PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM WELLS

PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM WELLS DO MANH HUNG, NGUYEN QUANG BAU Department of Physics, Colleg

PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM WELLS

DO MANH HUNG, NGUYEN QUANG BAU Department of Physics, College of Natural Science, Vietnam National University in Hanoi

Abstract The parametric transformation and parametric resonance of confined acoustic phonons and confined optical phonons in quantum wells in the presence of an external electromagnetic field are theoretically studied by using a set of quantum kinetic equations for phonons The analytic expression of the parametric transformation coefficient (K 1 ) and the threshold amplitude (E th )

of the field in quantum wells are obtained Unlike the case of unconfined phonons, the formula

of K 1 and contains a quantum number m characterizing confined phonons Their dependence

on the temperature T of the system and the frequency Ω of the electromagnetic field is studied Numerical computations have been performed for GaAs/AlAsAl quantum wells The result have been compared with the case of unconfined phonons which show that confined phonons cause some unusual effects.

Keyword: Parametric transformation and parametric resonance, quantum well

I INTRODUCTION

It is well known that in the presence of an external electromagnetic field (EEF),

an electron gas becomes non-stationary When the conditions of parametric resonance (PR) are satisfied, parametric interactions and transformations (PIT) of same kinds of excitations, such as phonon - phonon, plasmon - plasmon, or of different kinds of excita-tions, such as plasmon - phonon, will arise; i.e., energy exchange processes between these excitations will occur [1, 2] The PIT of acoustic and optical phonons has been considered

in bulk semiconductors [3 - 5] The physical picture can be described as follows: due to the electron - phonon interaction, propagation of an acoustic phonon with a frequency

ω− →q is accompanied by a density wave with the same frequency When an EEF with frequency Ω is presented, a charge density waves (CDW) with a combination frequency

ω− →q ± N Ω (N = 1, 2, ) will appear If among the CDW there exists a certain wave having

a frequency which coincides, or approximately coincides, with the frequency of the optical phonon, ν− →q, optical phonons will appear These optical phonons cause a CDW with a combination frequency of ν− →q ± N Ω, and when ν− →q ± N Ω ∼= ω− →q, a certain CDW causes the acoustic phonons mentioned above The PIT can speed up the damping process for one excitation and the amplification process for another excitation There have been a lot of works on the PIT for low dimensional semiconductors in the case of unconfined phonons [6 - 8] However, parametric transformation and parametric resonance of acoustic and optical phonons in quantum wells in the case of confined phonons have not been studied

yet Therefore, in this paper, we have studied parametric transformation and paramet-ric resonance of acoustic and optical phonons in quantum wells in the case of confined phonons The comparison of the result of confined phonons to one of unconfined phonons shows that confined phonons causes some unusual effects To this clarify, we estimate numerical values for a GaAs/AlAsAl quantum well, and we discuss the conditions under which the parametric resonance occurs

II THE PARAMETRIC RESONANCE OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM

WELLS

It is well known that the motion of an electron and phonon in a quantum wells is confined and that its energy spectrum is quantized into discrete levels In this paper, we assume that the quantization direction is the z direction The Hamiltonian of the electron

- confined acoustic (confined optical) phonon system in a quantum well in the second quantization representation can be written as:

H (t) = X

n,−→k⊥

εn−→k⊥− e

~c

−

→

A (t)a+

n,−→k⊥a

n,−→k⊥ + X

m,−→q

⊥

~ωm,−→q

⊥b+m,−→q

⊥

bm,−→q

⊥

m,−→q

⊥

~νm,−→q

⊥c+m,−→q

⊥

cm,−→q

⊥+ X m,−→q

⊥

X

n,n 0 ,−→k⊥

C− →mq

⊥

Im

n,n0a+

n 0 ,−→k⊥+−→q

⊥

an,− →

k⊥



bm,−→q

⊥ + b+m,−−→q

⊥



m,−→q

⊥

X

n,n 0 ,−→k⊥

D− →mq

⊥

Im

n,n0a+

n 0 ,−→k⊥+−→q

⊥

an,− →

k⊥



cm,−→q

⊥+ c+m,−−→q

⊥



here, n, n’ are denotes the quantization of the energy spectrum in the z direction (n, n’ = 1, 2, 3, ),



n,−→k⊥

 and



n0,−→k⊥+ −→q⊥

 are electron states before and after scattering, respectively; −→k⊥, −→q⊥ is the in-plane (x, y) wave vector of the electron (phonon); a+

n,−→k⊥a

n,−→k⊥, (b+m,−→q

⊥

, bm,−→q

⊥; c+m,−→q

⊥

, cm,−→q

⊥) are the creation and the annihila-tion operators of the electron (acoustic phonon; optical phonon), e is the charge of the electron, c is the of light velocity, −→A (t) is the vector potential of an EEF, respectively

