Theory of exciton doublet structures and …takaghra/tak2000.pdfTheory of exciton doublet structures and polarization relaxation in single quantum dots T. Takagahara* NTT Basic Research

PHYSICAL REVIEW B 15 DECEMBER 2000-IIVOLUME 62, NUMBER 24

Theory of exciton doublet structures and polarization relaxation in single quantum dots

T. Takagahara*NTT Basic Research Laboratories, 3-1 Morinosato Wakamiya, Atsugi-shi 243-0198, Japan

~Received 22 March 2000; revised manuscript received 25 August 2000!

The mechanism of exciton doublet structures in quantum dots is identified as the long-range part of theelectron-hole exchange interaction which is emphasized by the anisotropic shape of quantum dots. The physi-cal origin of the energetic order of the orthogonally polarized exciton states of each doublet is clarified byinspecting the spatial distribution of the exciton polarization. The key concepts to understand the energeticorder are the node configuration of the distribution function of exciton polarization and the dipole-dipoleinteraction energy originating from the long-range electron-hole exchange interaction. The population relax-ation and the polarization relaxation of excitons are studied and the extremely slow polarization relaxationwithin exciton doublet states is predicted. It is also found that the inter-doublet cross-relaxation betweenorthogonally polarized exciton states occurs as efficiently as the population relaxation.

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I. INTRODUCTION

The spatial confinement of electrons and holes alongthree dimensions leads to a discrete energy level strucwith sharp optical absorption lines. In this sense a semicductor quantum dot~QD! can be regarded as an artificisolid-state atom. The concentration of the oscillator strento sharp exciton transitions makes QD’s very attractiveelectro-optic and nonlinear optical applications. To be mspecific, QD’s are considered as promising elementsimplementing the coherent control of the quantum sta1

which is an essential function to achieve quantum informtion processing and quantum computation.2 The excitonicwave function has been manipulated by controlling the ocal phase of the pulse sequence through timing and polation and the wave-function engineering in semiconducQD’s is now being developed.1 In this wave-function ma-nipulation, important features are the sharp spectral linesdicating a long coherence time and well-defined polarizatcharacteristics. In this paper we discuss the origin ofrecently observed exciton doublet structures, their polartion characteristics and the dynamics of polarization relation among these exciton states.

Recently, very fine structures were observed in exciphotoluminescence ~PL! from GaAs quantum wells~QW’s!.3–5 These sharp lines are interpreted as luminescefrom localized excitons at island structures in QW. Theisland structures can be regarded as zero-dimensional qtum dots~QD’s!. Furthermore, in the photoluminescence ecitation spectra, doublet structures with mutually orthogolinear polarizations were discovered. The direction of linepolarization is closely related to the geometrical shape ofQW islands. As observed by the scanning tunnelmicroscopy,5 QW islands are elongated along the@110# di-rection. Correspondingly, the exciton states are polarialong the@110# or @110# direction. The splitting energy oeach doublet states is about several tens of micro eV anof the same order of magnitude as the exciton~longitudinal-transverse! splitting energy in bulk GaAs.From these experimental results, we can deduce that t

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splittings arise from the electron-hole exchange interactiespecially its long-range part which is sensitively dependon the geometrical shape of quantum dots. Recently theportant role of the electron-hole exchange interaction inexciton fine structures has been discussed for III-V and IIsemiconductor quantum dots.6–9 A theory of the exciton finestructure in quantum dots was presented taking into accoonly the heavy-hole band.10 But actually the heavy-hole banand the light-hole band are mixed together. In this simtheory the doublet lines have the same oscillator strengththe experimental details cannot be explained, e.g., the asmetric shape of the excited exciton doublets.

Thus the primary purposes of this paper are to develomicroscopic theory of the electron-hole exchange interacin semiconductor quantum dots and to clarify the exciton fistructures in detail, e.g., the relation between the energorder of the polarized doublet states and the node configtion of the distribution function of the exciton polarizationAt the same time, the exciton dynamics or the excitdephasing is the key issue in estimating the feasibility ofquantum coherent control in general. The exciton dephasin quantum dots was discussed by the present author11 inconjunction with the linewidth of the very sharp PL lineRecently Bonadeoet al.12 carried out the pump-probe spetroscopy on a GaAs single quantum dot and estimatedpopulation relaxation rates among individual exciton leveAlso Gotohet al.13 reported the exciton spin relaxation timin an InGaAs quantum dot. These dynamical studies areportant to clarify the relaxation mechanisms in quantum dand to estimate the ultimate limit of the exciton coherentime. Here we discuss in detail the population relaxation athe polarization relaxation among the exciton doublet strtures and investigate the efficiency of cross-relaxationtween the orthogonally polarized exciton states. A partthese results was preliminarily reported in Ref. 14.

This paper is organized as follows. In Sec. II, thelectron-hole exchange interaction in quantum dots is alyzed extensively with emphasis on its long-range part athe method of calculation is presented. In Sec. III, the callated exciton doublet structures are discussed concertheir dependence on the size, shape, and depth of the

16 840 ©2000 The American Physical Society

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PRB 62 16 841THEORY OF EXCITON DOUBLET STRUCTURES AND . . .

finement potential. In Sec. IV, the physical origin of the eergetic order of the orthogonally polarized exciton stateseach doublet is clarified by inspecting the spatial distributof the exciton polarization. In Sec. V, the population andpolarization relaxation processes among exciton doustructures are discussed and it is discovered that the crelaxation between the orthogonally polarized exciton staoccurs as efficiently as the population relaxation withoutchange of the polarization direction.14 In order to see theefficient polarization relaxation more clearly, we examineSec. VI the photoluminescence spectra under selective etation and the excitation spectra of photoluminescence.nally, our results and predictions are summarized in Sec.

