Influence of confined phonons on the hall coefficient in a Cylindrycal Quantum Wire with an infinite potential (for electron – acoustic optical phonon scattering)

VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 46-51

Original Article

Influence of Confined Phonons on the Hall Coefficient
in a Cylindrycal Quantum Wire with an Infinite Potential
(for Electron – acoustic Optical Phonon Scattering)
Pham Ngoc Thang1,*, Le Thai Hung2, Do Tuan Long1, Nguyen Quang Bau1
1

Faculty of Physics, VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam
University of Education, Viet Nam National University, 144 Xuan Thuy, Hanoi, Vietnam

2

Received 13 March 2019
Revised 15 June 2019; Accepted 01 July 2019

Abstract: The influence of confined acoustic phonons on the Hall Coefficient (HC) in a Cylindrycal
Quantum Wire (CQW) with an infinite potential (for electron – confined acoustic phonons
⃗ , a constant scattering). Consider a case where CQW is placed in a perpendicular magnetic field 𝐵
electric field ⃗⃗⃗⃗
𝐸1 and an intense electromagnetic wave 𝐸⃗ = ⃗⃗⃗⃗
𝐸0 𝑠𝑖𝑛 Ω𝑡. By using the quantum kinetic
equation for electrons interacting with Confined Optical Phonon (COP), we obtain analytical
expressions for (HC), which are different from in comparison to those obtained for the HC in the
case of normal bulk semiconductor and in the case of cylindrycal quantum wire with electron –
unconfined phonons scattering mechanism. Numerical calculations are also applied for
GaAs/GaAsAl cylindrycal quantum wire, we see the HC depends on magnetic field B, temperature T,
frequency Ω and amplitude E0 of laser radiation and especially quantum index m1 and m2 characterizing
the phonon confinement. This influence is due to the quantum index m1 and m2, which makes an increase

of Hall coefficient by 2,3 times in comparison with the case of unconfined phonons. When the quantum
number m1 and m2 goes to zero, the result is the same as in the case of unconfined phonons.
Keywords: Hall Coefficient, Quantum kinetic equation, Cylindrycal quantum wire, Confined
acoustic phonons.

1. Introduction
In recent years, the study of low – dimensional semiconductor systems has been increasingly
interested, include the electrical, the magnetic and the optical properties. In these systems, the motion
________
Corresponding author.

Email address:
https//doi.org/ 10.25073/2588-1124/vnumap.4333

46


P.N. Thang et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 46-51

47

of carriers is restricted, thus leading to their new properties under the action of external fields for
example: the absorption coefficient of an electromagnetic wave, the Hall effect, the Radioelectric effect,
the Acoustoelectric effect. The Hall effect — the effect of drag of charge carriers caused by the external
magnetic field has been studied extensively [1–3]. There have been study of the Hall effect in bulk
semiconductor in the presence of electromagnetic waves, in which classical theory of Hall effect in bulk
semiconductor when placed in electricity, the magnetic field is perpendicular to the presence of an
electromagnetic wave is built on the basis of Boltzman's classical kinetic equation, while quantum theory
is based on quantum-kinetic equation [4]. In two-dimensional semiconductor systems, there have been
studies on the Hall effect with the electronic – confined phonon scattering [5-9]. In one-dimensional

semiconductor system, there have been studies on the Hall effect with the confined electronics –
unconfined phonon [10]. But the influence of the confined phonons on the HC in one-dimensional
semiconductor system is not studied. In this work, we study new properties of the HC under the effect
of COP. Considering an infinite potential quantum wire subjected to a dc electric field 𝐸⃗ , a magnetic
⃗ and a laser radiation 𝐸⃗ = ⃗⃗⃗⃗
field 𝐵
𝐸0 𝑠𝑖𝑛 Ω𝑡. The article is organized as follows: in section 2 we present
the confinement of electron and optical phonons in a CQW. Thus, by using the quantum kinetic equation
method, we obtained analytical expressions for the Hall coefficient. Numerical results and discussions
for the GaAs/GaAsAl cylindrycal quantum wire are given in section 3. Finally, section 4 shows remarks
and conclusions.

