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MICROCAVITY POLARITONS IN QUANTUM DOT LATTICES

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ERIC MATTHIAS KESSLER

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6/07 p:/ClRC/DateDue.indd-p.1

MICROCAVITY POLARITONS IN QUANTUM DOT
LATTICES

By

Eric hila'ttihias Kessler

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

2007

ABSTRACT

MICROCAVITY POLARITONS IN QUANTUM DOT
LATTICES

By

Eric Matthias Kcssler

Exciton-polaritons are mixed modes resulting from the strong coupling of electron-
hole pairs in a semiconductor (excitons) and photons. In this thesis, the exciton-
polariton modes of a quantum dot lattice embedded in a planar optical microcavity
are studiedDue to the symmetry mismatch of the exciton state in the discrete lattice
with the continuous two dimensional photon modes, each exciton mode couples to
many cavity field modes. These additional coupling terms do not conserve momentum
and are called “umklapp” terms.

We provide a complete derivation of the system's Hamiltonian, which is subse-
quently investigated both by analytical and numerical methods. we focus our anal-
ysis on novel polaritons appearing at the edge of the Brillouin zone in the reciprocal
space. The large in-plane momentum of these polaritons can give rise to a total
internal reflection, which is expected to greatly enhance their spontaneous emission
lifetime. We show that at certain symmetry points in the Brillouin zone both of a
square and a hexagonal QD lattice this new kind of polaritons can be considered as
nearly bosonic quasiparticles with an exceptionally small, isotropic mass of the order
of 10—8 electron masses (7710). much smaller than both the exciton mass (~ 0.5 mg)
and the cavity photon mass (~ 10"5 mo). Polaritons with positive as well as negative
masses are found. The large lifetime and the extremely small mass suggest interesting

possibilities for the observation of polariton condensation effects.

ACKNOWLEDGMENTS

I would like to thank my advisor Professor Carlo Piermarocchi for his great support

in the past. year.

iii

TABLE OF CONTENTS

LIST OF TABLES ..............................
LIST OF FIGURES .............................
Introduction .................................
1 Core Concepts ..............................
1.1 Wannier-h'lott Exciton Theory ......................
1.2 Polariton Theory .............................
1.3 Confined Excitons .............................

2 The Model .................................
2.1 The Hamiltonian .............................
2.1.1 The Exciton Hamiltonian ....................

2.1.2 The Photon Hamiltonian .....................

2.1.3 Exciton-Photon Interaction ...................

2.2 Analysis of the full Hamiltonian .....................
2.2.1 Simplifications and Approximations ...............

2.2.2 The Exciton at Resonance and the Strong Coupling Regime

3 Numerical Calculation ..........................
3.1 Preliminary Discussion ..........................
3.2 Square Lattice: The X—Point ........................
3.3 Square Lattice: The M—Point ........................
3.4 Hexagonal Lattice: The VV—Point ....................
3.5 Dark Polariton States ...........................

Conclusion ..................................
APPENDICES ................................
A Derivation of the Effective Mass Equation in Bulk .........
B Evaluation of the Interaction Matrix Element ............

BIBLIOGRAPHY ..............................

vi

3.1

3.2

3.3

LIST OF TABLES

The Rabi splitting Q depends on the relative size of the wave vector (I
at the crossing point and the number of intersecting modes. .....

The effective masses for the upper two polariton modes at X. The
masses are expressed in units of the photon effective mass in a ideal
cavity of length L = 0.171 pm (mph :2 2.52 - 10-57710) ..........

The effective masses for the upper (UP) and the lower polariton (LP) at
Ml and M2. The masses are expressed in units of the photon effective

mass in a ideal cavity of length L = 0.171 ,um (mph = 2.52 . 10"5 m0).

\f

46

48

1.1

1.2

1.3

1.4

1.5

2.1

2.2

2.3

2.4

2.5

3.1

3.2

LIST OF FIGURES

The conduction and valence band of the semiconductor with a direct
bandgap of size A. ............................ 7

In the two particle picture the exciton states are found to be in the
gap between zero energy and the uncorrelated electron-hole energies
(shaded area). E1(q) is the exciton energy of the first excited exciton
state given in Eq. (1.21). ........................ 11

The Polariton dispersion. The dashed lines are the exciton and photon
dispersions without interaction. Note that this sketch is not to scale. 16

The exciton and photon shares, ull‘ and vi of the upper polariton mode 16

In zeroth order the bandstructure varies spatially like a step function
at the edge of two sen'iiconductors ................... 17

N identical quantum dots of disc-like shape in an ideal periodic array
embedded in a planar microcavity. ................... 21

The quantum dots we are considering have a cylindrical, disc-like
shape. The thickness of the dot is labelled by L; and the radius by Ito. 22

The in-plane dispersion modes (11:12.3) of a microcavity. The dashed
line is the two dimensional dispersion of a free photon. ........ 27

In the quantum well case there is a one to one correspondence of states
in the exciton—photon interaction (a). whereas the quantum dot exciton
couples to a continuous photon bath (b). The QD lattice presents an
intermediate case, where the exciton couples to a infinite but discrete
set of photon modes (c). In the latter two cases the coupling for large
q is suppressed by the form factor ){(q). ................ 33

The first four photon modes in the reduced zone scheme. ....... 34

The reciprocal lattice:(a) of a square lattice with lattice constant a
and (b) of a hexagonal lattice with constant. a. As usual F denotes the
origin q = O ................................. 44

The function f (,13, L). The light regions correspond to higher values. . 45

v i

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

Plot of the derivative of f (R0/ \/2, L) with respect to L for a fixed
R0 = 50 nm. Notice that the coupling is maximal for a value of
L = 0.171 pm ................................

The polariton modes (1)-(3) at the X—point along F7 (a) and along
W (b). The upper polariton (1) has a local minimum and can thus
be considered as a quasiparticle with a positive effective mass. The
dashed lines represent the unperturbed modes ..............

The upper polariton dispersion at the X-point. Because of the special
7r-rotational symmetry of this point the dispersion has a valley at the
edge of the Brillouin zone. ........................

2
The exciton component of polariton mode (1) lag)(q)| . Only in the
direct vicinity of qO the upper polariton becomes excitonic. .....

2
The exciton component of mode (2) |a£2)(q)' . Remarkably this mode

is purely photonic on the edge of the Brillouin zone ...........

(3) ’2

ox (q) . At resonance this mode

 

The exciton component of mode (3)
is half excitonic half photonic. ......................

At the M point we distinguish two cases. Ml labels the crossing, where
the closest photon modes intersect (black dots). At the M2 crossing the
second closest photon modes intersect (crosses). The different modes
fulfill le’ + qOI = (11/2 for the two cases, respectively. .........

The polariton modes at the lowest crossing point at M (MI). The
modes are displayed along the path W ................

The polariton modes at the lowest crossing point at M. The modes are
displayed along a path parallel to the qI-axis. Due to the symmetry
we find the same energy dispersions along the qy-axis .........

The upper polariton mode at MI. The energy dispersion is symmetric
in qx-and qy-direction. ..........................

2
The exciton component of mode (1) at. Ml lagsl)(q)| . At resonance
this mode is half excitonic half photonic. An almost identical figure is
found in the case of the lower polariton (5). ..............

vii

45

47

48

49

50

51

52

53

53

54

3.14

3.15

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

At M2 eight photon modes are crossing. The resulting nine polariton
modes are displayed along the qx-direction. Note that along this path
the unperturbed photon modes can be grouped in degenerate pairs. . 56

The highest polariton mode at M2. The large number of interacting
photon modes provides a high symmetry of the dispersion. ...... 57

The exciton component of the highest mode at M2 Iax(q)|2. At res-
onance this mode is half excitonic half photonic. We find an almost.

identical figure in the case of the LP. .................. 57
The polariton modes at the W—point of the hexagonal lattice along the

x-direction (a) and the y-direction (b) are displayed ........... 58
The highest polariton mode at the W—point of the hexagonal lattice. . 59

The exciton component of the highest mode at W in the hexagonal
lattice |a1(q)|2 . At resonance this mode is half excitonic half photonic. 60

The lowest polariton mode at the W—point of the hexagonal lattice.
Note that it is rotated by 7r with regard to the upper polariton mode. 60

The exciton component of the lowest mode at W in the hexagonal
lattice Iax(q)|2. It is rotated by 7r with regard to the excitonic part of
the upper mode ............................... 61

The dispersion for external photons for several arbitrary values of the
(continuous) parameter kg. The dispersion for k; = 0 imposes a lower
boundary for the energies accessible to free photons. ......... 63

The boundary between states accessible for external photons in the
reduced zone scheme for the unperturbed cavity modes in the cases (1)
next < am, (2) next 2 anr and (3) next > anr. Moreover, the figure
shows the lowest 2 cavity photon modes along 6)? of the square lattice. 64

If the condition next < anr is fulfilled there are polariton modes in
the dark region (shaded area). These states do not couple to external
photons and are thus expected to have a greatly enhanced lifetime. . 6)

viii

Introduction

Within the broad area of Solid State Physics the physics of semiconductor nanostruc-
tures is currently one of the most active research fields. Perhaps, the best examples
of novel systems in this blossoming area are given by low dimensional quantum struc-
tures such as quantum dots (QD) [36, 37, 38] and quantum wells (QW) [40, 49], which
are zero and two dimensional quantum heterostructures, respectively. Tremendous
efforts both to engineer and to understand the physics of these low dimensional sys-
tems have been made in the last few decades. The elementary electronic excitations
of a semiconductor can be treated in a quasiparticle picture. The properties of these
quasiparticles, called excitons, depend strongly on the dimensionality of the solid.
For instance, the many body behaviour of excitons ranges from an almost purely
bosonic behaviour in the bulk and QVV case [12] to purely fermionic behaviour in the
point-like QD case [50].

The most exciting features of low dimensional semiconductor structures become
manifest in the interaction with light. Light trapped in a microcavity - ideally given
by two facing mirrors— can strongly couple to excitons giving rise to a new kind of
quasiparticle called exciton-polariton {19, 40]. Polaritons are coupled half-matter,
half-light states with a photon-like mass - and thus a group velocity close to c - and
at the same time the ability to strongly interact with matter, due to their excitonic
part. These properties as well as recent advances in the experimental fabrication of

semiconductor nanostructures have suggested that polaritons could be applied to the

realization of scalable quantum information devices [51].

The concept of quantum computing was originally proposed by Feynman and
others in 1982 [17]. Feynman suggested that it might be impossible to simulate a
quantum mechanical system with an ordinary computer without an exponential slow-
down in the efficiency of the simulation. A new type of computers, based entirely on
the principles of quantum mechanics, would be necessary for these challenging tasks.
Nevertheless, it was not until Shor’s publication in 1994 [44], where he presented an
algorithm that demonstrated the outstanding potential of quantum computing, that
this novel field gathered momentum and turned into an active and promising area of
research. In a quantum computer the logical units, called qubits , can be realized
by impurity spin states or electrons in quantum dots. For the selective entanglement
of those states polaritons seem to be excellent candidates. Due to their photon-like
mass combined with the ability to interact with matter [35], they can be used as “me-
diating” particles in order to establishing an effective interaction between the qubit
states.

Furthermore, the possible observation of quantum phase transitions in strongly
coupled light-matter systems has attracted much attention lately. Polaritons be—
have in certain limits like a non interacting Bose gas and have an extremely light
mass. Therefore they are considered to be suited optimally for the realization of
quantum condensed phases in solids. In fact, recently several works have been pub-
lished claiming the experimental evidence of microcavity exciton-polariton condensa-
tion [3, 14, 24]. Disregarding the controversial theoretical discussion about the nature
of the observed condensation effects - we will briefly discuss some aspects of that sub—
ject at the end of the thesis - the field of polariton quantum condensation is highly
interesting because it brings pure quantum effects to a macroscopic scale.

In this thesis we will provide for the first time a theoretical discussion of a system

that can be considered as an intermediate case between a quantum well on the one

hand and a single quantum dot on the other. It consists of a periodic array of
semiconductor QD’S embedded in a planar microcavity. The invariance of the system
under translations by an arbitrary lattice vector partially restores the full translational
symmetry of a QW, which is completely destroyed in the single QD case.

Chapter 1 starts with a review of the theoretical core concepts of the physics
of semiconductor nanostructures. We will provide a quantum mechanical treatment
of excitons in QW’S as well as in confined structures like QD’s and discuss their
many body behaviour in these two cases. Furthermore, we will study the exciton-
photon interaction and present the polariton concept by introducing the Hopfield
transformations.

In Chapter 2, we will focus on the physics of the system under consideration,
which consists of a QD lattice in a semiconductor microcavity. We will see that the
exciton center-of-mass motion in the lattice is quantised and it can be characterized
by a quantum number q, restricted to the first Brillouin zone, which we identify as
the exciton in—plane momentum. In this in-plane motion the exciton behaves like a
quasiparticle with infinite mass. In the interaction with light the quantized in-plane
motion of the exciton gives rise to a quasi-momentum conservation. Unlike the case
of a QVV, there is not a one to one correspondence of an exciton mode with a photon
mode in the exciton-photon interaction, but an exciton with momentum q couples to
a discrete and yet infinite set of photons with in-plane wave vector q + Q, where Q
denotes an reciprocal lattice vector. Due to the finiteness of the quantum dot size,
the coupling to photon states with large momentum is suppressed by an exponential
factor, so that the set of photons has effectively a finite size. The chapter ends with
some preliminary considerations on the numerical calculations of Chapter 3

In Chapter 3, we will finally present the numerical calculations of the polariton
modes in our system. At certain symmetry points of the reciprocal lattice the po-

lariton can be considered as a quasiparticle with an exceptionally small and isotropic

mass, which is smaller than the mass for QW’ polaritons by a factor of 10—3. We will
study these novel kind of polaritons in a square and a hexagonal lattice at several
special symmetry points of the reciprocal space. Furthermore, we will see that in our
system polaritons with large in-plane momentum can be created. This is in contrast
with the QW case, where the polariton momentum is always close to zero due to the
steep photon dispersion. This large in-plane momentum can give rise to a total inter-
nal polariton reflection in the microcavity, which is expected to dramatically increase
the spontaneous emission lifetime of the QD lattice polaritons.

