## Electron-electron scattering in a double quantum dot: Effective mass

approach

S. Yu. Kruchinin,1,a兲A. V. Fedorov,1,b兲A. V. Baranov,1T. S. Perova,2and K. Berwick3 1

*Saint-Petersburg State University of Information Technologies, Mechanics and Optics,*
*49 Kronverksky Avenue, 197101 St. Petersburg, Russia*

2

*Department of Electronic and Electrical Engineering, Trinity College, University of Dublin, Dublin 2,*
*Ireland*

3

*Department of Electronic and Communications Engineering, Dublin Institute of Technology, Dublin 8,*
*Ireland*

共Received 10 June 2010; accepted 20 July 2010; published online 10 September 2010兲

We present a theoretical description of the first-order scattering of interacting electrons and holes in
a double quantum dot. Assuming infinitely high walls, strong confinement, and a two-band
approximation, we derive general expressions for the two-particle matrix elements of the screened
Coulomb potential. We also determine the selection rules for different scattering channels and
consider special cases where the corresponding matrix elements can be represented by simple
analytical expressions. Numerical calculations of the matrix elements and an analysis of their
dependence on the geometrical and material parameters of the double quantum dot have also been
performed. ©*2010 American Institute of Physics*.关doi:10.1063/1.3477766兴

## I. INTRODUCTION

Semiconductor nanocrystals, or the so-called quantum
dots, with the characteristic linear size*R*0comparable to the
exciton Bohr radius*rB*of the corresponding bulk crystal, are

of great interest because of unusual electronic structure and
optical properties.1 Depending on a relative magnitude
be-tween*R*0and*rB*, they may be classified into two regimes: for
*R*0Ⰷ*rB*, a weak-confinement regime where the exciton

center-of-mass motion is quantized, and for*R*_{0}Ⰶ*rB*, a strong

confinement one where the motions of the electron and the hole are separately quantized. In principle, due to the size-confinement effect in quantum dots, the energy spectra of quasiparticles, such as electrons, holes, phonons, and exci-tons, are transformed. Moreover, the confinement also modi-fies the interactions between the quasiparticles and external field and between the quasiparticles themselves. In this con-nection, studies of such interactions, together with the inher-ent electronic structure, are of considerable importance. In addition, the systems containing quantum dots are expected to be promising materials for some applications in optoelec-tronics.

Resonant energy transfer due to Coulomb interaction in ordered structures of quantum dots共QDs兲has received much attention in recent theoretical2–6and experimental7–9 investi-gations. This phenomenon has been studied using several different approaches, including tight-binding2 and pseudopotential5 calculations, a quantum-electrodynamical approach3,10 and using an effective mass approximation.4,6 Previous theoretical treatments have been primarily directed to the investigation of resonant energy transfer involving in-terband transitions, although variations in the size and

mate-rial used in creating the QDs can enable resonance between the states in the electron-hole pair and a single charge carrier, as well as between electron or hole states. Thus, interdot energy transfer processes accompanying the intraband tran-sitions can be induced by Coulomb coupling. A simple analysis of the electron-electron interaction Hamiltonian for a double QD shows that these processes do exist and that they will manifest themselves through the modification of the optical properties of the double QD. For the incoherent energy transfer regime,6,11 these processes result in line broadening and an intensity change in the optical spectra, e.g., either an increase or quenching of the luminescence. In the coherent energy transfer regime,12where the transfer rate overcomes the phase relaxation rate, the processes can lead to the appearance of fine structure in the optical spectra due to the quantum entanglement effect.13

Previous theoretical and experimental studies confirm the validity of the simple dipole-dipole approximation and use an effective screening constant for the calculation of the two-particle matrix elements of Coulomb interaction in spherical direct-band semiconductor QDs 共e.g., Refs. 2, 5, and7兲. For instance, the difference between squared matrix elements calculated with the point dipole approximation and that using a pseudopotential calculation is less than 3% even at quasicontact interdot distances.5Thus, one can expect that the problem symmetry will allow simple expressions de-scribing interdot energy transfer processes accompanied by intraband transitions to be obtained. Doing so will allow the theory of electron-electron interactions for QD systems using a simple, two-band effective mass approximation to be de-veloped.

In this paper, we present the theory of first-order
scatter-ing of charge carriers in a double QD. We perform our
cal-culations under the following assumptions: electrons and
holes are strongly confined in QDs with infinitely high
po-a兲_{Electronic mail: stanislav.kruchinin@gmail.com.}

b兲_{Electronic mail: a_v_fedorov@inbox.ru.}

tential walls, the interaction of electrons and holes in adja-cent QDs is described by the Coulomb potential with an effective screening constant, and the energy spectrum of the charge carriers is described by a two-band effective mass approximation. These simplifications allow us to derive gen-eral expressions for the matrix elements that define the prob-ability amplitudes for all significant two-particle processes. In Sec. II, we obtain expressions for the matrix elements of the electrostatic interaction potential. In addition, we discuss the selection rules defining various specific scattering chan-nels. In Sec. III, we provide the results of numerical calcu-lations of the matrix elements and their dependence on the geometric and material parameters for a double quantum dot. A short summary and concluding remarks are given in Sec. IV.

