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Theory of exciton fine structure in semiconductor quantum dots :
Quantum dot anisotropy and lateral electric field
Theory of exciton fine structure in semiconductor quantum dots: Quantum dot anisotropy
and lateral electric field
Eugene Kadantsev*and Pawel Hawrylak
Quantum Theory Group, Institute for Microstructural Sciences, National Research Council, Ottawa, Canada K1A 0R6
共Received 21 October 2009; revised manuscript received 14 December 2009; published 13 January 2010兲
Theory of exciton fine structure in semiconductor quantum dots and its dependence on quantum-dot aniso-tropy and external lateral electric field is presented. The effective exciton Hamiltonian including long-range electron-hole exchange interaction is derived within the k · p effective-mass approximation. The exchange matrix elements of the Hamiltonian are expressed explicitly in terms of electron and hole envelope functions. The matrix element responsible for the “bright” exciton splitting is identified and analyzed. An excitonic fine structure for a model quantum dot with quasi-two-dimensional anisotropic harmonic oscillator confining po-tential is analyzed as a function of the shape anisotropy, size, and applied lateral electric field.
DOI:10.1103/PhysRevB.81.045311 PACS number共s兲: 78.67.Hc, 73.21.La, 78.55.Cr I. INTRODUCTION
One of the promising applications1,2 of semiconductor
quantum dots共QDs兲 共Refs.3 and4兲 for quantum
cryptogra-phy is the generation of entangled photon pairs 共EPPs兲 on demand.5–8 EPPs can be generated with great efficiency via
the biexciton cascade process 共BCP兲 共Ref. 1兲 in which a
biexciton radiatively decays into the ground state via two different indistinguishable paths involving two intermediate dipole-active 共bright兲 exciton states. The major barrier to EPPs generation in BCP is the splitting of the intermediate “bright” exciton levels which distinguishes the paths of ra-diative decay and, as a result, destroys the entanglement. The splitting and mixing of the two bright exciton states is con-trolled by the long-ranged electron-hole exchange共LRE兲 in-teraction and depends on dot asymmetry and applied mag-netic field.6,8–10 Lateral electric fields have been applied in
the hope to control the dot anisotropy and hence anisotropic exchange splitting.11–13
Better understanding of the electron-hole exchange inter-action in QDs, particularly its LRE part, should help in the development of EPP generation schemes.
The exchange integral can be decomposed in real space into long-range part, i.e., exchange interaction between two “transition densities” localized in two different Wigner-Seitz 共WS兲 cells and short-range part, i.e., exchange interaction within a single WS cell. A closely related decomposition of exchange into analytical and nonanalytical part exists in the reciprocal space. In what follows, we will use the “real space” definition of long-range and short-range exchange 共SRE兲. We will reserve the use of words “local exchange” and “nonlocal exchange” to describe the type of integrals which contribute to the exchange matrix elements.
LRE electron-hole interaction in bulk semiconductors was investigated almost 40 years ago14–16 and it is now well
es-tablished that LRE splits the energy levels of bright excitons. For example, in zinc-blende direct band-gap semiconductors with fourfold-degenerate valence band ⌫8v and twofold-degenerate conduction band ⌫6c, the ground-state exciton is eightfold degenerate. The addition of SRE interaction into the excitonic Hamiltonian will split the eightfold-degenerate ground state into “dark” and bright multiplets with
degenera-cies of five and three. The addition of LRE interaction will further modify the fine structure by splitting threefold-degenerate bright exciton level into two transverse excitons with Jz= ⫾ 1 and one longitudinal exciton with Jz= 0.
Recent advances in single QD spectroscopies motivated reexamination of electron-hole exchange in systems with re-duced dimensionality within the framework of envelope function approximation17–23 and, more recently, from the
point of view of atomistic empirical pseudopotential and tight-binding approximation.24–28
Within the envelope function approximation, Takagahara17have shown that if the electron-hole pair
enve-lope contains only Y00共,兲 angular momentum component, then the long-range part of the electron-hole exchange van-ishes. In Refs. 18and20, fine structure of localized exciton levels in quantum wells was considered. Efros et al.19 and
Gupalov et al.22investigated band-edge excitonic fine
struc-ture of spherical CdSe nanocrystals. In Ref. 21, Takagahara derived effective eight-band excitonic Hamiltonian which takes into account electron-hole exchange interaction. In par-ticular, within the envelope function formalism of Refs. 17
and21, LRE is a dipole-dipole interaction. Takagahara21
ap-plied his scheme to investigate excitonic fine structure of disklike GaAs/AlGaAs QDs as a function of QD anisotropy and size. It was demonstrated numerically in Ref. 21 that LRE vanishes for the ground-state bright exciton doublet in QDs with rotationally symmetric three-dimensional confin-ing potential. A similar to that of Ref. 21 representation of the electron-hole exchange was discussed by Maialle et al.18
in connection with the exciton spin dynamics in quantum wells.
Within atomistic empirical approach it has been demon-strated that the physical origin of long-range exchange inter-action might be different from that of bulk24 and that LRE
has a nonvanishing magnitude even in “shape-symmetric” dots.26,27 The former and the latter stem from the loss of
local orthogonality on the unit-cell scale between electron and hole single-particle orbitals compared to the bulk case and “reduced” symmetry of the atomistic confining potential, respectively. For ⌫6⫻ ⌫7exciton, Gupalov et al.25identified
the dipole-dipole and monopole-monopole contributions to the LRE with the intra-atomic and interatomic transition den-sities, respectively.