The thesis ends with comments on possible applications of this novel system and
future directions of research that may expand the investigations presented in this

thesis.

CHAPTER 1

Core Concepts

By definition a polariton is an excitation that arises from the coupling of an
electromagnetic wave with the elementary excitations of a crystal, like for exam-
ples phonons, plasmons. excitons, magnons, but also coupled excitation modes like
phonon-plasmons, etc. In the further discussion we will focus on the case of exciton-
polaritons, i.e. the coupling of an electronic crystal excitation, the exciton, and the
radiation field.

In the description of polaritons there are two different ways to address the prob—
lem. On the one hand there is the semiclassical, macroscopic point of view, and on the
other hand there is the purely quantum mechanical, microscopic treatment. In the
macroscopic picture the electromagnetic wave propagates inside the crystal, which
reacts to the external fields according to a linear response theory. This response of
the crystal due to the electromagnetic field gives rise to a dielectric constant, that
finally alters the dispersion relation of the propagating wave. These altered electro-
magnetic modes are nowadays called polariton modes, although the name polariton
(a combination of polarisation and photon) had not been introduced until 1958 when
Hopfield established the quantum mechanical treatment of this issue [20].

Hopfield introduced the concept of polaritons, which are the normal modes of a
system of 2 coupled oscillations. namely the photon and the crystal excitation. Al-

though similar ideas have been discussed earlier by Fame [16] and by Born and Huang

[5, 21], Hopfield was the first one to realize that the classical picture of electromag-
netic radiation interacting with matter, in which the wave is partially absorbed and
partially propagates inside the specimen, must be fundamentally reviewed.

In his new idealised picture only polaritons exist inside the crystal. A photon on
the crystal surface can give rise to a polariton which propagates inside the crystal
until it arrives at another surface or decays, e.g. due to scattering processes. In the
following we are going to describe this quantum mechanical picture of a bulk semi-
conductor crystal coupled to photons. We assume excitons to be the only excitations
of the crystal, which is a good approximation in the vicinity of the exciton resonance

frequency.

1.1 Wannier-Mott Exciton Theory

According to the effective mass equation (which we will derive later) excitons are
quasiparticles in a solid that can be considered as a hydrogenic bound state of an
excited electron in the conduction band and the remaining hole in the valence band.

In this chapter we will review the theory of Wannier-Mott excitons. In contrast to
the so called Frenckel exciton the W—M exciton has a Bohr radius much larger than
the interatomic spacing of the crystal.

The kind of excitons in a solid is determined by the properties of the material
under consideration. in particular by the value of the dielectric constant. Typically,
the type of exciton in inorganic semicomluctors is the “'annier—lt'lott exciton due to
the large dielectric constant, which leads to a screening of the effective electron-hole
interaction and thus to a large Bohr radius.

Starting from first principles, an exciton is an excitation of the whole electron
many body system in the crystal including the electron—electron interaction. For
simplicity we only consider the valence and conduction band of the semiconductor

and we assume a quadratic dispersion for valence and conduction band electrons, as

 

conduction band

E(k)

valence band

 

 

 

l
T

k

Figure 1.1. The conduction and valence band of the semiconductor with a direct bandgap
of size A.

well as a direct bandgap of size A (Fig. 1.1).

In this choice the single electron energies in valence and conduction band are:

h2k2 r21?
mic) = — E(,(k) : A + '

2mv ‘

 

 

(1.1)

27:10
Here, me and —mv are the effective masses of conduction and valence electrons,
respectively. So, in second quantisation, the Hamiltonian without. interaction, H0,

takes the form

110 = Z (Ev(k)c:kc,,k + Ec(k)c:_kcck) , (1.2)
k

where Cit/ck and cchk are the creation and annihilation operators for Bloch electrons

in the valence and conduction band. respectively:

(1.1ch IO) = (r| 0k) = —Leikruak(r). (1.3)

\/ V
The development in the completely delocalized Bloch functions rather than in the
localized VVannier functions turns out to be convenient for the description of weakly

bound, i.e. Wannier-h‘lott excitons.

K]

As the intraband il’iteraction of electrons does not affect excitations in the optical
frequency range we are considering here [13], we exclusively take into account the
interaction between valence and conduction electrons. The general form for such an
interband interaction is

”I = Z fk1k2k3k4 C:.k167:k2Q-k3evk4- (1-4)
k1k2k3k4
Note, that we did not take into account spin interaction and therefore dropped the
related index.

In the usual approximation [25] the matrix element fk1k2k3k4
( klv, k2C] l7 [k3c, k4?) ), which arises from the repulsive Coulomb interaction
between electrons, can be evaluated as

_ 47m? 1 6
fk1k2k3k4 — éV |k1_ k4|2 ki-k4,k3~k2*

 

(1.5)

where e is a finite dielectric constant due to the effective screening of all not explicitly
included charges (electrons and ions) and V is the quantisation volume.

Hence, we can write the total exciton Hamiltonian as

h2k2 52k?
=Z<A+ acct-z: 25m
k

 

 

 

2”to k 27m,
47m 1 1t 1
+ Z (V k k anrklcckgf'ckgcvh 5k1—k4,k3_k2- (1.6)
k1k2k3k4 I 1 — 4]

The subsequent step is to diagonalize this Hamiltonian. The ground state of the

system can be found by occupying all electron states in the valence band
_ i
|<I>o> — 11c...k 10>. (1.7)
R
It is straightforward to verify that this state is is an eigenstate with eigenvalue

.Eb ==:E:.Et(k). (1.8)
k

In order to find the other eigenstates we define a trial excited state as a superposition
of all possibilities to promote one electron from valence to conduction band:
I ’r
= Z A(k, k )cckcvk, [‘1’0) (1.9)

kkl

It turns out to be convenient to introduce new variables, namely

'm-ck, + mvk
K = —— 1.10
M < )
q = k—k’. (1.11)
In this choice (1.9) becomes:
)zquAqm Cc,K+(mC/M)q CH—UK (ml/111))ql(p0> (1°12)

where we have changedq the summation index and renamed the coefficients
Aq(K) :2 A(K + (mC/M)q,K — (mu/AIM). This choice is strictly not necessary,
but turns out to be useful, as it transforms the electron-hole system in the center of
mass frame.

Now we try to determine the coefficients Aq(K) in such a way that [‘11) is an

eigenstate of the Hamiltonian (1.6). From the stationary Schrodinger equation
H [\11) : E|\II) (1.13)
we find by applying the Fermi commutation relations for the electron operators

T _
[Ct'((‘)k- CU(C)I(I] + — 6k3k’: (114)

that the coefficients obey the so called effective mass equation (EME) in momentum

space representation

 

h 471’62
-—E -——-It2 —— A( .A I = 0. 1.1r

where M 2 me + m.” and p— — fihgf-K are the total mass and the reduced mass

respectively and E = E — E0 — A. The complete derivation of the EME is given in

Appendix A.

Eq. (1.15) tells us that coefficients with different q are decoupled, which makes q
a good quantum number to label the eigenstates. In this spirit we find the exciton

states to be of the form

PDQ) : Z Aq(K)C:,K+(me/A[)qcv,K—(Tnv/Al)q [(D0) - (1-16)
K
If we now Fourier transform the coefficients in (1.15)

we, R) = 1172 Aq(K)ei<qR+Kr), (1.17)
q.K

we realize that Eq. (1.15) is equivalent to the Schrodinger equation of 2 particles of
positive masses rm and my in an attractive Coulomb potential in the center of mass

frame, which is the eflectz‘ve mass equation for excitons in real space representation:

(_§A_Iv§, _ 27v? —- e7) 11: = Exp. (1.18)

This is a remarkable result. The problem of solving the Schrodinger equation for
the intricate Hamiltonian (1.6) reduces to the elementary problem of two particles
in an attractive Coulomb potential. Naturally we interpret this result in the manner
that the promoted electron enters a hydrogenic bound state with the remaining hole
in the valence band. In that interpretation q is the total momentum of the electron
hole system, which is conserved in our idealised model and thus a good quantum
number to label the exciton eigenstates.

We should mention that the effective mass equation can be derived from first prin-
ciples and taking into account. the full electron-electron interaction [43]. By using a
Green’s—function formalism, Eq. 1.18 can be obtained in the lowest. order approxima-
tion to the effective electron-hole interaction.

Eq. (1.18) has the well known solutions

h2q2 11.64 1
2M 28h? 712’

 

‘Ilq,n(R, 1‘) : eiqR¢n(r). qun = (1.19)

10

E,.,(q)

 

 

 

 

 

Figure 1.2. In the two particle picture the exciton states are found to be in the gap
between zero energy and the uncorrelated electron-hole energies (shaded area). E1(q) is
the exciton energy of the first excited exciton state given in Eq. (1.21).

where ¢n(r) are the atomic orbital states of the hydrogen problem.

With the choice E0 = 0 we have finally found the eigenstates and energies of the

exciton Hamiltonian to be

leg) = ZAg(104,,“(me/M)qc,,K_("h/M),l (oo) (1.20)
K

h2q2 1164 1

2M _ 26W 3’ (1'21)

En(Q)

 

with the coefficients A3(K) obeying Eq. (1.15).

Note that the exciton energies are found to be below the uncorrelated electron-hole
energies (Fig. 1.2).

By defining the exciton creation operators

“It" = Z A3(K)CZ,K+(me/AHqCU-K’Unr/Anq (1'22)
K

these new states can be considered as quasiparticles. A calculation of the commutator

[00,ny can] = Z [143(k)]2 {1 - CI'KCC‘K — chCLK} , (1.23)
K

11

shows that these particles behave approximately as bosons and the deviation is pro-
portional to the density of electrons in the conduction and holes in the valence
band [12, 19], where for simplicity we conducted the calculation for excitons of zero
total momentum.

Thus it is clear that the Hamiltonian becomes diagonal in the basis of excitons
H I Z En(q)(lll’naqsn. (1.24)
QJI

In the following discussion we will focus on the lowest internal state ((1),, = (1913) only

and thus drop the index n.

1 .2 Polariton Theory

As mentioned earlier polaritons arise from the interaction of excitons with photons.
So in order to describe the polariton modes we have to construct the full Hamiltonian
of the exciton-photon system. This Hamiltonian consists of three parts, namely the

exciton, the photon and the interaction Hamiltonian.
H 2 Ha; + Hem + H] (1.25)
Let’s first consider the Hamiltonian without interaction
Ho = Ha: + Hem. (1.26)

\Ne have seen in the previous section that excitons are (in the low excitation limit)

bosonic quasiparticles with a quadratic dispersion

 

h2k2
E-k =fa2-k 2E . 1.27
...() z.() 0+2], ( )
Thus the second quantized form of the exciton Hamiltonian is
H1. = Z fwr(k)o[(ak. (1.28)

k

The photon Hamiltonian has the well known second quantized form [11]

Hem = 219;”: |k|A[(Ak, (1.29)
where A; and AR are the bosonic creation and annihilation operators for photons and
72,. is the refraction index of the semiconductor material. We neglected the polarisation
index, as for the moment we are only interested in a qualitative description. Obviously
the natural basis of H0 consists of the direct product states of exciton and photon

Fock states and consequently an eigenvector can be written as

 

h h h
72i1,72i2,nfi3,...> ® nil ,nfi 2.,..,72fi3 .>, (1.30)

where the quantities mi and nah stand for the exciton and photon occupation numbers
in the state k, respectively.

In the next step we are going to introduce the exciton-photon interaction. It is
crucial to realize where this interaction has its origin in. The exciton is, as we have
seen earlier, an excitation of the whole electron many body system and with that in
mind it is clear that the exciton—photon coupling arises from the Coulomb interaction
of every single electron with the radiation field.

The Hamiltonian for electrons interacting with light can be written using the

canonical momentum [31] and reads in first quantisation
H: 23—— ,m (p.- — .A(.,))2, (1.31)

where A(rz-) = 2k A0,k {ALeke ikr, + he} is the electromagnetic vector potential
and e and mo are the electron charge and mass, respectively. Moreover, ri and p,-
are the position and momentum of the 2th electron. In the Coulomb gauge the vector
potential commutes with the electron momentum [A(r,—).p,-] = 0, since VA' 2

Therefore, by expanding the square in Eq. 1.31 we obtain

2
6 8 2
H: E —A - - —A - . 1.32
{252:3 7220 (1.2)“ + 27720 (1.2)} ( )

 

13

The first term in the latter equation is part of the exciton Hamiltonian H1; and so we
identify the interaction Hamiltonian as
8 62 2 ‘
If] 2 ——- A(1‘z‘) - p,- + —- A (ri). (1.33)

7220 2' 27720 2'

Note that the summation in Eq. (1.33) runs over all the electrons of the system.
It is not straightforward to find the second quantized form of that operator, as one has
to perform the transition from electron to exciton operators [41]. After a somewhat

lengthy calculation the result is

H1 : 2'ng (AL + A—k) (ak — aik) + de (AIc + A_k) ( [2+ A—k) , (1.34)
k k

with the constants

 

_ 27TH ikl‘
gk —— wx(k) —ckV (\I/(k) [ere (1)0) (1.35)
_ , 27f ikr 2 .
dk — w$(k)ckv <‘Il(k) ere ¢0>[ . (1.36)

 

 

The first part in Eq. (1.34) originates from the A - p—term , whereas the second part,
arises from the A2sterm. e is the polarisation vector of the electromagnetic field,
which will be discussed later.

