THE PARAMETRIC TRANSFORMATION COEFFICIENT OF CONFINED ACOUSTIC AND CONFINED OPTICAL PHONONS IN THE RECTANGULAR QUANTUM WIRE LUONG VAN TUNG Department of Physics, Dong Thap University, 78
Trang 1THE PARAMETRIC TRANSFORMATION COEFFICIENT OF CONFINED ACOUSTIC AND CONFINED OPTICAL PHONONS
IN THE RECTANGULAR QUANTUM WIRE
LUONG VAN TUNG Department of Physics, Dong Thap University,
783 Pham Huu Lau, Cao Lanh, Dong Thap NGO THI THANH HA, NGUYEN THI THANH NHAN,
LE THI THU PHUONG, NGUYEN QUANG BAU Department of Physics, Hanoi National University, 334 Nguyen Trai, Hanoi
Abstract The parametric transformation of confined acoustic and confined optical phonons in the rectangular quantum wire is theoretically studied by using a set of quantum kinetic equations for phonons The analytic expression of parametersq transformation coefficient of confined acous-tic and confined opacous-tical phonons in the rectangular quantum wire is obtained The dependence
of the parametric transformation coefficient on the temperature T and parameters of the rectan-gular quantum wire is numerically evaluated, plotted and discussed for a specific quantum wire GaAs/GaAsAl All the results are compared with those for the unconfined phonons to show the difference.
I INTRODUCTION
It is well known that in presence of an external electromagnetic field, an electron gas becomes non-stationary When the conditions of parametric resonance are satisfied, para-metric resonance and tranformation (PRT) of same kinds of excitations such as phonon-phonon, plasmon-plasmon, or of different of excitations, such as plasmon-phonon will arise i.e , the energy exchange process between these excitations will occur [1-9] The physical picture can be described as follows: due to the electron-phonon interaction, propaga-tion of an acoutic phonon with a frequency ω~ accompanied by a density wave with the same frequency Ω When an external electromagnetic field with frequency is presented,
a charge density waves (CDW) with a combination frequency ν~± lΩ (l =1, 2, 3, ) will appear If among CDW there exits a certain wave having a frewuency which coincides,
or approximately coincides, with the frequency of optical phonon ν~, optical phonons will appear These optical phonons cause a CDW with a combination frequency of ν~± lΩ, and when ν~± lΩ ∼= ω~, a certain CDW causes the acoutic phonons mention above The PRT can speed up the damping 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 interactions and tranformation of unconfined acoustic and unconfined optical phonons has been considered in bulk semiconductors [1-5], for low-dimensional semiconductors (doped superlattices, quantum wells, quantum wire), the dependence of the parametric tranformation coefficient of unconfined acoustic and un-confined opitical phonons on temperature T [9] In order to improve the PRT theoretics
Trang 2for low-dimensional semiconductors, we, in the paper, examine dependence of the para-metric transformation coefficient of confined acoustic and confined optical phonons in the rectangular quantum wire