PHYSICAL REVIEW B 81, 045311共2010兲
In this work, we will reexamine electron-hole exchange within the framework of envelope approximation. We note that Takagahara’s condition17 for the vanishing of LRE in
spherically symmetric quantum dots constitutes only a
sufficient condition for quenching of LRE. It does not
ex-plain, for example, why LRE vanishes in disklike QDs with rotational共C⬁v兲 symmetry. These QDs are “squeezed” in one direction and the ground-state envelopes of single-particle states will contain angular momentum components that are higher than Y00共,兲. Motivated by recent experiments, we
will investigate, within our model, the effects of the external electric field and size scaling of LRE interaction.
We derive here effective “four-band” excitonic Hamil-tonian which includes effects of LRE interaction. The ele-ments of the effective Hamiltonian are expressed explicitly in terms of envelope functions. The “microscopic parts” of single-particle orbitals are integrated out and enter the effec-tive Hamiltonian as numerical parameters. The number of bands is truncated to two conduction and two valence bands for the simplicity of the interpretation. We present explicit expression for the matrix element responsible for the bright exciton splitting and, therefore, establish a new sufficient condition for LRE quenching.
An excitonic fine structure for a model system in which the confining potential has a form of two-dimensional-like anisotropic harmonic oscillator共2DLAHO兲 is considered as a function of lateral anisotropy. We present an explicit ex-pression for the bright splitting of excitonic ground state as a function of lateral anisotropy. It is found, in agreement with previous work,21 that, within the envelope approximation,
the bright ground-state exciton splitting vanishes in the case of laterally isotropic confining potential. The quenching of bright exciton splitting coincides with vanishing of the ma-trix element responsible for the bright exciton splitting in our effective Hamiltonian.
We also analyze the effect of the lateral electric field on the excitonic fine structure of our model quantum dot. We find that the bright exciton splitting decreases due to the spatial separation of electron and hole envelopes.
Finally, the scaling of the bright exciton splittings with system’s size is analyzed. It is found that the scaling of bright exciton splittings differ from the laws established us-ing simple dimensionality arguments.
Effective units of length and energy are used throughout unless otherwise specified. The lengths are measured in effective Bohr’s aeB=⑀ប2/共mⴱe2兲, where ⑀ is dielectric
constant and mⴱ is the conduction-band effective mass. Energies are measured in effective Hartree’s, 1 hartree = mⴱe4/共⑀ប兲2= 2 Ryⴱ. For example, using material param-eters for GaAs, aeB= 97.9 Å and 1 hartree= 11.86 meV.
II. THEORY OF EXCITON FINE STRUCTURE IN ENVELOPE FUNCTION APPROXIMATION
We now describe the single-particle states and the exciton fine structure in the envelope function approximation. The single-particle orbitals of the electron in a quantum dot are two-component spinors共two-row columns兲 written as
共r兲 =
冉
a共r兲 b共r兲冊
=a共r兲兩␣典 +b共r兲兩典, 兩␣典 =冉
1 0冊
, 兩典 =冉
0 1冊
. 共1兲A dagger sign †will denote complex conjugation for com-plex quantities.
With c† and h† 共c and h兲 electron and hole creation 共an-nihilation兲 operators, the interacting electron-hole Hamil-tonian can be written as
HˆX= Hˆe+ Hˆh+ Hˆint, Hˆe=
兺
i i ec i †c i, Hˆh=兺
j j hh j †h j, Hˆint=兺
ijkl ci†hj†hkcl冉
− 1 ⑀Vikjl C + Viklj E冊
, Vijkl= V共i, j;k,l兲 =冕冕
dr1dr2 i†共r1兲j †共r 2兲k共r2兲l共r1兲 兩r1− r2兩 , 共2兲 where Hˆe, Hˆh, and Hˆint are electron, hole, and electron-holeinteraction Hamiltonian, respectively. The electron-hole in-teraction Hamiltonian consists of two parts: VC/⑀ is the
rect electron-hole Coulomb attraction screened by static di-electric constant ⑀ and VE is the electron-hole exchange
interaction. The question of screening of electron-hole ex-change is a subtle one and it is generally agreed that, at least, parts of the electron-hole exchange should be screened. In Ref. 29 共page 252兲, the long-range electron-hole exchange
interaction in reciprocal space is screened by the high-frequency dielectric constant. We will assume, for now, that our exchange interaction VE contains screening implicitly.
To obtain excitonic states, Hamiltonian共2兲 is diagonalized
in the basis of all electron-hole pairs of the type c†h†兩g.s.典, where兩g.s.典 denotes a many-body state with fully occupied valence and empty conduction bands.
In this work, the hole and electron single-particle states are computed in the effective-mass approximation 共EMA兲 共Refs.29and30兲 which neglects band coupling in the
single-particle states. In EMA, the hole hand electron e
single-particle states are uniquely specified by band label 共valence
vj z
⬘
and conduction c⬘
兲 and envelop index 共r and p兲, forexample, h共r兲 = Fvj z ⬘ r 共r兲uvjz⬘共r兲, e共r兲 = Fc⬘ p 共r兲uc⬘共r兲. 共3兲
Here, u共r兲 is the periodic part of Bloch eigenstate at k=0. In what follows, u共r兲 is a two-component spinor. We will as-sume that “bra” electron-hole pair is described by indices
csand vjzq, respectively.
The excitonic Hamiltonian matrix element in the basis of electron-hole pairs is
具g . s.兩hvjzqccsHˆXcc⬘p † h vj z ⬘r † 兩g . s.典 =␦vjzq,vj z ⬘r␦cs,c⬘p共vjzq h + cs e 兲 − VC共cs,vj z
⬘
r;vjzq,c⬘
p兲/⑀ + VE共cs,vjz⬘
r;c⬘
p,vjzq兲. 共4兲The effective Hamiltonian which involves only envelope functions is obtained by integrating out “microscopic” de-grees of freedom共u兲. The derivation follows that of Ref.21. Assuming Tdsymmetry of the crystal, two conduction-band
microscopic functions兩c1/2典 and 兩c−1/2典 with spin projec-tions sz= 1 / 2 and sz= −1 / 2 are written as
uc1/2共r兲 = 具r兩c1/2典 =s共r兲
冉
Y00共rˆ兲 0冊
, uc−1/2共r兲 = 具r兩c − 1/2典 =s共r兲冉
0 Y00共rˆ兲冊
. 共5兲In Eq. 共5兲, we assumed that microscopic conduction-band
function are of pure s symmetry.
