e 1.02 eV for O
2
. Using this corrected value we
obtain the heat of formation as 12.05 eV (see Table C.1), in go o d agreement with the exp erimental
value.
C.2 The electro-chemical potential
In a semiconductor the electro-chemical p otential (
Fermi level
) can vary from the valence band
maximum (VBM) in
p
{typ e material to the conduction band minimum (CBM) in
n
{typ e material.
Defects usually lead to a broadening of the valence band. However, the Fermi level in Eq. (5.7) is
to b e taken with resp ect to the valence band maximum of a p erfect crystal. Therefore, one should
use the VBM calculated within a p erfect bulk cell, denoted by VBM(bulk cell). If charged defects
are treated within p erio dic cells a comp ensating uniform background charge is intro duced via the
Ewald{summation technique (see Eq. (A.11)) and gives rise to an articial p otential. This results in
a rigid shift of the sp ectrum so that the suitable VBM is not any more the same as VBM(bulk cell).
In order to get the value bywhich VBM(bulk cell) needs to b e shifted one can use a characteristic
bulk level in the sp ectrum of b oth, the charged defect cell and the uncharged p erfect bulk cell. Via
this level one can determine the shift which added to VBM(bulk cell) gives the appropriate VBM:
VBM(defect cell) = VBM(bulk cell) +
"
char
(defect cell)
;
"
char
(bulk cell)
:
App endix D
Structural Mo delling of Surfaces and
Defects
D.1 Surfaces
In this work
SCC{DFTB
is used to investigate surfaces. The surfaces were mo delled by ten mono-
layer thick slabs with periodic b oundary conditions in two dimensions. The rst six mono-layers
were allowed to relax, while the remaining atoms were xed to preserve the bulk lattice spacing.
In order to prevent articial charge transfer between the b ottom of the slab and the surface, we
follow the approach of Shiraishi [133] and saturate the dangling b onds on the b ottom with pseudo{
hydrogen. By demanding that the charge distribution in our slab mo del should not dep end on
whether we terminate our slab with an N or a Ga monolayer, one can derivean equation for the
charge contribution of the pseudo{hydrogen atoms, yielding 1.25 and 0.75 electrons p er H atom, for
the replacementofan N and a Ga atom resp ectively [59 ]. These charges corresp ond to the charge
per bond contributed from a tetrahedrally b ound N or Ga atom. For an illustration see Fig. D.1.
The vacuum region separating the slabs in the direction of the surface normal is chosen to b e 25
A.
This is sucient to reduce spurious elds in the vacuum region which can arise from a rep eated
conguration with dierent p olarities and might articially inuence the surface reconstructions.
Numerical tests for p olar surfaces have shown that the change in the surface energies is smaller
than 2 meV/
A
2
if another monolayer is added and smaller than 0.25 meV/
A
2
if the vacuum region
is doubled thus showing that our mo del is converged with resp ect to the slab thickness.
For calculations of the formation energies the lateral dimensions of the surfaces are chosen su-
ciently large (Fig. D.1) to justify the gamma p oint approximation implemented in the
SCC{DFTB
metho d. The sp ecic lateral dimensions chosen dep end on the surface orientation and the p erio dicity
to b e examined and will b e given in the corresp onding application chapters.
For a selected set of surfaces band structure calculations are p erformed. In contrast to geometry
optimisations and calculations of the surface formation energy the smallest lateral real{space unit
cell p ossible for the sp ecic reconstruction (for an example see Fig. D.1) is used in order to obtain
a large Brillouin zone in recipro cal{space.
95
96
APPENDIX D. STRUCTURAL MODELLING OF SURFACES AND DEFECTS
N
H
-- cut I
-- cut I
Ga
Ga
H
[10-10]
N
(N) I
[0001]
[11-20]
N
Ga
[10-10]
Figure D.1: Side view (along [1120]) and top view of the GaN (0001) surface. The b ottom of the
slab is saturated with pseudo{hydrogen of charge 3/4. Formation energies are evaluated within
the gamma{p oint approximation in a large sup ercell in order to compare dierent reconstructions.
Band structure calculations are p erformed within the smallest p ossible real space sup ercell so that
a large recipro cal space Brillouin zone is obtained.