Putting Eqs. (1.28), (1.29) and (1.34) together we realize that the full Hamiltonian

is separable in k.
H = 212(k) (1.37)
It

For each k and —k pair we have an equation that. describes a system of four coupled
harmonic oscillators {Ah A_k, ak, a_k}. Hopfield presented a scheme to diagonalize

this Hamiltonian exactly by introducing suitable operational transformations [20]

0,12, 011 C12 C13 C14 Ak

0k _ C21 C22 C23 C24 Gk (138)
I _ “v ' '
0-1, C31 C732 C133 C34 A—k

03k Chl CH2 CH3 CH4 a—k

14

011‘ and 012‘ are the polariton operators for the upper and lower polariton mode, re-

spectively. By replacing the former operators by the polariton creation and anni-

hilation operators and introducing a corresponding basis the polariton Hamiltonian

Hp = H; + Hem + H I is diagonal in terms of the new operators and has the form
Hp = Z ZE§(k)a§*a§. (1.39)

521,2 k

In general, the coefficients Ci]- of the Hopfield transformation in Eq. (1.38) have the

complicated form given in [20]. But if we apply the rotating wave approximation

and additionally drop the terms quadratic in the photon operator A (which is a good

approximation in the low intensity limit [11]), the polariton operators are found to

 

 

 

 

 

be
oi : uiak + viAk, (1.40)
where
22.5 = Qf‘ _ wph(k) (1.41)
“ 29f. -— phtk) — ate)
vi 2 ii Qf‘ — walk) (1.42)
2n§ — wph(k) — 823(k)
1 1 . .
eg 2 50.1412) + w,,,(k)) i Efiwgk) — w,,,(k))2 + (291.)? (1.43)

In the latter equations, the + and — signs correspond to the upper (g = l) and lower
(6 = 2) polariton modes respectively. Note that in the limit (a); — cum)? >> (29)2 we
regain the unaltered exciton and photon modes.

A plot of the Polariton energies E])(k) and E%(k) shows the anticrossing be-
haviour of the polariton modes at resonance (Fig. 1.3). In Fig. 1.4 the exciton and
photon shares of the upper polariton are depicted.

In the limits k —> 0 and k ——+ 00 the upper polariton becomes purely excitonic and
photonic,respectively. At resonance however, the polariton consist of an excitonic and

a photonic part of equal weight.

 

 

 

 

 

 

Figure 1.3. The Polariton dispersion. Figure 1.4. The exciton and photon
The dashed lines are the exciton and shares, 12], and 2211‘ of the upper polari-
photon dispersions without interaction. ton mode

Note that this sketch is not to scale.

1 .3 Confined Excitons

Up to now we only considered excitons in a bulk material. In order to describe
excitons in low dimensional quantum structures we have to find a way to confine
both conduction electrons and holes in the same region. With that in mind it is
plausible that conventional confinement potentials like e.g. in electric or magnetic
traps are not applicable, as they act on conduction and valence electrons in the same
way and consequently in an opposite manner on holes. In fact, a simple zeroth
order calculation shows that an electron potential V(r,-) acts on holes like a potential
—V(r,-). So it is obvious that it is impossible to confine both electrons and holes in
the same region by applying a conventional potential.

The method to arrange such a confinement for electrons and holes is to make use
of the band alignment of two different semiconductors. In this case a low dimensional
semiconductor structure is embedded in a bulk semiconductor of a different type,
with a conduction band structure like the one depicted in Fig. 1.5.

For the numerical calculations in Chapter 3 we will use a GaAs/AlggGaOJAs
system, i.e. the Gods structure is embedded in A10,3G(20,7As.

Although an exact solution for this problem has not been presented in literature

16

E<q=01_ . .

 

 

T II I l T II
We ,Typel, YPe Conduction
! ' band
I I
i i
7—"?
i 1
i u
. .
___i i___ Valence
1 I band
r

Figure 1.5. In zeroth order the bandstructure varies spatially like a step function at the
edge of two semiconductors

so far, in the effective mass equation (1.18) this particular band structure is usually
represented as an effective potential for both electrons and holes [34]. Thus, the

effective mass Hamiltonian (1.18) for the electron hole system reads as [46]

’12 2 ’12 2 82 ~

’fi; re‘fi; "h' (Ire—rm
As our aim is to describe excitons in a. nearly two dimensional structure, we assume
the z-direction confinement potentials to be strong enough to confine the exciton
in the x-y plane. Therefore the Coulomb potential only depends on the in-plane
distance [2 2 PC — pm. Moreover in case of a very strong confinement in z—direction

the potentials approximately separate in the z and in-plane components as

V},(r(.) : Ve(p€) + U€(z€), (1.45)
as well as
V}2(rh) : Vhfph) + L’fc‘(:}2)‘ (1'46)

This choice makes the Hamiltonian separable in z and p and the exciton wave function

can be written as
‘I’I(re~rh) : (pf/’0Phl‘bcfzcl‘f’clez): (1.47)

with the notation re”, 2 (l’e/h- ze/h).

17

In case of a strong and narrow confinement it it plausible to assume a rectangular

potential for both electrons and holes [7] and so the functions (be and (1)), are just the

eigenfunction for a particle in a box. The remaining problem is to solve the in-plane

Hamiltonian

( F22 V2 I12 V2 (72

 

After introducing the center of mass coordinates

 

.0 :Pe " ph
mepe + mhph
R = ,
A!

we expand the potentials in powers of p and find

mp.) = VAR + 1%) =ve(R) + vvam§§p + 0(2)?)
deR)
and
vary.) — vim {(1,522) =v1.(R) - vv1<R){’§p + 0922)
zVMR)

TI]- + Ve(/)e) + Vh(10h)) \le 2 E911.

(1.48)

(1.49)

(1.50)

(1.51)

(1.52)

Since the exciton Bohr radius is much smaller than the confinement radius (due to

the strong Coulomb potential). we only have to keep the zeroth order in the latter

expansion. In that limit the Hamiltonian separates in the relative and center-of—mass

iii-plane motion and finally the. envelope function can be written as

‘I’Ifrev rh) I X'fRIq’IPICbefzclééffzhL

where (I)(p) is the solution of the two dimensional hydrogen-like problem

(“ng — —’—) at» = eerie),

18

(1.53)

(1.54)

and x(R) is the solution of the problem

2
(‘fieva + vent) + mm) X(R) = Baum). (1.55)

The eigenenergies of a bound 2D exciton state consequently are
E Z A ‘1‘ E26 + EZ’I '1‘ Ec‘rn + ETEI’ (1.56)

where E 7,8 and E2: h are the quantized energies for an electron and a hole in a 1D box,
respectively.

We should note that Kumar et al. [26] showed that the in-plane confining potential
for both electrons and holes in a quantum dot (or the in—plane directions of a quantum
disc) can be approximated reasonably well by a parabolic potential. This has the
great advantage that the preceding approximations in Eqs. (1.51) and (1.52) are not
necessary, as the relative and center-of-mass motion separate exactly. The latter then
turns out to be harmonic oscillator like.

Like in the case of excitons in bulk material, we can define exciton creation and

annihilation operators,

(LI, : Z: Aa(k, k’)c:kcvkr (1.57)
k,k’
and
i

Here, (1 denotes a set of quantum numbers, which characterize the state of the in—
plane, center of mass as well as the electron and hole z-direction motion and Aa(k, k’)

is the Fourier transform of the exciton wave function in Eq. (1.53). The commutator

19

of this new defined operators is

[aatail] : Z A;(k11k2)Aa’(k31k4) [Clyk2cckleclk30vk4]
k1,k2,k3,k4
:ZA*( (k ,OIk’M (k k’) 2 AZ; (k1.k2) )4 a’(kltk‘1)cl’k4cf)k2
k,l(, k11k22k4
— Z A;(klak2)Aa’(k3ak2)C:k3Cck1- (1.59)
k11k21k3

The first term of the RHS gives a delta function due to the orthogonality of the
eigenstates, so that we get
[(20, 8:1,] :5 —Z .4;(k1,k2)Aa,(k1,k4)c,,k4c[,k2
klvk22k4

Z A;(k11 k2)Aa'(k32 k2)(":k36(3k1' (1'60)
k11kzik3

In the limit of an infinite Clot size we recover the relation (1.23) for the QW
case which implies a bosonic behaviour if the exciton density is much smaller than
a saturation density nsaturation m 1/(27ra%), where a B denotes the two dimensional
exciton Bohr radius [10, 9].

In contrast, in the limit of point-like dots excitons behave as fermions [50]. Being
in an intermediate regime we have to be careful about making assertions concerning
the behaviour of excitons on mesoscopic structures like the quantum dots we are
considering. However, according to reference [50], the large in-plane size of the dots
we are considering (approx. 50 nm compared to a the 2D exciton Bohr radius of

GaAs of a B = 5.8 nm) suggests that we can expect a bosonic behaviour.

20

CHAPTER 2

The Model

The system we are about to study consists of a periodic array of N identical semicon-
ductor quantum dots embedded in a planar microcavity (Fig. 2.1). It similar to the
system examined in reference [49], where we replaced the quantum well by a quantum

dot lattice.

—>
N)

 

   

Quantum
Dot

Mirror

ex I

Figure 2.1. N identical quantum dots of disc-like shape in an ideal periodic array embedded
in a planar microcavity.

The dots are assumed to have a cylindrical, disc—like shape, like the ones described

in reference [46] (Fig. 2.2).

21

 

(ph’zh)

Figure 2.2. The quantum dots we are considering have a cylindrical, disc-like shape. The
thickness of the dot is labelled by L; and the radius by R0.

We are going to study the interaction of the confined electromagnetic field with
the QD excitons.

Unlike in the quantum well case, the exciton—photon interaction does not conserve
the in-plane momentum, if the exciton is confined to a quantum dot structure. This
prevents obtaining an exact solution to the problem [47]. As we will see, the spa-
cial periodicity of our system restores a. quasi—momentum conservation. This feature
allows us to diagonalize the system hamiltonian with a few approximations and to

analyse the resulting polariton energy dispersions.

2.1 The Hamiltonian
The Hamiltonian of the system we consider consists of three terms:

H = H; + Hem + Hint» (2.1)
In the following we introduce each term separately before analysing the full Hamilto-
nian.
2.1.1 The Exciton Hamiltonian

The first term we are going to discuss is the exciton Hamiltonian, Hz.

Within the quasiparticle picture introduced in the previous chapter, a quantum dot

22

exciton can be in bound states with discrete energies

E0 = A + E361 + E"; + 133?. + E04 (2.2)

rel’

where a = (01,02, (13, 04) stands for a set of quantum numbers characterizing the
state of the in—plane, center-of-mass as well as the electron and hole z—direction motion.

The corresponding eigenfunctions are indicated by
I‘I’fD = all |<I>o>. (2.3)

As we cannot expect an exact. bosonic behaviour of the exciton operators we limit
ourselves to the case of a single exciton. Thus, in second quantisation we find the
Hamiltonian to be

111(1) = Z EaaLaa. (2.4)
O

For simplicity, we will only include the ground state and the first excited state in
the calculations and we thus drop the index 0.

Instead of a single dot we consider a system of N dots arranged in a periodic lattice.
If the separation between the dots is large enough (large enough that the center—of-
inass wave functions of excitons on two different dots have no significant overlap) we
can treat each dot separately. Besides, we assume the dots to be identical, so that
we find the same effective potential for electrons (and for holes) on every dot. This

implies that the effective potential on the ith (lot can be written as

ma) = VJ’m — R.) (2.5)
for electrons and
v,;-'(R) = v,?(R — 11,-) (2.6)

for holes, where R; identifies the center of the ith dot (index 0 indicates the dot at
the origin). It is obvious that this property of the potentials implies for the related

Center Of [[1358 “Fave fllnCtlUIlS
MIR) = X0(R — R-i)- (2-7)

23

Consequently the wave function for an exciton on the 'ith dot reads

‘I’Mre, rh) = X0(R - Rd)®(p)¢e(3c)¢e(zh)a (2-8)
and we can write the Hamiltonian for the quantum dot array as a sum over the

Hamiltonians from the N sites.
N

i=1 i
where the operator (1: creates an exciton (in the lowest excited state) on the dot 2'.

Consequently the eigenstates of the system can be written as Fock states:

In.1,n2, . . .) = ((10721 ((19112 . . . I920), (2.10)
where we restrict the occupation number 711-, i.e. the number of excitons on dot 2', to
{0,1}.