II. ELECTRON-HOLE EXCHANGE INTERACTIONIN QUANTUM DOTS

We formulate a theory of excitons in a quantum dot taing into account the multiband structure of the valence band the electron-hole exchange interaction. For the valeband, we take into account four states belonging to theG8symmetry (J53/2) and employ the full LuttingerHamiltonian15 without making the spherical approximatioWe neglect the spin-orbit-splitJ51/2 valence band becausthe splitting energy between theJ53/2 and theJ51/2 va-lence bands is much larger than the typical energy level sings to be discussed in this paper. Then we denote two cduction band Bloch functions byuct& wheret indicates thespin-up(a) or the spin-down(b) state and similarly four va-lence band Bloch functions byuvs& wheres stands for thez-component of the angular momentum~3/2, 1/2, 21/2,23/2!. Thus the exciton state is composed of 8 combinatiof the conduction band and the valence band, namely

uX&5(t,s

(r e ,r h

f ts~r e ,r h!actr e

† avsr hu0&, ~2.1!

where the Wannier function representation of the annihtion and creation operators is used andf ts is the envelopefunction. The eigenvalue equation forf ts is given as

(t8s8r e8r h8

H~ctr e ,vsr h ;ct8r e8 ,vs8r h8! f t8s8~r e8 ,r h8!

5E fts~r e ,r h! ~2.2!

with

H~ctr e ,vsr h ;ct8r e8 ,vs8r h8!

5dss8d r hrh8„ect,ct8~2 i¹e!1dtt8ve~r e!…

2dtt8d r ere8„evs8,vs~ i¹h!1dss8vh~r h!…

1V~ctr e ,vs8r h8 ;ct8r e8 ,vsr h!

2V~ctr e ,vs8r h8 ;vsr h ,ct8r e8!, ~2.3!

-f

neetss-s

e

ci-i-I.

-dce

c-n-

s

-

whereect,ct8(evs8,vs) is the band energy of the conductio~valence! band electron,ve(vh) is the confinement potentiafor the conduction~valence! band electron andV( i , j ;k,l ) isthe Coulomb or exchange matrix element defined by

V~ i , j ;k,l !5E d3r E d3r 8 f i* ~r !f j* ~r 8!

3e2

eur 2r 8ufk~r 8!f l~r ! ~2.4!

in terms of the Wannier functions$f i% and the dielectricconstante. ect,ct8 is simply approximated as

ect,ct8~k!52\2

2mek2dtt8 ~2.5!

with the effective massme and evs8,vs(k) is the LuttingerHamiltonian given by

2\2

m0F S g11

5

2g2D k2

22g2~kx

2Jx21ky

2Jy21kz

2Jz2!

2g3kykz~JyJz1JzJy!2g3kzkx~JzJx1JxJz!

2g3kxky~JxJy1JyJx!G ~2.6!

in terms of the Luttinger parametersg1 ,g2 andg3, the freeelectron massm0 and the angular momentum operatoJx , Jy, andJz for J53/2. Recently, the significance of thoperator ordering in the Luttinger Hamiltonian was pointout in conjunction with the interface-induced effects.16 Theexciton energy level structure on the order of a few mmay be affected by the operator ordering but be influenmore importantly by the actual morphology of the islandliquantum dot in a quantum well. However, the main issuethis paper is the doublet splitting of each exciton level duethe electron-hole exchange interaction which is of the orof tens of micro eV and is much smaller than the excitlevel spacing. Thus the operator ordering in the LuttingHamiltonian is not relevant and will not be taken into acount in this paper. The Coulomb matrix element canapproximated due to the localized nature of the Wannfunctions as

V~ctr e ,vs8r h8 ;vsr h ,ct8r e8!

'd r ere8d r hr

h8dtt8dss8

e2

eur e2r hu. ~2.7!

The exchange term can also be approximated as

V~ctr e ,vs8r h8 ;ct8r e8 ,vsr h!

'd r er hd r

e8rh8@d r er

e8V~ctr e ,vs8r e ;ct8r e ,vsr e!

1~12d r ere8!V~ctr e ,vs8r e8 ;ct8r e8 ,vsr e!#.

~2.8!

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16 842 PRB 62T. TAKAGAHARA

The first ~second! term in the parentheses is usually callthe short-range~long-range! exchange term. The long-rangterm can be decomposed into the multipole expansionthe lowest-order term is retained as

V~ctr e ,vs8r e8 ;ct8r e8 ,vsr e!'mW ct,vs

@123nW • tnW #

ur e2r e8u3

mW vs8,ct8

~2.9!

with

nW 5r eW2r e8W

ur e2r e8u, ~2.10!

mW ct,vs5E d3r fctR* ~r !~rW2RW !fvsR~r !, ~2.11!

wherefct(vs)R(r ) is a Wannier function localized at the siR. Hereafter the vector symbols will be dropped becauseno fear of confusion. Thus the long-range exchange termdescribed as the dipole-dipole interaction between the eton transition dipole moments. In the calculation of the mtrix elements of this long-range exchange term, the singnature in Eq. ~2.9! can be avoided by a mathematicartifice.17,11 As a result, we have

(r eÞr e8

f ts* ~r e ,r e! f t8s8~r e8 ,r e8!mct,vs

@123n• tn#

ur e2r e8u3

mvs8,ct8

524p

3~mct,vs•mvs8,ct8!

3E d3r f ts* ~r ,r ! f t8s8~r ,r !

1E d3r divr„f ts* ~r ,r !mct,vs…

3divrF E d3r 8mvs8,ct8

ur 2r 8uf t8s8~r 8,r 8!G . ~2.12!

It can be shown that this long-range exchange term vanisfor thes-like envelope functions.17 Thus the admixture of thehigher angular-momentum states is necessary to have amagnitude of the long-range exchange energy, which wobe induced by the anisotropic shape of quantum dots othe strained environment around the quantum dots.18 On theother hand, the matrix element of the short-range exchaterm is simply given as

V~ctr 0 ,vs8r 0 ;ct8r 0 ,vsr 0!E d3r f ts* ~r ,r ! f t8s8~r ,r !.

~2.13!

Now we proceed to a more detailed calculation of thematrix elements. The angular momentum and spin partthe J53/2 valence-band Bloch functions can be written a

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es

iteldy

ge

eof

U32 3

2L 5uv1&5Y11a,

U32 1

2L 5uv2&5A2

3Y10a1A1

3Y11b,

U32 21

2L 5uv3&5A1

3Y121a1A2

3Y10b,

U32 23

2L 5uv4&5Y121b, ~2.14!

wherea(b) denotes the up~down! spin state. Then we canidentify the nonzero matrix elements of the short-rangechange interaction at the same siter 0 as

V~car 0 ,v1r 0 ;car 0 ,v1r 0!5V~cbr 0 ,v4r 0 ;cbr 0 ,v4r 0!

