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

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VIE.TNAM
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HANOI

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- Physics24,No. lS (2008)240-243
VNU Journalof Science,Mathematics

The parametricresonanceof confinedacousticphononsand

confinedopticalphononsin quantumwells
NguyenThi Mai Nhien,Le

'Ihai

Hung*, Do Manh Hung,NguyenQuangBau

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
dependenceof it on the temperature T of the s;ystemand the frequecy O of the external
electromagneticfield is sfudied. Numerical compuLtations
and graphs are performed for GaAsGaAsAl quantum well. The results are compared wilh the case of unconfined phonons

1. Introduction
field (EEF), an electrongas
As we known that in the presenceof an extemalelectromagnetic
becomesnon-stationary.When the conditionsof parametricresonanceare satisfied,parametnc
( PIT) of somekinds of excitationssuchasphonon-phonon,
plasmoninteractionsandtransformations
plasmon-phonon
will
arise,
the
exchange
energy
kind

such
as:
different
of
excitations
plasmon,or of
processbetweentheseexcitationswill occur U-zl.Ttre PIT of acousticand opticalphononshasbeen
in bulk semiconductors
considered
[3-4].
There have been a lot of works on the PI'f for low dimentional semiconductors15-10].
However, the parametricresonanceof acousticand optical phononsin quantumwell in the caseof
confined phononshave not sfudied yet. Therefore,in this paper, we continue to study the the
parametricresonanceof acousticand opticalphononsin quantumwell in the presenceof EEF in the
caseof confinedphonons.This resulthasbeencomparedwith the caseof unconfinedphonons.
We usea simplemodelfor a quantumwell, in which a two-dimentionalelectron,phonongasis
well alongthe z directionand electronsand
confinedby the infinity potentialV(z) of a rectanguLlar
phononsare free on the x-y plane.A laserfield E ==E, sin(Q/) inadiatesthe samplein a direction
as a
which is normalto the x-y plane,its polarizationis arlongthe x axis and its strengthis expressed
and optical phononsinteraction
If the ele,;tron-acoustic
vector potential A0=lE".os(fU).
f)
potentialis used,the Hamiltonianfor the systemof the electronand the acousticand opticalphonons
in caseof confinedphononsis writtenas:
(1)
H = H " * H n r n + H " p n + H " - n p t+ H r - o p n
In order to establisha set o1'quantumkinetic equationsfor acousticand optical phonons,

usethe generalquantumdistnbutionfunctionsfor
(U^,r^l,and (c^.u^),*h..* (t4), a.notesa stastical
. Conesponding
I I 82@yahoo'com'\/n
E-mail:hunglethaiI9
author.

240
I

;
I
l

I


N-I.M. Nhien et al. / WU Journal of Science,Mathematics- Physics24, No. IS
e\Aq

phonons,
theconfined
average
at thernoment(W), =rrl*i),(,|

t

t

240-243

241

ir rt. densirymatrixoperator),

Hamintonianin Eqs(1) and realizingoperator
algebraiccalculatioru.
2. The parametric resonanceof confined acousticphononsand confined optical phononsin
quantum wells

usingFourier
rransforma
tion,(v^.a,)
,=+-Tr^,u,(ap-'^aat, vmdt(rd:)\r^u,),{^

(2)

_ _ €

_ @

We obtain the general dispersion equation for the parametric interaction and tranformation
betweenconfined acousticphonons and optical phononsin compositional surperlatice:

- rr)"1,.Ne),
- r,,, - :,,1'
(m,
a+n,")
(o -,,^,u,
Q,,

|ZV lryu.lIq I o^,i.n.
I
=#}Vr",l'|ry,l'lri,l'a^,4,e^4,n*(*,4,,r)rt*(m,Q',at+NO)

(3)

Here:

= .(*)r., (*
n,(*,d,,,ul
+,e),
Zt
)r,u,(ot

(4)

!r (a+rc)=

(s)

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

We assumethat electron- phononinteractionsthe condition:

.o
Vnl^lq,l'1ry,]'

(6)

If we write the dispersionrelationof acousticandopticalphononas:
to,(*,Q ) = o)oI ir o,

,o,(*,4,) = 0o + ir o

(7)

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

(8)

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

(e)

We obtaintheresonance
acousticphononmode:
1 I
*

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

withA:#zvl,,,|'t{,DflRer^,u(r^,u,\

I

(10)
(ll)

