phys. stat. sol. (c) 3, No. 9, 3347–3350 (2006) / DOI 10.1002/pssc.200567057

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Decoherence effects due to exciton-phonon interaction

in CdSe quantum dots: Contribution of the piezoelectricity

and deformation potential couplings

Karim Sellami *, 1

and Sihem Jaziri1,2

1 Laboratoire de Physique des Matériaux, Faculté des Sciences de Bizerte, 7021 Jarzouna, Tunisia 2 Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, Tunisia

Received 5 September 2005, revised 8 January 2006, accepted 22 May 2006

Published online 28 August 2006

PACS 63.22.+m, 71.35.–y, 78.67.Bf

We analyse decoherence effects of an exciton in II-VI semiconductor quantum dot. We calculate the scat-

tering rate of excitons by acoustic phonons laying a foundation for work on understanding carrier relaxa-

tion in quantum dots providing then a basis for studying decoherence phenomena in these systems. We

considered two exciton-acoustic phonons scattering mechanisms the standard deformation potential cou-

pling and the piezoelectric coupling. The latter was found to be relatively important in large dots. We dis-

cuss the influence of an external applied magnetic field and the dot size on the scattering rate. We show

that it depends strongly on the quantum dot size, then using a modulated external magnetic field the inter-

action of the exciton and acoustic phonon modes can be modified in a wide range.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Semiconductor quantum dots continue to receive a great deal of attention, largely due to their unique properties arising from δ -function like profile of the density of states [1]. These systems provide the opportunity to investigate semiconductor behaviour in the finite size regime. Due to quantum confine-ment, these materials have many fundamentally interesting and potentially useful optical properties. The effects of an applied magnetic field on the physical properties of quantum dots have been studied with interest from the theoretical and experimental points of view. These studies have been performed with the proposal of understanding the fascinating novel phenomena and of fabricating devices with new functions or to improve the performance of the existing devices [2]. The optical excitation of a semicon-ductor quantum dot structure like any other semiconductor structure with a short coherent light pulse results in the creation of a coherent superposition of valence-and conduction- band states [3]. Subse-quently, this phase coherence decays due to various interaction mechanisms of electrons and holes. For many applications such as optoelectronic devices (see, e.g., Ref. [4]) a good knowledge of the dephasing is of utmost importance. This holds most prominently is semiconductor quantum dots are to be used as basic building blocks for quantum information processing [5] where the operation completely relies on the presence of the coherence. The interaction of acoustic phonons with excitons in QDs has been found to controle their decoherence at low temperature [6] and plays a decisive role for understanding their optical properties, in particular the dephasing mechanisms. Therefore research on exciton energy relaxation in semiconductor quantum dots has attracted much interest due to the critical importance of carrier relaxation in the performance of novel

* Corresponding author: e-mail: karimsellami@yahoo.fr, Phone: +216 98 222 854, Fax: +216 72 590 566

3348 K. Sellami and S. Jaziri: Decoherence effects due to exciton-phonon interaction in CdSe quantum dots

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

semiconductor devices based on quantum dots. The physical mechanism underlying carrier relaxation are not yet fully understood, hence it is necessary that alternative carrier relaxation paths should be discussed systematically. In this work we study theoretically the exciton scattering rate (SR) which is studied as a basis for un-derstanding relaxation, and coherence effects in a CdSe semiconductor quantum dot. We considered two mechanisms for the interaction between the carriers and the exciton modes: (i) the macroscopic deforma-tion potential (DP) and (ii) the piezoelectric coupling. We examine theoretically the effect of an external magnetic field and the quantum dot size on the exciton phonon scattering rate.

2 Model and computation

The exciton Hamiltoninan in a disk like semiconductor quantum dot interacting with acousting phonons is written as: ex ph ex phH H H H

-

= + + . ex

H denotes the exciton Hamiltonian which is expressed as:

ex e h cH H H V= + + , where

eH and

hH are electron and hole Hamiltonian respectively, and

cV is the

Coulomb potential which describes the electrostatic interaction between the charge carriers (for more

details see, e.g., Ref. [7]). 1

2ph n n n

n

H b bω�+Ê ˆ= +Ë ¯Â q qq

q

is the phonon Hamiltonian, and the exciton-phonon

interaction term in the Hamiltonian is ( h.c.),+

n n

n

excH c g b +=Â q q

q

where c+ ( c ) are exciton creation

(anhilation) operators, and ( )n nb b+

q q are creation (anhilation) operators for phonons in branch n and

wave number q with a corresponding energy of n

ω�q

, and nq

g is the coupling constant which depends

on the specific configuration of the system and the type of interaction. We consider that the quantum dot is under an applied uniform magnetic field B in the z direction normal to the x-y plane. We calculate the transition rate Γ for an exciton to be scattered from an initial state i to a final state f, accompanied by the emission of an acoustic-phonon characterized by the frequency ω

qby using the

following expression:22

( ) ( )i f if i f qg q E Eπ

Γ δ ω��

Æ= - - . We consider two different exciton-acoustic

phonon coupling mechanisms: i) the deformation potential coupling and ii) piezoelectric coupling mechanism. The bulk matrix element for the DP coupling is given by:

3 3 *( ) ( , ) ( , )2

ei i

if e h e h f e h c vi

LA

qg q d r d r r r r r D e D e

c Vρψ ψ

�= -È ˘Î ˚Ú hqr qr , where

LAc and ( )c v

D are respectively

the longitudinal sound velocity, and the electron (hole) DP coupling constants. The piezoelectric cou-

pling constant is given by: 3 3 *

0

( ) d d ( , ) ( , )2

i i

if e h e h f e hi

s

eKg q r r r r r r e e

q Vε ε ρψ ψ

�= -È ˘Î ˚Ú e hqr qr , where K is

the electromechanical coupling coefficient [8] which represents the quantitative strength of the piezo-

electric scattering interaction, ρ =5.81 g cm–3 is the crystal density, 13.59

LAc cm s

-

= [9], and s

ε =9.56

[10] is the static dielectric constant. We have calculated the exciton SR Γ from the ground state S to the first exciton-excited state P

-

, with respectively the symmetry ne(h)=0, me(h)=0, and ne(h)=0, me(h)= -1, taking the function of the non interacting electron and hole system as the first approximation for the exciton wave function

( ) ( )

'( , ) ( ) ( )e e h h

e h

nn e h n m e n m hΦ Ψ Ψ=r r r r , with

2 2

2 2( )

2 ( )4( ) ( ) ( ) ( ) ( ) 2

2 1/ 4 1/ 2

( )

1( ) ( , ) ( ) ( )

2

i

r z

Ri i i i m im m h

nm nm nm n

i

rr z N r e L e e

R h

θΨ Ψ ϕ φπ

^- -

- ^

^ ^ ^= =r , (where i=e, h)

phys. stat. sol. (c) 3, No. 9 (2006) 3349

www.pss-c.com © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

where Re(h) is the effective disk radius under the applied magnetic field effect for the electron (hole). We consider only the first level for the quantization in the z-axis direction since

0i izω ω<< .

3 Results and discussion

In this work, we present calculations of exciton-acoustic phonons scattering rates considering both the scattering of excitons by phonons via the DP and piezoelectric potential coupling. The piezoelectric coupling process was considered to have a weak relaxation contribution in comparison with the main deformation potential scattering mechanism [11]. In Fig. 1(a), SR variation is plotted as function of the quantum dot size R, considering the DP mecha-nism. We notice that the SR strongly depends on the confinement, it diminishes considerably as the dot radius decreases. We considered in our calculations quantum dots larger than 40 Å since Knipp and Reinecke [12] have shown that for smaller dots the macroscopic deformation potential (ripple mecha-nism), which results from the time-dependent modulations of the interface become dominant by more than an order of magnitude. We note also that for such dot radius domain, the exciton level separation do not match the longitudinal optical phonons energy [13], so the excitonic polaron formation condition is not satisfied, therefore we focus our analysis only on the acoustic phonons. In Fig. 1(b), we notice that the effect of the magnetic field is more pronounced for large dots, which could be interesting since the dephasing due to acoustic phonons could be controlled through the application of the magnetic field.

40 50 60 70 80 90 100

104

105

106

107

108

109

1010

1011

Scatt

erin

g R

ate

(s-1

)

Dot Radius (A)

(a)

0 10 20 30 40 50 60

0

1x1010

2x1010

3x1010

4x1010

5x1010

6x1010

R=90 A

R=80 A

R=70 A

R=60 A

Sca

tteri

ng

rate

(s-1)

B (T)

(b)

Fig. 1 a) Exciton-acoustic phonons SR as function of the dot radius R considering the DP mechanism. b) Exciton-

acoustic phonons SR as function of the applied magnetic field B for different dot radius R, considering the DP

mechanism. In Fig. 2 the deformation potential and piezoelectric exciton phonon scattering rate ratio DP Piezo

/Γ Γ is calculated as function of the quantum dot radius.We observe that this ratio depends strongly on the dot size. It diminishes from the value 4×103 to almost unity when increasing the dot radius from 4 nm to 20 nm. Hence for small dots the DP in the dominant scattering mechanism, whereas for relatively large dots the piezoelectric process can not be neglected in the exciton acoustic phonon sacttering rate calculations since the ratio between the DP and piezoelectricity SR contribtutions becomes around the unity.

3350 K. Sellami and S. Jaziri: Decoherence effects due to exciton-phonon interaction in CdSe quantum dots

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

40 60 80 100 120 140 160 180 200

101

102

103

ΓDP/ ΓPiezo

Dot Radius (A)

Fig. 2 Deformation potential and piezoelectric exciton-phonon scattering rate ratio as function of the dot radius.

4 Conclusion

We presented calculations of the exciton-acoustic-phonon scattering rates for CdSe quantum dots. We investigated the effect of the nanocrystal size and the applied magnetic field on this rate. As we pointed out for small dots the DP is the main sacttering channel, however for large dots the piezoelectric mecha-nism should be considered since the both calculated scattering rate are of the same order of magnitude. Such result could be of great importance in the interpretation of the exciton relaxation processes in quan-tum dots and understanding the origin of decoherence so ultimtely it may be reduced or controlled.

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