II THE QUANTUM KINECTIC EQUATION FOR PHONONS
We use model for the rectangular quantum wire with electron gas is confined on the 0xy plane and electron is free along the 0z direction If laser field ~E(t) = ~E0sin(Ωt) irradiates the sample in direction which are along the 0z axis, the electromagnatic field
of laser wave will polarize parallels the z axis and its strength is expressed as a vector potenal ~A(t) = ΩcE~0cos(Ωt) (c is the light velocity, Ω is EMW frequency, ~E0 is amplitude
of the laser field)
The Hamiltonian of the electron-confined acoustic phonon-confined optical phonon system in the rectangular quantum wire can be written as (in this paper ~ = 1):
n,l,k z
εn,l
~kz−e c
~ A(t)
a+
m 1 ,m 2 ,q z
ωm1,m2,qzb+m1,m2,qzbm1,m2,qz+ X
m 1 ,m 2 ,q z
νm1,m2,qzc+m1,m2,qzcm1,m2,qz
m 1 ,m 2 ,q z
X
n,l,n 0 ,l 0 ,k z
Cm1 ,m 2
~ z In,l,n0 ,l 0a+n0 ,l 0 ,k z +q zan,l,kz(bm1,m2,qz+ b+m1,m2,−qz)
m 1 ,m 2 ,q z
X
n,l,n 0 ,l 0 ,k z
Dm1 ,m 2
q z In,l,n0 ,l 0a+n0 ,l 0 ,k z +q zan,l,kz(cm1,m2,qz + c+m1,m2,−qz)
Where εn,l ~kz − e
cA(t)~
is energy spectrum of an electron in external electromagnetic filed, a+n,l,kzan,l,k z is the creation (annihilation) operater of an electron for state |n, l, kzi,
b+m1,m2,qz, bm 1 ,m 2 ,q z (c+m1,m2,qz, cm 1 ,m 2 ,q z) is the creation operator and annihilation operator
of an confined acoustic (optical) phonon for state |m1, m2, qzi, m1, m2 are the index con-fined The electron- confined acoustic and optical phonon interaction coefficients take the form [10]:
Cm1 ,m 2
q z
2
2
2ρvsV
s
q2+
m1π
Lx
2
+
m1π
Ly
2
Dm1 ,m 2
q z
2
2
~ω0
2V0ε0
1
χ∞
− 1
χ0
qz2+
m1π
Lx
2
+
m2π
Ly
2−1
Here V, ρ, vs and ξ are the volume, the density, the acoustic velocity and the deformation potential constant, respectively, ε0 is the electronic constant, χ∞, χ0 are the static and high-frequency dielectric constants, respectively, e is the charge of the electron
Trang 3The electronic form factor, In,l,n0 ,l 0 is written as [11]:
4(qxLxnn0)21 − (−1)n+n0cos(qxLx)
(qxLx)4− 2π2(qxLx)2(n2+ n02) + π4(n2− n02)22 (3)
4(qyLyll0)21 − (−1)nl+l0cos(qyLy)
(qyLy)4− 2π2(qyLy)2(l2+ l02) + π4(l2− l02)22
Here, n, n0 is the position of quantum, l, l0 is the radial quantum number, Lx(Ly) is width (length ) of the rectangular quantum wire, qx, qy is wave vector
Energy spectrum of electron in the rectangular quantum wire [12]
εn,l(kz) = k
2 z
2m∗ +
π2
2m∗
n2
L2 x
+ l
2
L2
Here, m∗ is the effective mass of the electron
In order to establish a set of quantum kinetic equations for confined acoustic and confined optical phonons, we use equation of motion of statistical average value for phonons
i∂
∂t m1 ,m 2 ,q z
t = m 1 ,m 2 ,q z, H(t)
t; i∂
∂t m1 ,m 2 ,q z
t = m 1 ,m 2 ,q z, H(t)
t (5)
Where
t is means the usual thermodynamic average of operator X
Using Hamiltonian in Eq.(1) and realizing operator algebraic calculations, we obtain