(11)thesigns(t) in thesubcriptof atl correspond
ln equation
to theslgns(t) in thefrontof
to the othersignpairs.The signsdependon
theroot andthe sig"s (t) in subcriptof atl correspond
condition:
theresonant


242

N.T.M- Nhien et ol. / VNU Journr'il of Science,Mathematics- Physics24, No. lS (2005)
240-243

^r,

4!)' r^l ^.r,(,,,,,,-

[R"f'''a-,
@^JT-

For instance,the existenceof ir positiveimaginarypart of atf impliesa parametricamplication
of the acousticphonon.In suchcasethat .L <<1, .o.r"rio.rding to the maxiamlresonance,
we obtarn:
tro) +^- lt

(12)

r)

From equation (5) the condition for the resonantacousticphonon modesto have a positive
imaginarypart leads to Inl't 4ro'r'o.Using this conditionand equations(18-20),we yield the
intensityof the thresholdfield E1 for EEF:
I

(l3)

In equation(13), we can seethe markeddifferencebetweenthe caseof confinedphononsand
unconfined phonons, the formulla trf E6 contains a quanfum number m characterizingconfined
phonons.
3. Numericalresultsand discussions
In oder to clarify the mechapismfor parametricresonanceof acoustic- optical phononsin
case of the confined phonons,in this section,we considera GaAs/GaAsAlquantumwell. 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 vE, n = 1 0 6 vl m ,
ks : 1.3807
.10-23
J I K, € : 1.60219.1
O-re
C, h : | .05459.1
0{4-/s.
do thr Eth{r)

x 10?

2.5

i

i

i

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

i

; . F

i
i

i l i
i l i

i t T
i

fi rs

t

i

" " ' 1 "t " 7 :i" " " ' il ' - - -i - -

:
'

t

; :
l

:

i: il ii 3 i:" !t

/" . /

i
:

,r
/'
'-"'i!'
/'i'-"'-i"'-

l /i '

250
I

Fig. 1. The dependenceof the intensity of the

threshold field (kVcm-^) on temperature T(K) in
both casesofconfined phonons and unconfined
phonons.

^
xl09

Fig. 2. The dependenceof the intensity of the threshold
field (kVcm-t) on wave vector q (m-') in both casesof
confined phononsand unconfinedphonons.


N.T.M.Nhienet al. / VNUJournalof Science,
- Phvsics
Mathematics
24, No.1S(200& 240-243

243

kt fig 1, It shows that Es, as a function of temperatureT in both casesof confined phonons and
unconfined phonons. The graph shows that confined phonon increase the intensity of the threshold
field Eu,in comparisonwith the caseof unconfined phonons.Namely, at the sameteperatureT :200K,
(kvcm-'; in case of confined phonons,but E,,,:O.SS.tdt(kVcm-r)in case of unconfined
E6.,:2.6.107
phonons.
In fig 2 present E6 os a function of the wave vector at : 27K. The figure shows that the curve
hasgreatestmacximalvalue E15:7.3.10i-(V.cm-';.
Dift'eringfrom the caseoiunconfinedphonons,the
curve has an extra maximal E6,:6.1.10s(V.cm-'). This is due to the fact that confined ohonon has
quantum wave number following the confined axis.

4. Conclusions
In this paper, we analytically investigatedthe possibility of parametric resonanceof 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 equationscan only be
obtained within some limitations. Using these limitations for simplicity we obtained the parametnc
resonantcondition, the intensity of the threshold field En fbr acousticphonon parametricamplification
in quantum well in case of confined phonons. And we have also paid attention to E6 in case of
unconfined_phonons to compare with the result above. We numerical calculated and graped the
intensity of the threshold field for GaAslGaAsquantum well. The results show that confined phonons
cause some unusual effects. The threshold field .0.r,strongly depends on the temperatureT, and the
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
electromagneticfield is sometimeshigher than that of the intensrty of the threshold field. Especially,
in the caseof confined phonons, the curve of the intensity of the threshold field as a function of wave
number has severalmaxima and the confined phonons increasesthe intensity of the threshold field at
the sametemperaturecomparing with caseof unconfinedphonons;.
Acknowledgments. This work is completed with financial strpport from the Program of Basic
Researchin Natural Science405906 and QT.08.15.

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parametricresonanceof confinedacousticphononsand confinedoptical phonons,itl quantuln
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