Two valence-band microscopic functions 兩v−3/2典 and 兩v+3/2典 with hole angular momentum projections jz= −3 / 2
and jz= 3 / 2 are taken as
uv−3/2共r兲 = 具r兩v − 3/2典 =p共r兲
冉
Y11共rˆ兲 0冊
, uv+3/2共r兲 = 具r兩v + 3/2典 =p共r兲冉
0 Y1−1共rˆ兲冊
. 共6兲In Eq. 共6兲 we assumed that microscopic valence-band
func-tions are of pure p symmetry. The eigenfuncfunc-tions of angular momentump共r兲Y11共rˆ兲 andp共r兲Y1−1共rˆ兲 can be expressed in terms of Cartesian functions px and py. We, therefore, are
neglecting pz contribution to the microscopic valence-band
functions.
In what follows, we give a brief derivation of the effective excitonic Hamiltonian.
A. Calculation of Coulomb direct matrix elements
The unscreened Coulomb direct matrix element is given by VC共cs,vj z
⬘
r;vjzq,c⬘
p兲 =冕冕
dr1dr2 q1共r1兲q2共r2兲 兩r1− r2兩 , q1共r1兲 = Fcs†共r1兲uc†共r1兲Fc ⬘ p 共r1兲uc⬘共r1兲, q2共r2兲 = Fvj z ⬘ r† 共r2兲uvj z ⬘ † 共r2兲Fvjz q 共r 2兲uvjz共r2兲, 共7兲 where we explicitly presented electron and hole single-particle orbitals as a product of an envelope F and a micro-scopic part u. The unscreened Coulomb attraction matrix el-ement is just Coulomb interaction between two transition densities, where each “transition density” is a product of twoelectron orbitals or two valence orbitals. The matrix element is approximated as VC共cs,vjz
⬘
r;vjzq,c⬘
p兲 =␦cc⬘␦vjzvj z ⬘冕冕
dr1dr2 ⫻ Fcs†共r1兲Fc ⬘ p 共r1兲Fv⬘ r† 共r2兲Fv q 共r 2兲 兩r1− r2兩 . 共8兲B. Calculation of Coulomb exchange matrix elements
The exchange matrix element can be written as
VE共cs,vjz
⬘
r;c⬘
p,vjzq兲 =冕冕
dr1dr2 q1共r1兲q2共r2兲 兩r1− r2兩 , q1共r1兲 = Fcs†共r1兲uc†共r1兲Fvjz q 共r 1兲uvjz共r1兲, q2共r2兲 = Fc ⬘ p 共r2兲uc⬘共r2兲Fvj z ⬘ r† 共r2兲uvj z ⬘ † 共r2兲. 共9兲 The exchange matrix element can be thought of as a Cou-lomb interaction of two transition densities, where each tran-sition density is a product of electron-hole single-particle orbitals.We decompose exchange integral关Eq. 共9兲兴 into the
short-range and long-short-range contributions in real space. We will refer to the whole integration region as Born-von Karmen cell共BvK cell兲 and to the individual unit cell within BvK cell as WS cell. The double integration over BvK cell 共which consists of Ncell WS cells兲 is replaced by Ncell⫻ Ncell
inte-grals over WS cells,
冕
r1苸BvK冕
r2苸BvK dr1dr2 →兺
i=1 Ncell冕
r 1苸WS共Ri兲冕
r2苸WS共Ri兲 dr1dr2 +兺
i,j=1 i⫽j Ncell冕
r1苸WS共Ri兲冕
r2苸WS共Rj兲 dr1dr2, 共10兲where Ri and Rj label positions of the WS cells. The first
term in Eq.共10兲 consists of Ncellintegrals in which r1and r2 go over the same cell whereas the second term consists of
Ncell⫻共Ncell− 1兲 integrals in which r1 and r2 go over two distinct WS cells. The first sum in which r1 and r2 go over the same cell is the short-range exchange VSRE whereas the second sum in which r1 and r2 go over two different WS cells is the long-range exchange VLRE .
