Theory for Light-Matter Interaction in Semiconductor Quantum Dots

Theory for

Light-Matter Interaction

in Semiconductor

Quantum Dots

A dissertation submitted

for the degree PhD at

the Universität Bremen

with permissionof the Institutefor Physi s and Ele tri alEngineering, University of Bremen

Supervisor:

Prof.Dr. Frank Jahnke

Firstexaminer:

Prof.Dr. Frank Jahnke,UniversitätBremen

Se ond examiner:

Prof.Dr. Andreas Knorr,Te hnis he Universität Berlin

Dateof submission: April21, 2008

Oral Defense: July17, 2008

A mi ros opi theory is developed and applied to des ribe lumines en e from semi

ondu tor quantum dots (QDs). The radiative emission dynami s is studied by the

investigationof time-resolved photolumines en e. Spe ial emphasis is pla ed on the

role of arrier orrelations and the dieren es between QDs and atoms. From the

most general form of the theory a laser model for QDs in mi roresonators is devel

oped, whi h is the entral a hievement of this thesis. In this model semi ondu tor

ee ts anbein luded ina onsistentmanner. Goingbeyond therateequationlimit,

we al ulatethe rst-and se ond-order orrelationfun tionsto hara terizethe laser

threshold properties, whi h are, in the lassi al sense, no longer well dened in ur

rent state-of-the-artmi ro avity lasers withhigh spontaneous emission ouplinginto

the laser mode. To underlinethe lose onne tion toappli ations, allresults are pre

sented togetherwithresultsfromre entexperiments. Togainadeeperunderstanding

of the derived laser theory and of the dieren ebetween QDs and atoms, a detailed

omparison with quantum-opti al models is performed, namely with the rate equa

tions, a master equation approa h and the Liouville/von-Neumann equation for the

I amindebted to FrankJahnke, who has supervised my work and who ithas been a

pleasure towork with. I amespe iallygrateful that he madeit possible forme to go

totheplm n onferen einCuba,whi hwasveryenjoyable. AtthesametimeIwould

liketothank JanWiersig,with whomI spentmost ofmy time workingtogether. All

the best foryour newpositioninMagdeburg,and stay asrelaxedas youare and you

willbe agoodboss...

Many olleagues have made the past three and a half years a olourful experien e,

rsttomentiontheBremen-basedteam: PaulGartner,SandraRitter,Mi haelLorke,

JanSeebe k andFrithjofAnders. Ithasbeenmypleasuretobeabletoworktogether

with olleaguesinexperimentalestablishments,ofwhomIwouldliketomentionSven

Ulri h,ThomasAuer, Serkan Ates, Thorsten Berstermann and Manfred Bayer.

Whenit omestothesubje tof physi sand thePhDthesis,Ibelievethatmyfriends

andfamilyhavenotsueredtoomu hundermyprofession. Nevertheless,Iwouldlike

to happily thank them here: Linus and Jana, who have made it possible on e more

for us to spend some time of my graduation period in the same pla e, Norman and

Stefan for goodpart-time diversion at work and o work, Ivona for many su essful

evening events, Christian for lots and lots of distra tion, Maren for a heaps of fun

and some workshop time, aswell asSta ey and my familyfor always being there.

My a knowledgements alsoto Martin Ze h for the design of the thesis over.

FinallyI would like topointout that I am grateful toeverybody who doesnot stop

1. Introdu tion 1

2. Light-MatterCoupling inSemi ondu tor Quantum Dots 7

2.1. System and Hamiltonian . . . 9

2.2. Cluster ExpansionMethod . . . 16

2.3. Equations of Motion . . . 18

2.4. Stationary Photolumines en eSpe trum . . . 25

2.5. Time-Resolved Photolumines en e. . . 29

2.6. Time-Resolved Photolumines en eof Doped and Undoped QDs . . . 33

2.7. Time-Resolved Photolumines en eof QDs inMi roresonators . . . . 35

2.8. Con lusion . . . 42

3. Laser Theory for Quantum Dots inMi ro avities 45 3.1. Preliminary Considerations about QD Mi ro avity Lasers . . . 48

3.2. Dynami Laser Equations . . . 53

3.3. Numeri al Results. . . 64

3.4. Comparison with Experiments . . . 70

3.5. PulsedandContinuous-WaveEx itationinAtomi andSemi ondu tor QD Lasers . . . 74

3.6. Con lusion . . . 81

4. First-OrderCoheren e in Quantum-Dot Mi ro avity Lasers 83 4.1. First Order Coheren e . . . 83

4.2. Cal ulation of Two-Time Quantities . . . 84

4.3. Comparison with Experiments . . . 87

4.4. Con lusion . . . 89

5. Comparison withAtomi Models 93 5.1. Rate Equation Limitof the SLE . . . 94

5.2. The Master Equation . . . 96

5.3. Extended Jaynes-Cummings Model for a SingleEle tron System . . . 102

5.4. Con lusion . . . 118

Appendices

124

A. Appli ationof the Cluster-Expansion Method 127

B. Interplay of Time Constants 129

C. Operator Equationsof Motion in the Density MatrixFormalism 131

D. Obtaining the Rate Equations fromthe Liouville/von-NeumannEquation 135

E. Numeri alMethods 137

E.1. Solving the SLEinTime . . . 137 E.2. Solving the Laser Equations . . . 137

1.1. Lumines en e from olloidalquantumdotsofdierentsizeand ompo sition . . . 2 1.2. Twoexamplesformi roresonators: aVCSEL-pillarand diskmi ro avity 3

2.1. Bandstru tureforGaAsands hemati magni ationaroundthe

Γ

-point 10 2.2. S hemati drawing of the energy levelsof a oupled QD-WL system . 11 2.3. S hemati of the relevant pro esses inthe QD-WL system . . . 13 2.4. Time evolutionof the arrierpopulationintoa steady state after wet

ting-layerex itationat 10K . . . 27 2.5. Stationary PLspe tra for

T = 200

Kand

T = 77

K . . . 29 2.6. Logarithmi plot of the time evolutionof the quantum-dot photolumi

nes en e . . . 32 2.7. Time-resolvedPLfordopedandundopedQDsex itedintothe

p

-shell:

theory and experiment . . . 34 2.8. Time evolution of ele tron and hole populations in the

s

-shell of un

doped and doped QDs . . . 35 2.9. SEM image of apillar mi roresonator with InGaAsQDs . . . 36 2.10.SEM image of a typi al VCSEL stru ture with embedded II/VI QDs

and strength of the ele tromagneti eld forthe fundamental mode in a VCSEL-stru ture . . . 38 2.11.Low-ex itationtime-resolved PL for mi ropillarswith dierent diame

ters: theory and experiment . . . 40 2.12.PLde ayofQDsina5

µ

m avityatdierentex itationpowers: theory

and experiment . . . 41

3.1. Low-temperaturePL spe trumof twohigh-Q 4

µ

m mi ropillars . . . 48 3.2. S hemati of the QD laser model . . . 50 3.3. S hemati des riptionof athermal, oherent,and sub-poissonianlight

sour e . . . 51 3.4. Cal ulatedPLof anensembleof QDswithinhomogeneous broadening

for identi al QDs with maximum oupling strength and on-resonan e transitions, and for identi alQDs with averaged ouplingstrengths . 57 3.5. Cal ulated input/output urve and auto orrelation fun tion

g

(2)

_{(τ =}

3.6. Sub-threshold values of

g

(2)