## II. COULOMB MATRIX ELEMENTS IN A DOUBLE QD SYSTEM

The Hamiltonian for direct electron-electron interaction in a double QD with infinitely high potential walls can be written as

*H*int=

冕冕

*d*3

*r*I

*d*3

*r*III†共rI兲II†共rII兲

*V*共rI,

**r**II兲I共rI兲II共rII兲.

### 共1兲

Here, the integration is performed over the volume of the
quantum dots, *V*共rI,**r**II兲 is the potential of the
electron-electron interaction, the coordinates of the electron-electrons **r**_{␣}共␣
= I , II兲belong to the corresponding QDs, numbered I and II,
and_{␣}and_{}† are field operators, which can be represented
as the sum of the electron共*c*兲and hole 共*v*兲components

_{␣}†_{共r}

␣兲=␣,*c*

† _{共r}

␣兲+␣,*v*共r␣兲,

### 共2兲

␣,*c*

† =

兺

*k*
*ak*

†_{⌿}

*k*
ⴱ_{共r}

␣兲, ␣,*v*=

兺

*k*

*dk*⌿*k*共r␣兲.

Here 共*ak*, *ak*

†_{兲}

and 共*dk*, *dk*

†_{兲}

are the creation and annihilation
operators for electrons and holes, respectively, and ⌿*k*共r␣兲

are the electron and hole wave functions. Substitution of Eq. 共2兲into Eq. 共1兲 and rewriting Eq.共1兲using normal-ordered products of field operators14,15 give the sum of terms that describe a variety of processes, viz., scattering of electrons and holes, creation and annihilation of electron-hole pairs, forward scattering, and others. We do not consider the ex-change interaction since it is negligibly small in the QD-matrix systems16 studied. Real scattering processes can be represented by the diagrams depicted on Fig.1.

We assume that the electrostatic interaction of two spherical QDs can be described by the screened Coulomb potential

*V*共rI,**r**II兲=

*e*2
兩r+**r**I−**r**II兩

### . 共3兲

Here,**r**is the vector directed from the center of QD II to the
center of QD I, vectors**r**Iand**r**IIoriginate from the center of
the corresponding QD, and dielectric screening is taken into
account by the effective dielectric constant . In general,
should be treated as a fitting parameter associated with the

dielectric constant of the matrix*M*and the quantum dots,I
and II. In the following analysis, it should be understood
that the dielectric constants of the materials should be
re-placed by the squares of their refractive indices.

Approximate expressions for can be derived as
fol-lows. The dielectric screening of the *l*Ith and*l*IIth multipole
terms generated by interaction between the charge
distribu-tions within the spherical QDs I and II is given by the
fol-lowing expression:4,17,18

*l*_{I}*l*_{II}=

*M*
*fl*_{I}共I/*M*兲*fl*_{II}共II/*M*兲

### . 共4兲

Here,_{␣}and*M*are the dielectric constants of quantum dot
␣ and the matrix, respectively, *l*I and *l*II are the multipole
moments, and

*fl*共␣/*M*兲=

2*l*+ 1
共␣/*M*+ 1兲*l*+ 1

共5兲

is a field factor describing the screening of the*l*th pole by the
QD material and the matrix.4,17,18A consideration of*l*_{I}*l*_{II}as a

function of*l*_{I} and*l*_{II}, for typical values of the dielectric
con-stants_{␣}and*M*, shows that it varies slowly with*l*IIfor fixed
values of *l*_{I} and vice versa 共see Appendix for details兲. This
property of*l*_{I},*l*_{II}allows us to use a single effective dielectric

constant =*l*

I

⬘,*l*

II

⬘, where *l*_{I}

⬘

and*l*

_{II}

⬘

are the moments of the multipole term, which provide the main contribution to the electrostatic interdot interaction. In particular, for interband-interband dipole-allowed transitions, the main term in the multipole expansion corresponds to dipole-dipole interac-tion, so we have the moments*l*

_{I}

⬘

= 1 and*l*

_{II}

⬘

= 1, and the effec-tive dielectric constant should be taken as(a) (b)

[image:2.612.349.521.47.201.2](c) (d)

⬅11=

### 共I+ 2M兲共II+ 2M兲

9*M*

. 共6兲

If the interband transition in the first QD is dipole-allowed but in the second QD is dipole-forbidden, the leading term will correspond to dipole-quadrupole interaction, and we should use the following effective dielectric constant:

⬅12=

共I+ 2*M*兲共2II+ 3*M*兲

15*M*

. 共7兲

In the Appendix, we discuss the arguments for this model of dielectric screening in more detail and give explicit expres-sions for the effective dielectric constants for different types of interaction. We also show that when the values of the dielectric constants of the QDs and the matrix are close enough, the screening of dipole-dipole and dipole-multipole interactions between two spherical QDs can be approximated by the same dielectric constant, as it has been done in our previous work.6,11

To simplify the expression for the matrix element of the interaction potential, we start from the Fourier expansion

1

兩**r**+**r**_{I}−**r**II兩= 4

冕

*d*3*q*

### 共2兲3

*eiq*ជ共**r+r**I−rII兲

*q*2 ,

which allows us to rewrite the matrix element in the follow-ing form:

*M*⬅ 具2,4兩*V*兩1,3典= *e*
2

22

冕

*d*3

_{qe}i**qr**

*S*_{21}I 共q兲*S*_{43}II共−**q兲**, 共8兲

where

*Sij*␣共q兲=

冕

*d*3

*r*␣

*e*

*i*

**qr**

_{␣}

_{⌿}

*i*
ⴱ_{共r}

␣兲⌿*j*共r␣兲, ␣= I,II. 共9兲

In a two-band effective mass approximation,19 the wave function of strongly confined electrons and holes in a spheri-cal QD with infinitely high potential walls has the following form:20

⌿*k*共r␣兲 ⬅⌿,*i*共r␣兲=*u*共r␣兲*i*共*r*␣兲,

where*u*_{}is the Bloch amplitude共=*c*,*v*兲,

*i*共r␣兲=*Rn _{i}l_{i}*共

*r*␣兲

*Yl*共␣,␣兲,

_{i}m_{i}is the envelope wave function,

*Rn _{i}l_{i}*共

*r*␣兲=

冑

2

*R*_{␣}3

*jl _{i}*共

*n*␣/

_{i}l_{i}r*R*␣兲

*jl*+1共

_{i}*n*兲

_{i}l_{i}### 共10兲

is the radial part,*Yl _{i}m_{i}*共␣,␣兲is a spherical harmonic,

*jl*is a

_{i}spherical Bessel function, *n _{i}l_{i}* is the

*ni*th root of equation

*jl*共

_{i}*x*兲= 0, and

*R*␣ is the radius of the QD. It should be noted

that the important part of an intradot Coulomb interaction is taken into account by the effective mass approach.

Let us express**r**_{␣} as the sum of the radius vectors of an
elementary cell**r***k*_{␣} and of the electron inside the cell**r**_{␣}

⬘

. Byusing the following properties of the envelope and Bloch functions:

*i*共r*k*_{␣}+**r**_{␣}

⬘

兲 ⬇*i*共r

*k*

_{␣}兲,

*u*

_{1}共r

*k*

_{␣}+

**r**

_{␣}

⬘

兲=*u*

_{1}共r

_{␣}

⬘

兲,we can replace the integral over the volume,*V*_{␣}, of the
quan-tum dot, by the sum of the integrals over the volumes⍀of
elementary cells. This allows us to represent*Sij*␣ as the

prod-ucts of the integrals over coordinates in the envelope space and the Bloch space

*Sij*␣=*sij*␣*Uij*␣,

where

*sij*␣=⍀

兺

*k*

_{␣}

*ei***qr**_{k}

␣*i*ⴱ共**r***k*_{␣}兲*j*共**r***k*_{␣}兲,

*Uij*␣=

1 ⍀

冕

⍀*d*3_{r}

␣

⬘

*ei*

**qr**

_{␣}⬘

_{u}*i*

ⴱ_{共}** _{r}**
␣

⬘

兲*u*

_{}

*j*共**r**␣

⬘

兲.After the replacement of the sum over the elementary cells with the integral over the volume of the nanocrystal, we obtain

*sij*␣=

冕

*d*3

*r*␣

*e*

*i*

**qr**

_{␣}

_{}

*i*
ⴱ_{共r}

␣兲*j*共r␣兲. 共11兲

The expansion of the plane wave*ei***qr**␣_{in terms of }
cal waves, expressing the integral of product of three
spheri-cal functions through Clebsch–Gordan coefficients21and
not-ing the spherical symmetry of quantum dots, allows us to
transform the integration over the angular coordinates of
vector **r**_{␣}in Eq.共11兲into the following sum:

*sij*␣共⫾**q**兲=

冑

2*lj*+ 1

2*li*+ 1*l*

兺

_{␣}=0

*l _{i}*+

*l*

_{j}共⫾*i*兲*l*_{共}_{2}_{l}_{+ 1}_{兲}_{C}*l _{j}*0,

*l*

_{␣}0

*l _{i}*0

_{C}*l _{j}m_{j}*,

*l*

_{␣}0

*l _{i}m_{i}*

⫻

冕

0

*R*_{␣}

*dr*_{␣}*r*2*jl*_{␣}共*qr*␣兲*Rn*ⴱ* _{i}l_{i}*共

*r*␣兲

*Rn*共

_{j}l_{j}*r*␣兲, 共12兲

where*Cl _{j}m_{j}*,

*l*

_{␣}0

*limi* _{is the Clebsch–Gordan coefficient.}

In the long-wave approximation,*qa*Ⰶ1, where *a* is the
lattice constant of the semiconductor, we can decompose

*ei***qr**␣⬘_{into a Taylor series and estimate the integrals}* _{U}_{ij}*␣

_{of the}Bloch amplitudes. When the functions of the initial and final state belong to the same energy band, the first nonzero term of

*U*␣ is 1. Then we have

_{ij}*Uij*␣共*i*=*j*=兲 ⬇

1

⍀

冕

_{⍀}

*d*3

*r*␣

⬘

兩*u*共

**r**␣

⬘

兲兩2= 1. 共13兲Otherwise, for interband transitions, we obtain

*Uij*␣共*i*⫽*j*兲 ⬇
*i***q**

⍀

冕

_{⍀}

*d*3

*r*␣

⬘

*u*

*i*

ⴱ_{共}_{r}

␣

⬘

兲**r**␣

⬘

*u*

*j*共

**r**␣

⬘

兲=*i*

**qr**

*i*,

*j*

␣ _{.}

### 共14兲

The matrix element of the radius-vector inside the
elemen-tary cell between Bloch functions**r**_{}

2,1

␣ _{can be expressed via}

the Kane parameter *P*_{␣}=ប2/*m*0具*S*_{␣}兩/*z*兩*Z*_{␣}典=ប

冑

*Ep*,␣/共2

*m*0兲 and the energy gap

*Eg*,␣as follows:

兩r*c*␣*v*兩=*P*␣/*Eg*,␣, 共15兲

Using Eqs. 共13兲 and 共14兲, we obtain the following ex-pression for matrix element共8兲:

*M*_{ᐉ}␣= *e*
2

22

冕

*d*3*q*
*q*2*e*

*i***qr**_{}

ᐉ
␣_{s}

21
I _{共q兲}

*s*_{43}II共−**q兲**, ᐉ= 0,1,2. 共16兲

Here, we introduce the indexᐉthat determines the number of interband transitions involved in scattering, the upper index

␣in*M*_{ᐉ}␣denotes the QD共I or II兲number where the interband
interaction occurs,_{ᐉ}␣is the function that determines the
mu-tual orientation of**q**and**r***c*␣*v*,

␣0= 1, 1␣= ⫾*i***qr**␣*cv*, 2␣=共qr*cv*

I _{兲共qr}

*vc*

II_{兲}
,

where *e***r***c*␣*v* are the dipole moments of the interband
transi-tions. In this and other similar expressions for
interband-intraband transitions共ᐉ= 1兲, the plus sign is used when the
interband transition occurs in the first quantum dot 共␣= I兲
and a minus sign for the opposite case 共␣= II兲. Integration
over the angles between**q** and**r**and change in the
integra-tion order of*r*and*q*allow us to evaluate the integral over*q*

analytically. We obtain the following expression:

*M*_{ᐉ}␣=*e*
2

*X*_{ᐉ}␣

*r*ᐉ+1, 共17兲

*X*_{0}␣=*I*0, *X*1␣= ⫾*I*1**n***r***r***c*␣*v*,

*X*_{2}␣=*I*1**r***cv*

I
**r***cv*

II

−*I*2共n*r***r***cv*

I _{兲共n}

*r***r***cv*

II_{兲}
.

Here**n***r*=**r**/*r*,

*I*_{ᐉ}= 2

*l*

兺

_{I}=0

*l*_{1}+*l*_{2}

兺

*l*_{II}=0

*l*_{3}+*l*_{4}

### CI+CII−

⫻

冕

0

*R*_{I}

冕

0*R*_{II}

*dr*I*dr*II*r*I2*r*II2RIRII*Ql*ᐉ_{I},*l*_{II}, ᐉ= 0,1,2 共18兲

are the multipole amplitudes that come from the integration over the coordinates of the envelope wave functions,

