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The Dependence of the
Magnetoresistance in Quantum Wells
with Parabolic Potential on Some
Quantities under the Influence of
Electromagnetic Wave
a

a

b

Nguyen Dinh Nam , Do Tuan Long & Nguyen Vu Nhan
a

Department of Physics, College of Natural Science, Viet Nam
National University, Ha Noi, Viet Nam
b

Department of Physics, Academy of Defence force-Air force, Ha
Noi, Viet Nam

Published online: 23 May 2014.

To cite this article: Nguyen Dinh Nam, Do Tuan Long & Nguyen Vu Nhan (2014) The Dependence of the
Magnetoresistance in Quantum Wells with Parabolic Potential on Some Quantities under the Influence
of Electromagnetic Wave, Integrated Ferroelectrics: An International Journal, 155:1, 45-51, DOI:
10.1080/10584587.2014.905122
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Integrated Ferroelectrics, 155:45–51, 2014
Copyright © Taylor & Francis Group, LLC

ISSN: 1058-4587 print / 1607-8489 online
DOI: 10.1080/10584587.2014.905122

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The Dependence of the Magnetoresistance
in Quantum Wells with Parabolic Potential
on Some Quantities under the Influence of
Electromagnetic Wave
NGUYEN DINH NAM,1,* DO TUAN LONG,1
AND NGUYEN VU NHAN2
1

Department of Physics, College of Natural Science, Viet Nam National
University, Ha Noi, Viet Nam
2
Department of Physics, Academy of Defence force-Air force, Ha Noi, Viet Nam
The magnetoresistance is one of the important properties of semiconductors. Starting from the hamiltonian for the electron-acoustic phonon system, we obtained the
expression for the electron distribution function and especially the expression for the
magnetoresistance in quantum wells with parabolic potential (QWPP) under the influence of electromagnetic wave (EMW) in the presence of magnetic field. We estimated
numerical values and graphed for a GaAs/GaAsAl quantum well to see the nonlinear
dependence of the magnetoresistance on the temperature of the system T, the amplitude E0 and the frequency of the electromagnetic waves, the magnetic field B, the
parameters of the quantum well and the momentum relaxation time τ clearly.
Keywords Dependence of magnetoresistance

I. Introduction
In the past few years, there have been many scientific works related to the properties of
the low-dimensional systems such as the optical, magnetic and electrical properties [1–9].
These results show us that there are some differences between the low-semiconductor and
the bulk semiconductor that the previous work studied.

The magnetoresistance is also interested. However, it has not been resolved in the
quantum wells with parabolic potential under the influence of electromagnetic wave. The
calculation of the magnetoresistance in the QWPP in the presence of magnetic field under
the influence of EMW is done by using the quantum kinetic equation method that brings
the high accuracy and the high efficiency [1–4]. Comparing the results in this case with in
the case of the bulk semiconductors, we also see some differences.

II. The Magnetoresistance in Quantum Wells with Parabolic Potential under
the Influence of Electromagnetic Wave in the Presence of Magnetic Field
It is well known that in the quantum wells, the motion of electrons is restricted in one
dimension, so that they can flow freely in two dimensions [1, 2, 7]. We consider a quantum
Received July 23, 2013; in final form January 12, 2014.
∗
Corresponding author. E-mail:

45


46

N. D. Nam et al.

well with parabolic potential of GaAs embedded in AlAs. It subjected to a crossed electric
−
→
−
→
field E1 = (0, 0, E1 )and magnetic field B = (0, B, 0). In the presence of an EMW with
−
→ −

→
electric field vector E = E0 sin t (where E0 and are the amplitude and the frequency
of the EMW, respectively), the Hamiltonian of the electron-acoustic phonon system in the
above-mentioned QWPP in the second quantization presentation can be written as:
H =

e −
→
−
→
A (t) a + −
εN kx −
ω−
→
→
→
→+
→a −
q
q +b−
q b−
c
N, kx N, kx
−
→
−
→
q
N, kx

Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014

+

−
→−
→
q

N,N , kx ,

+

→
CN,N (−
+ b+−
q )a + −
→
→ b−
→ −
→
→ + aN,−
q
−q
kx
N , kx + qx

→
φ(−
q )a + −

→,
→ −
→a −
N, kx + qx N, kx
−
→
q

(1)

+
)
and b−
where N is the Landau level index (N = 0,1 ,2 . . . ), a + −
→
→ and a −
→ , (b−
→
q
q
N, kx
N, kx
−
→
are the creation and the annihilation operators of the electron (phonon), |N, kx > and
−
→ −
qx > are electron states before and after scattering, ω−
|N , kx + →
→

q is the energy of an
−
→
acoustic phonon; CN,N ( q ) = C−
→
→
q is the electron-phonon interacq IN,N (qz ), where C−
→
tion constant and IN,N (qz ) is the electron form factor [4], φ −
q is the scalar potential
−
→ −
→
of a crossed electric field E1 , A (t) is the vector potential of an external electromagnetic
−
→
−
→
wave A (t) = eE0 sin( t)/ . If the confinement potential is assumed to take the form
2
V (z) = mω0 (z − z0 )2 /2, then the single-particle wave function and its energy are given by
[1, 2]:

→
ψ(−
r )=

εN (kx ) = ωp N +

1

2

→−
→
1 i−
e k⊥ r ψ(kx , z),
2π

+

1
2m∗

2 2
kx

(2)

kx ωc + eE1
ωp

−

2

(3)

,

Here, m∗ and e are the effective mass and charge of conduction electron, respectively,

k⊥ = (kx , ky ) is its wave vector in the (x,y) plan, z0 = ( kx ωc + eE1 ) /mωp2 , ωp2 = ω02 +ωc2 ,
ω0 and ωc are the confinement and the cyclotron frequencies, respectively, and
ψm (z − z0 ) = Hm (z − z0 ) exp −(z − z0 )2 /2 ,

(4)

with Hm (z)being the Hermite polynomial of mth order.
From the quantum kinetic equation for electron in single scattering time approximation
and the electron distribution function, using the Hamiltonian in the Eq. 1, we find:
∂f −
→
N, kx
+
∂t

−
→
e E1

−
→ −
→
+ ωc kx , h

∂f −
→
2π
N, kx
−
→ =

∂ kx

N

−
→
,q

→
CN,N (−
q)

2

2N−
→
q +1

+∞

Jl2 (αqx ) × f
l=∞

−
→ −
→ δ εN (kx + qx ) − εN (kx ) − l
→ − fN,−
k

N , kx + qx

x

.

(5)


Dependence of Magnetoresistance in QWPP

47

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−
→ −
→
−
→
where kx = (kx , 0, 0), h = B /B is the unit vector in the direction of magnetic field,
f −
→ is an unknown distribution function perturbed due to the external fields, Jl (x) is the
N, kx
th
l - order Bessel function of argument x and N−
→
q is the time-independent component of the
electron distribution function.
For simplicity, we limit the problem to case of l = −1, 0, 1. If we multiply both sides
−

→
of the Eq. 5 by (e/m∗ ) kx δ(ε − εN (kx )), carry out the summation over N and kx and use
J02 (αqx ) ≈ 1 − (αqx )2 /2, we obtain:
−
→
R (ε)
−
→ −
→
−
→
−
→
+ ωc h , R (ε) = Q (ε) + S (ε),
τ (ε)

(6)

−
→
R (ε) =

(7)

where

2π e
−
→
S (ε) = −

4 m∗

e −
→
kx f −
→ δ (ε − εN (kx )) ,
∗
N, kx
−
→m
N, kx
2

−
→
q

2
CN,N (q) (2N−
→
q + 1)(αqx )

N ,

−
→

f

−

→
−
→ −
→ kx
→ − fN,−
k

N , kx + qx

x

N, kx

× 2δ εN (kx + qx ) − εN (kx ) − δ εN (kx + qx ) − εN (kx ) −
−δ εN (kx + qx ) − εN (kx ) +
e
−
→
Q (ε) = − ∗
m

−
→
N, kx

→
−
→ −
kx F ,

,

(8)

