The photostimulated quantum effect in rectangular quantum wire with an infinite potential for the case of electron-acoustic phonon scattering
TẠP CHÍ KHOA HỌC SỐ 20/2017
67
THE PHOTOSTIMULATED QUANTUM EFFECT IN RECTANGULAR
QUANTUM WIRE WITH AN INFINITE POTENTIAL FOR THE CASE
OF ELECTRON-ACOUSTIC PHONON SCATTERING
Nguyen Vu Nhan1, Hoang Dinh Trien2, Hoang Van Ngoc3
1Hanoi Metropolitan University
2The University of Da Nang
3Hanoi University of Sciences, Hanoi National University
Abstract: Based on the quantum kinetic equation for electrons under the action of a linearly
polarized electromagnetic wave, a dc electric field and an intense laser field, analytic
expressions for the density of the direct current in rectangular quantum wire with an
infinite potential for the case of electron - acoustic phonon scattering are calculated. The
current density is studied as a function of the frequency of the laser radiation field, the
frequency of the linearly polarized electromagnetic wave, the temperature of system and
the size of quantum wire. The analytic expressions are numerically evaluated and plotted
for a specific quantum wire, GaAs/AlGaAs. All these results of quantum wire are compared
with bulk semiconductors and superlattices to show the differences.
Keywords: Semiconductors, quantum wells, quantum wires, superlattices and quantum dot.
Email: nvnhan@daihocthudo.edu.vn
Received 02 December 2017
Accepted for publication 25 December 2017
1. INTRODUCTION
The photostimulated quantum effect by electromagnetic wave is explained by the
possibility of using this phenomenon for detecting intense electromagnetic radiation, as well
as for characterizing kinetic properties of semiconductors [1]. It is known that the presence
of intense laser radiation can influence the electrical conductivity and kinetic effects in
material. In recent years, it has been revealed that the photostimulated quantum effect in
superlattices and in quantum wells should be characterized by new features under the action
of strong fields [2-4]. However, in quantum wire, the photostimulated quantum effect still
opens for studying.
68
TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
In this work, we use the quantum kinetic to study the drag of charge carriers in
rectangular quantum wire with an infinite potential by a linearly polarized electromagnetic,
a dc electric field and a laser field. We obtained the density of the current for the case
electrons interacting with acoustic phonon.
2. CALCULATING THE DENSITY OF THE CURRENT BY THE QUANTUM
KINETIC EQUATION METHOD
We examine the electron system, which is placed in a linearly polarized electromagnetic
it
E(t) E(e eit ),H(t) n,E(t)
wave (
), in a dc electric field
and in a strong radiation field
E0
The Hamiltonian of the electron - phonon system in the quantum wire can be
F(t) Fsin t.
written as:
e
H = H0 + U =
C .I
+
b b
(pz A(t)).an,l,p .an,l,p
n,l,pz
q
q
q
z
z
c
n,l,pz
q
+
(q)an ,l ,p q .an,l,p (bq bq )
,
(1)
n,l,n ,l
q
s
z
n,l,n ,l pz ,q
where A t is the vector potential of laser field (only the laser field affects the probability
0
1
of scattering):
; a and an,l,p (bq and bq ) are the creation and
A(t) F sin t
n,l,pz
z
c
q
C
annihilation operators of electron (phonon);
is the frequency of acoustic phonon; q is
2q
2vsV
Cq2
the electron-acoustic phonon interaction constant:
, here V, , vs and
are
volume, the density, the acoustic velocity and the deformation potential constant; In',l',n,l (q)
is form factor.
The electron energy takes the simple:
p2z 22
n
l2
2
n,l,p
(
,
).
n 0, 1,2,... l 1, 2,3,...