The “effective” expression for the exchange matrix ele-ments in terms of envelop functions is obtained in a way which follows “standard” k · p derivation, the only difference is the nature of microscopic integrals to be parameterized. One postulates the “slowly varying nature” of envelop func-tions and factors them out of the integral expression. For
THEORY OF EXCITON FINE STRUCTURE IN… PHYSICAL REVIEW B 81, 045311共2010兲
example, the short-range exchange matrix element is ex-pressed as VSRE 共cs,vjz
⬘
r;c⬘
p,vjzq兲 =兺
i=1 Ncell Fcs†共Ri兲Fvjz q 共R i兲Fc⬘ p 共Ri兲Fvj z ⬘ r† 共Ri兲 ⫻ISR共c,vjz⬘
;c⬘
,vjz兲, 共11兲where microscopic integral
ISR共c,vjz
⬘
;c⬘
,vjz兲 =冕
r1苸WS共Ri兲冕
r2苸WS共Ri兲 dr1dr2 ⫻ uc†共r1兲uvj z ⬘ † 共r2兲uc⬘共r2兲uvjz共r1兲 兩r1− r2兩 共12兲 depends only on band labels. In the case of long-range ex-change, the expression for VSRE 共cs, vjz⬘
r; c⬘
p, vjzq兲in-volves the sum over Ncell⫻共Ncell− 1兲 cells and the
micro-scopic integral is over two distinct WS cells referenced by vectors Riand Rj, ILR共c,vjz
⬘
;c⬘
,vjz兲 =冕
r1苸WS共Ri兲冕
r2苸WS共Rj兲 dr1dr2 ⫻ uc†共r1兲uvj z ⬘ † 共r2兲uc⬘共r2兲uvjz共r1兲 兩r1− r2兩 . 共13兲 Then, with the help of the truncated multipole expansion1/兩r1− r2兩 = 4 r⬎ Y00共1,1兲Y00†共2,2兲 +4r⬍ 3r⬎2 m
兺
=−1 1 Y1m共1,1兲Y1m † 共 2,2兲, r⬍= min共r1,r2兲, r⬎= max共r1,r2兲, 共14兲 one evaluates microscopic ISR and ILR integrals and, effec-tively, parameterizes the short-range and long-range ex-change interactions. The higher-multipole contributions are zero due to the postulated s and p symmetries of the micro-scopic functions. As a result of this procedure, we obtain the following “master” expression for the exchange matrix ele-ment: VE共cs,vjz⬘
r;c⬘
p,vjzq兲 = VSRE 共cs,vjz⬘
r;c⬘
p,vjzq兲 + VLR E 共cs ,vjz⬘
r;c⬘
p,vjzq兲, VSRE 共cs,vjz⬘
r;c⬘
p,vjzq兲 = ESR共HSR int兲 c⬘,vjz⬘ c,vjz冕
drFcs†共r兲Fvjz q 共r兲F c⬘ p 共r兲Fvj z ⬘ r† 共r兲, VLRE 共cs,vjz⬘
r;c⬘
p,vjzq兲 = −4 3 2关共d cvjz 0 兲†· d c⬘vj z ⬘ 0 兴 ⫻冕
drFcs†共r兲Fvj z q 共r兲F c⬘ p 共r兲Fvj z ⬘ r† 共r兲 −2兺
␥,␦=1 3 共dc0vjz兲␥†共dc⬘vj z ⬘ 0 兲␦冕冕
冋
2F c s†共r 1兲Fvjz q 共r 1兲 r1␥r1␦册
⫻ Fc ⬘ p 共r2兲Fvj z ⬘ r† 共r2兲 兩r1− r2兩 dr1dr2, 共15兲where ESRand2 are two numerical constants parameteriz-ing short-range and long-range exchange interactions, re-spectively,共HSRint兲c
⬘,vjz⬘
c,vjz
is an element of short-range exchange microscopic matrix, HSRint=
冢
兩c⬘
,vjz⬘
典 兩c⬘
,vjz⬘
典 兩c⬘
,vjz⬘
典 兩c⬘
,vjz⬘
典冏
1 2,− 3 2冔
冏
1 2, 3 2冔
冏
− 1 2,− 3 2冔
冏
− 1 2,+ 3 2冔
兩c,vjz典冏
1 2,− 3 2冔
1 0 0 0 兩c,vjz典冏
1 2,+ 3 2冔
0 0 0 0 兩c,vjz典冏
− 1 2,− 3 2冔
0 0 0 0 兩c,vjz典冏
− 1 2,+ 3 2冔
0 0 0 1冣
共16兲 and dcvjz 0共dcvjz 0 兲 x 共dcvjz 0 兲 y 共dcvjz 0 兲 z 兩c,vjz典
冏
1 2,− 3 2冔
− 1 − i 0 兩c,vjz典冏
1 2,+ 3 2冔
0 0 0 兩c,vjz典冏
− 1 2,− 3 2冔
0 0 0 兩c,vjz典冏
− 1 2,+ 3 2冔
1 − i 0. . 共17兲The numerical parameters ESRand2can be determined as to reproduce the excitonic fine structure in bulk semiconductors and, therefore, implicitly contain screening effects.
Taking into account Eqs. 共16兲 and 共17兲, a “block” Hamiltonian corresponding to the exchange interaction between two
electron-hole pairs FcpFv r
and FcsFv q
can be presented in the form
冢
兩c⬘
,vjz⬘
典 兩c⬘
,vjz⬘
典 兩c⬘
,vjz⬘
典 兩c⬘
,vjz⬘
典冏
1 2,− 3 2冔
冏
1 2,+ 3 2冔
冏
− 1 2,− 3 2冔
冏
− 1 2,+ 3 2冔
兩c,vjz典冏
1 2,− 3 2冔
␦0 SRE,L+␦ 0 LRE,L+␦ 0 LRE,N 0 0 ␦ 12 LRE,N 兩c,vjz典冏
1 2,+ 3 2冔
0 0 0 0 兩c,vjz典冏
− 1 2,− 3 2冔
0 0 0 0 兩c,vjz典冏
− 1 2,+ 3 2冔
␦21 LRE,N 0 0 ␦ 0 SRE,L+␦ 0 LRE,L+␦ 0 LRE,N冣
, 共18兲where we separated different contributions to the exchange based on their origin 共SRE or LRE兲 and integral type 共“lo-cal” and “nonlo共“lo-cal”兲. The contributions are
␦0SRE,L= ESR
冕
drFc s†共r兲F v q 共r兲Fc p 共r兲Fv r†共r兲, ␦0LRE,L= −8 2 3冕
drFc s†共r兲F v q共r兲F c p共r兲F v r†共r兲, ␦0LRE,N= −2共R xx+ Ryy兲, ␦12LRE,N= VLRE 共1/2s,3/2r;− 1/2p,− 3/2q兲 =2共R xx− 2iRxy− Ryy兲, ␦21LRE,N= V LR E 共− 1/2s,− 3/2r;1/2p,3/2q兲 =2共Rxx+ 2iRxy− Ryy兲, R␦␥=冕冕
再
冉
2 r1␦r1␥冊
Fc s†F v q冎
Fc p共r 2兲Fv r†共r 2兲 兩r1− r2兩 dr1dr2. 共19兲With regard to the above exchange expressions we note the following: 共a兲 the short-range exchange causes splitting between bright 共兩12, −32典,兩−12, +23典兲 and dark doublets 共兩1 2, + 3 2典,兩− 1 2, − 3
2典兲 by moving bright doublet up in energy by
␦0SRE,L. SRE does not split the bright doublet. The integral describing SRE is local in nature and involves overlap be-tween two electron-hole pairs.共b兲 Long-range exchange con-tains expressions of two types: a local term ␦0LRE,L which arises from 2关共d
cv
0 兲†· d
c⬘v⬘
0 兴 and nonlocal terms ␦ 0 LRE,N,
␦12LRE,N, and ␦ 21
LRE,N which involve differentiation operators applied to electron-hole pair envelopes. The nonlocal terms describe dipole-dipole interaction between electron-hole transition densities. 共c兲 The splitting between bright-dark states is affected by LRE through the local LRE term␦0LRE,L and nonlocal LRE term ␦0LRE,N. LRE, in general, splits the bright doublet by coupling electron-hole pairs with “com-pletely opposite” z projections of angular momentum, for example, 兩−12,32典 and 兩12, −32典. The terms which are respon-sible for the bright exciton splitting are ␦12LRE,N and
␦21LRE,N—the nonlocal expressions arising from long-range exchange interaction. 共d兲 The splitting within the dark dou-blet is not described in our model.