a set of coupled quantum kinetic equations for phonons The equation for the confined acoustic phonons can be formulated as
∂
∂t m1 ,m 2 ,q z
t + iωm1,m2,qz m1,m2,qz
t = −1
~2 X
n,l,n 0 ,l 0 ,k z
+∞
X
ν,µ=−∞
(6)
× In,l,n0,l0
2
Jν
λ Ω
Jµ
λ Ω
h
fn0 ,l 0(kz− qz) − fn,l(kz)
i
×
Z t
−∞
dt1expni
~εn,l(kz) − εn0 ,l 0(kz− qz)(t1− t) − iνΩt1+ iµΩto
× n
Cm1 ,m 2
q z
2h
hbm1,m2,qzit1+ hb+m1,m2,−qzit1i + Cm1 ,m 2
−~ q Dm1 ,m 2
q z
h
hcm1,m2,−qzit1 + hc+m1,m2,−qzit1io
A similar equation for the optical phonons can be obtained in which hcm 1 ,m 2 ,q zit, hbm 1 ,m 2 ,q zit,
νm1,m2,qz, ωm1,m2,qz, Cm1 ,m 2
q z , Dm1 ,m 2
q z are replaced by hbm1,m2,qzit, hcm1,m2,qzit, ωm1,m2,qz,
νm 1 ,m 2 ,q z, Dm1 ,m 2
q z , Cm1 ,m 2
q z , respectively
In Eq.(6) fn,l(~k) is the distribution function of electrons in the state |n, l, ~ki, Jµ Ωλ
is the Bessel function, and λ = e ~E0 ~ z
mΩ
Trang 4III THE PARAMETRIC TRANSFORMATION COEFFICIENT OF ACOUSTIC AND OPTICAL PHONON IN RECTANGULAR
QUANTUM WIRE
In order to establish the parametric transformation coefficient of confined acoustic and confined optical phonon, we use standard Fourier transform techniques for statistical average value of phonon operators: hbm1,m2,qzit, hb+m1,m2,qzit, hcm1,m2,qzit, hc+m1,m2,qzit The Fourier transforms take the form:
Ψ~(ω) =
Z +∞
−∞
hΨ~iteiωtdt; hΨ~it = 1
2π
Z +∞
−∞
Ψ~(ω)e−iωtdω
One finds that the final result consists of coupled equations for the Fourier transformations
Cm1,m2,~q(ω) and Bm1,m2,qz(ω) of hcm1,m2,qzit and hbm1,m2,qzit
For instance, the equation for Cm 1 ,m 2 ,q z(ω) can be written as:
(ω − ωm 1 ,m 2 ,q z)Cm 1 ,m 2 ,q z(ω) =
n,l,n 0 ,l 0
|In,l,n0 ,l 0|2|Dm1 ,m 2
q z |2νm 1 ,m 2 ,q z
Cm 1 ,m 2 ,q z(ω)
ω + νm1,m2,qzΠ0(m1, m2, qz, ω)
n,l,n 0 ,l 0
|In,l,n0 ,l 0|2Cm1 ,m 2
q z Dm1 ,m 2
q z νm 1 ,m 2 ,q z
∞
X
s=−∞
Bm1,m2,qz(ω − sΩ)
ω − sΩ + ωm 1 ,m 2 ,q z
Πs(m1, m2, qz, ω)
(7)
In the similar equation for Bm1,m2,qz(ω), functions such as Cm1,m2,qz(ω), Cm1,m2,qz(ω−sΩ),
Bm 1 ,m 2 ,q z(ω − sΩ), νm 1 ,m 2 ,q z, ωm 1 ,m 2 ,q z, Cm1 ,m 2
q z , Dm1 ,m 2
q z are replaced by Bm 1 ,m 2 ,q z(ω),
Bm1,m2,qz(ω − sΩ),Cm1,m2,qz(ω − sΩ), ωm1,m2,qz, νm1,m2,qz, Dm1 ,m 2
q z , Cm1 ,m 2
q z , respectively
In Eq (7), we have:
Πs(m1, m2, qz, ω) =
+∞
X
v=−∞
Jv λ
ΩJv+s λ
ΩΓm1,m2,qz(ω + vΩ), (8)
Γm 1 ,m 2 ,q z = X
~ k
fn,l(kz) − fn0 ,l 0(kz− qz)
εn,l(kz) − εn0 ,l 0(kz− qz) − hvΩ − ~ω − i~δ. (9)
Where, the quantity δ is infinitesimal and appears due to the assumption of an adiabatic interaction of the electromagnetic wave (EMW)
In Eq (8), the first term on the right-hand side is significant just in case s = 0
If not, it will contribute more than second order of electron-phonon interaction constant Therefore, we have
Trang 5(ω − ωm 1 ,m 2 ,q z)Cm 1 ,m 2 ,q z(ω) =
n,l,n 0 ,l 0
|In,l,n0 ,l 0|2|Dm 1 ,m 2
q z |2νm1,m2,qz Cm1 ,m 2 ,q z(ω)
ω + νm 1 ,m 2 ,q z
Π0(m1, m2, qz, ω)
n,l,n 0 ,l 0
|In,l,n0 ,l 0|2Cm1 ,m 2
q z Dm1 ,m 2
q z νm 1 ,m 2 ,q z
∞
X
s=−∞
Bm 1 ,m 2 ,q z(ω − sΩ)
ω − sΩ + ωm1,m2,qzΠs(m1, m2, qz, ω).