∂f −
→
N, kx
−
→ δ (ε − εN (kx )),
∂ kx

(9)

with
ε − εN (kx )
−
→
−
→
∇T .
(10)
F = eE1 − ∇εF −
T
−
→
−
→
−
→
Finding R (ε) in term of Q (ε), S (ε)and through some computation steps, we obtain

the expression for conductivity tensor:
σim =

τ (εF )
τ (εF )
e
c0 δik +d0 d1
δik −ωc τ (εF ) εikl hl + ωc2 τ 2 (εF ) hi hk
∗
2
2
m 1+ωc τ (εF )
1+ωc2 τ 2 (εF )
+ d0 d2

)
τ (εF −
1 + ωc2 τ 2 (εF −

)

δik − ωc τ (εF −

) εikl hl + ωc2 τ 2 (εF −

) hi hk

+ d0 d3

)

τ (εF +
1 + ωc2 τ 2 (εF +

)

δik − ωc τ (εF +

) εikl hl + ωc2 τ 2 (εF +

) hi hk

× δkm − ωc τ (εF ) εkmn hn + ωc2 τ 2 (εF ) hk hm }

(11)

where
c0 =
N

eLx
π

0θ (

0 ),

(12)


48

N. D. Nam et al.
eLx ξ 2 kB T e2 E02 eE1 ωc
I
(qz ),
4π 2 m∗ ηυ 2 4 ω4 ω02 N,N

d0 =
N,N

d1 =

0

4

(

+3

0

1 )θ (

0 )θ (

1)

0

−2

1

N,N
0

−

(

0

2

+3

3 )θ (

0 )θ (

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2

d2 = √1

2

d3 = √1

=
≈

0

+3

2 )θ (

0 )θ (

1 ),

0 )θ (

2)

2

+ 2 √0

3)

−

2
1

0

1

3

0

(

(13)

2

eE1 ωc
ω02
2m∗ ωp2
2 ω2
0

1

2

=

3

=

4

=

5

=

−

−

2
4

4

1

−

2
5

5

1

4 )θ (

1 ),

(15)

θ(

5 )θ (

1 ),

(16)

2m∗ ωp3 N +

2 ω2
0

2 ω2
0

2m∗ ωp2
2 ω2
0

2m∗ ωp2
2 ω2
0

2m∗ ωp2
2 ω2
0

− e2 E12 − 2m∗ ωp2 εF

1
2

2 ω2
0

2m∗ ωp2

2m∗ ωp2

(14)

θ(

εF − ωp N +

=

θ(

1
2

(17)

,

εF − ωp N +

1
2

(18)

,

εF +

− ωp N +

1
2

,

εF −

− ωp N +

1
2

,

εF −

− ωp N +

1
2

,

εF +

− ωp N +

1
2

.

(19)

(20)

where Lx ξ, η, υ, kB , T , εF are the x-directional normalization lengths, the deformation
potential constant, the density, the acoustic velocity, the Boltzmann constant, the temperature of system and the Fermi energy, respectively.
In this work, we consider the case of electron-acoustic phonon scattering and the
−
→
presence of electric field E1 . Comparing with the case of electron-optical phonon scattering


Dependence of Magnetoresistance in QWPP

49

−
→
and no electric field E1 that was studied previously [10], we see some differences in the
expression of conductivity tensor and also in the expression of magnetoresistance.
The magnetoresistance is given by the formula:
ρ
σzz (H ) σzz (0)
−1
= 2
2 (H )
ρ
σzz (H ) + σxz

(21)

Using the Eq. 11, we obtain the explicit formula of the magnetoresistance as following:

Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014

ρ
=
ρ

N,N

+

e
τ (εF )

d0 d1 τ (εF )
c0 +
1 − ωc2 τ 2 (εF )h2
∗
2
m 1 + τ (εF )
1 + ωc2 τ 2 (εF )

d0 d2 τ (εF −
)
1 − ωc2 τ (εF )τ (εF −
2
2
1 + ωc τ (εF −
)