₍₀₎

for

β = 1

. . . 67 3.7. Swit h-on os illations of the arrier and photon population with and

without arrier-photon orrelations . . . 69 3.8. 3

µ

mmi ropillarwith InGaAsQDs as the a tive layer . . . 70 3.9. Experimental gure: Integrated intensities and auto orrelation fun 

tions for a 3 and

4 µ

m mi ropillar under non-resonant pulsed and w ex itation . . . 71 3.10.Input/output urves together with the se ond-order oheren e fun 

tion: theory and experiment . . . 72 3.11.Measured oheren e times for a3

µ

mmi ropillar . . . 73 3.12.Input/output urves for an atomi laser model under w and pulsed

ex itation . . . 76 3.13.Time evolution of the number of photons

n

and ex ited two-level sys

tems

N

after ex itationwith dierent pulse widths inanatomi system 76 3.14.The same asabove,but for xed pulsewidthsand dierentintegrated

ex itation intensities . . . 77 3.15.Input/output urves for the semi ondu tor ase for pulsed and w ex

itation,resultsare shown withand withoutthe suppressionof satura tion ee ts due toPauliblo king . . . 79 3.16.Results ofthe semi ondu tormodel: time evolutionof photonnumber

and arrierpopulations for dierent pulse durations . . . 80

4.1. Theoreti alinput/output urvesand oheren etimesforvariousvalues of the

β

parameter . . . 86 4.2. Spe trumandinput/output urvesforthefundamentalmodeemission

from w-ex ited 1.8, 2.6, and 6

µ

mpillars: theoryand experiment . . 90 4.3. First order oheren e fun tion as a fun tion of the delay time: theory

and experiment . . . 91 4.4. Pump-power dependen e of the oheren e time

τ

c

for three dierent

mi ropillars . . . 92

5.1. Four-level laser s heme des ribed by a set of rate equations . . . 94 5.2. Input/output urves obtained fromthe atomi rate equations. . . 96 5.3. S hemati representationoftherelevantpro essesinamasterequation

birth/death model . . . 98 5.4. Comparisonoftheresultsfromthe atomi masterequationandamod

ied single-ex itation version of the semi ondu tor theory: auto orre lation fun tion and input/output hara teristi s . . . 101 5.5. Leveldiagram of aone-atom laser system . . . 103

5.6. `Translation' between the atomi model introdu ed in the previous gure and the semi ondu tor model . . . 112 5.7. Comparisonof the exa t Liouville/von-Neumannmodel withthe trun

Whenever the dimensionalityof a system isredu ed itgives rise toa variety of stun ningnewee ts. Tonameanexamplefromatomi physi s, onsideradilutedatomi gas of bosons that an undergo aphase transitionand forma Bose-Einstein onden sate at nite temperatures [5, 36, 43℄. This is true if the gas is not restri ted to fewer than three spatial dimensions. A homogeneous Bose gas in two or one dimen sion, however, is subje t to enhan ed phase u tuations that make the formation of one oherentphase impossible,a fa tstated inthe Mermin-WagnerHohenberg theo rem: Spontaneous symmetrybreakingin ontinuousone-ortwo-dimensionaltheories annot o ur at nite temperatures [34, 60, 90, 91℄. Also in semi ondu tor systems lower-dimensionalee tsplayanimportantroleandareanintegralpartoftoday'sde vi eappli ations. The dis overyofthe integerandfra tionalquantumHallee t has wonNobelprizesinphysi s[75,80,107℄,andbandengineeringreliestoalargeextent onthe modi ations the density of statesexperien es under dimensionalredu tion.

Thisredu tionismostnaturallya hievedinnanos alesystems. Infa t,thephysi sof low-dimensional semi ondu tor nanostru tures is a more vivid and rapidly evolving eld of resear h than ever, owing to the advan es made in growth te hnologies and manipulationonsu hsmalllengths ales. Inaseriesofin reasing onnement,where quantumwellsand quantum wiresrestri tthe propagationof arrierstotwoand one singledimension,quantumdotsdenetheultimatelimitofnofree arrierpropagation atall. Quantumdotsarezero-dimensionalsemi ondu torstru turestypi ally onsist ingof thousandsup tohundreds ofthousandsof atomsand beingtens ofnanometers insize. Theprominentee tofthisredu tionindimensionalitymanifestsitselfinthe freedensity ofstates. The ontinuousbehavior typi alfor bulkmaterialbe omesdis rete. In ontrasttoatoms,however, thelevelspa ingisdeterminedbythesize,geom etryand ompositionof thequantum dots, makingitpossibletotailorthe ele troni andopti alproperties. Thedis reteandtunableopti alspe trumisthefoundationof quantumdotresear handexertsavastimpulseonoptoele troni devi eappli ations. With quantum dot emitters one an, in prin iple, over the whole visible spe trum andbeyond. AavorofthisisgiveninFigure1.1,wherelightemissionfrom olloidal quantumdotsisshown. Nexttotherangeofopti albertele ommuni ationnetworks (1300/1550nm),laserdiodeswith wavelengths inthe blueandgreenpart ofthespe  trumare ofgreat interestfor imagingdevi es. Laser diodes [8,31, 95, 104,138, 144℄,

Figure 1.1.: Lumines en e from olloidal quantumdots of dierent size and omposition. Thepeakwavelengthof the emission an be tuned all the waythrough the visiblerangeof the opti alspe trum and fartherintothe infrared. CourtesyofDr. Andrey Roga h[1℄.

single-photon dete tors and eld ee t transistors [48, 131, 132℄, and non- lassi al lightsour es [3, 15, 94, 96, 105, 112, 119, 126, 134, 158℄ for fundamentalstudies and quantum informationte hnologyare amongthe most relevant appli ations.

Inanalogytothe three-dimensional arrier onnementinquantumdots,light anbe onned in all spatial dire tions. This is a hieved in opti al mi ro avities, see [146℄ andreferen estherein. Asitispossibleforquantumdotstotailortheiropti alproper ties,the avity anbebroughtinlinewiththeemissionpropertiesofthegainmaterial by hoosing their stru turea ordingly. Thesemi rometer-s ale resonatorspossessa dis rete photoni density of states and a large free spe tral range due to their small size, whi h is omparable tothe wavelengthof the lightthey onne animportant quality for the a hievement of single-mode lasing. The photon onnement is based eitherontotalinternalorBraggree tion. A ombinationofboth onnementme h anisms is employed in the so- alled VCSEL (verti al avity surfa e emitting laser) stru turesusedformi ropillarlasers[151℄. Twoexamplesformi ro avitiesare shown inFigure 1.2, ami ropillarand a mi rodisk resonator.

By embedding quantum dots or other kinds of emitters in mi roresonators we enter the fas inating regimeof avity-quantum ele trodynami s(CQED). The ouplingof the emitter to a single mode (or more) of the resonator leads to a variety of new ee ts that an hangethe emissionrate fromthegain material,thedire tionalityof the emittedlightand the spe tralproperties. Ifthe ouplingis strong,emission into the avity mode is followed by reabsorption in a reversible y le of energy ex hange between the emitter and the avity. The strong oupling leads to the formation of polariton quasiparti les [113℄. This regime plays a role in fundamental studies of light-matterintera tion. A topi fervidlydis ussed isthe possibilityof Bose-Einstein ondensation of polaritons [39, 68, 133℄. From an appli ation point of view, the regime of weak oupling plays a more important role [93, 146℄. Photons emitted by

mi ro avity, taken from Refs. [113℄ and [94℄, respe tively. In the pillar ase the light is onned due to total internal ree tion at the sidewalls and due to alternating layers of materials with dierent refra tive indi es (distributed Bragg ree tors) along the verti al dire tion. In the maximum of the onned eld a layer of quantum dots is pla ed as the gain medium. In ase of the mi rodisk, long-lived modes are so- alled whispering gallery modesthatexist loseto the perimeter ofthe disk.

the gain material leave the avity at a rate determined by the quality fa tor of the avity. The spontaneous emission into the avity, however, is hanged due to the modied lo al density of opti al modes inside the resonator, a phenomenon alled the Pur ell ee t [110℄. In avities ontaininga lo alizedlong-lived mode, expressed by a large quality fa tor

Q

and a small mode volume

V

, the spontaneous emission rate be omes in reased

∝ Q/V

by fa tors of ten and more [82, 124℄. This ee t is extensively used in the fabri ationof e ient low-threshold laser devi es, where the spontaneousemissionintothelasermodeisenhan edovertheothernon-lasingmodes [144℄. The importantquantity hara terizing a laser is the

β

-fa tor that determines thefra tionofthetotalspontaneous emissionthatis hannelizedintothe lasermode. In onventional gas lasers typi ally only one out of

10

6

photons is emitted into the laser mode. In quantum-dot based semi ondu tor lasers values of

β

lose to unity have been obtained [31, 104, 134, 138, 144℄.