D.2 Point defects
Within the
SCC{DFTB
metho d we use sup ercells to mo del p oint defects. This allows to calculate
formation energies and compared with mo delling in a cluster avoids the interaction between the
defect and the cluster surface. The inconvenient of a sup ercell mo delling is that one do es not
describ e an isolated p oint defect but a rep eated sequence of defects. This could give rise to a
non{negligible disp ersion of the defect levels due to defect{defect interaction. The simplest way
to avoid this would be to increase the size of the sup ercells and thus the distance between the
point defects. Computationally exp ensive metho ds, as e.g. plane wave schemes, make use of the
fact that an average of the defect band is very close to the level of the isolated defect. Employing
sp ecial
k
{p oints one p erforms an integration over the Brillouin zone and thus achieves the desired
averaging even within a small 32 atom sup ercell [8 ].
SCC-DFTB
oers the p ossibility to treat large
systems. We therefore take the approach of using large sup ercells of 128 atoms to overcome the
defect{defect interaction problem. Another critical problem is caused by the Coulomb interaction
arising if charged defects are mo delled in sup ercells which are p erio dically rep eated. This is usually
solved by intro ducing a comp ensating background charge. As explained in chapter 4
SCC-DFTB
do es this automatically by making use of the Ewald{summation technique for the evaluation of the
Coulomb sums.
Within the
AIMPRO
metho d clusters are used to describ e p oint defects. The advantage of mo delling
point defects in clusters over a mo delling in p erio dic systems is that in clusters the defect{defect
interaction mentioned ab oveisavoided. Unfortunately,thisadvantage is in general comp ensated by
a p ossible interaction of the defect with the cluster surface if the cluster is not chosen very large.
Of course, as in the case for the b ottom of the surface slabs also the surfaces of the clusters should
b e saturated with pseudo{hydrogens.
D.3. LINE DEFECTS
97
D.3 Line defects
Line defects are defects which are p erio dic in only one dimension. The most interesting typ e of
line defects are dislo cations which represent the b oundary of a region where slip b etween adjacent
atomic planes has taken place. Thus, a single dislo cation must either be a closed lo op within the
crystal, or terminate on the surface at b oth ends. The displacement is given by the Burgers vector
b
.Away from the dislo cation line the crystal is lo cally only negligibly dierent from the p erfect
crystal, and near the line the atomic p ositions are substantially dierent from the original crystalline
sites. The resulting strain pattern is that of the dislo cation characterised jointly by its path through
the crystal and the Burgers vector
b
.In a crystal the Burgers vector must generally be equal to
a lattice vector in order to maintain the crystallinity of the material. Such dislo cations are called
p erfect.
Dislo cations are characterised by the angle
between the dislo cation line and the Burgers vector,
see gure D.2. Sp ecial cases are edge dislo cations where the Burgers vector
b
is p erp endicular to the
dislo cation line (
=90
), and screw dislo cations where
b
is parallel to the line (
=0
). In wurtzite
GaN the most interesting dislo cations have dislo cation lines parallel to the growth direction
c
and
are therefore called
threading dislocations
. See Fig. D.2 for an example.
In this work the initial p ositions of atoms are determined via linear elasticity theory. If the atoms
are emb edded in an innite, continuous, and isotropic medium, then a straight dislo cation will
displace an atom at (
x; y ; z
)by
u
x
=
b
e
2
tan
;
1
y
x
+
xy
2(1
;
)(
x
2
+
y
2
)
u
y
=
;
b
e
2
1
;
2
4(1
;
)
ln(
x
2
+
y
2
)+
x
2
;
y
2
4(1
;
)(
x
2
+
y
2
)
!
(D.1)
u
z
=
b
s
2
tan
;
1
y
x
;
where
is the Poisson's ratio of the medium, and
b
s
,
b
e
are the screw and edge comp onents of the
Burgers vector, resp ectively. The structures are then optimised using
AIMPRO
or
SCC{DFTB
in
the conjugate gradienttechnique.
In order to evaluate line energies (see equation (5.15)) it is suitable to use
SCC-DFTB
in the
sup ercell approach. To this end two dislo cations with opp osite Burgers vector are put together.
This results in a vanishing Burgers vector and thus p erfect crystalline p erio dicity. The arrangement
can therefore b e mo delled within a sup ercell (for an example see Fig. D.2). The dimensions of the
sup ercells used in this work will b e given in the corresp onding application chapters.