(10) Transforming Eq (10) and using the parametric resonant condition ωm1,m2,qz + N Ω ≈
νm 1 ,m 2 ,q z, the parametric transformation coefficient is obtained:
Cm1,m2,qz(ν0)
Bm 1 ,m 2 ,q z(ωm 1 ,m 2 ,q z) =
P
n,l,n 0 ,l 0 ,k z|In,l,n0 ,l 0|2Cm1 ,m 2
−q z Dm1 ,m 2
−q z Π−1(m1, m2, qz, ωm1,m2,qz)
δ − iP
n,l,n 0 ,l 0|In,l,n0 ,l 0|2|Dm1 ,m 2
q z |2ImΠ0(m1, m2, qz, νm 1 ,m 2 ,q z)
(11) Consider the case of N = 1 and assign
γ0 = X
n,l,n 0 ,l 0
|In, l, n0, l0|2|Dm 1 ,m 2
q z |2ImΠ0(m1, m2, qz, νm1,m2,qz) (12)
Note that δ γ0, we have
K1 =
P
n,l,n 0 ,l 0 ,~ k|In,l,n0 ,l|2Cm1 ,m 2
−q z Dm1 ,m 2
q z Π−1(m1, m2, qz, ωm1,m2,qz)
Using Bessel function, Fermi-Dirac distribution function for electron and energy spectrum
of electron in Eq (4), we have:
|K1| =
Γ 2γ0
(14) Where
n,l,n 0 ,l 0
|In,l,n0 ,l|2Cm1 ,m 2
q z Dm1 ,m 2
q z
λ
ΩReΓm1 ,m 2 ,q z(ωm1,m2,qz) (15)
ReΓm 1 ,m 2 ,q z(νm 1 ,m 2 ,q z) =
Lf 0
2π
q
2m ∗ π
β exph− β π 2
2m ∗
n 2
L 2
x +Ll22 y
i
π 2
2m ∗
n 02 −n 2
L 2
x +l02L−l22
y
+2mq2∗+ νm 1 ,m 2 ,q z
×nexp
h
− β π
2
2m∗
n02− n2
L2 x
+l
02− l2
L2
i
ImΓm1,m2,qz(νm1,m2,qz) =
−Lm
∗f0
qz exp
n β
h
−m
∗A2 2q2 − π
2
2m∗
n2
L2 x
+ l
2
L2
+νm1 ,m 2 ,q z
2
io sh
hβνm 1 ,m 2 ,q z
2
i (17)
A = π
2
2m∗
n02− n2
L2 x
+l
02− l2
L2
+ q
2 z
2m∗ + ωm1 ,m 2 ,q z (18)
Trang 6In Eqs.(16) and (17), β = 1/(kBT ) (kB is Boltzmann constant), L is depth of the rectan-gular quantum wire, f0 is the electron density in rectangular quantum wire
When the index confined m1, m2to tend to 0, parametric transformation coefficient
of confined acoustic and confined optical phonon in rectangular quantum wire be the same
as parametric transformation coefficient of unconfined acoustic and unconfined optical phonon[9]
K1is analytic expression of parametric transformation coefficient of confined acous-tic and confined opacous-tical phonon in rectangular quantum wire when the parametric resonant condition ωm 1 ,m 2 ,q z + N Ω ' νm 1 ,m 2 ,q z is satisfied
IV NUMERICAL RESULTS AND DISCUSSIONS
In order to clarify the mechanism for parametric transformation coefficient of con-fined acoustic and concon-fined optical phonon in rectangular quantum wire, in this section
we perform numerical computations and graph for GaAs/GaAsAl: be quantum wire The parameters used in the calculation [6,7]: ξ = 13, 5eV, vs=5378m/s, χ∞=10.9, χ0=12.9, ρ=5.32g/cm3, m∗=0.67×9.1 × 10−31kg, ~ω0 = 36.25eV, E0=106V/m, qz = 2 × 1051/m,
L = 10−7m, f0= 1023m−1, e = 1.60219 × 10−19C, ~ = 1.05459 × 10−34Js
0
5
10
15
20
25
Temperature T(K)
Photon Energy=36.2 meV Photon Energy=37.5 meV
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Temperature T (K)
Photon Energy=36.2 meV Photon Energy=37.5 meV
Fig 1 Dependence of K1 on T when confined phonon (left) and unconfined
phonon (right).