Typi ally, in the input/output power tra e of onventional lasers, the laser thresh old is developed as a sudden intensity jump over several orders of magnitude. In smalllaserswithsu he ientspontaneous emission oupling,however, thethreshold graduallydisappearstobe omeawidesmeared-outtransitionregion[118,152℄. Nev ertheless, even if the thresholdis notvisibleatallinthe intensityoutputof the laser devi e, the emitted light hanges qualitatively fromthermal to oherent as the dom inatingme hanism hanges fromspontaneous tostimulated emission. This behavior

is expressed in the photon statisti s, whi h is ree ted in the photon se ond-order orrelationfun tion. It servesas anunambiguousidentier forthe oheren e proper ties of the emittedlight,anda knowledgeof the oheren e propertiesisof uttermost relevan e for the development of high-

β

laser diodes.

The view of quantum dots as arti ial atoms ree ts the histori al development in the eld,and sodoesitseem tojustify the extensive use ofatomi modelstoexplain quantum-dot-relatedphenomenainthe literature. Quantum dots,however, are semi ondu tormaterialsandsemi ondu toree tsinuen etheirlumines en eproperties in several ways atomi models are in apable of a ounting for. It is the purpose of the work presented in this thesis to devise new models on whi h basis lumines en e from quantum-dot-based systems an be des ribed. In parti ular this in ludes the development of a laser theory for quantum dots in mi ro avities that allows for a systemati in lusion of semi ondu tor ee ts. The al ulation of the rst- and se  ond-orderphoton orrelationfun tionsisanimportantpartofthis thesisandthekey toabetterunderstanding of urrentstate-of-the-artlaserstru tures. Thetheoreti al models are presented alongside numerous results from experimental ollaborations. This ombinationyieldsa onsistent pi ture ofthe underlyingphysi s.

Outline

This thesis overs three major topi s: The fundamentaldes ription of photolumines en efromquantum dotstogetherwiththe introdu tionof theemployed mi ros opi model and the equation-of-motion te hnique, the development of a semi ondu tor laser theory, and an in-depth omparison to various atomi models with dierent degrees of sophisti ation.

The des ription of lumines en e from semi ondu tor quantum dots by means of a mi ros opi theory is the topi of the rst hapter, leading to the introdu tion of the semi ondu tor lumines en e equations. The theoreti al methods used in the re mainder of the thesis are explained and results are presented for the spe trally and time-resolved photolumines en e. Spe ial emphasis is pla ed onthe role of semi on du toree ts and dieren es toatomi models.

Basedonthetheoreti alba kgroundand insightobtained intherst hapter,alaser theory for quantum dots in mi ro avities is developed from the semi ondu tor lu mines en e equations in the se ond hapter. By in luding higher-order orrelation fun tions a ess is granted to the oheren e properties and the threshold hara ter isti s of the laser devi e. The appli ability of the developed theory is demonstrated

itly onsidering the time evolution, we point out dieren es in the output behavior of a laser pumped by pulsed and ontinuous wave ex itation. With respe t to fre quently used atomi systems, we dis uss the well-known relation of the

β

fa tor to the threshold in the input/output hara teristi s. This point is of importan e for the interpretation of experimental results, as vital parameters are frequently overes timateddue to the inappropriateusage of atomi models.

Inthe fourth hapter,our theory isextended tothe al ulationof expe tationvalues that depend on two time arguments. Together with experimental results, the rst order oheren e properties of semi ondu tor quantum-dot lasers are analyzed. This in ludesthe two-time rst-order orrelation fun tion,fromwhi hthe oheren e time of the emittedlight an be obtained.

Chapter5isadevotedtheory hapter,wherea`two-level'versionofthesemi ondu tor laser theory isopposed tothree dierentatomi models. Inherent to the many-body problemis the appearan e of aninnite hierar hy of oupled equations. To obtaina losed set of equations, this hierar hy must be trun ated by an appropriatemethod. Firstly,we showhow the well-established rate equations are obtained from the semi ondu tormodelintheatomi limit. Se ondly,by omparingtoabirth/deathmodel for the diagonal density matrix, we are able to verify the validity of the employed method used to trun ate the innite hierar hy of equations of motion. Finally, we onsider the single-atom laser, a system for whi h the exa t density matrix an be obtained. This parti ularsystem doesnotonlyoeron e morethe possibilitytotest the trun ation method,but also providesinsight into the role of dephasing and s at tering termsan important step towards the des ription of the single-quantum-dot laser.

Quantum Dots

Semi ondu tor quantum dots (QDs) are like atoms in many aspe ts, but not in all. Mostsigni antly,the band-likeenergy dispersionof the bulkmaterialis augmented by dis rete levels due to the islands of atoms we all QDs. Lying energeti ally be low the states of the bulk material, arriers an be aptured in the QD states and re ombine at a xed transition energy, like in an atom. The tunability of this tran sition energy is one of the biggest advantages in QD-based systems. In prin iple, optoele troni devi es with avast range of emissionwavelengths an be designed by hoosing a parti ular material system, geometry and surrounding. Appli ations in lude lasers and non- lassi al light sour es, as well as fundamental resear h in the eld of avity-quantum ele trodynami s[82, 93℄.

The foundationof allappli ationsis the understandingof the lumines en ebehavior of QDs. Due to their dis rete level spe trum, QDs are often onsidered as `arti ial atoms' and treated by means of atomi models. Predi tions from these models in ludeanexponentialde ay behavior ofthe photolumines en e (PL), whi hstandsin ontradi tion with many experimental ndings [17, 19, 77, 82, 101, 124℄. As many- parti le physi ists, we often use this behavior to stress that QDs are not arti ial atoms, but semi ondu tor systems with a large number of ele trons and holes that are subje t tomany-body ee ts and orrelations. In parti ular ases this an mean thatQDs behavelikewe expe t itfromatomi systems, but itis not truein general.

One promising approa h for the des ription of PL is the equation-of-motion te h nique [70, 71℄. The resulting semi ondu tor lumines en e equations (SLE) have previously been used to study PL spe tra [65℄ and ex iton formation [62℄ in quan tum-wellsystems,and the PL-de ay dynami sofanensemble of QDsembeddedinto a mi ro avity [124℄. It is well-known that the equation-of-motionte hnique leads to an innite hierar hy of orrelation fun tions due to the Coulomb and the light-mat ter intera tions. A systemati way to trun ate this hierar hy is found in the luster expansionmethod that is used throughoutthis thesis [45, 46, 63℄.

Fromanappli ationviewpointtheuseofQDsasana tivemediuminmi roresonators is of great relevan e, reating the possibility to ontrol the spontaneous emissionto

a large extent. If emitters are pla ed inside a avity, the modied opti al density of states an ause an in rease or suppression of the spontaneous emission time, an ee t rst predi ted by E. Pur ell in the 1940s [110℄. While the inuen e of Coulomb- orrelated multi-ex iton states on opti al spe tra of QDs has been investi gated by several groups [10, 12, 13, 16, 27, 37, 59℄, mu h less is known about the inuen e of orrelationson the spontaneous re ombinationdynami s. Time-resolved photolumines en e studiesprovidedire t a essto the e ien y of arriers attering pro esses after opti al ex itation with short pulses [97℄ and to the modi ation of the spontaneous emissionlifetimefor QDsinopti al avitiesduetothe Pur ellee t [77, 82, 101℄. The theoreti al des ription is rather hallenging, be ause it requires not only a omputation of arrier s attering and orrelations, but also a full quan tum-me hani al treatment of the light eld. The derivation of the semi ondu tor lumines en eequations omprises the quantizationofthe ele tromagneti eld. Con sequently,spontaneous emissionisnaturallyin luded andthisapproa hiswellsuited for the des ription ofthe dis ussed systems.

