DSpace at VNU: The parametic resonence of confined acoustic phonons and optical phonons in quantum wells

confined optical phonons in quantum wells Nguyen Thi Mai Nhien, Le 'Ihai Hung*, Do Manh Hung, Nguyen Quang Bau Faculty of Physics, Collelie of Science, WU, 334 Nguyen Trai, Hanoi, Vietn

DSl HQC QU0C GrA HA NQr

VIE.TNAM NATIONAL UNIVERSITY, HANOI

t s S N u B 6 6 ' 86l ?

r ; r n F e '

KHO

I

P,"

confined optical phonons in quantum wells

Nguyen Thi Mai Nhien, Le 'Ihai Hung*, Do Manh Hung, Nguyen Quang Bau

Faculty of Physics, Collelie of Science, WU, 334 Nguyen Trai, Hanoi, Vietnant

Received 17 November 2008

Abstract The parametric resonance of confined acoustic and confined optical phonons in

quantum wells by using a set of quantum kinetic equations for phonons is studied The analytical

expression of the tfueshold amplitude E6 of the field in quantum wells is obtained The

dependence of it on the temperature T of the s;ystem and the frequecy O of the external electromagnetic field is sfudied Numerical compuLtations and graphs are performed for GaAs-GaAsAl quantum well The results are compared wilh the case of unconfined phonons

1 Introduction

As we known that in the presence of an extemal electromagnetic field (EEF), an electron gas becomes non-stationary When the conditions of parametric resonance are satisfied, parametnc interactions and transformations ( PIT) of some kinds of excitations such as phonon-phonon, plasmon-plasmon, or of different kind of excitations such as: plasmon-phonon will arise, the energy exchange process between these excitations will occur U-zl.Ttre PIT of acoustic and optical phonons has been considered in bulk semiconductors [3-4].

There have been a lot of works on the PI'f for low dimentional semiconductors 15-10] However, the parametric resonance of acoustic and optical phonons in quantum well in the case of confined phonons have not sfudied yet Therefore, in this paper, we continue to study the the parametric resonance of acoustic and optical phonons in quantum well in the presence of EEF in the case of confined phonons This result has been compared with the case of unconfined phonons.

We use a simple model for a quantum well, in which a two-dimentional electron, phonon gas is confined by the infinity potential V(z) of a rectanguLlar well along the z direction and electrons and phonons are free on the x-y plane A laser field E == E, sin(Q/) inadiates the sample in a direction which is normal to the x-y plane, its polarization is arlong the x axis and its strength is expressed as a vector potential A0=lE".os(fU) If the ele,;tron-acoustic and optical phonons interaction

f) potential is used, the Hamiltonian for the system of the electron and the acoustic and optical phonons

in case of confined phonons is written as:

H = H " * H n r n + H " p n + H " - n p t + H r - o p n ( 1 )

In order to establish a set o1'quantum kinetic equations for acoustic and optical phonons, (U^,r^l ,and (c^.u^), *h * (t4 ), a.notes a stastical use the general quantum distnbution functions for

240

I

;

I

the confined phonons, average at the rnoment (W), =rrl*i),(,| ir rt densiry matrix operator),

t t

Hamintonian in Eqs (1) and realizingoperator algebraic calculatioru.

2 The parametric resonance of confined acoustic phonons and confined optical phonons in quantum wells

We obtain the general dispersion equation for the parametric interaction and tranformation between confined acoustic phonons and optical phonons in compositional surperlatice:

Here:

(o -,, ^,u, - rr)"1, .Ne), - r,,, - |ZV :,,1' lry u.l I q I o ^,i.n (m, Q,, a +n,")

I

=#}Vr",l'|ry,l'lri,l'a^,4,e^4,n*(*,4,,r)rt*(m,Q',at+NO)

!r (a+rc)=

We assume that electron - phonon interactions the condition:

If we write the dispersion relation of acoustic and optical phonon as:

to,(*,Q ) = o)o I ir o, ,o,(*,4,) = 0o + ir o

,o : -!ZV:,,1'lri,l' *r;,u (,^,,,\

', = - lZV i,^,1'ln;,1' u * r,,u (- ^,r,\

h u * ' '

We obtain the resonance acoustic phonon mode:

* 1 I

,ot = ,, *;L(u, t ,,)n(q)- i(r, + r")+

I

(3)

(4)

(s)

(6)

(7)

(8)

(e)

( 1 0 )

( l l )

ln equation (11) the signs (t) in the subcript of atl correspond to the slgns (t) in the front of the root and the sig"s (t) in subcript of atl correspond to the other sign pairs The signs depend on the resonant condition:

?b ^W ) -',,V, - q,) - nfi - tto +

242 N.T.M- Nhien et ol / VNU Journr'il of Science, Mathematics - Physics 24, No lS (2005) 240-243

^.r,(,,,,,,-[R"f'''a-,

@^JT-For instance, the existence of ir positive imaginary part of atf implies a parametric amplication

of the acoustic phonon In such case that L <<1, o.r"rio.rding to the maxiaml resonance, we obtarn:

From equation (5) the condition for the resonant acoustic

imaginary part leads to Inl't 4ro'r'o Using this condition and

intensity of the threshold field E1 for EEF:

t r o ) + ^ - l t ( 1 2 )

r )

phonon modes to have a positive equations (18-20), we yield the

I

( l3 )

In equation (13), we can see the marked difference between the case of confined phonons and unconfined phonons, the formulla trf E6 contains a quanfum number m characterizing confined phonons.

3 Numerical results and discussions

In oder to clarify the mechapism for parametric resonance of acoustic - optical phonons in case of the confined phonons, in this section, we consider a GaAs/GaAsAl quantum well The parametric used in the caflculation are following: 4 =13.5eV, p = 5.32gcm | ,

v , = 5 3 7 0 m s - ' , K , = 1 2 9 , K * = 1 2 9 , m ' = 0 0 6 6 m " c ) = 5 1 0 ' ' , f i a ) o = 3 6 2 5 m e v , E n = 1 0 6 v l m ,

k s : 1.3807 10-23 J I K, € : 1.60219.1 O-re C, h : | 05459.1 0{4 -/s

x 10? do thr Eth{r)

250

x l 0 9

Fig 2 The dependence of the intensity of the threshold field (kVcm-t) on wave vector q (m-') in both cases of confined phonons and unconfined phonons

Fig 1 The dependence of the intensity of the

threshold field (kVcm-^) on temperature T(K) in

both cases ofconfined phonons and unconfined

phonons

2.5

f i r s

do Ihi E{q}, nl=1.n2=2

i ; F

i i l i

i i l i

i t Ti t i

t i i i

" " ' 1 " " 7 : " " " ' l '

-: ; -: :

' t l

i l i i

: i i 3

-" t " / i

'-"'i!'

/'i'-"'-i"'-l / i '

kt fig 1, It shows that Es, as a function of temperature T in both cases of confined phonons and

unconfined phonons The graph shows that confined phonon increase the intensity of the threshold

field Eu, in comparison with the case of unconfined phonons Namely, at the same teperature T :200K,

phonons

In fig 2 present E6 os a function of the wave vector at : 27K The figure shows that the curve

quantum wave number following the confined axis

4 Conclusions

In this paper, we analytically investigated the possibility of parametric resonance of confined

acoustic and confined optical phonons We have obtained a set of quantum kinetic equations for

hansformation of phonons However, an analytical scllution applying to these equations can only be

obtained within some limitations Using these limitations for simplicity we obtained the parametnc

resonant condition, the intensity of the threshold field En fbr acoustic phonon parametric amplification

in quantum well in case of confined phonons And we have also paid attention to E6 in case of

intensity of the threshold field for GaAslGaAs quantum well The results show that confined phonons

wave vector Q in comparison with the case of unconfined phonons Confined phonons will increase

the values of the threshold field Eth The parametric amplification for acoustic phonons in quantum

well in case of confined phonons can occur under the condition that the amplitude of the external

in the case of confined phonons, the curve of the intensity of the threshold field as a function of wave

number has several maxima and the confined phonons increases the intensity of the threshold field at

Acknowledgments This work is completed with financial strpport from the Program of Basic

References

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[ 0] Tran Cong Phong, Le Dinh, Nguyen Quang Bau, Dinh Quoc Vuorrg J.Koz lhys.Soc 49 (2006) 2367

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