−

→

A (t) = Ωc−→E0cos (Ωt) and ~ωm,−→q

⊥



~νm,−→q

⊥



is the energy of the confined acoustic (op-tical) phonon, ωm.−→q (νm,−→q

⊥) is frequency confined acoustic (optical) phonon; m is the quantum number characterizing confined phonons εn−→k⊥



is the energy spectrum of the electron in quantum wells take the simple form [7]:

εn

−→

k⊥



= ~

2−→

k2

⊥ 2m∗ +~

2π2n2

where, L is the well width, m∗ is the effective mass C− →m

q ⊥



D− →m

q ⊥



is the electron - confined acoustic (electron - confined optical) phonon interaction constant take the form [9]

C− →m

q ⊥

2

2

~

ρυaV

r

−

→q2

⊥+

mπ L

2

(3)

D− →m

q ⊥

2

= e

2

~νm,−→q

⊥

V

1

ε0

 1

χ∞

− 1

χ0



1

−

→q2

Here, V, ρ, υa, and ξ are the volume, the crystal density, the acoustic wave velocity, and the deformation potential constant, respectively ε0 is the electronic constant; χ0 and χ∞ are the static and high - frequency dielectric constant, respectively The electron form factor In,nm 0(qz), is written as [10]

In,nm 0(qz) = 2

L

L Z

0

Nm(z) sin n0πz

L

 sinnπz L



Nm(z) = η (m) cos

mπz L

 + η (m + 1) sin

mπz L



(6) ( η (m) = 0 if m even and η (m) = 1 if m ext) In order to establish a set of quantum transport equations for confined acoustic and confined optical phonons in quantum wells,

we use the general quantum distribution function [11] for the confined acoustic (confined optical) phonons,Dbm,−→q

⊥

E

t and Dcm,−→q

⊥

E

t:

i∂

∂t

D

bm,−→q

⊥

E

= 1

~

Dh

bm,−→q

⊥, H (t)

iE

and

i∂

∂t

D

cm,−→q

⊥

E

= 1

~

Dh

cm,−→q

⊥, H (t)iE

Where hψit denotes a staticical average value at the moment t and hψit = T r(W∧

∧ ψ)(W∧ being the density matrix operator) Hamiltonian Eq (1), (7) and (8) and using the com-mutative relations of the creation and the annihilation operators, we obtain the quantum kinetic equation for confined acoustic (confined optical) phonon in quantum wells:

∂

∂t

D

bm,−→q

⊥

E

t+ iωm,−→q

⊥

D

bm,−→q

⊥

E

t= − 1

~2 X

n,n 0 ,−→k⊥

∞ X

ν,µ=−∞

In,nm 0

2

Jν

 λ Ω



Jµ

 λ Ω



×hfn0

−→

k⊥− −→q⊥− fn−→k⊥i

t Z

−∞

dt1e

i

~

h

ε n

 − →

k⊥



−εn0−→k⊥−−→q⊥

i (t 1 −t)−iνΩt 1 +iµΩt

×

(

|Cm,−→q

⊥|2 Dbm,−→q

⊥

E

t 1

+

b+m,−−→q

⊥

t 1

! + C−−m→q

⊥

Dm− →q

⊥

D

cm,−→q

⊥

E

t 1

+

c+m,−−→q

⊥

t 1

!)

(9)

∂

∂t

D

cm,−→q

⊥

E

t+ iνm,−→q

⊥

D

cm,−→q

⊥

E

t= − 1

~2 X

n,n 0 ,−→k⊥

∞ X

ν,µ=−∞

In,nm 0

2

Jν λ Ω



Jµ λ Ω



×hfn0

−→

k⊥− −→q⊥− fn−→k⊥i

t Z

−∞

dt1e

i

~

h

ε n

 − →

k⊥−εn0−→k⊥−−→q

⊥

i (t 1 −t)−iνΩt1+iµΩt

×

(

D−−m→q

⊥

C− →mq

⊥

D

bm,−→q

⊥

E

t 1

+

b+m,−−→q

⊥

t 1

! + |Dm,−→q

⊥|2 Dcm,−→q

⊥

E

t 1

+

c+m,−−→q

⊥

t 1

!)