Inthis hapterwebegin withthe derivationofthe semi ondu torlumines en eequa tions, starting with a des ription of the model system, the single parti le properties andthemany-bodyHamiltonianintherstse tion. InSe tion2.2theequation-of-mo tion te hnique is explainedand asystemati lassi ation-and trun ation pro edure ofoperatoraveragesisintrodu ed. Inthe derivationoftheSLEinSe tion2.3,spe ial emphasis ispla edonthe role of arrier orrelationsintrodu edby the Coulomband the light-matter intera tion. Correlations between arriers and photons are of key interestin thedes ription of the statisti alpropertiesof laser lightand are dis ussed in Chapter 3. Numeri al results are presented in Se tion 2.4, where the stationary spe trum is investigated. The ee t of orrelations on the time-resolved photolumi nes en e from QDs is studied inSe tion 2.5 and limiting ases are dis ussed. In the framework of the DFG resear h group Quantum Opti s In Semi ondu tor Nanos tru tureswehave ollaboratedwithtwoexperimentalgroups,sothatourtheoreti al analysis is omplemented by joint results: In Se tion2.6 the emissionintofreespa e from unstru tured QD samples is investigated. Comparing the emissionfrom doped and undoped QDs, on lusions are drawn about the inuen e of ele tron-hole orre lations. Finally,inSe tion2.7 we studyphotolumines en e fromQDsembeddedina mi ro avity. The relevantme hanismsare dis ussed and modiedSLE, that ontain feedba k and damping terms to a ount for a photon population in the avity, are introdu ed. The theoryis omparedwith resultsfromexperimentsperformedonQD inmi roresonators. A re apitulationof the maina hievementsofthis hapter an be found inthe on lusion.

2.1. System and Hamiltonian

The step required prior to being able to formulate the many-body approa h is the hoi eofthe single-parti lebasisof thenon-intera tingsystem. Intheformulationof our theoreti al model we assume the single-parti leproblemto be solved. Thus, our formalismis ompletelyindependentoftheexpli it hoi eofthesingleparti lestates. However, they are needed for the al ulation of the intera tion matrix elements and the free- arrierspe trumthat both enter intothe Hamiltonian. Before westart with the derivation of the dynami equations, the hoi e of the single parti le states is explained.

2.1.1. Single-Particle States

The al ulation of the single-parti le states for a given onnement geometry is all but trivial, and so is the al ulation of the bulk band stru ture. In the following we will give a brief overview that barely s rat hes the surfa e of band stru ture al ulations,but givesageneraloverviewofthe relevantphysi sthatdeterminesome of the propertiesof the single parti le states.

Due to the three-dimensional arrier onnement QDs possess lo alized states in ontrast tothe band stru ture of the bulkmaterial. Nevertheless, the single-parti le states are losely related to the properties of the bulk material, and of ourse they depend onthe QD size and geometry.

First-prin iple al ulations are available for the omputation of the ele troni stru  ture of small QD systems (200400 atoms) [111, 120℄. Due to the large number of atoms, together with the absen e of translational symmetry to greatly simplify bulk al ulations, density fun tional theory-based approa hes are omputationally too demanding to date. Semi-empiri al approa hes are used instead, either ontin uumapproa hes likethe

k

· p

-model[35℄orthe ee tive-massapproa h[56,130,153℄ that is used here. Alternatively, methods where the mi ros opi stru ture of the latti e enters, like in pseudo-potential[30℄ or tight-bindingmodels [121123℄ an be used. The hoi e of the ee tive mass approximation may be understood by look ing at the band stru ture of bulk material, whi h is basi ally that of GaAs, shown in the left panel of Figure 2.1. The suitability of Zin -blende material systems like GaAs, AlAs, InAs, InGaAs, InP, CdSe, and alsonitride-based systems grown in the Wurtzite stru ture for optoele troni devi e appli ation lies in the dire t band gap, allowingfor arriertransitionsbetween the ondu tionandvalen ebandsbyemitting orabsorbing photons. Thus, the relevant region of the Brillouinzone isrestri ted to

E

k

heavy holes

light holes

split-off holes

electrons

∆

so

1

Figure 2.1.: Left: Bandstru ture for GaAs al ulated using a pseudo-potential method. Thebandgapenergy is4.71eV.Thegureistakenfrom[30℄. Right: Simplieds hemati magni ation ofthe bulk band stru ture near the

Γ

point. The valen e-band edge is two- times degenerate. Due to the redu ed dimensionality in QDs, the degenera y is lifted as both the light-hold andthe split-o bandsshiftto lower energies.

its enter, the so- alled

Γ

-point. A simplied lose-up of this region is shown in the right panel of Figure 2.1. Three bands are relevant in the vi inity of the

Γ

-point: the degenerate heavy- and light-hole band, and the split-o band that o urs due to the spin-orbit oupling. In systems with redu ed dimensionality, the separation of the omponent in growth dire tion, together with strain-indu ed energy shifts of the bands, leads to a separation of the split-o band and the light-hole band from the heavy-hole band[141℄. The single-parti lewave fun tionsare determinedby the solutionof the S hrödinger equation.

Obviouslimitationsof ontinuummethodslikethesingle-bandee tivemassandthe multi-band

k

· p

approa hes arise, if the atomisti stru ture and symmetry be omes relevant over the global shape. For Zin -blende-QDs this is the ase if the QD size be omes too small (base diameter

/ 12

nm) [130, 136℄. For the work in this thesis, we negle t band mixing ee ts for the sake of simpli ity, so that only one valen e band and ondu tion band is onsidered. Furthermore, the physi s des ribed by an Hamiltonian that is based on a paraboli approximation of the band stru ture in the enter of the Brillouin zone an only be expe ted to be orre t, if the relevant pro esses involve single-parti le states in the vi inity of the

Γ

-point, whi h an be assumed for opti alpro esses in semi ondu tor materialswith adire t band gap.

p−shell

s−shell

wetting layer

quantum dot

holes

electrons

NIST Boulder Labs

Figure 2.2.: S hemati ofthe oupledQD-WL system. Two onned shellsare onsidered for both ele trons andholes,whi h lieenergeti ally belowa quasi- ontinuum of delo alized WL states, orresponding to the in-plane motion of arriers in the WL. The transmission ele tronmi rographimage oftheInGaAsQDonaGaAssubstrateistaken fromRef.[135℄.

We onsiderlens-shapedInGaAs/GaAsQDsgrownintheStranski-Krastanowgrowth mode[137℄,wherelatti e- onstantmismat handsurfa eenergyminimizationindu es the self-assembly of islands of atoms. In this pro ess athin lmof a few nanometer thi kness alledwettinglayerisformedbetweentheQDsandthesubstrate. TheQDs andthewettinglayer onstitutea oupledsystemwith ommonele troni states. Typ i aldimensionsofthe QDsare25nmbasediameter andless than10nm inheight,an example anbeseeninFigure2.2. Fromwhatwehavedis ussed above,thegeometry of this kindof QDs allows for ades ription withinthe ee tive-mass approximation, where a free arrier dispersion with ee tive masses for ele trons and holes is as sumed. In fa t, it has been shown that in the ase of at, ylindri ally symmetri InAs/GaAs QDs, the ee tive mass approximation yields results very lose to those obtainedfroma

k

· p

andatight-bindingmodel[130℄. Weseparate thewavefun tion intoanaxialand anin-planepart. The single-parti lebound-state wavefun tionsin the plane perpendi ular tothe growth dire tion are well approximated by those of a two-dimensional harmoni os illator [153℄. Due to the rotational symmetry around the QD axis, the orresponding angularmomentum is a good quantum number. We onsider the rst two onned shells of su h asystem, whi hare denoted by

s

and

p

a ording to their in-planesymmetry. The

s

-shell is onlyspin-degenerate, while the

p

-shellhasanadditionalangular-momentumtwo-folddegenera y. Toa ountforthe strong onnement in growth dire tion both for the QDs and the wetting layer, we useaninnitepotentialwelltomodelthe orrespondingniteextensionofthewave- fun tion. The spe trum of the potential well introdu es a splitting into subbands with a spa ing that depends onthe strength of the axial onnement, although the

termsubbandissomewhatmisleadingfortheQD ase,asthesepossesonlyadis rete spe trum due to the additional in-plane onnement. For an innite potential well, the spa ingbetween the rst two energy levelsisgiven by

∆E = 3h

2

_/8m

eff

L

2

,where

L

is the width of the well and

m

eff

is the ee tive mass of ele trons or holes. We study stru tures ofonly afewnanometers thi kness, orrespondingtoalarge energy spa ing for the ele trons of roughly 1eV for

L = 4

nm. For this reason, only the energeti ally lowest subband is onsidered.