The splitting of bright exciton levels vanishes provided that ␦12LRE,N and␦21LRE,N vanish which constitutes a condition
THEORY OF EXCITON FINE STRUCTURE IN… PHYSICAL REVIEW B 81, 045311共2010兲
for LRE quenching within our model. In the next section, we demonstrate numerically and analytically that this condition is fulfilled in the case of isotropic 2D HO confining potential.
III. EXCITON FINE STRUCTURE OF QUASI-2DLAHO QUANTUM DOT
In this section, we will investigate excitonic fine structure for a model quantum dot in which the confining potential is of 2D-like anisotropic harmonic-oscillator type. We note that HO spectrum has been observed in self-assembled quantum dots.31
The EMA equations for holes are
Hˆhh= − 1 2M储,hh 共ⵜx 2 + ⵜy 2兲 +1 2M储,hh共x,hh 2 x2+ y,hh 2 y2兲 − 1 2M⬜,hhⵜz 2+1 2M⬜,hhz,hh 2 z2, x,hh=hh 0 共1 + t兲, y,hh= hh0 1 + t, x,hhy,hh=共hh 0兲2 = const, z,hhⰇhh0 , 共20兲
where M储,hh and M⬜,hh are effective masses, ␥共␥= x , y , z兲
denotes confinement frequency in x, y, and z directions, re-spectively, hh0 determines the typical energy scale of the confining potential and, together with the mass and the con-finement. The energies and lengths are expressed in the ef-fective units.
EMA equations for electrons have a similar form
Hˆee= − 1 2共ⵜx 2 + ⵜy 2 + ⵜz 2兲 +1 2共x,ee 2 x2+ y,ee 2 y2+ z,ee 2 z2兲, x,ee=ee 0共1 + t兲, y,ee= ee 0 1 + t, x,eey,ee=共ee 0兲2 = const, z,eeⰇee 0 . 共21兲
The excitonic fine structure is studied as a function of anisotropy t. For example, for t = −0.5, x=共1/2兲0, and
y= 20, the confinement along x is weaker than along y. For t= 0.5,x=共3/2兲0, and y= 2 / 30, confinement along x is
greater than along y. In the case t = 0, the confining potential for holes and electrons is isotropic in the lateral direction. By changing anisotropy t, we control the shape of the confining potentials for holes 关Eq. 共20兲兴 and electrons 关Eq. 共21兲兴 and,
as a consequence, the single-particle energy spectra and eigenfunctions.
We have assumed that the confinement in the vertical di-rection z is much stronger than confinement in the lateral direction zⰇ0. Therefore, we will always assume the
ground-state solution in z direction and the single-particle energy spectra for holes关Eq. 共20兲兴 and electrons 关Eq. 共21兲兴 is
2D like, E共n,m,0兲 =
冉
n+1 2冊
x+冉
m+ 1 2冊
y+ 1 2z, 共22兲 where n and m are 2DLAHO quantum numbers. In what follows, we use numerical parameters M储,hh= 9.930853,M⬜,hh= 5.218662, M储,ee= 1.0, M⬜,ee= 1.0, z,hh= 20.155737, andz,ee= 105.185977.
The eigenfunctions are products of one-dimensional Her-mite polynomials and exponential function,
nm0共r兲 =n共x兲m共y兲0共z兲, 共23兲 where the eigenfunction in one of the Cartesian directions 共for example, x兲 is given by
n共x兲 =
冑
1 2nn!冉
mx 冊
1/4 exp冉
−mx 2 x 2冊
H n共冑
mxx兲, Hn共x兲 = 共− 1兲nexp共x2兲 dn dxnexp共− x 2兲. 共24兲Figures 1共a兲–1共d兲 show single-particle spectra for electrons and holes, respectively, for two different characteristic en-ergy scales 0. Solid horizontal line共around 52.6 and 10.1 hartree for electrons and holes, respectively兲 corresponds to the confinement energy in vertical direction z/2.
The single-particle energy spectra are given by Eq. 共22兲.
