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V N U Journal o f S cien ce, M ath em a tics - P h y sics 25 (2 0 0 9 ) 123-128

The dependence o f the parametric transformation coefficient o f acoustic and optical phonons in doped superlattices on

concentration o f impurities

**Hoang Dinh Trien*, Nguyen Vu Nhan, Do Manh Hung, Nguyen Quang Bau**

Faculty o f Physics, Coliege o f Science, VNU 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

Received 30 July 2008; received in revised form 15 August 2009

Abstract. The parameữic ứansformation of acoustic and optical phonuns in doped superlattices is theoretically studied by using a set of quantum kinetic equations for the phonons. The analytic expression of parametric transformation coefficient of acoustic and optical phonons in doped superlattices is obtained, that depends non-linear on the concenfration of impurities. Numerical computations of theoretical results and graph are performed for GaAs:Si/GaAs:Be doped superlattices.

Keywords: parametric transformation, doped superlattices.

1. Introduction

It is w e ll k n o w n th a t in p re s e n c e o f an e x te rn a l e le c tro m a g n e tic fie ld , an e le c tro n ga<5 b e c o m e s non-staticmary. When the conditions o f parametric resonance are satisfied, parametric resonance and Iransfoimation (PRT) o f same kinds o f excitations such as phonon-phonon, plasmon-plasmon, or o f different kinds o f excitations, such as plasmon-phonon will arise; i.e., the energy exchange process betw een these excitations will occur [1-9], The physical picture can be described as follows: due to the electron-phonon interaction, propagation o f an acoustic phonon with a frequency cOị accompanied by a density wave with the same frequency Q . When an external electromagnetic field with frequency is presented, a charge density waves (CDW) with a combination frequency Uị ±IQ. (/=1,2,3,4...) will appear. If among the CDW there exits a certain wave having a frequency which coincides, or approximately coincides, with the frequency o f optical phonon,L>- , optical phonons will appear.

These optical phonons cause a CDW with a combination frequency o f u - ± / Q , and when u. ± I Q ^ ũ ) ^ , a certain CDW causes the acoustic phonons mentioned above. The PRT can speed up the camping process for one excitation and the amplification process for another excitation. Recently, there have been several studies on parametric excitation in quantum approximation. The parametric

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interactions and ừansformation o f acoustic and optical phonons has been considered in bulk semiconductors [1-5], for low-dimensional semiconductors (doped superlattices, quantum wells, quantum wires), the dependence o f the parametric transformation coefficient o f acoustic and optical phonons on temperature T and frequency Q is has been also studied [6-9]. In order to improve the PRT theoretics for low-dimensional semiconductors, we, in the paper, examine dependence o f the parameừic ừ'ansformation coefficient o f acoustic and optical phonons in doped superlattices on concentration o f impurities.

2. Model and quantum kinetic equation for phonons

We use model for doped superlattice with electron gas is confined by the superlattice potential along the z direction and electrons are free on the xy plane. If a laser field ẼỌ) = ẼQSÌn(Qt) irradiates the sample in direction which is along the z axis, the electromagnetic field o f laser wave will polarize parallels the X a x is and y a x is, and its strength is ex p ressed as a v ecto r potential A { i ) = — ẽ qCo s{ Qí)

(c is the light velocity; Q is EMW frequency; Eq is electric field intensity).

The Hamiltonian o f the elecfron-acoustic phonon-optical phonon system in doped superlattice can be written as (in this paf>er, we select h =1):

Ỉ Ỉ ( 0 = + Ỵ y ^ c lC ị +

a Ọ 4

+ Ị Ị C 5Í + c 'ị ) (1)

Í aa aa

where £■„(/) + - A ( t ) ) is energy spectrum o f an electron in external electromagnetic field, a * , c

(a„) IS the creation (annihilation) operator of an elecưon for state \n ,k j^ ), b'ịị , b. { c - , Cị) is Ihc creation operator and annihilation operator o f an acoustic (optical) phonon for state have wave vector q .