The dis rete states are lo ated energeti ally below a quasi- ontinuum of delo alized states, orresponding to the two-dimensional motion of arriers in a wetting layer. At this pointit is worth mentioning that the a tual number of dis rete QD states is limited by the height of the onnement potential and the distan e in energy to the wetting layerstates. Lo alizedstates exist only belowthe quasi- ontinuum states of the wettinglayer. Inthe wetting layer, the in-planemomentum

k

is agoodquantum number if the ee t of the lo alizedstates on the ontinuum is negle ted. Thus, in the most simple approximation the quasi- ontinuum of the wetting layer states an be modeled by using plane waves with wave ve tor

k

. In Figure 2.2 a s hemati of the energylevelsofthe oupled QD-wettinglayersystemisshown. Furtherdetailsof the QD model are dis ussed inRef. [100℄. Stri tly speaking, the lo alizedstates and the wetting layer states are solutions of the single-parti leproblem for one ommon onnement potentialand must, therefore, formanorthogonal basis. Byperforming aseparateansatzfortheQDsandthewettinglayer,thisorthogonalityisnot ensured and anbeenfor ed,forexamplebyanorthogonalizationpro edureofthe ontinuum states. The obtained so- alled orthogonal plane waves (OPWs) represent a more orre tdes ription of the wetting layerstates. Fordetails we referto Ref. [100℄.

2.1.2. Many-Body Hamiltonian

Toinvestigatetheopti alpropertiesofQDs,thesystemmaybeo-resonantlyex ited by an opti alpulse. The ex itation reates arriers inthe barrier-, wetting layer-, or higherQD-states. The possibilityof this non-resonantkind of ex itationis afeature of self-assembled QDs and is not found in olloidal QDs where the wetting layer is missing, or even in atomi systems where only an ex itation of lo alized states is possible. In the ase of barrier ex itation, there are plenty of s attering hannels between the two quasi- ontinua, so that relaxation into the wetting layer states is fast. Fast s attering (relaxation)intothe lowerQD statesis fa ilitatedby s attering with LO-phonons and, espe ially at high arrier densities, arrier- arrier s attering [100,128℄. Bothme hanismsprovidee ients attering hannelsbothfortransitions betweenlo alizedstatesandbetweenlo alizedstatesandthewettinglayer. Figure2.3

delocalized states

delocalized states

localized states

localized states

p−shell

s−shell

optical processes

pumping

conduction band

capture

WL relaxation

valence band

QD relaxation

Figure 2.3.: S hemati of the relevant pro essesin the QD-WL system. O-resonant ex i tation reates arriers inhigherex itedlo alizedstatesorin theWL,followedbyrelaxation pro esses due to arrier- arrier and arrier-phonon s attering. Carrier re ombination via photon emissiontakespla ebetween the lo alizedstates.

s hemati allydisplays the relevant pro esses in the system for exemplary ex itation intothe wetting layer.

At lowtemperatures and low to moderate densities, the arriers solely populate the QDstates. Inthis asethe wettinglayerstates aremainlyimportantfor arrier-s at teringpro esses if the system is ex ited in the ontinuum states. Forthe re ombina tiondynami sdueto arrier-photonintera tion,theunpopulatedwettinglayerstates are of negligibleimportan e. Furthermore, if the wetting layer states are mainlyun populated,Coulomb orrelationsbetweenthedis reteQDstatesandtheenergeti ally displa ed quasi- ontinuum of the wetting layer are mu h weaker than those between QD states. Cal ulations presented in this thesis are performed atlowtemperatures, mostly at4K.For this reason,the wetting layer isnot in luded inour al ulations.

The driving pro ess for photolumines en e is the spontaneous re ombinationof ar riers in the valen e and ondu tion band. Spontaneous emission is a quantum-ele -trodynami al pro ess that is aused by eld u tuations. Thus we treat both the ele tromagneti eld and the arriersystem in the formalismof eld quantization.

1

1

Itis,however,possibletodes ribespontaneousemissionwitha lassi allighteldintheformalism ofquantum me hani sorquantumstatisti sby onsideringexternalu tuations,e.g.,usingthe Langevintheory[29℄.

The total Hamiltonianforthe system has the following ontributions:

H = H

_carr

0

+ H

Coul

+ H

ph

0

+ H

D

.

(2.1)

The Coulomb Hamiltonian whi h des ribes the intera ting system of valen e- and ondu tion-bandele trons has the two parts

H

_carr

0

=

X

ν

ε

c

_ν

c

†

_ν

c

ν

+

X

ν

ε

v

_ν

v

†

_ν

v

ν

,

(2.2)

H

Coul

=

1

2

X

α

′

_νν

′

_α

h

V

_α

cc

′

_ν,ν

′

_α

c

_α

†

′

c

†

_ν

c

ν

′

c

_α

+ V

_α

vv

′

_ν,ν

′

_α

v

†

_α

′

v

_ν

†

v

ν

′

v

_α

i

+

X

α

′

_νν

′

_α

V

_α

cv

′

_ν,ν

′

_α

c

†

_α

′

v

†

_ν

v

ν

′

c

_α

.

(2.3)

The free Hamiltonian

H

0

carr

ontains information about the single-parti le spe trum

ε

c,v

ν

and des ribes a system of non-intera ting harge arriers. The Coulombintera  tionbetweenthe arriersisa ountedforin

H

Coul

. FordetailswerefertoRefs.[10,58℄. Theoperators

c

ν

(

c

†

ν

)annihilate ( reate) ele tronsinthe one-parti lestates

|νi

of en ergy

ε

c

ν

. The orresponding operators and single-parti le energies for valen e-band ele tronsare

v

ν

(

v

†

ν

)and

ε

v

ν

,respe tively. Theexpli itformofthesingle-parti lewave fun tion

hr|ν, λi = ψ

λ

ν

(r)

entersthe des riptionviatheCoulombmatrixelements[10℄

V

λλ

′

α

′

_ν,ν

′

_α

=

Z

d

3

_r

Z

d

3

_r

′

_ψ

λ∗

α

′

(r)ψ

λ

′

_∗

ν

(r

′

)V (r − r

′

)ψ

λ

′

ν

′

(r

′

)ψ

_α

λ

(r)

(2.4)

with the band index

λ = c, v

and the Coulomb potential

V (r) = e

2

_/4πǫ

0

ǫr

, and via

the light-matterintera tion (see below). The diele tri onstants of the va uum and the ba kground materialare given by

ǫ

0

and

ǫ

, respe tively.

To obtain the expressions for the quantized Hamiltonian involving the ele tromag neti eld, the transverse ele tri eld

2

E

T

and the magneti eld

B

are expanded into modes, where ea h mode

ξ

is asso iated with a quantum me hani al harmoni os illatorwith the mode energy

~ω

ξ

. In the usual fashion, operators are introdu ed that reateordestroy aphotoninthe mode

ξ

,denoted as

b

†

ξ

and

b

ξ

,respe tively. For details we refer to [64, 87, 89℄. The total energy of the free ele tromagneti eld is given by the Hamiltonian[92℄

H

_ph

0

=

X

ξ

~ω

ξ



b

†

_ξ

b

ξ

+

1

2



.

(2.5) 2

For the quantization of the ele tromagneti eld, the Coulomb gauge is usually used. In the Coulomb gauge, the ele tri eld de ouples into transversal and longitudinal parts, su h that the longitudinalpartvanishes in theabsen e ofsour es in thesystem. The transversalpartis determinedbythedynami softheve torpotential,thusdes ribingele tromagneti waves.