The single-particle ground state is the nodeless s envelope which corresponds to n = m = 0. The “excited” levels are
or-52.00 54.00 56.00 58.00 60.00 62.00 −0.4 −0.2 0 0.2 0.4 Energy (H ) Anisotropy t Electron Energy vs. t (ωee0=1.63) 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 −0.4 −0.2 0 0.2 0.4 Energy (H ) Anisotropy t HH Energy vs. t (ωhh0=0.52) 64.50 65.00 65.50 66.00 66.50 67.00 −0.4 −0.2 0 0.2 0.4 Energy (H ) Anisotropy t Exciton Energy vs. t (ωhh 0=0.52,ω ee 0=1.63) (a) (c) 52.00 54.00 56.00 58.00 60.00 62.00 −0.4 −0.2 0 0.2 0.4 Energy (H) Anisotropy t Electron Energy vs. t (ωee0=2.30) 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 −0.4 −0.2 0 0.2 0.4 Energy (H) Anisotropy t HH Energy vs. t (ωhh0=0.73) 65.50 66.00 66.50 67.00 67.50 68.00 −0.4 −0.2 0 0.2 0.4 Energy (H) Anisotropy t Exciton Energy vs. t (ωhh 0=0.73,ω ee 0=2.30) (b) (d) (f) (e)
FIG. 1. Single-particle and exciton energy levels as a function of lateral anisotropy t of the confining potential.
ganized in shells referred to as p, d, f, and so on with char-acteristic “spatial” degeneracies of 2, 3, 4, etc., in the case of isotropic共t=0兲 confinement potential. The dispersion E共t兲 of
senergy level with anisotropy is weak compared to the dis-persion of excited envelopes. Strong anisotropy might lead to the level crossing where the single-particle energy of d-type envelope is below the single-particle energy of p-type enve-lope. The level crossing happens, for example, at t⬇−0.3 for the second excited state in Fig.1共a兲. Due to the smaller ef-fective mass of electrons, the spacing between electron single-particle levels is larger than that of the hole levels.
Figures1共e兲and1共f兲show “noninteracting” electron-hole and “interacting” exciton energies as a function of lateral anisotropy t for two different characteristic energy scales0. The black squares in Figs. 1共e兲 and 1共f兲 correspond to the noninteracting electron-hole pair energies
␦vq,v⬘r␦cs,c⬘p共vq h
+ cs
e 兲. The empty circles are obtained
by diagonalizing the full excitonic Hamiltonian. Each of the empty circles is actually a multiplet of four exciton states with fine structure determined by the electron-hole exchange interaction. The noninteracting electron-hole and excitonic spectra look quite similar. The Coulomb electron-hole attrac-tion simply decreases the exciton energy. The Coulomb attraction-induced mixing of electron-hole pairs in an exciton is small due to the large separation of single-particle levels compared with the magnitude of screened Coulomb attrac-tion.
In the lower part of the interacting excitonic energy spec-tra and for small anisotropies 兩t兩ⱕ0.2, the dispersion EX共t兲
strongly resembles that of the dispersion of single-particle levels. This happens because hh0 ⬍ee0 and the lowest-energy electron-hole pairs follow the order of hole levels
sesh, seph, and sedh.
A. LRE interaction for s-type envelopes
Consider the case when exciton is given by the product of ground-state envelopes of electron and hole single-particle states 共seshexciton兲. In this case,
Fc†共r兲Fv共r兲 = NcNvexp共−␣xx 2−␣ yy2−␣zz2兲, Nc=
冉
2␣xc 冊
1/4冉
2␣yc 冊
1/4冉
2␣zc 冊
1/4 , Nv=冉
2␣xv 冊
1/4冉
2␣yv 冊
1/4冉
2␣zv 冊
1/4 , ␣x=␣xc+␣xv, ␣y=␣yc+␣yv, ␣z=␣zc+␣zv, ␣xc=x,ee 2 , ␣yc= y,ee 2 , ␣zc= z,ee 2 , ␣xv= M储,hhx,hh 2 , ␣yv= M储,hhy,hh 2 , ␣zv= M⬜,hhz,hh 2 , 共25兲 where Nc and Nv are normalization constants, ␣␥c and␣␥v共␥= x , y , z兲 denote confinements of electrons and holes in
x, y, and z directions, respectively, and␣␥=␣␥c+␣␥v denote confinements of electron-hole envelope.
