Colloidal CdSe nanopa rticles: Linea r and
nonlinea r p rop erties of the electronic
system under high fields and high
intensities
vo rgelegt von
Master of Science
Ricca rdo Scott
geb o ren in München
von der F akultät I I – Mathematik und Naturwissenschaften
der T echnischen Universität Berlin
zur Erlangung des akademischen Grades
Dokto r der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
V o rsitzender: Prof. Dr. Thomas F riedrich
Gutachterin: Prof. Dr. Ulrik e W oggon
Gutachter: Prof. Dr. Celso de Mello Donegá
Gutachter: Dr. habil. V asily T emnov
T ag der wissenschaftlichen A ussp rache: 28.07.2017
Berlin 2017
Abstract
Photonics migh t prev en t the end of Mo ore’s la w and main tain the ev er-accelerating progress of
tec hnology , b eneficial for so ciet y and living standards in the w orld. Semiconductor nanoparticles
are an imp ortan t part of the photonic to ol-kit. Materials with high ligh t-matter in teraction,
they can tak e on new or greatly enhanced optical prop erties engineered b y comp osition, size and
shap e. Understanding their fundamen tal prop erties and ho w to manipulate them is the k ey to
put them to a successful use.
This w ork concen trates on the no v el tw o-dimensional (2D) CdSe nanoplatelets with zincblende
crystal structure. These nearly ideal quan tum w ells of finite size are an excellen t mo del system
for 2D semiconductors: atomically flat with lo w dephasing rates and inhomogeneous broadening,
and high oscillator strength and exciton binding energy . In con trast to spherical quantum dots,
nanoplatelets ha v e indep enden tly tunable ultra-strong quan tum confinemen t in one axis (z) and
w eak confinemen t in the other t wo (x and y). This enables in-depth studies of the linear and
non-linear optical prop erties of highly anisotropic quan tum confinemen t.
The effects of anisotropic confinemen t on the en v elop e and Blo c h part of the w a vefunctions
and the dielectric effects arising from the particle’s shap e in a surrounding dielectric medium are
studied in v arious w ays: temp erature and field-dep enden t photoluminescence (PL) dynamics,
angular dep enden t radiation and absorption c haracteristics (with 2D
k
-space sp ectroscop y) and
non-linear effects b y t w o-photon absorption exp erimen ts.
W e show that the ultra-strong confinemen t lifts the hea vy-hole
hh
and ligh t-hole
l h
degeneracy .
This leads to highly directed
hh
exciton emission from a brigh t plane of degenerate transition
dip oles orien ted in the platelet plane (far more directed than radiation from e.g. a linear dip ole).
By con trast, one-photon absorption at higher energies in v olves all three v alence bands and is
isotropic.
The anisotropic confinemen t lifts the en v elop e
p
-state degeneracy , pushing
p z
to m uc h higher
energies. W e demonstrate excitonic emission from lo w er
p
-states and the existence of a longi-
tudinal optical (LO) phonon b ottlenec k b et w een the ground (
s
) and excited (
p
) exciton state.
The in ter-scattering rate b et w een the t wo can be tuned by the lateral confinemen t with a high
impact on the recom bination dynamics. Lateral confinemen t also con trols the radiative rates of
the t w o states via the Giant Oscillator Strength Effect – an effect that also allo ws to extract
dephasing rates from a linear PL exp erimen t, for the first time. With this, CdSe nanoplatelets
form a three-lev el system where radiativ e and in ter-scattering rates can b e tuned b y the lateral
confinemen t, indep enden tly from the bandgap energy , whic h is giv en b y their thic kness.
iv
Ultra-strong confinemen t is also suggested to lead to the observ ed, unexp ected sup er linear
v olume scaling of the t wo-photon absorption (TP A) cross section. Main taining confinemen t
in one direction while b eing able to increase the area (i.e. volume) leads to record TP A cross
sections, exceeding an y rep orted v alues. Com bined with 2D
k
-space sp ectroscop y w e resolv e the
angular dep enden t TP A cross section and relate it to the microscopic effects of the anisotropic
confinemen t. The lifted degeneracy of the en v elop e
p
-states and of the v alence band leads
to a selection of ligh t p olarized in the platelet plane, for in traband and in terband transitions,
resp ectiv ely . This results in directional TP A, in con trast to the isotropic one-photon absorption.
The electron and hole exciton w a v efunction ov erlap can be manipulated by an external
electric field. The hereb y induced p olarization of the exciton has allo w ed to determine the first
exp erimen tal v alue of the high exciton binding energies in this system.
The dev elop ed concepts and metho ds can b e readily transferred to other ultra-strong anisotrop-
ically confined systems, suc h as 2D transition metal dic halc hogenides, p ero vkites or other I I-VI
materials.
Zusammenfassung
Die Photonik k önn te das Ende des Mo oresc hen Gesetzes v erhindern und den für Gesellschaft und
Leb ensstandard meist v orteilhaften F ortschritt v on T ec hnologie fortführen. Halbleiternanopar-
tik el sind ein wic h tiger Bestandteil photonischer An w endungen. A usgezeichnet durc h hohe
Lic h t-Materie W ec hselwirkung, nehmen sie neue o der stärk er ausgeprägte optisc he Eigensc haften
an. Diese können durc h ihre Zusammensetzung, Größe und F orm eingestellt w erden. V erständnis
ihrer fundamen talen Eigensc haften und wie diese manipuliert w erden k önnen, sind der Schlüssel
für ihren erfolgreic hen Einsatz.
Diese Arb eit b esc häftigt sic h mit den neuartigen zw eidimensionalen (2D) Zink Blende CdSe
Nanoplättc hen. Diese fast idealen Quan ten-T röge endlic her lateraler Größe sind ein ideales
Mo delsystem für 2D Halbleiter: Sie sind atomar glatt mit geringen Dephasierungsraten und
inhomogener V erbreiterung und hoher Oszillatorstärk e und Exziton bindungsenergie. Im V ergleic h
zu sphärisc hen Quan tenpunkten hab en Nanoplättc hen unabhängig einstellbare ultrastark e
Größenquan tisierung üb er ihre Dic k e (z) und sc hw ac he Größenquan tisierung in den anderen
b eiden A c hsen (x und y). Dies ermöglic h t prinzipielle Studien der linearen und nic htlinearen
optisc hen Eigensc haften ho c hanisotrop er Größenquan tisierung.
Der Einfluss ho c hanisotrop er Quan tisierung auf die Einh üllende und den Blo c h teil der W ellen-
funktion und die, durc h die F orm des P artik els und dessen dielektrisc hen K on trast zum umgeb en-
den Medium, v erursac h ten dielektrischen Effekte w erden un ter div ersen Asp ekten un tersuc h t:
T emp eratur- und F eldabhängige, zeitaufgelöste Photolumineszenz (PL) Spektroskopie, wink e-
laufgelöste Emission und Absorption (mit 2D
k
-Raum Sp ektrosk opie) und nic h tlineare Effekte
mit Zw eiphotonenabsorptionsexp erimen ten.
Wir zeigen, dass die ultrastark e Quantisierung die En tartung des Sc h w er- (
hh
) und Leic h tlo c hs
(
l h
) aufhebt. Dies führt zu extrem geric h teter Sc h werlochexzitonemission in einer Eb ene orien-
tierter en tarteter Üb ergangsdip olmomen te. Diese Emission ist viel geric h teter als b eispielsw eise
die eines linearen Dip ols. Dies steh t im K on trast zu der isotrop en Absorption in höherenergetisc he
Zustände, b ei der alle drei V alenzbänder teilnehmen.
Die anisotrop e Größenquan tisierung hebt eb enfalls die En tartung der
p
-artigen Exzitonzustände
der einh üllenden W ellenfunktion auf und sc hiebt den
p z
Zustand zu höheren Energien. Exzi-
tonisc he Emission der un teren
p
Zustände und die Existenz eines longitudinalen optisc hen
Phonon-Engpasses zwisc hen Grundzustand (
s
-Exziton) und des angeregten Zustands (
p
-Exziton)
wird gezeigt. Die Interrelaxationsrate zwisc hen den beiden Zuständen kann durch die laterale
Größe der Plättc hen eingestellt w erden mit großem Einfluss auf die Rek om binationsdynamik. Die
laterale Größenquan tisierung hat auc h erheblic hen Einfluss auf die radiativen Raten de r b eiden
vi
Zustände üb er den Gian t-Oscillator-Strength Effekt, ein Effekt der es auc h ermöglic ht zum ersten
Mal Dephasierungsraten aus einem linearen PL Exp erimen t zu extrahieren. Somit stellen CdSe
Nanoplättc hen ein Dreiniv eausystem dar, in dem die radiativ en Raten und Interrelaxationsraten
durc h die laterale Größe und die Üb ergangsenergie üb er die Dic k e unabhängig v oneinander
einstellbar sind.
Die ultrastark e transv ersale Quan tisierung ist v ermutlic h auc h für die b eobac h tete unerw artete
sup erlineare V olumenskalierung des Quersc hnitts der Zweiphotonenabsorption (TP A) v eran t-
w ortlic h. Der Erhalt der stark en Quan tisierung in z-Rich tung k om biniert mit der einstellbaren
Plättc henfläc he und damit dem T eilc hen v olumen führt zu rek ordhohen TP A Quersc hnitten,
die alle v eröffen tlic hten W erte üb ersc hreiten. Die K om bination mit 2D
k
-Raum Sp ektrosk opie
ermöglic h t es, die Wink elabhängigkeit des TP A Quersc hnitts zu un tersuchen und sie den
mikrosk opisc hen Effekten der anisotrop en Größenquan tisierung zuzuordnen. Die A ufhebung der
En tartung der
p
Exzitonzustände (En v elop e) für In trabandüb ergänge so wie die des V alenzbandes
für In terbandüb ergänge führen selektiv zur Absorption v on in der Plättc heneb ene p olarisiertem
Lic h t. Dies führt zu geric h teter Zw eiphotonenabsorption, in Kon trast zur isotrop en Einphoto-
nenabsorption.
Der Üb erlapp der exzitonisc hen Elektron und Lo c h W ellenfunktion kann durch ein äußeres
elektrisc hes F eld manipuliert w erden. Die dadurc h v erursac hte P olarisierung des Exzitons
erlaubt es, zum ersten Mal die hohen Exziton bindungsenergien dieses Systems exp erimen tell zu
b estimmen.
Die en t wic kelten K onzepte und Methoden können unmittelbar auf andere ultrastark anisotrop
größenquan tisierte Systeme wie z.B. 2D Üb ergangsmetaldic halc hogenide, Pero vskite o der w eitere
I I-VI Materialien üb ertragen w erden.
Con ten ts
Publications viii
In tro duction 1
I Linear Optical Pr oper ties of Nanop ar ticles
1 Basic Concepts 7
1.1 Electronic Prop erties of Lo w-Dimensional Semiconductors ..................................... 7
1.1.1 Bandstructure ............................................................................................... 7
1.1.2 Single-P article Electronic States ........ ........................................................... 8
1.1.3 Excitons ........................................................................................................ 10
1.1.4 Optical Absorption Sp ectra of T w o-Dimensional Semiconductors ................ 11
1.1.5 Exciton Emission .......................................................................................... 16
1.2 Lo cal Field F actors of Anisotropic Nanoparticles .................................................... 17
2 Directed Emission from Orien ted Nanoplatelets and Relation to Blo c h States 19
2.1 k -space Sp ectroscop y of Nanoplatelets Orien ted in a Monola y er ............................. 20
2.2 Nano crystal Monola y er as an Effective Medium ...................................................... 22
2.3 F rom 2D k -space Sp ectroscop y to T ransition Dip ole Momen t Distribution ............. 24
2.4 Radiation P atterns of Oriented Nanocrystals .......................................................... 30
2.5 Theoretical Considerations ....................................................................................... 33
2.6 APPENDIX: k -space Analysis ................................................................................. 39
I I
Contr olling the Ex citon Lifetime by Shape, External Fields
or Band Alignment
3 Excited-State Luminescence and an LO Phonon Bottlenec k in CdSe NPLs 49
3.1 Double Emission ...................................................................................................... 50
3.2 Dynamics ................................................................................................................. 51
3.3 A Three Lev el System - Rate Equation Mo del ........................................................ 52
3.4 Results ..................................................................................................................... 55
3.5 Exclusion of LO Phonon Replica, T rions, GS Saturation and Bi-excitons............... 56
3.6 APPENDIX: k · p en v elop e function theory ............................................................. 59
4 The Gian t Oscillator Strength Effect and Dephasing in NPLs 61
4.1 Linewidth and Oscillator Strength ........................................................................... 62
4.2 Three Lev el System Combined with Gian t Oscillator Strength Effect ..................... 65
4.3 Dephasing ................................................................................................................ 67
4.4 Exciton-Phonon Coupling Constan ts ....................................................................... 69
4.5 APPENDIX: Mo deling a Three-Lev el System with the GOST Effect ..................... 72
viii Contents
5 Field Con trol 75
5.1 Field-Dep enden t Sample Structure and Exp erimen tal Details ................................. 76
5.2 Exciton Binding Energy of Nanoplatelets Determined b y Exp erimen t .................... 76
5.3 T ransition Dipole Moment and its P olarizabilit y ..................................................... 82
5.4 APPENDIX: Field-Dep enden t Absorption of CdSe Nanoplatelets .......................... 90
6 Engineering the Dynamics of Excitons with a T yp e I I Band Alignmen t 91
6.1 Characterization of CdSe-CdT e and In v erse Hetero-Nanoplatelets .......................... 92
6.2 T emperature Dep endent Luminescence Dynamics of Hetero-platelets ..................... 93
6.3 Discussion ................................................................................................................ 97
III Non-Linear Pr oper ties: Tw o-Photon Absorption
Shape and Angular Dependence
7 T w o-Photon Absorption in CdSe Nano crystals: The Influence of Dimen-
sionalit y and Size 103
7.1 Theory...................................................................................................................... 105
7.2 Exp erimen tal T ec hniques ......................................................................................... 106
7.2.1 Nano crystal Solution Concen trations Determined b y Size and Shap e .......... 107
7.2.2 Z-Scan........................................................................................................... 110
7.2.3 T wo-Photon Photoluminescence E xcitation Sp ectroscop y ............................ 113
7.2.4 Sp ectral Pulse Width in TP A Exp erimen ts .................................................. 116
7.3 Size and Shap e Dep endence of T wo-Photon Absorption.......................................... 118
7.3.1 T wo-Photon Absorption Spectra of CdSe Nanoplatelets .............................. 118
7.3.2 V olume Scaling of the TP A Cross Section .................................................... 120
7.3.3 In trinsic TP A Cross Section: Electronic and Dielectric Con tributions ......... 122
7.3.4 Strong V ersus Intermediate Confinemen t: CdSe vs. CdS ............................. 124
7.3.5 TP A Efficiency.............................................................................................. 125
7.4 APPENDIX ............................................................................................................. 128
8 k -Space Resolv ed T w o-Photon Absorption in Orien ted Nanoplatelets 131
8.1 Sample Characterization, Exp erimen tal Setup and Mo del ...................................... 132
8.2 k -Space Resolv ed T w o-Photon Absorption: Data and Mo del .................................. 135
8.3 Theory...................................................................................................................... 136
8.4 Discussion ................................................................................................................ 139
8.5 APPENDIX ............................................................................................................ 143
9 Concluding Remarks 149
A c kno wledgmen ts 152
Bibliograph y 155
Publications and other con tributions
Articles:
* Equally con tributing authors
1. p -State Luminescence in CdSe Nanoplatelets: Role of Lateral Confinemen t
and a Longitudinal Optical Phonon Bottlenec k.
A. W. A c htstein*,
R. Scott
*, S. Kic khöfel, S. T. Jagsc h, S. Christodoulou, G. H. V.
Bertrand, A. V. Prudnikau, A. An tano vic h, M. Artemy ev, I. Moreels, A. Sc hliw a and U.
W oggon,
Phys. R ev. L ett. , 116(11), 116802 (2016).
2. Highly directed emission of CdSe nanoplatelets originating from strongly an-
isotropic 2D electronic structure.
R. Scott
*, J. Hec kmann*, A. V. Prudnikau, A. An tano vic h, A. Mikhailo v, M., N. Owsc himik o w,
M. Artem y ev, J. I. Climen te, U. W oggon, N. B. Grosse and A. W. A c h tstein
Natur e Nanote ch. A ccepted
3. Time-Resolv ed Stark Sp ectroscop y in CdSe Nanoplatelets: Exciton Binding
Energy , P olarizabilit y , and Field-Dep enden t Radiativ e Rates.
R. Scott
*, A. W. Ac h tstein*, A. V. Prudnikau, A. An tano vic h, L. D. A. Siebb eles, M.
Artem y ev and U. W oggon,
Nano L ett. 16, 6576 (2016).
4. T w o Photon Absorption in I I-VI Semiconductors: The Influence of Dimen-
sionalit y and Size.
R. Scott
*, A. W. Ac h tstein*, A. V. Prudnikau, A. An tanovic h, S. Christo doulou, I. Moreels,
M. Artem y ev and U. W oggon,
Nano L ett. , 15, 4985 (2015).
5. T emp erature dep enden t radiativ e and non-radiativ e recom bination dynamics
in CdSe-CdT e and CdT e-CdSe t yp e I I hetero nanoplatelets.
R. Scott
*, S. Kickhöfel, O. Sc ho eps, A. An tanovic h, A. V. Prudnikau, A. Chuvilin, M.
Artem y ev, U. W oggon and A. W. A c h tstein*,
Phys. Chem. Chem. Phys. , 18(4), 3197 (2016).
6. Linear Absorption in CdSe Nanoplates: Thic kness and Lateral Size Dep en-
dency of the In trinsic Absorption.
A. W. A c htstein , A. An tano vic h, A. V. Prudnikau, A.,
R. Scott
, U. W oggon and M.
Artem y ev,
J. Phys. Chem. C , 119(34), 20156 (2015).
7. One- and T w o-Photon Absorption in CdS Nano dots and Wires:
The Role of Dimensionalit y in the One- and T w o-Photon Luminescence Exci-
tation Sp ectrum.
A. W. A c h tstein, A. Ballester, J. L. Movilla, J. Hennig, J. I. Climen te, A. Prudnikau, A.
An tano vic h, R. Scott , M. V. Artem y ev, J. Planelles and U. W oggon,
J. Phys. Chem. C , 119(2), 1260 (2015).
x Contents
8. k -Space Resolv ed T w o-Photon Absorption in Orien ted Nano-Platelets
J. Hec kmann*,
R. Scott
*, A. V. Prudnikau, A. An tano vic h, N. Owschimik o w, M. Artem y ev,
J. I. Climen te, U. W oggon, N. B. Grosse and A. W. A c h tstein
Submitted to Nano L ett.
Conference con tributions and talks
• In vited talk: Exciton dynamics in CdSe nanoplatelets: The impact of an excited state
Seminar on nanoph ysics and related nanosciences,
BerliNano, Berlin, German y 2016
• In vited talk:
Excited Exciton State Luminescence in CdSe Nanoplatelets: Lateral
Confinemen t and an LO-Phonon Bottlenec k
Seminar at the c hair for photonics and opto electronics, F eldmann group,
LMU Munic h, German y 2016
• T alk:
p-State Luminescence in CdSe Nanoplatelets: The Role of Lateral Confinemen t and
an LO-Phonon Bottlenec k
A. W. A c h tstein,
R. Scott
, S. Kic khöfel, S. T. Jagsch, S. Christo doulou, G. H. V. Bertrand,
A. V. Prudnikau, A. An tano vic h, M. Artemy ev, I. Moreels, A. Sc hliw a and U. W oggon,
Nanax7, Marburg, German y 2016
• P oster:
Gian t T w o Photon Absorption in I I-VI Semiconductor Nanoplatelets: The
Influence of Dimensionalit y and Size
R. Scott
*, A. W. Ac h tstein*, A. V. Prudnikau, A. An tanovic h, S. Christo doulou, I. Moreels,
M. Artem y ev and U. W oggon,
Nanax7, Marburg, German y 2016
• P oster:
Linear Absorption and Photoluminescence of CdSe and CdSe-CdT e Nanoplatelets
A. W. A c h tstein, R. Scott , A. An tano vic h, A. V. Prudnikau, U. W oggen, M. Artem y ev
NanoGe, Berlin, German y 2016
• P oster:
T wo Photon Absorption in I I-VI Semiconductors: The Influence of Dimensionalit y
and Size.
R. Scott
, A. W. A c h tstein, A. V. Prudnikau, A. An tanovic h, S. Christodoulou, I. Moreels,
M. Artem y ev and U. W oggon,
NanoGe, San tiago de Comp ostela, Spain, 2015
• T alk: Linear and Non-linear Absorption in I I–VI Semiconductors
TUBerlin BMBF- Bilk en t TUBIT AK w orkshop,
Bilk en t Univ ersity , Ankara, T urkey 2015
Contents xi
• P oster:
One- and T wo-Photon Absorption in CdS Nanodots and Wires: The Role of
Dimensionalit y in the One- and T w o-Photon Luminescence Excitation Sp ectrum
A. W. A c h tstein, A. Ballester, J. L. Mo villa, J. Hennig, J. I. Climen te, A. V. Prudnikau,
A. An tano vic h, R. Scott , M. V. Artem y ev, J. Planelles, U. W oggon
EMRS, Lille, F rance 2015
• P oster:
Electroabsorption b y 0D, 1D and 2D Nano crystals: A Study of CdSe Colloidal
Quan tum Dots, Nanoro ds and Nanoplatelets
R. Scott
, A. W. A c h tstein, M. Artemy ev, U. W oggen, A. V. Prudnikau, L. Gurino vic h, S.
Gap onenk o
30 y ears of QDs, P aris, F rance 2014
In tro duction
In tro duced in 1947 the curren t t yp e of transistor w as 40
µ
m long [
1
]. Miniaturization has led to
the rev olution of electronics that has drastically c hanged the w orld in the last century . No w adays,
with structure sizes of around 14 nm, barrier conductivity , cross talk and decreased thermal
conductivit y limit the clo c k rate and th us the p erformance of electronic devices.
A mark et gro wing twice the global a v erage rate, photonics is in its early stage [
2
,
3
]. It
can pro vide the next dev elopmen tal leap to ov ercome the bandwidth limitations of electronics
b y h ybrid solutions or all-optical computing [
4
]. It is exp ected to ha ve a drastic impact in a
v ast range of applications. As is already the case in fib er-optic comm unication, new adv ances
in photonics will increase data transmission rates and capacit y . Laser pro cessing will pro vide
high-v olume, lo w-cost as well as individual production – think of 3D printing. By improving
diagnostics, prev en tion and treatments photonics will cut healthcare costs, enabling fast and
p ersonalized medicine. In agri-photonics imaging sensors on planes can b e used to map soils,
moisture and crop densit y , for efficient irrigation and fertilizer consumption, a task of increasing
relev ance [
5
]. Solid-state ligh ting com bined with in telligent ligh t managemen t is exp ected to cut
electricit y consumption b y ab out 70% [6] – and put resources to more efficien t use.
In researc h, adv ances in ultrafast science, quantum optics, and nano-photonics are bringing
new fundamen tal insigh ts and scien tific fields that driv e photonics further [
7
]. As for electronics,
the predominan t and successful materials in photonics are semiconductors. They satisfy the need
for materials with high ligh t-matter in teraction, further enhanced in semiconductor nanoparticles.
These can tak e on new or greatly enhanced (optical) prop erties engineered b y comp osition, size
and shap e, suc h as bandgap tunabilit y b y size quan tization [8–11].
Semiconductor quan tum dots form an established field since 30 y ears [
12
]. Particles of differen t
comp ounds and shap es ha v e emerged in the course of the last decades [
10
,
13
,
14
]. How ev er, ev en
in mono-comp ound zero-, one- and (recen tly) t w o-dimensional particles man y op en questions
remain to gain full understanding of the ligh t-matter in teraction, vital to harness their p oten tial.
Therefore it is necessary to study the effects of quan tization and shap e on the electronic system
and its coupling to the ligh t field, with resp ect to b oth linear and non-linear optical prop erties.
New t w o-dimensional (2D) materials include I I-VI nanoplatelets, nanob elts as w ell as 2D
transition metal dic halc hogenides. A tomically flat these systems exhibit exciting prop erties
of nearly ideal quan tum w ells[
15
–
30
] suc h as ro om temp erature exciton coherence[
31
], large
oscillator strength p er unit v olume compared to bulk, the Gian t Oscillator Strength Effect
(GOST)[
17
,
19
,
26
] and observ able excitonic features at ro om temp erature due to high exciton
binding energies [22, 24, 32].
2 Contents
This w ork concen trates on the linear and non-linear optical prop erties of 2D CdSe nanoplatelets
(NPLs), as a mo del system. Their thic kness of
∼
1 nm (3 to 5 CdSe monola yers) is m uc h smaller
than the deBroglie w a v elength of the electrons and holes, or the bulk exciton Bohr radius [
33
].
Their lateral extensions v ary from ten to ov er thousand square nanometers and are larger or
equal to the deBroglie w a v elength or Bohr radius. Both can b e tuned indep enden tly b y syn thesis.
The effects and phenomena arising from the ultra-strong quan tum confinemen t in one direction
and w eak quan tization in the lateral extension are studied under v arious aspects:
- Ho w do es the platelet area affect the p olarization and p opulation dynamics?
- What are the emitting states?
- Whic h role do dimensionalit y and particle size pla y for the tw o-photon absorption?
- Are emission, one- and t w o-photon absorption of nanoplatelets anisotropic and to what degree?
- T o what extent do dielectric and electronic effects determine all these properties?
- Is the exciton-phonon coupling dep enden t on the lateral size?
- What happ ens with the electronic system under an electric field?
- Do the spatially separated electron and hole in t yp e I I hetero platelets still correlate to an
exciton? And man y other exciting questions.
On the w a y w e extend several experimen tal techniques and point out to some generally
o v erlo ok ed, y et imp ortan t exp erimen tal parameters. W e dev elop analysis that tak e sets of data
of differen t exp erimen ts in to accoun t simultaneously in order to determine ph ysical prop erties,
suc h as, for example, the phonon coupling constan ts, quan tum yield, radiativ e rates, and the
exciton binding energy .
The w ork consists of eigh t c hapters organized in three parts: Linear prop erties; The effects of
shap e, field and band-alignmen t on the dynamics, and finally; Non-linear prop erties. App endices
are attac hed to eac h individual c hapter. In the first chapter the general theoretical concepts
are in tro duced suc h as confinemen t dep enden t effects on the electronic structure and the ligh t
matter in teraction – with an emphasis on 2D semiconductors.
The 2D shap e of nanoplatelets gran ts orien ted dep osition in to a monola y er, exploited in the
second c hapter. A metho d is dev elop ed that relates the fundamen tal prop erties of the electronic
system with the angle dep enden t radiation and absorption c haracteristics, i.e. 2D
k
-space
sp ectroscop y . The (isotropic) zincblende electronic system can b e altered to pro duce highly
directed emission using ultra-strong confinemen t. The absorption in to the energetically higher
states, ho wev er, remains isotropic. These con trasting prop erties are sho wn to arise from the
Blo c h parts of the w a v e functions inv olv ed in the studied transitions.
The second part in v estigates the manipulation of the dynamics of CdSe nanoplatelets b y
their area, an external electric field and the introduction of a type I I band alignmen t. First,
Contents 3
the temp erature dep enden t time-resolv ed photoluminescence is studied. An emitting excited
excitonic state is found and related to a
p
-lik e en v elop e function, observ able due to high radiative
rates and a lo w densit y of phonon mo des. A longitudinal optical phonon b ottlenec k b et w een the
ground and excited state go v erns their in ter-scattering rate as a function of the platelet size.
F urther, the GOST effect is shown to enable tunable radiativ e lifetimes of the order of 10 ps,
making CdSe nanoplatelets one of the fastest nano emitters. A three-lev el system with radiativ e
and in ter-scattering rates tunable b y the lateral size, while the emission energy is giv en b y the
thic kness, is in triguing and new. It should stimulate further studies on e.g. excited state related
lasing and size-tunable lasing thresholds.
The strong mo dulation of the dynamics under an external electric field is in v estigated b y
time-resolv ed Stark sp ectroscop y – a global analysis mo del dev elop ed to extract, for example,
the exciton transition p olarizabilit y and binding energy from photoluminescence data.
A ttac hing an outer ring of CdT e with band offsets differen t to those of CdSe forms a t yp e I I
junction at the in terface of the t wo materials. The spatial separation of the electron and hole
w a v efunction decreases the radiativ e rate of the emitting transition by t w o orders of magnitude,
compared to core only CdSe platelets men tioned ab o v e. Excitons are found to b e the prev ailing
sp ecies at the t yp e I I junction – a disadv an tage for solar cells y et promising for lasing devices.
Non-linear prop erties are discussed in the third part. The t wo-photon absorption (TP A)
dep ending on shap e, quantum confinemen t and size is in v estigated in CdSe platelets, ro ds
and quan tum dots. CdSe nanoplatelets ha v e the largest ev er rep orted TP A cross sections for
colloidal semiconductor nano crystals and sho w a step-lik e TP A spectrum. A comparison with
CdS nanoparticles suggests a univ ersal and material indep enden t mec hanism of tw o-photon
absorption enhancemen t related to ultra-strong confinemen t.
The final c hapter com bines TP A with
k
-space sp ectroscop y , shown to also be a p o werful
metho d to in v estigate the electronic nature of non-linear optical prop erties. In contrast to the
isotropic one-photon absorption in the con tin uum, orien ted nanoplatelets sho w a directional
t w o-photon absorption. It is sho wn to b e related to the p olarization selection of the Blo c h states
and en v elop e functions in v olved in the most lik ely TP transitions.
The concepts and fundamen tal insigh ts of, for example, relating
k
-space sp ectroscop y to
the Blo c h functions, studies on TP A enhancemen t and analysis of field-dep enden t data can
b e transferred to other I I-VI platelets or 2D transition metal dic halcogenides or p ero vskites.
Anisotropic confinemen t is a promising route to design the prop erties of new emitters for
photonics.
P a rt I
Linear Optical Pr oper ties of
Nanop ar ticles
1 | Basic Concepts
This c hapter is mean t to co ver the basic concepts needed throu ghout the thesis. The material
system, the semiconductor Cadmiumselenide (CdSe), and its bulk electronic band structure will
b e in tro duced. The effects of confinemen t in lo w er dimensional systems with resp ect to free c harge
carriers and excitons are discussed–with an emphasis on t w o-dimensional (2D) semiconductors
and their theoretical absorption sp ectra. Since 2D CdSe nanoplatelets are in the fo cus of this
w ork, the influence of their thic kness and size on exp erimen tal absorption sp ectra is discussed
together with an outlo ok to the c hapters treating the effects in detail. The coupling of this
electric system to the ligh t field is go v erned b y the shap e of the nano crystals and the dielectric
con trast to their surrounding and will b e discussed in the last section of this c hapter.
1.1 Electronic Prop erties of Lo w-Dimensional Semiconducto rs
1.1.1 Bandstructure
CdSe is a p olar I I-VI semiconductor that o ccurs in three differen t crystal phases: ro c k salt,
wurtzite WZ (hexagonal) and zincblende ZB (cubic). Since the ro c k salt structure is only stable
at high pressures w e concen trate on the latter t wo, sho wn in figure. 1.1. Dep ending on the
gro wth pro cedure and conditions, in e.g. colloidal syn thesis, WZ or ZB ma y b e the energetically
preferred structure.
The differen t symmetries of these crystal structures (ZB F m3m, WZ P6
3
mc space group)
pro duce differen t electronic bandstructures, see figure 1.1. As in other I I-VI semiconductors the
lo w est uno ccupied band (the conduction band CB) is formed b y the s-orbitals of the metal ions
and the highest o ccupied band (the v alence band VB) predominan tly b y the p-orbitals of the
group-VI anions [
34
]. F or CdSe these are the Cadmium 5s and Selenium 4p-orbitals, resp ectiv ely .
Optical transitions o ccur in the vicinit y of the Γ - p oin t (
k
= 0 ) since the w a v evector (k-v ector)
of a photon is negligible. The energetic separation of VB and CB at this p oin t defines the optical
bandgap
E g
of the material. In this region the bands can b e appro ximated as parab ola, so that
E ∝ ~ 2 k 2 /m e,h
, with the effectiv e mass
m e,h
of a conduction band electron (
m e >
0 ) or v alence
band hole ( m h < 0 ).
F or b oth, ZB and WZ, the sixfold degeneracy (
p x,y ,z
, eac h with spin up and down) at the
top of the v alence band at the Γ -p oin t is reduced b y spin–orbit coupling. In the case of CdSe
the t w ofold degenerate band with total angular momen tum
J
=
L
+
s
= 1
/
2 (
m j
=
±
1
/
2) , is
shifted to lo w er energies b y
∆ S O
[
34
]. It is referred to as the spin–orbit split-off (so) band in ZB
and as the C-band in WZ. In zincblende the remaining band with
J
= 3
/
2 (
m j
=
±
3
/
2
, ±
1
/
2)
8 1 Basic C oncepts
Zincblende W urtzite
E
k
hh
lh
so
Δ s o
cb E
k
A
B
C
Δ s o
cb
Δ c f
(a) (b)
(c) (d)
Figure 1.1: Bandstructure of bulk zincblende and wurtzite CdSe.
(a) Calculated band-
structure for zincblende from ref. [
35
]
©
(2006) Elsevier. (b) W urtzite bandstructure from ref. [
36
]
©
(1967) American Ph ysical So ciet y . (c) and (d) P arab olic band appro ximation around the Γ -p oin t.
The corresp onding total angular momen tum and magnetic quan tum n umber are given for eac h
band. The unit cells abov e are tak en from refs. [37] and [38].
is fourfold degenerate a t the Γ -p oin t. In WZ it is further split b y the crys tal field in teraction
leading to three bands: th e A, B and C-band, sho wn in figure 1 .1 (d). F or
k 6
=0 the degeneracy
of the
m j
subbands (
J
=3
/
2 ) in ZB is lifted forming the heavy-hole (hh) and the ligh t-hole (lh)
band, as seen in figure 1.1 (c).
1.1.2 Single-P a rticle Electronic States
In a crystal an electron exp eriences an effe ctiv e p erio dic lattice p oten tial
V 0
(
r
)=
V 0
(
r
+
R
)
giv en b y the n uclei of the atoms and all other electrons.
R
is a lattice v ector c onnecting t w o
iden tical latti ce sites. This transl ational symmetry leads to the w ell-kn o wn Blo c h theorem. In
bulk it is satisfied by the electronic w a v efu nction
ψ λ ( k , r )= e i k · r u λ ( k , r ) (1.1)
1.1 Electronic Prop erties of Lo w-Dimensional Semiconductors 9
if
u λ
(
k , r
) , kno wn as the Blo c h F unction, has the lattice p eriodicity in real space.
λ
is the energy
Eigen v alue.
u λ
(
k , r
) form a complete, orthonormal basis and are normalized to the unit cell.
The Blo c h F unctions v ary spatially on an atomic scale. The en velope function for an infinite
crystal has the form of a plane w a ve,
e i k · r
.[
32
] In a real crystal the quan tum coherence is not
main tained on a macroscopic scale so that, for small
k
-v alues, the en velope function v aries on a
mesoscopic scale.[32]
In mesoscopic particles the length of confinemen t L is still larger than the unit cell, y et
comparable to the spatial extension of the en v elop e function of excitons, electrons or holes
in the corresp onding bulk material. Here the p eriodicity is main tained. F or quan tum w ells,
wires or dots–or in their colloidal coun terparts: nanoplatelets, nanoro ds and quan tum dots–the
free motion of energetically lo w lying electron and hole states is confined in one, t w o or three
directions of space,resp ectiv ely . In this case (in an infinite well model) the env elop e function
m ust v anish at the interface of the particle. The plane w a v e in eq. 1.1 can b e replaced b y a
quan tized standing w a v e
ζ ν
. Similar to the particle in a b o x the quan tum n um b er
ν
giv es rise to
an energy lev el. Each lev el forms a subband. F or a quantum w ell (2D) confined in the z-direction,
a wire (1D) in x, y and a dot (0D) in x, y , z-direction w e find[32]:
2D ψ ( r ) = ζ n ( z ) e i ( k x x + k y y ) L − 1 u λ ( k ≃ 0 , r ) (1.2)
1D ψ ( r ) = ζ n ( x ) ζ m ( y ) e ik z z L − 1
2 u λ ( k ≃ 0 , r ) (1.3)
0D ψ ( r ) = ζ n ( x ) ζ m ( y ) ζ p ( z ) u λ ( k ≃ 0 , r ) (1.4)
The restriction of free motion in to D dimensions leads to strongly differing distributions of
a v ailable states
N
p er unit v olume at a giv en w a ve v ector or energy , see figure 1.2. The densit y
of states is giv en b y
ρ
(
E
) =
d N / d E
. In the parab olic band appro ximation for the holes in the
v alence band ( E < 0) it is:
for D=1 , 2 , 3 ρ D ( E ) ∝ − E D
2 − 1 (1.5)
for D=0 ρ D ( E ) ∝ δ ( − E ) . (1.6)
Electrons (with
E >
0 ) in the conduction band ha v e the same functional dep endence, for them
− E
is to b e replaced b y
E − E g
. The free carrier (one-particle) absorption follo ws the densit y of
states.
The quan tization of the en v elop e states for the 2D case (corresp onding to 1D confinemen t) in
eq. 1.2 is similar to the one-dimensional particle in a b o x. Assuming an infinite p oten tial barrier
10 1 Basic Concepts
Figure 1.2: Densit y of states ρ
(
E
) of an electron that can mo v e freely in three, t wo, one or
zero dimensions in space, i.e. in bulk, 2D, 1D and 0D systems, resp ectiv ely . 2D systems feature
a c haracteristic step-lik e increase at every subband. With decreasing dimensionalit y more states
concen trate at the edges of the subbands. In dots (0D) the restriction of motion in an y dimension
leads to a δ -lik e densit y of states.
at the in terfaces of the 2D structure the energies can b e found analytically [32]:
E n = n 2 ~ 2 π 2
2 m e L 2 (1.7)
for electrons with
n
= 1
,
2
,
3
. . .
forming a succession of alternating ev en and o dd states. The
lo w est state n=1, the ground state, corresp onds to an ev en w a vefunction. At the cen ter b et w een
the t w o in terfaces confining the 2D structure the env elop e functions of ev en states ha v e their
maxim um – o dd states a no de. Every quan tum n um b er corresp onds to a subband forming a step
in the 2D DOS in figure 1.2. The energies in eq. 1.7 scale in verse to the length of confinemen t L
squared. This allo ws con trol of the optical prop erties, e.g. the bandgap energy , b y the thic kness
of the 2D structure. As seen later in figure 1.5 (a) this thickness quan tization also holds for CdSe
nanoplatelets.
1.1.3 Excitons
Due to Coulom b in teraction an electron in the conduction band and an uno ccupied hole state
can form a h ydrogen-lik e b ound state called exciton. The attraction b et w een the t wo particles
correlates their motion. The exciton motion can b e decomp osed in to the unaffected cen ter of
mass (CM) motion and the relativ e motion of the t wo c harge carriers.
The exciton w a v e function can b e expressed as a linear com bination of the one particle w av e
functions (eq. 1.1):
Ψ ( r e , r h ) = X
k e , k h
ξ ( k e , k h ) ψ ( r e , k e ) ψ ( r h , k h ) (1.8)
1.1 Electronic Prop erties of Lo w-Dimensional Semiconductors 11
with the expansion co efficien t
ξ ( k e , k h )
. At the Γ -p oin t w e can assume the lattice p erio dic
functions to b e constan t in
k
,
u v ,c
(
k
=
0 , r
) for the v alence band (index
v
) and conduction band
(index c ), so that
Ψ ( r e , r h ) = u v ( 0 , r ) u c ( 0 , r ) X
k e , k h
ξ ( k e , k h ) e i k e r e e i k h r h
| {z }
φ ( r e , r h )
(1.9)
Here φ ( r e , r h ) is the exciton en v elop e function describing the electron-hole in teraction.
W e consider the case where the extension of the ground state exciton is considerably larger
than the lattice constan t. F or these W annier excitons the Coulom b p oten tial v aries little o ver
one unit cell and the effectiv e mass appro ximation is applicable [
32
,
39
]. Center of mass and
relativ e motion can b e separated in the Sc hrö dinger equation. F or the relativ e motion we obtain
the t w o-particle, W annier, equation:
− ~ 2
2 µ ∇ 2
r + e 2
4 π εr φ ( r ) = E r φ ( r ) (1.10)
With the reduced mass
1
µ
=
1
m e
+
1
m h
and relativ e electron-hole co ordinate
r
=
r h − r e
and
distance
r
=
| r |
. The CM motion is describ ed b y a free particle with the mass
M
=
m e
+
m h
,
k -v ector K and energy E C M . The total exciton energy is [39]:
E n = E g + E r + E C M = E g − E B
n 2 + ~ 2 K 2
2 M (1.11)
with the bandgap energy
E g
, the binding energy of the state
E r
, Rydberg energy
E B
=
e 2 µ/ (2 ~ 2 ε 2 ) and the exciton quan tum n um b er n .
The W annier equation in form of eq. 1.10 is v alid for three and t w o-dimensional semiconductors.
F or one-dimensional wires or ro ds along the z-direction of radius R the Coulomb potential m ust
b e replaced b y the quasi-1D Coulom b p oten tial
V 1 D
=
e 2 /
(
| z |
+
γ R
)
ε 0
with the empirical
parameter
γ
[
32
]. T o obtain exciton energies of zero dimensional systems the e-h Coulom b
in teraction can b e treated b y p erturbation theory [
34
,
40
,
41
]. All cases lead to discrete states
b elo w the band gap and a con tin uum of unbound states ab o ve it.
