CAOL 2008, September 29 - October 4, Alushta, Crimea, Ukraine

BINDING ENERGY AND EXCITON SPECTRUM IN DOUBLE CYLINDRICAL QUANTUM DOT

O.M. Makhanets, M.M. Dovganiuk, Ju.O. Seti

Chernivtsi National University58012, Kotsiubynsky Str. 2, Chernivtsi, Ukraine

e-mail: theorphys@chnu.cv.ua

Abstract-the spectral characteristics of exciton in combined nanoheterosystem consisting of semiconductor cy-

lindrical quantum wire containing two quantum dots separated by thin barrier-shell are investigated. It is shown

that the binding energy non monotonously depends on the geometrical characteristics of nanoheterosystem ap-

proaching several minimum and maximum magnitudes.

Keywords: exciton, quantum dot, quantum wire, binding energy.

INTRODUCTION

During the last decades, the multi well quantum nanoheterosystems are actively theo-

retically and experimentally studied. Such systems can be utilized as the base elements of dif-

ferent transistors, diodes, memory elements, also for the production of lasers, working at the

middle infrared range and as chemical or biological sensors and so on [1,2].

It is quite clear that the spectral characteristics of electrons and holes do not totally de-

fine yet the exciton spectrum in the nanoheterosystem under research. It is necessary to take

into account the interaction between these quasiparticles which, naturally, can be essential at

the condition when the quasiparticles are closely localized in the space.

The investigation of exciton binding energy for the cylindrical nanoheterosystems

meets some serious difficulties arising due to the demand of agree between the cylindrical

symmetry of the nanosysten itself and spherical symmetry of Coulomb potential. Therefore,

different variation methods [3] are used for the study of exciton spectrum in such systems.

These methods describe only the ground exciton state rather well [4] because the direct calcu-

lation of Coulomb potential matrix elements are too sophisticated.

In the paper it is proposed the new method to obtain the exciton binding energy. The

developed theory allows study the dependences of ground and higher exciton states on the

geometrical parameters of tunnel-bound quantum dots physically correct.

ANALYTICAL CALCULATION

The combined semiconductor cylindrical quantum wire containing two quantum dots

of the same material (“1”- HgS−β ), separated by thin barrier of the other material (“0”-

CdS−β ) is under study. The radius of

nanowire ( 0ρ ), heights of quantum dots

( 1h and 2h , respectively) and the thick-

ness of semiconductor shell separating

quantum dots (Δ ) are assumed as fixed

(Fig.1).

Taking into account the symmetry

considerations, all further calculations are performed in the cylindrical coordinate system with

OZ axis directed along the axis of quantum wire.

Fig.1. The nanoheterosystem spatial scheme.

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CAOL 2008, September 29 - October 4, Alushta, Crimea, Ukraine

The effective masses and potential energies of electron and hole in the combining

parts of nanosystem are the same as in the respective bulk crystals:

( )⎪⎩

⎪⎨⎧

μ

μ=μ

"1" ��������� ,

"0" ��������� , z

)h,e(1

)h,e(0)h,e( , ( )

⎩⎨⎧

=ϕρ "1"��������� ),(U U

"0"��������� 0,z,,U

(h)0

(e)0

h)(e, . (1)

In order to investigate the exciton states in semiconductor quantum wire with two

quantum dots it is necessary to solve the stationary Schrodinger equation:

( ) ( ) ( )heexexheexheex r,rEr,rr,rH�������

Ψ=Ψ , (2)

where

( ) ( )( ) ( )( ) ( ) g0hehh

ee

heex ErrUrHrHr,rH +−++=���������

, (3)

|rr|)r,r(

e|)rr(|U

hehe

2

he ������

−ε−=− (4)

- the potential energy of electron – hole interaction, 0gE - forbidden band width for the quan-

tum dot material, ( )he,H�

- the Hamiltonians of uncoupling electron and hole, ( )he r,r��

ε - dielec-

tric constant, which in general case is a complicated function of electron and hole spatial loca-

tion in the system under research. The equation (2) with Hamiltonian (3) can not be solved

exactly. Therefore, the perturbation method is further used taking into account the fact that

electron and hole are mainly localized in the space of quantum wells (media „1”) and that

their interaction energy is much smaller than the size quantization energy.