Note that we have set the ground state energy to zero.
According to our assumption that the exciton wave functions from different dots

(i)

have no significant overlap, the operators al-

[(1,- ai]=5i,j [(1, a1], (2.11)

where the commutator [mal] stands for the expression derived in Eq. (1.60) for

obey the commutation relations

a = a = 0. In order to exploit the periodicity of our system we are going to

introduce new operators

1 N
aq := f; (2.12)
and
aqzz (aq)— — 71:2: (126 iq Ri. (2.13)

These operators obey the same commutation relations as the localized operators

1 __ l iqR- —iq’R’ , It
[(i.q,(1q,] — N 26 16 J aha].

2.]
6i,j[a,aI]

: 7:.Zei(q-Q')Rd [mat] = 5q,q’ [mal] . (2.14)

24

Moreover, they reflect the periodicity of the lattice as for each reciprocal lattice

vector Q we have

q+Q Wu =“qa

z:

aft) __1___ agileiiqRieiiQRd (i) (215)

since by definition of the reciprocal lattice it is eiQR'i = 1. So we see that the quantum
number q can be restricted to the first Brillouin zone and if we use periodic boundary
conditions we find exactly N discrete, possible values for q. Accordingly, the inverse
expression for Eq. (2.12) is

1 _,- .
a: :2 -\/—7—V- Z (1118 zqu. (2.16)
qu.BZ

We would like to express the exciton Hamiltonian, H1, in terms of these new
operators and thus we insert the latter expression in Eq. (2.9). By exploiting the

useful property
Zei(q-q’)Ri : N6

2'

q,q’ (2.17)

we obtain

szhwz Z aLaq. (2.18)
qu.BZ

This implies that the excitons in our model are quasiparticles with a quantized
in-plane momentum q, restricted to the first Brillouin zone, and an infinite mass.
Furthermore, we have seen that these quasiparticles obey the same commutation
relations as the QD excitons in Section 1.3.

The natural basis for this Hamiltonian is

nql ' ”(12
l-nql,nq2, . . .) = (0.1“) (all?) ...|(f)0), (2.19)

where "(12‘ is a counter for excitons in the state (LL. |<I>0), restricted to {0, l}.
1
Please note that we changed our single particle basis from a set of completely

localized states a: |(I>0) to a set of completely delocalized states all [(D0)

2.1.2 The Photon Hamiltonian

In this section we are going to provide a quantum mechanical description of electro-
magnetic radiation inside a semiconductor microcavity. A microcavity is basically a
device to confine the electromagnetic field. The simplest structure for such a confine-
ment is a planar Fabry-Pérot resonator.

This structure consists of two parallel mirrors of a high reflectivity separated by
a dielectric material called the ’spacer’. Classically, an electromagnetic field can only
exist between the mirrors, if the successive passes of a propagating wave interfere
constructively. This leads to the condition that the wave vector perpendicular to the

mirrors kz has to obey the condition

where n 21,2,3... or
2
w
L 27712 - k? 2 mr, (2.21)

where w is the photon frequency, k” is the component of the photon wave vector
parallel to the mirrors, L is the cavity spacing and nr is the refraction index of the
dielectric spacer. The latter equation tells us that the main effect of a microcavity is
the quantization of the electromagnetic field in the direction of the confinement; in
our case the z-direction.

The second quantization form of the electromagnetic field inside a microcavity
is obtained using the standard procedure of replacing the amplitudes of the vector
potential in its expansion into plane waves by photon creation and annihilation op—
erators [11]. Because of the special symmetry of the microcavity, it turns out to be
convenient to define certain photon polarizations, which we call E—mode (E; =2 0)
and M-mode (Bz = 0). The definition of these modes is in analogy with the TE and

TM modes of a waveguide [23]. The field operators for the E—and M—modes can be

26

 

frequency (0

 

 

 

 

in-plane wavevector k"

Figure 2.3. The in-plane dispersion modes (n=1,2,3) of a microcavity. The dashed line is
the two dimensional dispersion of a free photon.

written as [4]

/ h . . . -
AkE(r) = Wcos(kzz)q X zequAkE + h.c. (2.22)
h C] . . .kz A 2’ A .
AkM(r) = —— ——sm(kzz)z — z—cos(kzz)q e qukM + h.c., (2.23)
6071.?ka k k

where AkE and Ak M are the destruction operators for photons in the E—and M-
mode, respectively. As usual the notation is k = (q, k2) as well as r = (p, 2). V is an
arbitrary quantisation volume, which we choose to be identical to the quantisation
volume that appeared in the treatment of QD lattice excitons in the previous section.
The operators Ak E and ilk M are normalized, so that energy of the field is

hwk(n + 1 / 2) when there are n photons in the corresponding mode. Consequently,
the photon Hamiltonian can be written in the form

Hem = Z Z fkaLV/iku. (2.24)

kg (IV

Since we will tune the cavity spacing in a way that the exciton energy matches the

energy of the first cavity mode (k2 = 1r / L) we can neglect the rest of the modes as

27

they are off-resonant. Therefore the Hamiltonian simplifies to
Hem = Z hkaLV/lku, (2.25)
(11/

with k : (q, 7r/L) and V = E, M.

2.1.3 Exciton-Photon Interaction

In this section we derive the interaction between the exciton on a quantum dot lattice
and cavity photons from first principles.

At first we have to describe the coupling of one quantum dot at Rj with the
radiation field. As we have seen in Section 1.2 the exciton-photon interaction has
to be expressed in terms of the electron-photon Hamiltonian in first quantisation
(Eq. (1.33)) 2

Him.) = ‘71"?“0 - p.- + 5,32%»,
where the sum runs over all electrons in the dot at RJ- and mo denotes the electron
mass. For low-intensity radiation the term quadratic in A becomes negligible [11].

This allows us to retain the linear term only and we can write the interaction as

H1<Rj>=—— . A(r.>.p.-. (2.26)

“"0 have seen earlier that the exciton states PD?) 2 (If: |(I>0) form a complete set
of basis vectors for the many body system in the quantum dot j. Therefore the

completeness relation
2 Iva (‘IISEI = I (2.27)
(1

holds and we can express the interaction in terms of the exciton functions by inserting

the unity operator twice as

8 - {3 3
11101)) = $62 W2) (‘PSIZAM-pil \I». > < a I- (228)
0,6 2'

28

Using pi— — ijfl[H1-(j),ri] and the fact that HA] )I‘Ilfg) 2 Eat leg) we can simplify

the. precedii’ig equation to

.6 wit/3
11,(R,-) z ngfi<E€rE13—)<‘I’3|ZA(W r,| 1112 ) (xi/g)” . (2.29)
0a

As we did before we limit our single exciton basis to the lowest two states, i.e. the
ground state I‘DO) and the first excited state, which we indicate by Him). Before we
proceed, we need to express the many electron operator 2,- A(r,-) - ri in terms of a
delocalized basis of Bloch functions (Eq. (1.3)). According to the second quantisation

formalism this expansion is given by

Z: A( (r,)r =2 tk k’Ck Mck/ (2.30)

kk’ a,’0
0,0’
where t stands for the matrix element
tkk, _ —(ka| A (r)r lk’a ’). (2.31)
0,0’

r

Using Eq.(l.o7’) it is straightforward to verify that Eq. (2.29) can be simplified to

H,(R,)=m.r Ztkkflm (,)akk' H121“ A(,)a,,kk’ , (2.32)
kk’ C‘ kk’ ”C

where we again used the orthogonality of the states CL 001‘; a, |<I>0) and identified the

1.
.7
him) (\Ill-I with the lowering operator (13-. Note that in Eq.(2.32) A(k, k’) indicates the

expression Nix) ((PQI with the creation operator a and consequently the expression
Fourier transform of the exciton wave function as defined in Section 1.3 and should
not be confused with the electromagnetic field in Eqs. (2.22) & (2.23).

At. this point it is important to notice that the matrix element t is a completely
delocalized quantity. The information that we are dealing with an exciton localized
on a disc at position R1 is entirely encoded in the coefficients A(k, k'). Note also that

the values of k and k’ are restricted to the first Brillouin zone.

29

Due to the property

2 1k,_kA(k, k’) = Z tk.k,A*(k,k’) , (2.33)

kk/ U,C kkl 0,”

the only thing that remains to be done is to evaluate the expression
I
Z ink/Mk, k ). (2.34)

After a somewhat lengthy calculation, which we present in Appendix B, the final

result for the interaction Hamiltonian for one dot is:

H,(R,) = h: {gg*(R,-)Ak0a} + g]: (Rh/11“,} , (2.35)
qa
where q denotes the in—plane component of the wave vector k = (q, kg) and the

coupling constant is defined as

1

9.15 (11,-) =-iewx<1>18(0)h

Cka(Q)I(kz)ucva (2.36)

for the E—mode. The coupling constant for the M-mode obeys the relation:

-. .k; - ,
142%.) = ~1ng (11,). (2.37)

The quantities in Eq. (236) are defined as

/ h.
C 2 —.——— 2.38
k 60711,?ka ( )

[(1.73) :/dzcfie(z)c‘)h(:)cos(k;z) (2.39)
em) = / acme—"q”, (2.40)

and 11.61. is the length of the in-plane component of

l
—/ dru;(r)ruc(r). (2.41)
UC’

11 cc 2
VUC

The function (1)18 is the hydrogenic ground state of the Hamiltonian in Eq. (1.54) and

VUC is the volume of the crystal elementary unit cell.

30

Earlier we have seen (Eq. (2.8)) that the functions describing the center-of-mass
motion for an exciton at site RJ- is related to that of an exciton at the origin by the
equation

XJ-(p) = X0(P - Rj)- (2-42)

In the lowest order approximation the center-of—mass potential is parabolic and the

solution of Eq. (1.55) is a Gaussian of the form

2
as

X0(P) = 2E8 - (2.43)

From here it is straightforward to derive that the Fourier transform of X j(p) reads
em) = e“"qR2ro(q) = (2*qu «am-WV, (2.44)
which leads to the final form of the coupling constant
@501.) = ie—iqu‘gut-flab...<0>Ckxo<qii<w/L>ua. (245)

Because of our assumption of a negligible overlap for the exciton wave function at
different sites the interaction Hamiltonian for the whole system is simply the sum of

the terms for the N single dots

H, = Z Him»
1'

l ' ' R. 1 l —i R.
2h E gfi*/lk0 — E (ll-elq 3 +gfi/l —— a-e q .7
,/- J k0 [— E 3 J
q0 1V j 1V j

:11: {9fl*Aka‘lh + QEALUU'Q} . (2.46)
qa

where the constant gf‘ is defined as

.- 9 _€ . -
91f; =\/1Tg|£(R = 0) : IEta-T«37(le((~))CkX0(Q)[(n/Lfl‘cc

. 27r 1 {3 _l 2.2
=2\/fiewx Wife ifq ’3 <I>IS(0)I(rr/L)ucv. (2.47)

31

where n— — 1%— stands for the number of dots per unit area. Moreover it is

.k; .
91:1: —z—k:-gE. (2.48)

Note that in Eq. (2.46) q runs over the full reciprocal space. Therefore each vector

q in the sum can be expressed as a sum of a reciprocal lattice vector Q and a vector

within the first Brillouin zone q' as

q=Q+d. am)

If we in addition take into account the periodicity of aq given in Eq. (2.15) we can

rewrite Eq. (2.46) as

H]: E Z Z Z {gait '+Q AQI+Q Gal, + gflthAL’aLQflaQI}, (2.50)
q.’El BZ QERLO: E,11I

. 1 _ 1 1 . , = .
where the notation Aq’+Q,a = Ak=(q’+Q,7r/L),o as well as ggurq _ gfl:(q’+Q,w/L) lS

understood. Hence, the interaction Hamiltonian is separable in q’ .

2.2 Analysis of the full Hamiltonian

By combining Eqs. (2.18). (2.25) & (2.50) we compose the full Hamiltonian of the
system. It can be separated in the variable q, which is restricted to the first Brillouin

zone of the two dimensional lattice of quantum dots.

Zjhm) am)

qu.BZ

where

h.(q) =hwxallaq + h 2 Z wq+QAL+anAq+Qfl

oxEJIQERL
i i
+ 5 Z Z {ggiqu+Q,0(lq + 93-1-Q‘4q1Q,aa<1}' (2.52)
a:EaIQeRL

32

In this form of the Hamiltonian the first remarkable feature of the system becomes
manifest. Unlike in the case of one quantum dot, where the exciton couples to a bath
of photon modes, or the opposite case of a quantum well, where there is a one to
one correspondence in the exciton-photon coupling, in our system an exciton state

Iq) = all [0) couples to a discrete and yet infinite set of so called umklapp photon

t
q+Q10

Similar umklapp photon processes and their role in polariton properties have been

states [1q+Q,o> = A [0), where Q denotes a reciprocal lattice vector (Fig. 2.4).

studied exhaustively in molecular crystals [8].

a) i b) c)
l

_\
‘1:

 

1‘ q
Exciton Photon [ 1

Figure 2.4. In the quantum well case there is a one to one correspondence of states in the
exciton-photon interaction (a), whereas the quantum dot exciton couples to a continuous
photon bath (b). The QD lattice presents an intermediate case, where the exciton couples
to a infinite but discrete set of photon modes (c). In the latter two cases the coupling for
large q is suppressed by the form factor 2(q).

117e, often will refer to the picture where we consider the different photon modes
as quasiparticles labelled by a quantum number Q, that have an in—plane momentum
q, which is restricted to the first Brillouin zone. I11 this spirit. the umklapp photon
Q has an energy dispersion wq(q) = wq+Q.