1.1.4 Optical Abso rption Sp ectra of T wo-Dimensional Semiconducto rs
Figure 1.3 sho ws absorption sp ectra of colloidal CdSe nanoplatelets, ro ds and quan tum dots.
Qualitativ ely they can b e describ ed b y the one-particle (uncorrelated carrier) absorption of
sev eral bands follo wing the densit y of states together with excitonic (correlated) features b elo w
eac h band.
12 1 Basic Concepts
E n e r g y
A b s o r b a n c e
E n e r g y E n e r g y
Figure 1.3: Ro om temp erature absorption sp ectra of 2D, 1D and 0D CdSe nanopar-
ticles
i.e. nanoplatelets, nanoro ds and quan tum dots. F or nanoplatelets the 2D densit y of states
steps of eac h band (hea vy-, light-, split-off hole band) plus the corresponding energetically low er
lying first excitonic p eak can b e seen. Nanoro d and quan tum dot sp ectra also sho w excitonic
features.
Since 2D CdSe nanoplatelets are the fo cus of this thesis w e will concen trate on the optical
absorption sp ectrum of t w o dimensional semiconductors. F or simplicit y the follo wing discussion
is restricted to t w o bands at zero temp erature–resulting in a fully o ccupied v alence band (index
v ) and an empt y conduction band (index c ).
T ransition dip ole matrix element
W e consider an optical transition of an electron from a VB state
| a i
to a CB state
| b i
with the
w a v efunction:
| Ψ λ i = | u λ ( r , k ) i | φ λ ( r ) i (1.12)
Where
u λ
and
φ λ
are the p erio dic Blo c h function and the en v elop e function of the electron in
the band
λ
=
c,v
. In the en v elop e function appro ximation the dip ole matrix elemen t can b e
expanded:
P ab = h b | p | a i = h φ b | h u b | p | u a i | φ a i = h φ b | φ a i h u b | p | u a i + h φ b | p | φ a i h u b | u a i (1.13)
p
is the momen tum op erator. The last term v anishes for in terband transitions because of the
orthogonalit y of the Blo c h functions
h u a | u b i
= 0 for
a 6
=
b
. So that for the in terband transition
P ab = h φ b | φ a i h u b | ~ p | u a i . (1.14)
This giv es rise to the follo wing selection rules: The en v elop e in tegral
h φ b | φ a i
=
C ab
only giv es
finite v alues if the en velope functions of
| a i
and
| b i
are of the same symmetry , resp ectiv ely , for
eac h direction in space. As w e will see later nanoplatelets are confined not only in the short axis.
Their w eak confinemen t in the longer lateral extensions also leads to further lo calization of the
en v elop e function. The selection of p olarization of ligh t (for in terband transitions) is giv en b y
the Blo c h functions
h u b | ~ p | u a i
and is discussed in detail in c hapter 2. The photon w a v e vector
1.1 Electronic Prop erties of Lo w-Dimensional Semiconductors 13
can again b e assumed negligible with resp ect to the carriers’
k e , k h k ph (1.15)
giving the
k
-conserv ation rule for an optical transition. The matrix elemen t can then b e written
as [42]:
| P ab | 2 = h| P cv | 2 i δ k e , k h C ab (1.16)
The momen tum matrix elemen t for a transition b et w een v alence and conduction band
h| P cv | 2 i
is a v eraged b ecause random p olarization of ligh t is assumed. It can b e expressed b y the electron
rest mass
m 0
, the electron mass in the conduction band
m e
, the crystal field splitting
∆ cf
and
band gap energy E g [42]:
h| P cv | 2 i = m 2
0 E g ( E g + ∆ cf )
3 m e ( E g + 2 ∆ cf / 3) (1.17)
Continuum
The optical absorption for photon energies
~ ω
exceeding the band-gap energy in a 2D semicon-
ductor is [32]:
α cont ( ~ ω ) = 4 π ε
m 2
0 cna 0 ω h| P cv | 2 i
| {z }
F ree carriers
· e π
√ ∆
cosh π
√ ∆
| {z }
Coulom b enhancement
(1.18)
with the p ermittivit y
ε
, refractiv e index n, sp eed of ligh t c, electron rest mass
m 0
and 3D exciton
Bohr radius
a 0
= 4
π ε ~ 2 /µe 2
. The detuning
∆
= (
~ ω − E g
)
/E B
is expressed as a function of the
bulk exciton binding energy
E B
. The contin uum absorption th us comprises of the band-band
con tribution from the free carriers and the Coulom b enhancemen t, also kno wn as the Sommerfeld
factor, whic h v anishes for high energies ∆ → ∞ .
Excitons
The binding energy of excitons in bulk (3D) and of ideal 2D excitons b elo w the bandgap is,
resp ectiv ely
E n
B , 3 D = ~ 2
2 µa 2
0 · ( n ) − 2 for 3D and n = 1 , 2 , 3 , . . . (1.19)
E n
B , 2 D = 2 ~ 2
µa 2
0 · n + 1
2 − 2
for 2D and n = 0 , 1 , 2 , . . . (1.20)
As indicated in figure 1.4 the 2D exciton biding energy is four times higher than in bulk for the
lo w est, ground state. Compared to bulk excitons, 2D excitons are th us sp ectrally far b etter
resolv ed and observ able also at ro om temp erature. The 2D Bohr radius is half that of bulk
14 1 Basic Concepts
- 6 - 4 - 2 0 2 4 6 8 1 0
E x c i t o n s C o u l o m b e n h a n c e m e n t
F r e e c a r r i e r s
Figure 1.4: Sc hematic absorption sp ectrum of a 2D semiconductor around the band
edge at 0 K.
Plotted as a function of the detuning
∆
= (
~ ω − E g
)
/E B
, normalized to the bulk
exciton binding energy E B . Note that the 2D exciton binding energy is E B , 2 D = 4 E B .
a 2 D
0 = a 0 / 2 [32, 42]. The oscillator strength of 2D excitons is cubic in n[42]:
f n = h| P cv | 2 i 2
m 0 π a 2
0 ~ ω n + 1
2 − 3
, n = 0 , 1 , 2 , . . . (1.21)
so that the lo w est, first exciton has b y far the highest contribution to the absorption spectrum
f 0 /f 1
= 27 . F urther it can b e sho wn that the transition oscillator strength of a 2D exciton is 16
times higher as compared to bulk [
42
]. The con tributions of free carriers as well as Coulom b
correlated b ound (excitons) and un b ound states to the ideal 2D absorption sp ectrum are plotted
in figure 1.4.
CdSe Nanoplatelets Abso rption Sp ectra
The ideal situation ab o v e has neglected sev eral effects. Thermal o ccupation of free carriers
results in o ccupied conduction band states and uno ccupied v alence band states. The underlying
F ermi-Dirac statistics leads to a smo other absorption edge at the bandgap. In ZB CdSe the
three v alence bands (heavy , light, and split-off ) lead to more than one band-edge. Exciton-
phonon scattering with acoustic and longitudinal optical phonons broadens sp ectral features,
con tributing to homogeneous broadening, (see also c hapter 4). Inhomogeneous broadening may
arise from fluctuations of confinemen t and material comp osition. F urther, for nano crystals, image
c harge effects from differen t dielectric surrounding [17, 43] as w ell as the (frequency and shap e
dep enden t) coupling to the ligh t field (sec. 1.2) add to the complexit y of measured absorption
sp ectra of e.g. colloidal CdSe nanoplatelets.
Figure 1.5 sho ws absorption sp ectra of differen t sizes of 2D CdSe nanoplatelets (NPLs). Their
lateral sizes are determined from TEM-analysis. The thickness is giv en in n um b ers of monola y ers
(ML) where
d M L
= 0
.
304 nm is half the lattice parameter of zincblende CdSe [
44
]. NPLs are
assumed to b e cadmium-terminated on b oth basal planes [
44
,
45
]. Atomically flat, they sho w
1.1 Electronic Prop erties of Lo w-Dimensional Semiconductors 15
so
hh lh
CB
VB
1.06 nm 1.37 nm 1.67 nm
3.5 ML
4.5 ML 5.5 ML
Figure 1.5: T ransv erse and lateral confinemen t in CdSe nanoplatelets.
(a) Absorption
sp ectra of 3.5, 4.5 and 5.5 monola y er CdSe nanoplatelets. The transverse confinemen t of NPLs
go es in in teger monola y er steps. (b) Lateral size dep endence: Emission and absorption sp ectra of
4.5 monola y er CdSe nanoplatelets with corresp onding transmission electron microscop e images.
Emission tak es place from the hea vy hole exciton. The platelet areas, ranging from 100 to
∼
500 nm
2
,
do not strongly affect the energetic p ositions of absorption features. 17x6 and 41x13 samples
sho w con tamination of 3.5 ML nanoplatelets with higher transition energies. (c) Sketc h of the
con tributions of hea vy-hole (hh), light-hole (lh) and split-off (so) bands and their corresponding
exciton to the absorpti on spectr a.
v ery small inho mogeneous broadening, since strong size q uan tization only o ccurs in the short
axis, also referred to as t he z-direction in the follo wing, figur e 1.5 (a). The abso rption sp ectra
can b e clearly assigned to the thickness b y the p osition of the first hea vy hole ( hh) exciton.
Platelets with the first exci tonic pe ak at around 460 nm (2.70 e V) corresp ond to three Se and
four Cd monola yers and are referred to as 3
.
5 ML NPLs. With decreasing transverse quan tum
confinemen t the energetic p osition of the first exciton decreases to redder w a v elengths: for 4
.
5 ML
NPLs to ∼ 512 nm (2.43 e V) an d 5 . 5 ML to ∼ 550 nm (2.25 e V), resp ectiv ely .
The extreme transverse confinemen t has a great influence o n the optical prop erties of nano-
platelets: As seen in figu re 1.5 (c) it lifts the degeneracy of th e hea vy-hole and the ligh t-hole
band leading to e.g. h ighly directed emission from hea vy-ho le excitons (c hapter 2). Str ong
16 1 Basic Concepts
confinemen t is further assumed to explain the record t wo-photon absorption cross sections of
NPLs (c hapter 7).
Lateral confinemen t has no large effect on the transition energies, figure 1.5 (b). It do es though
affect the lateral extension of the en v elop e function of the exciton. This leads to an observ able
luminescen t excited state of p-lik e c haracter and an energy spacing b et w een this excited state and
the ground state exciton that is tunable b y the lateral confinemen t (c hapter 3). The oscillator
strength is also found to increase with the NPL area (c hapter 4). Via b oth, excited state and
oscillator strength con trol, the lateral size of NPLs has great impact on the dynamics in this
system.
2D nanoplatelets th us com bine strong transv erse confinement that leads to the 2D properties
of the electronic with the c hance to alter the w eak confinemen t in the lateral extensions. This
con trol of a 2D system has so far b een neither p ossible in 2D quan tum w ells (confined only in
one direction) nor in quan tum dots – confined equally in all directions leading to a 0D electronic
system.
1.1.5 Exciton Emission
After photo-excitation the electron and hole relax non-radiativ ely to the band edge b y dissipating
their excess kinetic energy to the lattice via electron-phonon in teraction on a sub picosecond
time scale. The carriers can form excitons due to their Coulomb attraction. These band edge
excitons further relax either b y radiativ e or non-radiative recom bination of the electron and
hole. Radiativ e recom bination leads to the emission of a photon. The exciton can in teract with
a phonon b efore emitting. The resulting energetic shift b et ween absorption and emission peak,
the Stok es shift[
46
,
47
], dep ends on the exciton-phonon coupling in the system, discussed in
c hapter 4. Ha ving higher energies, longitudinal optical (LO) phonons predominantly determine
this shift. As seen in figure 1.5 (b) the emission of CdSe nanoplatelets stems from hea vy hole
excitons. Compared to quan tum dots the Stokes shift and th us the exciton coupling to LO
phonons is smaller in the platelets [17].
The radiativ e rate of the sp on taneous exciton emission is prop ortional to the transition matrix
elemen t b et w een the initial and final state:
| h 0 | p | Ψ X i | 2
. In the quasi-particle picture this is
the exciton
| Ψ X i
(eq.1.9) and the crystal ground state
| 0 i
, i.e. the presence of no exciton.
| 0 i
is
comp osed of all Blo c h functions and all o ccupied states [
42
]. As sho wn in c hapter 2 the density
of photon mo des in to whic h the ligh t field can couple as well as the strength of this coupling
determine the radiativ e rate [48, 49]:
Γ r ad ( ω ) = ω 3 n
3 π ε 0 ~ c 3 | h 0 | p | Ψ X i | 2 | f LF | 2 (1.22)
1.2 Lo cal Field F actors of Anisotropic Nanopa rticles 17
with the frequency of the transition
ω
, refractiv e index of the emitting material
n
, free space
p ermittivit y
ε 0
and sp eed of ligh t
c
. The coupling to the light field is expressed here b y the
factor | f LF | 2 whic h will b e discussed in detail in the follo wing section.
1.2 Lo cal Field F acto rs of Anisotropic Nanopa rticles
In an y optical exp erimen t with nano crystals the particles m ust couple to the external optical
field, b oth in absorption and emission. It is therefore imp ortan t to quan tify this coupling when
studying in ternal electronic prop erties. As seen in eq. 1.22 its kno wledge is essen tial for obtaining
transition dip ole momen ts from measured radiativ e rates (c hapters 4 and 5), or to extract
the in ternal transition dip ole distribution from particles arranged in a monola y er (c hapter 2).
The coupling also relates the in ternal to the externally applied static field strength in electric
field dep enden t studies (c hapter 5). F urther, it affects the intrinsic absorption cross sections
of particles. As seen in c hapter 7, absolute particle concen trations are imp ortan t for optical
in v estigations and applications and can b e determined from absorption sp ectra once the ligh t
coupling is quan tified.
Here w e in tro duce the concept of the Loren tz lo cal field theory applied to a dielectric general
ellipsoid. A homogeneous external electric field results in a p olarization that reduces the field
inside the ellipsoid, and vice v ersa for an in ternal field propagating out of the ellipsoid [
49
,
50
],
see figure 1.6. The p olarization is prop ortional to the depolarization factor
L i
that dep ends
purely on the geometrical shap e of the ellipsoid.
The dep olarization factors
L i
of a nano crystal can b e calculated assuming e.g. NPLs to b e
oblate and nanoro ds to b e prolate ellipsoids. Quan tum dots are assumed to b e spheres:
L i =
∞
Z
0
a i a j a k
2( s + a 2
i ) 3 / 2 ( s + a 2
j ) 1 / 2 ( s + a 2
k ) 1 / 2 d s (1.23)
L x,y ,z
are obtained b y cyclic p erm utation of the semi-axes
a i,j,k
of the ellipsoids [
51
–
54
]. F or
spheres (QDs)
L x
=
L y
=
L z
= 1
/
3 is found [
52
]. The lo cal field factor, the ratio of the resulting
and the driving field, in the direction i of a nanoparticle is given b y [49–54]:
f i ( ω ) = 1
1 + L i ( ε S ( ω ) /ε M ( ω ) − 1) (1.24)
with the relativ e dielectric functions
ε S,M
(
ω
) of the semiconductor inclusion (S) and the sur-
rounding, isotropic matrix (M) at a giv en frequency
ω
. Expressing the electric field amplitude of
18 1 Basic Concepts
Figure 1.6: Internal and external fields in an anisotropic nanoparticle
for absorption and
emission. The optical fields exp erience a stronger reduction in the short axis of the ellipsoid.
ε S,M
denote the dielectric functions of the semiconductor inclusion (S) and the surrounding, isotropic
matrix (M). Propagation go es along the pointing v ector ~
S = ~
E × ~
H .
a field ~
E in spherical co ordinates with azim uth angle φ and p olar angle θ :
~
E ( φ, θ ) = E 0 ·
sin θ cos φ
sin θ sin φ
cos θ
(1.25)
and taking the reduction b y the field factors in to accoun t we find:
~
E lf =
E x
E y
E z
=
f x 0 0
0 f y 0
0 0 f z
· ~
E ( φ, θ ) (1.26)
The lo cal field factor squared of a particle in an isotropic medium relates the reduced field
in tensit y I lf and I 0 :
| f | 2 ( θ ,φ ) = I lf ( φ, θ )
I 0
= | f x | 2 sin 2 θ cos 2 φ + | f y | 2 sin 2 θ sin 2 φ + | f z | 2 cos 2 θ (1.27)
This equation will b e the starting p oin t to obtain lo cal field factors for ensem bles of nanoparticles
that are either orien ted in one plane (c hapter 2) or randomly oriented in spac e e.g. in solution or
em b edded in a p olymer throughout this thesis.
2 | Directed Emission from Orien ted Nanoplate-
lets and Relation to Blo c h States
New 2D semiconductor materials suc h as transition metal dic halcogenides and I I-VI (CdSe)
nanoplatelets exhibit prop erties of nearly ideal quan tum w ells.[
15
–
30
] Ho w ev er, the extremely
anisotropic electronic system in these materials is less in v estigated and not fully understo o d
y et. The anisotropic shap e also alters the coupling of ligh t to a dielectric surrounding. T ogether
these electronic and dielectric effects are exp ected to yield anisotropic emission, in con trast to
highly symmetric spherical particles, for instance PbSe dots[
53
] that radiate isotropically . It will
b e sho wn in the follo wing that CdSe nanoplatelets ha v e highly directional emission. Dielectric
effects further increase this directionalit y .
Ligh t-matter in teraction is mediated b y the transition dip ole matrix elemen t. In semiconductors
it is giv en b y the Blo c h functions asso ciated to the bands in volv ed in an optical transition and
the en v elop e functions defining the selections rules.[
42
] The Blo c h functions determine the
p olarization of the ligh t emitted or absorb ed in the transition, i.e. the orien tation of the
transition dip ole momen t (TDM). The orien tation or distribution of TDMs and their coupling to
the densit y of photon states determine the emission c haracteristics and radiativ e rates.[55, 56]
F or CdSe nanoplatelets as a mo del system it will b e sho wn that b y measuring the angular
distribution of an emitter’s radiation the in ternal transition dip ole distribution and subsequen tly
the underlying Blo c h functions can b e accessed.
T o do this tw o-dimensional k -space sp ectroscop y is applied to a monola y er of oriented zinc-
blende CdSe nanoplatelets. They are excited in the higher lying 2D con tin uum states where the
absorption is sho wn to ha ve an isotropic transition dipole distribution. Surprisingly the finally
emitting hea vy hole exciton has a highly anisotropic transition dip ole distribution – c haracterized
b y a brigh t plane that coincides with the platelet plane. It will we sho wn that this relates to
the in trinsic prop erties of the semiconductor Blo c h functions. They are anisotropic for hea vy
hole excitons and isotropic in the con tin uum where
hh
,
l h
and the
so
bands con tribute. This
electronic effect leads to a strongly directed emission p erp endicular to the platelet plane. CdSe
nanoplatelets are th us in trinsically directional emitters.
Theoretical predictions sho w that the dielectric con trast b et w een an anisotropic nanoparticle
and its surrounding ligands and matrix strongly alters the angular emission prop erties.[
49
]
Man y emitters used in mo dern photonics sho w no directional emission, the directionalit y is then
imprin ted b y external dielectric effects or a manipulation of the external densit y of states for the
ligh t field. CdSe nanoplatelets, ho w ev er, are directional based on their electronic system. Their
20 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
anisotropic shap e induces dielectric effects that further concen trate emission p erp endicular to
the platelet. The brigh t plane of TDMs is aligned to the axis fa v ored b y dielectric effects.
CdSe nanoplatelets are th us in teresting, realistic and efficien t directional emitters for photonic
applications lik e in displa y tec hnology . This study ma y help design directional emitters, optimiz-
ing the in terpla y of the confinemen t and shap e dep enden t anisotropic transition dip ole momen t
with dielectric screening.
2.1 k -space Sp ectroscop y of Nanoplatelets Oriented in a Monola y er
Zincblende (ZB) CdSe nanoplatelets of 4.5 monola y er (1.37 nm) thickness and 19
.
6
×
9
.
6 nm
2
lateral size w ere dep osited on 170
µ
m thic k fused silica substrates and a TEM grid b y a Langm uir
tec hnique. This w a y a monola yer of nanoplatelets is obtained that lie flat on the substrate
(platelet plane parallel to substrate plane) with random lateral orien tation. A reference sample
of a monola y er of CdSe dots (ZB structure) with similar emission energy w as pro duced as an
isotropic emitter. The in-plane orien tation of the platelets in a monola yer can be seen in the
TEM image in figure 2.1 (a). The samples on fused silica are in tro duced in an in v erted microscop e
lik e setup with a high n umeric ap erture (1.49) immersion oil infinit y corrected ob jectiv e (Nikon
1.49 NA 60x). This forms a three lay er system consisting of substrate/nanocrystals with ligands
/air (figure 2.1 c). The samples are excited by the second harmonic of a titanium sapphire laser
(Coheren t Mira, FWHM 150 fs, 75.4 MHz, 0.1 W/cm
2
excitation densit y) in the con tin uum of the
nanoparticle absorption at 2.99 e V ( 415 nm). Angle dep enden t confo cal excitation is realized b y
parallel displacemen t of the b eam with resp ect to the optical axis of the ob jectiv e. The F ourier
plane at the rev erse side of the ob jectiv e is imaged b y a 1/4.3 magnification to the CCD. The
bac k ap erture of the ob jectiv e, and th us the k-v ector distribution is recorded b y a CCD camera.
F or stray ligh t suppression and definition of the sampling area (
≈
20
µ
m diameter on sample) a
b eam blend (BB) is placed b et w een L3 and L4. The setup allows an angle and subsequen tly
k
-v ector dep enden t detection of the (sp ectrally in tegrated) photoluminescence (PL) signal from
the nanoparticle monola y er ensem bles, see figure 2.1 (b).
Figure 2.2 a sho ws the concept of the exp erimen t p erformed at ro om temp erature. An emitter
placed in the sample plane (
x
-
y
plane) ma y ha ve out-of-plane (OP)
z
-orien ted dip oles and
in-plane (IP) dip oles. The Hertzian radiation pattern of IP or OP dip oles in the fo cus of the
ob jectiv e can b e related to a w a v e-v ector distribution in the back fo cal plane. Thus the emitted
radiation can b e decomp osed in to its Cartesian comp onen ts
~
k
=
~
k x
+
~
k y
+
~
k z
leading to a t w o
dimensional pro jection of the radiation patterns in the k x - k y -space.
By in tro ducing a p olarizer w e select electric fields oscillating parallel to the x-z plane, defining
radiation where
~
E
and
~
k
coincide in the x-z-plane as p-p olarized. Radiation with
~
E
p oin ting in
2.1 k -space Sp ectroscop y of Nanoplatelets Oriented in a Monola y er 21
Figure 2.1: Monola y er of orien ted CdSe nanoplatelets and the 2D k -space sp ec-
troscop y setup
(a) T op: TEM image of 4.5 ML nanoplatelets brough t to a TEM grid by Langm uir
tec hnique. Bottom: Gaussian fits to the distribution of short (
l y
) and long (
l x
) lateral length.
The platelets are
l z
=
d N P L
= 4
.
5
·
0
.
301 nm thic k. (b) 2D k -space sp ectroscop y setup. F2: lo w
pass edge filter to suppress the excitation at 415 nm, P2: p olarizer transmitting ligh t p olarized
in the x-z plane. (c) Sc heme of k -space coupling geometry . Glass coupling la y er consisting of
substrate, immersion oil and ob jectiv e (
ε 1
), sample la y er (
ε 2
) and surrounding medium (air,
ε 3
).
The sample la y er is also pro duced b y Langmuir tec hnique, its thic kness
D M L
is determined b y the
NPLs’ thic kness and their ligand lengths.
x-direction and
~
k
lying in the y-z-plane is th us s-p olarized. The selection of cuts in the bac k fo cal
plane, dashed white lines in figure 2.2 (a), reduces the analysis to a 2D problem:
k x
pro jections
con tain information ab out dip oles lying in- and out-of-plane (p-p ol pro jection);
k y
pro jections
(s-p ol) con tain only radiation from IP dip oles. Figure 2.2 (b) sho ws a
k
-v ector resolv ed emission
of the orien ted CdSe nanoplatelets (excited at normal incidence). Figure 2.2 (c) sho ws the CdSe
quan tum dot reference. The measured in tensit y profiles of the
k x
- and
k y
-pro jections are plotted
next to the CCD images. Dashed lines indicate the selected regions (cuts). The
k
-scale is
normalized to the w a v e-v ector in air. Radiation with a w av e v ector
| k x,y /k 0 | >
1 cannot couple
in to air resulting in increased reflected in tensit y collected by the objective for these k -v ectors.
The
k
-dep enden t emission profile is considerably differen t for platelets and dots. It can
b e mainly related to a differen t fraction of IP and OP transition dip oles. The s-p olarization
pro jections (
k y
-cut) originate only from IP dip oles, their in tensit y profile merely dep ends on
the dielectric function and thic kness of the nano crystal monola y er. Details on s-p ol pro jections
are found in app endix 2.6. In the following w e will concen trate on cuts in
k x
direction as only
p-p olarization con tains signal from in- as w ell as out-of-plane dip oles.
22 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
b
a
c
p-pol
k x k y
k x
s-pol
IP+OP
IP
s-pol
IP+OP
IP
p-pol
s-pol
x
y
z
OU T -O F-P LAN E D IPO LE IN -PL ANE DIP OLE IN-P LAN E D IPO LE
R a d i a t i on p a t t e r n
k-space pr o j e c t i on
P o l a r i z er
I n t e n s i ty i n k - s p a c e
Figure 2.2: Momen tum ( k -space) resolv ed photo-luminescence
(a) By c ho osing one p o-
larisation axis, the bac k ap erture image of a microscop e ob jectiv e can b e analyzed with resp ect to
in-plane (IP) and out-of-plane (OP) transition dip ole emission. Cuts in
k x
or
k y
direction are w ell
defined and reduce the problem to a 2D geometry . (b)
k
-v ector dep enden t emission of a monola yer
of orien ted nanoplatelets and (c) a monola yer of dots. The profiles attac hed to the images display
cuts in
k x
(p-p ol) and
k y
(s-p ol). The latter con tain only emission from IP oriented dipoles. Cuts in
k x
rev eal the relativ e o ccurrence of IP and OP dip oles, i.e. the anisotrop y of the dipole distribution.
The
k
-v ector dep enden t platelet emission (at the lo w est 2.42 e V heavy hole exciton transition
of CdSe platelets) and the absorption (in to the 2D con tinuum at 2.99 e V) are mo deled b y a
m ultila y er system (glass - effective medium of oleic acid and nano crystals - air). The orien ted
CdSe platelets are surrounded b y a ligand shell, consisting of oleic acid ligands. The nanoplatelet
monola y er samples form an anisotropic effectiv e medium on top of the fused silica substrate
whic h will b e discussed no w.
2.2 Nano crystal Monola y er as an Effective Medium
The dep olarisation factors
L x,y ,z
of the platelets are calculated using eq. 1.23 and the results of
the TEM analysis (figure 2.1 a) on the lateral platelet dimensions:
l x /
2 ,
l y /
2 for the azim uthal
axes and thic kness
d N P L /
2 for the short axis of the ellipsoid.
L x,y ,z
= 0.04, 0.11, 0.85 are giv en b y
cyclic p erm utation of the semi-axes. F or spheres (QDs) w e take
L x
=
L y
=
L z
= 1
/
3 . F ollo wing
2.2 Nano crystal Monola yer as an Effective Medium 23
Sih v ola et al.[
52
] the effectiv e p ermittivit y of an anisotropic medium of aligned ellipsoids at the
frequency ω in the direction i is:
ε i
ef f ( ω ) = ε ( ω ) + F V [ ε s ( ω ) − ε ( ω )] ε ( ω )
ε ( ω ) + (1 − F V )[ ε s ( ω ) − ε ( ω )] L i
(2.1)
where
F V
is the v olume fraction of the semiconductor particles with p ermittivit y
ε s
in the host
material with the relativ e p ermittivit y
ε
. Inserting
L i
=
L z
yields the out-of-plane p ermittivit y
ε O P
ef f
=
ε z
ef f
. T o tak e the lateral random in-plane azimuthal orien tation of the nanoplatelets in
the monola y er in to account w e analyze the exp ectation v alue of the effectiv e p ermittivit y o v er
all in plane directions. In tegration o v er a quarter circle (
π /
2 ) exploiting the symmetry of the in
plane dep olarisation giv es:
ε I P
ef f ( ω ) = h ε i
ef f ( ω ) i φ = 2
π
π
2
Z
0
ε ( ω ) + F V [ ε s ( ω ) − ε ( ω )] ε ( ω )
ε ( ω ) + (1 − F V )[ ε s ( ω ) − ε ( ω )]( L x sin φ + L y cos φ ) d φ
(2.2)
The dielectric function
ε s
(
ω
) of ZB CdSe bulk at 415 nm for excitation and at 515 nm (525 nm)
for emission of our NPLs (QDs) is tak en from Ninomiy a et al.[
57
]. The optical p ermittivit y
of oleic acid surrounding is assumed to b e constan t
√ ε
=
n O A
= 1
.
46 [
58
]. As Ro wland et
al.[
59
] sho w, the dielectric function of nanoplatelets at their excitonic resonance do es not differ
strongly from that of bulk whic h can th us b e used, in our case, for effectiv e medium mo deling.
T o express the volume fraction
F N P L
in the monola y er w e use an effective lateral extension
h l N P L i
=
√ A N P L
(for lateral azim uthal random orien tation in the monolay er) with
A N P L
the
platelet area. F urther, the NPLs are assumed to b e co v ered on all facets b y their oleic acid
ligands. The v olume fraction
F N P L
is giv en b y the NPL v olume divided by the v olume of NPL
and ligands:
F N P L = d N P L A N P L
( h l N P L i + 2 l OA + l O A,xy ) 2 ( d N P L + 2 l O A + l OA,z )
| {z }
D M L,N P L
(2.3)
Here
l O A
= 2
.
1 nm[
60
] is the length of an oleic molecule ,
l O A,xy
and
l O A,z
are additional ligand
excess lengths in the in-plane (index xy) and p erp endicular (index z) direction, resp ectiv ely .
They are related to the fact that there is a sligh t ligand excess in the la yer.
F or our quantum dots of 4 nm diameter d QD the v olume fraction in the monolay er is:
F QD =
4 π
3 d QD
2 3
( d QD + 2 l O A + l OA,xy ) 2 ( d QD + 2 l OA + l O A,z )
| {z }
D M L,QD
. (2.4)
24 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
With
l O A,xy
= 3
.
2 nm and
l O A,z
= 0 nm for NPLs and
l O A,xy
= 0 nm and
l O A,z
= 0 nm for
QDs w e find v olume fractions to b e 16 % for NPL and 13 % for the quan tum dot monola y ers. The
thic kness of the monola yers of NPLs (
D M L,N P L
) and QDs (
D M L,QD
) needed for the k -space
analysis (describ ed in detail in app endix 2.6) are hereb y determined b y the length of the ligands
and the thic kness (diameter) of the nanoparticles. It should b e noted that the estimate of the
surface co v erage from the TEM image in figure 2.1 (a) of
∼
40% is in go o d agreemen t with the
41% resulting from the ab o v e used excess ligand length and platelet size.
With this w e are set to mo del the dielectric function of the nano crystal monola y ers and
can pro ceed with describing the heterogeneous three la y er system giv en b y the exp erimen tal
geometry .
2.3 F rom 2D k -space Sp ectroscop y to T ransition Dip ole Moment
Distribution
In a heterogeneous system a transition dip ole momen t couples in to a differen t optical density of
states as compared to air. F or the coupling of IP and OP transition dip oles in the nano crystal
monola y ers w e therefore use an effective densit y of states. A ccording to F ermi’s Golden R ule,
the radiativ e rate
Γ r
of an emitter is prop ortional to the pro duct of the Einstein co efficien t
A
(prop ortional to the transition dip ole momen t (TDM)
µ 2
) and the densit y of (photon) states
ρ
.
This yields
Γ r
(
ω
) =
A
(
ω
)
˜ ρ
(
ω
) =
π ω
3 ~ | µ ( ω ) | 2
ε ρ
(
ω
) with
ε
=
ε r ε 0
.
˜ ρ
(
ω
) is the normalized densit y of
states and
ρ
(
ω
) the optical densit y of states in a giv en medium. Using the densit y of states in
a homogeneous medium
ρ 0
(
ω
) =
ω 2
(
εµ
)
3 / 2 /π
the total radiativ e deca y rate and the TDM are
decomp osed in to IP and OP comp onen ts. With the IP and OP Einstein co efficien ts[
30
,
48
,
61
]:
A I P ( ω ) = ρ 0 ( ω ) π ω
3 ~ N r ,I P | µ I P ( ω ) | 2
ε
A O P ( ω ) = ρ 0 ( ω ) π ω
3 ~ N r ,O P | µ O P ( ω ) | 2
ε
(2.5)
the PL emission originating from sp on taneous hea vy hole exciton recom bination can b e re-
lated to the densit y of photon states in the en vironmen t of the emitter.
N r ,I P | µ I P
(
ω
)
| 2
and
N r ,O P | µ O P
(
ω
)
| 2
are the IP and OP pro jections of the dip ole strengths with respect to the
principle axes of the transition dip ole ellipsoid.
N r ,I P
and
N r ,O P
are the relativ e w eigh ts, whic h
dep end on its eccen tricit y .
˜ ρ p,s
(
ω , k x,y
) =
ρ p,s ( ω,k x,y )
ρ 0 ( ω )
are the relativ e (photon) densit y of states
for p and s p olarization. They accoun t for the alteration of the radiativ e rate in our heterogeneous
system with resp ect to free space.
2.3 From 2D k -space Spectroscopy to T ransition Dip ole Moment Distribution 25
100% OP dipoles
100% IP dipoles
Intensity (arb. units)
1
0
Excitation
-1 0 1
-1
0
1
k out
x /k 0
k in
x /k 0
a
-1 0 1
-1
0
1
k out
x /k 0
k in
x /k 0
b
c
d
Figure 2.3: Calculated k -space sp ectra of pure in-plane (a) and out-of-plane (b) tran-
sition dip oles and an isotropic distribution of them (c).
Left column: p-polarized sp ectra
in
k
-represen tation. Righ t column: corresp onding p olar plots as a function of the angle
θ
with
resp ect to the surface normal of the sample plane. Angles and w a ve v ectors are related b y
k a
x
=
sin
(
θ a
)
· √ ε a
1 ω a /c
and the p ermeabilit y
ε a
1
of the glass substrate at the emission (
a
=
out
) or
excitation (
a
=
in
) frequency
ω a
. The Jacobian con v ersion satisfying
R I ( k ) d k
=
R I ( θ ) d θ
is tak en
in to accoun t.[
62
] The case for emission is sho wn. Excitation is very similar as dispersion effects
do not strongly alter the sp ectra. Compared to (a) the OP dip ole con tribution in the isotropic
case (c) leads to an increase of the heigh t difference
∆
of the outer and inner maxima and a less
pronounced minim um at
| k x /k 0 |
= 1 (the angle of total in ternal reflexion). (d) Excitation sc heme,
sk etc hes of the in-plane and out-of-plane dip ole radiation patterns. The calculated 2D k -space
sp ectra, false color plots, displa y the mo deled p-p olarized emission
I
(
k out
x
) as a function of the
excitation w a ve v ector ( k in
x ). The case for resonan t excitation is sho wn.
As the radiativ e rates are related to the TDMs via eq. 2.5 w e can mo del the measured angularly
dep enden t emission and absorption c haracteristics. This allows direct access to the in ternal
distribution of TDMs in IP and OP directions. F or a detailed discussion of F resnel terms and
photon densit y of states see app endix 2.6.
Figure 2.3 sho ws the theoretical predictions for pure IP transition dip oles (ro w (a)), pure OP
transition dip oles (b) at the emission energy . F or an isotropic TDM distribution, sho wn in (c),
67% IP to 33% OP is exp ected, due to t w o degenerate IP dip ole orien tations in
x
and
y
-direction
and one OP dip ole orien tation in
z
-direction. The formalism for emission and excitation sp ectra
is analogous.[
63
] In our case – of con tin uum excitation and lo w est exciton emission – the
k
-space
sp ectra (for the same TDM distribution) v ary only sligh tly due to the materials disp ersion
relations at the excitation and emission frequencies. Therefore the follo wing discussion is v alid
for the PL emission in tensit y as a function of k in
x as w ell as for the emission sp ectra I ( k out
x ) .
26 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
The left column of figure 2.3 (a-c) displays the k -space dep enden t luminescence in tensity
I
(
k
) .
F or comparability the total in tensit y
R I
(
k
)
d k
is the same in all three cases (a-c). The right
column sho ws the corresp onding p olar represen tation (angle
θ
with resp ect to the surface normal
of the sample plane). T ypical features of the k -space sp ectra translate in to characteristic angular
emission patterns in the righ t hand column.
The t w o side lob es in the pure IP dip ole emission pattern corresp ond to modes b ey ond the
angle of total in ternal reflection (TIR) of the glass to air in terface (
| k x | /k 0
= 1 ). They can
b e observ ed with our ob jectiv e due to the index matc hing immersion optics used. A p erfectly
in-plane orien ted transition dip ole has no electric field comp onen t in the z direction and w e
th us exp ect no emission or absorption around the TIR angle, as seen in (a). In con trast, pure
OP dip oles can only in teract with the ligh t field in a small region of
k
-v ectors (angles) around
| k x /k 0 | = 1 (TIR angle), seen as t w o lob es in (b).
An imp ortan t c haracteristic of k -space sp ectra is the heigh t difference
∆
b et w een the side
lob es and the lo cal maxim um at
| k x | /k 0
= 0 , sho wn in figure 2.3 a and c. It increases with
increasing OP con tributions. F urther, the con tribution of OP dip oles in an isotropic distribution
of transition dip oles (figure 2.3 c) leads to less deep dips at
| k x | /k 0
= 1 as compared to the pure
IP case.
In our 2D k -space sp ectroscop y w e measure the emission
I
(
k out
x
) at differen t excitation k -
v ectors
k in
x
. F or addressing IP and OP dip oles the excitation b eam is p-p olarized and v aries in
k x
, as indicated in figure 2.3 (d). P anel (d) also displa ys t w o dimensional plots of the
k
-v ector
dep enden t luminescence in tensit y of pure IP or OP dip oles, whic h dep ends b oth on the excitation
and emission
k
-v ector,
k in
and
k out
, resp ectiv ely . F or resonan t excitation the 2D
k
-space images
are symmetric.[63]
The false color plots in figure 2.4 a and b sho w the measured p-p olarized emission
I
(
k out
x
) as a
function of the excitation w a v e vector (
k in
x
) for platelets (a) and for the QD reference sample (b).
These 2D k -space maps are obtained b y plotting the cut in
x
direction of the bac k ap erture image
(see figure 2.2 b and c) for ev ery excitation angle
k in
x
. QDs’ 2D k -space maps are symmetric,
the NPLs’ less so – a first indication that NPLs ha v e different transition dipole distributions
for absorption and emission. (Absorption and emission are measured differen t energies, see
figure 2.5 (b) and (d)).
The cross sections for emission sp ectra
I
(
k out
x
) are mark ed as v ertical dotted lines for
k in
x /k 0
= -
1.23; -1 and 0 (red, blue and gra y resp ectiv ely). They are plotted in tensit y normalized on
the righ t side of the false color plots. The curv es on top of the false color plots sho w the
(in tensit y normalized) cross section
I
(
k in
x
) at
k out
x /k 0
= 0 , marked as dash-dotted horizon tal
white lines. F rom the horizon tal (excitation) and vertical (emission) cross sections w e find that
2.3 From 2D k -space Spectroscopy to T ransition Dip ole Moment Distribution 27
- 1 0 1
- 1
0
1
c
I ( k
i n
x
)
I ( k
o u t
x
)
k
o u t
x
/ k
0
k
i n
x
/ k
0
N P L s
b a
- 1 0 1
- 1
0
1
k
o u t
x
/ k
0
k
i n
x
/ k
0
Q D s
- 1 0 1
- 1
0
1
d
k
i n
x
/ k
0
0
1
N o r m . i n t e n s i t y
I ( k
i n
x
, k
o u t
x
) / ( a . u . )
- 1 0 1
k
i n
x
/ k
0
9 5 % I P
- 1 0 1
k
i n
x
/ k
0
I n t e n s i t y n o r m . c u t s
M o d e l
D a t a
7 0 % I P
- 1
0
1
k
o u t
x
/ k
0
I ( k
o u t
x
)
6 8 % I P
- 1 0 1
I ( k
i n
x
)
6 9 % I P
k
i n
x
/ k
0
Figure 2.4: 2D k -space sp ectra of CdSe nanoplatelets (a) and quan tum dots (b)
The
false color plots are obtained b y plotting the measured p-p olarized emission
I
(
k out
x
) for ev ery
excitation w a ve v ector (
k in
x
). Crossectional plots on top of 2D plots: Intensit y normalized horizon tal
cross section, i.e. emission at
k out
x /k 0
= 0 as a function of excitation. Plots to the righ t: In tensit y
normalized v ertical cross sections at:
k in
x /k 0
=-1.23, -1 and 0, red, blue and gra y , respectively . Fit
curv es are added as blac k lines as well as the resulting in-plane (IP) transition dipole contributions.
