THE PHOTOSTIMULATED QUANTUM EFFECT IN RECTANGULAR QUANTUM WIRE WITH AN INFINITE POTENTIAL FOR THE CASE OF ELECTRON-ACOUSTIC PHONON SCATTERING Nguyen Vu Nhan 1 , Hoang Dinh Trien 2 , Ho
Trang 1THE PHOTOSTIMULATED QUANTUM EFFECT IN RECTANGULAR QUANTUM WIRE WITH AN INFINITE POTENTIAL FOR THE CASE
OF ELECTRON-ACOUSTIC PHONON SCATTERING
Nguyen Vu Nhan 1 , Hoang Dinh Trien 2 , Hoang Van Ngoc 3
1
Hanoi Metropolitan University
2
The University of Da Nang
3 Hanoi 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
Trang 2In 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
E(t) E(e e ), H(t) n, E(t)
), in a dc electric field E 0
and in a strong radiation field
F(t) Fsin t.
The Hamiltonian of the electron - phonon system in the quantum wire can be
written as:
H = H0 + U =
z
e
c
+
z
n ,l,n ,l p ,q
, (1)
where A t
is the vector potential of laser field (only the laser field affects the probability
of scattering): 1A(t) F sin0 t
c
;
z
n,l,p
z
n,l,p
a (bq and bq) are the creation and
annihilation operators of electron (phonon); q is the frequency of acoustic phonon; Cqis
the electron-acoustic phonon interaction constant:
2 2 q s
q C
2 v V
, 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:
z
z
(n 0, 1, 2, , l 1, 2, 3, ).
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:
z
z z
n,l,p
t
Trang 3f (t) a a is distribution function. From Eqs. (1) and (2), we obtain the
quantum kinetic equation for electrons in quantum wire (after supplement: a linearly
polarized electromagnetic wave field and a direct electric field E0
):
z
eE q 2
m
fn ,l ,p zq z(t) f n,l,p z(t) n ,l ,p zq z n,l,p z L
where h H
H
is the unit vector in the magnetic field direction, 0
eE q
m
is the Bessel function
of real argument; Nq is the time-independent component of distribution function of phonon:
B
q
s z
k T
N
v q
; c is the cyclotron frequency, ( )is the relaxation time of electrons with
energy .
For simplicity, we limit the problem to the case of l 0, 1 We multiply both sides Eq.
(2) by ( e / m)p (z n,p )
are carry out the summation over n, l and p z
, we obtained:
1
, (4)
1
, (5)
* 0
R ( )
( )
with:
z
z
n,l,p
e
m
, (7)
z
2
2
2
n ,l,p B
e E
m k T
, (8)
Trang 4
z z
2
n,l,n ',l',p ,q
e F q
2 e
z
n,l,p
)
(10)
where
z
0
n,l,p
f
f (p ) p
e E m
B
k T
; n0 is particle
density; kB is Boltzmann constant; T is temperature of system.
z z
2
n ,l,n ',l ',p ,q
e F q
2 e
z
n ,l,p
)
(11)
with
z
0
n ,l,p
f
f (p ) p
z
z
n,l,p
n ,l,p
e E
.
Solving the equation system (4), (5), (6), we obtain:
2
2 c
S, h
. (12)
The density of current:
2 2
0
, (13)
n,l,n ',l '
n,l,n ',l '
7 / 2
7/ 2
(15)
Trang 52 2 2 2
2m
2m
2
0
F
n ,l
, (18)
(a ,b,z)
(a ) 0
1
We obtain the expressions for the current density j0
, and select: E 0x
; h0y
:
j AC D E ; j 0 y AC D E 0 y (19)
2 2
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
In this section, we will survey, plot and discuss the expressions for j for the case of a 0z 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;
0
n 10 m ; 5.3 10 kg / m 3 3; 2.2 10 J 8 ; E = 106
V/m; E0 = 5.106V/m; F = 105N. Fig.1 shows the dependence of j on the frequency of the intense laser radiation. 0z
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 j
Trang 6
Fig.1 The dependence of j z on the frequency of the laser radiation with different values of T
Fig.2 The dependence of j z on the frequency of the electromagnetic wave with different values of T
Fig.2 shows the dependence of j on the frequency of the electromagnetic wave. 0z
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.
j0z
j0z
Trang 7
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 expressions for j The expressions of 0z j show the dependence of 0z j on the frequency 0z
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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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