(10) Where, fn−→k⊥



is the distribution function of the electron in the state

n,−→k⊥

E

; Jν Ωλ

is the Bessel function; λ = e~

−

→

E 0 − →q

⊥

mΩ One finds that the final result consists of an infinite set of coupled equations for the Fourier transformations Bm,−→q

⊥(ω) and Cm,−→q

⊥(ω) of D

bm,−→q

⊥

E

t and Dcm,−→q

⊥

E

t, respectively For instance, the equations for Bm,−→q

⊥(ω) and

Cm,−→q

⊥(ω), can be written as:



ω − ωm,−→q

⊥



Bm,−→q

⊥(ω) = 2

~ X

n,n 0

In,nm 0

2

C− →mq

⊥

2

ωm,−→q

⊥

Π0 m, −→q⊥, ω

ω + ωm,−→q

⊥

Bm,−→q

⊥(ω)

+2

~

X

n,n 0

In,nm 0

2

C−−m→q

⊥

Dm− →q

⊥

∞ X

s=−∞

νm,−→q

⊥

Πs m, −→q⊥, ω

ω − sΩ + νm,−→q

⊥

Cm,−→q

⊥(ω − sΩ) (11) and



ω − νm,−→q

⊥



Cm,−→q

⊥(ω) = 2

~ X

n,n 0

In,nm 0

2

D− →mq

⊥

2

νm,−→q

⊥

Π0 m, −→q⊥, ω

ω + νm,−→q

⊥

Cm,−→q

⊥(ω)

+2

~

X

n,n 0

In,nm 0

2

C− →mq

⊥

D−−m→q

⊥

∞ X

s=−∞

ωm,−→q

⊥

Πs m, −→q⊥, ω

ω − sΩ + ωm,−→q

⊥

Bm,−→q

⊥(ω − sΩ) (12) From Eq (11), we have:



ω2− ω2

m,−→q⊥−2

~ X

n,n 0

In,nm 0

2

C− →mq

⊥

2

ωm,−→q

⊥Π0 m, −→q ⊥, ω



Bm,−→q

⊥(ω)

= 2

~

X

n,n 0

In,nm 0

2

C−−m→q

⊥

Dm− →q

⊥

νm,−→q

⊥



ω + ωm,−→q

⊥

X

s=−∞

Πs m, −→q⊥, ω

ω + sΩ + νm,−→q

⊥

Cm,−→q

⊥(ω + sΩ)

(13) From Eq (12), after some mathematical transformations, and for ω = ωm,−→q

⊥ and s = N,

we find the expression

⊥



ωm,−→q

⊥+ N Ω



=

2

~

P

n,n0

I m n,n0

2

C m

−−→q ⊥D m

−

→

q ⊥ωm,−→q ⊥ΠN(m,−→q⊥,ω+N Ω)Bm,−→

q ⊥



ωm,−→

q ⊥







 ω+N Ω−νm,−→

q ⊥−

2

~ P n,n0

Im n,n0

Dm → −

q ⊥ 2

νm,−→

q ⊥Π0

 m,−→q ⊥,ωm,− →

q ⊥+N Ω



ωm,−→

q ⊥+N Ω+νm,−→q ⊥





 2ωm,−→

q ⊥

We obtain equations dispersion describe interaction between confined acoustic phonon and confined optical phonon in quantum wells:



ω2− ω2m,−→q

⊥− 2

~ X

n,n 0

In,nm 0

2

C− →mq

⊥

2

ωm,−→q

⊥Π0 m, −→q⊥, ω





×



(ω + N Ω)2− ν2m,−→q

⊥−2

~ X

n,n 0

In,nm 0

2

Dm− →q

⊥

2

νm,−→q

⊥Π0 m, −→q⊥, ω + N Ω





= 4

~2

X

n,n 0

In,nm 0

4

C− →mq

⊥

2

D− →mq

⊥

2

ωm,−→q

⊥νm,−→q

⊥

∞ X

s=−∞

ΠN m, −→q⊥, ω
ΠN m, −→q⊥, ω + N Ω

(14)