C_{␣}⫾=共⫾*i*兲*l*␣_{共}_{2}_{l}_{␣}_{+ 1}_{兲}

冑

2*lj*+ 1 2

*li*+ 1

*Cl _{j}*0,

*l*

_{␣}0

*l _{i}*0

*Cl _{j}m_{j}*,

*l*

_{␣}0

*l _{i}m_{i}*

, 共19兲

R_{␣}=*R*** _{n}**
␣

⬘*l*_{␣}⬘
ⴱ _{共}

*r*_{␣}兲*Rn*_{␣}*l*_{␣}共*r*␣兲, 共20兲

*Ql*ᐉ_{I},*l*_{II}=*r*ᐉ+1

冕

0

⬁

*dqq*ᐉ*j*_{ᐉ}共*qr*兲*jl*_{I}共*qr*I兲*jl*_{II}共*qr*II兲

= 2ᐉ−3_{}3/2

冉

*r*I

*r*

冊

*l*

_{I}

冉

*r*II

*r*

冊

*l*

_{II}

⫻⌫

冋

共2ᐉ+*l*I+

*l*II+ 1兲/2

*l*_{I}+ 3/2,*l*_{II}+ 3/2,1 −共*l*_{I}+*l*II兲/2

册

⫻*F*4

冉

*l*I+*l*II
2 ,

2ᐉ+*l*I+*l*II+ 1
2 ;*l*I+

3
2,*l*II

+3 2;

*r*_{I}2
*r*2;

*r*_{II}2

*r*2

冊

, 共21兲where*F*4共*a*,*b*;*c*,*c*

⬘

;*x*;

*y*兲is the Appel’s fourth hypergeomet-ric function,22

*F*4共*a*,*b*;*c*,*c*

⬘

;*x*;

*y*兲=

兺

*k*,*l*=0

⬁ _{共}

*a*兲*k*+*l*共*b*兲*k*+*l*

共*c*兲*k*共*c*

⬘

兲*l*

*xkyl*

*k*!*l*!, 共22兲

共*a*兲*k*=⌫共*a*+*k*兲/⌫共*a*兲is a Pochhammer symbol.

Next, we introduce orientational factors that are indepen-dent of the values of the transition dipole moments,

0=*I*0, 1␣= ⫾*I*1cos␣, 2=*I*1sinIsinIIcos
+共*I*1−*I*2兲cosIIcosII; 共23兲

the matrix elements can be rewritten in the following way:

*M*_{0}=*e*
2

0

*r*, *M*1
␣_{=}*e*2

1␣

*r*2*rcv*

␣_{,} _{M}

2=

*e*2

2

*r*3*rcv*

I _{r}

*cv*

II_{.}

Hereafter, we omit the superscripts ␣in the expressions for intraband-intraband 共ᐉ= 0兲 and interband-interband 共ᐉ= 2兲 transitions.

It should be noted that the infinitely high potential walls
approximation significantly reduces the probability of
transi-tion between the states with different principal quantum
numbers*ni*, so in this paper, our consideration is restricted to

transitions between the states, which differ only by the
an-gular moment values. This limitation stems from the
or-thogonality of the radial wave functions共10兲by*ni*and thus

from the elimination of the terms proportional to共*r*_{␣}/*r*兲2*k*_{,}_{k}

= 0 ,⬁ 关see Eqs.共21兲and共22兲兴. The transitions between the
states with different principal quantum numbers can be
con-sidered more precisely in the approximation of finite
poten-tial walls by replacing the radial parts of the envelope wave
functions and extending the integration limits in Eq.共18兲, as
discussed in our previous work.6 In order to maintain the
spherical symmetry of the task, one must neglect the
pres-ence of the wave function of the first QD in the second QD
and vice versa. For example, the integration over*r*Ishould be
performed inside the QDs共0⬍*r*I⬍*R*I兲and inside the
spheri-cal layer of matrix共*R*I⬍*r*I⬍*r*−*R*II兲.

When the quantum numbers of the initial and final states
are the same 共1=2,3=4,*i*=兵*ni*,*li*,*mi*其兲, we have a

dipole-allowed transition, and the function*I*_{ᐉ} is reduced to

*I*_{ᐉ}_{,a}=共2ᐉ− 1兲!!, 共24兲

where 共2ᐉ− 1兲! ! = 1 · 3 · 5 · . . . ·共2ᐉ− 1兲 is a double factorial
function. For instance, Eq. 共24兲 gives *I*0,a= 1, *I*1,a= 1, and

*I*2,a= 3, and the matrix element for dipole-allowed,
interband-interband transitions takes the well-known form4,12

*M*2,a=

*e*2
*r*3关r*cv*

I
**r***cv*

II

−共3**n***r***r***cv*

I _{兲共n}

*r***r***cv*

### II兲兴

. 共25兲

Let us consider the selection rules defined by expres-sions共17兲and共18兲. Condition

*Cl _{j}*0,

*l*

_{␣}0

*l _{i}*0

*Cl _{j}m_{j}*,

*l*

_{␣}0

*l _{i}m_{i}*

_{⫽}

0 共26兲

implies the following selection rules for the transitions due to Coulomb interaction:

兩*lj*−*l*␣兩ⱕ*li*ⱕ兩*lj*+*l*␣兩,

*mi*=*mj*.