Infreespa e,the modelabel

ξ

ontainsthewaveve tor

q

andthepolarizationve tor of the ele tromagneti eld

e

p

(q)

, with the index

p = ±

. The mode frequen ies are then given by

ω

ξ

= c|q|

,with being the speed of light,and the expli it form of the modes is

U

ξ

(r) = e

p

(q)e

i

√

ǫ qr

[87℄.

The light-matterintera tion Hamiltonianindipoleapproximationreads [6,33,70℄

H

D

= −i

X

ξ, αν



g

ξαν

c

†

α

v

ν

b

ξ

+ g

ξαν

v

†

α

c

ν

b

ξ



−

h. . (2.6)

Theresonantelementarypro essasso iatedwiththisHamiltonianisthetransitionof anele tron from the valen e intothe ondu tion band (or vi e versa) by absorption (emission) ofa photon. The non-resonant terms ontained inEq. (2.6) are negle ted inthe rotatingwave approximation

3

(RWA) [58℄. As anexample for a non-resonant pro ess, onsider the absorption of a photon while a transition fromthe ondu tion to the valen e band takes pla e (inEq. (2.6) the se ond term), and vi e versa. The matrixelements

g

ξαν

des ribethe ouplingbetween themode

ξ

oftheele tromagneti eld and the arrier transitionbetween states

|αi

and

|νi

and are given by

g

ξαν

= E

ξ

Z

d

3

r ψ

_α

c∗

(r)erU

ξ

(r)ψ

ν

v

(r) ,

(2.7)

where

E

ξ

=

p~ω

ξ

/2ǫǫ

0

V

is the va uum eld amplitude [92℄ and

V

is the normaliza tionvolume. Usingtheenvelope-fun tionapproximation[58℄thewave-fun tion

ψ

c

α

(r)

and

ψ

v

ν

(r)

an now be de omposed into an envelope part, whi h varies only slightly overaunit ell, andthe rapidlyos illatingBlo h-fa tor

u

k

≈0

(r)

. Takingintoa ount that the ele tromagneti eld is approximately onstant over the extent of a QD and onsidering for simpli ity equal envelopes for the ondu tion- and valen e-band ele trons (equal-envelope approximation [58℄), one nds

g

ξαν

= E

ξ

d

cv

U

ξ

(r

0

)δ

αν

≡ g

ξ

δ

αν

.

(2.8) Here,

d

cv

are the interbandmatrix elements and

r

0

is thepositionof the QD.In this form, the band indi es o ur only in the interband matrix elements, whi h are the same for identi al QDs. Thus, the light-matter oupling onstant depends only on the mode of the ele tromagneti eld. If dierent QDs were onsidered,the index

ν

must be kept with the ouplingmatrix element to a ount for the varying oupling strengths in the ensemble, as it ontains both the single-parti le state and the QD position. From Eq.(2.8) itfollows that,withinthe envelope-fun tionapproximation, opti al transitions o ur only between the

s

-shells or the

p

±

-shells of ele trons and

holes. 3

Therotatingwaveapproximationis ommonlyappliedinquantum-opti alproblems. IntheRWA, o-resonantterms o urring in the equations of motionare dropped. For simpli ity, this may already be done in the Hamiltonian. These terms posses a phase fa tor ausing aos illation rapidin omparisonto thedynami s ausedbytheopti alpro esses losetoresonan e.

2.2. The Hierarchy Problem and Truncation of Correlations (Cluster

Expansion Method)

The time evolution of the single arrier and photon operators is obtained by using Heisenberg's equations of motion together with the Hamiltonian of the intera ting system. Foran operator

a

it isgiven by

i~

d

dt

a = [a, H] .

(2.9)

From this wederive oupled equationsfor operator averages, likethe arrierpopula tion or photon number ina avity mode orin a ontinuum mode of freespa e. The many-body problem, and in parti ular the equation-of-motion approa h inherently bears a hierar hy problem, aused by the intera tion parts of the Hamiltonian. In ordertoa hievea onsistentformulationofthisproblem,the lassi ationandtrun a tionof orrelationfun tionswillbeaddressedbeforethesemi ondu tor lumines en e equationsare introdu ed.

Theappearingoperatoraveragesare lassiedintosinglets,doublets,triplets, quadru-plets, et ., a ording to the numberof parti lesthey involve. Considering interband transitions, itmust beborne in mind thatthe ex itationof one ele tron isdes ribed as the destru tion of a valen e band arrier and the reation of a ondu tion band arrier. For the orresponding intera tion pro esses, a photonoperator is onne ted totwo arrieroperators[11,69℄. Formally,this an beseenfromintegrating thetime evolution of aphoton operator,readily obtained from Eq. (2.9),

i~

d

dt

b

†

ξ

(t) = −~ω

ξ

b

†

ξ

(t) + i

X

ν

g

ξ

c

†

ν

(t)v

ν

(t) ,

(2.10) whi h yields

b

†

_ξ

(t) = b

†

_ξ

(0)e

iω

ξ

t

₊

1

~

X

ν

Z

t

0

dt

′

g

ξ

c

†

ν

(t

′

)v

ν

(t

′

)e

iω

ξ

(t−t

′

₎

.

(2.11)

This fa t isused to lassifymixed expe tation values with photonand arrier opera tors. For example, the ele tron population

f

e

ν

= hc

†

ν

c

ν

i

is a singlet ontribution, the sour eterm ofspontaneous emission

hc

†

α

v

α

v

ν

†

c

ν

i

andthe photon-assistedpolarization

hb

†

ξ

v

†

ν

c

ν

i

are doubletterms.

Inthefollowing,

N

-parti leaverages,s hemati allydenotedas

hNi

and ontaining

2N

arrieroperatorsoranequivalentrepla ementbyphotonoperators,arefa torizedinto allpossible ombinationsofaveragesinvolvingone upto

N − 1

parti leaverages. For

the dieren e between the full operator average and this fa torization, we introdu e a orrelation fun tion of order

N

, denoted as

δhNi

. S hemati allythe fa torization of singlets,doublets, triplets,and quadruplets isgiven by

h1i = δh1i ,

(2.12a)

h2i = h1ih1i + δh2i ,

(2.12b)

h3i = h1ih1ih1i + h1iδh2i + δh3i ,

(2.12 )

h4i = h1ih1ih1ih1i + h1ih1iδh2i

+ h1iδh3i + δh2iδh2i + δh4i .

(2.12d)

Looking at the last equation, the rst four terms on the right hand side represent allpossible ombinationsof singlets, singletsand doublets, singletsand triplets, and doublets,respe tively. Thelasttermistheremainingquadruplet orrelationfun tion. Continuing the series(2.12a)(2.12d)leads toquintupletterms and soon. Notethat singlets annot be fa torizedany further.

The essential idea of what has be ome known as the luster expansion method [45, 46, 63℄ is to repla e all o urring operator expe tation values

hNi

a ording to the Eqs. (2.12) so that equations of motion for the orresponding orrelation fun tions

δhNi

are obtained. Then the hierar hy of orrelation fun tions is trun ated rather thanthe hierar hyof expe tationvaluesitself. Thisallowsthe onsistentin lusionof orrelationsup toa ertainorderinallof the appearing operatorexpe tationvalues. Noting that terms of in reasing order in the luster expansion des ribe orrelated events betweenmoreand moreparti lesjusties atrun ation,asthese eventsare the lessprobablethemoreparti lesareinvolved. Thedes ribedtrun ationpro edure has previouslybeenusedtodes ribethelumines en edynami sofquantumwells[62,70℄ and re ently also of QDs [11, 44, 124℄. If the hierar hy is trun ated at the level of two-parti le orrelationfun tions,theso- alledsemi ondu torlumines en eequations for the oupled arrier and photon populations emerge, whi h onsistently in lude arrier- arrier orrelations up to the doublet level. When we dis uss the photon statisti sin Chapter 3, orrelations up to the quadruplet level must be in luded, as the photonse ond-order orrelationfun tion

g

(2)

_{(0) ∝ hb}

†

_b

†

_bbi

isof this orderitself.