2. The Influence of Confined Phonons on the Hall Coefficient in a Cylindrycal Quantum wire
with an infinite potential
Consider a cylindrycal quantum wire with an infinite potential V= R2L subjected is placed in a
⃗ , a constant - electric field ⃗⃗⃗⃗
perpendicular magnetic field 𝐵
𝐸1 and an intense electromagnetic wave 𝐸⃗ =
⃗⃗⃗⃗
𝐸0 𝑠𝑖𝑛 Ω𝑡. Under the influence of the material confinement potential, the motion of carriers is restricted
in x,y direction and free in the z one. So, the wave function of an electron and its discrete energy now
becomes:
Ψn,l,k⃗ (𝑟, , 𝑧) =

1

⃗

√𝑉0

𝑒 𝑖𝑚 𝑒 𝑖𝑘𝑧 𝑛,𝑙 (𝑟) , where 𝑛,𝑙 (𝑟) =

1
𝑟
𝐽 (𝐵
)
𝐽𝑛+1 (𝐵𝑛,𝑙 ) 𝑛 𝑛,𝑙 𝑅
2

2 2

𝑛
𝑙
1
ℏ 𝑘
1 𝑒𝐸1
𝜀𝑛,𝑙 (𝑘⃗𝑧 ) = (𝑁 + 2 + 2 + 2) ℏ𝜔𝑐 + 2𝑚 − 2𝑚 ( 𝜔 )

(1)
(2)

𝑐

where k, m is the wave vector and the effective mass of an electron, R being the radius of the CQW,
n = 1,2,3,… and l = 0, ±1, ±2, … being the quantum numbers charactering the electron confinement,
𝑒𝐵
is the Planck constant, 𝜔𝑐 = 𝑚 is the cyclotron frequency.
When phonons are confined in CQW, the wave vector and frequency of them are given by [11,12]:
2
𝑞 = (𝑞𝑚1 𝑚2 , 𝑞𝑧 ), 𝜔𝑚1 ,𝑚2 ,𝑞⃗⊥ = √𝜔02 − 𝛽 2 (𝑞𝑚

+ 𝑞𝑧2 )
1 𝑚2

(3)

Where  is the velocity parameter, m1, m2 = 1,2,3,…being the quantum numbers charactering phonon
confinement. Also, matrix element for confined electron – confined optical phonon interaction in the
CQW now becomes [11]
𝑚 ,𝑚

𝑚1 ,𝑚2

𝐷𝑛11,𝑙1 ,𝑛22 ,𝑙2 ,𝑞𝑧 = 𝐶𝑞⃗

𝑚 ,𝑚

𝑚1,𝑚2 2

𝑚 ,𝑚

∗ 𝐼𝑛11,𝑙1 ,𝑛22 ,𝑙2 where |𝐶𝑞⃗
2

R

| =

e2 ω0 1
(
2ε0 𝑉 χ∞

−

1
1
)
χ0 q2z +q2𝑚1 ,𝑚2

I𝑛11,𝑙1 ,𝑛22 ,𝑙2 = R2 ∫0 𝐽|𝑛1 − 𝑛2 | (𝑞, 𝑅)φ∗n2 ,l2 (r)φn1 ,l1 (r)r dr.

(4)
(5)


P.N. Thang et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 46-51

48

Though equations (1-5), it has been seen that the CQW with new material confinement potential
gives the different electron wave function and energy spectrum. In addition, the contribution of confined
phonon could enhance the probability of electron scattering. As a result, the Hall Coefficient in a CQW
under influence of confined optical phonon and laser radiation should be studied carefully to find out
the new properties. The effect of confined optical phonons and the laser radiation modify the Hamitonian
of the confined electron – confined optical phonons system in the CQW. This leads the quantum kinetic
equation for electron distribution. Using Hamiltonian of the confined electrons — confined optical
phonons in a CQW, we establish the quantum kinetic equation for electron distribution function. After
some manipulation, the expression for the conductivity tensor is obtained:
τ
2 2
2 2

σie =
(6)
2 2 {δik − ωc τεijk hk + ωc τ hi hj }{aδeị + b(δje − ωc τεjef hf + ωc τ he hf )}
1+ωc τ

here  ik is the Kronecker delta; 𝜀𝑖𝑗𝑘 being the antisymmetric Levi-Civita tensor; symbols 𝑖, 𝑗, 𝑘, 𝑙, 𝑝
corresponding the components x, y, z of the Cartesian coordinates. From this we obtain the expression
for the hall coefficient
σyx
1
R H = − B σ 2 +σ
(7)
2
With σxx =