5EXS ,


52

3EX

S ,


51

3EX

S ,

V~car 0 ,v2r 0 ;cbr 0 ,v1r 0!5V~cbr 0 ,v1r 0 ;car 0 ,v2r 0!

5A1

3EX

S ,


52

3EX

S ,


5A1

3EX

S , ~2.15!

with

EXS5

4pe2

3 E dr r 2E dr8 ~r 8!2 fs~r !fp~r 8!

3r ,

r .2

fs~r 8!fp~r !, ~2.16!

where r ,5min(r ,r 8), r .5max(r ,r 8) and fs(fp) denotesthe radial part of thes(p)-like Wannier function at a site.

As mentioned before, the exciton is a linear combinatstate with respect to the electron-hole pair indexts in Eq.~2.1!, namelyav1 ,av2 ,av3 ,av4 ,bv1 ,bv2 ,bv3, and bv4

@v1;v4 are defined in Eq.~2.14!#. The short-range exchangterm in this 838 matrix representation is given a


~2.17!

heocp

qar

where the first line in the matrix is inserted to indicate tBloch function bases and each matrix element is a blmatrix having the dimension of the number of the envelofunction bases.

The matrix elements of the interband transition in E~2.11! which appear in the long-range exchange termgiven as

^caurWuv1&5~21,2 i ,0!m, ^caurWuv2&5S 0,0,2

A3D m,

^caurWuv3&5S 1

A3,2 i

A3,0D m, ^caurWuv4&5~0,0,0!,

ke

.e

^cburWuv1&5~0,0,0!, ^cburWuv2&5S 21

A3,2 i

A3,0D m,

^cburWuv3&5S 0,0,2

A3D m, ^cburWuv4&5~1,2 i ,0!m ~2.18!

with

m5A2p

3 E0

`

dr r 2fs~r ! r fp~r !, ~2.19!

wherefs and fp have the same meaning as in Eq.~2.16!.Then the factor (mct,vs•mvs8,ct8) in Eq. ~2.12! can be writ-ten in the 838 matrix representation as

~2.20!

As can be guessed from Eq.~2.18!, the second term of the long-range exchange energy in Eq.~2.12! has a form given by


~2.21!

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where * indicates a nonzero block matrix whose dimensis given by the number of the envelope function bases. Slar matrix representations of the exciton exchange interacin QW’s were discussed by Maialleet al.19

Here we consider localized excitons at quantum-dot-lislands in QW structures which are formed by the latefluctuations of the QW thickness. In these islandlike strtures, the confinement in the direction of the crystal growis strong, whereas the confinement in the lateral directiorather weak. Furthermore, the island structures were founbe elongated along the@110# direction.5 Thus these islandstructures can be modeled by an anisotropic quantum disorder to facilitate the calculation, the lateral confinementtential in thex andy directions is assumed to be Gaussian

ve(h)~r !5ve(h)0 expF2S x

aD 2

2S y

bD 2G , ~2.22!

where the lateral size parametersa and b can be fixed inprinciple from the measurement of morphology by, e.STM but the actual size would be different from the morphlogical size because of the presence of the surface deplelayer and so on. Thus the parametersa and b are left asadjustable parameters. The exciton envelope function in san anisotropic quantum disk can be approximated as

f ~r e ,r h!5 (l e ,l h ,me ,mh

C~ l e ,l h ,me ,mh!

3S xe

a D l eS xh

a D l hS ye

b D meS yh

b D mh

•expF21

2 H S xe

a D 2

1S xh

a D 2

1S ye

b D 2

1S yh

b D 2J 2ax~xe2xh!2

2ay~ye2yh!2Gw0~ze!„w0~zh! or w1~zh!…

~2.23!

with

ni-n

el-histo

In-s

,-ion

ch

w0~z!5A 2

LzcosS pz

LzD ,

w1~z!5A 2

LzsinS 2pz

LzD , ~2.24!

whereC( l e ,l h ,me ,mh) is the expansion coefficient,Lz theQW thickness, the factor 1/2 in the exponent is attachedmake the probability distributionu f (r e ,r h)u2 to follow thefunctional form of the confining potential andax and ayindicate the degree of the electron-hole correlation anddetermined variationally. Concerning the envelope functin the z-direction, the lowest two functions are taken inaccount for the hole to guarantee sufficient accuracy. Telectron-hole relative motion within the exciton state is nmuch different from that in the bulk because the lateral cfinement is rather weak. As a result, the parametersax anday are weakly dependent on the lateral size. Because ofinversion symmetry of the confining potential, the parity isgood quantum number and the wavefunction can be clafied in terms of the combination of parities ofxl e1 l h, yme1mh

and w0 or w1. In actual calculations, terms up to the sixpower are included, namely, 0< l e1 l h , me1mh<6 to en-sure the convergence of the calculation. The potential defor the lateral motion of the electron and the hole canguessed from the splitting energies of the exciton statesto the monolayer fluctuation of the QW thickness. The vaof uve

02vh0u is typically about 10 meV for the nominal QW

thickness about 3 nm.1 Of course, even ifve02vh

0 is fixed,each value ofve

0 andvh0 cannot be determined uniquely. He

we employve0526 meV andvh

053 meV in most parts ofthis paper, referring to the experimental results on the eton energy spectra and assuminguve

0u:uvh0u52:1, but we will

later examine several cases of the potential depth.It is important to examine the size-dependence of the

change energy. For that purpose, we note first of all that

f ~r e ,r h!}1/L3~quantum dot!,1/L2~quantum disk!,~2.25!

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where L is the characteristic size of a quantum dot or tlateral size of a quantum disk with a fixed disk height. Thscaling comes simply from the normalization

E d3r eE d3r h u f ~r e ,r h!u251. ~2.26!