W e find highly anisotropic distribution of emitting NPL dipoles. (c) and (d): Calculated 2D k -space
sp ectra for platelets and dots.
the angular emission c hanges in o v erall intensit y but its shape is indep enden t of the excitation
angle (
k in
x
). A ccording to the discussion ab o v e this indicates that differen t dip oles are in v olved
in the absorption pro cess at 415 nm and the emission at 513 nm. Th us the electronic system has
no ’memory’ of the excitation field’s p olarization.
As seen in figure 2.4 (a) the heigh t difference
∆
b et w een the maxim um of the cen ter and of
the side lob es is smaller for emission compared to excitation. F urther the dip around the TIR
angle
| k x | /k 0
= 1 for platelet emission
I
(
k out
x
) is more pronounced than for excitation. F ollo wing
the discussion in figure 2.3 nanoplatelets m ust ha ve a lo w er fraction of emitting OP transition
dip oles, compared to absorption.
This conclusion can b e confirmed b y comparing the exp erimen tal results to our theory .
All mo del parameters are determined b y literature v alues and exp erimen tal conditions, e.g.
nano crystal sizes, monola y er thic kness and resulting dielectric functions of effectiv e medium at
the absorption and emission photon energy . They are listed in T able 2.1 in app endix 2.6. The
28 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
only free parameters (determined b y least square routine) are the dip ole distributions, i.e. the
ratio of IP to OP dip ole momen ts in absorption and emission as describ ed in app endix 2.6.
The fit-mo del results are sho wn as blac k solid curv es in figure 2.4 a and b, and as 2D k -space
maps
I
(
k in
x ,k out
x
) in c and d for NPLs and dots, resp ectiv ely . F or b oth NPLs and dots w e
observ e an excellen t agreement of the model with the exp erimen tal results for emission and
excitation. The fit of the absorption c haracteristics of nanoplatelets (dots) yields 70% (69%) IP
and 30% (31%) OP con tributions. Also for the emission of CdSe dots w e observ e a nearly ideal
isotropic 68% IP to 32% OP ratio. The angular emission from dots th us do es not differ from
their excitation.
The observ ed small deviations of the excitation c haracteristics of platelets and dots (top panels
of figure 2.4 (a) and (b)) are the result of dielectric effects. As men tioned ab o v e our nanoplatelet
monola y er samples form an anisotropic effectiv e medium differing from the QD monolay er.
Ho w ev er, the angular dep enden t emission of CdSe nanoplatelets is considerably distinct from
their absorption. F rom our mo del w e deduce a 95% IP to 5% OP distribution of the transition
dip ole for the CdSe nanoplatelets’ hea vy hole emission. All v alues given abov e ha v e an absolute
error margin determined as ± 5 %.
A more in tuitiv e represen tation of
k
-sp ectra are p olar plots. They allo w for an estimate
of the radiated p o w er p er solid angle in a giv en direction. Figure 2.5 (a) and (c) display the
angular dep enden t c haracteristics of platelets and dots for p-p olarized excitation (blue frame)
and emission (green frame). Figure 2.5 (b) sho ws that the CdSe platelets are excited in the 2D
absorption con tin uum, as the exp ected step lik e, constan t density of state s is observed near the
excitation laser energy . F or the used dots an excitation energy of 2.99 e V also leads to con tinuum
state absorption, figure 2.5 (d). Here the excited carrier states do not dep end on the spatial
quan tum confinemen t. As discussed later the TDM distribution of the QD ensem ble is isotropic.
Due to the high con tribution of IP transition dip oles the emission of CdSe platelets is
concen trated strongly along the surface normal. The ratio of the in tensit y within the angle of
total in ternal reflexion compared to the in tensit y of the side lob es increases from 3.5 (excitation) to
5.0 (emission) in figure 2.5 (a). Therefore NPLs can b e considered as highly directional emitters.
The strong anisotropic confinemen t in the CdSe nanoplatelets has induced a predominan t
orien tation of the TDM in a brigh t plane that coincides with the nanoplatelets’ plane. Dip ole
transitions in the orthogonal direction (OP) are absen t as discussed later.
The insets in figure 2.5 (b) and (d) sho w the actual distributions of the transition dip ole
momen ts of nanoplatelets and quan tum dots in emission and con tinuum absorption. The
measured 95% IP con tribution for CdSe platelets leads to a strongly oblate shap e, reflecting the
brigh t plane.
2.3 From 2D k -space Spectroscopy to T ransition Dip ole Moment Distribution 29
2 . 2 5 2 . 5 0 2 . 7 5 3 . 0 0 3 . 2 5
P L L a s e r
S a m p l e p l a n e
E n e r g y ( e V )
N P L s Q D s
E m i s s i o n E x c i t a t i o n
T r a n s i t i o n d i p o l e d i s t r i b u t i o n
A b s o r p t i o n
N P L s Q D s
b d
E m i s s i o n
9 5 % i n - p l a n e
E x c i t a t i o n
I s o t r o p i c
2 . 2 5 2 . 5 0 2 . 7 5 3 . 0 0 3 . 2 5
I s o t r o p i c
E n e r g y ( e V )
I s o t r o p i c
E m i s s i o n E x c i t a t i o n
- 9 0
- 6 0
- 3 0
0
3 0
6 0
9 0
0
1
1
a
o u t
N o r m . I n t e n s i t y
( a r b . u n i t s )
- 9 0
- 6 0
- 3 0
0
3 0
6 0
9 0
0
1
1
i n
- 9 0
- 6 0
- 3 0
0
3 0
6 0
9 0
0
1
1
c
N o r m . I n t e n t s i t y
( a r b . u n i t s )
o u t
E m i s s i o n E x c i t a t i o n
- 9 0
- 6 0
- 3 0
0
3 0
6 0
9 0
0
1
1
i n
T I R
A n g l e o f t o t a l i n t e r n a l r e f l e c t i o n E x p e r i m e n t T h e o r y
Figure 2.5: Angular dep enden t p-p olarized emission and excitation of CdSe nano-
platelets and spherical quan tum dot monola y ers.
(a) and (c): P olar plots of the angular
dep enden t emission (
θ out
, green frame) and the excitation (
θ in
, blue frame) along with theory
curv es. Data and mo del are normalized to their total in tensit y , i.e. area under curv e. The emission
is measured for
θ in
= 0 . The excitation curves are obtained b y in tegrating the detected emission
o v er
θ out
. (b) and (d): Absorption and photoluminescence sp ectra of the studied nanoplatelets and
QDs along with the exciting laser. Insets: distributions of dip ole momen ts for random orientation
and for the strong anisotropic case of the emission of the orien ted NPL monola yer calculated from
our mo del.
This has a great tec hnological adv antage. F or example, ligh t extraction efficiency in quan tum-
w ell LED tec hnology suffers from high contrib utions of OP dip oles.[
64
] In AlGaN-based LEDs,
for instance, this limits the total extraction efficiency to b elo w 10 % [
65
]. Zincblende NPLs
ha v e (nearly) only IP TDMs in emission. Th us the out-coupling of radiation (within the TIR
angle) is strongly enhanced and non out-coupled radiation efficien tly suppressed compared to, for
example, an isotropic emitter (see figure 2.5 a and c). Hence NPLs and comparable 2D materials
are promising candidates for efficien t LED tec hnology , b oosting the out-coupling efficiency . In
particular ZB CdS[
19
] nanoplatelet based LEDs with adjustable UV emission ma y o v ercome the
fundamen tal ligh t-extraction efficiency limitations[65] of AlGaN-based LEDs.
30 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
2.4 Radiation P atterns of Oriented Nano crystals
Up to no w the external emission c haracteristics of the nanoparticle monola yers em b edded in the
three la y er system ha ve been discussed. W e no w consider to whic h exten t the directionalit y of
dip oles can b e translated to the radiated field of CdSe nanoplatelets, and ho w it is affected b y
the surrounding dielectrics. The in ternal distribution of transition dip ole momen ts in the CdSe
nanoplatelet and QD monola y ers could b e extracted with the presen ted mo del. This allo ws
the reconstruction of radiation in tensit y patterns of an orien ted platelet monolay er in differen t
dielectric surroundings.
The dip ole distribution in the NPL or QD monola y er ensem bles can b e defined as an ellipsoid
with the semi-axes
˜ µ r,I P
=
N r ,I P | µ I P
(
ω
)
| 2
for b oth in-plane directions (x and y axes) and
˜ µ r,OP = N r,OP | µ O P ( ω ) | 2 for the z-axis:
| µ ( θ ,φ ) | 2 = ˜ µ 2
r ,I P ˜ µ r,OP
q ( ˜ µ r ,I P ˜ µ r,OP cos φ sin θ ) 2 + ( ˜ µ r ,I P ˜ µ r,OP sin φ sin θ ) 2 + ( ˜ µ 2
r ,I P cos θ ) 2 (2.6)
The radiance of a Hertzian dip ole of strength
| µ | 2
can b e expressed in spherical co ordinates for
θ
from 0 to
π
and
φ
from 0 to 2
π
as
˜
D
(
θ ,φ,r
) =
| µ | 2 R
(
r
)
D
(
θ ,φ
) . Its angle dep enden t emission
c haracteristics are giv en as:
D ( θ ,φ ) = C 2 1 − cos 2 θ (2.7)
The constan t
C 2
is c hosen so that
D
(
θ ,φ
) is in units of W/sr(Cm)
2
, i.e radian t intensit y p er
dip ole momen t squared.
Let us consider a unit sphere (with radius r) con taining a monolay er ensem ble of NPLs or
QDs in its cen ter. The irradiance transmitted through the surface of the sphere at a giv en solid
angle is then
I r
(
θ ,φ
) . T o compare the radiation c haracteristics of the NPL and QD monolay er
ensem bles their total emitted p o w er
I tot
=
R R I r ( θ ,φ ) d Ω
with
d Ω
=
sin θ d θ d φ
is assumed to b e
iden tical.
First, w e study the case of a CdSe nanoplatelet emitting in to a material without dielectric
con trast to CdSe. This resem bles excitons radiating in to a virtually infinite CdSe medium. This
in tra platelet radiation pattern
I r
(
θ ,φ
) is giv en b y the con volution of the dipole distribution
ellipsoid
| µ
(
θ ,φ
)
| 2
with the angle dep enden t radiation pattern of a Hertzian dip ole
D
(
θ ,φ
) –as
indicated in figure 2.6 (a).
I r ( θ ,φ ) =
π
Z
0
2 π
Z
0 | µ ( θ − θ 0 ,φ − φ 0 ) | 2 · D ( θ 0 ,φ 0 ) d Ω 0 (2.8)
2.4 Radiation Patterns of Oriented Nanocrystals 31
This pure electronic con tribution to the radiation pattern is sho wn in blue for NPLs and in
orange for dots in figure 2.6 (b). The radiation emission pattern of a single transition dip ole
(lik e in man y organic molecules[
63
]) is isotropic in the plane p erp endicular to the dipole axis
and hence not directional, see Hertzian radiation pattern in figure 2.6 (a). Clearly , the emission
resulting from t w o degenerate dip oles forming a brigh t plane is b y far more directional, see
figure 2.6 (b).
No w w e consider an isotropic surrounding with dielectric con trast to the CdSe nano crystals.
The refractiv e index of a matrix in whic h nano crystals are t ypically em b edded, oleic acid
n=1.46[
58
], is c hosen for the surrounding. This sim ulates a realistic en vironmen t. The dielectric
effects are describ ed b y lo cal field factors
f x,y ,z
of the NPLs[
54
] at the exciton emission energy .
By a v eraging o ver the t w o in-plane comp onen ts on the long axes of the platelets
f x
and
f y
w e
tak e the azim uthal random orien tation of the NPL monolay er ensem ble in to accoun t:
D | f | 2 ( θ ) E φ = R 2 π
0 | f | 2 ( φ, θ ) d φ
R 2 π
0 d φ (2.9)
using the field factor ellipsoid | f | 2 ( θ ,φ ) as defined in eq. 1.27.
F or directions parallel (IP) and p erp endicular (OP) to the monola y er of orien ted NPLs the
in-plane a v eraged lo cal field factors at the frequency ω in an isotropic surrounding are:
D | f ( ω ) | 2 π
2 E φ = | f ( ω ) | 2
I P
D | f ( ω ) | 2 (0) E φ = | f ( ω ) | 2
O P
(2.10)
Similarly to the transition dip ole ellipsoid w e can define the lo cal field factor ellipsoid
| f | 2
(
θ ,φ
)
– obtained b y replacing
˜ µ r,I P
and
˜ µ r,OP
in equation 2.6 b y
| f ( ω ) | 2
I P
and
| f ( ω ) | 2
O P
, resp ectiv ely .
The pro duct[
49
,
66
] of the field factor ellipsoid and the dip ole distribution
| µ
(
θ ,φ
)
| 2
is con v o-
luted with the radiation pattern of a Hertzian dip ole
I r
lf ( θ ,φ ) =
π
Z
0
2 π
Z
0 | µ ( θ − θ 0 ,φ − φ 0 ) | 2 · | f ( θ − θ 0 ,φ − φ 0 ) | 2 · D ( θ 0 ,φ 0 ) d Ω 0 , (2.11)
with
d Ω 0
=
sin θ 0 d θ 0 d φ 0
. The obtained angular dep enden t radiation in tensit y of an emitter in an
isotropic dielectric medium is sho wn in figure 2.6 (c) for NPLs (magen ta) and QDs (orange).
Platelets ha v e a highly anisotropic shap e and high dielectric con trast to a t ypical ligand
or p olymeric surrounding. Their in-plane lo cal field factors (
| f I P | 2
) exceed the one in the
out-of-plane direction (
| f O P | 2
) b y a factor
∼
9 . The radiation from in-plane dip oles, i.e. the
brigh t plane, is th us also fa vored b y the dielectric effects. This leads to an ev en more directed
32 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
0
9 0
1 8 0
2 7 0
1 3
1 3
( i n o l e i c a c i d )
E n h a n c e d d i r e c t i o n a l i t y I n t r i n s i c d i r e c t i o n a l i t y
( i n C d S e )
D i e l e c t r i c c o n t r a s t
t o t h e s u r r o u n d i n g
z
y
x
z
y
x
N P L s
Q D s
y
z
x
N P L s
Q D s
c
D i p o l e r a d i a t i o n
p a t t e r n
x - z p l a n e
a b d
T r a n s i t i o n d i p o l e
d i s t r i b u t i o n
Figure 2.6: Radiation patterns of nanoplatelet and quan tum dot monola y er ensem bles
in isotropic media. (a) The transition dip ole distribution | µ | 2 ( θ ,φ ) (sho wn here for platelets as
deriv ed from the fits in figure 2.5) is con voluted with the radiation pattern of a Hertzian dipole. The
nanoparticle monola y ers are oriented in the x,y plane. (b) The obtained radiation in tensity pattern
of nanoplatelets (blue) and dots (orange) of the pure electronic con tribution, as if the emitters w ere
em b edded in a medium without dielectric con trast to CdSe. (c) The external radiation patterns
of platelets (magen ta) and dots (orange) are calculated for an isotropic dielectric surrounding of
oleic acid, taking lo cal field factors in to account. The data in (b) and (c) is normalized to the
total in tensit y
I tot
emitted o v er 4
π
. (d) Comparison of radiation patterns of dots (orange) and
NPLs with and without dielectric effects (blue and magen ta) in the x,z plane. Clearly , platelets
emit predominan tly in direction of the surface normal, which corresponds to 0 and 180 degrees.
Dielectric effects (magen ta) further enhance this directionality .
emission in a realistic en vironmen t, as seen in figure 2.6 (c). On the other hand, QD monola y er
ensem bles emit isotropically since they ha v e b oth an isotropic orien tation and lo cal field factors.
The cut through the x-z plane of the radiation patterns is plotted in figure 2.6 (d). In a
surrounding with no dielectric con trast to CdSe the in tensit y emitted p erp endicular to the
platelet is
∼
4 times higher as in to the x-y plane (blue curv e). When considering dielectric
effects in a t ypical surrounding (magen ta) this v alues reac hes ∼ 16 .
W e conclude from figure 2.6 that the emission of the CdSe nanoplatelets is highly directed
normal to the platelet plane due to the existence of a brigh t plane of IP dip oles. A brigh t plane
results in m uc h more directional radiation as a linear dip ole. This electronic effect is further
in tensified b y the shap e of the platelets in a dielectric surrounding.
The relev ance of alignmen t of transition dip ole momen ts with the dielectrically fa v ored axes of
the crystal and the influence of TDM distributions is illustrated in figure 2.7. It giv es an o v erview
of the transition dip ole distributions, lo cal field factor ellipsoids and resulting radiation patterns
for CdSe wurtzite ro ds and ZB nanoplatelets. ZB dots are added for comparison. Ro ds gro w
only in WZ structure, the c-axis coincides with the long axis of the ro d. The TDM is along this
axis. The lo cal field factors are highest along this axis so the out-coupling efficiency is–as for the
NPLs–dielectrically fa v ored. The pattern here is calculated for ro ds aligned along their c-axis
with a similar disorien tation as our NPL monola yer allo wing for some IP dip ole con tribution
(80% OP 20% IP). The radiation pattern is v ery similar to the Hertzian radiation pattern of a
2.5 Theoretical Considerations 33
Z B P l a t e l e t s W Z R o d s Z B D o t s
N o r m a l i s e d i r r a d i a n c e
o f o r i e n t e d m o n o l a y e r
L o c a l f i e l d f a c t o r
( C d S e i n o l e i c a c i d @ 5 1 2 n m )
H e r t z i a n d i p o l e
r a d i a t i o n p a t t e r n
T r a n s i t i o n d i p o l e d i s t r i b u t i o n
z
y
x
Figure 2.7: Alignmen t of brigh t plane and lo cal field factors and resulting radiation
patterns of CdSe zincblende dots and platelets and wurtzite ro ds.
T op ro w: The transi-
tion dip oles of the emitting exciton in ZB platelets form a brigh t plane and wurtzite nanoro ds a
brigh t axis. Middle ro w: Lo cal field factors obtained from mo deling the nano crystals as ellipsoids
in oleic acid. Dots of 4 nm diameter, ro ds with 2.6 nm diameter and 6 nm length and the 4.5 ML
20x10 nm
2
NPLs used ab o v e. F or NPLs the lo cal field factor ellipsoid coincides with the brigh t
plane, for NRs the long axes of TDM and lo cal field factors also coincide. This leads to a quasi
Hertzian emission pattern from nanoro ds in a dielectric surrounding (oleic acid) compared to the
directed emission from nanoplatelets.
linear dip ole. As mentioned abov e it is not isotropic but also not directed. In a more realistic
situation, e.g. obtained from Langmuir tec hnique, ro ds w ould lie flat on a substrate[
67
]. As a
result the c-axis w ould b e randomly orien ted in the substrate plane and the emission isotropic.
2.5 Theo retical Considerations
The microscopic origins of the formation of a brigh t plane in CdSe nanoplatelets are discussed
in the follo wing. The presen ted concepts are based on the w ork of Juan I. Climente (Univ ersitat
Jaume I, Castelló de la Plana, Spain).
The optical anisotrop y of ZB NPLs can b e explained from their electronic structure. One can
describ e the NPL as a cub oid with dimensions
L x ≥ L y L z
. F or a qualitativ e study , w e can
restrict to a simple effectiv e mass description, disregarding band coupling. The w a ve function of
confined carriers is then giv en b y
Ψ i
=
f i
ν,n | u i i
, where
i
=
e,hh,l h,so
indicates electron, hea vy
34 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
hole, ligh t hole or split-off hole, resp ectiv ely .
f i
ν,n
is the en v elop e function, defined o ver the en tire
NPL, whose p oin t symmetry is
ν
and its main quan tum n um b er is
n
.
| u i i
is the band-edge
p erio dic function (Blo c h function) at the Γ p oin t, defined ov er the unit cell.
P erio dic functions of electrons and holes
F or ZB crystals with spin-orbit interaction, it is customary to define the p eriodic Blo c h functions
in a basis of spherical harmonic functions with w ell defined total (orbital plus spin) angular
momen tum and its z-pro jection,
| J,J z i
.[
68
] F or conduction electrons, whic h ha v e
s
-lik e symmetry ,
the orbital angular momen tum is
l
= 0 . Hence,
J
=
l
+
σ
= 1
/
2 .
| u e i
(including spin) can then
tak e the follo wing forms:
| 1
2 , 1
2 i e = | S i| ↑i | 1
2 , − 1
2 i e = | S i| ↓i
F or v alence band holes, whic h ha v e
p
-lik e symmetry , the orbital angular momen tum is
l
= 1 .
Hence,
J
= 3
/
2
,
1
/
2 . F or
J
= 3
/
2 ,
J z
=
±
3
/
2 pro jections corresp ond to the hea vy hole subband.
| u hh i can then tak e the follo wing forms:
| 3 / 2 , 3 / 2 i = − 1
√ 2 | ( X + iY ) i| ↑i | 3 / 2 , − 3 / 2 i = 1
√ 2 | ( X − iY ) i| ↓i .
In turn,
J
= 3
/
2 with
J z
=
±
1
/
2 pro jections corresp ond to ligh t hole subband.
| u lh i
can then
b e:
| 3 / 2 , 1 / 2 i = r 2
3 | Z i| ↑i − 1
√ 6 | ( X + iY ) i| ↓i | 3 / 2 , − 1 / 2 i = r 2
3 | Z i| ↓i + 1
√ 6 | ( X − iY ) i| ↑i .
Last, J = 1 / 2 corresp onds to the split-off subband. | u so i can then b e:
| 1 / 2 , 1 / 2 i = 1
√ 3 | Z i| ↑i + 1
√ 3 | ( X + iY ) i| ↓i | 1 / 2 , − 1 / 2 i = − 1
√ 3 | Z i| ↓i + 1
√ 3 | ( X − iY ) i| ↑i .
Strong confinement lifts heavy-light hole degeneracy at the Γ -P oint
Considering the strong confinemen t in z-direction [001] (p erp endicular to the platelet plane), the
energy of a carrier
j
is roughly giv en b y
E j ≈ E j
Γ
+
~ 2 π 2
2 m j
[001] L 2
z
. Here
E j
is the energy splitting
from the top of the resp ectiv e band at the
Γ
p oin t, and
m j
[001]
the effectiv e mass in z-direction.
Unlik e in wurtzite semiconductors, the top of the v alence band in bulk ZB semiconductor has
degenerate
hh
and
l h
subbands (
E hh
Γ
=
E lh
Γ
), see figure 2.8 (a). As the names infer the
hh
effectiv e mass (
m hh
[001]
= 0
.
33 ) exceeds the
l h
’s (
m lh
[001]
= 0
.
13 ) in z-direction[
69
] so that the top
of the v alence band is formed almost exclusiv ely by
hh
states.
l h
states and split-off holes are
2.5 Theoretical Considerations 35
Figure 2.8: The con tribution of differen t Blo c h states determines the isotropic ab-
sorption and anisotropic emission of zincblende CdSe nanoplatelets
(a) Bulk zincblende
(ZB) semiconductor has degenerate hh and lh subbands. The effectiv e masses along the gro wth
direction are hea vier for hh than for lh, so the top of the v alence band in NPLs is formed almost
exclusiv ely b y hh states. (b) Schematic of the conduction band electron and v alence band hole
energy lev els in a rectangular (
D 2 h
) NPL with ZB crystal structure. Symmetry lab els for en v elop e
| ν i
and p erio dic parts
| u j i
of a few states,
| ν i| u j i
, are giv en. The en v elop e functions, c haracterized
b y their basic symmetries (
| A g i
and
| B 2 u i
), are shown on the left. The Blo c h states are giv en
color-co ded at the b ottom of (b). ( hh ): Dip ole-allo wed near-band-edge transitions (emission).
( c ont. ): Dip ole-allo w ed high energy transitions in absorption. In emission only hh lev els (red) are
in v olved, resulting in prev ailing out-of-plane emission. In con tin uum absorption ( c ont. )
hh
,
lh
and
split-off holes are equally in v olved, resulting in isotropic absorption. The corresp onding transitions
of ( hh ) and ( c ont. ) are indicated in the NPL absorption sp ectrum in (c). In con trast to the bulk
case sho wn in (a) the confinement lifts the degeneracy of the hh and l h .
split b y h undreds of me V, as noted in differe nt experiments[
17
,
19
] and indicated in figure 2.8 (c).
Owing to the large energetic separation b et w een
hh
and
l h
states, lo w-energy lev els are exp ected
to ha v e w eak hh − l h coupling.
A diagram of the electron and hole energy lev els resulting from the ab o v e considerations is
plotted in figure 2.8 (b). The lev els are lab eled b y the corresp onding p erio dic function as w ell as
the en v elop e function symmetry related to the
D 2 h
p oin t group of the rectangular nanoplatelet.
P ola rization of inter-band transitions
The oscillator strength of a transition b et w een a conduction and v alence band lev el is prop ortional
to
| µ ej | 2
=
|h f e
ν e ,n e | f j
ν j ,n j i h u e | µ | u j i| 2
. The env elop e in tegral pro vides selection rules, as only
transitions fulfilling
δ ν e ,ν j
will b e allo w ed. The in tegral o v er the unit cell
h u e | µ | u j i
defines the
p olarization of the emitted/absorb ed ligh t, as w e sho w next.
µ
=
q ~ r
stands for the dip ole momen t of the electromagnetic radiation, with
q
the electric
c harge.
| u j i
functions are giv en in the previous section. The cubic symmetry of zincblende
36 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
crystals leads to
h S | x | X i
=
h S | y | Y i
=
h S | z | Z i
=
K
, with all other in tegrals b eing zero. Note
that, while in bulk
x
,
y
and
z
directions are equiv alen t, in the presence of confinement
z
is tak en
as the ([001]) direction. This is in line with the surface normal [001] of the platelets. With these
considerations, one can compare h u e | µ | u j i for different kinds of in terband transitions.
F or transitions inv olving hh lev els, |h u e | µ | u hh i| 2 can tak e the follo wing v alues:
|h 1 / 2 , 1 / 2 | µ | 3 / 2 , 3 / 2 i| 2 = |h 1 / 2 , − 1 / 2 | µ | 3 / 2 , − 3 / 2 i| 2 = q 2 K 2
1
2
1
2
0
(2.12)
|h 1 / 2 , 1 / 2 | µ | 3 / 2 , − 3 / 2 i| 2 = |h 1 / 2 , − 1 / 2 | µ | 3 / 2 , 3 / 2 i| 2 = ~
0 .
F or transitions inv olving l h lev els, |h u e | µ | u lh i| 2 can tak e the follo wing v alues:
|h 1 / 2 , 1 / 2 | µ | 3 / 2 , 1 / 2 i| 2 = |h 1 / 2 , − 1 / 2 | µ | 3 / 2 , − 1 / 2 i| 2 = q 2 K 2
0
0
2
3
(2.13)
|h 1 / 2 , 1 / 2 | µ | 3 / 2 , − 1 / 2 i| 2 = |h 1 / 2 , − 1 / 2 | µ | 3 / 2 , 1 / 2 i| 2 = q 2 K 2
1
6
1
6
0
.
Last, for transitions in v olving so levels, |h u e | µ | u so i| 2 can tak e the following v alues:
|h 1 / 2 , 1 / 2 | µ | 1 / 2 , 1 / 2 i| 2 = |h 1 / 2 , − 1 / 2 | µ | 1 / 2 , − 1 / 2 i| 2 = q 2 K 2
0
0
1
3
, (2.14)
|h 1 / 2 , 1 / 2 | µ | 1 / 2 , − 1 / 2 i| 2 = |h 1 / 2 , − 1 / 2 | µ | 1 / 2 , 1 / 2 i| 2 = q 2 K 2
1
3
1
3
0
.
Equation 2.12 sho ws that transitions b et ween electron and
hh
are enabled only b y in-plane (
x,y
)
p olarized ligh t. Therefore, a hea vy hole exciton Ψ = Ψ e Ψ hh forms an in-plane electronic dip ole,
iden tified as the brigh t plane. Since the lev els near the top of the v alence band in NPLs are of
hh
c haracter, this explains the fact that most photoluminescence emission propagates orthogonal to
the NPL plane. By contrast, eqs. (2.13) and (2.14) sho w that transitions b et w een electron and
l h
or
so
holes are enabled b y an y ligh t p olarization. F or high energy in terband transitions, lik e those
in the absorption measuremen ts ab o v e, the final state can b e of
hh
,
l h
or
so
c haracter (or mixed
2.5 Theoretical Considerations 37
c haracter) with similar lik eliho o d. In suc h a case
| µ ej | 2
=
P j | µ ej | 2 | /
3 =
q 2 K 2
(1
/
3
,
1
/
3
,
1
/
3) .
That is, the p olarization and the propagation of ligh t is isotropic, as indeed observ ed in our
exp erimen t.
In photoluminescence exp erimen ts, electron-hole recom bination tak es place b et ween states
near the band edges, as p opulation excited in the con tin uum relaxes fast and radiationless to
the band-edge. As sho wn in figure. 2.8 (b) column ( hh ), this inv olv es only emission from the
hea vy hole to whic h all excited higher excitons and e-h pairs co ol do wn. The emission dip oles of
these
hh
excitons lie in the x-y plane and are th us IP . This theoretical exp ectation is in excellent
agreemen t with our measured 95% IP dip ole orien tation, considering our error margin of 5%.
Consequen tly , emission is mostly orthogonal to the NPL surface. Notice this is indep enden t of
the en v elop e function symmetry , and hence compatible with previous works suggesting emission
can ha v e finite con tribution from excited (
B 2 u
) excitons. (Ref. [
27
] and Chap. 3). On the other
hand, our absorption exp erimen ts are p erformed at high energy . As sho wn in Figure 2.8 (b)
column ( c ont. ), optical transitions include an y kind of hole state (or their mixture). On a v erage,
this means that the con tin uum absorption in our CdSe nanoplatelets is isotropic with resp ect to
the in ternal dip ole distribution. This again is in agreemen t with our exp erimen tal results of a
nearly isotropic transition dip ole distribution (70% in-plane to 30% out-of-plane) in absorption.
Conclusion
Using t w o-dimensional
k
-space sp ectroscop y w e ha v e demonstrated the directed emission of
highly orien ted CdSe nanoplatelet monola y ers. The presen ted analysis sho ws that the heavy-hole
(
hh
) exciton transition dip oles are orien ted in a brigh t plane that coincides with the platelet
plane. The resulting out-of-plane directed emission is additionally fa v ored b y dielectric and lo cal
field effects of the platelets.
It has further b een sho wn that the observ ed anisotropic transition dip ole distribution for
the lo w est
hh
excitons is directly related to the basic anisotrop y of the electronic Blo c h states
go v erning the dip ole momen t. It is observ able b ecause the ultra-strong confinemen t lifts the
hea vy-hole ligh t-hole degeneracy . Ultimately ,
k
-space sp ectroscop y allo ws direct access of the
in ternal Blo c h states. The transition dip ole distribution in the platelets’ 2D con tin uum, ho wev er,
has no preference: As the absorption and emission of quan tum dot monola y ers it is isotropic.
The com bination of isotropic absorption and highly anisotropic emission mak es CdSe nano-
platelets in teresting for photonic applications lik e in displa y tec hnology or lasing, where directed
emitters are desirable. As losses to non out coupled mo des (under oblige angles) are suppressed
due to the dominating IP emission w e sho w a route to ov ercome basic limitations to ligh t
extracting efficiency in devices suc h as displa ys or LEDs.
38 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
Gain media of orien ted platelets in sup er structures are exp ected to outp erform random
orien ted or spherical nano emitters. Impro v ed in v ersion conditions and reduction of losses
through sp on taneous emission migh t b e ac hiev ed with high directional gain and flexible pump
directions.
Exciton funneling in a solar cell funnels a broad energy sp ectrum of absorb ed radiation to a
narro w energy band. In NPLs the contrasting properties of isotropic absorption in the broad
con tin uum, yet anisotropic emission of the hea vy hole exciton migh t giv e funneling a new asp ect:
Energy and
k
-space funneling. Energy in the whole con tin uum is collected isotropically in
k
-space. It is then concentrated to the lo w est states, where it is directionally re-emitted – an
energy and k -space concen trator.
Finally , the presented concept of com bining 2D
k
-space sp ectroscop y and the assessmen t of
the underlying crystal Blo c h functions can b e applied to other systems suc h as 2D transition
metal dic halcogenides or p ero vskites to engineer their prop erties and in v estigate their electronic
system and effects of confinemen t.
2.6 APPENDIX: k -space Analysis 39
2.6 APPENDIX: k -space Analysis
In this section w e dev elop the theory describing the
k
-v ector and optical densit y of states
dep enden t absorption and emission.
F or an isotropic dip ole transition, the emission rate p er unit frequency (
δ ω
) and in-plane
momen tum ( δ k || ) can b e written as [30, 61]
Γ s,p = A ( ω ) ˜ ρ s,p ( ω , k x ) (2.15)
with the normalized electric lo cal densit y of states
˜ ρ s,p ( ω , k x ) = ρ s,p ( ω , k x )
ρ 0 ( ω ) (2.16)
The total sp on taneous emission rates (i.e. Einstein coefficients) for electric dipoles within a
homogeneous isotropic medium ( ε = ε r ε 0 ) are given b y F ermi’s golden rule [48]:
A ( ω ) = ρ 0 ( ω ) π ω
3 ~ | µ ( ω ) | 2
ε (2.17)
Here,
ρ 0
(
ω
) =
ω 2
(
εµ
)
3 / 2 /π
is the densit y of electromagnetic mo des in the emitter medium and
ω
is the emission frequency . T o distinguish betw een transition dip oles lying in-plane and those
that lie out-of plane, w e write the resp ectiv e rates as:
A I P ( ω ) = ρ 0 ( ω ) π ω
3 ~ N r ,I P | µ I P ( ω ) | 2
ε
A O P ( ω ) = ρ 0 ( ω ) π ω
3 ~ N r ,O P | µ O P ( ω ) | 2
ε
(2.18)
where
N r ,I P | µ I P
(
ω
)
| 2
and
N r ,O P | µ O P
(
ω
)
| 2
are in-plane and out-of-plane pro jections of the dip ole
strengths with resp ect to the principle axes of the dip ole ellipsoid.
N r ,I P
and
N r ,O P
are the
relativ e w eigh ts, which depend on orientation. As we are using cuts in the
x
-direction of the
x − y
plane in our exp erimen ts w e analyze only the
x
comp onen t of the rotationally symmetric
external radiation pattern, whic h is shown in figure 2.2. This wa y w e obtain the ratio of the
IP transition dip ole con tribution in x direction to the OP con tribution from z -orien ted dip oles
from our mo del fits in figure 2.5. F rom this tw o dimensional (
x
-
z
–plane) IP to OP ratio w e can
calculate the ratio taking also the (IP) dip oles orien ted in the
y
direction in to accoun t. This
corresp onds to the real con tribution of all OP and IP con tributions to the total emission in three
dimensions. The fraction
R x − z
of IP and OP dip ole con tributions to the emission in the
x
-
z
40 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
plane is defined as
R x − z
IP = N r, I P | µ IP ( ω ) | 2
N r, I P | µ IP ( ω ) | 2 + N r, O P | µ OP ( ω ) | 2
R x − z
OP = N r, O P | µ OP ( ω ) | 2
N r, I P | µ IP ( ω ) | 2 + N r, O P | µ OP ( ω ) | 2
(2.19)
and the 3D fraction of OP a nd IP con tributions as:
R 3 D
OP = 1
2
R x − z
OP − 1 (2.20)
R 3 D
IP =1 − R 3 D
OP (2.21)
With these equations we calculate the fractional IP and OP con tri butions with resp ect to 3D
space.
The geometry used for
k
-space analysis consists of three dielectric la y ers sc hematically sho wn
in Figure 2.9. Glass substrate, immersion oil and microscop e ob jectiv e are tak en as one la yer
with (
ε 1
) follo w ed by a thin sample la y er (
ε 2
) of thic kness
D ML
, that con tains e ither NPLs or
QDs. The third la yer is air ( ε 3 ). The plane of incidence is chosen to b e the x − z -plane , whic h
defines the s-p olarized field to p oin t in the y -direction.
With the v acuu m w a v e-v ector
k 0
=
ω
c
and the w a v e-v ector comp onen t parallel to the interface
k x
=
√ ε 1 k 0 sin
(
θ in
) , whic h is conserv ed b et ween eac h la y er, w e get for the p erp endicular
comp onen ts
k i,z = q k 2
0 ε i − k 2
x (2.22)
in media i=1,2,3, resp ectively .
If the emitter is placed i n an uniaxial material as describ ed ab o v e by the lo cal field approac h
to the effectiv e me dium, w e can calculate the effective perm ittivities with eqs. 2.1 and 2.2 for
the axis parallel (
ε IP
ef f
) and the axis p erp endicular to the interface (
ε OP
ef f
) . In the case of pure
e 1
e 3
e 2 D ML
k z 1,
k 1,x
k 1
x
z
y
m OP
m IP
k
z
1,
Substrate
Immersion oil
High N.A. objective
Figure 2.9: Scheme of k -space coupling geometry .
2.6 APPENDIX: k -space Analysis 41
s-p olarization, the electric field directly follo ws the dielectric function along the in terface and its
w a v e-vector is
k s
2 ,z = q k 2
0 ε s
2 − k 2
x , where ε s
2 = ε I P
ef f (2.23)
F or p-p olarized ligh t, we can define an angle-dependent dielectric function
ε p
2 ( θ in ) = ε I P
ef f ε O P
ef f
q ( ε I P
ef f sin θ in ) 2 + ( ε O P
ef f cos θ in ) 2
(2.24)
to tak e in to accoun t the anisotrop y of the material. The resp ectiv e w av e-v ector is then giv en b y
k p
2 ,z = q k 2
0 ε p
2 − k 2
x (2.25)
F or recipro cal (linear, time-in v arian t) media, the linear densit y of states is prop ortional to the
absorption rate for an iden tically orien ted dip ole [70].
˜ ρ s,p ( ω , k x ) = C 0
E x ( ω , k x )
E s,p
0
2 + C 0
E y ( ω , k x )
E s,p
0
2 + C 0
E z ( ω , k x )
E s,p
0
2 (2.26)
with
C 0 = 1
8 π k 2
0
k 0
k 3 ,z
(2.27)
No w w e can distinguish b et w een in-plane and out-of-plane contributions to the densit y of
electromagnetic mo des in the emitter medium:
˜ ρ s
I P ( ω ,k x ) = C 0 ¯
t s
12 e 1
2 ik s
2 ,z D M L (1 + r s
23 e ik s
2 ,z D M L )
2
˜ ρ p
I P ( ω ,k x ) = C 0
k p
z 2
p ε p
2 k 0
¯
t p
12 e 1
2 ik p
2 ,z D M L (1 − r p
23 e ik p
z 2 D M L )
2
˜ ρ p
O P ( ω ,k x ) = C 0
k x
p ε p
2 k 0
¯
t p
12 e 1
2 ik p
z 2 D M L (1 + r p
23 e ik p
z 2 D M L )
2
(2.28)
where m ultiple reflections are tak en in to account b y a F abry-P érot mo del.
¯
t s
12 = t s
12
1 − r s
23 r s
21 e 2 ik s
z 2 D M L ¯
t p
12 = t p
12
1 − r p
23 r p
21 e 2 ik p
z 2 D M L
D M L is giv en in eqs. 2.3 and 2.4. The F resnel co efficien ts are defined as
r s
ij = k z ,i − k z ,j
k z ,i + k z ,j , t s
ij = 2 k z ,i
k z ,i + k z ,j ,
r p
ij = k z ,i ε j − k z ,j ε i
k z ,i ε j + k z ,j ε i , t p
ij = 2 √ ε i √ ε j k z ,i
k z ,i ε j + k z ,j ε i .
for s- and p-p olarization, resp ectiv ely .
W e can now fit a superp osition of the in-plane and out-of-plane excitation and emission rates
to the observ ed exp erimen tal signal
S
from our samples for s-p olarization and p-p olarization
42 2 Directed Emission from Oriented Nanoplatelets and Relation to Blo ch States
T able 2.1:
P ermittivities used in the mo del.
D M L
is the thic kness of the effectiv e medium (E.M.)
of ligands and nano crystals, see eqs 2.3 and 2.4.
Material ε ( ω exc. ) ε ( ω em. ) D M L (nm)
Bulk CdSe[57] 7.9+3.5i 7.9+2.6i
Oleic acid (O A) ligands[58] 2.129 2.129
Eff. medium NPLs+O A ordinary ( ε I P
ef f ) 2.663+0.263i 2.652+0.192i 5.6
Eff. medium NPLs+O A extraordinary ( ε O P
ef f ) 2.337+0.033i 2.332+0.026i 5.6
Eff. medium QDs+O A ( ε ef f ) 2.335+0.058i 2.327+0.044i 8.2
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
-90
-60
-30
0
30
60
90
0.00 0 .01 0.01
Angle of tot al inte rnal re flexion
θ in θ ou t
(b)
Norm . total PL inte nsity ( θ in ) / a .u.
(a)
TIR TIR
p - pol s - pol p - pol
Exci tation Em ission
s - pol
Norm . total PL inte nsity ( θ in ) / a .u.
Exp eriment T heory
Angu lar emis sion inte ntsity ( θ out ) / a.u .
NPLs
QDs
Angu lar emis sion inte ntsity ( θ out ) / a. u.
Figure 2.10: P olar plots of the measured external excitation (a) and emission (b)
c haracteristics in the x - z plane for CdSe nanoplatelets and dots
along with the fits to the
presen ted theory .
and deduce the fraction of IP and OP dip ole con tributions.