Additional selection rules arise from the condition ⌫共1 −共*l*I
+*l*II兲/2兲⫽⬁ 关see Eq. 共21兲兴, which implies 共*l*I+*l*II兲/2
= a half-integer number. This condition will be satisfied
only when *l*I and *l*II have opposite parities. Since Eq. 共26兲
should also be satisfied, we conclude that the matrix element
is nonzero when the sums of the angular moments of the
initial and final states have opposite parities, i.e., in two
cases: 共1兲 *l*1+*l*2= an odd number and *l*3+*l*4
= an even number and 共2兲 *l*1+*l*2= an even number and *l*3
+*l*4= an odd number.

It is useful to derive expressions for the squared modulus of matrix elements averaged by directions of transition di-pole moments and their maximal values,

兩*M*0兩2_{= max}_{兩}_{M}_{0兩}2_{=}*e*
4_{兩}_{I}_{0兩}2

2* _{r}*2 , 共27兲

兩*M*_{1}␣兩2_{=}1
3

*e*4

2
兩*I*1兩2

*r*4 兩**r***cv*

␣_{兩}2_{,} _{max}_{兩}* _{M}*
1

␣_{兩}2_{=}*e*
4

2
兩*I*1兩2

*r*4 兩**r***cv*

### ␣兩2, 共28兲

兩*M*2兩2=1_{3} *e*
4

2* _{r}*6兩r

*cv*

I _{兩}2_{兩r}

*cv*

II_{兩}2

兵

_{兩}

_{I}_{1兩}2

_{−}1

3共*I*1ⴱ*I*2+*I*1*I*2ⴱ兲+
1
3兩*I*2兩

2

其

_{,}max兩

*M*2兩2=

*e*

4

2* _{r}*6兩r

*c*I

*v*兩2兩rII

*cv*兩2兩

*I*1−

*I*2兩2. 共29兲

Conditions for 兩*M*_{ᐉ}兩2 maximality over the angular variables
are determined by the orientation factors _{ᐉ} 关see Eq. 共23兲兴.
Intraband-intraband transitions共ᐉ= 0兲 do not depend on
lat-tice orientation because they do not include any coordinates
of Bloch space and because QDs are spherically symmetric.
Forᐉ= 1, there are two conditions:共1兲_{␣}= 0 and共2兲_{␣}=,
i.e., 兩*M*1兩2 is maximal when the transition dipole moment is
parallel to the vector **r** connecting the centers of the QDs.
For ᐉ= 2, we have four orientations of the transition dipole
moments, which maximize the matrix element: 共1兲 _{I}= 0,

II= 0; 共2兲I= 0,II=;共3兲I=,II= 0; and共4兲 I=,II
=, i.e., both vectors**r**_{I} and**r**_{II}should be parallel to**r**.

It is convenient to designate the real transitions in a double QD by a notation commonly used in the theory of Auger processes in bulk semiconductors and heterostructures23,24 共CVVC, CVCC, CCCC,…兲. The letters C and V denote the conduction and the valence bands, re-spectively, but in this context the first two letters describe the transition in the first QD and the last two in the second QD. The matrix elements共17兲, with respective values for the in-dex ᐉ, describe the probability amplitudes of the processes listed in TableI.

The expressions for the matrix elements obtained in this section allow the determination of important parameters that characterize the resonant interaction of two QDs. For ex-ample, the rate of resonant energy transfer between two non-degenerated states is given by the expression

### ␥= 2 ប2兩Mᐉ兩

2 ⌫

⌫2_{+}_{⌬}2=

⍀_{ᐉ}2
2

⌫

⌫2_{+}_{⌬}2, 共30兲

where⌫is the transition dephasing rate and⌬is the detuning between the transition frequencies of two interacting

sys-tems. The parameter⍀_{ᐉ}= 2兩*M*_{ᐉ}兩/ប, following from a density
matrix description of interacting two-level systems,12 plays
an important role in the theory of resonant energy transfer.
This is the frequency characterizing the oscillations of the
excited states population in resonantly interacting two-level
systems. The relationship between⍀_{ᐉ}, the population
relax-ation rate ␥_{II} of the final state, and the transition dephasing
rate ⌫determines the regime of resonant coupling:
incoher-ent irreversible共⍀_{ᐉ}Ⰶ␥_{II},⌫兲, incoherent reversible共␥_{II}⬍⍀_{ᐉ}
⬍⌫兲, or coherent共⍀_{ᐉ}Ⰷ␥_{II},⌫兲. Notice that Eq.共30兲based on
the Fermi’s Golden Rule is valid for the description of the
energy transfer rates in the following cases: ⍀_{ᐉ}Ⰶ␥_{II},⌫ and

␥II⬍⍀ᐉ⬍⌫.

In the next section, we perform numerical calculations to
determine max⍀_{ᐉ}, assuming that the mutual orientation of
the dipole moments of the interband transitions satisfies one
of the conditions for maximality of *M*_{ᐉ}.