Itisworthpointingoutthatthe hoi eofatrun ationmethodisnotunique. Further more, atrun ation at a onsistent levelin the number of involved parti lesdoes not ne essarily introdu e physi al ee ts onsistently. One example is the s reening of the Coulombpotential o urring in the equations of motion for singlets. There, the s reening is introdu ed by ele tron-hole orrelations, whi h are doublet quantities. However, the s reeningof the Coulombpotentialin the equations of motionfor dou bletquantities isintrodu edonthe tripletlevel,sothatatrun ation ondoubletlevel

leaves the Coulomb potential relevant for intera tion pro esses between orrelated arriers uns reened. Thus, if s reening of arrier- arrier orrelations was required, the onsisten y must either be violated to in lude only the s reening terms of the triplet level, or the triplet level must be in luded all together. Then the problem arries over to the next higher level, where s reening of triplet quantities is intro du ed onthe quadruplet level. This standsin ontrast to the diagrammati Green's fun tion approa h, where physi al pro esses are always onsidered for all o urring quantities via the hoi e of an appropriate self energy. The introdu ed renormaliza tionsor s reeningee ts are then onsidered up toinniteorderdue tothe re ursive stru ture of the Dyson equation. However, by the hoi e of ertain diagramsfor the self energy orthe polarizationfun tion in the ase of Coulomb s reening, the theory isthen restri ted to ertain lasses of intera tion pro esses.

The orrelation fun tions dened a ording to Eq. (2.12) show pe uliarities when algebrai manipulations are performed on them. This is illustrated in the following example,where we onsider the operator identity

c

†

_ν

c

ν

c

†

ν

c

ν

= c

†

ν

c

ν

.

(2.13)

Fora orrelation fun tion,however,

δhc

†

ν

c

ν

c

†

ν

c

ν

i 6= δhc

†

ν

c

ν

i ,

(2.14) as an be seen by expanding the leftand righthand sides of this equationa ording toEqs. (2.12b)and (2.12a), respe tively, i.e.

hc

†

ν

c

ν

c

†

ν

c

ν

i − hc

†

ν

c

ν

i

2

+ hc

†

ν

c

ν

ihc

ν

c

†

ν

i 6= hc

†

ν

c

ν

i .

(2.15)

This impliesthatalgebrai operationsinside the

δ

-symbol of the orrelationfun tion arenotpermitted. OtherwiseusinganidentitylikeEq.(2.13) ouldbeusedto hange the order of orrelations,e.g.

δhc

†

ν

c

ν

c

†

ν

c

ν

i

useEq.(2.13) inside

δh...i

=

_δhc

†

_ν

c

ν

i ,

(2.16)

whi h stands in ontradi tion toEq.(2.14).

Examplesforthefa torizationofoperatoraveragesaregiveninAppendix Awiththe intention to larify the appli ation of the lusterexpansion method tothe reader.

2.3. Equations of Motion

With the lassi ation and trun ation method at hand, we now derive the so- alled semi ondu tor lumines en e equations that ontain operator averages onsistently

up to the doublet level. This in ludes arrier- arrier orrelations that arise from the sour e term of the spontaneous emission. Amongst these are ex itoni populations that play an important role in the understanding of the de ay behavior of the pho tolumines en e, and we go into detail about this in Se tion 2.5. With the theory on this level we des ribeboth the spe trum and time-resolved photolumines en e.

We assume in the whole of this thesis that the lumines en e takes pla e inthe in o herentregimewhere theinuen e ofa oherentpolarization an benegle ted. Exem plarysituationsin ludein oherent arrierex itationsor oherentex itationofhigher states (barrier, wetting layer or higher lo alized states) with rapid dephasing and arrier relaxation, leading to a quasi-equilibrium distribution of the arriers at the latti etemperature. The absen e of a oherent external eld isexpressed inthe van ishingexpe tation values of the eld operator

hb

ξ

i = 0

and the oherent polarization

hv

†

ν

c

ν

i = 0

. True at the initial time

t = 0

, it an be shown to be preserved also during the time evolution: Taking the operator average of Eq. (2.11) we see that the evolution of the eld operatoris driven by the oherent polarization. Vi eversa, the oherent polarizationremains zero at all times without a driving oherent eld [70℄. However, for situationswhere a oherent polarizationis present, like resonan e uores en e [2, 7072℄, these additionaltermsmust be onsidered inallequations of motion.

Operator averages are kept within the rotating wave approximation, implying that termswith a rapidly os illatingphase are negle ted.

2.3.1. QD Semiconductor Luminescence Equations

Photon and carrier numbers.

WritingdownHeisenberg'sequationofmotionforthe photonnumber, we nd

i~

d

dt

hb

†

ξ

b

ξ

i = 2i Re

X

ν

g

_ξ

∗

_hb

†

_ξ

v

_ν

†

c

ν

i ,

(2.17)

whi h ouples to the photon-assisted polarization amplitude

hb

†

ξ

v

†

ν

c

ν

i

.

4

The orre spondingequation of motionis given by

i~

d

dt

hb

†

ξ

v

†

ν

c

ν

i = (˜ε

c

ν

− ˜ε

ν

v

− ~ω

ξ

− iΓ) hb

†

_ξ

v

†

ν

c

ν

i

+ (f

_ν

c

_{− f}

_ν

v

)

X

α

V

νανα

hb

†

ξ

v

α

†

c

α

i

+ i g

ξ

f

ν

c

(1 − f

ν

v

) + i

X

α

g

ξ

C

αννα

x

.

(2.18) 4

Inthein oherentregime

hb

†

ξ

i = 0

andthus, a ordingtoEq.(2.12b),

hb

†

ξ

v

†

ν

c

ν

i ≡ δhb

†

The evolution is determinedby the Coulomb-renormalizedenergies

˜

ε

c

ν

= ε

c

ν

−

X

α

V

νανα

f

α

c

,

˜

ε

v

_ν

= ε

v

_ν

₋

X

α

V

νανα

f

α

v

,

(2.19)

the resonan efrequen y

ω

ξ

of theopti almode

ξ

,andaphenomenologi aldephasing

Γ

ausing a broadening of the spe tral lines. The relevant me hanism for dephasing insemi ondu tor systemsis Coulombs atteringand s attering witha ousti oropti alphonons. This ould bein luded ona mi ros opi levelby adding aHamiltonian analogous to Eq. (2.6) for the arrier-phonon intera tion [62, 127, 128℄. Predomi nantly interested in the lumines en e and ee ts introdu ed by the Coulomb and light-matterintera tion, we restri t ourselves to a phenomenologi al dephasing on stant in asso iation with arrier interband transitions. For further dis ussion, see Se tion 5.3.5. The term in the se ond line is analogous to the quantum well ase, whereit givesrise to the ex itoni photolumines en e belowthe band gap [71℄. Here it introdu es the orresponding ex itoni resonan es for the QD states due to the interband Coulomb ex hange intera tion. The sour e term of spontaneous emission

hc

†

α

v

α

v

ν

†

c

ν

i

enters the theory naturally due to the quantization of the light eld. In its fa torized form it appears in the last line of Eq. (2.18), with the ele tron-hole orrelationfun tion

C

x

α

′

_νν

′

_α

= δhc

†

_α

′

v

_ν

†

c

_ν

′

v

_α

i

. The spontaneous emissionsour e termis of parti ularinterest, asitdeviates fromthe sour eterm obtained inatomi models. Whileforatomsonlythenumberofele trons ontributes, hereatwo-parti leaverage ontainingele tron- and hole operators o urs. We haveomitted the term represent ingstimulatedemission/absorption,whi h ontributesforexampleifanexternaleld isresonant with the onsidered transitions orif aresonator providesfeedba k due to the emittedphotons [124℄. This feedba k term is of extreme relevan e when onsid ering QDs inmi ro avities and will be dis ussed in Se tion 2.7, where the theory is presented for open- avity systems. Before evaluatingthe orrelation termin the last lineof Eq. (2.18),we givethe time evolutionof the arrierpopulation

i~

d

dt

f

c

ν

= −2i Re

X

ξ

g

_ξ

∗

_hb

†

_ξ

v

_ν

†

c

_ν

_i

(2.20)

+ 2i Im

X

αα

′

_ν

′

V

να

′

_αν

′

(C

_να

c

′

_αν

′

− C

_α

x

′

_ναν

′

) ,

i~

d

dt

f

v

ν

= 2i Re

X

ξ

g

_ξ

∗

_hb

†

_ξ

v

†

_ν

c

_ν

_i

(2.21)

− 2i Im

X

αα

′

_ν

′

V

να

′

_αν

′

(C

_να

v

′

_αν

′

− C

_α

x

′

_ναν

′

) .