τ
{a +
1+ω2c τ2

xx

yx

b[1 − ω2c τ2 ]} ; σyx =

−τ
(a
1+ω2c τ2

+ b)ωc τ


0
Lz e
n l 1  e2 E12  
 
  2m 

exp




N

  
  F
c
2 
2 
2 m2 1  c2 02
2
2
2

 2mc  

 
  

a

b

 

e
τ
2 ie
b ; b0 
I m1 ,m2
2 2 0
m 1  ωc τ
m 1 , 2, m1 ,m2 1 , 2,




n

l

3/2

(8)



(9)

2

 A  A  A  A  A  A  A  A 
2

1

2

3

4

5

6

7





2
1  1  eE1  


 1
Lz  e 0 e
1 



e
2
X0 
4 2 0V 2  X 
2

E02
2



n

l

2
1  1  eE  

1
 
  F c  N      

  
2
2
2
2



c

 







Lz e 4 E02 k B T  1
1 
 


 exp    F   1 
4 8 0V  X 
X0 

 




3/ 2


K 3/ 2 





1
2mA11  2m K1 

2


 q

2
m1 , m2

2

(12)

2

A11

4


 

   K 0 
2 




2

2

qm
A11   
,m
2
 qm1 ,m2  1 2 K 0 
2
2
m




A11   
  A11 K 0 
2




2
 A11 qm
K 3/ 2 
1 , m2






2

2

2

(11)

 qm2 ,m
  A11 
 1 2 K 2 

 2 
 2m

  A11 
  A11 
  A11  
K0 
 A11 
 A11 



2m
 2 

 2 
 2 


1
 
2
mA
11  2 m



2

qm21 ,m2

A3 

(10)

    N      

F
c 1

2 2 2  2  c   1 k T

L me 0  1
1  
B


A1  z

e


2 2 0V  X 
X0 
2 
  A11 
  A11 
  A11 
  A11 
2
4
 2qm2 ,m m 2 A11
exp  
K 2 
 8 A11
m 4 qm21 ,m2 exp  
K 3 




1
2

2 
2 


 2 

 2 
  A11 
  A11 
  A11 
  A11  
3 3
 mA11 exp  
K 1 
  4m A11 exp   2  K 2  2  
2 

 2 





2

A2 

8

2

A11   


2


A11   

2


2


A11    
1
  

2
2
mA

2
m

11

 


A11    1
  2mA11  2m K1 
2

 2


2

A11    
 
2
 

(13)
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P.N. Thang et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 46-51

A4 





 
e 2 E02 Lz  1
m2 e2 kB T
1 


 exp     F  1 
2 3

2 
2 0V
X0 
4m  4  X 
 

2

A5 


A11   
  2m A11   K1 


2



2
A11
   2

  qm , m  A11   K 0
4
2    1 2





A11   
  qm21 , m2  A11 qm21 , m2 K 1 


2







 
e 2 E02 Lz  1
m 2 e 2 k BT
1 


 exp     F  1 
2
3
2 
2 0V
4m  4  X  X 0 
 

2 2
1
qm1 ,m2
 

2
4mB1


A6 

2

2


 2
 K1 
 2


 qm1 ,m2
 2

K

2 
4 B1

 2
2



 



2

A11    


2


 2
B1    2
K
 

0 
2    2m
 2 

2

(15)

E02
2
4

B1  


2  

2

(16 (16)

3/2

 2 B1
 1 
  B1  qm1 ,m2
  B1 
  B1  
K1 
K0 
 B1qm21 ,m2 
 K 3/2 




 2  2m
 2 
 2 
 2 B1 
 2m
2
e2 k B T e2 E02 Lz  1
 
1 


 
A7 


exp





B


 1    m B1   K1
  F


1
2 0V
2
44 4 2  X  X 0 
   2m
 
2


  B1        B1 


2

2 2
qm1 ,m2

2m

(14)