Fig 1 shows the parametric transformation coefficient K1 as a function of temper-ature T It is seen that the parametric transformation coefficient of acoustic and optical phonons in rectangular quantum wire depends non-linearly on temperature T The results are compared with those for the unconfined phonons to show the bigger When the tem-perature increases 250K : 350K, the parametric transformation coefficient K1 reduced Fig 2 shows the parametric transformation coefficient K1 as a function of width Lx of the rectangular quantum wire It is seen that, when width Lx less than about 20nm or width Lx greater than about 70nm, the parametric transformation coefficient of acous-tic and opacous-tical phonons in rectangular quantum wire independents on width Lx When 20nm < Lx< 70nm, with Ly = 40nm, the parametric transformation coefficient of acous-tic and opacous-tical phonons in rectangular quantum wire is some maximum value The results are compared with those for the unconfined phonons to show the bigger
Trang 70 0.2 0.4 0.6 0.8 1
x 10−7 0
1
2
3
4
Lx (m)
Photon Energy=39.5 meV Photon Energy=13.2 meV
x 10−7 0
0.2 0.4 0.6 0.8
Lx (m)
Photon Energy=39.5 meV Photon Energy=13.2 meV
phonon (right).
V CONCLUSION
In this paper, we obtain analytic expression of the parametric transformation co-efficient of acoustic and optical phonons in rectangular quantum wire in presence of an external electromagnetic field K1, Eqs (14)-(18) It is seen that K1 depends on tem-perature T and parametric of rectangular quantum wire Numerical computations and graph are performed for GaAs /GaAsAl be quantum wire Fig 1 shows the parametric transformation coefficient K1 depends non-linearly on temperature T Fig 2 shows, when 20nm < Lx< 70nm, with Ly = 40nm, the parametric transformation coefficient of acous-tic and opacous-tical phonons in rectangular quantum wire is some maximum value The results are compared with those for the unconfined phonons to show the bigger
ACKNOWLEDGMENT This work is completed with financial support from the Program of Basic Research
in Natural Sciences, NAFOSTED (103.01.18.09)
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Received 10-10-2010
... density in rectangular quantum wireWhen the index confined m1, m2to tend to 0, parametric transformation coefficient
of confined acoustic and confined optical. .. the parametric transformation coefficient of acoustic and optical phonons in rectangular quantum wire depends non-linearly on temperature T The results are compared with those for the unconfined... m∗ is the effective mass of the electron
In order to establish a set of quantum kinetic equations for confined acoustic and confined optical phonons, we use equation of motion of statistical