In Eq (14), the first terms describe the interaction between phonons that belong to the same kind (acoustic - acoustic phonon or optical - optical phonon) while the second terms describe interaction between phonons that belong to different kinds (acoustic - optical phonon) We limit our calculation to the case of the first order resonance, ωm,−→q ± νm,−→q =

Ω Because the solution to the general dispersion equation, Eq (14), is complex Here,

we assume that the electron - phonon interactions satisfy the condition

In,nm 0

4

C− →mq

⊥

2 ,

D− →mq

⊥

2

 1 approximation can be regarded as 

In,nm 0

4

C− →mq

⊥

2

Dm− →q

⊥

2 zero, and the solution Eq (14) by means of the disturbance, we obtain:



ω2− ω2m,−→q

⊥− 2

~ X

n,n 0

In,nm 0

2

C− →mq

⊥

2

ωm,−→q

⊥Π0 m, −→q⊥, ω





(ω + N Ω)2− ν2m,−→q

⊥− 2

~ X

n,n 0

In,nm 0

2

D− →mq

⊥

2

νm,−→q

⊥Π0 m, −→q⊥, ω + N Ω



= 0 (16)

In these limitations, if we write the dispersion relations for confined acoustic and confined optical phonons as ωac(m, −→q⊥) = ωa+ iτa and νoc(m, −→q⊥) = ω0+ iτ0 with conditions

|ωa|  |τa| and |ω0|  |τ0|, and consider the case of N = 1, we obtain:

ωa≈ ωm,−→q

⊥ +1

~ X

n,n 0

In,nm 0

2

C− →mq

⊥

2 ReΓm,−→q

⊥



ωm,−→q

⊥



(17)

ω0≈ νm,−→q

⊥ +1

~ X

n,n 0

In,nm 0

2

Dm− →q

⊥

2 ReΓm,−→q

⊥



νm,−→q

⊥



(18)

τa= −1

~

X

n,n 0

In,nm 0

2

C− →mq

⊥

2 m3/2f0

2p2πβ~2−→q

⊥ exp



−

βm∗εn,n0



ωm,−→q

⊥



2~−→q2⊥



e−βεnh1 − eβ~ωm,−→q ⊥i

(19)

τ0 = −1

~

X

n,n 0

In,nm 0

2

Dm− →q

⊥

2 m3/2f0

2p2πβ~2−→q

⊥ exp



−

βm∗εn,n0



νm,−→q

⊥



2~−→q2

⊥



e−βεn

h

1 − eβ~νm,−→q ⊥

i

(20)

Λ = λ

~Ω X

n,n 0

In,nm 0

2

C− →mq

⊥D− →mq

⊥ReΓm,−→q

⊥



ωm,−→q

⊥



(21) and

ReΓm,−→q

⊥



ωm,−→q

⊥



= f0m

∗ 2πβA~2

 exp



−βπ

2

~2n02 2m∗L2



− exp



−βπ

2

~2n2 2m∗L2



(22)

A = π

2

~2 n02− n2
2m∗L2 +~

2−→q2

⊥ 2m∗ + ~ωm,−→q

ReΓm,−→q

⊥



νm,−→q

⊥



= f0m

∗ 2πβA1~2

 exp



−βπ

2

~2n02 2m∗L2



− exp



−βπ

2

~2n2 2m∗L2



(24)

A1 = π

2

~2 n02− n2
2m∗L2 +~

2−→q 2

⊥ 2m∗ + ~νm,−→q

εn,n0



ωm,−→q

⊥



= εn− εn0 −~

2−→q2

⊥

εn,n 0



νm,−→q

⊥



= εn− εn0 −~

2−→q2

⊥

We obtain the resonant acoustic phonon modes

ω±±= ωa+1

2

 (υa± υ0) ∆ (q) − i (τa+ τ0) ±

q [(υa∓ υ0) ∆ (q) − i (τa− τ0)]2± Λ2

 (28)

In Eq (27) the signs (±) in the sub-script of ω±(±) correspond to the signs in front of the root and the sings (±) in the superscript of ω(±)± correspond to the other sign pairs These signs depend on the resonance condition νm,−→q