According to the perturbation theory, the whole exciton energy and wave function is

written in the form: eee

z

hhhz

hhhz

eeez

eeez

hhhz

mnn

mnnmnnmnn0g

mnn

mnnEEEEE ρ

ρρρ

ρ

ρΔ+++= , (5)

( ) ( )hmnnemnnhe

mnn

mnnrr)r,r( hhh

zeee

z

eeez

hhhz

����

ρρ

ρ

ρΨΨ=Ψ . (6)

Here eeez mnn

Eρ

( hhhz mnn

Eρ

), ( )emnnreee

z

�

ρΨ ( ( )hmnn

rhhhz

�

ρΨ ) – the energies and wave functions

of electron (hole), defined as the solutions of the respective Schrodinger equations for these

quasiparticles and eee

zhhh

z

mnn

mnnE ρ

ρΔ - exciton binding energy, calculated as diagonal matrix element

of operator (4) at the wave functions (6) using the expansion in the Fourier range

( )∑ −π=

− q

rrqi

2he

heeq

1

V

4

rr

1

�

���

�� . (7)

ANALYSIS OF THE RESULTS

The numeric calculation of exciton energy spectrum was performed for the nanohetero-

system created at the base of HgS,CdS −β−β semiconductors with the known physical pa-

rameters. The results are the following.

In fig.2 the dependences of exciton binding energy (fig.2) and energy of its excitation

(fig. 2b) in several lowest states at m=0 and m=1 on the height of one quantum dot ( 2h ) at

fixed parameters: CdSHgS1HgS0 a2,a15h,a10 =Δ==ρ are shown.

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CAOL 2008, September 29 - October 4, Alushta, Crimea, Ukraine

0 10 20 30 40 50-40

-35

-30

-25

-20

-15

-10

-5

0

1111

1010

2020

2121

a

h2(aHgS)

ΔEn ze m

e nρe

n zh m

h nρh,mev

0 10 20 30 40 50800

1000

1200

1400

1600

1800

2000

b1010

2020

1111

2121

Enze m

e nρe =1

n zh m

h nρh =1,mev

h2(aHgS) Fig.2 Dependences of exciton binding energy (fig.2) and energy of its excitation (fig. 2b) in several lowest

states at m=0 and m=1 on the height of one quantum dot ( 2h ) at fixed parameters:

CdSHgS1HgS0 a2,a15h,a10 =Δ==ρ

Figure proves that the exciton binging energy for all states non monotonously depends

on 2h height, approaching several minimum and maximum magnitudes. Such behavior is

clear from physical considerations: really when electron and hole are located in their ground

states (fig. 2), their binding energy 101101EΔ would be maximal at 0h2 = . Since, the both qua-

siparticles are localized in quantum dot with the height HgS1 a15h = and the overlap of their

wave functions is essential. When the other quantum dot appears and its height 2h increases,

the binding energy decreases because the electron and hole wave functions penetrate into the

quantum dot with 2h height. The binding energy minimum is approached at

HgS21 a15hh == , when the both quasiparticles with equal probability are in the same quan-

tum dots. Further 101101EΔ is again increasing due to the increase of electron and hole location in

quantum dot with 2h height. Passing the maximum, the binding energy is decreasing due to

electron and hole localization in the space of wide quantum well becomes smaller.

The non monotonous dependence of exciton binding energy in the higher states is ex-

plained according to the analogous considerations.

The absolute magnitude of exciton binding energy is two orders smaller than the ener-

gies of electron or hole ( ( )he

zmnnE

ρ), since, the dependences of exciton energies on 2h heights

(fig.2b) is generally caused by the peculiarities of electron and hole states behavior. It is

proven by the anti crossings in the energy dependences of free electron and hole on the quan-

tum dot height ( 2h ).

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