Then the short-range exchange term in Eq.~2.13! scales as

L31

L3

1

L3}

1

L3~quantum dot!,

L21

L2

1

L2}

1

L2~quantum disk!, ~2.27!

where the first factorL3 or L2 comes from the volume integral. The first term of the long-range exchange energy in~2.12! has the same size-dependence. In the second terthe long-range exchange energy, the operator div andur2r 8u give rise to a factor of 1/L and the integral in Eq.~2.12!scales as

L31

L

1

L3

1

LL3

1

L

1

L3}

1

L3~quantum dot!,

L21

L

1

L2

1

LL2

1

L

1

L2}

1

L3~quantum disk!.

~2.28!

Thus the exchange energy in quantum dots scales as 1/L3 forboth the short-range and the long-range parts. On the ohand, in quantum disks, the short-range and the long-raparts of the exchange energy depend on the size differeas 1/L2 and 1/L3, respectively. Of course, this sizedependence will hold asymptotically only in the strong cofinement regime. In general, details of the envelope funcare substantially dependent on the size through, for examthe variational parametersax and ay , leading to possibledeviation from the rough scaling laws estimated above.

It is important to note that the long-range part of telectron-hole exchange interaction is not absent for the oparity exciton states. For these exciton states,

E d3r f ~r ,r !50 ~2.29!

and the oscillator strength vanishes. However, the integin Eq. ~2.12! do not vanish in general.

III. EXCITON DOUBLET STRUCTURES

Now we discuss the schematic exciton energy level strtures in an anisotropic quantum disk as shown in Fig. 1. Tconduction band levels are doubly degenerate due to thedegree of freedom. The valence band levels are also dodegenerate owing to the Kramers theorem in the absencemagnetic field even when the valence band mixing is tainto account. Thus each exciton state is fourfold degenein the absence of the electron-hole exchange interaction~leftcolumn!. When the short-range exchange term is includ~middle column!, the quadruplet is split into two doublet

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The first doublet consists of almost degenerate levels whare mainly composed of states having the z-component oftotal angular momentumJz of 2 and –2, namelyuts&5uav4& or ubv1& in Eq. ~2.1! and is denoted as,62. inFig. 1. Another doublet is mainly composed of the stahaving theJz-component of –1, 0 and 1 and is denoted^61&. This doublet is split slightly by the short-range echange term. When the long-range exchange term is ad~right column!, the levels denoted by62& remain almostthe same because the exchange interactions are absent fJz562 states. On the other hand, the levels denoted^61& are affected rather strongly due to the mixing amothemselves. Namely, the doublet splitting increases andlowest doublet is pushed down by the exchange interacwith excited exciton states. This level restructuring is dpicted by solid arrows in Fig. 1. As a result, the excitoground state can become an optically active state depenon the magnitude of the long-range exchange energy.

This schematic understanding can be confirmed bymerical calculations in which we employ a quantum dimodel to simulate a localized exciton in a QW island. In tfollowing, material parameters of GaAs~Ref. 20! are em-ployed ~see Appendix for details!. The chosen size parameters in Eq.~2.23! area520 nm,b515 nm, andLz53 nm.The exciton level structure is shown in Fig. 2 for the casewhich only the short-range exchange energy is included.exciton ground state is optically dark and the first opticaactive exciton doublet is lying about 0.3 meV above the eciton ground state. The splitting energy of the first opticaactive exciton doublet is much smaller than the obsersplitting. Also for the higher-lying exciton doublets the splting energy is very small (,1 meV!. On the other handwhen both the short-range and the long-range exchaterms are included as shown in Fig. 3, the exciton groustate is optically active and its doublet splitting is 29meV inagreement with experiments.5 The optically dark excitonstates are mainly composed of theJz562 components andits first doublet is lying about 0.8 meV above the excitground state. Thus the long-range exchange term is imtant to understand the experimental results. So far we hconsidered the even-parity exciton states which include bthe optically active and the dark states. In addition to thethere are odd-parity exciton states, all of which are optica

FIG. 1. Exciton energy level structures in an anisotropic qutum dot are shown schematically for the cases with no exchainteraction~left column!, with only the short-range exchange inteaction ~middle column! and with both of the short-range and thlong-range exchange interactions~right column!.

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dark. They are plotted by diamonds in Fig. 3 to show thenergy positions. They do not contribute to the luminescebut would play the role as the intermediate states of reation processes as will be discussed in Sec. VI.

Furthermore, the importance of the long-range exchainteraction can be seen in the dependence of the dousplitting energy on the aspect ratio of the confinement potial ellipse. The variation of the doublet splitting energyshown in Fig. 4 for the lowest three optically active excitdoublets when the semimajor axis lengtha is fixed at 20 nmand the semiminor axis lengthb is varied around 20 nm witha fixed value ofLz53 nm. When the potential ellipse approaches a circular shape, the doublet splitting energy

FIG. 2. Exciton energy level structures are shown for a Gaelliptical quantum disk with the size parametersa520 nm,b515nm andLz(disk height)53 nm, in which only the short-range exchange term is taken into account. Thex(y)-direction of the polar-ization is along the semimajor~semiminor! axis of the potentialellipse. For the first and the fourth doublets, the energy positionthe intensity are almost degenerate. The triangles indicate theergy positions of the even-parity dark states.

FIG. 3. Exciton energy level structures are shown for a Gaelliptical quantum disk with the size parametersa520 nm,b515nm andLz(disk height)53 nm, in which both of the short-rangand the long-range exchange terms are taken into account.meaning of thex(y)-direction and the triangles are the same asFig. 2. The diamonds indicate the energy positions of the odd-paexciton states.

re-

eletn-

e-

creases to zero. This demonstrates the sensitivity of the lorange exchange interaction to the quantum disk shape ansignificance in determining the doublet splitting.