⊥

± ωm,−→q

⊥ = N Ω For instance, the existence of a positive imaginary part of ω+(−) implies a parametric amplification of the confined acoustic phonons ωa and ω0 are the renormalization (by the electron - phonon interaction) frequency of the acoustic phonon and optical phonon; ∆ (q) = q − q0, the distance to the intersection of dispersion curves, q0 being the wave number for which the resonance is satisfied;υa (υ0) is the group velocity of the acoustic (optical) phonon;τa τ0, are electronic decrease constant of the acoustic and optical phonons, β = 1/kBT , kBis the Boltzmann constant, f0is the density of electron In such case that λ  1, the maximal resonance, q = qx (qy = qz=0), we obtain:

F = Imω+−= Im



ωa+1 2



−i (τa− τ0) +

q [− (τa− τ0)]2− Λ2



(29)

The condition for the resonant acoustic phonon modes to have a positive imaginary part

F > 0 so 12



−i (τa− τ0) +

q (τa− τ0)2+ |Λ|2



> 0 leads to |Λ|2 > 4τaτo, Using this condition and Eqs (19) - (21), yields the threshold amplitude for EEF:

Eth= 2m

∗Ω2 e~−→q⊥







ImΓm,−→q

⊥



ωm,−→q

⊥

 ImΓm,−→q

⊥



νm,−→q

⊥



h ReΓm,−→q

⊥



ωm,−→q

⊥

i2







1

(30)

ImΓm,−→q

⊥



ωm,−→q

⊥



∗f0 2π~3q⊥

r 2m∗π

βm∗A 2~2 −→q

⊥

2

!

× exp



−βπ

2

~2n2 2m∗L2

 exp β~ωm,−

→q

⊥

2

!

sh β~ωm,−

→q

⊥

2

!

(31)

ImΓm,−→q

⊥



νm,−→q

⊥



∗f0 2π~3q⊥

r 2m∗π

βm∗A21 2~2 −→q

⊥

2

!

× exp



−βπ

2

~2n2 2m∗L2

 exp β~νm,−

→q

⊥

2

!

sh β~νm,−

→q

⊥

2

!

(32)

Equation (30) means that parametric amplification of the confined acoustic phonons is achieved when the amplitude of the EEF is higher than some threshold amplitude and easy to come back to the case of unconfined phonons [7] when m → 0

III THE PARAMETRIC TRANSFORMATION OF CONFINED

ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN

QUANTUM WELLS Parametric transformation of confined acoustic phonons and confined optical phonons

in quantum well is determined by the formula:

KN =

Cm,−→q

⊥



νm,−→q

⊥



Bm,−→q

⊥



ωm,−→q

⊥



Cm,−→q

⊥



νm,−→q

⊥

 are determined from Eq (13) Using the parametric resonant conditions

ωm,−→q

⊥+ N Ω ≈ νm,−→q

⊥, the parametric transformation coefficient is obtained:

KN =

1

~ P n,n 0

Im n,n 0

2

C−→m q

⊥

Dm−→ q

⊥

ΠNm, −→q⊥, ωm,−→q

⊥



δ + iγ0

(33)

where, the quantity δ is infinitesimal Consider the case of N = 1 and note |δ|  γ0, we get:

K1 =

Γ 2γ0

(34) with

Γ = λ

~Ω X

n,n 0

In,nm 0

2

C− →mq

⊥

D− →mq

⊥

ReΓm,−→q

⊥



ωm,−→q

⊥



(35)

γ0= −1

~ X

n,n 0

In,nm 0

2

Dm− →q

⊥

2 ImΓm,−→q

⊥



νm,−→q

⊥



(36)

Where, ReΓm,−→q

⊥



ωm,−→q

⊥

 and ImΓm,−→q

⊥



νm,−→q

⊥

 are determined by the formula (22), and (32) γ0 is the decline electronic constant of the optical phonon Equation (34) means that parametric transformation coefficient of confined acoustic phonons and confined opti-cal phonons in quantum well is achieved when the amplitude of the EEF is higher When

m → 0, easy to come back to the case of unconfined phonons, is determined by the formula:

K =

Γ∗ 2γ∗

(37) Where,

Γ∗ = λ

~Ω X

n,n 0

D −→q⊥
C −→q⊥
ReΓ− →q

⊥



ω− →q

⊥



(38)

γ∗ = −1

~ X

n,n 0

D −→q ⊥

2 ImΓ− →q

⊥



ν− →q

⊥



(39)