In this quasiparticle picture it is easy to prove that the quantum number q is
conserved. It is important to notice that this is a quasi-momentum conservation
only, as the introduction of the umklapp modes is nothing but a relabeling of the
modes in the reduced zone scheme of the photon dispersion (Fig. 2.5). Obviously the

conservation of q does not imply a iii-plane momentum conservation.

33

flu) [a.u.]

   
    

O Qy [ZR/a]
—0.5

0 —0.5
q)( [271/8] 0.5

Figure 2.5. The first four photon modes in the reduced zone scheme.

2.2.1 Simplifications and Approximations

As we limit our basis to the states with one excitation only, the basis we choose
consists of the ground state [0) : [0) X [0) as well as the exciton state |q) : [q) ® [0)
and the umklapp photon states [1q+Q70) = [0) ® [1q+Q,a>1 where Q runs over the
reciprocal lattice and a = E. M.

In this basis we state that the total number operator

I‘V : [NZ]; '1‘ lhenl : Z "(La-q + Z Z AL+Q30Aq+Q’a‘ (253)
qu.BZ qu.BZ QERL,U

commutes with the full Hamiltonian (2.51)
[H, v] : 0. (2.54)

Thus the eigenvalues of N, ie the total number of excitations, is conserved.
The first. approximation affects the coupling constant for large wave vectors. Due

to the finiteness 0f the quantum dots the coupling constant is suppressed by the form

34

factor flq) = 6—311qu} 2. Thus we introduce a cutoff wavelength Q0 and neglect in
Eq. (2.52) all parts in the sum over the reciprocal lattice vectors with [Q] > Q0. This
approximation reduces the dimension of our system to a finite value.
Note that for point-like oscillators (6 ——+ 0), where this cutoff is not present the
contributions from all vectors Q can be summed up analytically in a series [29].
Using the basis a = {(0) , Iq) , {|1q+Q,a) , |Q| < Q0,a = E, M}} the Hamilto-

nian h(q) can be written in the matrix form

 

 

, E 11* E M M
f on: 911* 9%; 9.,qu 9,,qu gq+*Qn\
gE w 0 0 0 0
q q
gal 0 wq 0 0 0
g§+Ql 0 0 0 wq+Q1 0
M. . . . . . .
\gQ'th 0 0 0 0 01¢qu

where we have chosen an arbitrary numeration for the reciprocal lattice vectors in
which Qn denotes the last vector that fulfills the condition [Q] < Q0.

It is possible to block diagonalize this matrix by changing the basis in the 11
subspaces, each spanned by the two polarisation vectors ]1<1+inE> and I1q+QiJ1>,

respectively. The new basis vectors

 

TQ+Q1> and ]L9+Qi> mix the E and M polari-

sations and are defined as

 

IT<1> Z‘lq llq,E> + fiq [111,110 (2.56)
14(1) Zfiq [1q.E> + “q llq,11l>.~ (2.57)
where
1'.

“<1 Z—T— (2.58)

1+ (7:1)2
13c: =-£/fi—- (2.59)

1+ (1:1)?

So the transformation matrix between the two basis systems has the block diagonal

form

, (2.60)

 

 

) I 8%an

where the two dimensional blocks Bq are defined as

3,, = (“q [39) . (2.61)

,Bq aq
In this new basis the Hamiltonian fz(q) = Sah(q)@q splits up into two blocks
of dimension n + 1 and n, of which the latter is diagonal. It turns out that this
diagonal block belongs to the space spanned by the n L-polarisation vectors so that
these polarisations are completely decoupled from the system and thus neglected in
the following.

The remaining 71 + 1 dimensional Hamiltonian has the form

 

 

( “1:2:th 9q+Q1 9q+Q2 grim.)
gq Wq 0 0 "° 0
1 0 w 0 0
h(q)=h 9q+Q1 “Ql (2.62)
99+Q2 0 0 L”(WC-22 ' ' ' 0
\9q+Qn 0 0 0 wQ+Qn)

where the new coupling constant is

k? 2 E
gq= 1+ (7.7) [_(/q]. (2.63)

In the next step we show that we can safely neglect terms in the Hamiltonian that
are off-resonant. i.e. for which the condition [4.1),- — 1121-] >> 29,- holds. where 1'. stands
for the expression q — Q,- for a fixed q.

We will calculate the energy shift due to the off—resonant modes using a pertur—

- e 2 I yo -
bative approach on the quantity :17,- := ;—_ng—;. \1‘ 1thout loss of generality we assume
1 .

36

 

all off-resonant modes to have a higher energy than the exciton. The treatment of
the modes below the exciton energy is analogous, with the only difference that the
respective energies are shifted in the opposite direction.

In the first order in :13,- the energy shift of the exciton mode can be calculated as

1
A111,. = — 2 59113 + 0 (as?) , (2.64)
i
and the shift in the photon energy (12,- is
1 2
Aw,- = +5911; + O (513,-) , (2.65)

Denoting with Q0 the off—resonant photon mode with the lowest energy and thus
smallest deviation from the exciton energy, we notice that 11:0 = __22(L_ represents a

(JO—LL11:

upper boundary for the set {117,-}
(Bi S 170 VI. (2.66)

Since also 0 < 9) < gizo Vi, we can obtain an upper bound for the exciton energy

shift as:

 

[A1113] 3 lgggzrg + O (:18) , (2.67)
where m denotes the total number of off-resonant modes which is finite due to the
earlier introduced cutoff. This result holds in the case where we take into account
photon modes below the exciton energy.

This approximation shows that in the system we are considering we can safely ne-
glect the off resonant terms, as for the parameters we are using the energy deviation
can be estimated to be of smaller than 0.1 meV. Note that this is a very rough esti—

mation as we approximated the :12,- by their upper bound 2:0. Numerical calculations

show that the actual effect of the off-resonant modes is much smaller.

2.2.2 The Exciton at Resonance and the Strong Coupling Regime

In this section the focus lies 011 examining the Hamiltonian given in Eq. (2.62) at

special symmetry points of the Brillouin zone. The feature that makes these points

37

 

of special interest for us is that here two or more photon modes are intersecting.
Let qO be such a symmetry point where n photon modes intersect at energy
11). Then, by neglecting the off-resonant modes according to the previous section and
assuming the system to be tuned in such a way that the exciton has an energy resonant
with the intersecting photon modes, the corresponding matrix has dimension n + 1

and the simple form

 

 

{w y g g g)
g 11) 0 0 0
.. g 0 w 0 0
h(q6)=h 9 0 0 w 0 , (2.68)

where g E gq+Q and Q is the quantum number of one of the considered photon
modes.
In a procedure similar to the one used when we were dealing with the E—and

M-polarisation we can reduce this matrix to the form

{(1) fig
fig w

oEoo
Eooo

(2.69)

 

 

\ 0 0 0 0 - -- at)
We realize that in the new basis at the crossing point only one mode couples to the

exciton while the remainder of the modes is left unaltered. The two dimensional

matrix can easily be diagonalized and we find the eigenvalues of the whole matrix to

be

2112 = w :1: fig, (2.70)

as well as the (n - l)-fold degenerate eigenvalue

)3 = u). (2.71)

38

 

That means that the Rabi splitting between the highest and the lowest mode Q =
Qfig is proportional to the square root of the number of intersecting photon modes.
The factor \ffi gives an important enhancement of the Rabi splitting at the lattice
symmetry points. According to our assumptions at a given point go in the first
Brillouin zone and for a given radius r we only have to consider those umklapp terms

for which the Bragg-condition
T-5< IQO+Qil <T+5> (2'72)

holds. Here 6 denotes a small deviation tolerance. The number of modes that fulfill
this condition for a large 1' in a regular lattice is proportional to r. In the limit of
point-like quantum dots the form factor )Z(q) reduces to 1 and the coupling constant
is proportional to 71;. Thus we see that for large T the factor due to the number of
modes cancels the factor arising from the coupling constant.

So far we did all the calculations under the assumption that we are dealing with
an ideal cavity and an exciton of infinite lifetime. We now want to include the effects
of a finite lifetime of both excitons and cavity photons, due to e.g. exc.iton-phonon
scattering processes and the imperfection of the cavity mirrors.

We therefore introduce decay factors K, for photons and '7 for excitons in the Hamil-
tonian, which become manifest as an imaginary part of the diagonal elements [33] and
subsequently solve the resulting non-hermitian eigenvalue problem.

At the crossing point of n modes the Hamiltonian then has the form

5((10) Z (w “All fig) , (2'73)
fig (4) + 2h.

so that the eigenvalues can be calculated as

 

1 l .
A13 : w + 5117+ I.) :t §\/(2\/7_Ig)2 — (f.- — 7)? (2.74)

Note that additionally there is the (n — l)-fold degenerate eigenvalue A3 = w + in.

39

Fr0m Eq. (2.74) it follows that there is only a Rabi splitting if the condition
(N59)2 > (K - 7')? (2-75)

is fulfilled. In the literature, this case is called the strong coupling regime. The other
case, where the root becomes purely imaginary and the real part of the eigenvalues
is left unaltered is called the weak coupling regime. We assume the decay rates to
be small enough to be in the strong coupling regime and neglect splitting reductions

due to decay processes as they can assumed to be small.

40

CHAPTER 3

Numerical Calculation

In this chapter we will present the numerical calculations we conducted in order to
study the energy dispersion of the QD lattice polaritons.

In the analytical expression we provided for the coupling constant gk in Eq. (2.63)
and (2.47) there are certain constants of unknown magnitude like the relative motion
wave function (1)1 3(0), the integral I (1r/ L) and the dipole moment ucv. Instead of
conducting analytical estimations for these quantities we will determine them by
linking our theoretical results to experimental data.

The system we consider in the following is GGAS/AngGaOJAS. This III-V com-
pound has a refraction index of nr 2 3.55 in the considered wavelength range [1] and
a bulk exciton resonance energy of fin);- = 1.515 eV [28]. Due to the z-confinement
the exciton energy in a 25 nm quantum well is increased to the value hug W = 1.525
eV [27]. In such a QVV the Rabi splitting if found to be $2 = 3.6 meV [6].

In order to exploit these data we use the fact that we can recover the QW coupling
constant [41] from Eq. (2.63) in the limit of one dot (N = 1) which covers the whole

quantisation area ( \/§ —+ 1). Moreover, it is necessary to replace the QW by the

 

QD exciton energy, which has been found for lens shaped dots of radius R0 = 50 nm
to be MED = 1.68 eV [18].

Therefore, using the fact that the coupling is proportional to the exciton energy,

41

we find the relation

Q0
(4);, 132 2 g u/
gk : “JCS”, «EV 277,663" q [[3 , (3.1)
SE

Q ”I

where ( can be written as
k

. )2 V
k x/E,/—_Leff
Here, u stands for all the ren'iaining constants, including the unknown quantities

(Db-(0), 1(7r/L) and um.

(3.2)

Note that the cavity length I; has to be replaced by an effective width Le” = 2L+
L DB R which arises from the imperfection of the cavity mirrors and the resulting finite
penetration depth of the electric field in the mirrors [39]. L denotes the resonance
length of an ideal x\/ 2 cavity. L D B R can be calculated by an exact transfermatrix
calculation for the modes of a real cavity. we use the value L D B R from [39] which is
sufficiently accurate for our purpose.

By comparison with the experimental data we find a value for the effective constant
I/ of hz/ : 8.09 meV.

In the following sections we are going to examine the dispersion modes for a square
and a hexagonal lattice, which are obtained by a numerical diagonalization using
generalized Hopfield transformations. These transformations are implicitly applied in
the diagonalization of the matrix in Eq. ( 2.62). After having calculated the eigenvalues

numerically the Hamiltonian can be written in the form

H = Z hwf;<q>pl,,pq,g. (3.3)
q.€

(i)

where hwgm) denotes the {St eigenvalue and p q.§

is the polariton creatirm (annihila—
tion) operator.
We have to stress that. the latter form of the Hamiltonian was derived under the

assumption of a system with a single excitation. In order to expand this result to the

42

many body problem of a highly excited system, a careful investigation of the polariton
commutator relations is necessary.

Since the polariton operator is composed by the exciton and photon operators as
pT = 0me + Z aphqfllq, (3.4)
Q
the commutator can be calculated as

[19. pl] 2 [OJ-[2 [mal] + Z [ath]? (3.5)
Q

Therefore, in the case of very large dots, in which the bosonic approximation
holds and [mal] % 1 the polaritons approximately behave as bosons too and their

operators fulfill
lpqéle : hwg(9)pq,€' (3'6)

Thus, the Hamiltonian is diagonal in terms of the polariton operators Pq,§ and

1,

p q 6 and the eigenstates of the system can be written as a Fock state

1 n . n
[72.01.1102, . . .) : I (p11) 01 (72:12) 02 mlcbo), (3.7)

”01 lila2. . . .

 

      

where a,- denotes an arl_)itrary pair (q,£).
This is not true if the exciton operators show a remarkable deviation from bosonic

behaviour.

3.1 Preliminary Discussion

In this section we will exemplarily present the conducted preparatory calculations
for the case of a quantum dot square lattice, before we turn to the actual numerical
calculation of the polariton modes. An analogous calculation has been carried out for
the hexagonal lattice.

The simple square Bravais lattice, with square primitive cell of side a, has as its

reciprocal another simple square lattice, but with square primitive cell of side 271' / a.

43

 

.m M as Ms]
/ w

 

T V 3 V 15/; O U C:
2% \ Wigner-Seitz ' \Wigner-Seitz
Cell Cell

,J

Figure 3.1. The reciprocal lattice:(a) of a square lattice with lattice constant a and (b) of
a hexagonal lattice with constant a. As usual I‘ denotes the origin q = O.