2
2
z
2m 2m Lx Ly
In order to establish the quantum kinetic equations for electrons in quantum wire, we
use general quantum equations for the particle number operator or electron distribution
function:
fn,l,p (t)
z
i
an,l,p an,l,p ,H t
,
(2)
z
z
t
TẠP CHÍ KHOA HỌC SỐ 20/2017
69
with fn,l,p (t) an,l,p an,l,p t is distribution function. From Eqs. (1) and (2), we obtain the
z
z
z
quantum kinetic equation for electrons in quantum wire (after supplement: a linearly
polarized electromagnetic wave field and a direct electric field E0 ):
fn,,lp (t)
fn,l,p (t)
z
z
e.E(t)e.E p ,h(t)
0
c
z
t
pz
2
2
eE0q
(q) . J2 (
)Nq
f
n ,l ,p q (t)fn,l,p (t) .(
n,,lp L)
D
n,l,n ,l
L
n ,l ,pz qz
m2
z
z
z
n ,l ,q
L
fn ,l ,p q (t) fn,l,p (t)
n,l,p L
(3)
n ,l ,pz qz
z
z
z
z
h
eE0q
H
H
J (
)
is the Bessel function
where
is the unit vector in the magnetic field direction,
L
m2
Nq
of real argument;
is the time-independent component of distribution function of phonon:
k T ; c is the cyclotron frequency, () is the relaxation time of electrons with
B
Nq
vsqz
energy .
For simplicity, we limit the problem to the case of
We multiply both sides Eq.
l 0, 1.
(e / m)p (
)
are carry out the summation over n, l and
(2) by
p
z , we obtained:
z
n,p
1
(i
(i
)R Q() S() R (),h ,
(4)
(5)
(6)
c
0
()
*
*
1
)R Q() S () R (),h ,
c
0
()
*
R0 ()
Q0 () S0 () c R() R (),h ,
()
with:
e
R()
p f (p )
,
(7)
(8)
(9)
z
1
z
n,l,pz
m
n,l,pz
e2E
m2k T n,l,p
p2f (
)
,
Q()
z
0
n,l,pz
n,l,pz
B
z
e2E0
m2k T n,l,p
p2f (
)
,
Q0 ()
z
0
n,l,pz
n,l,pz
B
z
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TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
e2F2q2z
2
2e
m
S0 ()
C2q In,l,n',l' (q) Nqqzf10 (pz )
4m24
n,l,n',l',pz ,qz
(
n,l,p q ) (n',l',p q n,l,p q )
n',l',pz qz
z
z
z
z
(10)
(n',l',p q n,l,p q ) (n',l',p q n,l,p )
q
z
z
z
z
z
z
( n,l,p ),
z
e
n,l,pz
f0
f0 n0 exp(
)
; n0 is particle
E
f10 (pz ) pz
where
;
;
0
0
n,l,pz
0
kBT
m
n,l,pz
density; kB is Boltzmann constant; T is temperature of system.
e2F2q2z
4m24
2
2e
m
S()
Cq2 In,l,n',l' (q) Nqqzf1(pz )
n,l,n',l',pz ,qz
(
n,l,p q ) (n',l',p q n,l,p q )
n',l',pz qz
z
z
z
z
(11)
(n',l',p q n,l,p q ) (n',l',p q n,l,p )
q
z
z
z
z
z
z
( n,l,p ),
z
E
f0
e
n,l,pz
f1 (pz ) pz
with
;
.
m
1i
n,l,pz
n,l,pz
Solving the equation system (4), (5), (6), we obtain:
2c2 ()
1 ()
S,h
2
Q,h 2 ()Re
.
(12)
R0 () ()(Q0 S0 )
c
2
2
1 i
The density of current:
2
2
0
2
1
F
F
F
c
AC D E,h
j R ()d AC D E
,
(13)
(14)
0
0
1 22 1 22
F
0
n e3F22 2
2
2
n
l2
F
0
where A
I2n,l,n',l' exp
,
2
32m4vs22
2m Lx L2y
n,l,n',l'
C 4N17/2 12
24
N1
N1
N1
(4,9/2;
)
(3,7/2;
)
(2,5/2;
2m
2m
2m )
(15)
4N72/2 12
24
(2,5/2;
,
N2
N2
N2
(4,9/2;
)
(3,7/2;
)
2m
2m
2m )
TẠP CHÍ KHOA HỌC SỐ 20/2017
71
(16)
(17)
(18)
2
2
2m
22
2mLy
2
2
2
2
N1
(n n )
(l l ) ,
2 2mL
2
x
2
2
2
2m
22
2
2
2
2
N2
n02e2
(n n )
(l l ) ,
2 2mL
2mLy
2
x
2
2
2
2
n
l2
2
2
D
exp
,
F
4m2k T 2m
2m Lx Ly
2
n,l
B
1
(a,b,z)
ezx xa1(1 ax)ba1dx
is the Hypergeometrix function.