ReΓ− →q

⊥



ω− →q

⊥



= f0m

∗ 2πβA2~2

 exp



−βπ

2

~2n02 2m∗L2



− exp



−βπ

2

~2n2 2m∗L2



(40)

ImΓ− →q

⊥



ω− →q

⊥



∗3/ 2f0

2√2πβ~2q⊥

exp −βm

∗A22 2~2−→q2

⊥

! exp



−β~2π2A22 2m∗L2





1 − expβ~ω− →q

⊥



(41)

A2 = π

2

~2 n02− n2
2m∗L2 + ~

2−→q2

⊥ 2m∗ + ~ω− →q

IV NUMERICAL RESULTS AND DISCUSSIONS IV.1 In the case parametric resonance

In order to clarify the mechanism for parametric resonance of acoustic and optical phonons in the case of confined phonons, we consider a AlAs/GaAsAl quantum well The parameters used in this calculation are as follows [12]: χ∞ = 10.9, χ0 = 12.9, L = 100A0, m = 0.067m0, (m0 being the mass of free electron), ~ν0 = 36.25mev, Ω = 2 ×

1014Hz, ξ = 13.5ev ρ = 5.32g.cm−3υs= 5370m.s−1, E0= 106v/m, e = 1.60219×10−19C,

~ = 1.05459 × 10−34J.s

Figure 1 show the dependence of the threshold amplitude Eth on the magnitude of wave vector −→q at temperature T = 72K As shown in, the threshold amplitude reaches

Fig 1 The dependence of E th v.cm−1
on the q m −1
with T = 72K

Fig 2 The dependence of E th v.cm−1
on the T with q = 2.8 × 10 8 m−1

the maximum value when q = 1.2 × 108 m−1
Other cases of unconfined phonon, the curve has a sub-maximal when q = 2.5 × 108 m−1
The cause of this difference is due to the wave vector of phonon quantum chemical confined phonon Because the wave vector

of phonon is quantized of the energy in the confined phonon direction

Figure 2 (solid line - confined and dot line - unconfined) show the dependence of the threshold amplitude on the temperature T for both the confined phonon and unconfined phonon From the graph shows, at the same temperature, the confined phonons makes the threshold amplitude increases

IV.2 In the case parametric transformation

In order to clarify the mechanism for the parametric transformation of acoustic and optical phonons in the case of confined phonons, in this section, we will consider quantum wells The parameters used in this calculation are as follow [12]:χ∞= 10.9, χ0= 12.9, L = 100A0, m = 0.067m0, (m0 being the mass of free electron), ~ν0 = 36.25mev,

Ω = 2 × 1014Hz, ξ = 13.5ev ρ = 5.32g.cm−3 υs = 5370m.s−1, E0 = 106v/m, e = 1.60219 × 10−19C, ~ = 1.05459 × 10−34J.s

Figure 3, and Figure 4 shows the influence of confined phonon on the changing phe-nomenon of the parameter between the acoustic phonon and optical phonon Concretely,

the confinement of phonon makes an increase of the coefficient-changing parameter be-tween the acoustic phonon and optical phonon in quantum well In a same range of temperature T, (in the case confined phonon) the coefficient oscillation around the unit, with the case unconfined phonon the threshold amplitude is very small Because, when phonon is confined, the energy bands of phonon are divided into mini-bands like electrons

in potential well Therefore, the probability of occurrence greater resonance conditions

In other words, the chance of changing acoustic phonon into optical phonon and vice versa becomes bigger In short, the coefficient of parametric transformation between acoustic phonon and optical when phonon is confined is more stronger than unconfined phonon From all figures above, we can see clearly the effect of confined phonons on the parameter transformation coefficient Namely, the confined phonons increase the phonon transfor-mation coefficient in quantum wells

Fig 3 The dependence of K1on the T (In the case confined phonon)

V CONCLUSIONS

In this paper, we analytically investigated the possibility of parametric transforma-tion and parametric resonance of confined acoustic phonons and confined optical phonons

We obtained a general dispersion equation for parametric amplification and transformation

Fig 4 The dependence of K 1 on the T (In the case unconfined phonon)

we use the general quantum. .. com-mutative relations of the creation and the annihilation operators, we obtain the quantum kinetic equation for confined acoustic (confined optical) phonon in quantum wells:

∂

∂t... function [11] for the confined acoustic (confined optical) phonons, Dbm,−→q

⊥

E

t and Dcm,−→q