We have examined the dependence of the doublet sting energy on the size of a quantum disk with a fixed aspratio a/b. The results are shown in Fig. 5 fora/b54/3. Thesize-dependence of the doublet splitting energy for the loest doublet is rather weak, suggesting that the exciton wafunction is localized in the interior of the confinement potetial and is not strongly affected by the potential size. Onother hand, for the higher-lying exciton doublets, the spting energy shows a dependence of 1/an(n51.3;1.5) as de-picted by solid lines in Fig. 5. A simple argument basedEqs.~2.27! and ~2.28! indicates the 1/a2 dependence for theshort-range exchange term and the 1/a2 or 1/a3 dependencefor the long-range exchange term. The calculated sdependence is deviating from these simple predictions.mentioned before, this deviation might be ascribed tosize-dependence of details of the exciton wavefunctions.

s

dn-

s

he

ty

FIG. 4. The variation of the doublet splitting energy in a Gaelliptical quantum disk is shown for the lowest three optically actexciton doublets when the semimajor axis lengtha is fixed at 20 nmand the semiminor axis lengthb is varied around 20 nm with a fixedvalue ofLz53 nm.

FIG. 5. The doublet splitting energy in a GaAs elliptical quatum disk is plotted for the lowest four exciton doublets as a functof the semimajor axis lengtha with a fixed aspect ratioa/b54/3and a fixed value ofLz53 nm. The solid lines show a dependenof 1/an(n51.3;1.5).

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heen-akens

zo-lf


So far we have employed the values ofve0526 meV and

vh053 meV for the lateral confinement potential in E

~2.22!. We examined, however, the dependence of the dblet splitting energy on the potential depth. The splitting eergy of the ground exciton doublet is 27meV for (ve

0 ,vh0)

5(24 meV, 2 meV!, 29 meV for (ve0 ,vh

0)5(26 meV, 3meV!, 34meV for (ve

0 ,vh0)5(29 meV, 4 meV! and 46meV

for (ve0 ,vh

0)5(215 meV, 10 meV!. The trend of these numerical results can be understood as follows. When theeral confinement is strengthened, both of the electron andhole wavefunctions shrink spatially and the overlap betwthem increases, resulting in the increase of both the shrange and the long-range exchange interactions and thehancement of the doublet splitting energy.

IV. POLARIZATION CHARACTERISTICSOF EXCITON DOUBLETS

The polarization characteristics of exciton doubletsquite interesting and reveal an important physical aspect.the lowest doublet, the lower exciton state is polarized alothe x-direction, namely along the semimajor axis of the ptential ellipse. On the other hand, the upper exciton statpolarized along they-direction ~the semiminor axis!. How-ever, for the second and third doublets the relation betwthe energetic order and the polarization direction is reversAccording to our calculation, the polarization direction of tlower exciton state isx, y, y, and x for the lowest fourexciton doublets in complete agreement with texperiment.5 These features can be understood in view ofspatial distribution of exciton polarization.

The transition dipole moment of an exciton state is givby

Pi5(r

(t,s

^ctum i uvs& f ts* ~r ,r !, ~ i 5x,y,z!, ~4.1!

wherem i is the dipole moment operator in thei-direction. Inthe following we interpret(t,s^ctum i uvs& f ts* (r ,r ) as thepolarization density in thei-direction at the positionr. Butthis quantity is in general complex and we consider thesolute value to understand the qualitative features. Thelarization distribution calculated in this way is shown in Fi6 for the first exciton doublet in a confinement potential w(a,b)5(20 nm, 15 nm!. For the lower exciton state~1 state!,the distribution function of thex-polarization is single-peaked@Fig. 6~a!# corresponding to a uniform distribution othe polarization direction. On the other hand, the distributfunction of they-polarization@Fig. 6~b!# is four-peaked, in-dicating a staggered distribution of the polarization directiFor the upper exciton state of the first doublet~18 state!, thespatial distribution functions of thex- and y-polarization@Figs. 6~c! and 6~d!# are interchanged relative to the casethe lower exciton state. These features of polarization disbution are schematically depicted in Figs. 7~a!–7~d!.

Now we consider the dipole-dipole interaction originatifrom the long-range exchange interaction. The dipole-dipinteraction energy changes its sign according to the polartion configuration as shown in Fig. 8. Then we see thatthe x-polarized exciton state~1 state! the negative contribu-tion is significant, whereas for they-polarized exciton state

u--

t-henrt-en-

eorg-is

nd.

e

n

-o-

n

.

fi-

lea-r

~18 state! the positive contribution is substantial. For thstaggered configuration of polarizations like in Figs. 7~b! and7~c!, the positive and negative dipole-dipole interactiolargely canceled each other and the resultant contribuwas rather small compared to that from Figs. 7~a! and 7~d!.As a consequence, thex-polarized exciton state has a loweenergy than they-polarized state.

The same argument can be extended to the higher-lydoublet states. The distribution functions of thex- andy-polarization are depicted in Figs. 9, 10, and 11 for tsecond, third, and fourth exciton doublets, respectively.the nth doublet, the exciton levels are named asn and n8corresponding to thex- andy-polarized state, respectively. Ithe excited states, the number of nodes of the exciton wafunction increases and correspondingly the polarizationtribution becomes more staggered and complicated. Forsecond and third doublets, there are two nodes inx-direction, whereas for the fourth doublet there are tnodes in they-direction. These polarization distribution functions are schematically depicted in Figs. 12, 13, and 14.sign of the dipole-dipole interaction energy deduced froFig. 8 are also given in the figures. Then we see that forsecond and third doublets they-polarized state~28 or 38state! is energetically lower than thex-polarized state~2 or 3state!. On the other hand, for the fourth doublet the energythe x-polarized state~4 state! is lower than that of they-polarized state~48 state!. Thus the key concepts to undestand the energetic order of the orthogonally polarized eton doublet states are the node configuration of the polartion distribution and the dipole-dipole interaction originatinfrom the long-range exchange term.

We have examined the dependence on the lateral sizthe quantum disk of the energetic order of the orthogonapolarized exciton doublet states. For the quantum disks wa fixed aspect ratio, namelya/b54/3, we examined the caseof (a,b)5(12 nm, 9 nm!, ~20 nm, 15 nm!, ~28 nm, 21 nm!and~36 nm, 27 nm! with a fixed disk height ofLz53 nm andfound that the polarization direction of the lower excitostate of the lowest four exciton doublets is commonlyx, y, y,andx. This reflects the same node configuration of the exton polarization for these quantum disks having the commvalue of the aspect ratio.