(a)
0
We obtain the expressions for the current density j0 , and select: E 0x ; h 0y :
j AC D E
0x ; j AC D E
(19)
(20)
0x
0y
0y
2
2
2
1
F
F
F
c
j AC D E
AC D E
0z
0z
1 22 1 22
F
Equation (13) shows the dependent of the direct current density on the frequency of
the laser radiation field, the frequency of the linearly polarized electromagnetic wave, the
size of the wire. We also see the dependence of the constant current density on characteristic
parameters for quantum wire such as: wave function; energy spectrum; form factor In,l,n’,l’
and potential barrier, that is the difference between the quantum wire, superlattices, quantum
wells, and bulk semiconductors.
3. NUMERICAL RESULTS AND DISCUSSION
j0z
In this section, we will survey, plot and discuss the expressions for
for the case of a
specific GaAs/GaAsAl quantum wire. The parameters used in the calculations are as follows
[2-12]: m = 0,0665m0 (m0 is the mass of free electron); F = 50meV; (F ) 10-11s-1;
n0 1023 m3 ; 5.3103 kg / m3 ; 2.2108 J ; E = 106 V/m; E0 = 5.106V/m; F = 105N.
j0z
Fig.1 shows the dependence of
on the frequency of the intense laser radiation.
From these figure, we can see the nonlinear dependence of j0z on the frequency of the
intense laser radiation, when the frequency of the intense laser radiation increases joz
increases.
72
TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
j0z
Fig.1. The dependence of jz on the frequency of
the laser radiation with different values of T
j0z
Fig.2. The dependence of jz on the frequency of
the electromagnetic wave with different values of T
j0z
Fig.2 shows the dependence of
on the frequency of the electromagnetic wave.
From these figure, we can see the nonlinear dependence of j0z on the frequency of the
electromagnetic wave, when the frequency of the electromagnetic wave increases joz
decreases and j0z will have a stable value when có giá trị cỡ 1013 .
Fig.3 shows the dependence of j0z on the size of the wire. From this figure, when radius
increase joz increases, when Lx, Ly continue to increase then j0z will have a stable value.
TẠP CHÍ KHOA HỌC SỐ 20/2017
73
j0z
Fig. 3. The dependence of
on the size of the wire
4. CONCLUSION
In this paper, we have studied the photostimulated quantum effect in rectangular
quantum wire with a infinite potential for the case of electron – acoustic phonon scattering.
In this case, one dimensional electron systems is placed in a linearly polarized
electromagnetic wave, a dc electric field and a laser radiation field at high frequency. We
obtain the expressions for current density vector j0 , in which, plot and discuss the
j
j0z
j0z
show the dependence of on the frequency
expressions for 0z . The expressions of
of the linearly polarized electromagnetic wave, on the size of the wire, the frequency of
the intense laser radiation; and on the basic elements of quantum wire with a infinite
potential. The analytical results are numerically evaluated and plotted for a specific quantum
wire GaAs/AlGaAs.
Acknowledgment: This work was completed with financial support from project
B2016.DNA.25, thanks also basic research program of the Hanoi Metropolitan University.
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TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
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HIỆU ỨNG KÍCH THÍCH QUANG LƯỢNG TỬ TRONG DÂY
LƯỢNG TỬ HÌNH CHỮ NHẬT VỚI HỐ THẾ CAO VÔ HẠN
TRONG TRƯỜNG HỢP TÁN XẠ ELECTRON – PHONON ÂM
Tóm tắt: Thu được biểu thức giải tích cho mật độ dòng điện trên cơ sở phương trình động
lượng tử cho các eletrons dưới ảnh hưởng của các trường sóng điện từ phân cực thẳng,
điện trường không đổi và laze cường độ mạnh trong dây lượng tử hình chữ nhật với hố thế
cao vô hạn trong trường hợp tán xạ electron-phonon âm. Mật độ dòng điện là một hàm số
phụ thuộc vào tần số của laze, tần số của sóng điện từ phân cực thẳng, nhiệt độ hệ và kích
thước của dây lượng tử. Biểu thức giải tích của mật độ dòng được đánh giá số và vẽ đồ thị
cho dây lượng tử đặcbiệt GaAs/AlGaAs. Các kết quả nhận được trong dây lượng tử được
so sánh với các kết quả tương ứng trong bán dẫn khối và siêu mạng cho thấy sự khác biệt.
Từ khóa: Bán dẫn, hố lượng tử, dây lượng tử, siêu mạng, chấm lượng tử.
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