S s ( ω ,k x ) = C 1 ˜ ρ s
I P ( ω ,k x ) A I P ( ω )
S p ( ω ,k x ) = C 1 h ˜ ρ p
I P ( ω ,k x ) A I P ( ω ) + ˜ ρ p
O P ( ω ,k x ) A OP ( ω ) i (2.29)
The s-p olarization pro jections (
k y
-cut) originate only from IP dip oles, their in tensit y profile
merely dep ends on the dielectric function and thic kness of the nano crystal monola y er. The
parameters are listed in T able 2.1. The fits to s-p ol and p-p ol excitation and emission from NPL
and QD samples are sho wn in figure 2.10.
P a rt I I
Contr olling the Ex citon Lifetime by
Shape, External Fields or Band
Alignment
45
st
1 approximation:
Thermal occupation + constant rates
Exciton polarisation:
3 Level system LO phonon bottleneck
Resonant
∆ E
Off-resonant
∆ E
T emperature dependence Field dependence
Stark shift:
Dephasing rates
Exciton - phonon coupling
Dipole polarisability
Exciton binding energy
Dynamics: Rate equation model
2.4 2.5
Energy (eV)
T ransition dipole polarisability
T ransition dipole moment
Field control of bottleneck
GOST ef fect in 2D
nd
2 approximation:
Figure I I A: A three lev el system with a longitudinal optical phonon b ottlenec k go v-
erns the recom bination dynamics of CdSe nanoplatelets.
T emperature dep enden t studies
rev eal the in trinsic radiative rates of the excited and ground state exciton and their in ter-scattering
rate
γ 0
as a function of the platelet size. Applying an external field allows for manipulation of these
rates within one platelet size. Man y other prop erties can be quantified b y rate equation mo deling
of static and dynamic photoluminescence data.
So far the (directed) em ission prop erties at ro om temp erature ha v e given insigh t to the effect
of transv erse co nfinemen t on the electronic system of Cd Se nanoplatelets. No w w e study their
dynamics. This will, amongs t other findings, giv e insigh t into the effects of lateral confinemen t.
Time-resolv ed photoluminescence (TR-PL) measuremen ts allo w to access v arious aspects of
the studied electronic s ystem[
71
], such as the exciton fine structure [
72
,
73
], trio ns [
74
], biexcitons
[
75
,
76
], amplified sp on taneous emission and lasing dynamics [
75
,
77
]. Sin ce the coupling to
the ligh t field also gov erns the dynamics, dielectri c effects can b e studied with TR-PL [
48
,
49
].
F urther it reveals non-radiativ e pro cesses suc h as A uger recom bination and energy transfer
pro cesses [59, 78].
As men tioned in chapter 1 the radiativ e ra te of an exciton is prop ortional to its transition
dip ole momen t. In a bulk sem iconductor it is determined b y the ba nd-to-band dip ole matrix
elemen t – a prop ert y giv en b y the material system. In lo wer-dimensional systems, ho w ev er, the
confinemen t indu ced redistribution of oscillator strengt h allo ws to con trol and manipulat e the
dynamics b y thei r shap e.
In this part of the w ork th e radiativ e rates will b e manipulated in three differ en t w a ys:
in trinsically b y the shap e of the n anoplatelets [
27
], expl oiting the Gian t Oscillator Strength
46
(GOST) effect; externally by applying an electric field [
24
]; and by spatially separating the
electron and hole w a v efunction with an in-built type I I band-alignmen t [79].
In c hapters 3 and 4 the temp erature dep enden t time-resolv ed PL of core only CdSe nanoplatelets
is studied. As illustrated in figure I I A a double emission is observed and related to t w o emitting
excitonic states. This ground state (GS) and excited state (ES) are iden tified as an s-state
and a p-state, with energies confirmed by theory . Their energetic separation decreases with
increasing platelet area and is of the order of the CdSe longitudinal optical (LO) phonon energy
– a situation that leads to an LO phonon b ottlenec k go v erning the in ter-relaxation b et ween the
t w o states. In a first appro ximation this three lev el system of GS, ES and the crystal ground
state is mo deled b y a rate equation system with temp erature indep enden t radiativ e rates in
c hapter 3.
The mo del is refined in c hapter 4 with the in tro duction of the GOST effect. It connects the
radiativ e rates with the homogeneous transition linewidths leading to temp erature dep enden t
radiativ e rates. Careful analysis sho ws that the radiative rate is either determined b y the ph ysical
platelet size or b y a smaller area giv en b y the homogeneous linewidth. W e extract, for the first
time, dephasing rates from a PL exp erimen t, see figure I I A.
In c hapter 5 the radiative rates in the platelets are externally manipulated b y a static electric
field. As indicated in figure I I A the presen ted time-resolved stark spectroscopy allo ws to deriv e
the transition dip ole momen t of the excited and ground state exciton and their transition
p olarizabilities – to our kno wledge the ev er rep orted for colloidal nano crystals. F urther, the first
exp erimen tal pro of of the high exciton binding energy in CdSe nanoplatelets is giv en [24].
In the final c hapter 6 type I I CdSe-CdT e Core-wing nanoplatelets are studied. It will b e sho wn
that the spatial separation of electron and hole w a v efunction decreases the radiativ e rate of the
emitting transition b y t w o orders of magnitude, compared to core only platelets. Electron and
hole are iden tified as Coulom b correlated excitons. This rules out un b ound electron-hole pairs as
the emissions’ origin – an imp ortan t implication for the applicabilit y of these hetero-structures
in solar cells or lasing devices.
Throughout this part the global analysis of static and dynamic PL data will pro v e a p o w erful
to ol, for example, for reducing fitting am biguities when w orking with rate equation systems.
47
Time-integrated and Time-resolved Photoluminescence Setup
Setup for time resolv ed and in tegrated photoluminescence exp erimen ts.
By remo ving
mirrors M
2
to M
5
differen t excitation sources and w av elengths can b e used. The emission of the
sample is collected b y an ob jectiv e and can b e brough t to a sp ectrometer with attac hed CCD for
time in tegrated measuremen ts or to a streak camera or av alanc he photo dio des for time resolv ed
measuremen ts.
The exp erimen tal setup used for TR-PL It allo ws the consecutiv e measuremen t of time-
in tegrated and -resolv ed fluorescence of a sample in a cry ostat or at ro om temp erature. The
samples are excited in a confo cal configuration and their emission detected through an ob jectiv e
(N.A.=0.4). A sp ectrometer with an attac hed CCD (Horiba IHR550 and Rop er Sp ec10) is
used for time-in tegrated measuremen ts. F or fast dynamics (c hapters 3 to 5) a streak camera
(Hamamatsu C5680) is op erated with 3.5 ps (120 ps) and 23 ps (2.2 ns) time resolution (range).
As excitation source the second harmonic (SHG) of a titanium sapphire laser at 420 nm is used.
The slo w dynamics of t yp e I I systems in c hapter 6 is recorded with av alanc he photo dio des
(APDs) with 400 ns time range, the temp oral resp onse function is giv en in c hapter 6. Here a
pulsed laser dio de, GaN-dio de excites the samples. A con tin uous w a v e (CW) laser dio de at
409 nm is used for lo w excitation density measur ements. The excitation source of time in tegrated
field dep enden t PL measuremen ts (c hapter 5) is a HeCd laser (CW at 441 mn).
3 | Excited-State Luminescence and an LO Phonon
Bottlenec k in CdSe NPLs
This c hapter mainly consists of w ork published as Ac h tstein, A. W., Scott, R. et al.
(2016)
, Phys.
R ev. L ett. , 116(11). Figures are adapted with p ermission from ref. [
27
],
©
2016 the American
Ph ysical So ciet y .
In this c hapter the effects of the lateral CdSe platelet size on the temp erature dep enden t
photoluminescence and its dynamics are studied. At first, emission from an excited excitonic
state is evidenced w ell b elo w ground state saturation. It will b e demonstrated in the follo wing
that the energy difference
∆E
b et w een the ground state (GS) and this excited state (ES) strongly
increases with the lateral platelet quan tization. The observ ed bi-exp onen tial photoluminescence
deca y of NPLs is connected to the v ery large radiativ e rates of the ES and GS and a longitudinal
optical phonon b ottlenec k go v erning their in ter-relaxation. In nanoplatelets longitudinal optical
(LO) phonon mo des are suppressed due to the strong transv ersal confinemen t [
17
]. This leads to
a slo wdo wn of the ES–GS exciton transfer rate and subsequen tly to observ able ES luminescence
w ell b elo w ground state saturation.
The ES–GS in ter scattering rate is size dep enden t. This size-tunable three lev el system is,
for instance, ideal to tune lasing prop erties – c hanging the in ter-scattering rate b et ween the
t w o states is exp ected to influence the in v ersion conditions of this system. CdSe platelets are
p erfectly suited to con trol the exciton-phonon in teraction b y c hanging their lateral size while
the optical transition energy is determined b y their thic kness.
The size dep enden t energies as w ell as the p opulation dynamics of ES and GS will b e determined
b y four indep enden t metho ds: time-resolv ed photoluminescence (PL), time-in tegrated PL, rate
equation mo deling and Hartree renormalized
k · p
calculations – all in v ery go o d agreemen t.
P ossible scenarios suc h as biexcitons, trions and LO phonon replica will b e excluded as the origin
of the observ ed double emission. The
k · p
calculations indicate that the ground and excited
state are related to an s- and p-state, resp ectiv ely .
F or the in vestigated CdSe nanoplatelets figure 3.1 (a) sho ws the ev olution of the low est exciton
s and p-states with increasing transv ersal confinemen t and anisotropy . Their (single particle)
w a v e functions and allow ed transitions are indicated in panel (b).
50 3 Excited-State Luminescence and an LO Phonon Bottleneck in CdSe NPLs
ΔE
(a) (b)
Figure 3.1: Anisotropic confinemen t lifts the p-state degeneracy .
(a) Ev olution of the
electron
p
-shell degeneracy when an isotropic quan tum b o x ev olves in to a platelet with unequal side-
lengths. (b) Ov erview of the allow ed optical (coulom b correlated) transitions observ ed in exp erimen t
along with w a ve-function probabilit y densit y plots in real space. A measured ground-state (GS)
and excited-state (ES) PL sp ectrum is pictured in the cen ter.
Samples
The in v estigated 4.5 monola yer (ML) core-only CdSe NPLs with the firs t exciton absorption
bands around 512 nm (4.5 ML) and 3.6x8.1, 17x6, 29x8, 30x15 and 41x13 nm
2
lateral size w ere
syn thesized as describ ed in ref.
75
. The sizes 21x7 and 30x10 nm
2
w ere syn thesized as in ref.
80
.
Exemplary TEM images are sho wn in figure 3.2. The particles w ere redissolv ed in toluene and
em b edded in PMA O p olymer (from Aldric h) on thin fused silica substrates. They w ere moun ted
in a Cry o v ac Con ti IT cry ostat (3.5-300 K) in the setup sho wn in figure 2.12. The v olume fraction
in the p olymer w as k ept b elo w 1 % to av oid an y aggregation or FRET effects [81].
3.1 Double Emission
The samples w ere excited at 420 nm b y a 150 fs 75.4 MHz Ti:Sa laser with
∼
0.2 W/cm
2
(CW
equiv alen t). The dual photoluminescence (PL) emission of CdSe NPLs is shown in figure 3.2 (a).
W e see an ov erall (sligh t) red-shift with increasing NPL size – as exp ected for decreasing lateral
confinemen t. Also the in tensities and energetic separation of the t w o p eaks c hanges with lateral
size. The p eak cen ters and integrated peak areas of the excited state (ES) and ground state
(GS) are quan tified b y V oigt-fits. In figure 3.2 (b) the ES to GS p eak area ratio is plotted v ersus
the ES-GS energy difference
∆E
. A clear minim um close to the LO phonon energy in CdSe of
25.4 me V is observ ed as a first hin t to lateral confinemen t con trolled LO-phonon coupling. F rom
figure 3.2 (c) it can b e seen that the energy difference b et w een the t w o states
∆E
increases from
18 to 38 me V with decreasing platelet size, i.e. with increasing lateral confinemen t. The results
of
k · p
en v elop e function theory are added in figure 3.1 (c) and discussed in app endix 3.6. W e
find go o d agreemen t b et w een calculated and exp erimen tal energy spacings.
3.2 Dynamics 51
8 . 1 x 3 . 6 n m ²
( a )
4 1 x 1 3 n m ²
3 0 x 1 5 n m ²
2 1 x 7 n m ²
2 9 x 8 n m ²
1 7 x 6 n m ²
8 . 1 x 3 . 6 n m ²
3 0 x 1 5 n m ²
2 . 4 5 2 . 5 0 2 . 5 5
G S
E n e r g y ( e V )
E S
5 0 n m
5 0 n m
0 2 0 0 4 0 0 6 0 0
1 0
2 0
3 0
4 0
5 0
( c )
E
E S - G S N P L a r e a ( n m ² )
E
E S - G S
( m e V )
P L
C a l c .
( 1 7 x 6 )
( 8 x 3 . 6 )
( 3 0 x 1 0 )
( 2 1 x 7 )
( 2 9 x 8 )
( 3 0 x 1 5 )
( 4 1 x 1 3 )
E
L O
- 1 0 - 5 0 5 1 0 1 5
0
1
2
3
( 2 9 x 8 )
( 8 x 3 . 6 )
( 4 1 x 1 3 )
E S / G S I n t . r a t i o
( b )
E
E S - G S
- E
L O
( m e V )
E
L O
= 2 5 . 4 m e V
Figure 3.2: Excited and ground state emission in CdSe nanoplatelets.
(a) Time-
in tegrated PL emission at 4 K of 4.5 monola yer (ML) CdSe NPLs with increasing lateral size from
top to b ottom (blue lines). The ground-state (GS) and excited-state (ES) emission p eaks are fitted
with V oigt-profiles. (b) LO-Phonon b ottlenec k: ES–GS PL in tensit y ratios at 4 K vs. the detuning
of the ES–GS energy spacing
∆E
to the LO phonon energy . Op en dots are deduced from (a), solid
dots from time-resolv ed sp ectra using data from fig. 3.3 and eq. 3.8. (c)
∆E
from exp erimen t (PL)
and calculations (Calc.) plotted vs. the NPLs’ area (sizes in nm in parenthesis, dotted line only a
guide to the ey e).
3.2 Dynamics
T o supp ort the h yp othesis of p-state emission and con trol of LO-phonon coupling by lateral
size w e in v estigate the luminescence dynamics. Streak camera measurements on t w o time scales
of b oth emissions are p erformed. A short time scale streak image is sho wn in figure 3.3 (a). A
rate equation mo del describing the dynamics of a three lev el system will b e compared to the
data. F rom the differing dynamics of the ES and the GS we will exclude the double emission to
originate from a zero-phonon p eak and its first LO phonon replica. Biexciton and T rion emission
as w ell as GS saturation will also b e ruled out b y CW and p o w er dep enden t measuremen ts
in section 3.5. A table listing the exp erimen tal energy spacings can b e found in table 3.1 in
app endix 3.6. Figure 3.3 (a+b) displa ys a streak measuremen t of 29x8 nm
2
(lateral dimensions)
platelets along with sp ectral cuts in time sho wing the ev olution of the dual ES-GS emission.
A fast ES and a slo w er GS PL deca y can b e observ ed and separated b y fitting the sp ectral
con tributions vs. time (fig. 3.3 (b)).
The ES fills the GS, while the ground state then deca ys on a longer time scale. The detuning of
the ES-GS energy spacing to the LO phonon energy (
∆E − E LO
) v aries for the differen t samples
(c-f ). In the resonant case (d+e) an ultra fast ES recom bination and GS filling is observ ed.
52 3 Excited-State Luminescence and an LO Phonon Bottleneck in CdSe NPLs
0 2 0 4 0 6 0 8 0
3 0 x 1 5 n m
2
/ 2 0 m e V ( c )
T i m e ( p s )
( a )
P L I n t e n s i t y ( a . u . )
2 9 x 8 n m
2
/ 2 7 m e V ( e )
0 2 0 4 0 6 0 8 0
2 1 x 7 n m
2
/ 3 2 m e V
( f )
T i m e ( p s )
3 0 x 1 0 n m
2
/ 2 4 m e V ( d )
E S G S
2 . 4 5 2 . 5 0
( b )
1 1 0 p s
9 0 p s
2 p s
4 p s
E n e r g y ( e V )
6 p s
8 p s
1 2 p s
6 0 p s
G S E S
0 p s
1 1 0 p s
C o u n t s ( a . u . )
Figure 3.3: Excited and ground state dynamics in CdSe nanoplatelets at 4 K.
(a)
T ransien t PL deca y and ev olution of the ES and GS emission with time for a platelet size of
29x8 nm
2
(b) Sp ectral cuts with exemplary V oigt fits at different times for 2 ps temp oral bins.
(c)-(f ) ES (red) and GS (blac k) transien ts obtained from V oigt fits as in (b) for differen t NPL sizes
and corresp onding ES-GS spacings
∆E
. The fits to the rate equation mo del (solid lines, eqs. 3.6
and 3.7) include con v olution with the instrument response (IRF).
In the off-resonan t case (c+f ) the ES dynamics slo w do wn. This effect is directly related to
the existence of an LO phonon b ottlenec k b et w een the excited and the ground state. When
∆E 6
=
E LO
the ES-GS in ter-relaxation is suppressed. Correspondingly the ES/GS emission
in tensit y ratio increases with increasing detuning from the minim um at the LO phonon resonance,
as seen in figure 3.2 (b).
This trend is indep enden tly confirmed b y b oth dynamic and static PL data: Time-in tegrated
ES/GS in tensit y ratios can b e deduced from biexp onen tial fits to the ES and GS deca y transien ts
(solid lines in figure 3.3 (c-f ) to b e reasoned in the follo wing). This ES/GS in tensit y ratio deriv ed
from dynamic PL is in go o d agreemen t with that deriv ed from the time-in tegrated PL data, see
op en and solid dots in figure 3.2 (b).
3.3 A Three Level System - Rate Equation Mo del
T o analyze the dynamics of the double emission in our CdSe nanoplatelets w e employ a three
lev el system. As depicted in figure 3.4 it consists of the crystal ground state
h 0 |
(no exciton) and
ground
h GS |
and excited
h E S |
exciton states. GS and ES can deca y radiativ ely in to the crystal
ground state or non radiativ ely in to trap states. They are connected to eac h other by an LO
3.3 A Three Level System - Rate Equation Mo del 53
Figure 3.4: Sc heme of a three lev el system
with all the rates and energy lev els used in the
rate equation mo del.
phonon mediated scattering rate. This rate equation system is describ ed b y:
˙ n E S = − n E S ( Γ r
E S + Γ nr
E S + γ 0 ( n ∆ + 1)) + n GS γ 0 n ∆ (3.1)
˙ n GS = − n GS ( Γ r
GS + Γ nr
GS + γ 0 n ∆ ) + n E S γ 0 ( n ∆ + 1) (3.2)
Here (
i
=
E S, GS
) denotes the excited and ground state.
n i
and
Γ r
i
are the p opulation and
the in trinsic, temp erature indep enden t radiativ e rate.
Γ nr
i
=
Γ nr , 0
i · exp ( − ∆E t
i /k b T )
is the
non-radiativ e deca y rate with the trapping frequency factor
Γ nr , 0
i
and activ ation energy
∆E t
i
.
γ 0
is the zero temp erature scattering rate b et w een ES and GS. The energy spacing
∆E
b et w een ES
and GS and the temp erature dep endence of the ES
↔
GS scattering rate are tak en in to accoun t
b y a Bose statistics factor n ∆ = e ∆E / ( k b T ) − 1 − 1 .
W e identify eqs. 3.1 and 3.2 as a system of coupled first order differen tial equations, whic h w e
solv e b y calculating the state Eigen vectors. W e tak e
˙ n E S = A n E S + B n GS (3.3)
˙ n GS = C n E S + D n GS
with the matrix elemen ts:
A = − Γ r
E S − Γ nr
E S − (1 + n ∆ ) γ 0 (3.4)
B = n ∆ γ 0
C = (1 + n ∆ ) γ 0
D = − Γ r
GS − Γ nr
GS − n ∆ γ 0
and eigen v alues:
λ 1 / 2 = A + D
2 ∓ r A 2
4 + D 2
4 + B C − AD
2 (3.5)
54 3 Excited-State Luminescence and an LO Phonon Bottleneck in CdSe NPLs
Since b oth
λ 1
and
λ 2
, are negativ e and
λ 1 < λ 2
, w e iden tify
λ 2
as the slo w and
λ 1
as the fast
deca y rate, referred to as − λ S and − λ F .
After photo excitation, carrier co oling and exciton formation ha v e b een found to o ccur on a
time scale faster than our time resolution limit [
82
]. As initial condition to find our Eigen v ectors
w e can th us assume
n ( t =0)
GS
=
n ( t =0)
E S
. The solution for the in tensit y evolution of ES and GS is
then:
I E S ( t ) = S 0 Γ r
E S n E S ( t ) = S 0 Γ r
E S n ( t =0)
E S
e − λ F t
f
z }| {
−
C
λ F + D + 1
1 + C
λ S + D
e − λ S t
(3.6)
I GS ( t ) = S 0 Γ r
GS n GS ( t ) = S 0 Γ r
GS n ( t =0)
GS
− C
λ F + D
| {z }
g
e − λ F t + C
λ S + D
C
λ F + D + 1
1 + C
λ S + D
| {z }
h
e − λ S t
(3.7)
Here
S 0
is a prop ortionalit y factor giv en b y the sensitivit y of the setup. ES and GS deca y
bi-exp onen tially with iden tical deca y constan ts
λ F
and
λ S
, but with differen t amplitudes ( 1 ,
f
,
g
and
h
). The fits sho wn in figure 3.3 (c)-(f ) are global bi-exp onen tial fits (for eac h sample) with
shared deca y constan ts. The amplitudes are not alw a ys p ositiv e, esp ecially for NPLs where
∆E GS − E S
is nearly resonan t to the LO-phonon energy . Here a negativ e amplitude of the fast
comp onen t of the GS reflects its filling from the ES.
A t elev ated temp eratures it is not p ossible to resolv e ES and GS sp ectrally from streak images
as w ell as at 4 K. W e therefore sp ectrally bin the ES and GS emission. The long time range
measuremen ts (figure 3.5 b and d) are fitted mono exp onen tially to extract the slo w comp onen t
λ S
. It is then tak en as fixed in the biexp onen tial fits of the fast time range PL transien ts
(figure 3.5 a and c). The resulting slo w deca y comp onen t
λ S
and fast deca y comp onen t
λ F
are
plotted in the upp er panel of figure 3.5 (e), for 29x8 nm 2 and 41x13 nm 2 sized platelets.
The exp erimen tally accessible time in tegrated PL ratio of ES to GS emission is sho wn in
in the lo w er panel of figure 3.5 (e). It can also b e expressed with the parameters of the rate
equation system. F or this
I E S ( t )
and
I GS ( t )
are in tegrated from zero to infinit y . T aking the
ratio
R T I − P L
E S,GS
(
T
) of these t w o quan tities eliminates the prop ortionalit y factor
S 0
and w e obtain:
R T I − P L
E S,GS ( T ) = R ∞
0 I E S ( t ) d t
R ∞
0 I GS ( t ) d t = Γ r
E S
Γ r
GS · λ − 1
F + f λ − 1
S
g λ − 1
F + hλ − 1
S
(3.8)
The temp erature dep endence of the fast and slo w deca y constan ts (
λ F
and
λ S
) as w ell as
the time in tegrated PL in tensity ratio
R T I − P L
E S,GS
(
T
) can then b e fitted sim ultaneously . This is
done b y sharing the same temp erature indep enden t radiativ e rate (
Γ r
E S/GS
), the non-radiativ e
3.4 Results 55
Figure 3.5: T emp erature dep enden t deca y dynamics of CdSe nanoplatelets.
(a-d) Blue
lines: PL decay curv es at 4, 35 and 100 K of CdSe NPLs with lateral size of 29x8 nm
2
(a and b) and
41x13 nm
2
(c and d). The bi-exponential fits (on top of data) to the fast time-range deca ys (a and
c) use the slo w deca y rate derived from mono-exponential fits in (b and d) recorded in a wider time
windo w. The instrumen t resp onse function (grey line) is used for decon v olution. Inset: Sp ectrally
disp ersed Streak Camera image of the PL deca y in the first 0
.
5 ns of 29x8 nm
2
CdSe NPLs at 4 K.
The excited state emission is clearly visible. The fast time range is indicated b y a gra y frame. (e)
Deca y rates of 29x8 nm
2
and 41x13 nm
2
NPLs from time-resolv ed PL measuremen ts (top panel)
and time in tegrated ES/GS PL in tensity ratios (lo w er panel) vs. temperature. The ES-GS energy
spacing
∆E E S − GS
w as held fixed when sim ultaneously fitting eq. 3.5 to the fast and slo w decay
rates, λ F and λ S , and eq. 3.8 to the in tensit y ratios (solid lines).
(
Γ nr
E S/GS
) rates and in ter-scattering rate
γ 0
. Note that the radiativ e and non-radiativ e rates
are differen t for ES and GS. T ogether with
γ 0
they determine the deca y constan ts (
λ F
and
λ S
)
that are iden tical for ES and GS. The energy spacing
∆E
obtained from time in tegrated PL
measuremen ts is held fixed. The inclusion of the ES/GS ratio in the fit in tro duces a further
b oundary condition that reduces the fit am biguit y , see app endix 3.6 figure 3.7. F urther, the used
metho d of fitting the temp erature dep endence of the fast and slo w comp onen ts and PL in tensit y
ratio do es not include assumptions ab out the relev an t rates.
3.4 Results
Figure 3.5 (e) sho ws the excellen t agreemen t of the fit with the exp erimen tal fast and slo w decay
rates and with the PL emission in tensit y ratio, for the 29x8 and 41x13 nm
2
samples. The
follo wing radiativ e rates are obtained: for 29x8 nm
2
NPLs (
Γ r
E S
= 16 ns
− 1
,
Γ r
GS
= 3
.
7 ns
− 1
),
and for 41x13 nm
2
(
Γ r
E S
= 50 ns
− 1
,
Γ r
GS
= 6
.
4 ns
− 1
). With increasing lateral platelet size the ES
and GS PL deca y tends to b ecome faster (higher rates). This is in line with the predictions of
the Gian t Oscillator Strength (GOST) effect [
83
]. The v ery large ES radiativ e rates enable the
observ ation of the dual ES and GS PL emission. This finding is accompanied b y an LO phonon
mediated ES
↔
GS scattering rate
γ 0
. It is higher in the near resonan t case of 29x8 nm
2
NPLs
56 3 Excited-State Luminescence and an LO Phonon Bottleneck in CdSe NPLs
with (
∆E E S − GS − E LO
= 1
.
5 me V)
γ 0
= 101 ns
− 1
. It slo ws do wn in the off-resonant case of
41x13 nm
2
NPLs with (
∆E E S − GS − E LO
=
−
7
.
9 me V)
γ 0
= 25 ns
− 1
. The rate equation system
th us indep enden tly confirms the manifestation of an LO phonon b ottlenec k previously deduced
from the ES/GS in tensit y ratios in figure 3.2 (b).
The PL quan tum yield is not unit y in our samples. Th us the in tro duction of the non-radiativ e
deca y c hannels in the rate equation mo del is justified. It further allo ws to calculate the quan tum
yield of our samples – giv en b y the sum of all radiativ e deca y rates divided by the sum o v er
all radiativ e and non-radiativ e rates. The fits in figure 3.5 (e) pro duce a trapping frequency
factor
Γ nr , 0
E S
0
.
15 ps
− 1
/ 0
.
28 ps
− 1
and
Γ nr , 0
GS
0
.
03 ps
− 1
/ 0
.
06 ps
− 1
, resp ectiv ely , for our 29x8
/ 41x13 nm
2
samples. The activ ation energies of the trap states
∆E t
i
are shared b y ES and
GS to reduce fitting am biguities. They are found to b e 11.0 and 12.0 me V for our 29x8 and
41x13 nm
2
samples. Using the radiative rates for ES and GS and the non-radiativ e rates (eq.
giv en ab o v e) the calculated quan tum yield is 14 and 20 % at 300 K, for 29x8 and 41x13 nm
2
NPLs, resp ectiv ely . This is in very goo d agreemen t with the off-resonant quan tum yield at ro om
temp erature measured for CdSe NPLs[84] (c hapter 7.4).
3.5 Exclusion of LO Phonon Replica, T rions, GS Saturation and
Bi-excitons
The excitation densit y of the pulsed frequency doubled Ti:Sa laser w as held b elo w 0.2 W/cm
2
(CW equiv alen t) to av oid an y heating, ground state saturation and the presence of higher order
pro cesses suc h as biexcitons or trions. They will b e discussed no w.
Biexcitons:
Using the absorption cross sections of the CdSe NPLs (section 7.2.1 and ref. [
54
])
w e calculate that only
<
0
.
1 p ercen t of the platelets are excited within one laser pulse. This
results in an a v erage n umber of excitons p er platelet p er pulse
< N >
b elo w 10
− 3
at the used
excitation in tensit y , see green arro ws in figure 3.6 (c) and (d). The probabilit y that a platelet
undergo es t w o subsequen t photo excitations or t w o excitations within one pulse is
<
10
− 6
and
th us negligible. This excludes relev an t bi-excitonic effects.
Photogenerated T rions:
F or trion formation, the excess carrier added to the generated
exciton has to stem from an ionized exciton from the previous laser pulse. The v ery lo w excitation
densit y used here rules out an efficien t trion generation. The efficiency of suc h a com bined
pro cess m ust b e ev en lo w er than the one for biexcitons. The rep etition time of the laser (13 ns)
is to o long for an efficien t trion formation. The probabilit y of trion formation b y the collision of
t w o excitons generated within one pulse and platelet[
85
] will also b e considerably lo w er than
10
− 6
. A kinetic argumen t against trions holds as w ell, since the ratio of the exciton to trion
3.5 Exclusion of LO Phonon Replica, T rions, GS Saturation and Bi-excitons 57
2 . 4 2 . 5 2 . 6
0 . 0
0 . 5
1 . 0
2 . 4 2 . 5 2 . 6
0 . 0
0 . 5
1 . 0
2 . 4 2 . 5 2 . 6
0 . 0
0 . 5
1 . 0
3 0 x 1 0 n m ²
G S
P u l s e p e a k i r r a d i a n c e i n
( a ) a n d ( b ) ( k W / c m ² ) :
2 . 8 7
1 4 2 8
P L I n t . ( a . u . )
E n e r g y ( e V )
E S
I
( a )
1 0
- 4
1 0
- 3
1 0
1
1 0
2
1 0
3
P u l s e d v s . C W e x c i t a t i o n
P e a k a r e a ( a . u . )
G S E S
< N >
P o w e r d e p e n d e n c e
2 . 4 2 . 5 2 . 6
0 . 0
0 . 5
1 . 0
( b )
P L i n t . ( a . u . )
1 7 x 6 n m ²
E n e r g y ( e V )
G S
E S
1 0
- 4
1 0
- 3
1 0
1
1 0
2
1 0
3
1 0
4
( c )
P e a k a r e a ( a . u . )
< N >
( d ) ( f )
n o r m . P L I n t . ( a . u . )
1 4 K W / c m ² ( T i : S a )
0 . 1 4 W / c m ² ( C W )
E n e r g y ( e V )
( e )
n o r m . P L I n t . ( a . u . )
E n e r g y ( e V )
Figure 3.6: P o w er dep endence of the excited and ground state emission at 4 K.
(a) and
(b) PL sp ectra of 30x10 and 17x6 nm 2 CdSe NPLs for excitation densities (pulse p eak irradiance)
ranging o v er an order of magnitude. (c) and (d) The p eak areas of the excited and ground state
(ES and GS) resulting from V oigt fits to the spectra in (a) and (b) versus the fraction of excited
platelets within one laser pulse
<
N
>
. The excitation densit y used for the measuremen ts and
analysis ab o v e is indicated b y green arrows. (e) and (f) show the indistinguishable normalized
sp ectra after CW (at 409 nm, blac k) and pulsed (at 420 nm, gra y) excitation.
lifetime is found to b e univ ersally
τ X /τ T
= 2
.
2 in CdSe nanoparticles [
74
]. In our case the GS
and ES deca y rates differ b y a factor of 4.3-7.8.
A dditionally , the linear p ow erdependence seen in figure 3.6 (a-d) do es not comply to the
non-linear p o w erdep endence exp ected for trion and bi-exciton emission.
GS saturation:
In figure 3.6 (e)-(f ) w e observe iden tical normalized time in tegrated ES-GS
PL sp ectra for b oth CW and fs pulsed excitation. This rules out that the observ ed ES emission
is due to GS saturation, biexcitons or trions.
LO-phonon replica:
The energy difference
∆E
b et w een the ES and the GS increases from
18 to 38 me V with increasing lateral platelet confinemen t. This excludes that the energy spacing
is related to a zero-phonon p eak and its first LO phonon replica. In this case it should ha ve a
nearly confinemen t indep enden t energy spacing of the LO-phonon energy 25.4 me V. F urther, the
p eaks featuring the double emission should ha v e iden tical dynamics. As seen ab o v e in figure 3.3
this is clearly not the case. The double emission has also b een reported by Biadala et al.[
72
] in
time-in tegrated PL sp ectra and had b een in terpreted as the zero phonon line with its first LO
phonon replica. By chance the observ ed energy spacing there nearly matc hed the LO phonon
energy of CdSe.
58 3 Excited-State Luminescence and an LO Phonon Bottleneck in CdSe NPLs
Conclusion
It has b een sho wn that CdSe nanoplatelets exhibit not only lo w est s-exciton (ground state GS)
related photoluminescence (PL) up on con tin uum excitation. Also an excited state is observ able,
far b elo w GS saturation. Theoretical considerations (
k · p
theory) indicate that this excited state
is p-state related, th us an exciton. Other p ossible origins of the observ ed double emission suc h
as trions, bi-excitons and longitudinal phonon replica could b e excluded from con tin uous w a v e
and p o w er dep enden t excitation as w ell as from the PL dynamics.
Time in tegrated PL and
k · p
calculations, in go o d agreemen t, sho w a strong dep endence of
the ES-GS energy spacing on the lateral quan tization of the exciton w a v efunction. The spacing
increases from
∼
18 to 38 me V with lateral platelet sizes decreasing from 30x15 to 8.1x3.6 nm
2
–
at a constan t NPL thic kness of 4.5 CdSe monola y ers. These energy spacings are in the order of
the longitudinal optical (LO) phonon energy in CdSe NPLs of 25.4 me V. In fact, an LO phonon
b ottlenec k b et w een the t w o states is observed. The in ter-scattering rate b et ween the ES and GS
dep ends strongly on the detuning of the ES-GS energy spacing to the LO phonon energy , with a
high impact on the dynamics of these NPLs. F or platelets with an energy spacing of ES and GS
resonan t to the LO phonon energy the in ter-scattering rate b et ween the t w o states is high. It
is suppressed in off resonan t cases. The existence of this LO phonon b ottlenec k is confirmed
b y four metho ds: (1) A rate equation mo del for the temp erature dep endence of the emission
dynamics, (2) the temp oral course of sp ectrally resolv ed ES and GS emission at 4 K, sho wing
e.g. ES
→
GS filling, (3) the observ ation of a minimum in the ES/GS emission intensit y ratio
deriv ed from the deca y kinetics as w ell as (4) from time-integrated PL.
In a first appro ximation the used rate equation mo del assumes temp erature indep enden t
radiativ e rates for the t wo luminescen t states. They are sho wn to increase with the lateral
platelet size. This phenomenon is attributed to the Gian t Oscillator Strength Effect in 2D and
will b e addressed with a more sophisticated mo del in the follo wing c hapter.
T o conclude, w e ha v e no w demonstrated that exciton energy on the one hand, and LO-phonon
coupling and oscillator strength on the other, can b e con trolled separately in CdSe nanoplatelets.
The former is tuned b y the thic kness (z-direction) whereas the lateral size (x,y-direction) allo ws
for a sensitiv e con trol of the LO phonon coupling and dynamics.
3.6 APPENDIX: k · p envelop e function theo ry 59
3.6 APPENDIX: k · p envelop e function theo ry
Energies
This section is based on the con tribution of Andrei Sc hliwa (TU Berlin) to the article in ref. [
27
].
As in previous w ork [
17
] the electronic structure on the nanoplatlets is obtained using a 3D
implemen tation of eigh t-band
k · p
en v elop e function theory . F ollo wing the discussion in ref. [
22
]
the self-energy term for the Coulom b term is implemen ted in this Hartree approac h. Both, the
effects arising from the dielectric en vironmen t and the electron and hole self-energy are included.
The used dielectric constan ts are:
ε P latelet
r
= 9
.
4 and
ε out
r
= 2
.
3 [
86
]. T able 3.1 summarizes the
exp erimen tal and theoretical GS energies
E GS
and the corresp onding ES-GS energy spacings
∆E
. The resolution of the n umerical grid is half a monola y er in all three directions. Parameters
T able 3.1:
Sizes, hh ground state emission
E GS
at 4 K, and energy difference b et w een ground state
and excited state emission
∆E
of the in v estigated CdSe core NPLs extracted from PL measuremen ts
(PL) and calculated v alues (Theo).
Size E ( P L )
GS ∆E ( P L ) E ( T heo )
GS ∆E ( T heo )
(nm 2 ) (e V) (me V) (e V) (me V)
17x6 2.486 36 2.485 41
21x7 2.480 32 2.477 35
29x8 2.469 27 2.465 27
30x10 2.470 24 2.454 24
30x15 2.466 20 2.447 22
41x13 2.468 18 2.449 22
of the calculations (5 K) are giv en in T able 3.2. The artificial bandgap of the outer material
(ligand oleic acid) is tak en as 8 e V whic h is in go od agreement with the HOMO-LUMO gap of
the similar fatt y acid stearic acid (7.76 e V) [
87
]. Therefore the used CB and VB-offsets are 3.5 e V
and 2.494 e V.
T able 3.2: P arameters of CdSe for zinc-blende structure.
P arameter CdSe Reference
a (nm) 0.6052 [88]
E G (e V) 1.766 [89]
E p (e V) 16.5 [19]
∆ S O (e V) 0.42 [89]
γ 1 4.4 [90]
γ 2 1.6 [90]
γ 3 2.68 [90]
m e ( m 0 ) 0.12 [91]
60 3 Excited-State Luminescence and an LO Phonon Bottleneck in CdSe NPLs
Radiative Rates
F urther we performed theoretical calculations of the radiativ e rates of ES and GS and obtained
go o d agreemen t for the ground state, and deviations for the ES. F or the GS of the 29x8 nm
2
platelet w e obtain Γ r,calc
GS = 3 . 8 ns − 1 , whic h is in go o d agreemen t to our exp erimen tal fit result
of 3.7 ns
− 1
. The theoretical ES rate is also 3.8 ns
− 1
, whic h deviates from our exp erimen tal result
of 16 ns
− 1
. The ES deviations ma y b e related to a differen t impact of the Gian t Oscillator
strength effect on s and p-states. Even for fixed (exact) energy lev els, the transition dip ole
momen ts dep end on further parameters suc h as the cen ter of mass exciton motion. It is clear
that calculations of the state energy and its oscillator strength / transition rate are t w o differen t
issues. Ho w ev er w e wan t to p oin t out that a theoretical analysis of this effect is b ey ond the
scop e of this article/w ork and ma y b e sub ject of a separate, demanding theoretical in v estigation.
Ho w ev er, there is no doubt, that the ES radiative rate is greater th an the GS rate. In figure 3.7
w e fit the deca y dynamics (eq. 3.5) as w ell as the time-integrated PL in tensit y ratios of ES and
GS (eq. 3.8) in the global mo del with shared parameters. Blac k lines indicate restricted fits with
equal ES and GS radiativ e rates (
Γ r
E S
=
Γ r
GS
), the blue lines sho w the fit results for the case
where
Γ r
E S
and
Γ r
GS
are indep enden t from eac h other. Clearly , (nearly) equal GS and ES rates
are not p ossible, as they do not allo w for fitting the temp erature dep enden t ES/GS in tensit y
ratio. Hence the excited state radiativ e rate has to b e larger than the ground state’s.
1 0 1 0 0
1 0
- 2
1 0
- 1
1 0
0
1 0
1
1 0
2
1 0
3
1 0 1 0 0
1 0
- 2
1 0
- 1
1 0
0
1 0
1
1 0
2
1 0
3
o f f - r e s o n a n t t o L O p h o n o n
4
1 x 1 3 n m ² / - 7 . 9 m e V
r e s o n a n t t o L O p h o n o n
2 9 x 8 n m ² / 1 . 5 m e V
R a t i o / R a t e s
E S / G S i n t . r a t i o
r a d
E S
,
r a d
G S
i n d e p e n d e n t
r a d
E S
r a d
G S
c o n s t r a i n t
T e m p e r a t u r e ( K )
F
( n s
- 1
)
S
( n s
- 1
)
G l o b a l r a t e e q . m o d e l f i t :
T
e m p e r a t u r e ( K )
Figure 3.7:
F ast and slo w PL deca y rates globally fitted together with the time in tegrated ES and
GS emission ratio. Thin blac k lines indicate the restricted fits. F or the tw o upp er curv es the blac k
and blue curv es coincide for b oth platelet sizes. This also sho ws to what exten t the consideration
of the ES/GS time in tegrated in tensity ratio reduces the fitting am biguities.