K is Bessel fouction type 2

 
e
m e k BT
Lz  1
1 


 exp     F  1 
2 
2 0V
X0 
4m  4  X 
 
2

2 2

49


1
 K 0   B1     qm21 ,m2    B1  *
K 1   B1   

m
B


1


E2 L  1
e4 kBT
 
1 


2
 0 3 z2 

 exp    F  1   B1        B1   
2 0V
2
4 4  X  X 0 

 
2 2
2


qm1 ,m2 
qm ,m
 K 0    B1     1 2 K 1   B1   
K1    B1         B1 

2m 
m


(17)

A8 



A11    2   1  m1 ,m2 ,qz ; B 

qz2
   2   1  m1 ,m2 ,qz ; B1    2   1  0   qm21 ,m2
2m

(18)

2

(19)

The expression (7) is analytics expression of the Hall coefficient in CQW with an infinite potential
(for electron – confined optical phonons scattering). From this expression we see, the HC dependent on
the magnetic field B, frequency  and amplitude E0 of laser radiation, temperature T of system and

specially the quantum numbers m1, m2 characterizing the phonon confinement effect. Where m1, m2 goes
to zero, we obtain results as case of unconfined phonons [10].
3. Numerical results and discussions
In this section, we present the numerical evaluation of the Hall conductivity and the HC for the
GaAs/GaAsAl quantum wire. Parameters used in this according to the result in Ref. [11,12]: 𝑚𝑒 =
0.067𝑚0 , (𝑚0 is the free mass of an electron), χ∞ = 10.9, χ0 = 12.9, 𝜀𝐹 = 8. 10−21 𝐽, 𝜏 = 10−12 𝑠,
𝑚

𝜈 = 8.73 × 104 𝑚𝑠 −1 , 0  36.25meV , V = 1, E0  105V / m , E1  5.105V / m 𝑐 = 3. 108 𝑠 ,
, 𝑘𝑩 = 1.38. 10−23 𝐽/𝐾


50

P.N. Thang et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 46-51

Figure 1. The dependence of the conductivity tensor σxx on the cyclotron energy for confined phonon
(solid curve) and unconfined phonon (dashed curve), here E1  5.10 V / m and 𝐿 = 30 𝑛𝑚
5

In figure 1, we can see clearly the appearance of oscillations and oscillations are controlled by the
ratio of the Fermi energy and energy of cyclotron. First, phonons are confined in 2 dimensions x, y, only
motion free in the z one (quantum wires), therefore, The power spectrum of the external phonon depends
on the normal effects of free movement, depending on the confined index of phonon m1, m2
corresponding to the x and y directions. In case confined phonon get more two resonance peaks
comparing with that in case of unconfined phonons. When phonons are confined, specially the confined
2
optical phonons frequency is now modified to 𝜔𝑚1 ,𝑚2 ,𝑞⃗ = √𝜔02 − 𝛽 2 (𝑞𝑧2 + 𝑞𝑚
). Hence, confined
1 ,𝑚2

optical phonons make remarkable contribution on the resonance condition.

Figure 2. The dependence of the Hall coefficient on the laser amplitude for unconfined phonon
(dotted curve) and confined phonon m1 = 2, m2 = 2 (dashed curve)

m1 = 0, m2 = 0

Figure 2 shows the nonlinear dependence of the Hall coefficient on the laser amplitude at different
values of number m1, m2 characterizing the phonon confinement. When the laser amplitude has been
valid small, which makes an increase of Hall coefficient by 2,3 times in comparison with the case of
unconfined phonons. It has been seen that the HC decreases as the increasing of the laser amplitude and


P.N. Thang et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 3 (2019) 46-51

51

the HC reaches saturation when this amplitude is large. When the quantum number m1 and m2 goes to
zero, the result is the same as in the case of unconfined phonons [10].
4. Conclusions
In this article, the influence of confined optical phonons on the Hall coefficient in a quantum wires
with infinite potential (for electron – confined optical phonons scattering) has been theoretically studied
base on quantum kinetic equation method. We obtained the analytical expression of the Hall coefficient
in the CQW under the influence of COP. Numerical calculations are also applied for GaAs/GaAsAl
cylindrycal quantum wire, we see the HC depends on magnetic field B, temperature T, frequency Ω and
amplitude E0 of laser radiation and especially quantum index m1 and m2 characterizing the phonon
confinement. This influence is due to the quantum index m1 and m2, which makes an increase of Hall
coefficient by 2,3 times in comparison with the case of unconfined phonons.
Acknowledgments

This work was completed with financial support from the QG.17.38
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