The matrix elements responsible for bright exciton split-ting for sesh exciton are given by
␦12LRE,N=2共R xx− 2iRxy− Ryy兲, ␦21LRE,N=2共R xx+ 2iRxy− Ryy兲, 共26兲 where Rxx=
冕冕
冉
2F c †F v x 1 2冊
Fc共r2兲Fv †共r 2兲 兩r1− r2兩 dr1dr2= −共NcNv兲 2 ␣x␣y␣z 冑
Ix共␣x,␣y,␣z兲, Ix共␣x,␣y,␣z兲 =冕
0 /2冕
0/2 sin3cos2dd冋
sin2冉
1 2␣x cos2+ 1 2␣y sin2冊
+ 1 2␣z cos2册
3/2, Ryy=冕冕
冉
2Fc†Fv y 1 2冊
Fc共r2兲Fv †共r 2兲 兩r1− r2兩 dr1dr2= − 共NcNv兲 2 ␣x␣y␣z冑
Iy共␣x,␣y,␣z兲, Iy共␣x,␣y,␣z兲 =冕
0 /2冕
0/2 sin3sin2dd冋
sin2冉
1 2␣x cos2+ 1 2␣y sin2冊
+ 1 2␣z cos2册
3/2, Rxy=冕冕
冉
2F c †F v y 1x1冊
Fc共r2兲Fv †共r 2兲 兩r1− r2兩 dr1dr2= 0. 共27兲THEORY OF EXCITON FINE STRUCTURE IN… PHYSICAL REVIEW B 81, 045311共2010兲
It follows from Eq. 共27兲 that ␦12LRE,N and ␦ 21
LRE,N vanish provided that the “confinement” of electron-hole pair enve-lope is identical in x and y directions 共␣x=␣y兲. Figure 2
illustrates this. Figure 2 shows bright exciton doublet split-ting as a function of lateral anisotropy t. The doublet splitsplit-ting energy is defined as EX y − EX x , where EX x and EX y are ground-state bright exciton energy levels polarized along x and y directions, respectively. For t ⬍ 0共x⬍y兲, EX
y
⬎ EXx and the bright ground state is dipole active along the x axis whereas for t ⬎ 0共x⬍y兲 the bright ground state is dipole active
along the y axis. In our model, the bright ground state is always dipole active along the axis of weaker confinement. The inset of Fig.2shows Rxxand Ryynonlocal contributions
to LRE exchange as a function of anisotropy computed from Eq.共27兲. We can see that Rxxincreases and Ryydecreases in
magnitude as t goes from −0.5 to 0.5. The inset also shows the difference Rxx− Ryywhich determines bright exciton
split-ting as a function of anisotropy. We can see that Rxx− Ryy
steadily decreases as a function of t and passes through zero at t = 0. t = 0 corresponds to the laterally isotropic confining potential. In this case, Ryy= Rxxand the exchange matrix
el-ement which couple two bright exciton levels vanishes. As a result, the splitting between bright excitons vanishes as well. We note, once again, that this result is obtained assuming rotational symmetry C⬁v of the confining potential as well the dipole-dipole nature of the LRE interaction which stems from orthogonality of the electron and hole functions on the unit-cell scale in bulk.
B. Application of lateral electric field
In the previous section, we have demonstrated that the magnitude of the bright exciton exchange splitting can be controlled through the shape of the confining potential. In practice, one may, for example, try to quench the bright ex-citon splitting by picking “symmetric” dots from a large
number of samples grown under different conditions.6
Nev-ertheless, the growth of QD is essentially a random process and control through the QD shape is hard to achieve.
It is, therefore, of great interest to examine the effects of external fields on the excitonic fine structure. For example, QDs can be placed2between Schottky gates for the
applica-tion of vertical and lateral electric fields. Recently, it has been demonstrated theoretically32 and experimentally13 that
by applying an in-plane electric field it is possible to fine-tune photon cascades originating from recombination of mul-tiexciton complexes in QDs.
The electron-hole exchange interaction was treated in Ref.
32 using empirical exchange Hamiltonian. This section dis-cusses the effects of the lateral electric field on the exchange matrix elements of our effective EMA Hamiltonian共18兲. We
will focus on the bright exciton splitting as a function of the lateral electric field.
Within our model, application of lateral electric field
Fជ= Fex+ Feyof magnitude兩F兩 displaces the “origin” of
elec-tron and hole envelopes in xy plane from共0,0兲 to 共x0e, y0e兲 and 共x0h, y
0
h兲, respectively. The single-particle energy levels are
rigidly shifted by the Stark shift whereas the spacing be-tween the single-particle energy levels of HO is not affected. The “separation” of electron and hole envelopes in xy plane is determined by equations ⌬x0= x0 e − x0 h = eF
冉
1 x2,ee + 1 M储,hhx2,hh冊
, ⌬y0= y0 e − y0h= eF冉
1 y2,ee+ 1 M储,hhy,hh 2冊
. 共28兲 Evaluating the integrals which control bright exciton splitting Rxx共F兲, Ryy共F兲, and Rxy共F兲 as a function of field Ffor seshelectron-hole envelope, we obtain Rxx共F兲 = Rxx共0兲exp
冉
− 2␣xc␣xv ␣x ⌬x02−2␣yc␣yv ␣y ⌬y02冊
, Ryy共F兲 = Ryy共0兲exp冉
−2␣xc␣xv ␣x ⌬x0 2−2␣yc␣yv ␣y ⌬y0 2冊
, Rxy共F兲 = 0, 共29兲where Rxx共0兲 and Ryy共0兲 are integrals 关Eq. 共27兲兴 in the
ab-sence of the electric field 共F=0兲. Field dependence of
Rxx共F兲−Ryy共F兲 which determines the bright exciton splitting
is shown in Fig. 3 for five different initial values of lateral anisotropy. At F = 0, the bright exciton splitting attains maxi-mum value which is determined by the initial shape aniso-tropy of the confining potential. The larger the magnitude of the anisotropy, the larger is the initial 共F=0兲 bright exciton splitting. As F increases in magnitude, the electron and hole envelopes are pulled out in opposite direction and the mag-nitude of the splitting is reduced. Since the separation be-tween the envelopes depends on F2, the splitting is indepen-dent of the sign of the field F.
It is interesting to note that the “rate of quenching” of the splitting depends on the initial anisotropy—the stronger the
−4.00 −2.00 0.00 2.00 4.00 6.00 −0.4 −0.2 0 0.2 0.4 Splitting (10 −3 H) Anisotropy t
Bright Exciton Splitting vs. t (ωhh0=0.73,ωee0=2.30)
−5.0 −3.0 −1.0 1.0 3.0 −0.4−0.2 0 0.2 0.4 Rxx Ryy Rxx−Ryy
FIG. 2. Bright exciton doublet splitting as a function of lateral anisotropy t of the confining potential. Inset: Rxx共dark squares兲 and Ryy共dark circles兲 contributions to nonlocal LRE matrix elements as
a function of anisotropy of the confining potential; Rxx− Ryy共empty
circles兲 difference of two nonlocal LRE contributions that deter-mines the bright exciton splitting. For the isotropic confining poten-tial t = 0 and Rxx− Ryy= 0.
confinement, the larger is the drop in the magnitude of the splitting. One can use lateral electric field to produce two identical bright exciton splittings for two dots with different initial anisotropies. This happens, for example, at 兩F兩⬇0.6 for two dots with initial anisotropies of t = −0.5 and t = −0.4, respectively.