We examined also the cases with different aspect ratnamely the cases of (a,b)5(30 nm, 10 nm! and~40 nm, 10nm!. The polarization direction of the lower exciton statethe lowest four exciton doublets isx, y, y, and y for (a,b)5(30 nm, 10 nm! andx, x, y, andy for (a,b)5(40 nm, 10nm!, respectively. These features can be understood byspecting the polarization distribution functions, althouthey are not shown. Thus the polarization characteristicsthe exciton doublet states depend sensitively on the asratio of the lateral confinement potential.

V. POLARIZATION RELAXATION OF EXCITONS

Now we discuss the polarization relaxation by telectron-phonon interaction. In this system the relevantergy level spacings are about a few meV and thus we tinto account only the interactions with acoustic phonothrough the deformation potential coupling and the pieelectric coupling.21,11 The electron-phonon interaction itse

he size


FIG. 6. The polarization distribution function is plotted for the first exciton doublet states in a GaAs elliptical quantum disk with tparametersa520 nm,b515 nm, andLz(disk height)53 nm. ~a! x-polarization distribution for thex-polarized exciton state~1 state!, ~b!y-polarization distribution for the 1 state,~c! x-polarization distribution for they-polarized exciton state~18 state!, ~d! y-polarizationdistribution for the 18 state. The weight (z-axis! is plotted in the arbitrary units but its scale is common for all the cases.

acni-

er-

ernons

io

ri-

does not flip the electron spins. The exciton eigenstatesmixed states of all the eight combinations of the Bloch funtions as in Eq.~2.1!. Thus it is possible to make a transitiobetween the dominantlyx-polarized states and the dom

FIG. 7. Schematic representation of the polarization distributin Fig. 6 for the first exciton doublet.~a! x-polarization distributionfor the 1 state,~b! y-polarization distribution for the 1 state,~c!x-polarization distribution for the 18 state,~d! y-polarization distri-bution for the 18 state.

re-nantly y-polarized states through the electron-phonon intaction.

For the inter-doublet transitions, it is sufficient to considonly the one-phonon processes because the acoustic pho

n

FIG. 8. The sign of the dipole-dipole interaction energy for vaous configurations of polarization direction.

um disk


FIG. 9. The polarization distribution function is plotted for the second exciton doublet states in the same GaAs elliptical quantas in Fig. 6.~a! x-polarization distribution for thex-polarized state~2 state!, ~b! y-polarization distribution for the 2 state,~c! x-polarizationdistribution for they-polarized state~28 state!, ~d! y-polarization distribution for the 28 state.

vtrbs

nopt

a

an-theuan-

on

resonant with the relevant transition energy have a walength comparable to the quantum disk size and the maelement of the electron-phonon interaction has a sizamagnitude. On the other hand, for the intra-doublet trantions, we have to consider multiphonon, at least two-phoprocesses because the contribution from the one-phononcesses is vanishingly small. In general, the matrix elementhe electron-phonon interaction is proportional to

E d3r e E d3r h F f* ~r e ,r h!eiqW r eW

~or eiqW r hW

!Fi~r e ,r h!,

~5.1!

whereqW is the phonon wavevector andFi(F f) is the enve-lope function of the initial~final! exciton state. Since thedoublet splitting energy is about several tens ofmeV, thecorresponding wavevector of phonons is rather sm(;105 cm21) and we can approximate as

e-ixlei-nro-of

ll

E d3r e E d3r h F f* ~r e ,r h!S 11 iqW •r eW ~ iqW •r h

W !

11

2~ iqW •r e

W !2„~ iqW •r h

W !2…1••• DFi~r e ,r h!

5E d3r e E d3r h F f* ~r e ,r h!

3S 1

2~ iqW •r e

W !2„~ iqW •r h

W !2…1••• DFi~r e ,r h!,

~5.2!

where the first term in the parentheses of the first line vishes due to the orthogonality of the eigenfunctions andsecond term also vanishes because the parity is a good qtum number for the confinement potential with inversi

disk as


FIG. 10. The polarization distribution function is plotted for the third exciton doublet states in the same GaAs elliptical quantumin Fig. 6. ~a! x-polarization distribution for thex-polarized state~3 state!, ~b! y-polarization distribution for the 3 state,~c! x-polarizationdistribution for they-polarized state~38 state!, ~d! y-polarization distribution for the 38 state.

ron

no

eroa

nses

ro-n intheingareu-s be-

esaAsin

are

symmetry. As a result, the probability of one-phonon pcesses is rather small and we have to consider two-phoprocesses. The probability of the Raman-type two-phoprocesses is given by

(v1 ,v2

U(j

^ f uHe2ph~v2!u j &^ j uHe2ph~v1!u i &Ei2Ej7\v1

U2

3d~Ef2Ei6\v17\v2!, ~5.3!

whereHe2ph is the electron-phonon interaction,j denotes theintermediate exciton states and the summation over theergies (v1 ,v2) and the wave vectors of two phonons is caried out. In Table I, the calculated transition rates at 30 Kthe one-phonon and the two-phonon processes are comp

-onn

n--fred

for the intra-doublet transitions of the lowest four excitodoublets. It can be confirmed that the two-phonon procesare dominantly contributing.

Combining the one-phonon and the two-phonon pcesses, the calculated relaxation times at 30 K are showFig. 15 for the lowest four exciton doublet states havingeven parity. In this figure the dark exciton states consistof the even-parity dark states and the odd-parity statesnot included. The relaxation time within each exciton doblet states is rather long and about several nanosecondcause of the small energy difference. These relaxation timare not inconsistent with a recent measurement on InGQD.13 On the other hand, the typical relaxation times withexciton states having the same polarization (x- ory-polarization! are about 100 ps. These relaxation times

m disk


FIG. 11. The polarization distribution function is plotted for the fourth exciton doublet states in the same GaAs elliptical quantuas in Fig. 6.~a! x-polarization distribution for thex-polarized state~4 state!, ~b! y-polarization distribution for the 4 state,~c! x-polarizationdistribution for they-polarized state~48 state!, ~d! y-polarization distribution for the 48 state.

e

b

vtuep

on

ndncitio

atio

ob-

ax-nalig.ag-av-of

betheates

by

re-Inpro-be

in good agreement with the recently measured values12 usingthe single-dot pump-probe spectroscopy.