4 | The Gian t Oscillator Strength Effect and De-
phasing in NPLs
The p olarization dynamics of a semiconductor reflects the coherence of the exciton w a v efunction
with resp ect to an exciting photon. Ro om temp erature exciton coherence [
31
], and the Gian t
Oscillator Strength Effect (GOST)[
17
,
19
,
26
] ha v e b een demonstrated in CdSe NPLs. The
GOST effect in 2D directly relates the exciton coherence and dephasing with the radiativ e
lifetime [83, 92].
In the follo wing this un usual coupling of radiative lifetime and exciton dephasing in 2D will
b e in v estigated. This requires to unify the ES and GS p opulation relaxation mo del discussed in
the previous c hapter (and in ref. [
27
]) with the o ccurrence of the GOST effect. F or this a new
refined rate equation mo del will b e in tro duced that also tak es the linewidths of the luminescence
in to accoun t.
It will b e sho wn that the exciton lifetimes, dephasing rates and phonon coupling can b e tuned
b y the platelets’ lateral size o v er more than an order of magnitude. The radiative lifetime
of excitons can b e con trolled b y the exciton coherence v olume. It is either go v erned b y the
ph ysical lateral exten t of the nanoplatelets or b y the exciton’s coherence length whic h in turn is
determined b y the homogeneous linewidth. When the exciton coherence is limited b y the lateral
platelet size, dephasing and subsequen tly the radiativ e lifetime can b e con trolled via the lateral
exten t. This clear con trol of the 2D exciton coherence area, no w p ossible in colloidal quan tum
w ells of finite size, is en tirely new.
The ES-GS energy separation in our CdSe nanoplatelets is also lateral size dep enden t and
con trols the ES-GS in ter-scattering. A djusting the ES-GS energy separation and exciton
coherence area b y a c hange in lateral confinement allo ws to suppress or enhance the coupling to
LO-phonons as w ell as to tune the radiativ e rates of ES and GS (c hapter 3). T unable radiativ e
lifetimes of the order of 10 ps can b e ac hieved. This makes CdSe nanoplatelets one of the fastest
nano emitters with strong application p oten tial in photonics.
F urther it will b e sho wn, for the first time, that dephasing rates and homogenous linewidths
of quan tum w ell excitons can b e deduced from the temp erature dep enden t time-in tegrated and
-resolv ed PL data. The results of our global analysis mo del are in go o d agreemen t with classical
F our W a v e Mixing (FWM) results [26, 31].
62 4 The Giant Oscillator Strength Effect and Dephasing in NPLs
E nerg y
A p
A c
A p e -
h A c
A c >A
p
A p e -
h
A c
A c <A
p
T c
AC
L O+AC
W ave vector K
a b d
E x cit on st at e
Cryst al grou nd
s ta te
c
E x ci t on coherence area
T emp er at ure
T emp er at ure ( K)
E nerg y
Energy
A p
A c
A p e -
h A c
A c >A
p
A p e -
h
A c
A c <A
p
T c
AC
LO+AC
W ave vector K
a b d
Exciton state
Crystal ground
state
c
Exciton coherence area
T emperature
T emperature (K)
Figure 4.1: Exciton dephasing and the Gian t Oscillator Strength (GOST) effect.
(
a
)
GOST effect: Carriers are thermalized to w a v evectors outside the radiativ e cone and cannot deca y
there radiativ ely leading to a prolongation of lifetime with temp erature. ( b ) The fraction r ( T ) of
excitons that can deca y radiativ ely is gov erned b y the in trinsic homogeneous linewidth, e.g. for
∆ 0
=5 me V (solid blue line) or
∆ 0
= 10 me V (red line). The dashed lines sho w
r
(
T
) when the
coupling to acoustic (A C) and longitudinal optical (LO) phonons is added to the homogeneous
linewidth
∆
(
T
) .(
c
) Homogeneous linewidth
∆
(
T
) , radiativ e rate
Γ
and dephasing rate
˜ γ
for an
exciton state. (
d
) Sk etc h of the tw o cases to b e considered in colloidal quan tum w ells: the 2D
exciton coherence area is either limited b y the physical lateral size (
A p
) or b y the homog eneous
linewidth limited exciton coherence area (
A c
). F or temp eratures b elo w a critical temp erature
T c
(at whic h the tw o areas matc h
A c
=
A p
) the platelet area limits the spatial exciton coherence. A t
elev ated temp eratures the homogeneous exciton linewidth limits the spatial coherence of a 2D
exciton in CdSe nanoplatelets to an area
A c
smaller than the platelet area. Therefore the radiativ e
rate is quasi constan t for lo w temp eratures (as
r
(
T
) is quasi constan t at lo w temp eratures). Ab o v e
T c the radiativ e rate decreases ∝ r ( T ) /∆ ( T ) .
4.1 Linewidth and Oscillato r Strength
The homogenous linewi dth
∆
of an exciton transition and its temp oral coherence are c haracterized
b y the total depha sing rate
γ ∗
. Th e popula tion deca y rate
Γ
(in v ers e to the lifetime) and the
pure dephasing
˜ γ
originating e.g. from dec oherence due to phonon scattering are relate d to the
total homogeneous lin ewidth via[93]
∆ = γ ∗ / 2=( Γ/ 2+ ˜ γ ) / 2 (4.1)
see figure 4.1 (c). If w e consider a system, where the exciton lifetime dep ends on the homogeneous
exciton linewidth, s o that
τ → τ
(
∆
) , w e in tro duce a dep endence of th e popula tion lifetime and
the pure dephasing rate
˜ γ
via the equation ab o ve. This idea stim ulated the following in v estigation.
It will b e sho wn that the life time and therefore the transition oscillator s trength of 2D excitons
in CdSe nanoplatelet s is transition linewidth and subsequently dephasing dep enden t. Th us, the
oscillator strength an d the pure dephasing are no longer indep enden t of each other, allo wing to
retriev e the pure de phasing rates from lifetime measurements. T he transition oscillator strength
4.1 Linewidth and Oscillator Strength 63
F is related to the exciton in trinsic radiativ e rate via [83, 92]
Γ r = | f lf | 2 ˜ ne 2 ω 2 F
2 π ε 0 m 0 c 3 (4.2)
where
˜ n
is the refractiv e index,
m 0
the electron mass and
f lf
the lo cal field factor. The other
sym b ols ha v e their usual meaning.
The Gian t Oscillator Strength Effect in semiconductors at lo w temp eratures o ccurs due to the
fact that only excitons with small cen ter of mass
K
-v ector (
K < K 0
) can deca y radiativ ely .[
83
,
92
]
K 0
is the w a v e ve ctor of light in the crystal with the same energy as the exciton. Excitons with
large cen ter of mass w a ve v ector
K > K 0
cannot deca y radiativ ely b ecause of the
K
conserv ation
rule in the direction of free exciton motion.
When the homogenous linewidth
∆
of excitons is large so that the uncertain t y of the w a vev ector
is comparable to
K 0
it limits the spatial coherence of the 2D excitons to the coherence area
A c
= 2
π ~ 2 / M ∆
(with the exciton mass
M
). In this case the
K
= 0 exciton lifetime is replaced
b y a longer lifetime determined b y
A c
. The maxim um wa v e v ector of radiativ e exciton transitions
K 0
is replaced b y
π p π / A c
.[
83
,
92
] Only the fraction of excitons
r
(
T
) ha ving energies (and
related
K
v ectors) smaller than the homogenous linewidth
∆
can then deca y radiativ ely .[
83
]
The maxim um exciton momen tum and
∆
are related b y the 2D parab olic exciton disp ersion
E
(
K
) =
~ 2 K 2 /
2
M
, figure 4.1 (a). Thermally activ ated scattering processes (e.g. with phonons)
lead to an increasing n um b er of excitons outside the radiativ e cone in thermal equilibrium.
Defined b y Maxw ell statistics, the fraction
r
(
T
)=1
− e − ∆ ( T ) /k B T
of excitons that can deca y
radiativ ely th us decreases with increasing temp erature, as illustrated in figure 4.1 (b). This
(GOST) effect leads to a prolongation of the radiativ e lifetime with temp erature.
The homogenous linewidth
∆
(
T
) is itself temp erature dep enden t. The broadening of the
exciton resonance due to coupling to acoustic (A C) phonons and longitudinal optical (LO)
phonons can b e expressed as:[94]
∆ ( T ) = ∆ 0 + ∆ AC T + ∆ LO
e E LO /k B T − 1 (4.3)
Where
∆ AC
and
∆ LO
are, resp ectiv ely , the coupling constants to A C and LO phonons,
∆ 0
is
the 0 K homogeneous linewidth and E LO the LO phonon energy .
With the in tro duction of
r
(
T
) the exciton transition oscillator strength has to b e replaced
with the effectiv e 2D oscillator strength according to:
F 2 D
x,ef f = F 2 D
x r ( T ) (4.4)
64 4 The Giant Oscillator Strength Effect and Dephasing in NPLs
Nanoplatelets are quan tum w ells of finite lateral size. Figure 4.1 (d) illustrates the t w o cases to
b e considered here: either the exciton’s spatial coherence and th us the 2D oscillator strength
F 2 D
x is limited b y the nanoplatelets’ finite area A p or b y a smaller coherence area
A c = 2 π ~ 2 / ∆ ( T ) M , (4.5)
determined b y the homogeneous linewidth. The effectiv e oscillator strength can no w b e written
as [83]
F 2 D
x,ef f = r ( T ) f 0
A p
A x
, for A p < A c (4.6)
F 2 D
x,ef f = r ( T ) f 0
A c
A x
, for A p > A c (4.7)
f 0
is the oscillator strength p er unit cell and
A x
=
π a 2 D
B 2
the 2D exciton area.
a 2 D
B
is the 2D
exciton Bohr radius. Using the (2D) exciton binding energy E 2 D
B = ~ 2 / 2 µ ( a 2 D
B ) 2 w e obtain
F 2 D
x,ef f = A p · r ( T ) f 0 2 µE 2 D
B
π ~ 2 , for A p < A c (4.8)
F 2 D
x,ef f = A c · r ( T ) f 0 2 µE 2 D
B
π ~ 2 , for A p > A c (4.9)
With eq. 4.2 the temp erature dep enden t radiativ e rate can no w b e expressed as:
Γ r ( T ) = τ − 1 ( T ) = ˜ ne 2 ω 2 f 0 µE 2 D
B | f lf | 2
π 2 ~ 2 ε 0 m 0 c 3 · r ( T ) A X C (4.10)
where
A X C
denotes the exciton coherence area, whic h is either the platelet area
A p
for
A p < A c
or the linewidth limited coherence area A c (eq. 4.5) for A p > A c .
Figure 4.1 (d) shows a sk etc h of the coherence area. At lo w temp eratures it is limited b y the
ph ysical size of the nanoplatelet and th us constan t. F or temp eratures ab o v e a certain transition
temp erature
T c
(where
A c
=
A p
) the coherence area decreases due to more dephasing b y phonon
scattering increasing the exciton linewidth. In b oth cases the radiativ e rate (
Γ r ∝ F
) relates the
homogeneous linewidth
∆
(
T
) via
r
(
T
) . F or
T < T c
the radiativ e rate scales with the nanoplatelet
area
A p
. A t higher temp eratures
T > T c
an additional dep endence on
∆
(
T
) is in tro duced b y
the linewidth limited coherence area of the 2D exciton. The homogeneous linewidth is related
to the dephasing rate. In a 2D system suc h as nanoplatelets, the exciton lifetime, oscillator
strength and pure dephasing are th us not indep enden t an ymore and connected b y the GOST
effect. Therefore measuremen ts of the radiativ e deca y times in quantum w ells are also sensitiv e
to the pure dephasing.
4.2 Three Level System Combined with Giant Oscillator Strength Effect 65
4.2 Three Level System Combined with Giant Oscillato r Strength
Effect
In order to extract the radiativ e rates from PL deca y measuremen ts it is crucial to ha v e detailed
kno wledge ab out the recom bination dynamics of the studied system. F ollo wing c hapter 3 the
p opulation dynamics in CdSe nanoplatelets is go v erned b y an LO-phonon b ottlenec k b et ween
an excited (ES) and a ground (GS) exciton state. The three lev el system is sho wn here again in
figure 4.2 (b). The solutions of the underlying coupled differen tial equation system are giv en in
eqs. 3.6 and 3.7 in c hapter 3. There temp erature indep enden t, constan t radiativ e rates
Γ r
i
are
assumed for the excited and ground state ( i = E S,GS ). No w w e identify them as proportional
to the effectiv e oscillator strength
F 2 D
x,ef f
. T o connect this p opulation model with the Giant
Oscillator Strength Effect the constan t
Γ r
i
are replaced b y the temp erature and linewidth
dep enden t radiativ e rates for ES and GS Γ r
i ( T ) (eq. 4.10).
The non-radiativ e rates are assumed to b e thermally activ ated according to
Γ nr
i
=
Γ nr , 00
i
+
Γ nr , 0
i e − ∆E t
i /k B T
with a zero temp erature non-radiativ e rate
Γ nr , 00
i
, a trapping frequency factor
Γ nr , 0
i and an activ ation energy ∆E t
i , assumed to b e pairwise iden tical for ES and GS.
In the previous c hapter the exp erimen tally accessible quan tities used w ere the time-integrated
in tensit y ratio of the ES and GS emission and the decay constan ts
λ F /S
of the biexp onen tial
deca y of the sp ectrally binned ES and GS emission. Now, with the in tro duction of the GOST
effect, they can b e connected to the linewidths
∆ tot
(
T
) of ES and GS. F rom the time in tegrated
PL measuremen ts w e can also asses
∆
(
T
) . Due to the exact thic kness quantization to n+1/2
CdSe monola y ers and no or very w eak lateral confinemen t nanoplatelets sho w only a small
(temp erature-indep enden t) inhomogeneous broadening
∆ 00
. The exp erimen tally accessible
emission linewidth is the total linewidth
∆ i
tot ( T ) = ∆ i
00 + ∆ i ( T ) (i=ES,GS). (4.11)
It is obtained from V oigt fits to the time-integrated photoluminescence peaks of ES and GS
(example sho wn in figure 4.2 (a)). Th us the linewidths of ES and GS can b e included in the fit.
No w w e ha ve fiv e exp erimen tal curv es addressing the parameters within completely differen t
functional dep endencies on them: the ES/GS time-in tegrated emission intensit y ratio, the
linewidths
∆ tot
(
T
) of the t w o states and the PL deca y constants
λ F /S
. They are plotted in
figure 4.2 (d) and (e) for 29x8 and 41x13nm
2
nanoplatelets. The quan tities are presen ted on a
logarithmic scale due to their strongly differing scale v alues.
66 4 The Giant Oscillator Strength Effect and Dephasing in NPLs
∆ tot , TI intensity ratio , λ slow , λ fast
log 10 ( )
4.5 K
35 K
100 K
λ fast λ slow
4.5 K
ES
GS
b d
A p = 41x13 nm 2
= 29x8 nm 2
∆ E
e
A p
4.5 K
35 K
133 K
100 K
∆ ES
tot
∆ GS
tot
∆ E
GS
∆ tot (eV)
ES
λ fast (ns -1 )
λ slow (ns -1 )
TI intensity ratio
ES/GS (a.u.)
a
c
50 nm
50 nm
Figure 4.2: Three lev el system in CdSe nanoplatelets and global analysis of photolu-
minescence data. (a)
Time-in tegrated photoluminescence (TI-PL) at 4, 35, 100 and 135 K of
29x8 nm
2
CdSe platelets rev eals excited (ES) and ground state (GS) emission separated b y
∆E
.
(b)
An LO-phonon b ottlenec k couples the ES and GS with a zero temp erature in ter scattering
rate
γ 0
. The three level system of
| E S i
,
| GS i
and crystal ground state (zero exciton state)
| 0 i
leads to a biexp onen tial deca y .
(c)
Sp ectrally binned PL deca y transien ts featuring a fast (
λ F
)
and a slo w (
λ S
) deca y comp onen t. Inset: Sp ectrally resolv ed PL transien t (29x8 nm
2
). (
d + e
)
The temp erature dep endence of the measured linewidths
∆ tot
, the ES/GS PL in tensit y ratio and
the comp onen ts of the PL deca y for 29x8 nm
2 (d)
and 41x13nm
2 (e)
CdSe platelets are plotted on
a decadic logarithmic scale and mo deled sim ultaneously b y our global analysis scheme (solid lines).
They can no w b e mo deled with the same set of parameters. W e globally fit the temp erature
dep endence of these exp erimen tal quan tities with link ed mo del parameters resulting in a m ulti-
dimensional analysis. The mo del (implemen ted in C-co de) solv es the transcenden tal equation
A p
=
A c
using eq. 4.5 in ev ery fit iteration. This w a y the temp erature
T c
is obtained at whic h
the transition from platelet limited to linewidth limited exciton coherence area o ccurs. In the
t w o temp erature regimes
A X C
in eq. 4.10 is replaced by
A p
for
T < T c
and b y
A c
(
T
) for
T > T c
.
A more detailed description is found in app endix 4.5. Since the zero temp erature homogeneous
linewidth and phonon scattering constan ts can differ b et w een ground and excited state this
pro cedure is done separately for the GS and ES in the analysis routine.
T c
is found to decrease
from 92 K to b elo w 4 K with increasing platelet size.
Figure 4.2 (d) and (e) sho w the results of this analysis as solid curves. Fit and data are in
go o d agreemen t, esp ecially when considering that all curv es are mo deled with the same set of
parameters (for eac h platelet size). The rate equation mo del sho wn in figure 3.5 (e) in c hapter 3
assumed constan t radiativ e rates. This first appro ximation led to a less go o d agreemen t of data
and mo del. The more sophisticated mo del used here tak es the GOST effect into accoun t – an
apparen tly b etter description of the system.
The other studied lateral platelet sizes range from 17x6 to 41x13 nm
2
core only and an 8x8 nm
2
CdSe core with CdS lateral wings (14x14 nm
2
) platelet. The fits for all sizes are shown in the
app endix 4.5 of this c hapter.
4.3 Dephasing 67
Figure 4.3: P opulation and p olarization deca y in CdSe nanoplatelets. (a)
Radiativ e
rates of excited (ES) and ground state (GS) excitons scale linearly with the exciton coherence area
at 4 K. (
b
and
c
) Con tributions to the total dephasing in terms of linewidth
∆
(left hand scale)
and corresp onding rate
γ ∗
(righ t scale). Width and rate related b y
∆
=
γ ∗ ~ /
2 : Pure dephasing is
b y far the largest con tribution to the total dephasing rate
γ ∗
(and to the homogeneous linewidth)
at all temp eratures. Th us
γ ∗
and
˜ γ
coincide in the plots. Due to scattering with AC and LO
phonons the pure dephasing remains the dominan t comp onen t at higher temp eratures. As
k B T
approac hes the ES-GS spacing
∆E E S − GS
, the ES
↔
GS relaxation and scattering (dashed line)
outpaces the radiativ e con tributions (dotted lines) for platelets resonant (29x8 nm
2
)
(b)
and off
resonan t (41x13 nm
2
)
(c)
to the LO-phonon b ottlenec k (
δ
=
∆E E S − GS − E LO
). Dephasing widths
rep orted b y others ha v e b een inserted for comparison:
.
T essier et al. [
95
],
Naeem et al.[
26
],
◦
Cassette et al. [31] and our single particle measuremen t on the same NPLs × .
4.3 Dephasing
F rom the results of this global mo del analysis w e can calculate the radiativ e rates
Γ r
at 4 K,
presen ted in figure 4.3 (a). As exp ected a linear dep endence of the radiativ e rates of excited and
ground state on the exciton coherence area
A X C
is observ ed. F or larger platelets the exciton
coherence area at this temp erature is already determined b y the linewidth limited area
A c
,
mark ed b y
•
. F or the smaller platelets the critical temp erature
T c
is higher. F or them at 4 K the
exciton coherence area is limited b y their ph ysical lateral extent, the platelet area
A p
, mark ed
b y 4 .
The radiativ e rates of ES and GS, and the dephasing rates and coherence areas can no w b e
calculated as a function of temp erature. The con tributions asso ciated to p opulation deca y are
the (non) radiativ e deca y
Γ r ,nr
E S,GS
and in ter-lev el scattering, i.e.
γ 0 n ∆
for GS and
γ 0
(
n ∆
+ 1) for
ES. By subtracting them from the total dephasing using eq. 4.1, the pure dephasing rates
˜ γ
of
the excited and the ground exciton state in the nanoplatelets are obtained.
Figure 4.3 (b) and (c) displa y the temp erature dep enden t con tributions of pure dephasing,
radiativ e p opulation deca y and ES
↔
GS in ter-lev el scattering to the total dephasing rate (righ t
scale). The corresp onding linewidth is displa y ed in the left scale. Clearly , the pure dephasing
˜ γ
68 4 The Giant Oscillator Strength Effect and Dephasing in NPLs
asso ciated with acoustic and optical phonon scattering is the dominan t comp onen t. The radiativ e,
and ES-GS in ter-lev el scattering related constituen ts ha ve only a (v ery) lo w con tribution to the
total homogenous linewidth. Therefore the Gian t Oscillator Strength effect is most sensitiv e on
the dominating pure dephasing.
Nonetheless, these dephasing results m ust b e compared to classically obtained results b y F our
W av e Mixing (FWM). Naeem et al.[
26
] found 5 K homogeneous linewidths of 4.5 ML 350 nm
2
CdSe platelets of ab out 0.8 and 4 me V. Identified from biexponential fits to the p olarization
deca y , ho w ev er, their analysis do es not consider the presence of the ES and GS. The t w o states
ma y ha v e b een addressed sim ultaneously by the exciting laser pulse generating the population
grating. Nevertheless, the results are inserted as
in figure 4.3 (b) (similar sized platelet) and
sho w go o d agreemen t with ours.
Cassette et al.[
31
] recen tly rep orted ro om temp erature FWM data for one monola y er thic k er
5.5 ML CdSe platelets of 33x12 nm
2
size. The resulting
T 2
time corresp onds to a dephasing rate
of 54 ps
− 1
, sho wn as
◦
in figure 4.3 (b). This is comparable to our results on 4.5 ML platelets at
ro om temp erature.
W e also find go o d agreemen t of our dephasing related linewidths with rep orted single platelet
emission linewidths. Single platelet emission is, ob viously , not sub ject to inhomogeneous
broadening. F or one monola yer thic k er 5.5 ML 35x8 nm
2
CdSe platelets T essier et al.[
95
]
measured 3 me V at 20 K and
∼
40 me V at ro om temperature for comparable excitation densities,
see
.
in figure 4.3 (b). A single platelet emission spectrum at 10 K of the same 29x8 nm
2
nanoplatelets as used here is found in app endix 4.5. The obtained linewidth is mark ed b y × in
figure 4.3 (b) and in very goo d agreemen t with the GS.
The ab o v e men tioned comparisons infer that the global analysis metho d deliv ers dephasing
results in go o d agreemen t with published FWM and single particle emission results – b oth at
lo w and high temp eratures. Therefore it has b een sho wn that dephasing rates can b e deduced
from luminescence exp erimen ts on quan tum w ells. This is essen tially new.
The dep endency of the radiativ e lifetime, homogeneous linewidth and dephasing o ccurs within
the GOST effect in 2D and allo ws to extract pure dephasing rates from PL exp erimen ts. Normally
it is imp ossible to deriv e the homogeneous linewidth from PL exp erimen ts b ecause it cannot
b e separated from the inhomogeneous con tribution in an ensem ble. Here we made use of the
correlation of
∆
(
T
) and the radiativ e rate
Γ r
. This pro vides a new metho d for the analysis of
dephasing and p opulation deca y in 2D semiconductors.
4.4 Exc iton-Phonon Coupling Constants 69
E LO
Detuning
+
-
Optical phonon transition
Acoustic phonons Longitudinal optical phonons
a b
c d
d
g 0
(meV) d
)
g 0
(ns -1
0 200 400 600
10
100
Platelet area (nm 2 )
∆ AC (µeV/K)
GS
ES
-20 -10 0 10 20
0
10
20
30
40
(17x6)
(29x8)
GS
ES
∆ LO (meV)
(41x13)
d (meV)
-20 -10 0 10 20
1
10
100
E LO =25.4 meV
Figure 4.4: Phonon coupling constants and an LO-phonon bottleneck.
Lateral size
dep endence of the coupling constan ts to acoustic (
a
) and LO-phonons (
b
). (
c
) Detuning dep enden t
LO phonon b ottlenec k. (
d
) The obtained rate constan t
γ 0
with fit to eq. 4.12 sho ws a clear resonance
to the LO-phonon energy .
4.4 Exciton-Phonon Coupling Constants
The presen ted glob al PL data analysis allo ws to extract further i nformation b ey ond the exciton
dephasing rates and rad iativ e lifetimes. Figure 4.4 sho ws the exciton- phonon coupling constan ts
and the zero temp erature ES-GS sc attering rate. Since the a ddition of a shell of another material
influences the phonon coup ling in nano crystals [
96
,
97
], the t yp e I 8x8 nm
2
CdSe core / CdS
lateral shell nanoplate let is excluded from the follo wing discussion .
F or th e core platelets the coupling to acoustic p honons (
∆ AC
) v ersus platelet are a is sho wn in
figure 4.4 (a). A first in terestin g observ ation is the increase of th e coupling to acoustic phonons
∆ AC
with decreasing latera l size. This trend is in line with calcula tions b y Kelley[
98
] and
T akagahara et al .[
43
] for spherical CdSe quantum dots. T here, an incre ase of deformation
p oten tial coupling wi th confinemen t lead to an observed increasing coupling of excitons to
acoustic phonons. The results for large platelets are comparable to large dots, where T akagahara
et al. found the coupling to b e ab out 7
µ
e V/K. F or small quan tum dots of diameter comparable
to the thic kness of ou r 4.5 ML platelets (1.4 nm), they obtai ned
∼
100
µ
e V /K. This is in go od
agreemen t with t he v alues of smaller platelets in fi gure 4.4 (a).
The LO phonon coupling constan t ( ∆ LO ) is plotted versus the detuning b et w een the ES-GS
energy spacing and the LO -phonon energy
δ
=
∆E ES − GS − E LO
in 4.4 (b). A slight minimum
in the coupling to LO-pho nons is observ ed when the ES-GS energy spaci ng approac hes the LO
phonon energy of 25.4 me V [
27
] (zincblende CdSe). Y et the reason for this minim um is un clear.
70 4 The Giant Oscillator Strength Effect and Dephasing in NPLs
The minim um v alue corresp onds to a
∼
300 nm
2
sized platelet with 15 me V LO-phonon coupling
(GS). F or similarly sized 289 nm
2
CdSe nanoplatelets A c h tstein et al.[
17
] found comparable
12 me V. In general the obtained coupling constan ts are in the same range as rep orted b y Chia et
al. on thic k er CdSe epila y ers ( ≈ 20 me V) [99].
An LO-phonon b ottlenec k exists b et w een ES and GS. The detuning
δ
=
∆E − E LO
dep enden t
transition rate
γ 0
is exp ected to ha v e a maxim um for
δ
= 0 . Using F ermi’s Golden R ule [
100
,
101
]
the detuning dep endence of the transition rate of this phonon transition can b e expressed as
γ 0 = C ∗ ∆ t
( ∆E − E LO ) 2 + ( ∆ t / 2) 2 (4.12)
It has a Loren tzian shap e and a distinct resonance for
∆E
=
E LO
at the LO phonon energy . C
is a constan t and ∆ t the phonon mediated ES-GS transition linewidth.
Figure 4.4 (d) sho ws the results based on our global fits on platelets of differen t lateral size, i.e.
differen t ES-GS energy spacing
∆E
and th us detuning to the LO phonon energy . As exp ected,
there is a strong detuning dep enden t LO-phonon b ottlenec k resonan t to the LO-phonon energy .
It can b e w ell fitted with eq. 4.12. The resulting transition linewidth is 6.9 me V. This is a clear
pro of of the coupling of ES and GS in CdSe nanoplatelets via an LO-phonon b ottlenec k that is
con trollable b y the lateral size.
Conclusion
The p opulation and p olarization dynamics in CdSe nanoplatelets are go v erned b y the 2D Gian t
Oscillator Strength Effect together with a size-dep enden t longitudinal optical phonon b ottlenec k
b et w een an excited and ground state exciton.
The radiativ e rates of the t w o emitting states are determined b y the 2D exciton coherence area.
In these quan tum w ells of finite size this area is either limited b y the lateral platelet dimensions
or b y a smaller coherence area related to the homogeneous exciton linewidth. The homogeneous
linewidth and coherence area are determined b y the dephasing in the system. This un usual
coupling of radiativ e lifetime and linewidth in 2D th us leads to a dephasing-limited radiativ e
lifetime. A phenomenon not encoun tered up to no w.
F or the first time it has b een sho wn that it is p ossible to deriv e pure dephasing rates from
time-resolv ed and time-in tegrated photoluminescence measuremen ts on 2D quantum w ells of
finite size. The used global analysis scheme includes measured deca y constan ts, the excited to
ground state emission in tensit y ratio and the PL linewidth. The obtained results are in go o d
agreemen t to classical FWM measuremen ts. Hence w e can measure dephasing in a linear PL
exp erimen t, an en tirely new regime. The approac h in tro duces a new w a y to connect the radiativ e
4.4 Exciton-Phonon Coupling Constants 71
deca y rates to the coherence v olume and the dephasing rate reflected in the homogeneous
linewidth of the nanoplatelets.
In addition, the mo del allo ws to extract the coupling constan ts of the emitting excitons to
acoustic and longitudinal optical (LO) phonons – in go o d agreemen t with quan tum dots and
quan tum w ells of comparable size. F urther, the obtained in ter-scattering rate b et ween the
excited and ground exciton state is found to follo w F ermi’s Golden R ule – a further pro of for
the LO phonon b ottlenec k in CdSe nanoplatelets. It sho ws the exp ected maxim um when the
energetic separation of the t w o states approac hes the LO phonon energy of CdSe. The width of
this LO-phonon transition is found to b e 6.9 me V.
T unable radiative lifetimes of the order of 10 ps are feasible, making CdSe nanoplatelets one
of the fastest nano emitters with strong application p oten tial in nanoph ysics and photonics.
The pro vided insigh ts to fundamen tal asp ects of the opto electronic prop erties of colloidal
quan tum w ells might serv e as rational guidelines for the future dev elopmen t of 2D colloids in
ligh t emitting tec hnology . They can b e transferred to other 2D materials.
72 4 The Giant Oscillator Strength Effect and Dephasing in NPLs
4.5 APPENDIX: Mo deling a Three-Level System with the GOST
Effect
F or clarity w e list all parameters in v olv ed in our global analysis mo del. Some parameters are
in trinsically equal for ES and GS prop erties suc h as:
E LO : LO phonon energy of 24.5 me V in zincblende CdSe [102]
∆E E S − GS : Energy spacing b et w een ES and GS obtained from time in tegrated PL
γ 0 : Zero temp erature ES-GS in ter-lev el scattering rate
A p : Nanoplatelet area obtained from analyzing TEM images.
M : Exciton mass using m e = 0 . 12 m 0 and m h = 0 . 48 m 0 . [27]
The follo wing quan tities are treated separately for ES and GS:
ζ E S,GS : T emp erature indep enden t prop ortionalit y factor of the radiativ e rate (see eq. 4.13)
∆ E S,GS
0 : 0 K homogeneous linewidth
∆ E S,GS
AC : A coustic phonon coupling constan t
∆ E S,GS
LO : LO phonon coupling constan t
Γ nr , 00
E S,GS : Non-radiativ e rate at 0 K
Γ nr , 0
E S,GS : T rapping frequency factor
∆E t
E S,GS : T rap activ ation energy (assumed iden tical for ES and GS)
Γ r ( T ) = τ − 1 ( T ) = ˜ ne 2 ω 2 f 0 µE 2 D
B | f lf | 2
π 2 ~ 2 ε 0 m 0 c 3
| {z }
ζ
· r ( T ) A X C (4.13)
The exp erimen tal data w ere mo deled in a global fitting approac h for eac h platelet size sharing
this set of parameters. The fast and slo w deca y comp onen ts
λ F ,S
are fitted with eq. 3.5, the
ES to GS time-in tegrated PL in tensity ratio
R T I − P L
E S,GS
(
T
) with eq. 3.8 and the total linewidths
(obtained from PL) with eq. 4.11 and 4.3. The formula are implemen ted in C-co de. As men tioned
ab o v e, it is imp ortan t to distinguish b et w een ph ysical size- and homogeneous linewidth- limited
exciton coherence area. The platelet area
A p
is constan t in temp erature whereas the linewidth
limited coherence area
A c ∝ ∆
(
T
)
− 1
decreases with increasing temp erature predominan tly
due to phonon scattering, see Fig. 4.1 (d). In these t w o temp erature regimes the functional
dep endencies of the linewidths on the radiativ e rates (eq. 4.10) differ strongly . This strongly
influences the mo deled quan tities, e.g. deca y rates and ES/GS in tensit y ratio, via the underlying
rate equation system. Therefore the fitting pro cedure b egins with starting v alues for the mo del
4.5 APPENDIX: Mo deling a Three-Level System with the GOST Effect 73
parameters (listed ab o v e) and the transcenden tal equation
A p − A c ( T ) = A p − 2 π ~ 2
M ∆ 0 + ∆ AC T + ∆ LO
e E LO /k B T − 1 = 0 (4.14)
is solv ed (separately for ES and GS) for
T
to obtain the temp eratures
T c
at whic h the ES and
GS exciton coherence areas are limited b y the platelet’s ph ysical size
A p
(
T < T c
) or (at higher
temp eratures,
T > T c
) b y the smaller exciton linewidth limited coherence area
A c
(
T
) . At a giv en
temp erature one of the four differen t cases can o ccur: (1) b oth ES and GS are limited b y the
platelet area, (2) b oth are limited b y their linewidth-dep enden t coherence area
A E S/GS
c
or (3-4)
one state is limited b y
A p
while the other b y
A c
. An if, else if, else if, else structure c hec ks whic h
case o ccurs to whic h temp erature (using the giv en starting v alues or later the v alues from the
last iteration). Accordingly
A X C
in equation 4.10 is replaced b y
A p
or
A E S/GS
c
(
T
) for ES and
GS separately , the mo deled quan tities (
λ F ,S
(
T
) ,
R T I − P L
E S,GS
(
T
) and
∆ E S/GS
tot
(
T
) ) are calculated for
eac h temp erature, compared to the exp erimen tal data, optimized to all data sim ultaneously and
the fit go es in to the next iteration with new mo del parameters un til it con v erges. The results of
this pro cedure for all in v estigated samples are plotted in Fig. 4.6.
2 . 4 6 2 . 4 8 2 . 5 0 2 . 5 2 2 . 5 4
0
1 0 0
2 0 0
3 0 0
I n t e n s i t y ( C o u n t s )
E n e r g y ( e V )
4 . 5 M o n o l a y e r 2 9 x 9 n m
2
C d S e N a n o p l a t e l e t s
1 0 0 m W / c m ² @ 4 4 2 n m C W
@ 1 0 K
2 . 6 m e V
Figure 4.5:
Single particle PL sp ectrum of 29x8 nm
2
platelets at 10 K with a Loren tzian p eak fit
yielding 2.6 me V FWHM.
74 4 The Giant Oscillator Strength Effect and Dephasing in NPLs
1 0 1 0 0
- 2
- 1
0
1
2
3
1 0 1 0 0
- 2
- 1
0
1
2
3
1 0 1 0 0
- 2
- 1
0
1
2
3
1 0 1 0 0
- 2
- 1
0
1
2
3
1 0 1 0 0
- 2
- 1
0
1
2
3
1 0 1 0 0
- 2
- 1
0
1
2
3
l o g
1 0
( F W H M
E S , G S
, T I - P L I n t . r a t i o , T R - P L A m p . r a t i o ,
s l o w
,
f a s t
)
T e m p e r a t u r e ( K )
4 1 x 1 3 n m
2
L o g
1 0
o f
D a t a
f a s t
( n s
- 1
)
s l o w
( n s
- 1
)
T I - P L I n t . r a t i o ( a . u . )
F W H M E S ( e V )
F W H M G S ( e V )
F i t
3 0 x 1 5 n m
2
T e m p e r a t u r e ( K )
2 9 x 8 n m
2
T e m p e r a t u r e ( K )
2 1 x 7 n m
2
T e m p e r a t u r e ( K )
1 7 x 6 n m
2
T e m p e r a t u r e ( K )
C d S e - C d S w i n g 8 x 8 n m
2
T e m p e r a t u r e ( K )
Figure 4.6:
T emperature dep endence of deca y rates
λ f ast
(blue) and
λ slow
(green) in ns
− 1
, ES
to GS ratio of time-in tegrated PL emission
R T I − P L
E S/GS
(
T
) (grey) and the total full width at half
maxim um (FWHM)
∆ tot
in e V of ES (blac k) and GS (red) of all in v estigated samples (with
decreasing size from top left to b ottom righ t). Symbols denote exp erimen tal data, solid lines are
the results of the global fitting pro cedure describ ed abov e. As the abov e men tioned parameters
span man y orders in magnitude the decadic logarithm of the quan tities is plotted.
5 | Field Con trol
The w ork presen ted in the following c an also b e found in ref. [
24
] Scott, R. and A c htstein, A. W. et
al.
(2016)
, Nano L etters 16(10), 6576–6583. All figures adapted with p ermission,
©
2016 American
Chemical So ciet y .
2.1
2.3
Energy (eV)
0
kV/cm
1 75
kV/cm
2.1 2.3
0
Time (ps)
125
E-Field
+
-
NPLs in polymere
E-Field (kV /
cm)
Figure 5.1: Time-Resolv ed Stark
Sp ectroscop y in CdSe Nanoplate-
lets
allo ws to deduce the exciton bind-
ing energy , polarizability and the field
dep enden t radiativ e rates. These in
turn giv e access t o the exciton tran-
sition dip ole momen t
| µ |
and its p olar-
izabilit y .
Excitons with high bind ing energies are desirable for man y photo nic applications (e.g. lasing
[
75
]) and are of current in terest. They o ccur in 2D semiconductor nan omaterials with strong
confinemen t and hi gh dielectric mismatc h to the surround ing [
25
,
103
]. These robust excitons are
exp ected to allo w high sp ectral m odula tion while prev en ting field ioni zation in field con trolled
nano emitters and nanostructu re based mo dulators. Th is has great application p oten tial in
miniaturized and integrated photonics suc h as mo dulate d emitters, switc hable sin gle photon
sources or ultra high ba ndwidth, field-con trolled mo dulators [8 0, 104–111].
While field-controlled electroabsorption based mo dulators hav e b een studied in tensiv ely b oth
in exp erimen t and theor y [
108
,
109
,
112
–
121
], field-dep enden t pho toluminescence nano emitters
[
111
,
122
–
125
] ha v e b een mostly dis cussed on a qualitativ e or semi quantitativ e lev el. Ha ving a
strong electro optic resp onse [
80
] CdSe nanoplatelets (NPLs) offer a go o d mo del system for field
dep enden t photolumi nescence (PL) studies.
An external field c ha nges the static exciton dip ole momen t and the excit on’s transition dip ole
momen t. The Quan tum Confined Sta rk Effect and the F ranz-Kel dysh Effect lead to a strong
alteration of the excit on w a v efunction resultin g in a c hange of the in tensit y , lifetime, linewidth
and energy of the PL emission.[109, 112–116]
Based on the rate equation mo del for re com bination dynamics in CdSe nanoplat elets at
zero field in c hapt er 3, a formalism is dev elop ed to conne ct these field-induced c hanges to the
microscopic pro cesses. This allo ws to q uan tify the exciton transition dip ole mome n t
µ
, its
p olarizabilit y
X
as w ell as the exciton ’s pol arizabilit y
α
from time-in tegr ated and -resolv ed field
dep enden t PL studies.
76 5 Field Control
F urther, for the first time, the high exciton binding energies of
≈
170 me V of CdSe nanoplatelets
could b e deduced exp erimen tally – in line with theoretical predictions.
Up to no w it is still an op en question to what exten t the exciton binding energy and the
field-induced c hanges of the radiativ e rates affect the field-dep enden t emission in tensity and
broadening of nanoplatelets. An important question for spectrally broad mo dulators that
Time-Resolv ed Stark Sp ectroscop y can answ er.
5.1 Field-Dep endent Sample Structure and Exp erimental Details
CdSe core only NPls with the first exciton absorption bands around 550 nm (5.5 Monola yers
(ML)) and 42
±
7 nm x 8
.
7
±
2
.
1 nm lateral size w ere syn thesized as in Ref.
80
. A TEM image of
the sample is sho wn in figure 5.2 (b). The NPLs w ere em b edded in thin p olymeric films (made of
PMA O (p oly maleic anh ydride-alt-o ctadecene) and dep osited on the surface of indium-tin-o xide
(ITO)-coated glass. Their v olume fraction in the p olymer w as held well below 5 % to a v oid
F örster Resonant Energy T ransfer (FRET) effects[
81
]. A sandwich-lik e ITO/P olymer:CdSe-
NPLs/ep o xy/ITO structure is obtained b y attac hing a second ITO electro de to the p olymeric
film with ep o xy glue, see figure 5.2 (c). A DC v oltage p o w er supply is used to apply the external
electric field, calculated via the exact distance b et w een the ITO electro des that is measured to
b e 48 µ m.
F or time integrated field dependent PL measuremen ts a HeCd laser (CW, con tin uous w a ve
at 441 mn) is used as excitation source. The photoluminescence signal is collected b y an
N.A.