C. Scaling of bright exciton splitting
The question of size scaling of exchange interactions in nanosystems has received a lot of attention, particularly, in connection with the Stokes共“red”兲 shift of resonant PL spec-tra with respect to the absorption edge 共see, for example, Ref. 19兲. In this section, we examine the size scaling of
bright exciton splitting as a function of system’s size. Suppose that R is a characteristic size of a 2D-like system. From the normalization condition
冕
dr兩Fc共r兲兩2= 1,冕
dr兩Fv共r兲兩2= 1, 共30兲
envelope functions scale as 1 / R2. Based on this “normaliza-tion” argument, the nonlocal LRE integrals Rxx, Ryy, and Rxy
that determine bright exciton splitting scale as 1 R2 1 R2 1 R 1 R2R 2R2⬀ 1 R3. 共31兲
The dependence 关Eq. 共31兲兴 is expected to hold in the
strong confinement regime. Of course, one has to keep in mind that the exchange matrix element responsible for the bright exciton splitting is a linear combination of nonlocal LRE integrals Rxx, Ryy, and Rxy. Therefore, the actual size
dependence of bright exciton splitting might be different from 1 / R3 and depend strongly on the “nature” of the enve-lope functions involved in the LRE integrals.
We will examine the bright exciton splitting as a function of the confinement length squared l2= 1 /共M
储0兲. The ratio of
electron and hole confinement frequencies is kept constant
ee0 /hh0 = const and the lateral anisotropy t is fixed. Since
ee0 / hh 0 = const, by increasing/decreasing ee 0 we automati-cally increasing/decreasinghh0 .
Figure4shows the scaling of the splittings for the ground and excited bright exciton levels with size. The lateral aniso-tropy is fixed to t = −0.1 in all the calculations. We find that splittings decay as the size of the system increases. We find that the ground-state exciton splitting is size insensitive whereas the splittings of the excited bright excitons scale ⬀1 / R1.3. This scaling is different from 1 / R3dependence ex-pected from the normalization arguments关Eq. 共31兲兴. The
de-viation might be due to the fact that the exchange matrix elements that determine the bright exciton splitting ␦12 and
␦21 involve a linear combination of nonlocal exchange inte-grals Rxx, Ryy, and Rxy. Moreover, Rxxand Ryy enter the
ex-pression for␦12and␦21with different sign and some cancel-lation of terms may occur. Therefore, the size dependence of bright exciton splittings may depend strongly on the details of the excitonic wave function.
IV. CONCLUSIONS
A four-band electron-hole excitonic Hamiltonian is de-rived within the EMA which takes into account the electron-hole exchange interaction. The matrix elements of the effec-tive Hamiltonian are expressed explicitly in terms of electron and hole envelopes. The microscopic parts of single-particle orbitals are integrated out and enter our Hamiltonian implic-itly through numerical parameters. The matrix element re-sponsible for the bright exciton splitting is identified and analyzed. An explicit expression for this matrix element in terms of electron and hole envelopes is presented. An exci-tonic fine structure for a model system with 2D-like
aniso-0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Rxx −R yy (Eff.Units) [Field F (Eff.Units)] Field−dependence of LRE (ωhh 0 =0.73,ωee 0 =2.30) t=−0.5 t=−0.4 t=−0.3 t=−0.2 t=−0.1
FIG. 3. Nonlocal LRE Rxx− Ryywhich determines bright exciton
splitting as a function of lateral electric field F. At F = 0, Rxx− Ryy
attains maximum determined by the initial anisotropy of the confin-ing potential. As magnitude of field F increases, Rxx− Ryydecreases
due to the separation of electron and hole envelopes.
0.00 5.00 10.00 15.00 20.00 0.3 0.35 0.4 0.45 0.5 0.55 Splitting (10 −3 H) 1/(Meeω ee 0 ) (t=−0.1) Bright Exciton Splitting vs. 1/(Meeωee0)
sesh sedh
sedh
FIG. 4. Size scaling of the bright exciton doublet splittings. Lateral anisotropy is kept constant t = −0.1. The doublet splittings are plotted as a function of 1 /共M储,eeee
0兲. The ratio ee 0/ hh 0 is kept constant. The ground exciton is of se− shtype. Two excited bright
excitons correspond to se− dhelectron-hole pairs. The solid lines are
fit to the power law C / Rn, where C is a constant, n⬇1.30.
THEORY OF EXCITON FINE STRUCTURE IN… PHYSICAL REVIEW B 81, 045311共2010兲
tropic HO confining potential is considered. It is found that in the case of 2D isotropic potential, the bright exciton split-ting vanishes. Within the formalism of our effective exci-tonic Hamiltonian, the effects of the lateral electric field on the excitonic fine structure were considered. It is found that the excitonic structure can be tuned with lateral electric field and that the magnitude of the exchange splitting is reduced. The origin of this reduction is the spatial separation of electron-hole pairs by electric field. Finally, size dependence of the bright exciton splittings for the ground- and excited-state excitons was investigated and it was found that the size
scaling of bright exciton splitting can be different from the laws established using normalization conditions.
ACKNOWLEDGMENTS
The authors acknowledge discussions with M. Korku-sinki, M. Zielinski, R. Williams, and D. Dalacu and support by the NRC-NSERC-BDC Nanotechnology project, Quan-tumWorks, NRC-CNRS CRP, and CIFAR. The authors thank E. L. Ivchenko for reading the manuscript and making useful suggestions.
*ekadants@babylon.phy.nrc.ca
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