In the reciprocal space there are two symmetry points that are of special interest,
the X-point, which has a w-rotational symmetry and the M-point, which has a 7r / 2-
rotational symmetry (Fig. 3.1 a)).

After having linked the constants in our theory to the experimental data in

the previous section, we can finally write down the coupling constant for the

GaAS/Alo,3Ga0_7/ls system as

If

 

«as 1 k~2 r22
7 1+(4)e‘50, (a&

where f‘w : 8.09 meV.

At first, we try to optimize the system parameters with regard to the coupling
strength at resonance. The free parameters in our system are the quantum dot radius
R0 = [3/\/2, the length of the ideal cavity L and finally the lattice constant a, which
determines the dot density n as well as the size of the Brillouin zone.

Generally, at a point in the reciprocal lattice qO where two or more photon modes

Q,- intersect the Bragg—condition

[Q + (10] : (1 V7} (3.9)

is satisfied for a constant (1 (Fig. 39). Our purpose is to maximize the coupling

constant at this distance d in order to achieve the largest possible Rabi splitting.

44

Exploiting the fact that the photon energy has to match the exciton energy at the
crossing point

c 7r 2
he, —1.68 meV _ a; 012 + (I) , (3.10)

and by expressing d in units of the reciprocal lattice constant (1 = (fa-T we find a
functional dependence of the coupling constant for a fixed exciton energy hwx =

1.68 meV which is independent of the lattice constant a

9d = 2761.11), (3.11)

where we excluded the special case (i = 0.

Thus we only have to optimize the function f with regard to the remaining free
parameters of the system, which are the dot radius R0 = x/2fi and the cavity length
L. A two dimensional plot of f(fi, L) is given in Fig. 3.2.

 

 

 

 

 

 

3: 1--
.‘E.
D)
g p/
‘3 ~1 ‘
0'1 02 L23“) 0'4 0‘5 0.15 0.25 0.35 0.45
Lluml
Figure 3.2. The function f(fl,L). Figure 3.3. Plot of the derivative
The light regions correspond to higher of f (R0 / J2, L) with respect to L for
values. a fixed R0 = 50 nm. Notice that
the coupling is maximal for a value of
L = 0.17] pm.

For a fixed radius of R0 = 50 run we find the optimal value for the cavity length
to be L = 0.171 pm (Fig. 3.3). According to Eq. (3.11) this value for L is optimal

independently of the crossing point under consideration. We find a simple expression

45

for the coupling constant at the resonance point that only depends on (f

1

g 2:909
d d

(3.12)
where go 2 0.387 meV for the optimized values. me this equation we can calculate
according to Eq. 2.70 the expected half Rabi splitting f2 / 2 for different symmetry
points of the 13‘ BZ by multiplying gd with the square root of the number of inter-
secting photon modes. In Tab. 3.1 the calculated values for the Rabi splitting at the

symmetry points of the square lattice are listed.

 

 

 

 

X K1 K2
J 1/2 \/§ 5/2
#modes 2 4 8
Q 2.19meV 2.19meV 1.385 meV

 

 

 

 

 

 

Table 3.1. The Rabi splitting (2 depends on the relative size of the wave vector cf. at the
crossing point and the number of intersecting modes.

In the following we are going to turn our attention to the polariton modes in the

vicinity of a high symmetry point.

3.2 Square Lattice: The X-Point

At first, we will focus on the X-point at q0 = (7r / (1,0) of the reciprocal space of a
square lattice with constant a (Fig. 3.1). For the calculation we included all umklapp
terms in the Hamiltonian of Eq. (2.62) up to a certain cutoff length due to the form
factor flq). Furthermore, we dropped the completely decoupled L-photon modes.
At the X-point, the lowest photon modes crossing are arising from the original
photon mode (Q = 0) and the umklapp term with Q = (—27r/a, 0). Speaking in the
quasiparticle picture, these two photon modes have the same energy at the edge of

the Brillouin zone at (10 = (7r / a, 0). We will present the calculations of the polariton

46

 

 

  

 

 

 

 

 

 

 

 

 

1.684 a) 1) / 1.684
g \\ ” g
3 1'68' (2) 1 3’. 1.68 7:2“
UJ I, \\ LU
1.676 ‘3) \ 1.676»
0,495 05 0504 —0.1 —0.05 0 0.05 0.1
Clx [ZN/a] qY [ZN/a]

Figure 3.4. The polariton modes (1)-(3) at the X-point along F)? (a) and along X M (b).
The upper polariton (1) has a local minimum and can thus be considered as a quasiparticle
with a positive effective mass. The dashed lines represent the unperturbed modes.

dispersions and properties using the optimal parameters calculated in the previous
section.

For the lowest. crossing at the X-point, where (f = 1 / 2, the corresponding lattice
constant we have to choose according to Eq. (3.10) is a = 0.131 pm. Using these
parameters we find the polariton dispersions in the vicinity of qO in Fig. 3.4, which
shows the modes along the qx-and qy-axis, respectively.

As predicted, we find at the crossing point a Rabi splitting of about 2.2 meV
between the upper (UP) and the lower (LP) polariton mode.

The energy of the lower polariton mode has a saddle point at qO and thus the
assignment of an effective mass is not well defined. The upper (1) as well as the
central (2) polariton, however, Show a local minimum at the X-point. The upper
polariton mode is shown in a two dimensional plot in Fig. 3.5.

The 7r—rotational symmetry of the X—point is reflected in the energy dispersion.
which has a very different appearance along the qx—and qy—directions. This property
can be checked numerically by calculating the effective masses of the polariton modes.
The effective mass of a quasiparticle is defined only at a local minimum or maximum

of the mode as
__‘1 l 82E(q)
(mi) : ————’.
h2 dq

It describes the behaviour of the mode in the direct vicinity of the extremum. The ca]-

(3.13)

47

 

Qy [Zn/a]

 

 

 

0.1.__________.__H_.
0.4 0.45 0.5 0.55 0.6
QxIZH/al

Figure 3.5. The upper polariton dispersion at the X-point. Because of the special 7r—
rotational symmetry of this point the dispersion has a valley at the edge of the Brillouin
zone.

culated effective masses for the polariton modes (1) and (2) can be found in Tab. 3.2.

The masses are expressed in units of the of the cavity photon in—plane effective mass

 

UP (1) CP (2)
rum/mph 1.7- 10—3 0.44
myy/mph 3.32 1.24

 

 

 

 

 

 

 

Table 3.2. The effective masses for the upper two polariton modes at X. The masses are
expressed in units of the photon effective mass in a ideal cavity of length L = 0.171 um
(mph 2 2.52 - 10—51710).

in the lowest mode ( .z : %) at q z 0 in an ideal cavity of length L. This mass can

be calculated as mph : 525: = 2.52 - 10—5 mo, where m0 denotes the electron rest
mass.

As suggested by Fig. 3.5 the upper polariton mass is extremely small in the qx—
direction (about 10—3 mph), while in the qy—direction it is of the order of the photon
effective mass. The latter also is of the same order of magnitude as the effective mass

of quantum well polaritons [15].

48

An explanation of this behaviour is given in the expansion of the unperturbed
cavity photon modes at the point qO 2 (7r/a,0). The leading term in the cavity
modes along the qy—axis is (disregarding the constant term) is of quadratic order,
while along the (LT-direction the linear term dominates. This accounts for the fact
that the effective mass in (Ix-direction is approximately by a factor fig smaller than
the mass in qy-direction, as an expansion of the polariton energies in q; and qy
shows. g is defined as g = 93(1er and has for typical constants for the system under
consideration a value of the order of 10‘3.

The anisotropy of the polariton modes is a direct consequence of the 7r—rotational
symmetry of the X—point. So, in order to find polariton states with an isotropic mass,
which is important with regard to condensation effects we will discuss later, we have
to consider a point with higher symmetry like the M—point of the square lattice or the
W-point of the hexagonal lattice, which have an 1r/2- and 27r/3-rotational symmetry,

respectively. Before we are discussing these highly symmetric points we will closer

 

0.1

0.505 _

2
Figure 3.6. The exciton component of polariton mode (1) 0&1)(q)[ . Only in the direct

 

vicinity of (10 the upper polariton becomes excitonic.

examine the polariton modes at. the X-point.

49

 

”Ill ll

I
‘-

'u 'I I
l'l
.llll
,i
as

 

0.5
qx 0.505

. 2
Figure 3.7. The exciton component of mode (2) a-‘fl(q)[ . Remarkably this mode is

 

purely photonic on the edge of the Brillouin zone.

Since in the vicinity of qO the polariton states can be expressed as a linear com-

bination of the exciton state and the two photon states,
40 : —1.0 ,
[1%) = a§(Q)IQ> +ef,,,1(q) [Té2 >+ (if)h2(q) [T3 ( . l), (3.14)

we can keep track of the excitonic and photonic parts in the new eigemnodes.

5

2
In Figs. 3.6-3.8 the excitonic part ar(q)[ of the three polariton modes (g =

 

1, 2,3) is depicted. Since the polariton state is normalized to one, the photonic part

2
appears as the negative of the Figs. 3.6—3.8.

 

E 2 5

aph1<q)| + |a,,,,2(q>

We see that near resonance we have a strong mode mixing between photons and
excitons. At resonance, the upper and the lower polariton are exactly half excitonic
and half photonic. whereas the center mode is purely photonic. This is a remarkable
result, because although the center mode has exactly the exciton energy at resonance
the corresponding eigenstate is purely photonic.

Note that the center mode’s deviation from the unperturbed exciton mode in

Fig. 3.4 along the qI direction is too small to be seen on this scale. But in fact

50

 

 

0.505

2
(1(13)(q)[ . At resonance this mode is half

 

Figure 3.8. The exciton component of mode (3)
excitonic half photonic.

the mode has a local minimum at (10 with photon-like effective masses in qI—and
qy—direction, respectively (Tab.3.2).

Moreover the x—y asymmetry of X is once again reflected in Fig. 3.7. \Vhile in
(Jr-direction the central mode returns to its excitonic nature very quickly. it remains
purely photonic in the qy—direction on the edge of the Brillouin zone.

Note that except for the well localized region around qO, Fig. 3.7 is the negative
of Fig. 3.8. At the edges where the two modes change their nature from excitonic
to photonic and vice versa the unperturbed exciton mode intersects a single pho—
ton mode. At this intersection points. the polariton modes behave similar to bulk
polaritons depicted in Fig. 1.3.

We summarize that we have created a novel kind polariton states at the edge of the
Brillouin zone, i.e. they have, unlike quantum well polaritons, a finite momentum.
This is associated with interesting effects in the interaction with external photon

modes. These effects will be discussed later in section 3.5. Moreover, we found that

51

 

.---....-.~/ / 7 d2
‘ o o 7
. {a

Figure 3.9. At the M point we distinguish two cases. Ml labels the crossing, where the
closest photon modes intersect (black dots). At the M2 crossing the second closest photon
modes intersect (crosses). The different modes fulfill |Q,~ + q0] = (11/2 for the two cases,
respectively.

the upper polaritons have a photon like effective mass along the zone boundary, as
usual QW polaritons, but along the qty-direction we find an exceptionally small mass,

which is by a factor of 10"3 smaller than the QVV polariton mass.

3.3 Square Lattice: The M-Point

At the M-point we are going to examine not only the lowest crossing (Ml) but also
the second lowest crossing (M2) (Fig. 3.9).

The M-point has a 7r/2—rotational symmetry with regard to the reciprocal lattice.
We therefore expect the effective mass tensor to be isotropic. Moreover the higher
symmetry leads to the fact that at the lowest crossing point (Ml) four photon modes
(Q1 = 0, Q2 = (—27r/a,0), Q3 2 (—27r/a,—27r/a), Q4 = (O, ~27r/a)) are intersect-
ing instead of two at the X-point. In order to tune the intersection to the exciton
energy we have to choose a lattice spacing of a = 0.185 pm if we keep the remaining
parameters in their optimal ratio (R0 = 0.05 pm and L 2 0.171 pm).

Fig. 3.10 displays the polariton modes along the path FM . Note that the central

 

 

1.682 ’ (l)

 

 

 

 

 

(2)

E 1.68 ’ (3)

w (4)
1.678 r (5)

 

 

 

 

0.496 0.498 0.5 0.502 0.504
QX:Qy [27f/al

Figure 3.10. The polariton modes at the lowest crossing point at M (Ml). The modes are

displayed along the path W

mode (3) is an unperturbed photon mode, as along this path the energy of photon
Q2 and Q4 is identical and thus we can introduce a basis in the respective subspace

with one completely decoupled state (Section 2.2.1).

 

 

  

 

 

 

 

 

1.682 *
PS 3
2. 1.68’ ( )
UJ

1.678 '

A A (5) n A
0.496 0.498 0.5 0.502 0.504
(lx [ZN/a]

Figure 3.11. The polariton modes at the lowest crossing point at. M. The modes are
displayed along a path parallel to the qI-axis. Due to the symmetry we find the same

energy dispersions along the qy-axis

Along a path parallel to the qx-axis we find two unperturbed photon modes ((2)

and (4)) which arise because of the degeneracy of energy of the photons Q2 and Q3

as well as Q1 and Q4 (Fig. 3.11).