=0.4 ob jectiv e and detected with a liquid Nitrogen co oled CCD (Rop er Sp ec10) attac hed to
a sp ectrometer (Horiba IHR550). F or time resolv ed measuremen ts the sample is excited b y the
second harmonic of a titanium sapphire laser at 440 nm (Coherent Mira 900F, FWHM 150 fs,
75.4 MHz) and detected b y a streak camera (Hamamatsu C5680). The excitation density of
0.7 W/cm
2
(CW equiv alen t) in b oth excitation configurations results in an a v erage p opulation of
< 10 − 3 excitons p er platelet, excluding m ulti-excitonic effects. See also Ref. 27 and section 3.5.
5.2 Exciton Binding Energy of Nanoplatelets Determined b y
Exp eriment
Changing the applied field in the nanoplatelet field-effect structure alters the emission strongly ,
figure 5.2 (e). Decreasing emission in tensit y , line broadening and increasing spectral shifts are
observ ed with increasing applied electric fields. T o analyze the c hanges in more detail we use
5.2 Exc iton Binding Energy of Nanoplatelets Determ ined b y Exp eriment 77
IT O-coated glass-substr ates
Spacer
Nanoplatelets in polymeric film
E-Field
U
20 nm
d
E
d d
Figure 5.2: Field-effect structure and field dep enden t time in tegrated photolumines-
cence of CdSe nanoplatel ets.
(a) Lev el sc heme of the CdSe platelets’ electronic structure of
crystal ground state
| 0 i
, ground
| GS i
and excited state
| ES i
exciton, radiativ e transition rates
Γ r
E S/GS
, ES
↔
GS zero temp erature scattering or relaxation rate
γ 0
and nonradiativ e rates
Γ nr
E S/GS
.
(b) TEM image of the platelets em b edded in p olymer in the sandwic h-lik e field-effect stucture
sho wn in (c). (d) Mo del sk etch of the dual ES and GS emission of CdSe nanoplatelets centered
at
E c, 0 /F
E S/GS
and separated b y
δE 0
for zero and
δE F
for finite field
F
. V oigt emission profiles
(total FWHM
w F
E S/GS
) with natural Loren tzian linewidth
w L,F
E S/GS
and Gaussian inhomogeneous
linewidth
w G
E S/GS
(field-indep enden t inhomogeneous broadening) are used to fit the exp erimen tal
field-dep enden t emission sp ectra in (e). The area under curve
A F
E S/GS
of eac h emission is prop or-
tional to the emission in tensity
I F
E S/GS
. (e) Exp erimen tal data: Zero-field and field-dep enden t
time in tegrated PL along with V oigt fits as w ell as the field-dep enden t differen tial emission sp ectra.
Thin grey dotted line: W eak, red shifted defect emission.
the differen tial emission defined as
∆I
I E ,F
= I E ,F − I E, 0
I E, 0 (5.1)
sho wn in the low er part of figure 5.2 (e) where
I E ,F
is the emission at a sp ectral p osition (energy
E
) at an external field
F
. A strong electro-mo dulation behavior is observ ed. The lo cal minim um
and maxim um of th e differen tial emission sp ectrum corresp ond to t he ES and GS exciton
transition energies [
109
] of the CdSe nanoplatel ets. Quan tities related to the ex cited state or
ground state are indicat ed b y a subscript ES or GS, resp ectiv ely . See illustration in figure 5.2 (d)
for the used nomenclature. Both field
F
dep enden t emissions are mo deled b y Loren tzian profiles
f L
(
E
) of width
w L,F
E S/GS
con v oluted with a normalized Gaussian inhomogeneous broadening
f G
(
E
) of width
w G
E S/GS
. The inhom ogeneous broadening is assumed to b e field indep endent
78 5 Field Control
and has th us no sup erscript F. The resulting emission profiles are V oigt profiles
V A,E c ,w L ,w G
E S/GS
of
area under curv e A and center energy E c . W e use
V A,E c ,w L ,w G
E S/GS ( E ) = const + f L ( E ) ⊗ f G ( E ) (5.2)
f L ( E ) = 2 A
π
w L
4 ( E − E c ) 2 − ( w L ) 2
f G ( E ) = r 4 l n 2
π
e
4 ln 2 E 2
( w G ) 2
w G
to fit the field dep enden t emission sp ectra of ground and excited state. T o a v oid to o man y
indices, the field dep endence in the form ula ab o v e has b een omitted. The homogeneous linewidth
w L,F
E S/GS
, the emission cen ter energy
E c,F
E S/GS
and the sp ectrally in tegrated emission
A F
E S/GS
are
field dep enden t. With this definition the total emission of ES and GS and its field dep enden t
differen tial c hange (eq. 5.1) can b e expressed as:
I ( E ) = V E S ( E ) + V GS ( E ) (5.3)
so that
∆I
I E ,F
=
V E ,F
GS + V E ,F
E S − h V E , 0
E S + V E , 0
GS i
V E , 0
E S + V E , 0
GS
(5.4)
Up on application of an external field the parameters of the V oigt profiles are altered. The
follo wing con v entions are used:
- P eak areas A E S/GS → A F
E S/GS
- P eak cen ters E c
E S/GS → E c, 0
E S/GS + ∆E c,F
E S/GS
- Loren tzian widths w L
E S/GS → w L, 0
E S/GS + ∆w F
E S/GS
A w eak, broad defect emission at lo wer energies is tak en in to accoun t b y a Gaussian p eak,
seen as dotted lines in figure 5.2 (e). Figure 5.2 (e) also sho ws the corresp onding global fits to the
field-dep enden t emission, (upp er panels) and differen tial emission (lo w er panels), using eqs. 5.3
and 5.4, resp ectiv ely . F or each field-s trength the field-dep enden t and zero field emission and their
differen tial sp ectrum are fitted sim ultaneously sharing the set of parameters listed ab o v e. The
ES-GS energy spacing
δ E F
can b e sensitiv ely determined due to the inclusion of the differen tial
emission sp ectra. The fits for zero-field yield:
E c, 0
E S
= 2
.
239 e V,
E c, 0
GS
= 2
.
215 e V and
w L, 0
E S
=
46 me V,
w L, 0
GS
= 51 me V. The zero field ES-GS energy spacing of
E c, 0
E S − E c, 0
GS
=
δ E 0
= 23
.
9 me V
is close to the LO phonon energy in zincblende CdSe of 25.4 me V[27].
The fit-results for the field-induced c hanges of the transition energies
∆E F
and homogeneous
(Loren tzian) broadening
∆w L,F
E S/GS
are summarized in figure 5.3. The observed redshift of the
5.2 Exciton Binding Energy of Nanoplatelets Determined by Experiment 79
excitonic emissions, accompanied b y a significan t broadening, is a c haracteristic of the Quan tum
Confined Stark and F ranz-Keldysh effect. It has previously b een observ ed in v arious quantum
w ells and quan tum dots [
109
,
112
,
114
,
116
,
126
–
128
]. The quantum confined Stark shifts
mostly originate from a distortion of the square w ell confinemen t p oten tial reducing the energetic
difference b et w een hole states in the v alence band and electron states in the conduction band.[
112
]
The broadening is mostly due to the F ranz-Keldysh effect for platelets with a preferential parallel
orien tation to the electric field[
112
]. They are p olarizable due to the formation of spatially
distorted excitons in the presence of an external field. The platelets are randomly orien ted in the
field effect sample, th us the observ ed sp ectral shifts and broadening corresp ond to the angularly
a v eraged in ternal electrical field-strengths in the platelets.
DC Lo cal Field F acto r
In order to quan tify the observ ed sp ectral shifts, kno wledge of the in ternal field inside the
platelets is required. The applied external field is defined as
F
=
U /d
.
U
is the applied v oltage
and
d
the distance b et w een the ITO electro des of the field-effect structure.
F
is lo w ered in
the p olymer b y:
F poly .
=
F /ε r ,poly
. This field is further reduced inside the platelets b y lo cal
field effects due their shap e and dielectric con trast to the p olymer. By replacing the optical
p ermittivit y
ε M
and
ε S
in equation 1.24, resp ectiv ely , with the static p ermittivit y of the host
p olymer and CdSe, the static lo cal field factors
f D C
x,y ,z
of the platelets can b e calculated. The
static dielectric constan t of the platelet en vironmen t is taken as
ε r ,poly
= 2
.
5 [
129
], while the
CdSe platelets ha v e
ε r
= 10
.
2 [
130
]. With the basis v ector in spherical co ordinates and with
~
E poly . = E poly . · ˆ e r ( φ,θ ) , the field strength inside the particle is:
E int ( φ, θ ) = E poly . q | f D C
x | 2 sin 2 θ cos 2 φ + f D C
y 2 sin 2 θ sin 2 φ + | f DC
z | 2 cos 2 θ (5.5)
Here
| f D C |
(
φ,θ
) =
E int. /E poly .
is the lo cal field factor for the electric field strength in spherical
co ordinates. A ccoun ting for the random orien tation of the platelets in the p olymer w e calculate
its exp ectation v alue f l f o v er the whole solid angle, using d Ω = sin θ d φ d θ :
f lf = | f D C | ( φ,θ ) Ω = R Ω | f D C | ( φ,θ ) d Ω
R Ω d Ω (5.6)
In order to relate the applied external field
F
directly to the (a v erage) in ternal field inside the
platelets F int the effectiv e lo cal field factor ˜
f lf = F int /F = f lf /ε r,poly is in tro duced.
80 5 Field Control
0 1 0 0 2 0 0
- 4
- 2
0
0 1 0 0 2 0 0
0 . 8
0 . 9
1 . 0
0 1 0 0 2 0 0
0
4
8
E x t e r n a l F i e l d ( k V / c m )
c
S h i f t E ( m e V )
E S
G S
N o r m . t o t a l P L i n t . ( a . u . )
E S + G S
b
E x t e r n a l F i e l d ( k V / c m )
w ( m e V )
E x t e r n a l F i e l d ( k V / c m )
a
E S
G S
Figure 5.3: External electric field induced c hanges of the photoluminescence of CdSe
nanoplatelets.
(a) ES and GS transition energy shifts
∆E F
E S/GS
and fits according to eq. 5.7. (b)
Field-dep enden t total time in tegrated PL in tensity
A F
GS
+
A F
E S
, normalized to the zero field emission
in tensit y (
A 0
GS
+
A 0
E S
). (c) Field-induced homogeneous (Loren tzian) broadening
∆w L,F
E S/GS
. The
dashed lines in (c) are guides to the ey e.
Exciton P ola rizabilit y and Binding Energy
The field-dep enden t energetic shift ∆E F of the PL can b e mo deled as[110, 122]
∆E F = ( p + p ind ) F int = ( p + α F int ) F int = p + α ˜
f lf F ˜
f lf F = p ˜
f lf F + α ˜
f lf 2 F 2
(5.7)
with a linear and quadratic dep endence on the external field
F
. The p ermanen t exciton dip ole
momen t p and its p olarizabilit y α will b e discussed now in detail.
The exciton p olarizabilit y
α
can b e in terpreted as the difference of the p olarizabilities of the
final and initial state of the exciton transition. In analogy , the p ermanen t dip ole momen t
p
of
an exciton can b e in terpreted as the difference in dip ole momen ts of the final and initial state of
the exciton [
118
]. No (measurable) p ermanen t dip ole momen t
p
in the CdSe nanoplatelets is
observ ed, as the induced a v erage shift in a random distribution of dip oles is exp ected to b e zero.
F or further discussion it is conv enien t to estimate the (external) field for exciton ionization
in CdSe nanoplatelets. Exciton ionization is assumed to o ccur once the field induced drop of
p oten tial energy across its spatial exten t (exciton Bohr radius
a B
) equals its binding energy:
F ion,int
=
E B /
2
ea B
. With the effectiv e lo cal field factor and
E B ∼
200 me V [
22
], this results
in appro x. 2.6 MV/cm external field strength for ionization. A factor of
∼
15 higher than the
highest applied field strength, so that w e are not in a regime where exciton ionization con tributes
to the observ ed energy shift. This appro ximation is in line with the observ ed small induced shifts
of the exciton transition energies with resp ect to the exciton binding energy (
∆E F << E B
)
obtained from analysis of the data in figure 5.2 (e) and plotted in figure 5.3 (a).
The measured field-dep endence of the transition energies sho wn in figure 5.3 (a) is well fitted
with the mo del equation 5.4 (solid lines). F or the GS a dip ole p olarizabilit y
α
of 8
.
6
±
1
.
6
·
10
− 8
e V
cm
2
/k V
2
is found. It equals 1
.
4
±
0
.
3
·
10
− 36
Cm
2
/J in SI units. F or the ES
α
is found to b e
5.2 Exciton Binding Energy of Nanoplatelets Determined by Experiment 81
1
.
8
±
0
.
2
·
10
− 6
e V cm
2
/k V
2
( 3
.
0
±
0
.
3
·
10
− 35
Cm
2
/J). These v alues can no w b e compared with
previous theoretical and exp erimen tal w ork.
A rough estimate of the p olarizabilit y can b e obtained from p erturbation theory:
α
=
2
e 2 r 2 / ∆E
[
131
,
132
]. Here
r 2
is the exp ectation v alue of the p osition op erator squared,
e
the elemen tary c harge and
∆E
an a v erage energy spacing of all allo w ed transitions. The spatial
exten t of the exciton limits the dip ole length. Th us the volume of the exciton is estimated
as a cylinder with the heigh t
L w
= 1
.
67 nm (w ell width in p erp endicular direction) and a
radius equal to the Bohr radius
a B
= 1
.
41 nm. The exp ectation v alue of the p osition operator
squared is then
r 2
=
π L w a 2
B 2 / 3 /
4 .
∆E
is appro ximated b y a typical exciton binding
energy in CdSe nanoplatelets of
E B ∼
200 me V[
22
]. The same v alue w as also used to calculate
a B = ~ / (2 µ r E B ) 1 / 2 = 1 . 41 nm with the reduced exciton mass µ r .[17] This p erturbation theory
appro ximation yields
α
= 1
.
9
·
10
− 36
Cm
2
/J , in go o d agreemen t with the exp erimen tal ground
state v alue giv en ab o ve.
Kunneman et al.[
78
] measured an imaginary part of the mobilit y of
µ I
= 2
.
0 cm
2
/V s in CdSe
nanoplatelets. As their exp erimen t cannot distinguish b et w een the con tributions of ES and GS,
the rep orted v alue corresp onds to the sum of ES and GS p olarizabilities. The imaginary mobilit y
can b e related to the p olarizabilit y b y
α
= 2
π f µ I /e
where f is the used THz prob e frequency in
the exp erimen t in ref. [
78
]. A v alue of
∼
1
.
0
·
10
− 35
Cm
2
/J is obtained, in go o d agreemen t with
the sum of our ES and GS results, i.e. with the ES due to its m uc h larger v alue.
Compared to the ES, the GS p olarizabilit y is found to b e substan tially smaller (b y more
than one order of magnitude). This is in line with calculations of the in-plane p olarizabilit y of
infinite quan tum w ells by P edersen [
133
]. It m ust b e noted that reducing the dimensionalit y of
an exciton from bulk (D=3) to D=2 in a quan tum well increases the energy lev el spacings and
limits the dip ole length due to the spatial confinemen t. In general this strongly reduces the
exciton p olarizabilit y . This confinement-related decrease of the polarizability is stronger for the
GS [133], resulting in a lo w er p olarizabilit y of 2D GS excitons with resp ect to ES excitons.
The presen ted p olarizabilities extracted from our mo del th us are in go o d agreemen t with
p erturbation theory for the GS, with mobilit y measuremen ts [
78
] for the sum of ES and GS. Also
the trend of a smaller GS p olarizabilit y compared to ES is in-line with theoretical w ork[133].
The measured ground state exciton p olarizabilit y
α
can b e related to the p olarizabilit y v olume
V pol
via
V pol
=
α/
4
π ε 0
. This v olume allo ws an estimate of the exciton binding energy . W e
assume that the p olarizabilit y v olume of a quasi 2D exciton is determined b y its ph ysical size
(exciton v olume). Appro ximating the exciton again as a cylinder of height
L w
and radius
a B
(with E B = ~ 2 / 2 µ r a 2
B ), w e obtain
E B ≈ ~ 2 π L w / 2 µ r V pol = 170 ± 31 me V. (5.8)
82 5 Field Control
This exp erimen tally obtained exciton binding energy is in go o d agreemen t with the men tioned
theoretical results of Benc hamekh[
22
]
E B
= 193 me V, and the predictions in Ref.
17
for the used
platelet thic kness. This pro vides the first exp erimen tal pro of for the predicted high exciton
binding energies in colloidal CdSe nanoplatelets.
The prop osed appro ximation of the exciton binding energy via the p olarizabilit y also w orks
on rep orted field dep enden t data of CdSe quan tum dots. Menéndez-Proupin and T rallero-Giner
[
134
] rep ort data on the field dep enden t Stark shift of the lo w est exciton transition of CdSe QDs
of 4 nm radius. W e extract
α
from the rep orted shift. Using our metho d and assuming a spherical
p olarizabilit y v olume for the dots, the exciton binding energy is approximated to be 75 me V.
This v alue obtained with our metho d is in go o d agreemen t with calculations on the same sized
dots. F or the same quan tum dot size Laheld and Einev oll [
135
] calculate exciton binding energies
b et w een 65 and 75 me V, dep ending on the used confinemen t p oten tial. This indicates that the
prop osed metho d for appro ximating exciton binding energies from the measured p olarizabilit y
α
is a v aluable to ol, also for other systems.
Summarizing, the GS p olarizabilit y presen ted here is in go o d agreemen t with p erturbation
theory appro ximation, as is the sum of ES and GS p olarizabilities with THz exp erimen ts.
The p olarizabilit y v olume allo ws an approximation for the exciton binding energy in CdSe
nanoplatelets:
E B ≈
170
±
31 me V. This first exp erimen tal v alue is found to b e in v ery go od
agreemen t with rep orted predicted theoretical exciton binding energies for CdSe nanoplatelets.
5.3 T ransition Dip ole Moment and its P ola rizabilit y
Figure 5.3 (b) sho ws field-induced mo dulation of the total PL in tensity of up to 22 % . In this
case the magnitude of the PL reduction is only limited to the exp erimen tally accessible range of
fields. There is p oten tial to reac h an ev en higher mo dulation depth as w ell as sp ectral shifts.
A t this p oin t it is w orth men tioning that the total amount of absorption in the system is
unaffected b y the external field. Oscillator strength is only redistributed b y the field. The
so-called f-sum rule applies[
32
,
136
], see figure 5.7 in the app endix 5.4. A t the sp ectral p osition
of the exciting laser in the con tin uum, though, the field-induced c hanges of absorption are b elo w
0
.
1 % . Th us the c hange of the emission in tensit y under field is clearly not related to field-induced
c hanges of absorption.
The observ ed high mo dulation of the total emission suggests a strong alteration of the transition
dip ole momen ts of ES and GS under an applied external field whic h will b e in v estigated in the
follo wing.
5.3 T ransition Dip ole Moment and its P ola rizabilit y 83
Figure 5.4: Time-resolv ed field-dep enden t photoluminescence of CdSe nanoplatelets.
(a) Time resolv ed photoluminescence and temp orally binned PL as w ell as sp ectrally binned
transien ts (in (b)) for 0 k V/cm and 175 k V/cm. The binning ranges are indicated as white dashed
lines in (a). The instruments response function (IRF) used for conv olution is displa y ed in (b).
T ransien ts in (b) are fitted with biexp onen tials consisting of a fast and a slo w PL deca y comp onen t
as inferred from our rate equation mo del due to b oth the emissiv e GS and ES.
The field-dep enden t transition dip ole momen ts can b e understo o d as an addition of a p ertur-
bation to the zero field transition dip ole momen t µ [109]
µ F
E S/GS = µ + X F int (5.9)
with
X
the transition dip ole momen t p olarizabilit y and
F int
the field in the nanoparticle. The
radiativ e transition rate Γ r relates to the transition dip ole momen t µ F via[50]
Γ r ,F = ω 3 n | f opt | 2
3 π ε 0 ~ c 3 µ F 2 . (5.10)
Inserting eq. 5.9 in to eq. 5.10 giv es us:
Γ r ,F = ω 3 n | f opt | 2
3 π ε 0 ~ c 3 µ + X | ˜
f lf | F 2
= ω 3 n | f opt | 2
3 π ε 0 ~ c 3 µ 2 + 2 µX ˜
f lf F + X 2 ˜
f lf 2 F 2 . (5.11)
The transition dip ole momen t p olarizabilit y
X
is negativ e as indicated b y the decrease of the
radiativ e rates with field. The induced c hange of the dip ole momen t though is small with
resp ect to the zero-field case, so that
µ
+
X | ˜
f lf | F >
0 for all field strengths. With this the last
simplification in equation 5.11 can b e made.
f opt
is the optical lo cal field factor at the transition
energy , n the refractive index of the surrounding matrix and ω the transition frequency .
The rate equation mo del in tro duced in c hapter 3 allo ws to extract radiative rates of ES and GS
as w ell as their in ter-relaxation rate
γ 0
from the PL dynamics of the three lev el system. Time-
84 5 Field Control
resolv ed field-dep enden t PL measured at ro om temp erature is sho wn in figure 5.4. Biexp onen tial
fits to the sp ectrally binned deca y transien ts in (b) yield the fast and slo w decay rates (
λ f/s
from
eq. 3.5) as w ell as the ratio of p opulation deca ying through these tw o c hannels. Time in tegrated
PL yields ES and GS shifts and in tensit y ratios as w ell as relative c hanges of their in tensit y
under electric field. In the follo wing this data will b e analyzed in order to retriev e the ES and
GS transition dip ole momen t, its p olarizabilit y and the in ter relaxation rate γ 0 .
The field dep enden t radiativ e rates
Γ r
E S/GS
of eq. 5.11 can b e inserted as the radiativ e rates
in the rate equation system discussed in section 3.3. The in ter-relaxation rate
γ 0
can also b e
expressed as a function of the external electric field. The LO-phonon mediated transition b et w een
ES and GS follo ws F ermi’s Golden rule[100]:
γ 0 ( ∆E ) = C LO
δ LO
( ∆E − E LO ) 2 + ( δ LO / 2) 2 (5.12)
where
C LO
is a coupling strength constan t,
δ LO
the transition width, and
∆E
the detuning to
the phonon energy E LO .
The energetic spacing b et w een ES and GS at zero field
δ E 0
for the nanoplatelets that w ere
used here is nearly resonan t to the LO-phonon energy . ES and GS ha v e a differen t shift with
applied field. As shown in figure 5.3 (a) the ES-GS energy spacing
δ E
=
δ E F
=
E F
E S − E F
GS
=
δ E 0 −
(
∆E F
E S − ∆E F
GS
) is field-dep enden t and decreases with increasing field. Figure 5.5 (b)
sho ws the dep endence of the zero temp erature ES-GS scattering rate
γ 0
on the detuning
dE
.
A field-induced c hange in the ES-GS energy spacing
δ E F
mak es
γ 0
(
δ E F
) field-dep enden t and
externally con trollable b y the applied v oltage:
γ F
0 ( δ E F ) = C LO
δ LO
( δ E 0 − ( ∆E F
E S − ∆E F
GS )
| {z }
shifts, obtained from figure 5.3 (a)
− E LO ) 2 + ( δ LO / 2) 2 (5.13)
F or the analysis shown in the follo wing
C LO
= 0
.
131 me V/ns and
δ LO
= 6
.
9 me V are used from
c hapter 4.
The shifts are obtained from the quadratic fit curv es in figure 5.3 (a). It turns out that the
ES-GS relaxation rate is strongly field-dep enden t and v aries b y a factor of
∼
3 in the range of
applied fields, figure 5.5 (b). No w the parametrized field dep enden t ES-GS transition rate
γ F
0
can b e included in to the rate equation mo del. The biexp onen tial decays of ES and GS in eqs. 3.6
and 3.7 dep end only on the radiativ e, non-radiativ e and in ter relaxation rates. As in c hapter 3
n ( t =0)
GS
=
n ( t =0)
E S
is c hosen as b oundary condition. The amplitudes
f
,
g
, and
h
are functions of
the rates. The ratio
r T R,F
of the total p opulation deca ying through the fast and the slo w deca y
5.3 T ransition Dip ole Moment and its P ola rizabilit y 85
0 1 0 0 2 0 0
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
- 0 . 0 2 0 . 0 0 0 . 0 2
1 0
1 0 0
( I
F
/ I
0
)
G S
b
I
0
E S
/ I
0
G S
( r / r )
T R , F
( R / R )
T I , F
( I
F
/ I
0
)
E S
E x t e r n a l F i e l d ( k V / c m )
a
1 7 5 k V / c m
0
( n s
- 1
)
D e t u n i n g d E ( e V )
0 k V / c m
Figure 5.5: Time resolv ed stark sp ectroscop y analysis.
(a) Field-dep enden t relativ e
c hange of the time in tegrated ES
I F
E S /I 0
E S
and GS
I F
GS /I 0
GS
emissions and the differen tial c hange
(
∆R T I /R
)
T I ,F
of the time in tegrated in tensity ratio of the ES to GS emission. (
∆r /r
)
T R,F
is the
differen tial c hange of the ratio of the p opulation deca ying through the fast and slow PL deca y
comp onen ts. The ratio of the ES and GS emission at zero-field
I 0
E S /I 0
GS
b efore eac h measuremen t
at a finite field
F
is also sho wn. It is constan t, indicating the absence of degradation or memory
effects. (b) Field-dependent c hange of the ES-GS scattering rate
γ 0
: At zero-field the ES-GS energy
spacing,
δ E 0
=
E 0
E S − E 0
GS
of the measured CdSe nanoplatelet sample is near resonan t to the
LO-phonon b ottlenec k at
E LO
= 25
.
4 me V in zincblende CdSe[
27
]. An increasing field leads to
an alteration of the energy spacing (Fig. 5.3 (a)) and larger detuning
dE
=
δ E F − E LO
to the
LO-phonon resonance.
γ 0
is th us reduced with applied field. The accessible field range is indicated
b y a gra y shaded area. The displa y ed curv e is taken from c hapter 4 using a F ermi’s Golden rule
mo del for an LO-phonon mediated transition[100].
comp onen t can b e expressed as:
r T R,F = ( Γ r
E S + g Γ r
GS ) λ − 1
f
( f Γ r
E S + hΓ r
GS ) λ − 1
s =
j
z }| {
λ − 1
f Γ r
E S +
k
z }| {
λ − 1
f Γ r
GS g
λ − 1
s Γ r
E S f
| {z }
l
+ λ − 1
s Γ r
GS h
| {z }
m
= j + k
l + m (5.14)
in tro ducing the new placeholders j,k,l,m . Sup erscript F is in tro duced to denote the field
dep endence giv en b y the transition and in ter-relaxation rates. F urther the ratio of time integrated
ES and GS PL emissions R T I ,F
E S/GS = R T I ,F is obtained as:
R T I ,F = R I E S ( t ) d t
R I GS ( t ) d t = Γ r
E S
Γ r
GS · λ − 1
f + f λ − 1
s
g λ − 1
f + hλ − 1
s =
j
z }| {
Γ r
E S λ − 1
f +
l
z }| {
Γ r
E S f λ − 1
s
Γ r
GS g λ − 1
f
| {z }
k
+ Γ r
GS hλ − 1
s
| {z }
m
= j + l
k + m (5.15)
This directly defines our time in tegrated zero field (
F
= 0 ) ratio
I 0
E S /I 0
GS
=
R T I ,F =0
E S/GS
plotted in
figure 5.5 (a). The zero-field ratio is obtained b efore eac h measuremen t of finite field. No changes
are observ ed from measuremen t to measuremen t, indicating that there is no degradation with
time.
86 5 Field Control
F rom the time-resolved, field-dep enden t PL w e get
( ∆r /r ) T R,F
, the relativ e change of the
ratio
r T R,F
with resp ect to zero field.
j F
,
k F
,
l F
,
m F
with sup erscript
F
mark the quan tities
with field on. Sup erscript 0 indicates the zero field case:
∆r T R,F
r T R, 0 = ∆r
r T R,F
= j F + k F
l F + m F
j 0 + k 0
l 0 + m 0 − 1 (5.16)
The exp erimen tal v alues of (
∆r /r
)
T R,F
are depicted in figure 5.5 (a) as blac k dots. F rom time
in tegrated PL w e obtain the ratio of time integrated ES and GS emission
R T I ,F
. Its relativ e
c hange with resp ect to zero field is then:
∆R T I ,F
R T I , 0 = ∆R
R T I ,F
= j F + l F
k F + m F
j 0 + l 0
k 0 + m 0 − 1 (5.17)
The exp erimen tally obtained (
∆R/R
)
T I ,F
are sho wn as green dots in figure 5.5 (a). It should
b e noted that (
∆r /r
)
T R,F
and (
∆R/R
)
T I ,F
are differen t quan tities, whic h in general do not
coincide due to differen t dep endencies on the rates.
The field-induced c hanges of the ES and GS emissions
I F
E S/GS
=
R ∞
0 I F
E S/GS ( t ) d t
are sho wn
normalized to the resp ectiv e zero-field emissions as blue and red dots in figure 5.5 (a).
All ab o v e men tioned quan tities:
I F
E S/GS
, (
∆r /r
)
T R,F
(eq. 5.16) and (
∆R/R
)
T I ,F
(eq. 5.17)
are analytical functions of the radiativ e, non-radiativ e and in ter-scattering rates. They are
fitted together with the zero-field ES/GS in tensit y ratio sharing the same set of parameters:
Zero field transition dip ole momen t
µ E S,GS
, its p olarizabilit y
X E S,GS
and the field indep enden t
non-radiativ e rates.
Including time in tegrated and resolv ed PL data in this global approach mak es the fit v ery
sensitiv e to field-induced c hanges in all three rates:
γ F
0
whic h is parametrized and
Γ r ,F
E S,GS
whic h
only dep end on µ E S,GS and X E S,GS .
Figure 5.6 sho ws the results of the analysis for the field-induced changes in the radiativ e rates
Γ r ,F
E S/GS
. W e observ e b oth a reduction of the radiativ e rates and of the in ter-relaxation rate
γ 0
with increasing field. The decrease of the radiativ e rates can b e understo o d within the Quan tum
Confined Stark and F ranz-Keldysh effect. The electric field p olarizes the electron and hole w av e
functions in the exciton. This reduces the spatial o v erlap in tegral to whic h the transition dip ole
momen t and th us the radiativ e rate is prop ortional in an en velope function approximation [
42
].
The reduction of the ES-GS relaxation rate on the other side is a consequence of the decrease of
the ES-GS energy spacing with applied field. In zero-field the used platelets are nearly resonan t
to the b ottlenec k. The applied field detunes the energy spacing from the resonance to the
5.3 T ransition Dip ole Moment and its P ola rizabilit y 87
0 5 0 1 0 0 1 5 0 2 0 0
0 . 0 5
0 . 1 0
2 0
4 0
6 0
8 0
r , F
E S
r , F
G S
F
0
R a t e ( n s
- 1
)
E x t e r n a l F i e l d ( k V / c m )
T h e o r y
Figure 5.6: Field-dep enden t radiativ e rates of CdSe nanoplatelets.
ES-GS scattering
rate
γ F
0
and in trinsic radiativ e rates
Γ r ,F
E S/GS
of ES and GS as inferred from our global fit mo del.
The red dot represen ts the ground state radiative rate at zero field obtained from the transition
dip ole momen t calculated using a k · p based simple approximation.
LO-Phonon energy . This reduces the transition rate b et w een ES and GS according to F ermi’s
golden rule, eq. 5.13.
Via eq. 5.11 the GS and ES transition dip ole momen ts and their p olarizabilit y are directly
obtained from the global fit. This results in
| µ GS | 2
= 9
.
0
±
2
.
3
·
10
− 58
C
2
m
2
and
| µ E S | 2
=
5
.
5
±
1
.
4
·
10
− 58
C
2
m
2
for the zero-field transition dip ole momen t (squared). F or s-states,
i.e. the ground state, the squared exciton transition dip ole momen t can b e compared to an
estimate of the transition dip ole momen t using a k
·
p based simple appro ximation [
137
]. The
material constan ts from ref.
27
are used for the CdSe nanoplatelets. The rough theory estimate
| µ | 2 ≈
9
.
3
·
10
− 58
C
2
m
2
is sho wn as a red dot in figure 5.6. This is in excellen t agreemen t with
our analysis for the GS.
T able 5.1: Results from Time-resolv ed Stark sp ectroscop y .
Exciton p olarizabilit y
α
, the
transition dip ole momen t
| µ | 2
, and its p olarizabilit y
X
of the ground and excited state of 5.5 ML
CdSe nanoplatelets with 42 ± 7 nm x 8 . 7 ± 2 . 1 nm lateral size.
α | µ | 2 X
(Cm 2 /J) (C 2 m 2 ) (Cm 2 /V)
GS 1 . 4 ± 0 . 3 · 10 − 36 9 . 0 ± 2 . 3 · 10 − 58 − 6 . 9 ± 1 . 8 · 10 − 37
ES 3 . 0 ± 0 . 3 · 10 − 35 5 . 5 ± 1 . 4 · 10 − 58 − 4 . 1 ± 1 . 0 · 10 − 37
The fitting pro cedure also allo ws to deriv e the transition dip ole momen t p olarizabilit y
X E S/GS
via eq. 5.11. As seen there the linear and quadratic terms in
F
do not ha v e indep enden t co efficien ts.
The p olarizabilit y of the transition dip ole momen t leads to a linear and quadratic c hange of
Γ r
E S/GS
with applied field. F rom the fit in figure 5.5 we obtain:
X GS
=
−
6
.
9
±
1
.
8
·
10
− 37
Cm
2
/V
and
X E S
=
−
4
.
1
±
1
.
0
·
10
− 37
Cm
2
/V. T o the b est of our kno wledge these are the first rep orted
88 5 Field Control
v alues of the p olarizabilit y of the transition dip ole momen t in colloidal nano crystals. V alues
in the same order of magnitude ha v e b een rep orted for the transitions of F renk el excitons of
similar binding energy in h ydrazones [138].
The mo del for the field-dep enden t PL emission also allo ws, for the first time, to discriminate
b et w een the effects of transition rate c hanges and field-induced broadening, and to quan tify
their con tributions. Both reduce the radian t flux p er unit w a v elength. Consider first a narro w
band PL detection of e.g. 1 nm, m uc h smaller than the FWHM of the zero-field emission. F or
this narro w band the GS emission (at
E
= 2
.
215 e V) can b e lo wered b y 28 % in total detection
at a field of 175 k V/cm. 29 % of this reduction are related to broadening. The other 71% are
related to the transition dip ole p olarizabilit y , i.e. to the c hanges of transition oscillator strength.
F or a broad detection (m uch bigger than the spectral zero-field FWHM) only the change in the
transition oscillator strength leads to a field-dep enden t alteration of
≈
22% of the total emission.
Conclusion
It has b een sho wn that CdSe nanoplatelets are a go o d mo del system to study the effect of
external fields on colloidal 2D quan tum w ells.
Time-resolv ed Stark sp ectroscop y is pro v en to b e a p o w erful to ol for field dep enden t studies.
Photoluminescence in tensit y c hanges, broadening, sp ectral shifts and recom bination dynamics
are mo deled sim ultaneously . Both the excited state and the ground state exciton are addressed
b y this metho d. The static exciton p olarizabilit y , the exciton transition dip ole momen t and its
p olarizabilit y can b e retriev ed from the data.
The static exciton p olarizabilit y
α
is found to b e 8.6
·
10
− 8
e V cm
2
/k V
2
for the GS and 3
.
0
±
0
.
3
·
10
− 35
e V cm
2
/k V
2
for the ES – in excellen t agreemen t with theory and other rep orts. The
v alue of the GS is in go o d agreemen t with p erturbation theory . The sum ES and GS is in
go o d agreemen t with THz exp erimen ts[
78
] (where ES and GS are indistinguishable). The trend
α E S > α GS is in line with theoretical predictions for lo wer dimensional systems[133].
The presen ted analysis further allo ws an estimate of the exciton binding energy – giving the
first exp erimen tal pro of of the high exciton binding energies in CdSe nanoplatelets. F or the
studied 5.5 ML platelets w e find
E B
= 170 me V, in go o d agreemen t with theoretical predictions.
Th us CdSe NPLs exhibit highly robust excitons, whic h are stable even at room temp erature.
The metho d is also sho wn to b e applicable to other nanoparticles.
The zero-field radiativ e rate is determined b y the exciton transition dip ole momen t, found to
b e 3
.
0
·
10
− 29
Cm for the ground state exciton. The field dep enden t radiativ e rate is determined
b y the p olarizabilit y
X
of the transition dip ole momen t. It is found to b e in the order of
5.3 T ransition Dip ole Moment and its P ola rizabilit y 89
magnitude of similarly b ound F renk el excitons in organic molecules [
138
]. T o the best of our
kno wledge w e rep ort, for the first time, v alues for X for a colloidal nano crystal.
The field dep enden t sp ectral c hanges of the PL in tensit y are related to c hanges in b oth
the radiativ e rates and the emission linewidth. The presen ted global analysis also allows a
quan titativ e discrimination of the t wo phenomena. F or narro w band mo dulation at an external
electrical field of 175 k V/cm the PL emission is reduced b y 28 % (of whic h 29 % deriv e from
broadening, the rest from exciton p olarization). F or broad band detection the 22% mo dulation
is only due to exciton p olarization.
These results sho w that an efficien t field con trol ov er the exciton recom bination dynamics,
emission linewidth and emission energy in these nanoparticles is feasible and op ens up application
p oten tial as field-con trolled emitters.
90 5 Field Control
5.4 APPENDIX: Field-Dep endent Abso rption of CdSe Nanoplatelets
2 . 0 2 . 5 3 . 0
f
t o t .
E n e r g y ( e V )
E x c i t a t i o n :
A b s o r p t i o n ( a . u . )
T R - P L / T i : S i
T I - P L / H e C d
2 . 7 5 2 . 8 0 2 . 8 5
1 7 5 k V / c m
0 k V / c m
H e C d
T i : S a
I
F
/ I
0
N o r m . l a s e r a b s o r p t i o n
( F i e l d / z e r o f i e l d ) ( % )
E x t e r n a l F i e l d ( kV / c m )
N o r m . t o t a l P L
0 5 0 1 0 0 1 5 0 2 0 0
- 1
0
1
( c )
( b )
E x t e r n a l F i e l d ( kV / c m )
f
t o t .
/
f
t o t .
( % )
F - s u m r u l e
( a )
Figure 5.7:
(a) Field dep enden t absorption sp ectra of CdSe nanoplatelets with the laser p eaks of
the HeCd laser used for time in tegrated and the Ti:Sa used for time-resolved field dependent PL
studies. (b) The total oscillator strength (in tegrated absorption sp ectrum) is unc hanged under the
applied field strengths. The differential c hanges to zero field are far b elo w 1 %, confirming the f-sum
rule[
32
]. The applied field is only a small p erturbation to the electronic system. (c) Normalized
total PL in tensit y under field (right axis), compared to the normalized change of absorption of the
exciting lasers (left axis). No c hanges of the absorption around 440 nm can b e seen.
6 | Engineering the Dynamics of Excitons with
a T yp e I I Band Alignmen t
The results of this c hapter are found in Ref. [
79
] Scott, R. et al.
(2016)
, Phys. Chem. Chem. Phys. ,
18(4), 3197. All figures are reproduced and adapted b y p ermission of the PCCP Owner So cieties.
Electron and hole center of mass
Nanoplatelet
Electron and hole center of mass
Nanoplatelet
coherence volume
T ype II - platelet Core-only platelet
CdT e CdSe
Indirect exciton
Slow
PL
t
Fast
PL
t
CdSe or CdT e
Direct exciton
Figure 6.1: Indirect and direct ex-
citons in hetero CdSe-CdT e and
core only platelets.
Due to the spa-
tial separation of the electron and hole
w a vefunction indirect excitons at a t yp e
I I hetero junction sho w prolonged radia-
tiv e deca y compared to direct excitons
in core only platelets.
Bey ond the p ossibilit ies to alter ph ysical prop erties and recombination dynamics b y the ab o ve
men tioned appro ac hes (size and external fields), hetero na noparticles offer a further p ossibilit y
to tune their dynamics by an in-built t yp e I I transition.[139–141 ]
The prop erties of CdT e-CdSe hetero nano dots and -ro ds ha v e b een in v estigated in systems of
spherical or axial symmetry in the last y ears [
142
–
148
]. Recent studies on the synthesis of t yp e II
CdSe-CdT e and CdT e-CdSe core-lateral shell nanoplatelets ha v e in v estigated their morphology ,
the t yp e I I band offsets as well as the prolongation of the PL deca y tim e compared to core only
nanoplatelets at ro om temp erature [149–151].
In the follo wing the te mper ature-dep enden t deca y dynamics of type I I CdSe- CdT e and CdT e-
CdSe core-lateral she ll nanoplatelets are in v estigate d. By co m bining the temp erature-dep enden t
deca y dynamics an d the quan tum yield w e deduce the no n-radiativ e and radiativ e lif etimes of
hetero nanoplatelet s. In line wit h the predictions of the Gian t Oscilla tor Strength Effect in 2D
the radiativ e life time increases with temp erature. Comparing core only and hetero pla telets a
significan t prol ongation of the radiativ e lifetime by t w o orders of magnitude in typ e II platelets is
observ ed. Th e quan tum yield is barely affected. In a careful analysis of t he PL deca y transien ts
differen t recombination mo dels are compared – including elect ron-hole pair and exciton deca y .