The elecứon-acoustic and optical phonon interaction coefficients take the forms[ 10]

| O i P = r ^ ( — - - ) (2)

2 p v ỵ 2 V x q ' z . Zo

here V, p , and ệ are the volume, the density, the acoustic velocity and the deformation potential constant, resj>ectively. X is the elecfronic constant, . / o high-frequcncy dielectric constants, respectively. The elecfron fonn factor, ly. The elecfron fonn factor, I ,{q)I , (q) is written as [11]: is wr

^ ' (^ ) = (2 - j d ) d z

j-iỏ

(3) here, is the eigenfunction for a single potential well, and is the number o f doped superlattices periods, d is the period.

Energy specfrum o f electron in doped superlattic [12]:

2m

2 (4)

(3)

here, Hjj is c o n c e n tra tio n o f impurities, m and e are the effective mass and the charge o f the electron, respectively and are the energy levels o f an individual well.

In order to establish a set o f quantum kinetic equations for acoustic and optical phonons, we use equation o f motion o f statistical average value for phonons

i j < b , >=<[ố,-,//(/)]),; >=<K-,//(/)]>, (5)

where {X) i means the usual thermodynamic average o f operator X

Using Hamiltonian in E q .(l) and realizing operator algebraic calculations, we obtain a set of coupled quantum kinetic equations for phonons. Tbe equation for the acoustic phonons can be formulated as:

ị { b , ) , * w , ( b , ) , = - Y ^ Ỹ j > Ả ị : ) j Á ) Ư Á Ì ,

nnk^

X k d Q / • I' + < 6 -V ,)+ C + (c :,),)} x

• nn W/I

xexp{i[s . ( k ^ ) - ( k ^ - ợ)](/, - 0 - ( 6 )

ft

A similar equation for the optical phonons can be obtained in which Uạ, cOị, , D- are replaced by {b -),, ( c ^ ) , , ^{C O -}, V - , D - , , respectively.

In Eq.(6), f„ { k ^ ) is the distribution function o f electrons in the state I , J^{— ) is the Bessel function, and Ằ = — ^sE Q

mCl

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3. The parametric transformation coeflident of acoustic and optical phonon in doped supcrlattices

In order to establish the parametric ữansformation coefficient o f acoustic and optical phonon, we use standard Fourier transform techniques for statistical average value o f phonon operators: (6^),,

(blị)r y {<^q)t» • The Fourier transforms take the form :

( T - ) , ]^.Ậ co )e* ‘^dco (7)

—flO —00

3ne finds that the final result consists o f coupled equations for the Fourier fransformations C-(ứí) m ả B Ậ c o ) o f <c->, and <ồ->,.

For instance, the equation for C-(<y) can be written as:

„ ^ , , C g {e o -lĩì)

(ũ)-U,)CJCO) = 2 Y y \Ị . I' D ĩ v , - ^ ----p ,(q ,ũ )) +

’ ’ nn ’ ^ c o - i n + u, 'Q

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* 2 ỵ p l j D _ , C ,

nn

B Ặ c o - i n )

’ iU - /Q + iy. P,{q,co) ( 8 )

In the similar equation for B .( c o ) , functions such as c ^ (d )), C Ạ c o - ì í ì ) , B ^ ( ú ) - I Q ) , ưị , Củ^, C -, D- are replaced by B-{co), B Ạ c ứ - l ĩ ì ) , C Ậ co -lO .), ũ)-, ^V ^{- ,} D -, C - , respectively.

In Eq.(8), we have:

P,(q,oj)= Ỳ j d ) j Á r , ( c o + ^ ) (9)

where, the quantity Ổ is infinitesimal and appears due to the assumption o f an adiabatic interactiion of the electromagnetic wave (EMW).

In Eq.(8), the first term on the right-hand side is significant ju st in case 1=0. If not, it will confribute more than second order o f elecfron-phonon interaction constant. Therefore, we have

V- , „ C A to -IC i)

( c o - u ^ ) C Ặ c o ) = 2 ỵ \ I ^ ^ . p ,iq ,c a )

C O -I Q + U .

nn ¥

"" ’ ’ ^ c o - i a + c o , ' ^

nn H

(1 1)

(1?) Transforming E q .( ll) and using the parametric resonant condition 0)^ the parametric transformation coefficient is obtained

ỵ \ I . f D _ , C - P , { q , w )

nn

Consider the case o f / = 1; and assign /0 ” y • I ^ H A Note that í5 < , we have