Note that the increased number of intersecting modes compensates exactly the
effect of a decreased coupling constant, when we go from the X to the M point. Thus,
we find a Rabi splitting of Q = 2.19 meV like in the case of the X-point. Our main
interest lies on the first polariton mode because it has a positive isotropic effective

mass. This mode is depicted in Fig. 3.12 in the vicinity of C10-

 

 

 

 

0.504

0.502
E
Q 0.5
c?

0.498 ..

0.496

0.496 0.5 0.504
qx [277/3]

Figure 3.12. The upper polariton mode at M1. The energy dispersion is symmetric in
qI-and qy-direction.

The mode is half excitonic and half photonic at resonance (qo) and becomes purely
photonic away from of qo, as can be seen in Fig. 3.13.

The LP mode (5) has a strong resemblance with the UP mode (1). The energy
dispersions show the same shape and symmetry, with the difference that at qO we
find a local maximum for the LP mode, while there is a local minimum for the UP
mode. The excitonic part of the LP in the vicinity of the resonance point is almost
identical to the one of the UP 3.13.

As we expected the polaritons show a dispersion that is symmetric in rotations
about 7r / 2 like the lattice itself and we find an isotropic mass tensor as the calculation

of the effective masses at the Ml point. shows (Tab. 33). Note that all the polariton

54

 

0.505

0.5 qy

0
0.495

  

495

(Ix 0.50 I

2
Figure 3.13. The exciton component of mode (1) at Ml agl)(q)[ . At resonance this

 

mode is half excitonic half photonic. An almost identical figure is found in the case of the
lower polariton (5).

masses at M are of the order of 10—3 mph, which is 10—3 times smaller than the usual
QVV polariton mass.

Next we will turn to the M2 crossing where eight photon modes intersect as
depicted in Fig. 3.9. we calculated that in this case the increase of modes is not
able to compensate the reduced coupling constant completely (Tab. 3.1). Instead we
expect a Rabi splitting reduced by the factor V? of Q : 1.385 meV. This is depicted
in Fig. 3.14.

Again we focus on the upper polariton mode for which we find a very isotropic
dispersion relation (Fig. 3.15) with an positive effective mass of m = 2.08 - 10—3 mph,
which is even smaller than the mass at M 1 (Tab. 3.3). By keeping track of the exci—
tonic part of the mode in Fig. 3.16 we find that the magnitude of exciton component
is equal to the one of the photon component at resonance but then drops quickly to

zero at higher distances.

 

Ml M2
UP (1) LP (5) UP (1) LP (9)
3.43-10—3 —3.4-10—3 2.08-10—3 —2.06-10—3
3.43-10-3 —3.4-10—3 2.08-10“3 —2.06-10—3

 

 

mxz/mph

Table 3.3. The effective masses for the upper (UP) and the lower polariton (LP) at M1
and M2. The masses are expressed in units of the photon effective mass in a ideal cavity of

length L = 0.171 nm (mph 2 2.52 - 10—5 mo).

 

 

 

 

 

 

 

 

 

1.681

 

1.68

E [eV]

1.679

 

 

 

 

0.496 0.498 0.5 0.502 0.504
Clx [Zn/a]

Figure 3.14. At M2 eight photon modes are crossing. The resulting nine polariton modes
are displayed along the qI—direction. Note that along this path the unperturbed photon

modes can be grouped in degenerate pairs.

Like in the case of the M] point the LP mode bears a strong resemblance to the
UP mode, which is indicated by the almost identical magnitude of the effective masses
in both cases 3.3. Also, the lower mode shows the same shape and symmetry as the

UP in both the energy dispersion and the excitonic component. (Figs. 3.15 & 3.16).

56

 

qy [27r/a]

       

0.46

 

 

\

0.46 0.5 0.54
Qx [Zn/a]

 

Figure 3.15. The highest polariton mode at M2. The large number of interacting photon
modes provides a high symmetry of the dispersion.

 

0.505 0.495

Figure 3.16. The exciton component of the highest mode at. M2 [(2I(q)|2. At resonance

this mode is half excitonic half photonic. “79 find an almost identical figure in the case of
the LP.

57

3.4 Hexagonal Lattice: The W—Point

In this section we will consider a lattice of quantum dots in a hexagonal arrangement.
A two dimensional hexagonal lattice is made up of equilateral triangles and can thus
be characterized by the two primitive vectors a1 = ((1,0) and a2 = (a/Z, fia/ 2),
where the lattice constant (1 denotes the distance between two neighboring lattice
points. The reciprocal lattice is again a hexagonal lattice with lattice constant a* 2
£752. We are going to examine the lowest crossing at the so called W point of the
reciprocal lattice (Fig. 3.1), which has coordinates (1/2,1/(2\/§)), in units of a*. At
this point there are three photon modes crossing.

Also for the hexagonal lattice the coupling has a maximum for a dot radius of 50
run, if the cavity length has the value L = 0.171 nm. This corresponds to a lattice

constant of a = 0.175 mm and thus a optimal dot density of n = BIB-fig. For these

optimized values we find a Rabi splitting of Q = 2.16 meV. The polariton modes along

1.682 1.682 \

/\ \

0.485 0.5 0.515 0.28 0.288675 0.3
qx arr/(flan qy lain/(«San

 

 

1.68

E [eV]
E [eV]

 

 

 

 

 

 

 

 

Figure 3.17. The polariton modes at the W-point of the hexagonal lattice along the
x-direction (a) and the y—direct ion (1)) are displayed.

the qx-and (lg—axis, respectively, are depicted in Fig. 3.17. Note that the dispersion
relations along the qx-axis are even functions, while in qy-direction they are odd.
The highest polariton mode has an local minimum at the resonance point
(Fig. 3.18) and we find isotropic effective masses of 2.52 - 10—3 172.1,),2 and —2.51 -
10—3 mph for the highest and lowest mode, respectively. The excitonic character of

the mode has a strong resemblance to the modes in the previous cases (Fig. 3.19).

58

 

 

0.2895

 

'5' 0.289
F
E
O>_~ 0.2885
0.288
0.499 0.5 0.501
(lx [277/3]

Figure 3.18. The highest polariton mode at the \V-point of the hexagonal lattice.

One can clearly see that the excitonic component survives over a long range in the
directions where there is no nearest neighbor reciprocal lattice vector present. As
expected the 27r/3—rotational symmetry is reflected both in Figs. 3.18 and 3.19.

In the case of the \V—point of the hexagonal lattice another interesting effect
appears. While at the symmetry points of the square lattice the lower mode beared
a strong resemblance (in shape and symmetry) with the upper mode. in the case of
the hexagonal lattice the lower mode, still having a similar shape. is rotated by an
angle 7r (Fig. 3.20). The same effect can be seen for the excitonic part of the LP in
Fig. 3.21.

This mismatch in the upper and lower dispersion as well as the excitonic com-
ponents is likely to suppress the polariton scattering between these two modes as
polaritons can only scatter to states with a significant exciton component. In order
to confirm this assumption detailed investigations on the polariton dynamics of the

system are necessary.

59

 

2n

 

Figure 3.19. The exciton component of the highest mode at W in the hexagonal lattice
lar(q)|2. At resonance this mode is half excitonic half photonic.

qy [ZN/a]

    

0.499 0.5 0.501
Qx [22r/a]

Figure 3.20. The lowest polariton mode at the W-point of the hexagonal lattice. Note
that it is rotated by 7r with regard to the upper polariton mode.

60

 

 

2
qylfl

 

a 0.505

Figure 3.21. The exciton component of the lowest mode at W in the hexagonal lattice
|o-I(q)|2. It is rotated by 7r with regard to the excitonic part of the upper mode.

61

3.5 Dark Polariton States

Unlike quantum well polaritons, which can have a momentum in the direct vicinity
of zero only, the novel microcavity polaritons described in the previous sections have
a large in-plane momentum, being located at the edge of the Brillouin zone of the
reciprocal lattice. This leads to interesting effects that become manifest in the in-
teraction with external photons in a material of refraction index next, smaller than
the refraction index of the cavity spacer nr. Due to a total internal reflection of the
polaritons at the edge of the Brillouin zone their spontaneous emission lifetime is
expected to be greatly enhanced.

In the following we keep the basis determined by the microcavity symmetry, where
q and kz are the components of the wave vector parallel and perpendicular to the

mirrors, respectively. Since the external photons have an energy dispersion

V (12 + kg, (3.15)

”C (q represents a lower boundary for the
(’1‘.

E25

 

next

it. is clear that the function E (k: = 0) = h

 

energies accessible to the external photons (Fig. 3.22).
Let us consider polaritons at a symmetry point qO on the edge of the Brillouin
zone. These polariton states are partially in the non accessible region of the external

photons, if the polariton energy at the edge E p(C]0) is smaller than the lowest external

 

photon energy Ee$t(q0, k; = 0) = hnfu‘m'

By approximating the polariton energy with the unperturbed cavity photon energy
EP(QO) 6" Eintmo) 1' hfifiqg + (%)2, this condition reads

7?

h.

 

C .
q03> fi-- 08'+‘(—'
um], nr L

)2: (3-16)
which can be rewritten as a condition on the refraction indices as

next < am, (3.17)

62

E [a.u.]

 

 

 

q [a.u.]

Figure 3.22. The dispersion for external photons for several arbitrary values of the (con-
tinuous) parameter k,. The dispersion for [C2 = 0 imposes a lower boundary for the energies
accessible to free photons.

Where (1 =1/ 1+ (Lg—0)?

If this condition is fulfilled there are polariton modes in the vicinity of go that do
not couple to the external photons due to selection rules imposed by the conservation
of energy and in—plane momentum (Fig. 3.24). This effect can be seen as a total
internal reflection for microcavity polaritons for a large in—plane momentum q.

The primary decay process of polaritons is the coupling to the external photon
bath, which gives rise to a short polariton lifetime in quantum wells of the order of
a few picoseconds. If polaritons are unable to decay to external photons due to their
total internal reflection, we expect their spontaneous emission lifetime to be greatly
enhanced.

Although the spontaneous emission lifetime of these QD lattice polaritons can be
greatly enhanced using this total internal reflection. we have to note critically that
there are other decay processes that come into play. such as polariton-phonon or

polariton—polariton scattering.

63

 

 

 

 

Figure 3.23. The boundary between states accessible for external photons in the reduced
zone scheme for the unperturbed cavity modes in the cases (1) next < am, (2) next 2 om,
and (3) next > 012,. Moreover, the figure shows the lowest 2 cavity photon modes along
67 of the square lattice.

In fact, the upper polariton modes we are discussing do not represent the lowest
state of the system. While the scattering processes into the central modes are likely to
be strongly suppressed due to the purely photonic nature of these modes at resonance,
the scattering to the lower polariton mode may significantly reduce the polariton
lifetime.

This effect is not present if we consider polaritons in the lowest mode. These
polaritons have a negative isotropic mass of the same magnitude of the upper polari-
tons. The lower polaritons do not relax to lower and lower energies, but they tend to
accumulate at a certain value of q. This is due to the so called bottleneck effect, which
is a consequence of the vanishing excitonic part in the polariton state away from the
resonance point [48]. The polaritons lose their ability to scatter when their excitonic
nature decreases. At these accumulation points (e.g. along the excitonic branches in
Fig: 3.21) we expect a very long polariton lifetime and a population buildup.

In order to study these accumulation effects or to determine the actual lifetime of
QD lattice polaritons a careful study of the polariton dynamics is necessary and we

expect this to be subject of future investigations.

64

qxl27r/al

 

Figure 3.24. If the condition next < an, is fulfilled there are polariton modes in the dark
region (shaded area). These states do not couple to external photons and are thus expected
to have a greatly enhanced lifetime.

65

Conclusion

In recent years, the field of quantum phase transitions in solids (such as Bose-Einstein
condensation (BEC), Berezinskii-Kosterlitz—Thouless transition (KBT), or superflu-
idity) has attracted remarkable attention as it brings quantum mechanical effects to a
macroscopic scale. In particular, microcavity polaritons are promising candidates for
the observation of a phase transition into a coherent light-matter condensed phase,
because in certain limits they behave as a non-interacting Bose gas. Moreover, their
extremely light mass of the order of 10‘5 electron masses, suggests the possibility of
high temperature condensation. In fact, the critical temperature T c of a BBC in an
ideal, non-interacting Bose gas is inversively proportional to the particle mass [42]

27Th2( " )2/3. (3.18)

kBTc = —— m

As discussed by Imamoglu et al. in Ref. [22], microcavity polaritons might undergo
BEC. This has stimulated extensive experimental studies [3, 14, 24, 32] and controver-
sial theoretical discussions [42, 45]. Undoubtedly, some recent experiments show evi-
dence of some type of condensation of polaritons in semiconductor heterostructures,
but up to now it is unclear what the nature of the observed coherence phenomena is.
Most theoretical works agree that the ideal BEC theory is not applicable for several
reasons. Aside from the fact that the claim of the bosonic nature for polaritons
holds only in the low excitation limit (which is highly questionable in the context
of condensation effects), the most striking argument is that in a two dimensional

system, like the ones studied in the cited experiments, the critical density above

66

which BEC occurs diverges for every finite temperature. For interacting particles this
is a direct consequence of the Mermin—Wagner theorem [30]. Thus, theoretically BBC
in a two dimensional system occurs at zero temperature only. Moreover, BEC theory
describes always a system in thermal equilibrium, which is in direct contradiction with
the short lifetime of polaritons (of the order of a few picoseconds). Snoke therefore
proposed [45] to consider the observed effects rather within the more general concept
of a spontaneous emergence of coherence. Nevertheless, disregarding this questions
of terminology, the mentioned experimental data clearly proves the existence of some
kind of condensation effect for microcavity QW polaritons in quantum well structures.