A mo delling of dislo cations in a cluster like conguration is esp ecially useful if one wants to deter-
mine the geometries at the dislo cation core and wants to exclude any articial distortion arising from
the interaction of the dislo cations in the sup ercell. Within
SCC{DFTB
a very convenient mo del
can b e constructed by placing the dislo cation into a cluster which is p erio dic along the dislo cation
line, i.e. along [0001] for threading dislo cations in wurtzite GaN. The dangling b onds at the sides
of these one dimensionally p erio dic clusters are saturated with pseudo{hydrogen as describ ed in
section D.1. Since the currentversion of
AIMPRO
do es not supp ort p erio dic b oundary conditions
full clusters havetobeusedtorepresent dislo cations.
98
APPENDIX D. STRUCTURAL MODELLING OF SURFACES AND DEFECTS
[10-10]
BE
L
[1-210]
Figure D.2: View along the dislo cation line ([0001]) of a wurtzite sup ercell containing a dip ole of
threading edge dislo cations with Burgers vector
b
=
BE
=
1
3
[1210]. The cell contains 312 atoms
and has a p erio dicity of [0001] along the dislo cation line. Cutting out one dislo cation and saturating
the dangling b onds with pseudo{hydrogens gives acluster periodic along the dislo cation line (see
text). Position L is used as a bulk reference p osition to mo del p oint defects (see section 7.4).
D.4 Domain b oundaries
In this thesis domain b oundaries are investigated by the
SCC-DFTB
metho d using sup ercells with
two domain b oundaries p er cell. The sup ercells were chosen suciently large to reduce the interac-
tion b etween the domain b oundaries and to justify the gamma p oint approximation (see Fig. D.3).
[11-20]
[10-10]
Figure D.3: Top view (in [0001]) of a sup ercell mo delling a dip ole of domain b oundaries.
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Danksagung
An dieser Stelle mochte ich all jenen danken, die mir b ei der Arb eit an dieser Dissertation mit Rat
und Tat zur Seite standen. Mein Dank gilt insb esondere Prof. Dr. Thomas Frauenheim, Dr. Bob
Jones und Dr. Malcolm Heggie, die sichfur meine Arb eit sehr interessiert und sie mit Inspiration
b etreut hab en. IntensiveUnterst utzung erfuhr ichauchvon der fruchtbaren und freundschaftlichen
Zusammenarb eit mit Michael Haugk. Nat urlichdurfen die Hilfe und Diskussionen mit allen Mit-
gliedern der DFTB Grupp e in Paderb orn und Chemnitz nichtvergessen werden. Insb esondere nen-
nen mochte ichPaul Sitch, Alexander Sieck, Gerd Jungnickel, Dirk Porezag, Rafael Gutierrez, Silke
Uhlmann, Markus Elstner, Zoltan Ha jnal und Jorg Widany. Die engagierten Diskussionen mit den
genannten hab en dieser Arb eit gut getan. Michael Sternb erg sei an dieser Stelle f ur seinen grossen
Einsatz in der Systemverwaltung gedankt. Prof. Dr. M. Schreib er danke ich fur die wohlwollende
Forderung dieser Arb eit an seinem Lehrstuhl und die Bereitstellung von Arb eitsmitteln.
Wahrend meines Aufenthalts in Exeter mochte ich der AIMPRO Grupp e, Paul Leary (PL abroad),
Ben Hourahine, Chris Latham, Chris Ewels, Antonio Resende, Patrick Briddon, Jonathan Goss und
James Buttler sowie allen weiteren des DepartmentofPhysics f ur die fachlichen Ratschlage und die
gute Unterhaltung danken. Stellvertretend seien hier J. P.Srivastava, Simon Gay, Mehmet Kaknak,
Julia Gasson, Caroline Hoad, Mauro Bo ero und Hussein genannt.
Auch Mitglieder anderer Arb eitsgrupp en hab en sehr zu dieser Arb eit b eigetragen. Insb esondere
seien hier Michael Natusch, Alexander Kley, Markus Kaukonen, Yan Xin, Zusanna Liliental{Web er,
Tosja Zywietz, Jorg Neugebauer, Sven
Ob erg und Niclas Lehto erwahnt.
Schliesslich mochte ich Stefan Matzig und meinem Vater fur das Lesen meiner Dissertation und
Thomas Kohler f ur die Einarb eitung zu Beginn danken.
105