This is a relev ant issue for the applicabilit y of thos e structures in photonic devices lik e solar c ells
(e-h pair) or lasers (exc iton). The obser v ed PL deca y in hetero platelets i s biexpo nen tial. It is
sho wn to o ccur predomi nately due to spatially indirect exciton s presen t at the hetero junction,
see figure 6.1, and not ionized e-h pair recom bination.
92 6 Engineering the Dynamics of Excitons with a T yp e I I Band Alignment
Figure 6.2: Ro om temp erature absorption, PL and PLE sp ectra of 4.5 ML CdSe (a)
and 3.5 ML CdT e (f ) core only platelets as w ell as 4.5 ML CdSe-CdT e (b) and inv erse
3.5 ML CdT e-CdSe (h) hetero platelets.
The PLE sp ectrum is recorded at the hetero platelet
emission maxim um at 680 nm (b) and 550 nm (h), resp ectiv ely . Inset left: HAADF STEM images of
(c) the 16x8 nm
2
CdSe core and (d) 27x47 nm
2
CdSe-CdT e hetero nanoplatelets. (e) A color co ded
STEM-ED X elemen tal map (T e red, Se green and Cd blue) resolves the CdSe core (aquamarine) and
CdT e lateral shell (pink). Insets righ t: (g) Core only CdT e plates with lateral size 42x35 nm
2
. (i)
and (j): TEM and color co ded STEM-ED X elemen tal map of CdT e and CdSe of 3.5 ML CdT e-CdSe
hetero platelets with total lateral size of 70x58 nm 2 and the same core size as in (g).
6.1 Cha racterization of CdSe-CdT e and Inverse Hetero-Nanoplatelets
4.5 monola y er (ML) CdSe core only , 4.5 ML CdSe-CdT e nanoplatelets and in v erse type I I CdT e-
CdSe nanoplatelets w ere syn thesized according to recent literature.[
149
] Figure 6.2 sho ws the
absorption, PL and photoluminescence-excitation (PLE) sp ectra of 4.5 ML CdSe core only (a)
and CdSe-CdT e core lateral shell platelets (b) as w ell as their transmission electron microscop y
(TEM) c haracterization (insets (c)-(e)). The righ t side of figure 6.2 displa ys the corresp onding
data for the 3.5 ML CdT e only and the in v erse 3.5 ML CdT e-CdSe samples. Due to the stronger
transv erse confinemen t (thinner platelets) the corresp onding CdT e hh and CdSe hh transitions
are blue shifted with resp ect to the left side of figure 6.2. The actual thic kness of the CdSe
and CdT e plates and lateral wings is n +1/2 times the thic kness of one monolay er[
44
,
45
,
152
]
(Zincblende:
d C dS e
M L
= 0
.
304 nm,
d C dT e
M L
= 0
.
324 nm). Figure 6.2 (a) sho ws the c haracteristic
4.5 ML hea vy hole (hh) exciton luminescence (orange) around 512 nm, whereas the PL p eak
of the hetero system is red shifted to 680 nm (b). F rom the corresp onding absorption and PL
sp ectra the hh absorption p eaks of b oth, the CdSe core at 512 nm and CdT e shell at 555 nm,
6.2 T emp erature Dep endent Luminescence Dynamics of Hetero-platelets 93
- 1 012345
1 0
- 3
1 0
- 2
1 0
- 1
1 0
0
L i f e t i m e ( n s )
T i m e ( n s )
5 . 5 K 7 5 K 1 0 0 K
N o r m . I n t . ( a . u . )
I R F
( a )
0 . 1
1
1 0
1
2
L i f e t i m e ( n s )
( c )
( b )
1 0 1 0 0
0 . 1
1
1 0
r a d
n r
T ( K )
D i r e c t e x c i t o n
Figure 6.3: Normalized time-resolv ed photoluminescence of 4.5 ML CdSe core only
platelets
(a) recorded at the sp ectral maxim um of the emission for selected temp eratures (5.5,
75 and 100 K). The biexp. fits (solid curv es) tak e the con volution with the instrumen t response
function (IRF, green curv e) into accoun t. (b) Short and long time constants,
τ 1
and
τ 2
, deduced
from biexp. fits in (a) together with the a verage lifetime
τ
. (c) Radiative and non-radiativ e lifetimes
τ r and τ nr calculated from equation 6.7 and 6.8.
can b e iden tified. The PLE sp ectrum recorded at the type I I transition w av elength of 680 nm
clearly pro v es the formation of a CdSe-CdT e hetero junction. The PLE sp ectrum resem bles
the cum ulativ e absorption sp ectrum of the CdSe-CdT e nanoplatelets.[
149
–
151
] The samples
w ere em b edded in a p oly(lauryl acrylate) host matrix on quartz substrates for lo w temp erature
measuremen ts.
6.2 T emp erature Dep endent Luminescence Dynamics of
Hetero-platelets
Figure 6.3 sho ws that the 4.5 ML CdSe core only platelets are c haracterized b y relativ ely fast PL
deca y times. The PL decay is bi-exp onen tial[
27
], as seen b y the bi-exp onential fits in figure 6.3 (a)
and discussed in section 3.2. The con volution with the instrumen t resp onse IRF (green curv e) is
tak en in to accoun t. The IRF is recorded at the emission w a ve length of the excitation laser (Laser
dio de: 409 nm, 50 ps pulsewidth, 1 MHz rep etition rate). The excitation densit y is 0.2 W/cm
2
(CW equiv alen t). The PL decay components of CdSe core only nanoplatelets are fast and of the
order of 200 ps for the short and 1-2 ns for the long comp onen t, figure 6.3 (b). The radiative an d
non-radiativ e rates in panel (c) are discussed later.
In con trast to the fast recom bination of core only CdSe nanoplatelets t yp e I I hetero nanoplate-
lets sho w a strongly prolonged PL deca y , see figure 6.4 for 4.5 ML CdSe-CdT e (a) and 3.5 ML
CdT e-CdSe (b). F or comparison the 5.5 K deca y curv e of 4.5 ML core only platelets is sho wn in
panel (a). The prolongation of the PL lifetime in the hetero platelets is due to the formation of
a c harge transfer state at the hetero in terface.[
144
] After a carrier pair is optically generated
94 6 Engineering the Dynamics of Excitons with a T yp e I I Band Alignment
2 0 0 3 0 0 1 0 0
0 . 0 1
0 . 1
1
C o r e o n l y
4 . 5 M L C d S e
4 . 5 M L C d S e - C d T e
t ( n s )
N o r m . I n t e n s . ( a . u . )
( c ) ( b ) ( a )
I n d i r e c t e x c i t o n
1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
0 . 0 1
0 . 1
1
t ( n s )
3 0 0 K
2 0 0 K
1 0 0 K
4 K
4 K
1 0 0 K
2 0 0 K
2 0 0 3 0 0 1 0 0
0 . 1
1
3 . 5 M L C d T e - C d S e
A d j . R
2
:
0 . 8 0 6
0 . 9 9 4
0 . 9 9 7
D a t a 1 0 0 K
B i e x p . F i t
S e c . O r d . F i t
M o u r a d e t a l . F i t
t ( n s )
Figure 6.4: Normalized time-resolv ed photoluminescence of 4.5 ML CdSe-CdT e (a)
and 3.5 ML CdT e-CdSe (c) nanoplatelets
recorded at the sp ectral maxim um of their resp ectiv e
t yp e I I emission bands sho wn in figure 6.2 (b) and (h). Grey lines are biexp onen tial fits for exemplary
temp eratures taking the con v olution with the instrumen t resp onse in to account. (b) sho ws a
comparison of a biexp onen tial fit, a second order deca y fit and one according to Mourad et al.[
153
]
for the 4.5 ML CdSe-CdT e platelets at 100 K. The adjusted R
2
v alues clearly show that the biexp.
mo del (red curv e) repro duces the data with a v ery high accuracy .
in the CdSe or CdT e it diffuses fast to the hetero in terface. The electron w a v efunction is then
mainly lo calized in the CdSe, the hole w a v efunction in the CdT e part of the junction.[
149
] This
results in a spatially indirect state. The corresp onding v alence band (VB) and conduction band
(CB) offsets at the junction ha v e b een determined to b e 0.56 and 0.36 e V for CdSe-CdT e and
CdT e-CdSe platelets, resp ectiv ely .[149, 151]
Ho w ev er, it is still rather unclear whether the observ ed type I I luminescence is related to an
uncorrelated e-h pair or to a coulom b correlated exciton state. This question can b e answ ered b y
the PL deca y dynamics of the excited sp ecies. A free e-h pair is exp ected to deca y with a Deb ye
second order deca y for the electron ( n ) and hole ( p ) densities according to
d n/ d t
=
k 2 · n p
.
Under the v alid assumption
n
=
p
for optical excitation this results in a h yp erb olic PL deca y
according to
˜
I
(
t
) =
k 2 ∗ n 2
0 /
(1 +
k 2 ∗ n 0 ∗ t
)
2
with initial concen tration
n 0
and deca y rate
k 2
[
154
].
The fit obtained with this function in figure 6.4 (b) shows rather limited agreemen t. Systematic
deviations on long timescales with resp ect to the resulting
χ 2
indicate that this mo del cannot
repro duce the data.
It has b een p oin ted out that the p olarization field generated b y carrier separation at the
junction ma y alter the deca y kinetics and its rate constan t
k 2
. F ollo wing Mourad et al.[
153
] the
rate constan t
k 2
=
γ n
is then prop ortional to the carrier densit y with a prop ortionalit y constan t
γ
. The resulting altered PL deca y function
˜
I
(
t
) =
γ ∗ n 2
0 /
(1 +
γ ∗ n 0 ∗ t
)
3 / 2
do es not fit the data
appropriately either (figure 6.4 (b)). In con trast to these e-h pair recom bination based mo dels all
6.2 T emp erature Dep endent Luminescence Dynamics of Hetero-platelets 95
0
2 5
5 0
7 5
1 0 0
1
2
L i f e t i m e ( n s )
( a )
1 0 1 0 0
0
1 0 0
2 0 0
( b )
r a d
n r
L i f e t i m e ( n s )
T ( K )
Figure 6.5: Deca y and radiativ e lifetimes of 4.5 ML CdSe-CdT e hetero nanoplatelets.
(a) Time constan ts
τ 1
and
τ 2
from biexp onen tial fits to figure 6.4 (a), as w ell as the a v erage lifetime
τ
deriv ed from equation 6.2. (b) Radiativ e and non-radiativ e lifetimes calculated from equations 6.7
and 6.8.
PL deca ys ˜
I ( t ) can b e w ell fitted b y bi-exp onen tials whic h are exp ected from exciton deca y .
˜
I ( t ) = A 1 · e − t/τ 1 + A 2 · e − t/τ 2 . (6.1)
They are indicated b y ligh t gra y lines in figure 6.4 (a) and (c) for sev eral temp eratures.
The comparison of the three fit mo dels for an exemplary temp erature of 100 K in figure 6.4 (b)
clearly indicates the sup erior agreemen t of the biexp onen tial fit (red) with the measured data.
F rom the viewp oin t of the
χ 2
h yp othesis test it is the most suitable mo del. Therefore it can b e
concluded that the predominan t sp ecies causing the observ ed (red shifted) t yp e I I emission is
related to exciton recom bination and not e-h pairs.
Figure 6.5 (a) sho ws the resulting time constants
τ 1
and
τ 2
of the biexp onen tial fits for the
temp erature range from 4 to 300 K. The shorter time constan t seems to b e practically temp erature
indep enden t, whereas the longer one exhibits a significan t increase with temp erature. Using
these results w e calculate the a v erage lifetime of the tw o comp onen ts[155]
τ = R ∞
0 t · ˜
I ( t ) d t
R ∞
0 ˜
I ( t ) d t = P i A i τ 2
i
P i A i τ i
(6.2)
as plotted in red in 6.5 (a).
The observ ed PL lifetime
τ
(
T
) is related to the quan tum yield
η
(
T
) , the in trinsic radiativ e
lifetime τ r ad and the non-radiativ e lifetime τ nr b y
τ ( T ) = 1
τ r ad ( T ) + 1
τ nr ( T ) − 1
= τ r ad ( T ) · η ( T ) . (6.3)
96 6 Engineering the Dynamics of Excitons with a T yp e I I Band Alignment
Kno wing the temp erature dep enden t quan tum yield
η
(
T
) mak es it p ossible to deriv e the
radiativ e and non-radiativ e lifetimes from the measured
τ
(
T
) . With
η
(300 K) determined b y a
dy e referencing metho d (to Rho damin 6G), the temp erature dependent quan tum yield can b e
obtained from a measuremen t of the temp erature dep enden t time-in tegrated PL I ( T ) .
η ( T ) = I ( T )
I (300 K ) η (300 K ) (6.4)
Figure 6.6 sho ws the temp erature dep endence of the time in tegrated luminescence for the CdSe
core only and the t w o hetero platelet samples. The temp erature dep endence can b e w ell fitted
with a thermally activ ated quenc hing mo del according to [156]:
I ( T ) = I (0 K )
1 + C · e − E A /k B T (6.5)
Using
η
(
T
) =
η
(0
K
)
I
(
T
)
/I
(0
K
) the temp erature dep enden t quan tum efficiency is obtained:
η ( T ) = η (0 K )
1 + C · e − E A /k B T (6.6)
with the activ ation energy
E A
and constan t C. Figure 6.6 sho ws the results of the temp erature
dep enden t quan tum yield and the fits to equation 6.6.
A t lo w temp eratures the quan tum yield of our hetero platelets approac hes 70 %. It drops
to 19 % and 13 % for the CdSe-CdT e and CdT e-CdSe hetero platelets at ro om temp erature,
comparable to core only platelets – in the order of 20 %. The corresp onding activ ation energies
for the PL quenc hing deduced from the fits in figure 6.6 are 16 me V (4.5 ML core only), 50 me V
(4.5 ML CdSe-CdT e) and 31 me V (3.5 ML CdT e-CdSe). With the knowled ge of
η
(
T
) w e can
calculate the temp erature dep endence of the radiativ e
τ r ad ( T ) = τ ( T )
η ( T ) (6.7)
and non-radiativ e lifetime
τ nr ( T ) = 1
τ ( T ) − 1
τ r ad ( T ) − 1
= τ ( T ) (1 − η ( T )) − 1 . (6.8)
The resulting temp erature dep endence of
τ r ad
and
τ nr
for the 4.5 ML core only and CdSe-CdT e
hetero nanoplatelets is sho wn in figures 6.3 (c) and 6.5 (b), resp ectiv ely .
6.3 Discussion 97
0 . 0 0 . 1 0 . 2
1
2
3
1 / T ( 1 / K )
1 7
3 4
5 1
6 7
Q Y ( % )
4 . 5 M L C d S e - C d T e
0 . 0 0 0 . 0 5 0 . 1 0
1 . 0
0 . 3
5 . 0
P L I n t e n s i t y ( a . u . )
3 . 5 M L C d T e - C d S e
2 8
4 9
7
7 0
1 / T ( 1 / K )
Q Y ( % )
1
2
3
4 . 5 M L C d S e
P L I n t e n s i t y ( a . u . )
2 6
5 2
7 7
( c )
( b )
( a )
Figure 6.6: Arrhenius plots of the sp ectrally and time-in tegrated photoluminescence
of (a) 4.5 ML CdSe core only , (b) 4.5 ML CdSe-CdT e and (c) 3.5 ML CdT e-CdSe nanoplatelets.
The righ t hand scales sho w the corresp onding temp erature dep enden t quan tum yield. Excitation
conditions are iden tical to figure 6.4.
6.3 Discussion
T wo trends can be identified in figures 6.3 (c) and 6.5 (b): While the non-radiativ e lifetime
decreases, the radiativ e lifetime increases with temp erature. The increase of the radiativ e lifetime
with temp erature in a 2D system is related to the Gian t Oscillator Strength Effect (GOST) in
2D [
17
,
26
,
83
] as discussed in c hapter 4. With increasing temp erature, a thermal redistribution
of the generated excitons leads to a higher fraction of excitons with a
k
-v ector outside the
radiativ e ligh t cone that cannot radiativ ely decay . Since the out-of-ligh t-cone scattering pro cesses
are thermally activ ated, this phenomenon prolongs the ov erall deca y of the photo-generated
excitons with increasing temp erature. The effect leads to an increase of the radiativ e lifetime
with temp erature from 70 to 270 ns for hetero nanoplatelets. Core only nanoplatelets sho w a
sixfold increase of the lifetime. As in c hapter 4, here the strong increase of the radiativ e lifetime
with temp erature in our core and core-lateral shell platelets can b e understoo d as a suppression
of the radiativ e recom bination due to the increase of the homogeneous exciton linewidth with
temp erature.
The ab o v e men tioned effect of a mo derate increase of the radiativ e lifetime with temp erature
further indicates that excitons and not e-h pairs are presen t in the hetero platelets. In case of
free electron-hole pair recom bination a far stronger temp erature dep endence of the bimolecular
recom bination co efficien t
k 2
according to
k 2 ∝
(
k B T
)
− 3 / 2
w ould b e exp ected[
157
]. F or example
an increase from 4 to 200 K w ould result in a decrease of
k 2
b y a factor
∼
350 . This would
translate in to a prolongation of the radiativ e deca y time
τ r ad
(
∝
1
/k 2
) b y more than t w o orders
in magnitude – clearly not reflected in the exp erimen tal results on the radiativ e lifetime. In line
with the findings ab o v e, this rules out the presence and deca y of electron-hole pairs.
In con trast to the men tioned radiativ e pro cesses also non-radiativ e pro cesses suc h as exciton
trapping and deca y at defects tak e place. These non-radiativ e c hannels can b e seen as temp erature
98 6 Engineering the Dynamics of Excitons with a T yp e I I Band Alignment
0
2 5
5 0
( a )
1
2
L i f e t i m e ( n s )
1 0 1 0 0
0
5 0
1 0 0
1 5 0
T ( K )
r a d
n r
( b )
L i f e t i m e ( n s )
Figure 6.7: Deca y and radiativ e lifetimes of 3.5 ML CdT e-CdSe hetero nanoplatelets.
(a) Time constan ts
τ 1
and
τ 2
from biexp onen tial fits to figure 6.4 (c) f or 3.5 ML CdT e-CdSe
nanoplatelets. (b) Radiativ e and non-radiativ e lifetimes calculated from equations 6.7 and 6.8.
activ ated pro cesses. The mobile excitons ha v e to ov ercome some activ ation energy to mo v e to a
defect or quenc hing site. Thermally activ ated, the non-radiativ e rate increases and equiv alen tly
its lifetime decreases with temp erature as observ ed in figure 6.5 (b). Since a larger fraction of
excitons is deca ying non-radiativ ely a decreasing luminescence in tensity is observ ed in figure 6.6.
Ha ving discussed 4.5 ML CdSe-CdT e platelets w e no w in v estigate the temp erature dep endence
of the radiativ e and non-radiativ e lifetime of the in verse 3.5 ML CdT e-CdSe platelets in figure 6.7.
The corresp onding a v erage lifetime in panel (a) is again deriv ed b y bi-exp onen tial fits, and the
temp erature dep endence of the radiativ e and non-radiativ e lifetimes in (b) from eqs. 6.7 and 6.8.
With ∼ 120 ns the non-radiativ e lifetime is comparable to the 4.5 ML CdSe-CdT e nanoplatelets
at lo w temp eratures. It decreases more strongly ab o v e 40 K (
∼
0.025 k
− 1
) leading to an o v erall
decreasing a v erage lifetime with temp erature. This stronger impact of non-radiativ e pro cesses in
the in v erse structure is also reflected in the in tensit y drop of the PL emission and quantum yield
in figure 6.6 (c). An explanation for this observ ed difference b et w een the normal and in v erse
structure ma y b e syn thesis-related. A higher in terfacial defect densit y in the in v erse type I I
structure or more defects at the edges of the outer CdSe lateral shell are exp ected. A higher
electron trap densit y in the CdSe lateral shell compared to the hole trap densit y in the CdT e
lateral shell ma y result in differen t non-radiativ e lifetimes for the CdSe-CdT e and CdT e-CdSe
heterostructures.
A stronger con tribution of thermally activ ated non-radiativ e channels in 3.5 ML CdT e-CdSe
plates leads to a strongly decreasing a v erage lifetime with temp erature, figure 6.7 (a). The m uc h
w eak er thermally activ ated non-radiativ e con tribution in the 4.5 ML CdSe-CdT e platelets leads
to an a v erage lifetime that is dominated b y the prolongation of the radiativ e decay rate due to
the Gian t Oscillator Strength Effect (figure 6.5 (b)).
6.3 Discussion 99
Ho w ev er, ev en with the o ccurrence of relev an t non-radiativ e recom bination c hannels the
CdSe-CdT e and inv erse CdT e-CdSe hetero platelets sho w muc h longer a v erage (10-60 ns) and
radiativ e (50-270 ns) lifetimes as compared to core only plates (sub nanosecond) and CdSe-CdS
t yp e I nanoplates [
149
]. Therefore it is p ossible to engineer the deca y dynamics in these 2D
CdSe nanoparticles b y the in tro duction of a lateral hetero transition. The exp erimen tal results
indicate that the
∼
100 fold lifetime prolongation is not related to uncorrelated electron hole
pairs but to the formation of spatially indirect excitons.
Conclusion
The temp erature-dep enden t deca y kinetics of t yp e I I CdSe-CdT e and CdT e-CdSe core-lateral shell
nanoplatelets ha v e b een in v estigated. Their quan tum yield is comparable to core-only platelets.
A kinetic analysis of the photoluminescence (PL) deca y is com bined with a measurement of
the temp erature dep enden t quan tum yield. This approac h rev eals the temp erature dep endence
of the radiativ e and non-radiativ e lifetime of hetero nanoplatelets. The observed increase of
the radiativ e lifetime with temp erature is in line with the predicted increase due to the Gian t
Oscillator Strength Effect in 2D. It is th us attributed to an increase of the homogeneous transition
linewidth with temp erature. This temp erature dep endence indicates that predominan tly spatially
indirect excitons are presen t at the hetero junction and not ionized e-h pairs.
In comparison to core only platelets w e observ e a significan t prolongation of the radiative
lifetime b y t w o orders of magnitude in t yp e I I nanoplatelets. A t ro om temp erature, for example,
the radiativ e lifetime of hetero plates is found to b e
∼
300 ns – v ery slo w compared to the
sub-nanosecond deca y of core-only platelets. It is exp ected that the radiativ e lifetime can b e
further tuned b y a hetero-junction via confinemen t and band offsets.
The analysis of the CdSe-CdT e and CdT e-CdSe hetero platelet PL deca y sho ws that it is
biexp onen tial. A bimolecular e-h pair recom bination can b e excluded b y the deca y kinetics and
its temp erature dep endence. This kinetic argumen t also indicates the presence of c harge carriers
in form of spatially indirect excitons at the t yp e I I hetero-junction of these ultra-strong confined
nanoplatelets.
Stable ev en at ro om temp erature with long (carrier) lifetimes these robust excitons are desirable
for photonic applications. These findings suggest that hetero nanoplatelets are less suitable
for solarcells, since the generated carriers are coulom b correlated. On the other hand, lasing
applications migh t strongly b enefit from a broad gain sp ectrum generated b y the excitonic t yp e
I I transition.
P a rt I I I
Non-Linear Pr oper ties: Tw o-Photon
Absorption
Shape and Angular Dependence
7 | T w o-Photon Absorption in CdSe Nano crys-
tals: The Influence of Dimensionalit y and Size
The results sho wn in this c hapter are presented in Ref. [
84
] Scott et al.
(2015)
, Nano L etters , 15,
4985. Figures adapted with permission, © 2017 American Chemical Society .
2P-PLE
z-scan
local field
electronic effects
Figure 7.1: T w o-photon absorption
cross sections of CdSe nano crystals
re-
v ealed b y Z-scan and tw o-photon excitation
sp ectroscop y . Both electronic confinemen t
and lo cal field effects fa v or the platelets –
reac hing 10
7
GM, the largest ev er rep orted
cross sections for (colloidal) semiconductor
nano crystals – ideally suited for t w o-photon
imaging and broadband nonlinear opto elec-
tronics.
T wo-photon absorption (TP A) is the sim ultaneous absorption of t wo photons of an energy
b elo w the bandgap of a material. Their com bined energy induces a transition from a ground
state in to an excited electronic state. In 1931 Göpp ert-Ma y er[
158
] predicted this pro cess that
scales with the square of the photon flux. The in tro duction of the laser pro viding high enough
ligh t in tensit y enabled Kaiser and Garrett[159] to measure TP A for the first time in 1961.
T wo-photon absorption of semiconductor nano crystals is of curren t in terest tow ards applications
suc h as non-linear gain media[
45
], optical p o w er limiting, photo-dynamic t w o-photon (TP) cancer
therap y[
160
] or bio-lab eling[
161
]. Confo cal TP imaging in the range of 650-1350 nm is ideally
suited for cell and animal in-viv o imaging[
162
], com bining high spatial resolution and deep tissue
p enetration[
163
]. F urther applications, e.g. in micro-fabrication, lithograph y , p olymerization,
data storage and sp ectroscop y ha v e b een demonstrated [164–166].
The TP A cross section of molecules and nano crystals is t ypically given in Göppert Ma yer:
1 GM= 10
− 50
cm
4
s photon
− 1
particle
− 1
. It can b e seen as the product of t wo one-photon cross
sections and an in teraction time. Organic TP absorb ers typically ha v e cross sections smaller than
1000 GM[167], although cross sections of up to 10 6 GM hav e b een reported[168, 169]. Ho wev er,
I I-VI semiconductor based nano-particles offer significan tly higher TP A cross sections with
resp ect to the particle v olume.[
84
,
170
,
171
] The higher the cross section p er particle or v olume,
the lo w er the excitation in tensities needed to obtain the same resp onse in TP A applications with
less amoun t of material.
It is th us desirable to understand the influence of the semiconductor material as well as the
shap e and size of the nano crystals on their t w o-photon absorption cross section. Suc h a study
104 7 T w o-Photon Absorption in CdSe Nano crystals: The Influence of Dimensionality and Size
E
1PA T PA
(a)
(b)
(c)
400 600 800
0
1
2
3
4
5
6
7
Intrinsic absorption coeff. � (10 5 cm -1 )
Wavelength (nm)
4.5 ML NPLs
molecule
Cumarin 309
Gold clusters
Dyes
CdSe
H 2 O
(2)
10 -1 10 1 10 3 10 5 10 7
Unit cell
(1)
Figure 7.2: T w o-photon absorption and co efficien t
(a) Sc hematic illustration of a one-photon
absorption pro cess compared to a (degenerate) t w o-photon absorption in v olving tw o photons with
half the energy via a virtual state sho wn as a dashed line. (b) One-photon absorption sp ectrum of
4.5 ML CdSe nanoplatelets with t ypical (Ti:Sa laser) one- and corresp onding t w o-photon excitation
w a velengths. (c) The range of absolute TP A cross sections (in GM) spans o ver sev eral orders
of magnitude, from small molecules o ver dy es (1)[
172
] to mesoscopic particles suc h as e.g. gold
clusters (2) [173] and (as sho wn here) CdSe nano crystals [84].
w ould help to engi neer efficien t t w o-phot on absorb ers for v arious opto-elect ronic applications
and giv e insight in to the fundamen tal pro cesse s of TP A.
Here w e in vestigate the influence of v olume and shap e of col loidal CdSe nano-platelets, -ro ds
and -dots on their TP A cross section. Platelets combine large particle v olumes with ultra stron g
confinemen t in on e dimension and will b e in the fo cus of this w ork. In co n trast to w eakly
confined nano crystals, the TP A cross sections of CdSe nanopla telets scale sup er-linearly with
their v olume (
V ∼ 2
) and sho w ten times more efficien t TP A than nano-rods or dots. This strong
shap e dep endence go es w ell b ey ond the effect of lo cal fields. The larger the particles’ asp ect ratio,
the greater is the electro nic con tribution to the increased TP A. Both, electronic confi nemen t
and lo cal field effects fav or the platelets and mak e them uni que t w o-photon absorb ers with
outstanding cross sec tions of up to 10
7
GM, the largest ev er rep orted for (colloidal) semiconductor
nano crystals, see figure 7.2
Before presen ti ng these results in detail the concept of TP A will b e briefly in tro duced together
with the analytical framew ork and linear measuremen ts needed to obtain TP A cross sections
from the t w o non-linear tec hniques used: z-scan and t wo-photon excitation spectroscopy .
7.1 Theo ry 105
7.1 Theo ry
A t lo w field strengths the light-induced dipole moment per unit volume in a medium, the
p olarization
P
(
t
) , dep ends linearly on the electric field strength
P
=
ε 0 X E
.
ε 0
is the p ermittivit y
of free space,
X
the linear susceptibilit y . In a semi classical approac h the p olarization can b e
expanded in a p o w er-series of E to describ e the resp onse for high fields[174]:
P = ε 0 X (1) E + X (2) E E ∗ + X (3) E E E ∗ + · · · (7.1)
X ( n ) are the n-th order optical susceptibilities and tensors of rank (n+1). They ha v e generally
complex en tries. Second order pro cesses suc h as second harmonic and sum-frequency generation
or third order pro cesses lik e F our-W a v e-Mixing where photon energy is conserv ed are related
to the real part of
X (2)
and
X (3)
, resp ectiv ely , and referred to as parametric pro cesses[
175
].
T wo-photon absorption is a non-parametric process where p opulation is transferred from the
ground to an excited state. The linear absorption co efficien t
α
is prop ortional to the imaginary
part of the linear susceptibilit y
X (1)
. Similarly the TP A co efficien t
β
is prop ortional to the
imaginary part of
X (3)
. F or the case of degenerate TP A inv olving t w o photons of iden tical
frequency ω [174]:
β ≈ 8 π 2 ω
ε 0 c 2 n 2
0
Im X (3) ( ω ,ω , − ω ) (7.2)
with
c
the v acuum sp eed of ligh t and
n 0
the (linear) refractiv e index of the medium. Clearly ,
β
and X (3) are sp ectral quan tities, as are P and E .
β
is prop ortional to the probabilit y
W TPA
of a TP A pro cess taking place via the square of
the ligh t in tensit y:
W TPA ∝ β I 2
. Using an in termediate state that is related to all Eigenstates
of the electronic system – as done in the original w ork of Göpp ert-Ma y er [
158
] – the probabilit y
of a TP A pro cess is giv en b y second order F ermi’s golden rule:
W TPA = 2 π
~ X
i,f
X
m
h i | ~ e · ~ p | m i h m | ~ e · ~ p | f i
E m − E i − ~ ω
2
δ ( E f − E i − 2 ~ ω ) (7.3)
A deriv ation can b e found in e.g. R. Bo yd: Nonlinear Optics[
175
]. Here
| j i
and
E j
are the w a v e
function and energy of the electron in the initial (
j
=
i
) , in termediate (
j
=
m
) or final (
j
=
f
)
state.
~ e
is the ligh t p olarization v ector and iden tical for b oth photons in the degenerate case that
is considered in this w ork.
~ p
is the momen tum op erator. Concen trating on the electronic system
of semiconductor nano crystals, initial states
| i i
are in the v alence band, final states
| f i
in the
conduction band. In termediate states
| m i
can b e in an y band. F or zincblende CdSe platelets or
106 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
Energy
b
| u e | u hh | u lh | u so
a c
Final
Intermediate
Initial state
Figure 7.3: T ransitions of a degenerate t wo-photon absorption process
(a) Lev el sc heme
of a t w o-photon transition with an initial state
| i i
in the v alence band (comprised of heavy-, ligh t-,
and split-off hole band) to a final state
| f i
in the conduction band.
| u n i
denote the corresp onding
blo c h states of the lev els. (b) The in termediate state
| m i
can b e in an y band. The transition
probabilities of eac h paths add to the TP A transition probabilit y (summation o v er
m
in eq. 7.3).
(c) The TP A transition can start from differen t initial states as long as
E f − E i
= 2
~ ω
(sum o v er
i
and f in eq. 7.3).
dots the v alence bands are hea vy hole (HH), light hole (LH) or split-off (SO) subband as sho wn
in figure 7.3. F or wurtzite ro ds they consist of the A, B and C band.
The matrix elemen ts in the n umerator giv e the optical selection rules. T w o-photon transitions
tak e place through the virtual in termediate state
| m i
, whic h is short-liv ed and th us energetically
not w ell defined.[
174
] The efficiency of the transition is prop ortional to the sum of p erturbational
terms in v olving all p ossible non-resonan t states
| m i
, illustrated for a fixed initial and final state
| i i
and
| f i
in figure 7.3 (b). The denominator in eq. 7.3 giv es a rough estimate of the in tensity of
a giv en pro cess, maximized when there is an in termediate state near the laser energy
~ ω
so that
(in the degenerate case)
E m ≈ E f − E i
2
. Ho w ev er,
| m i
should nev er b e exactly resonan t, as this
w ould resem ble sequen tial TP A (
∝ I
) a first-order transition, a fundamen tally differen t pro cess
than non-resonan t TP A (
∝ I 2
). Figure 7.3 (c) sho ws that the delta condition for initial and final
state matc hing the t w o-photon energy can b e met for differen t
| i i
and
| f i
, tak en in to accoun t by
the outer sum in equation 7.3.
7.2 Exp erimental T echniques
A v ariet y of exp erimen tal metho ds has b een used to obtain TP A cross sections of I I-VI semi-
conductor nano crystals, non-linear transmission exp erimen ts suc h as z-scan [
176
,
177
] as w ell
as indirect metho ds suc h as t w o-photon induced fluorescence (TPIF)[
178
,
179
]. Degenerate or
non-degenerate excitation[
180
,
181
] has b een p erformed and pump-probe measurements[
176
].
Blan ton et al.[
96
] presen ted TPIF of single particles and relativ e TP A spectra of CdSe QDs[
182
].
7.2 Exp erimental T echniques 107
More exotic particles suc h as dop ed[
183
] or surfactan t-capp ed[
184
] QDs and quasi t yp e I I
CdSe/CdS dots in ro ds [
185
] with TP A cross sections up to 10
5
GM also ha v e b een in v estigated.
Our goal is to study the influence of dielectric and electronic confinemen t effects on the TP A
cross section and sp ectra of nano crystals. F or this fundamen tal, systematic in vestigation w e
concen trate on CdSe NPLs, NRs and QDs of different sizes. T o determine their TP A cross
sections w e use op en ap erture z-scan[
186
] at 800 nm. F or NPLs the TP A b eha vior is further
in v estigated for w av elengths resonan t to the first excitonic one-photon transition up to the 2D
con tin uum b y t wo-photon photoluminescence excitation spectroscopy (2P-PLE) from 750 nm to
950 nm. W e also presen t a new, highly efficient method to obtain TP A sp ectra.
7.2.1 Nano crystal Solution Concentrations Determined b y Size and Shap e
A study of the influence of shap e and geometry , i.e. the lo cal field, on the TP A cross-sections of
CdSe nanoparticles requires kno wledge of the particle size and concen tration in solution (for
z-scan). This will w e b e treated here briefly b efore presen ting the z-scan and 2P-PLE tec hnique.
Sizes and asp ect ratios
The set of samples for the TP A studies consists of CdSe zincblende nanoplatelets (NPLs)
and quan tum dots (QDs) and hexagonal (wurtzite) nanoro ds (NR). F or NRs the asp ect ratio
(AR) is defined as
AR
=
l /d
, with length
l
and diameter
d
. Since some studied NPLs ha v e
non-iden tical lateral sizes in x and y direction (edge-lengths
l x
and
l y
), w e use the geometrical
a v erage
L
=
p l x l y
as the NPL’s effectiv e edge-length to c haracterize them. Their asp ect
ratio is then defined as
AR
=
n · d M L /L
. The lateral sizes are determined from TEM-analysis,
d M L
= 0
.
304 nm is half the lattice parameter for zinc blende CdSe[
44
] (1 monola y er, ML) and
n
the n um b er of CdSe monola yers. W e assume NPLs to b e cadmium-terminated on b oth basal
planes as sho wn b y Li et al. [
44
] and She et al..[
45
] NPLs emitting at a w a velength of around
460 nm with three Se and four Cd monola y ers will b e referred to as 3
.
5 ML NPLs, NPLs emitting
at 512
nm
and 550
nm
as 4
.
5 ML and 5
.
5 ML, resp ectiv ely . TEM images of NPLs and ro ds are
sho wn in figure 7.4. The o ccurring of asp ect ratios in the ensem bles can b e fitted with log-normal
distributions (solid lines).
Determining P a rticle Concentration in Solution
W e will now discuss the influence of the nanoparticles’ shape on its one-photon absorption cross
section and ho w it can b e used to determine the particle concen tration of a colloidal solution.
The in trinsic one-photon absorption cross section
µ i
can b e deriv ed b y Loren tz lo cal field theory
com bined with an effectiv e medium Maxw ell-Garnett approach in the con tin uum absorption
108 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
( 2 . 9 0 . 7 ) x ( 2 4 . 6 5 . 4 ) n m
C o u n t s
A s p e c t r a t i o
( b )
Figure 7.4: Asp ect ratios and TEM images of CdSe nanoplatelets (a) and ro ds (b)
used in TP A exp erimen ts
region. Here the bulk dielectric function can b e used since it is unaffected b y size quan tization
effects.[54, 187–190]
µ i ( ω ) = σ (1) ( ω )
V P
= 2 ω n S ( ω ) k S ( ω )
3 n M ( ω ) c D | f ( ω ) | 2 E Ω (7.4)
Here,
σ (1)
(
ω
) is the one-photon absorption cross section of the particle and
V p
its v olume.
n
and
k
are the refractiv e index and absorption co efficien t of the matrix (M) or the semiconductor
inclusion (S).
D | f ( ω ) | 2 E Ω
is the exp ectation v alue o v er all angles of the lo cal field factor squared
| f ( ω ) | 2 ( φ, θ ) (see eq. 1.27 in sec. 1.2), accoun ting for random ensemble orien tation:
D | f ( ω ) | 2 E Ω = R 2 π
0 R π
0 | f ( ω ) | 2 ( φ, θ ) sin θ d θ d φ
R 2 π
0 d φ R π
0 sin θ d θ (7.5)
Figure 7.5 sho ws the calculated shap e dep endency of the in trinsic absorption cross section
for CdSe nanoparticles in the con tin uum at 4 e V. Due to the t w o in-plane field factors close to
unit y , nanoplatelets ha v e the largest one-photon absorption cross section p er v olume. Because of
t ypically larger particle v olumes
V p
NPLs also ha v e larger absorption cross sections p er particle
σ (1) (eq. 7.4).
The extinction co efficien t of a solution with absorb ers can b e expressed b y Beers La w
˜ ε
(
ω
) =
A
(
ω
)
/
(
C m L
) . Here
C m
,
A
,
L
are the molar concen tration, measured absorbance and sample
7.2 Exp erimental T echniques 109
r = c / a
R = b / a
Figure 7.5: Shap e dep endence of the in trinsic linear absorption cross-section µ i .
The
randomly orien ted zinc-blende CdSe nanoparticles at 4 e V (310 nm) are modeled as general ellipsoids
with semi-axes a, b, c with a>b>c.
R
=
b
a
is the lateral asp ect ratio and
r
=
c
a
the ratio of the
shortest to longest semi-axis.
length (e.g. cuv ette con taining nano crystal solution), resp ectiv ely . It can also b e expressed b y
˜ ε ( ω ) = N A σ (1) ( ω )
1000 ln (10) (7.6)
with the A vogadro constan t
N A
and the linear absorption co efficien t of the absorb er
σ (1)
(
ω
) .
Equating the t w o expressions for ˜ ε ( ω ) yields the molar concen tration C m :
C m = A · 10 3 ln (10)
Lµ i N A V p
. (7.7)
whic h is related to the particle concen tration
C p
b y the n um b er of particles p er mole (scaling
with its v olume V p ), the molar mass M and densit y ρ of the material, in this case CdSe.
C p = C m
M C dS e
ρ C dS e V p
. (7.8)
Summarizing, TEM images yield the particle size from whic h the linear absorption cross
section in the con tin uum can b e calculated. In connection with an absorbance measuremen t
the nano crystal concen tration in solution can then b e obtained. This pro cedure has pro ven to
b e applicable to CdSe dots, ro ds and platelets of v arious sizes and confirmed b y concen tration
determination via inductiv ely coupled plasma atomic emission sp ectroscop y (ICP-AES) in
Ref. [
54
]. It is used to determine the concen tration of sample solutions in order to derive absolute
t w o-photon absorption cross sections in the course of this c hapter. Concen trations of sev eral
nano crystal samples used in z-scan exp erimen ts are giv en later together with the exp erimen tal
details on the z-scan.
110 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
7.2.2 Z-Scan
In the follo wing w e will deriv e the mo del fit function to obtain the TP A co efficient
β
of a sample
from data obtained b y an op en ap erture (O A) z-scan (figure 7.6).
