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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

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

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

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.

→
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


→,
→ −
→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 + →
→

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,

∂ 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

→
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) = −

→
−
→ −
→ 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 ,


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

)


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



=
≈

0

+3

2 )θ (

0 )θ (

1 ),

0 )θ (

2)

2

+ 2 √0

3)

−

2
1

0


=

5

=

−

−

2
4

4

1

−

2
5

5

1

4 )θ (

1 ),


− e2 E12 − 2m∗ ωp2 εF

1
2

2 ω2
0

2m∗ ωp2

2m∗ ωp2

(14)

θ(

εF − ωp N +

=

θ(

1
2

(17)

,



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

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 +

×

⎧⎧
⎨⎨
⎩⎩


+

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
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.


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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