Here additionalintraband orrelation fun tions

C

c

α

′

_νν

′

_α

= δhc

†

_α

′

c

†

_ν

c

_ν

′

c

_α

i

and

C

v

α

′

_νν

′

_α

=

δhv

†

α

′

v

_ν

†

v

_ν

′

v

_α

i

appear. Restri tingourselvesto

s

-and

p

-shellsforthe lo alizedstates, we onsider only

s

-states with zero angular momentum and

p

-states with angular momentum of

±1

, whi h, as we now explain, allows us to take

ha

†

ν

a

ν

′

i = f

_ν

a

δ

_νν

′

. Initiallyallexpe tationvaluesbut for thepopulationinthe

s

- and

p

-shells areset to zero. Therotationalsymmetryofthesystemandtheresulting onservationofangular momentum ensures that allo-diagonalterms

ha

†

ν

a

ν

′

i

with

ν 6= ν

′

des ribeforbidden transitions and remain zero during the time evolution. Therefore, in all equations expe tationvaluesof two arrieroperatorsare restri tedtopopulations, i.e.averages ofvalen e-or ondu tion-band- arrier reatorandannihilatorwithequal indi es. An in lusionof higher angular momentum states is straightforward, but unne essary at lowtemperatures andleftout fortransparen y. Alsorememberthatpolarization-like averages of the form

hv

†

ν

c

ν

i

vanish in the in oherent regime, as is explained in the beginningof this se tion.

Carrier correlations.

We now turn to the interband, or ele tron-hole orrelations that o ur inEqs. (2.18), (2.20)and (2.21), namely

C

_α

x

′

_νν

′

_α

= δhc

†

_α

′

v

_ν

†

c

ν

′

v

_α

i

= hc

†

α

′

v

_ν

†

c

ν

′

v

_α

i − hc

†

α

′

v

_ν

†

c

ν

′

v

_α

i

_S

(2.22)

= hc

α

†

′

v

_ν

†

c

ν

′

v

_α

i + f

_ν

c

′

f

_ν

v

δ

_να

δ

_ν

′

_α

′

a ordingtothegeneraldenitionofatwo-parti le orrelationfun tioninEq.(2.12b). Here, the fa torizationintosinglets isequivalenttothe Hartree-Fo k approximation.

Withthefollowingdis ussion weoeranintuitiveinterpretationoftheinterband or relations

C

x

α

′

_νν

′

_α

. They des ribe apro ess where a ondu tion-band- arriertransition fromastate

ν

′

toa state

α

′

isa ompaniedby a valen e-band- arriertransitionfrom astate

α

toa state

ν

. It isimportanttounderstand that these transitionsare orre lated and donot oin identallyo uratthe same time,ree ted by the fa tthat the un orrelatedpart of theoperatoraverage

hc

†

α

′

v

_ν

†

c

ν

′

v

_α

i

has alreadybeen subtra tedin the denition of

C

x

inEq. (2.22).

A spe ial role inhere the matrix elements

C

x

αννα

, as they also ontain ontributions

orresponding toex itoni population. This an be inferred fromthe spontaneous-e mission ontribution in Eq. (2.18). The term

∝ f

c

ν

(1 − f

ν

v

)

is the fa torized part

hc

†

α

v

†

ν

c

ν

v

α

i

S

and des ribes anun orrelated ele tron-hole plasmafor the state

ν

. The orrelated part is given by the sum over the matrix elements

C

x

αννα

. If the matrix elements are non-zero, orrelations between ele trons and holes are introdu ed into the ele tron-hole system. The orrelation fun tion with four identi al indi es may

beinterpreted asex itoni arriero upations,but alsoo-diagonal orrelations on tribute towards the lumines en e. The formation of ex itoni populations has been studiedin quantum wellsin Ref.[62℄.

Note that a generalizationof the sour e term of spontaneous emission in Eq. (2.18) to QDs at dierent positions an be formulated to a ount for oupling of arrier transitionsindierent dots. Inthat ase,the state indi esareunderstoodtoin lude the position indi es of ea h dot, and the sum runs over all QD positions. Then, in additiontothe ele tron-hole orrelation(2.22) withinone QD,

R

1

= R

2

, one obtains orrelation fun tions of the type

δhc

†

α,R

1

v

†

ν,R

2

c

ν,R

2

v

α,R

1

i

with

R

1

and

R

2

referring to the spatial positions of two dierent QDs. It an be shown that the Coulomb inter a tion an els for QDs at dierent positions

R

1

6= R

2

, leaving only the light-matter intera tion as the driving me hanism for these orrelations. A straightforward in terpretation lies in the radiative, or more pre isely, superuores ent oupling. This ee t refers to the olle tive emission of radiation afteran in oherent ex itation. In ontrast to stimulated emission, however, this ee t is not mediated by a photoni population in a resonator, but is aused by a dire t radiative oupling between the emitters. While superuores ense and superradian e isintensely studiedin the liter ature [2224, 40, 103, 139, 140, 147, 148℄, we do not onsider this ee t within this thesis under the assumption that it is small in an ensemble of self-organized QDs. The oupling an onlybeee tiveiftheemitters ouple onstru tively. This iseither the ase for very few emitters, or for an array with a regular spa ing of half of the emission wavelengthbetween single emitters. In an irregulararray of self-assembled QDs,the ee t is expe ted toaverage to zero.

The time evolution of the interband orrelations isgiven by the equation of motion

i~

d

dt

C

x

α

′

_νν

′

_α

= − ε

c

_α

′

+ ε

v

_ν

− ε

c

_ν

′

− ε

v

_α

hc

†

α

′

v

†

_ν

c

ν

′

v

_α

i

−

X

ν

2

ν

3

ν

4

h

V

ν

1

ν

2

ν

3

α

′

hc

†

ν

4

(c

†

ν

2

c

ν

3

+ v

†

ν

2

v

ν

3

)v

†

ν

c

ν

′

v

_α

i

+ V

ν

4

ν

2

ν

3

ν

hc

†

α

′

v

_ν

†

₄

(c

†

_ν

₂

c

ν

3

+ v

ν

†

2

v

ν

3

)c

ν

′

v

α

i

− V

ν

′

_ν

₂

_ν

₃

_ν

₄

hc

†

α

′

v

†

_ν

(c

†

_ν

₂

c

ν

3

+ v

ν

†

2

v

ν

3

)c

ν

4

v

α

i

− V

αν

2

ν

3

ν

4

hc

†

α

′

v

_ν

†

c

ν

′

(c

†

_ν

2

c

ν

3

+ v

†

ν

2

v

ν

3

)v

ν

4

v

ν

4

i

i

− i

X

ξ

h

g

∗

_ξ

_hb

†

_ξ

v

_α

†

′

v

_ν

†

c

ν

′

v

_α

i − g

_ξ

∗

hb

†

_ξ

c

†

α

′

v

_ν

†

c

ν

′

c

_α

i

+ g

ξ

hb

ξ

c

†

α

′

v

_ν

†

v

ν

′

v

_α

i − g

_ξ

hb

_ξ

c

†

_α

′

c

†

_ν

c

ν

′

v

_α

i

i

− i~

_dt

d

hc

†

α

′

v

_ν

†

c

ν

′

v

_α

i

_S

.

(2.23)