## III. NUMERICAL RESULTS

Let us consider the transition rate dependencies on
geo-metric parameters for double QDs of two different materials:
CdSe and InSb. The relevant material parameters are listed in
Table II. Hereafter, we assume that the matrix material is
SiO2, so the dielectric constant at typical transition
frequen-cies is*M*= 2.13共Ref. 28兲.

We perform the calculations for materials with different
band gaps because it is expected that the energy gap value
will significantly affect the relative magnitudes of the matrix
elements for different types of transitions. Among the
intraband-intraband processes mentioned in Table I, only
CCCC and VVVV are of interest for certain QD materials.
The effective masses of electrons and holes in CdSe and
InSb differ significantly, *mv*/*mc*⬃10, so the energy

reso-nance conditions for CCVV and VVCC processes lead to unrealistic values for quantum dot radii for the lowest al-lowed transitions. Due to the similarity of the expressions for the energy spectrum in the conduction and valence bands in

TABLE I. Correspondence between the matrix elements共17兲and the tran-sitions in a double QD.

Intraband-intraband transitions

### M0: CCCC, CCVV, VVCC, VVVV

Interband-intraband transitions

*M*1I: VCCC, CVCC, CVVV, VCVV

*M*_{1}II_{: CCVC, CCCV, VVCV, VVVC}

Interband-interband transitions

*M*_{2}: CVVC, VCCV

TABLE II. Material parameters of CdSe共Refs.25and26兲and InSb共Ref.

27兲.*mhh*=*mv*is the heavy hole effective mass.
*mc*共*m*0兲 *mhh*共*m*0兲

*Ep*

共eV兲

*Eg*

共eV兲 _{␣}

a two-band approximation, CCCC and VVVV processes will have the same values for the interaction matrix elements, and so they will be considered together.

Figures2共a兲and2共b兲show the dependencies of max⍀_{ᐉ}
on the distance between the surfaces of interacting QDs共*R*

=*r*−*R*_{I}−*R*II兲 for the lowest allowed transitions. The
depen-dency on*R* will not be the same as that on center-to-center
distance *r*, but it is more convenient for comparison of the
calculated results for QD pairs with different radii. For CdSe
QDs, we see that the dominant process is CVCC. Thus, the
interband-intraband processes can be considered as another
effective mechanism of resonant energy transfer as well as
interband-interband processes. For InSb, the frequencies⍀_{ᐉ}
for CVVC and CVCC have much higher values than in CdSe
since InSb has higher values of transition dipole moments
than CdSe. At the same time, the rates of intraband-intraband
processes in InSb QDs are lower than in CdSe.

It should be noted that resonant electrostatic interaction of charge carriers inside QDs cannot be considered as a re-laxation mechanism itself since the total energy of the inter-acting particles is conserved, and so these transitions are fully reversible in the absence of the bath. Transitions be-tween single-particle states due to electrostatic correlations play an important role as relaxation mediators because they can bring the system into the states with a much faster relax-ation rate than the initial state.

Let us consider the dependency of max⍀_{ᐉ}on the radius
of the first QD at the fixed intersurface distance depicted in

Fig. 3. This figure shows that the max⍀_{ᐉ} for
intraband-intraband transitions decreases much more slowly than that
for other processes with increasing*R*_{I}. We also observe the
domination of dipole-allowed CVVC process in InSb QDs,
but CVCC process is also effective and overcomes
dipole-forbidden CVVC process in QDs with*R*_{I}ⱗ6 nm.

## IV. CONCLUSION

In this paper, we have studied the probability amplitudes for electron-electron scattering processes in a double QD un-der strong confinement. We assume that the electrostatic in-teraction between the QDs is described by the Coulomb po-tential and dielectric screening is taken into account by the effective dielectric constant. We discuss the expressions for this effective dielectric constant and determine the bound-aries of its possible variations if used as a fitting parameter. Numerical calculations have been performed for the depen-dencies of the matrix elements on interdot distance, QD size, and material parameters. It has been shown that in some cases, the interband-intraband and intraband-intraband tran-sitions can have higher rates than the commonly considered interband-interband transitions.

**ACKNOWLEDGMENTS**

Three of the authors 共S.Yu.K., A.V.F., and A.V.B.兲 are grateful to the RFBR 共Grant Nos. 00333 and 09-02-01439兲 and the Ministry of Education and Science of the Russian Federation 共Grant Nos. 2.1.1/1933 and 2.1.1/1880兲 for partial financial support of this work.