The relaxation time from the third exciton level to thsecond exciton level is rather long for both thex- and they-polarized states, reflecting the small energy differencetween the second and the third exciton levels (, 1 meV!.This is explained by the same argument given at Eq.~5.2!. Itis to be noted that the inset in Fig. 15 shows the exciton lestructures only schematically and does not reflect the aclevel spacings accurately. The relaxation time from the sond exciton level to the exciton ground state is about 100whereas that from the third exciton level to the excitground state is about several tens of picoseconds. Thusrelaxation from the third exciton level to the exciton groustate occurs directly instead of the two-step sequethrough the second exciton state. In general, the transbetween exciton levels whose energy difference is smwould be often bypassed by some other efficient relaxa

e-

elalc-s,

the

enlln

paths. This kind of phenomenon seems to have beenserved experimentally.22

Now we discuss the cross-relaxation or polarization relation between exciton states having the mutually orthogopolarizations. The calculated results at 30 K are given in F16. The cross-relaxation times are of the same order of mnitude as the relaxation times within the exciton states hing the same polarization. Thus the polarization relaxationexcitons occurs rather efficiently. At the same time, it is tonoted that the cross-relaxation rate is rather small fortransition between the third and the second exciton sthaving the orthogonal polarizations. This is also causedthe small energy difference.

The temperature dependence of the relaxation rate islated to the number of phonons involved in the transition.the case of one-phonon processes, the relaxation rate isportional to the phonon occupation number which canapproximated as

anionm

bu

bu

bu

elly

tionam-

e not

sttum

of


TABLE I. Comparison between the rates of the one-phononthe two-phonon processes at 30 K for the intra-doublet transitof the lowest four exciton doublets in a GaAs elliptical quantudisk with the size parametersa520 nm, b515 nm andLz(disk height)53 nm.

1-phonon 2-phonon

1218 1.42•1028meV 0.071meV2228 0.017 0.1563238 0.068 0.4184248 0.021 0.227

FIG. 12. Schematic representation of the polarization distrition in Fig. 9 for the second exciton doublet.~a! x-polarizationdistribution for the 2 state,~b! y-polarization distribution for the 2state, ~c! x-polarization distribution for the 28 state, ~d!y-polarization distribution for the 28 state.

FIG. 13. Schematic representation of the polarization distrition in Fig. 10 for the third exciton doublet.~a! x-polarization dis-tribution for the 3 state,~b! y-polarization distribution for the 3state, ~c! x-polarization distribution for the 38 state, ~d!y-polarization distribution for the 38 state.

FIG. 14. Schematic representation of the polarization distrition in Fig. 11 for the fourth exciton doublet.~a! x-polarizationdistribution for the 4 state,~b! y-polarization distribution for the 4state, ~c! x-polarization distribution for the 48 state, ~d!y-polarization distribution for the 48 state.

ds

-

-

-

FIG. 15. Population relaxation times at 30 K for thpolarization-conserving processes within the lowest four opticaactive exciton doublet states and for the intra-doublet relaxaprocesses in a GaAs elliptical quantum disk with the size paretersa520 nm,b515 nm andLz(disk height)53 nm. It is to benoted that the energy level structures are only schematic and arreflecting the actual energy spacings.

FIG. 16. Polarization relaxation times at 30 K within the lowefour optically active exciton doublet states in the same quandisk as in Fig. 15.

FIG. 17. Population relaxation rates are plotted as a functiontemperature for the successive transitions within thex-polarized ex-citon states in the same quantum disk as in Fig. 15.

inxen, tnlr

n

tiesd

ro

rehiesd

caonr

sela

tinan. Wtiod-intet co

e

thtetag.thomffi

learly

ctrafor

hetiotence

ig.oto-ar-lar-husis

ing

etex-piceticchoftheionr-

oletionletss-oc-ion

l

ed

theole


n~v!51

e\v/kBT21;

kBT

\v~5.4!

at high temperatures, such that\v!kBT. In Fig. 17, therelaxation rates for a few transitions within thex-polarizedexciton levels are shown as a function of temperature. Sthe energy difference between the second and the third eton states is rather small, the linear temperature dependis clearly seen. On the other hand, as mentioned beforerelaxation within the exciton doublet states occurs maithrough two-phonon processes and the rate is typically pportional to a factor given by

(v1 ,v2

M ~v1 ,v2!n~v1!„11n~v2!…, ~5.5!

where M contains the matrix elements of the electrophonon interaction. According to Eq.~5.4!, the rate in Eq.~5.5! is expected to be proportional toT2 at high tempera-tures. In Fig. 18, the temperature dependence of relaxarates is shown for the intra-doublet transitions of the lowfour exciton doublets. TheT2-dependence can be confirmeat high temperatures. In fact, in the temperature range f30 K to 50 K, the calculated results can be fit well byTn

(n51.98;2.24).

VI. PHOTOLUMINESCENCE SPECTRUMUNDER SELECTIVE EXCITATION

In the last section, it was found that the polarizationlaxation occurs efficiently in our system. In order to see teffect more clearly, we have calculated the photolumincence spectra under selective excitation. We have so farcussed the even-parity exciton states because the optiactive exciton states have the even parity. There are, hever, odd-parity exciton states which are optically dark atheir energy positions are shown by diamonds in Fig. 3 foGaAs elliptical quantum disk with the size parametersa520 nm,b515 nm andLz ~disk height!53 nm. The relax-ation times within the odd-parity exciton states and thobetween the even-parity exciton states and the odd-parityciton states are of the same order of magnitude as the reation times within the even-parity exciton states, suggesthat the dark exciton states would play possibly importroles as the intermediate states of relaxation processeshave formulated a set of rate equations for the occupaprobability of relevant exciton levels which include 12 odparity and 14 even-parity exciton levels in the ascendorder of energy up to 7 meV from the exciton ground sta

The photoluminescence spectrum under the resonanexcitation is obtained by a stationary solution of the setrate equations. The results for thex-polarized excitation atthe fourth exciton level are shown in Fig. 19 for 30 K. Thpolarization-conserving luminescence from thex-polarizedexciton ground state is strong but the luminescence fromy-polarized exciton ground state has also substantial insity. The situation is similar also for the resonant cw excition of they-polarized fourth exciton level as shown in Fi20. In this case, however, the luminescence fromx-polarized exciton ground state is stronger than that frthe y-polarized exciton ground state, indicating more e

ceci-ce

heyo-

-

ont

m

-s-is-lly

w-da

ex-x-gte

n

g.wf

en--

e

-

cient relaxation from they-polarized states to thex-polarizedstates than the reverse process. These results indicate cthe presence of efficient paths of polarization relaxation.