× 1 − ωc2 τ (εF )τ (εF +

)h2

×
N,N

× 2m∗ εF − ω0 N +

×

⎧⎧
⎨⎨
⎩⎩

N,N

1
2

)h2 +

e2 τ (εF )Lx
1 + τ 2 (εF )

m∗ π 2

θ εF − ω0 N +

d0 d2 τ (εF −
)
× 1 − ωc2 τ (εF )τ (εF −
1 + ωc2 τ 2 (εF −
)

+

d0 d3 τ (εF +
)
1 − ωc2 τ (εF )τ (εF +
1 + ωc2 τ 2 (εF +
)

N,N

+

1
2

e
τ (εF )
d0 d1 τ (εF )
c0 +
1 − ωc2 τ 2 (εF )h2
∗
2
m 1 + τ (εF )
1 + ωc2 τ 2 (εF )

+

+

)
d0 d3 τ (εF +
2
2
1 + ωc τ (εF +
)

)h2
2

)h2

e
τ (εF )
d0 d1 τ (εF )2ωc τ (εF )h2
ω
τ
(ε
)h
+
c
0
c
F
m∗ 1 + ωc2 τ 2 (εF )
1 + ωc2 τ 2 (εF )

)
d0 d2 τ (εF −
× ωc [τ (εF ) + τ (εF −
1 + ωc2 τ 2 (εF −
)

)
d0 d3 τ (εF +
ωc [τ (εF ) + τ (εF +
+
1 + ωc2 τ 2 (εF +
)

)] h2
2

)] h

2

−1

−1

(22)

Eq. 22 is the analytical expression of the magnetoresistance in the QWPP. It shows
the dependence of the magnetoresistance on the external fields, including the EMW. In the
next section, we will give a deeper insight into this dependence by carrying out a numerical
evaluation. In Eq. 22, we can see that the formula of the magnetoresistance is easy to come
back to the case of bulk semiconductor when ω0 reaches to zero [11, 12].

III. Numerical Results and Discussion
For the numerical evaluation, we consider the model of a quantum well of GaAs/GaAsAl
with the following parameters: εF = 50 meV , kB = 1.3807 × 10−23 J K −1 , υ = 5220 m/s


Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014

50

N. D. Nam et al.

Figure 1. The dependence of the magnetoresistance on the temperature.

and m∗ = 0.0067m0 with m0 is the mass of a free electron. For the sake of simplicity, we
also choose N = 0, N = 1, τ = 10−12 s [1, 2].
Figure 1 shows the magnetoresistance as a function of the temperature. The value of
the magnetoresistance increases sharply when the temperature is low, after that it decreases
steadily. With the different values of the electric field E1 , we get the resonant peaks at the
different points of temperature.
Figure 2 shows us the dependence of the magnetoresistance on the amplitude of the
EMW. The higher the amplitude of the EMW is, the faster the magnetoresistance grows
up. The line of the dependence of the magnetoresistance on the amplitude E0 of EMW also
changes when we change the value of the frequency of the EMW. We see that there are
some differences in the dependence of the magnetoresistance on the temperature and the
amplitude from the case of electron-optical phonon scattering [10]. We also get the same
graphs as in the case of bulk semiconductor [11, 12] when the confinement frequency ω0
reaches to zero.

Figure 2. The dependence of the magnetoresistance on the amplitude of the EMW.


Dependence of Magnetoresistance in QWPP

51

IV. Conclusions

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In this paper, we obtain the analytical expression of the magnetoresistance in QWPP under
the influence of EMW in the presence of magnetic field. We see that the magnetoresistance

in this case depends on some quantities such as: the magnetic field B, the temperature T, the
parameters of QWPP, the momentum relaxation time τ , the amplitude E0 and the frequency
of EMW. Estimating numerical values and graph for a GaAs/GaAsAl quantum well to
see this dependence clearly. Looking at the graph, we see that the magnetoresistance gets
the negative values and the dependence of the magnetoresistance on the temperature, the
amplitude and the frequency of the EMW are nonlinear. When ω0 reaches to zero, we obtain
the results as the case of bulk semiconductor that was studied [11, 12].

Funding
This research is completed with financial support from Vietnam NAFOSTED (103.012011.18) and TN13-04.

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