K. =

'>0

(13) Using Bessel function, Fermi-Dirac distribution function for elecừon and energy spectrum o f electron in Eq.(4), we have

• (14)

=12rc

Where r = 4 ^y ^{I /} 1^'D .C s R e T s( < y . )

nn

r o = Ị | / ; Ỹ \ D , ẹ i m T , { o , )

(15) (16)

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H.D. Trien e t a l / VNU Journal o f Science, M athem atics - Physics 25 (2009) Ỉ2 3 -I2 8 ¹²⁷

**RerẠio,) = ,fi^íexpl-P{ĩĩ^rịnV2)]-apí-pậĩ^r(.nim**

’ ’ 27T/3A, x m xm

(17)

Bv- Bv-

° ) '''( « + 1/2)] > ^ e x p i ^ ) s ỉ n h ạ ^ ) (18)

2/« ỵ m 2m x>n

In Eqs.(17) and (18), P = M k J ' (Ấ:^ is Boltzmann constant), /o is the elecứon density in doped superlattices.

A'l is analytic expression o f parametric ữansformation coefficient o f acoustic and optical phonon in doped superlattices when the parametric resonant condition Ũ)-±IQ = U- is satisfied

4. Num erical results and discussions

In order to clarify the mechanism for parametric transformation coefficient o f acoustic and optical phonons in doped superlattices, in this section we perform numerical computations and graph for GaAs:Si/GaAs:Be doped superlattices. The parameters used in the calculation [6,7] ^ = i3.5 eV ,

p = 5 3 2 g cm \ = 5378ms ' , ;i'„ = 1 0 .9 , ^ 0 = 1 2 .9 , d = \ồ n m , OT = 0.066/Mo, Wo being the mass o f free electron, a f free electron, ticử. = 36.25m eF , ti(0^= 36.25m eF , £^0 ~ = 10*F/m elecfric field intensity), elecfric field intensity), « 4 = =1.3807 X 1.3 ,

= e = 1 .6 0 2 1 9 x l0 '’ c , ft = 1 . 0 5 4 5 9 . T=300K, ạ = 3 .2 x lO 'w

Fig. 1. Dependence of the K. on .

Fig 1 shows the parametric transformation coefficient K1 as a function o f concenfration of impurities . It is seen that the parametric transformation coefficient o f acoustic and optical phonons in doped superlattices depends non-linearly on concentration o f impurities n ^. Especially, when concentration o f impurities tend toward zero, value o f the parametric transformation coefficient o f acoustic and optical phonons in doped superlattices will turn back nearly equal that in bulk

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128 H.D. T h en et al. / VN U J o u rn a l o f Science, M athem atics - Physics 25 (2009) 123-128

semiconductors [5] (Ẩ ”, 0.3 5 ), w hen concenữation o f impurities raises, the parametric transfonnation coefficient also to increase. This can be explained as follows: when concentration of impurities tend toward zero, the doped superlattices as a normal bulk semiconductors. When concentration o f im purities raises large enough, the doped superlattices is as low-dimensional semiconductor.

5. Conclusions

In this paper, we obtain analytic expression o f the paramefric fransformation coefficient of acoustic and optical phonons in doped superlattices in presence o f an external electromagnetic field K ị , Eqs.(14)-(19). It is seen that Kị depends on concenfration o f impurities. Numerical computations and graph are perfonned for GaAs:Si/GaAs:Be doped superlattices, fig 1. The results is seen non

linear dependence o f param etric transformation coefficient on concentration o f impurities, when concentration o f im purities tend toward zero, value o f the parametric transformation coefficient of acoustic and optical phonons in doped superlattices will turn back nearly equal that in bulk semiconductors [5] (Ả", ~ 0 .3 5 ), when concenứation o f impurities raises, the parametric ưansíormation coefficient also to increase. This can be explained as follows: when concentration o f impurities tend toward zero, the doped superlattice as a normal bulk semiconductors. When concentration o f im purities raises large enough, the doped superlattice is a low-dimensional semiconductor.

Acknowledgments. This w ork is com pleted with financial support from NAFOSTED and QG.09.02

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