β
and the cross section of
a particle
σ (2)
are prop ortional via the concen tration of TP absorb ers in the sample and the
photon energy ~ ω :
β = σ (2) C
~ ω (7.9)
F or a collimated b eam the atten uation of the photon flux
d φ/ d z
induced b y t wo-photon
absorption[186, 191] is:
dφ
dz = − ~ ω β φ 2 = − σ (2) C φ 2 (7.10)
In a z-scan exp erimen t the irradiance or photon flux
φ
(0) at the fron t face of a sample con taining
TP absorb ers is v aried as it passes through the w aist of a fo cused laser b eam, see figure 7.6. If
the sample length L is short compared to the Ra yleigh length
z 0
, a constan t b eam w aist along
the sample can b e assumed. If, further, the sample sho ws no one-photon absorption at the giv en
w a v elength (see e.g. figure 7.2 b) and w e assume negligible ground state depletion (small fraction
of absorb ers excited b y TP A) the photon flux at its exit face is giv en as[191]:
φ ( L ) = φ (0)
1 + σ (2) C Lφ (0) (7.11)
In order to deriv e a fit-function to our exp erimen tal z-scan transmittance curv es that also takes
the sec h
2
(
t/τ 0
) temp oral pulse shap e of our Ti:Sa regenerativ e amplifier system in to accoun t w e
c ho ose eq. 34 from R umi and Perry[
191
] as a starting p oin t. There the b eam profile in space and
time is giv en b y a Gaussian profile. By replacing the Gaussian temp oral profile
e − t 2 / ˜ τ 2
with
sec h 2
(
t/τ 0
) =
2
e
t
τ 0 + e − t
τ 0 2
the instan taneous photon flux at a giv en time t (within a pulse)
and radial distance r from the optical axis at the p osition z b ecomes:
φ z ( r ; t ) = φ 0 ( r = 0; t = 0) w 2
0
w 2
z
e − 2 r 2 / w 2
z sec h 2 ( t/τ 0 ) (7.12)
with the p eak photon flux φ 0 ( r = 0; t = 0) in the fo cal sp ot at z = 0 and for the time t = 0 :
φ 0 ( r = 0; t = 0) = E
E ph · 1
π w 2
0 τ 0
. (7.13)
E
is the in tegrated pulse energy ,
E ph
the energy of one-photon, w
0
the b eam radius at
z
= 0
and w
z
is the z-p osition dep enden t b eam radius. Using eqs. 7.11 and 7.12 the energy transmitted
b y a sample of length L at the p osition z o v er the whole b eam w aist and pulse duration is given
7.2 Exp erimenta l T echniques 111
Figure 7.6: Z-Scan setup
(a) A sample (e.g. a cuv ette with collo dial nano crystals in solution)
is passed through the w aist of a fo cused laser b eam along the z-axis. As a function of the position
z the b eam w aist (b) decreases, the irradiance (c) increases leading to more t w o-photon absorption
pro cesses (
∝ I 2
). The transmittance (d) is obtained b y dividing the non-linear transmission (NL T)
b y the reference signal (Ref.). It reac hes a minim um in the fo cal p osition. As the sample is mo v ed
out of the fo cus p osition the trends are re versed.
by
E z ( L )= E ph
+ ∞
Z
−∞
d t
∞
Z
0
2 πr d r φ z ( r ; t )
1+ σ (2) C Lφ z ( r ; t ) (7.14)
In tegration ov er r yields:
E z ( L )= E ph
π w 2
z
2 σ (2) CL
+ ∞
Z
−∞
d t ln 1+ q z sec h 2 t
τ 0 (7.15)
with
q z ( L )= φ 0 σ (2) CL
1+ z
z 0
(7.16)
By expanding the logar ithm in a p o w er series
ln(1 + x )= ∞
X
m =1
1
m x m ( − 1) m +1 (7.17)
112 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
and in tegrating eac h series term o ver time, the z-position dep enden t transmittance
T z
, i.e. the
ratio of the energy inciden t on the sample
E
and the energy transmitted through it
E z
is giv en
b y:
T z = E z
E =
m
X
n =0
√ π
2( n + 1)
Γ ( n + 1)
Γ ( n + 3
2 ) ·
− 2 arcsec h 1
√ 2 · σ (2) LC P av g
hω τ F W H M f π
q w 2
x + z − b
2 θ x 2 q w 2
y + z − d z − b
2 θ y 2
n
(7.18)
P av g
is the a v erage p o w er incident on the sample,
f
the rep etition rate of the laser. The
full width at half maxim um of a
sec h 2
(
t/τ 0
) profile is defined as
τ F W H M
= 2
arcsec h 1 / √ 2 τ 0
.
Due to sligh tly differen t div ergence angles of the laser b eam in
x
and
y
direction w e define w
x,y
as the b eam radius and
θ x,y
as the div ergence angle of the fo cused b eam in the (
x,z
) -plane
(subscript
x
) and (
y ,z
) -plane (subscript
y
), see fig. 7.6. The distance b et w een the t w o fo cal
p ositions is measured to b e
d z
= 0
.
3 mm, m uch smaller than the focal length of the fo cusing
lens ( 300 mm). Beam profile measuremen ts follo wing the proto col describ ed in ref.
192
obtained
w
x
= 27
µ
m; w
y
= 30
µ
m with corresp onding Ra yleigh lengths of
z r ,x
= 2
.
7 mm;
z r ,y
= 3
.
1 mm.
In our notation
z
is the z-p osition of the sample along the Gaussian b eam w aist with resp ect
to the b eam w aist at
z
=
b
.
Γ
(
n
) denotes the Gamma function. Z-scan curves are fitted with
eq. 7.18, the only free parameters are b and the TP A cross section σ (2) .
Exp erimental z-scan setup and curves
The z-scan curv es in figure 7.7 (a) w ere recorded using a Titanium:Sapphire (Ti:Sa) regenerativ e
amplifier laser system (Spitfire, Sp ectra Ph ysics) with 1 kHz rep etition rate, 1 W a verage output
p o w er at 800 nm (corresp onding to a single photon energy of 1
.
55 e V) and a sech
2
(t) temp oral
profile (90-150 fs FWHM). As sho wn in figure 7.6 the b eam is split up. One b eam passes the
fo cusing lens and the sample. The colloidal solutions are held in 1 mm fused silica cuv ettes
attac hed to a 10 cm tra vel (z) translation stage. The pulse energy transmitted through the
sample (
E T
) and the reference pulse energy (
E r ef
) from the other b eam are recorded b y t w o
energy detectors. The transmittance (
T
=
E T /E r ef
) is calculated p er pulse. This eliminates
noise caused b y pulse-to-pulse energy v ariations. A pulse-length of
τ F W H M
= 110 fs was held
constan t throughout all measuremen ts, monitored with an auto-correlator. Ev ery sample w as
measured with p o w er-densities in the fo cal plane (z=0) ranging from 50 to 170 GW/cm
2
. These
are b elo w rep orted TP A saturation p o w er-densities in I I-VI semiconductor QDs [176].
Figure 7.7 (a) sho ws z-scan curv es of 3
.
5 (17 x 11) , 4
.
5 (16 x 8) and 5
.
5 (82 x 22) CdSe nano-platelets,
sizes giv en in n um b er of monola yers (lateral extensions in nm), 25 x 3
nm
CdSe nano-ro ds and
7.2 Exp erimental T echniques 113
- 2 - 1 0 1 2
9 7 . 0
9 7 . 5
9 8 . 0
9 8 . 5
9 9 . 0
9 9 . 5
1 0 0 . 0
1 0 0 . 5
R e l a t i v e T r a n s m i t t a n c e ( % )
z ( c m )
( a ) ( b )
4 . 5 M L
3 . 5 M L
5 . 5 M L
2 5 x 3 n m
2 . 7 n m
N P L s
R o d
D o t
2 3 2 4 2 5
- 1 0
- 8
- 6
- 4
- 2
l n ( 1 - T
z
)
l n ( I
z
)
Figure 7.7: Z-Scan curves of CdSe nanoparticles
(a) Represen tativ e z-scan curves of
3
.
5 (17 x 11) , 4
.
5 (16 x 8) and 5
.
5 (82 x 22) nano-platelets, 25 x 3
nm
nano-ro ds and 2
.
7
nm
diameter
quan tum-dots. The curv es are stac k ed by 0.5 % for clarity . The solid lines represen t fits using
our TP A model. (b) ln (1
− T z
) o v er ln (
I z
) plot as prop osed in Ref.
193
. A slop e of one indicates
t w o-photon absorption.
2
.
7
nm
diameter CdSe quan tum-dots. The particle concen trations are, resp ectiv ely , 5
.
9
·
10
13
cm
− 3
,
1
.
7
·
10
14
cm
− 3
, 2
.
5
·
10
14
cm
− 3
, 1
.
3
·
10
14
cm
− 3
and 5
.
7
·
10
16
cm
− 3
. The TP A mo del, eq. 7.18,
fits the data excellen tly . In panel (b) ln (1
− T z
) is plotted o v er ln (
I z
) as prop osed in Ref.
193
. A
slop e of one indicates t w o-photon absorption. The slop es of the linear fits are: 1.07 (4.5 ML), 1.01
(3.5 ML), 1.1 (5.5 ML), 1.12 (25 x 3 nm nano-ro ds), 0.95 (2.7 nm quantum-dots). TP A saturation
w ould lead to a deviation of the linear b eha vior for higher
ln
(
I
) [
194
] of the data sho wn in
figure 7.7 (b). Since this is not the case, saturation effects can b e also b e excluded at this p oin t.
7.2.3 T w o-Photon Photoluminescence Excitation Sp ectroscop y
T wo-photon ph otoluminescence excitation sp ectroscop y (2P-PLE) is chosen to gain insigh t to the
con tributions of the electronic confinemen t to the TP A cross-section from wa v elengths resonan t
to one-photon excitonic transitions up to the 2D con tin uum (for NPLs). Based on t w o-photon
induced luminescence studies b y Xu and W ebb[
195
] com bined with a reference dy e, as shown in
Ref. [
171
], this metho d has an adv an tage to w ards the z-scan b ecause it is indep enden t of sp ectrally
v arying b eam parameters and th us v ery well suited for w a v elength dep enden t measuremen ts
of TP A cross sections. When normalized to one-photon PLE data, this metho d also tak es the
samples’ sp ectrally v arying quan tum yield into accoun t and the absolute TP A cross section is
obtained.
The 2P-PLE setup is describ ed in figure 7.8. The laser dy e LDS698 (purchased from Sirah) is
used here as reference since its TP A sp ectrum has a go o d sp ectral o v erlap with our region of
in terest. Both
σ (2)
LD S 698
and
σ (1)
LD S 698
, are w ell c haracterized by Makaro v et al.[
167
] b y the means
114 7 T w o-Photon Absorption in CdSe Nano crystals: The Influence of Dimensionality and Size
T i:Sa
240 fs k 74p5 MHz
750C960 nm
2
L 1 L 2
L 3
M 2
L 4
M 1 M 3
BBO ND
1
ND 2 P
BS
Obj
To
Powermeter
Autocorrelator
Mirror
Removable mirror
Dichroic mirror
DM
Laser
Cuvette
To
Spectrometer
Streak camera
Figure 7.8: T w o-photon excitation sp ectroscop y set up.
The fundamen tal b eam of a Ti:Sa
laser for t wo-photon excitation and the second harmonic ge nerated in a BBO crystal for one-photon
excitation are brough t to the same optical axis with a remo v able mirror (M3) and coupled in a
microscop e ob jectiv e. This objective (n umerical aperture
N.A.
=0
.
4 ) allo ws for confo cal excitat ion
and detection. Its fo cal plane coincides with the cen ter of the 1 mm optical path fused silica
cuv ettes con taining samples or a reference dye to ensure a reproducible collection volume. A b eam
splitter is used to div ert the fluorescence after one- or tw o- photon excitation to the optical axis
of a sp ectrometer (Jobin-Y v on IHR 460) with an attached
LN 2
co oled CCD. A streak camera for
time-resolv ed measuremen ts is on the sam e optical axis. This geometry allows the consecutiv e
measuremen t of time-in tegrated and time-resolved one- and t w o-photon related fluorescence of
sample and reference. The pulse energy and its temp oral width are determined online b y an
energy-meter and an auto-correlator.
of z-scan and 2P-PLE measuremen ts. W e will no w discuss the concepts needed to deriv e TP A
cross sections from 2P-PLE data.
Referencing to Dy e
The one- and t w o-photon induced fluorescence is giv en as[195]:
F (1)
b ∝ σ (1)
b η (1)
b φ (1)
b C b I 1 (7.19)
F (2)
b ∝ σ (2)
b η (2)
b φ (2)
b C b I 2
2 . (7.20)
Where
σ (1 , 2)
b
and
η (1 , 2)
b
are, resp ectiv ely , the one- or tw o-photon absorption cross section of
the analyte and its PL qu an tum yield for one- or t wo-photon absorption.
φ (1 , 2)
b
,
C b
and
I 1 , 2
are the photon collection efficiency of the setup for one- or t w o-photon related fluorescence,
analyte concentration and the inciden t a verage one- or t w o-photon excit ation in tensit y . The
index
b
denotes the sample or the ref erence dy e (index ’
x
’ or ’
r ef
’ in the follo wing). A quadra tic
excitation in tensit y dep endence of the fluorescence indicates TP excited emission – as can b e
seen in figure 7.9.
If w e assume that[171]:
(1) the quan tum y ield of sample or reference is iden ti cal after one- and t w o-photon a bsorption,
η (1)
b = η (2)
b and
(2) the photon collecti on efficiency of the setup
φ
is equal for sample and ref erence after one-
or t w o-photon absorption
7.2 Exp erimental T echniques 115
1 0
3
1 0
4
0 1 2
1 0
4
0 2 0 4 0 6 0
( c )
Q D
3 . 5 M L
I n t e g r a t e d T P A - P L ( a . u . )
I r r a d i a n c e ( G W / c m ² )
3 . 2 n m 4 . 5 M L
N P L s
( a ) ( b )
2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 3 . 2
P L E m i s s i o n ( a . u . )
E n e r g y ( e V )
1 P A @ 4 0 0 n m
1
= 4 0 . 5 p s
2
= 2 8 8 p s
D a t a
F i t
T P A @ 8 0 0 n m
T i m e ( n s )
1
= 4 1 . 5 p s
2
= 3 1 5 p s
3 . 5 M L ( 1 7 x 1 1 n m
2
)
2 . 5 2 . 6 2 . 7 2 . 8
0 . 0
0 . 5
1 . 0
E n e r g y ( e V )
1 P A
T P A
N o r m . P L I n t e n s i t y
3 . 5 M L ( 1 7 x 1 1 n m
2
)
E x
=
7 5 0 n m
7 6 6 n m
7 8 0 n m
8 0 0 n m
8 8 0 n m
Figure 7.9: One- and t w o-photon excited photoluminescence.
(a) PL emission sp ectra
with Loren tzian fits (dashed lines) of 3
.
2 nm CdSe quan tum-dots and 4
.
5 (27x23) and 3
.
5 (17x11)
CdSe NPLs excited at 800 nm. The in tegrated t wo-photon induced PL emission (solid dots) is
prop ortional to the excitation densit y . The solid lines are the b est fit
I T P A − P L
=
A · I 2
exc
where
A is a constan t and I the laser in tensity . (b) Time-resolv ed PL deca ys of 3
.
5 (17x11) CdSe NPLs,
recorded with a Hamamatsu, C5680 streak camera after one and t wo-photon excitation. The deca y
times
τ 1
and
τ 2
are deriv ed from bi-exp onen tial fits (gra y curves) taking the instrumen t resp onse
in to accoun t. (c) Normalized PL sp ectra of the same sample as in (b) after one and t w o-photon
excitation at differen t w av elengths. All PL spectra coincide in the tested range of 375-440 nm (one
photon) and 750-880 nm (t w o-photon) excitation w av elengths.
w e can express equation 7.19 and 7.20 for sample and reference dye and solv e the absolute TP A
cross section of the sample:
σ (2)
x = σ (2)
r ef σ (1)
x
σ (1)
r ef
D F (2)
x ( t ) E D F (1)
r ef ( t ) E
D F (2)
r ef ( t ) E D F (1)
x ( t ) E (7.21)
where
D F (1 , 2)
x,r ef ( t ) E
is the time a v eraged detected flux of the sample or reference-dye after one-
or t w o-photon excitation.
Since w e measure the sample and dy e under iden tical conditions, criterion (2) is directly met.
Therefore w e ha v e to discuss criterion (1) in more detail.
The time-resolv ed PL deca ys in figure 7.9 b as w ell as the normalized PL sp ectra after one-
and t w o-photon excitation (c) are practically indistinguishable. W e conclude that the one- and
t w o-photon related fluorescence of our samples originates from the same lo west emitting state.
F rom the PL decay in figure 7.9 b w e can further assume equal quan tum yield after one- and
t w o-photon absorption at a giv en wa v elength. The relaxation to the band edge o ccurs to the
same finally emitting states hence ha ving the same PL lifetime and quan tum yield. Criterion (1)
is th us also met and w e can use eq. 7.21 to calculate the TP A cross sections of our samples using
the kno wn cross sections[167] of the reference dy e LDS698.
116 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
The linear absorption cross section
σ (1)
x
and the concen tration of our nanoparticles is determined
from the Loren tz Lo cal field theory (see. section 7.2.1). The fluorescence
F (1 , 2)
x
of the samples
is giv en b y the area of Loren tzian fits to their PL emission, dashed lines in Figure 7.9 (a). The
broad PL of the reference dy e LDS698 cen tered at ∼ 698 nm is obtained by in tegrating the PL
p eak after bac kground subtraction.
With the presen ted dy e-referenced 2P-PLE metho d it is sufficien t to kno w the linear absorption
cross section of the sample and reference-dy e
σ (1)
x,r ef
and the TP A cross section of the reference-
dy e
σ (2)
r ef
and measure their one- and t w o-photon related fluorescence to derive
σ (2)
x
. All other
quan tities can b e eliminated.
Self-referencing
No w w e replace the 1P A and TP A fluorescence and absorption cross sections of the reference
dy e in eq 7.21 with the ones of the sample at a sp ecific, fixed wa v elength
λ a
;
D F (1 , 2)
x ( t,λ a ) E
,
σ (1)
x
(
λ a
) ,
σ (2)
x
(
λ a
) . The TP A cross section at another w av elength
λ b
can b e determined once
σ (2)
x ( λ a ) is kno wn, e.g. from z-scan or one dy e-referenced v alue:
σ (2)
x ( λ b ) = σ (2)
x ( λ a ) · σ (1)
x ( λ b )
σ (1)
x ( λ a ) D F (2)
x ( t,λ b ) E D F (1)
x ( t,λ a ) E
D F (2)
x ( t,λ a ) E D F (1)
x ( t,λ b ) E (7.22)
This has quite an adv an tage tow ards a reference dy e based metho d described ab o ve. It do es
not require a fully , sp ectrally pre-c haracterized reference sample that has to b e measured at
ev ery w a velength. It simply requires the knowledge of the sample’s TP A cross section at one
w a v elength (e.g. from z-scan).
7.2.4 Sp ectral Pulse Width in TP A Exp eriments
When p erforming t w o-photon related measuremen ts – z-scan or 2P-PLE – the exp erimen tally
accessible quan tit y is the pro duct of the TP A co efficien t
β
or cross section
σ (2)
and pulse p eak
in tensit y
I
squared:
σ (2) · I 2
. The fact that the pulse energy is distributed in time via its
temp oral profile has to b e alw a ys tak en in to accoun t, for example b y integrating o v er time in
eq. 7.14. Less ob vious is the distribution of the pulse energy ov er its sp ectral width as illustrated
in figure 7.10 (a).
A sp ectro-temp oral in tensit y profile
I
(
t,λ
) can b e defined with p eak in tensit y in time at
t
= 0
and its sp ectral p eak cen tered at
λ 0
. The commonly used quantit y is the temp oral peak intensit y
I 0 ,λ = R I (0 ,λ ) d λ .
Z-scan measuremen ts are p erformed with a
sec h 2
(
t
) temp oral profile, 2P-PLE measuremen ts
with a Gaussian. Since
sec h 2
and Gaussian are in v ariant under F ourier transformation their
7.2 Exp erimental T echniques 117
- 2 0 - 1 0 0 1 0 2 0
0 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5
1 . 3 1 . 4 1 . 5 1 . 6 1 . 7
1 0
2
1 0
3
1 0
4
( b )
T i m e
I
P L E , 0 , 0
W a v e l e n g t h ( n m )
R e l a t i v e s p e c t r a l i n t e n s i t y ( a . u . )
G a u s s ( M i r a , 2 P - P L E )
S e c h ² ( S p i t f i r e , Z - S c a n )
1 2 n m
7 n m
I
z - s c a n , 0 , 0
F W H M :
A r e a = 1
W a v e l e n g t h
( a ) ( c )
u n c o r r e c t e d
E n e r g y ( e V )
( 2 )
/ V ( G M / n m ³ )
4 . 5 M L ( 2 5 x 2 0 n m
2
) N P L s
z - s c a n
c o r r e c t e d f o r
s p e c t r a l w i d t h s
Figure 7.10: The influence of sp ectral widths
(a) Sc heme of the in tensity distribution of
a laser pulse in energy/w av elength and time. (b) Sp ectral profiles of the pulses used for z-scan
(sec h
2
(
t
) ,
F W H M
= 12 nm, blac k) and 2P-PLE (Gaussian,
F W H M
= 7 nm, red) plotted with the
same area, i.e. sp ectrally in tegrated intensit y , with the corresp onding p eak in tensities
I z − scan, 0 , 0
and
I P LE , 0 , 0
. (c) V olume-normalized TP A cross sections deriv ed from 2P-PLE and z-scan (blac k
dot) of 4.5 ML and 25x20 nm
2
lateral extension NPLs. Open circles represent TP A cross sections
deriv ed directly from the dy e based 2P-PLE metho d using eq 7.21, solid red circles after correcting
for the differen t sp ectral laser widths used in z-scan and 2P-PLE exp erimen t with the factor of
1 . 81 2 from eq. 7.25. The curves serv e as guide to the ey e.
sp ectral profiles are again
sec h 2
and Gaussian. The differen t sp ectral profiles and widths of the
pulses used for z-scan (
sec h 2
,
λ F W H M
= 12 nm) and 2P-PLE (Gaussian,
λ F W H M
= 7 nm) lead
to a differen t sp ectral distribution of the pulse energy . The t w o sp ectral profiles
I 0 ,λ
are plotted
with the same pulse energy i.e. area in figure 7.10 (b). This reveals that the t w o pulse p eak
in tensities in time and wa v elength I P LE , 0 , 0 and I z − scan, 0 , 0 differ b y a factor of 1 . 81 .
Since
I 0 , 0
is the more relev an t quantit y for the TP A signal in theory , as referring to the
mo deling in eq 7.18 and in ref.
191
, the exp erimen tally accessible quan tit y
σ (2) · I 2
0 , 0
should b e
constan t for b oth tec hniques. In this appro ximation the obtained
σ (2)
v alues will differ b y a
factor of 1 . 81 2 accordingly to:
σ (2)
P LE · I 2
P LE , 0 , 0 = σ (2)
z − scan · I 2
z − scan, 0 , 0 (7.23)
I P LE , 0 , 0 = 1 . 81 · I z − scan, 0 , 0 (7.24)
σ (2)
z − scan = σ (2)
P LE · 1 . 81 2 (7.25)
Because of the go o d agreemen t in our z-scan measuremen t of CdS bulk
β 0
with ref.
176
(see
App endix 7.4), we normalize our 2P-PLE data to the z-scan v alues, b y multiplying the TP A cross
sections obtained from the reference-dy e based metho d b y eq. 7.21 with the correction factor
of 1
.
81
2
. This leads to excellent agreemen t betw een corrected TP A cross sections of 2P-PLE
(solid dots in figure 7.10 (c)) and the z-scan v alue measured at 800 nm (blac k dot) – esp ecially
considering the published error margin giv en for the TP A cross sections of LDS698[
167
]. All the
118 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
dy e-deriv ed 2P-PLE results sho wn in the following (figures 7.12 to 7.15) ha v e b een corrected
with resp ect to pulse width effects. All measured TP A cross sections of z-scan and 2P-PLE and
nano crystal sizes are listed in T able 7.1 in the App endix.
It m ust b e noted that it is arbitrary to whic h sp ectral pulse width the v alues are corrected
and this ma y b e the reason for differences in rep orted TP A cross sections, for example for
3
.
2 nm CdSe QDs at 800 nm Nyk et al.[
179
] ( 20
·
10
3
GM), Pu et al.[
196
] ( 3
.
0
·
10
3
GM) or here
( 3 . 7 · 10 3 GM).
7.3 Size and Shap e Dep endence of T w o-Photon Abso rption
7.3.1 T w o-Photon Abso rption Sp ectra of CdSe Nanoplatelets
The TP A sp ectra of 3
.
5 and 4
.
5 monola y er NPLs of different aspect ratios are shown in
figures 7.11 (a) and (c). F or comparison, the corresp onding linear absorption sp ectra are plotted
o v er the one-photon energy in 7.11 (b) and (d).
Since it do es not alter the sp ectral course, w e can plot the v olume-normalized (VN) TP A cross
section (in GM/nm
3
) rather than the absolute particle cross section. Being the more in trinsic
v alue the former allows to study the influence of shape and geometry on the TP A absorption
directly . The exp erimen tal error of the TP A cross sections is estimated to b e 20 %. This t ypical
uncertain t y for TP A cross sections is mainly related to the TEM-based sizing. It results only
in a c hange in the absolute cross-sections and do es not alter the relativ e sp ectral course of the
measured TP A sp ectra. The op en circles in figures 7.11 (a) and (c) hav e b een calculated via
the self referencing (SR) metho d b y normalizing to the NPLs’ TP A cross sections at 800 nm
obtained from z-scan. Owing to the lac k of z-scan data, the sp ectral course for 3
.
5 (77 x 22) NPLs
is related to a dy e-deriv ed v alue (using LDS698 at 880 nm, larger blac k dot).
Comparing the TP A sp ectra determined by reference-dy e at eac h w a v elength (figure 7.11a,
solid dots) with the self-referenced sp ectra (op en circles) sho ws that the sp ectral course is barely
altered b y the SR metho d. Due to the correction to sp ectral laser widths it is indep enden t from
whic h tec hnique the kno wn TP A cross section needed for self-referencing is deriv ed from. Mainly
b ecause reference-dy e obtained v alues are in go o d agreemen t to z-scan results at 800 nm.
The ma jor observ ations in the TP A spectra of 3
.
5 ML NPLs in figure 7.11 (a) are: an almost
step wise increase of o v er three orders of magnitude, no pronounced sp ectral features and a v ery
w eak first hea vy hole excitonic resonance. Strong con tin uum TP A and a w eak first excitonic
resonance ha v e also b een rep orted for CdSe QDs [
179
,
182
], CdS NRs and QDs[
171
] as w ell as
for CdSe/CdS dots in ro ds[185].
7.3 Size and Shap e Dep endence of T w o-Photon Abso rption 119
1 . 3 1 . 4 1 . 5 1 . 6
1 0
0
1 0
1
1 0
2
1 0
3
1 0
4
1 . 2 1 . 3 1 . 4 1 . 5 1 . 6
1 0
2
1 0
3
7 7 x 2 0 n m ²
S R t o
( 2 )
2 P - P L E
( 8 8 0 n m )
E n e r g y ( e V )
E n e r g y ( e V )
( 2 )
/ V ( G M / n m ³ )
3 . 5 M L C d S e N P L s
( a )
1 . 6 x 1 0
7
G M
1 7 x 1 1 n m ²
S R t o
( 2 )
z
( 8 0 0 n m )
2 . 8 x 1 0
5
G M
2 . 6 2 . 8 3 . 0 3 . 2
A b s o r b a n c e ( a . u . )
( b )
3 2 x 8 n m ²
1 6 x 8 n m ²
2 5 x 2 0 n m ²
S R t o
( 2 )
z
( )
( d )
E n e r g y ( e V )
E n e r g y ( e V )
( 2 )
/ V ( G M / n m ³ )
4 . 5 M L C d S e N P L s
( c )
1 1 . 2 x 1 0
5
G M
3 . 4 x 1 0
5
G M
2 . 0 x 1 0
5
G M
2 . 4 2 . 6 2 . 8 3 . 0 3 . 2
A b s o r b a n c e ( a . u . )
Figure 7.11: One and t w o-photon absorption sp ectra of 3.5 and 4.5 monola y er CdSe
nanoplatelets.
V olume-normalized TP A cross sections (a,c) and linear absorption (b,d) of 3
.
5
and 4
.
5 ML CdSe NPLs. Solid sym b ols denote v alues obtained from referencing to
σ (2)
LD S 698
at eac h
w a velength. The op en circles sho w the self-referenced (SR) sp ectral TP A course pinned to the
dy e-deriv ed v alue
σ (2)
2 P − P LE
at 880 nm (large blac k dot in (a)) for 3
.
5 (77 x 20) and to the z-scan
v alues
σ (2)
z
at 800 nm for 3
.
5 (17 x 11) (large blue dot in (a)) and 4
.
5 (25 x 20) (large blac k dot in
(c)). T w o-photon excitation energies b elo w 1
.
3 e V are outside the tuning range of the setup. The
absolute TP A cross sections measured with 2P-PLE at 800 nm are giv en in panels (a) and (c). The
curv es in (a) and (c) serve as guides to the ey e.
The NPL’s t w o-photon absorption increases extremely from TP energies resonan t to the first
excitonic transition to the con tin uum absorption, note the logarithmic scale. The stepwise
c haracter o ccurs b ecause of their step-lik e 2D densit y of states leading to 3 steps for the HH, LH
and SO con tin ua. They are shifted to higher energies with resp ect to the first HH, LH and SO
exciton transitions b y the exciton binding energies, estimated to the order of 200 me V[
17
,
22
,
24
].
Esp ecially the steps related to the HH and LH con tin uum transitions can b e seen in figure 7.11 (a).
The observ ed increase of the transition strength ma y b e explained b y a confinemen t related
larger densit y of states (DOS) and a w eakening of selection rules in the con tin uum. Ho wev er,
sophisticated theoretical mo deling of the TP A sp ectra, b ey ond the scop e of this w ork, will b e
necessary for a definite attribution of this effect. The sp ectral course suggests the step-lik e 2D
DOS of the hea vy- and ligh t-hole, split-off band con tinuum to result in the largest con tributions.
F urther, at higher photon energies more t w o-photon transition paths contain in termediate states
120 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
near the photon energy . As discussed in sec. 7.1 and figure 7.3 this enhances the t w o-photon
transition probabilit y and th us the cross section at higher energies.
Since w e ha v e evidence for this increase from t wo independent methods – referencing to a dye
of kno wn TP A cross section per wa v elength and the self-referencing metho d related to z-scan
results – this strong effect cannot b e an y kind of measuremen t artifact.
TP A sp ectra of 4
.
5 ML NPLs are sho wn in figure 7.11 (c). This plot spans o v er less orders of
magnitude b ecause a resonan t excitation to the first excitonic transition at 512 nm ( 2
.
42 e V),
1024 nm t w o-photon w av elength, where TP A cross sections are smaller, is not p ossible with the
giv en tuning range of our setup.
W e see a volume scaling and a shap e dep endence for b oth 3
.
5 and 4
.
5 ML NPLs; the smaller
the asp ect ratio (thic kness to a v erage length) the larger the volume-normalized TP A in the
con tin uum. In other w ords; particles with larger lateral extensions of the same thic kness ha ve
larger TP A cross sections p er v olume or unit cell. These effects will b e in v estigated in the
follo wing section together with the z-scan results.
7.3.2 V olume Scaling of the TP A Cross Section
There is still discussion whether TP A cross sections of I I-VI materials scale with the particle
v olume, esp ecially with resp ect to the strong and in termediate confinemen t regime. Some authors
rep ort v olume scaling [
178
–
181
,
185
,
193
,
196
–
198
] others not [
199
]. F urther the p o w er of the
v olume scaling is still under discussion.
The TP A cross sections of all samples measured at 800 nm ( 1
.
55 e V) with z-scan and 2P-PLE
(corrected for pulse width effects) are plotted o v er the particle volume in figure 7.12. T o in terpret
the results in a greater con text exp erimen tal results of CdSe QDs from Pu et al.[
196
] and
theoretical results of F eng and Ji[200] are added.
The orange colored slop e obtained from a p o w er-la w fit (
σ (2)
=
A · V n
) to all data p oin ts
rev eals a sup er-linear v olume scaling (
n
= 1
.
93 ), which ma y b e attributed to a confinemen t
related increase of transition dip ole momen ts, a larger 2D densit y of states (DOS), band-mixing
or a w eak ening of selection rules. Esp ecially for NPLs and NRs w e observe a clear deviation
from a linear (bulk-lik e) v olume scaling (dotted slop e,
n
= 1 ). Man y authors ha ve reported
bulk-lik e v olume scaling (
σ (2) ∝ V 1
) of the TP A cross-sections[
178
–
181
,
185
,
193
,
196
–
198
] in
CdSe QDs and CdS QDs and NRs. This migh t b e related to the smaller degree of confinemen t
in the systems studied there. The exciton Bohr-radii of CdS and CdSe are (
a C dS
B
= 2
.
9 nm,
a h − C dS e
B
= 5
.
2 nm [
33
]). Previously in v estigated CdS ro ds are w eakly confined with diameters of
4.4 nm[
178
] and 5 nm[
171
], i.e. greater than the CdS Bohr radius. In con trast the CdSe ro ds
7.3 Size and Shap e Dep endence of T w o-Photon Abso rption 121
1 0
1
1 0
2
1 0
3
1 0
4
1 0
2
1 0
4
1 0
6
1 0
8
4 . 5 ( 2 5 x 2 0 )
3 . 5 ( 5 x 5 )
N P L s
N R s
Q D s
Q D s e x p . ( P u e t a l . )
C u b o i d s t h e o . ( F e n g )
( 2 )
( G M )
V o l u m e ( n m ³ )
3 . 5 ( 7 7 x 2 0 )
5 . 5 ( 8 2 x 2 2 )
~ V
1
~ V
1 . 9 3
Figure 7.12: V olume scaling of t w o-photon absorption cross section.
TP A cross-sections
of CdSe nano-platelets, -ro ds and -dots at 800 nm vs. particle-volume. NPLs (dark blue: 2P-PLE,
pale blue: z-scan), NRs (oliv e) and QDs (wine color: 2P-PLE, red: z-scan) represent experimental
results. Grey dots are experimental results from Pu et al.[
196
] and crossed dots theoretical results
from F eng and Ji[
200
] for CdSe cub oids forming dots or ro ds, dep ending on their asp ect-ratio.
Sev eral CdSe NPL sizes are lab eled with n um b er of monola yers (lateral extensions in nm). The
dash-dotted orange slop e represen ts a p o w er-la w fit (
σ (2)
=
A · V n
) to all data p oin ts (
n
= 1
.
93 ).
F or comparison a linear v olume scaling slop e ( n = 1 ) is inserted.
studied here (
d
= 4
.
1 and 2
.
9 nm) ha v e diameters smaller than the CdSe Bohr radius. (see also
figure 7.14 a, later)
Our QDs do not ha v e an ultra strong confined direction like the platelets, but are rather in
the strong to in termediate isotropic confinemen t regime. The absence of anisotropic confinemen t
ma y lead to the fact that CdSe dots sho w a more linear v olume scaling while the ultra strong and
anisotropically confined CdSe platelets and strongly confined ro ds sho w a sup er-linear v olume
dep endence of their TP A cross-section.
Ho w ev er, for particles with 2 to 3 nm diameter it is difficult to determine the exact size b y
TEM (as tak en from scaling curv e of Ref.
201
). It is and has b een therefore difficult to attribute
a clear v olume scaling to these small CdSe (and CdS) QDs.
CdSe NPLs, on the other hand, represent nano-particles with an ultra strong confined electronic
system (1D confinemen t do wn to b elo w 1
.
5 nm) and a v ailable particle sizes ranging ov er three
orders of magnitude in v olume. This allo ws for the first time an in vestigation of the v olume
dep endence of TP A cross sections in confined electronic systems o ver sev eral orders of magnitude.
This go es w ell b ey ond the in v estigated v olume range in publications having observ ed a linear
increase, ho w ev er, with particles that were not in suc h an ultra strong confinemen t regime.
NPLs reac h TP A cross sections of up to 5
·
10
7
GM for 5
.
5 ML (82 x 22) . T o our knowledge the
largest ev er rep orted for colloidal semiconductor nano crystals. This record v alue is confirmed b y
t w o indep enden t metho ds.
122 7 T w o-Photon Abso rption in CdSe Nano crystals: The Influence of Dimensionalit y and Size
Also when considering the action cross section (
η σ (2)
) p er atom this still holds. Other materials
suc h as gold clusters exhibit TP A cross sections of up to 17
·
10
3
GM p er atom[
173
]. Ho w ev er,
the v ery lo w quan tum yield (
η ∼
10
− 5
to 10
− 6
%) leads to TP A action cross sections p er atom of
η · σ (2)
per atom
= 1
.
7
·
10
− 3
GM/atom. With 45 GM/atom our CdSe NPLs exceed this v alue b y o v er
four orders of magnitude. (Calculated for 5
.
5 (82 x 22) assuming a quan tum yield of
≈
15 % at
800 nm as seen in App endix 7.4.) F or our QDs and NRs w e measure ab out 1
.
1 and 2
.
5 GM/atom.
7.3.3 Intrinsic TP A Cross Section: Electronic and Dielectric Contributions
In order to study the influence of the nanoparticles’ shap e, the v olume-normalized (in trinsic)
TP A cross section (in cm s photon
− 1
) is plotted o v er the asp ect ratio of the particles in figure 7.13.
It features a strong U-shap ed asp ect ratio dependence with a minimum for spher ical particles,
i.e. QDs (
AR
= 1 ). Nanoro ds (
AR ≥
1 ) exhibit larger and NPLs (
AR ≤
1 ) b y far the largest
v olume normalized TP A cross sections. Exp erimen tal data of Pu et al.[
196
] for CdSe QDs
additionally confirm the existence of a minim um at
AR
= 1 . The agreemen t for our CdSe QDs
with the results of Pu et al.[
196
] is go o d, esp ecially when considering the follo wing: the error
in determining the particle v olume has significan t influence on the TP A cross sections of QDs
(
V QD ∝ d 3
), the influence of the sp ectral widths of the laser pulses used in TP A exp erimen ts is
discussed ab o v e and can lead to a v ariation of cross section b y a factor of
∼
5 . F urthermore,
concen tration determination metho d used here differs from Pu et al.[
196
]. In addition, imp erfect
spheres – i.e. cub oids and spheroids – will hav e higher v olume-normalized TP A cross sections,
as calculations of F eng et al.[200] and ours indicate.
TP A lo cal field theory is used to obtain the v olume normalized cross section. A dapted from
Blan ton[96] and A c htstein et al.[170] it assumes a bulk lik e TP A co efficien t.
σ (2)
lf /V = 2 π ~ ω β bul k
3
X
i =1
1
3 | f ( ω ) i | 4 (7.26)
Where
~ ω
is the energy of the exciting (single) photons (800 nm / 1.55 e V). The lo cal field
factors
f i
(
ω
) for the nanoparticle axes
i
=
x,y ,z
are calculated as in eq. 1.24 in section 1.2. The
curv es in figure 7.13 sho w the results of this mo del using
β bulk
C dS e
= 10
.
1 cm/GW for zincblende
CdSe[
57
,
177
]. The lo cal field related alteration of the in trinsic TP A cross sections is displa y ed
for differen tly anisotropic particles.
L y /L x
= 1 represen ts p erfect spheres and NPLs with equal
lateral extensions.
L y /L x <
1 represen ts NPLs with differen t lateral extensions, imp erfect
spheres for whic h the asp ect ratio
AR
=
l z / p l x · l y
equals unit y or ro ds with a non-circular
cross-section.
7.3 Size and Shap e Dep endence of T w o-Photon Abso rption 123
Electronic effects Dielectric effects
Figure 7.13: Shap e dep endence of the v olume normalized t w o-photon absorption cross
section.
TP A cross-sections of CdSe nano-platelets, -ro ds and -dots at 800 nm normalized to
the particle v olume and plotted o ver their aspect ratio. Green (2P-PLE) and blue (z-scan) dots
sho w exp erimen tal results. Grey dotted curves are results from TP A lo cal field theory for zinc
blende CdSe for differen tly anisotropic particles.
L y /L x
= 1 represen ts a p erfect sphere or platelets
with iden tical lateral extensions and
L y /L x <
1 e.g. NPLs of non-identical lateral extensions.
Exp erimen tal results from Pu et al.[
196
] (grey dots) and theoretical results of F eng and Ji[
200
] for
CdSe cub oids (crossed dots) are added.
The mo del predicts a factor of
≈
2 b et w een the v olume normalized TP A cross sections of
the extreme cases of v ery flat NPLs (small AR) and v ery long NRs (large AR). The mo del also
sho ws that the TP A cross section of a CdSe nanoparticle increases with its asp ect ratio, either in
the form of a ro d or a platelet. Ho w ev er, it cannot entirely explain the high TP A cross sections
of platelets in the con tin uum absorption region.
W e observe an increasing difference betw een the mo del and experimental results, especially
for v ery flat NPLs. While our simple bulk-lik e TP A lo cal field model predicts a flattening out of
the v olume-normalized TP A cross-section for long ro ds and NPLs of large extent, the measured
in trinsic TP A cross-section of NPLs surp asses the mo dels’ v alues. Small asp ect ratios can only
b e ac hiev ed b y larger NPL areas. Due to the v ertical size quan tization in in teger monola yers,
NPLs of smaller AR will generally ha v e larger v olumes – same thickness but larger lateral exten t.
Th us, the observ ed deviation from the lo cal field mo del is partly an effect of the sup er-linear
v olume scaling discussed ab o v e, since an y scaling
V n
with
n >
1 will not result in a flat line in
the v olume-normalized plot of figure 7.13.
The observ ed enhancemen t of the v olume-normalized TP A cross section ma y b e ascrib ed to
the high 2D densit y of states and the ultra strong anisotropic confinemen t resulting in a c hange
of the transition dip ole momen ts, a confinemen t induced c hange of selection rules or band mixing.
[Document text truncated for crawler view.]
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