In this thesis we presented a study of polariton modes of a novel kind, for which we
propose the name of umklapp polaritons, as they arise from and umklapp scattering
of excitons and photons. The investigated umklapp polaritons are the eigenstates of a
periodic array of lens shaped GaAs quantum dots within a microcavity in the strong
coupling regime. This system is of special interest as it represents an intermediate
case between the limits of a zero dimensional quantum dot on the one hand, and of a
two dimensional quantum well on the other. The translational symmetry of the QVV
case, which is lost in the case of one QD, is partially recovered in a QB lattice.

Before turning to the actual physical system we reviewed in Chapter 1 the funda-
mental concepts needed throughout this thesis.

In Chapter 2, we provided a full quantum mechanical treatment of the system un-
der consideration. It turned out that, due to the periodicity of the system, the exciton
states have a quantized center-of-mass motion with an in-plane momentum which is
restricted to the first Brillouin zone of the reciprocal QD lattice. We have seen that
these new exciton states behave like quasiparticles of infinite mass. The quantized
center-of-mass motion accounts for the fact that a quasi in-plane momentum conser—
vation (which is lost in the case of one QD) is restored, as the exciton states couple

to a discrete and yet infinite set of umklapp-photon states. These umklapp states

67

turned out to be the different photon branches in the reduced zone scheme of the
QD lattice Brillouin zone. The resulting umklapp polariton modes have a quantized
in-plane motion labelled by the quantum number q, which is restricted to the 13‘
BZ. Due to the quasi-momentum conservation a polariton q is a mixed state of one
exciton mode of in-plane momentum q and a set of photon states with wave vectors
q + Q, where Q denotes a reciprocal lattice vector. In a good approximation this set
of states is finite, as the coupling to large wave vectors is strongly suppressed by an
exponential form factor due to the finite size of the dots.

QVV polaritons always have a in-plane momentum close to zero due to the steep
dispersion of the photon modes [40]. In contrast, in Chapter 3 we demonstrated that in
our scheme we can create polaritons with both positive and negative effective masses
and a large in-plane momentum at the edge of the Brillouin zone. We calculated
that in the majority of cases, namely the cases that have at least a 27r/3—rotational
symmetry, this new type of polariton has an extraordinary small, isotropic effective
mass of the order of 10—8 electron masses.

We studied the umklapp polariton dispersions for different symmetry points of the
square and hexagonal lattice, respectively. In all cases a Rabi splitting of approx. 2
meV was found. Since this value may appear to be very small, we should mention that
in II-IV compounds like CdTe, a much larger Rabi splitting at the edge of the Brillouin
zone is expected. Moreover in QW’s a significant increase of the Rabi splitting can
be achieved by the use of stacked layers of QW’S. The increase is approximately
proportional to the square root of the number of QW’s and thus huge values for
the QW vacuum field Rabi splitting (in CdTe up to 26 meV) are realizable [2, 39].
The investigation of the effect of stacking layers of QD lattices (i.e. effectively an
3D lattice with small extension in z-direction) could be an interesting project for the
future.

Furthermore, we showed in Section 3.5 that the system can be designed in such

68

a way that the polaritons accumulate in regions where the spontaneous emission
lifetime is greatly enhanced due to the inability to fulfill both energy and momentum
conservation in the interaction with external photons. We pointed out that for both
the positive and negative mass polaritons a population buildup is likely to happen,
due to the special dispersion profile and a bottleneck effect.

Certain properties of the umklapp polaritons (as the small mass and the enhanced
spontaneous emission lifetime) make them attractive for quantum condensation ef-
fects, suggesting very high critical temperatures (Eq. (3.18)). However, we have to
stress the fact that the calculations were conducted under the premise of a single
excitation and, as we stated above, the presented model only applies to many-bodyr
problems, if the lateral exciton confinement does not interfere with the bosonic char-
acter of the quasiparticles. Only in this pure bosonic case the statistical properties
can be studied exactly. Although Ref. [50] suggests that for a ratio between the dot
size and the exciton Bohr radius larger than 5 (for our system we find this ratio to
be 211% z 10) the bosonic behaviour is recovered, further investigations are necessary
in order to extend our model to a analysis the statistics of a many polariton system.

Another application for the system studied here could be found in the field of

quantum information. The natural exciton states are superpositions of localized states

on the different QD’S
]q) = Zciqu‘ pf). (3.10)

i
Thus in the superposition of multiple polariton states interference effects may lead
to a controlled pattern in the occupation of QDs in the lattice. This can be used
for a selective entanglement of quantum dot qubit states in the realization of scalable
quantum information devices [35, 51].

We conclude that the novel type of quasiparticle called umklapp polaritons, which

we studied in this thesis, has many properties. which are attractive for both the field

of quantum computing and the field of quantum condensation. Further theoretical

69

as well as experimental investigations of these novel polaritons seem to be interesting

and promising research projects for the future.

70

 

APPENDICES

APPENDIX A

Derivation of the Effective Mass Equation
in Bulk

In the follow 111g we are going to show how to derive the efftctlxe mass equation (1 18)
from the many-body Hamiltonian (1 6)

_Z(A+’

 

h2k2
W :—

 

 

2mv CIR cm
X B
1 1
+ Z fk1k2k3k4cvlqCck2cck3cbk4 (A'l)
k1k2k3k4
25

 

“e nerd to find an equation that the ((infi( rents A( ,k') have to fulfill in order
to make [‘III) 2 Zkk’ A(k k ) ckcvk’ [(190) an eigenstate of 1155 i C [\II ) has to satisfx
the stationary Schrédinger equation

ILJWI)=I?NQ).

(A.2)

Therefore we are gomg to examine the action of H J; on our trial exr iton start piece

by piece “e start with the part due to conduction ele( trons A

.4|\le)= :2: (A+ Hhk12)c]

’r ,
‘ZmC Ck1CCk1CckCek’ W0) A09 k )
k1 k,k’

 

(A53)

Kl
l0

The product of creation and annihilation operators can be simplified by applying the

fermion commutation relations

[CUk’ Cllkl] + : 6k,kl6a,al. (A.4)

Using these rules we find:

CIRICCklcikak’ '2 5kl,kC:klcvk’ 1” CZklclkcvk’CCkl' (A.5)
If we take into account that the last term on the RHS vanishes, when acting on the
ground state [(DO), Eq. (A.3) becomes
hk 2
A [4%) = Z {(A + (2 ) )A(k, k')} Clkcvk, [@0) (A6)

k k’ me

To deal with term B we have to simplify a similar term.

 

’r . f _ 1 t t
C>L'k1(’l’klcckc'z,ik’ —Cckcvk' + CUleckcvklc-vk’
=0ch +6 c c)r —c c]c 0] (A7)
ck vk' k1,k' ”k1 ck ”kl ck ’vk' vkl’ '
where again, the last term produces zero acting on the ground state. With this

simplification the action from B on the trial state reads as

(hk1)2 hk’ 2
B [411.) 2 Z Z W _ (37% A(k,k’) clkcvk/ |<I>0) (A.8)
k,k’ 1‘1

The calculation of the third term is slightly more complicated as we have to calculate

a product of six operators.
cT c]L c c (.1 c [@)——6 c] cT c c [(D) (A9)
“'ka 'ck2 ‘Ck3 ‘Uk4 'ck 'vk’ O — k33k ‘vkl Ck2 ”1‘4 ‘vk’ 0 '

The remaining product of four electron operators was already calculated in (A.7) an

so we finally find
0 Wm)

, I f - .t
=2 2 fk1k2k3k4’](k’k){6kyk36klak4cckgcvk’T6kk3ok1k’cckgc‘t’k4}ITO)
k,k, kl vk2’k3ak4

=2 Z(A(k2ak,)fkl,k,k2,k1_A(k17k2)fk2,k,k1,k’) ClkCz-k’]¢0> (A10)
k,k’ k1.k2

73

The first term on the RHS plays the role of a constant, diverging self-energy we are
going to suppress in the following. By plugging in the expression for the matrix
element f given by (1.5) and by exploiting the fact that the functions Clkcvk' [(130) are
linearly independent we end up with an equation for the coefficients A(k, k’) when

we put all the three parts together.

2 » I 2 7re2
(E — E0 — A) A(k, k’) = ((33)— + (119—) A(k, k’) — Z :Vq2A(k + q, k’ + q)

2mc 2mg q

 

(A.11)
By finally introducing the new coefficients

Aq(K) := A(K + (mC/M)q,K — (~772..L./ill)q) we obtain Eq. (1.15).

APPENDIX B

Evaluation of the Interaction Matrix

Element

In this appendix we are going to calculate the expression (2.34)
X = Z tk2.k1A(k1.k2),
klkg W

in order to complete the exact derivation of the interaction Hamiltonian (2.46) in
Section 2.1.3. We carry out the calculation for the E—mode only and present the solu-
tion for the M-mode witch is obtained in an analogous manner. We start by writing
the explicit expression for the matrix element t using the real space representation of
the Bloch functions in Eq. (1.3) and the form of the vector potential for the E—mode
(Eq. (2.22))

tk2.k1 = (kgv |A(r)r|k1c)

v,c

1 ‘ . _ . .
Z Z V /drezuq_k2)erfE(’0'9(kZz) (ALEe—MP + AkEe’qp) 111:2,1,(r)ruklc(r).
q

(13.1)

where Ck = M6 nikv, (E = (j x :3 and as before 1‘ = (p,z) and k = (q, kg).
0 1"

Using the Rotating Wave Approximation we can drop the term proportional to

ARE because in the Hamiltonian it would appear as a fast oscillating term of the form

 

AkEa. Inserting the latter relation to Eq. (2.34) we find

. . 1 'k _ _'
X = ZCkEEALE / dr V Z A(k1, 1.2).)“ 1 k2)r cos(kzz)e "qp‘uizv(r)ruklc(r).
q k1,l(2

(8.2)
At this point we will exploit the fact that the sum in k and k' are restricted to
the first Brillouin zone of the quantum dot lattice. This Brillouin zone is very small
compared with the crystal Brillouin zone, in which the Bloch functions are defined.
In fact for reasonable values for the two lattice constants the volumes of the two zones

differ by a factor of approx. 109. Therefore to a very good approximation we can

substitute the k dependent Bloch parts by the functions for k =-.. 0

ukc(v)(r) % U’Oc(v)(r) E uc(v)(r) (B3)

Within this approximation the only It dependence is inside the curly braces and one
can easily identify this expression as the exciton envelop function taken with the

electron and the hole at the same position r

$ 2 A(k1,k2)ci(k1‘k2)’ = Mr, r) = x<p><1>13<ma(new). (8.4)
klvk2

Now, in order to evaluate the remaining integral

I = fdrx(p)<I>1s(0)099(z)05,,(z)cos(k;z)e_iqpu:(r)ruc(r), (B5)

we decompose the integral over the whole space into a sum of integrals over the atomic

/ dr = )1: fwd”, (8.6)

where r,- = r — r? and r? denotes the ith atom in the crystal. Note that r,- runs only

unit cells as

over a simple crystal unit cell and thus its magnitude has an upper limit defined by

the interatomic spacing (a: 0.1 nm). None of the functions appearing in Eq. (B5)

76

changes noticeably within a unit cell except for the Bloch parts u; and uc.
0
I = Z [W draw.- + [1883(0sz mac? - zacoswz? - 2.))
i

x e’zQ(pi_p1)'u;(r?+ r,)(r? + ri)uc(r? + I‘i)

_' 0 *
zZx(p?)<f>13(0)d)e(z?)¢h(z?)cos(kzz?)e qul fUCdri'uv(ri)ri’“'6(ri)a (B7)

2
where we used the periodicity (ac/”(r9 + r,) = tic/”(13D and the orthogonality [13]
(f dru;(r)uc(r) = 0) of the Bloch functions. The remaining integral is independent

of the index i and we define this quantity as

1

= — dru.:(r)ruc(r), (38)
V00 UC

”on

where VUC is the volume of a crystal unit cell. By retransforming the sum into an

integral 2, ——> Vii—C f dr we find
I : (pls(0)92(9)1(kz)ucv» (B-Q)

in which we defined [(153) : fdzc§€(z)¢h(z)cos(kzz) and X(q) = fdpx(p)e_iqp. Note
that all the information about the position of the dot is contained in flq) = )Z'J-(q).
Gathering the results we finally can write down a compact form for the expression
X
X :

 

ie

)1 ~ .
0J2: :ng(R,-)ALE (13.10)
‘1

where we defined the coupling constant

- . 1 - -
9kE(Rj) : zC'v‘l’.’r-(I)1s(0)ECka((I)I('Iiz)(€Eucv)- (13-11)

Note that E E ucv is a constant independent of the unit polarization vector 6121101: 2 um,
In a equivalent calculation one finds the relation between the coupling constants for
E- and M-mode to be

- .kz -
ng(Rj) = 979mm). (13-12)

77

where we neglected the polarization component perpendicular to the mirrors, as the

corresponding coupling term is proportional to

11(3) = dwe<z>¢h<z>smflzt (13.13)
L L

and is zero if the z-direction confinement is symmetric. This is equivalent to the state-
ment that in this limit we exclusively couple to excitons that are polarized parallel

to the mirrors.

78

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79

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83

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