In the same way, we can calculate the excitation speof the exciton photoluminescence. The excitation spectrathe detection of luminescence from thex-polarized excitonground state are shown in Fig. 21 for both thex- andy-polarized excitation. This spectrum is very similar to tabsorption spectrum in Fig. 3, although the intensity rabetween thex- and they-polarized components is differenfor the fourth exciton doublet. Similar behaviors can be sein the excitation spectra for the detection of luminescenfrom the y-polarized exciton ground state as shown in F22. This indicates that the relaxation processes after phexcitation are not sensitively dependent on the initial polization direction because of the presence of efficient poization relaxation through the excited exciton states and tthe emission intensity from the exciton ground stategrossly determined by the absorption intensity of the excitlight.

VII. SUMMARY

In summary, we identified the origin of exciton doublstructures as the long-range part of the electron-holechange interaction which is manifested by the anisotroshape of quantum dots. The physical origin of the energorder of the orthogonally polarized exciton states of eadoublet is clarified by inspecting the spatial distributionthe exciton polarization. The key concepts to understandenergetic order are the node configuration of the distributfunction of exciton polarization and the dipole-dipole inteaction energy originating from the long-range electron-hexchange interaction. We studied the population relaxaand predicted a long relaxation time for the intra-doubtransitions about several ns. We also found that the crorelaxation between orthogonally polarized exciton statescurs, in general, as efficiently as the population relaxatwithout the change of the polarization direction.

ACKNOWLEDGMENTS

The author would like to thank Dr. Al. L. Efros for usefudiscussions at the initial stage of this work.

APPENDIX: PARAMETERS RELATEDTO THE ELECTRON-HOLE EXCHANGE ENERGIES

Here the determination of parametersEXS in Eq. ~2.16! and

m in Eq. ~2.19! is discussed. These quantities can be fixfrom the singlet-triplet splitting energyDST and thelongitudinal-transverse splitting energyDLT in bulk materi-als. In bulk, the exciton state can be written as

uX&K51

AN(t,s

(n,m

eiKRmFts~Rn! actRn1Rm

† avsRmu0&,

~A1!

whereK is the center-of-mass wave vector,F the envelopefunction describing the electron-hole relative motion andWannier function representation is used. The electron-hexchange energy is given by

er

8pe

in

n otu

na

nant

dere


(tst8s8

Fts~0!Ft8s8~0!V~ctR0 ,vs8R0 ;ct8R0 ,vsR0!

1 (tst8s8

Fts~0!Ft8s8~0! (m8Þm

e2 iK (Rm2Rm8)

3V~ctRm ,vs8Rm8 ;ct8Rm8 ,vsRm!, ~A2!

where the first~second! term is the short~long!-range ex-change term. By the multipole expansion, the second tcan be rewritten as

24p

3v0(

tst8s8Fts~0!Ft8s8~0! tmW ct,vs

3@ 123KW tKW # mW vs8,ct8 , ~A3!

FIG. 18. Polarization relaxation rates are plotted as a functiotemperature for the intra-doublet transitions in the same quandisk as in Fig. 15. The solid lines show a dependence ofTn(n51.98;2.24).

FIG. 19. The photoluminescence spectrum under the resocw excitation at thex-polarized fourth exciton level at 30 K in aGaAs elliptical quantum disk with the size parametersa520 nm,b515 nm, andLz(disk height)53 nm.

m

whereK is a unit vector in the direction ofK. In GaAs thetop valence band is the fourfold degenerateG8(J53/2)band. Carrying out a straightforward calculation in the38 matrix form and assuming the ground-state envelofunction as

F~r !5A v0

paB3

e2r /aB, ~A4!

whereaB is the exciton Bohr radius andv0 is the volume ofa unit cell, we find the energy level structure as shownFig. 23 with

ES5v0

paB3

EXS and EL5

4m2

3aB3

. ~A5!

The lowest state is the fivefold degenerateJ52 dark excitonstate, the next lowest state is the doubly degenerateJ51

fm

nt

FIG. 20. The photoluminescence spectrum under the resocw excitation at they-polarized fourth exciton level at 30 K in thesame quantum disk as in Fig. 19.

FIG. 21. Photoluminescence excitation spectrum at 30 K unthe detection of emission from thex-polarized exciton ground statin the same quantum disk as in Fig. 19.

lerac-thers

S6

an

.

D

,k

ys

tt.

andser

m-

.

ys.

.

ra,

de

e innce


transverse exciton state and the uppermost state is theJ51longitudinal exciton state. Then we can identify as

*Present address: Department of Electronics and Informationence, Kyoto Institute of Technology, Matsugasaki, Kyoto 6008585, Japan.

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FIG. 22. Photoluminescence excitation spectrum at 30 K unthe detection of emission from they-polarized exciton ground statin the same quantum disk as in Fig. 19.

DST54v0

3paB3

EXS , DLT5

32m2

3aB3

. ~A6!

The parametersEXS andm can be fixed from the experimenta

values ofDST and DLT .20 It is to be noted that the subtlproblem of screening of the electron-hole exchange intetion is circumvented in the above formulation becauseeffect of screening is implicitly included in the parameteEX

S andm.

ci--

d

.

,

.

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er

FIG. 23. Schematic level structure of the exciton ground statbulk materials like GaAs having the fourfold degenerate valeband. The definitions ofES andEL are given in the Appendix.

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Theory of exciton doublet structures and …takaghra/tak2000.pdfTheory of exciton doublet structures and polarization relaxation in single quantum dots T. Takagahara* NTT Basic Research