FIG. 2. Parameter max⍀_{ᐉ} as a function of the intersurface distance for
different types of electronic transitions in a double QD: CCCC, VVVV,
CVCC, and CVVC. We consider that the first QD radius has a fixed value

*R*I= 4 nm, and the second QD radius *R*II always satisfies the resonance

condition I共*R*I兲=II共*R*II兲, where ␣ is the corresponding transition

fre-quency. CCCC transition:兵*c*110;*c*100其_{I},兵*c*100;*c*120其_{II},*R*_{II}= 6 nm; CVCC
transition: 兵*c*100;*v*100其I, 兵*c*100;*c*110其II, *R*II= 3.85 nm; dipole-allowed

CVVC transition:兵*c*100;*v*100其I,兵*v*110;*c*110其II,*R*II= 5.72 nm; and

dipole-forbidden CVVC transition:兵*c*100;*v*100其I,兵*v*110;*c*100其II,*R*II= 5.68 nm.

FIG. 3. Parameter max⍀_{ᐉ}as function of the first QD radius*R*Iat

intersur-face distance of*R*= 1 nm. As before,*R*IIis calculated from the resonance

## APPENDIX: DIELECTRIC SCREENING OF TWO ELECTROSTATICALLY COUPLED QDS

The assumption that the interaction between two spheri-cal QDs is described by the Coulomb potential with effective dielectric constant offers many advantages. In particular, it simplifies the consideration of higher multipoles and allows a straightforward generalization of many-body theory for or-dinary solids14,29to QD solids. The simplicity of the expres-sions for probability amplitudes of electron-electron scatter-ing makes the ordered structures of spherical QDs a good model system for the investigation of various effects con-nected with electronic correlation, such as resonant energy transfer and carrier multiplication.30However, the applicabil-ity of the effective dielectric constant derived for dipole-dipole interactions when transitions in one of the quantum dots are not dipole-allowed, i.e., the envelope angular mo-menta of the initial and final states are different, is question-able.

The first reason for the usage of the same effective di-electric constant is that the charge distribution inside the QD remains spherically symmetric for all envelope states. So, according to classical electrostatics,31the potential generated by two oppositely charged, spherically symmetric charge dis-tributions can still be described by the point dipole located at the center of the sphere. Possible deviations from the point dipole approximation, arising from the asymmetry between the envelope wave functions of the electrons and holes, will be small if the transition occurs between states with the same principal quantum numbers.

A second reason is that the effective dielectric constant
describing the screening of the interaction between the *l*Ith
and*l*_{II}th multipoles in spherical QDs,4,17,18

*l*_{I}*l*_{II}共I,II,*M*兲=

*M*

*fl*_{I}共I/*M*兲*fl*_{II}共II/*M*兲

,

varies slowly over the lowest multipole moments*l*Iand*l*IIfor
typical values of the dielectric constants of semiconductors
and glasses. This is clearly seen in Fig.4, which depicts the
ratio of effective dielectric constants for dipole-multipole
and dipole-dipole interactions 1*l*/11 共*l*= 1 , . . . , 20兲. Note
that the closer the values of_{␣}and*M*, the slower the value

of1*l*will vary over*l*. The upper limit of1*l*/11is given by
the expression

lim

*l _{→⬁}*

1*l*

11

### =3共II+M兲 2II+ 2M

,

so we have1⬁/11⬇1.18 for a double QD made from CdSe
and1⬁/11⬇1.34 for a double QD made from InSb. Thus,
in order to approximately consider the static screening of
interactions in spherical QDs formed from direct-band
semi-conductor, it is sufficient to assume that*fl*_{␣}for any*l*␣is equal

to the field factor of the first leading term, e.g., *fl*_{␣}⬇*f*1 for
dipole-allowed transition and *fl*_{␣}⬇*f*2 for dipole-forbidden
transition.

If both transitions are dipole-allowed, the leading term in the multipole expansion of the potential 共3兲 corresponds to dipole-dipole interaction. Thus, the effective constant is de-termined by the expression

11=

### 共I+ 2M兲共II+ 2M兲

9*M*

. 共A1兲

This situation is realized for interband-interband transitions. If one of the transitions is dipole-forbidden共i.e., angular mo-ments of initial and final states are different兲, the leading term will correspond to dipole-quadrupole interaction, so one should use the following effective dielectric constant:

12=

### 共I+ 2M兲共2II+ 3M兲

15*M*

. 共A2兲

Finally, if both transitions are dipole-forbidden, we have

22=

共2I+ 3*M*兲共2II+ 3*M*兲

25*M*

.

The effective dielectric constant for different multipole
terms varies between 12 and1⬁=共I+ 2*M*兲共II+*M*兲/6*M*,

so when_{␣}Ⰷ*M*, one may treatas an adjustable parameter

to achieve a better agreement between calculations and ex-perimental results for dipole-multipole interactions.

Similarly, in case of real intraband-intraband transitions, the leading term will correspond to quadrupole-quadrupole interaction, so the effective screening constant varies be-tween22and

2⬁=共

### 2I+ 3M兲共II+M兲

10*M*

.

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