DSpace at VNU: Influence of a Strong Electromagnetic Wave (Laser Radiation) on the Hall Effect in Quantum Wells with a Parabolic Potential

Journal of the Korean Physical Society, Vol. 60, No. 1, January 2012, pp. 59∼64

Influence of a Strong Electromagnetic Wave (Laser Radiation) on the Hall
Effect in Quantum Wells with a Parabolic Potential
Nguyen Quang Bau and Bui Dinh Hoi∗
Department of Physics, College of Natural Science, Vietnam National University, Hanoi, Vietnam
(Received 17 October 2011, in final form 30 November 2011)
Based on the quantum kinetic equation for electrons, we theoretically study the influence of an
electromagnetic wave (EMW) on the Hall effect in a quantum well (QW) with a parabolic potential
V (z) = mωz z 2 /2 (where m and ωz are the effective mass of electron and the confinement frequency
of QW, respectively) subjected to a crossed dc electric field E1 = (0, 0, E1 ) and magnetic field
B = (0, B, 0) in the presence of a strong EMW characterized by electric field E = (E0 sin Ωt, 0, 0)
(where E0 and Ω are the amplitude and the frequency of EMW, respectively). We obtain analytic
expressions for the components σzz and σxz of the Hall conductivity as well as a Hall coefficient with
a dependence on B, E1 , E0 , Ω, temperature T of the system and the characteristic parameters of
QW. The results are numerically evaluated and graphed for a specific quantum well, GaAs/AlGaAs,
to show clearly the dependence of the Hall conductivity and the Hall coefficient on above parameters.
The influence of the EMW is interpreted by using the dependences of the Hall conductivity and the
Hall coefficient on the amplitude E0 and the frequency Ω of EMW and by using the dependences
on the magnetic field B and the dc electric field E1 as in the ordinary Hall effect.
PACS numbers: 72.20.My, 73.21.Fg, 78.67.De
Keywords: Hall effect, Quantum kinetic equation, Parabolic quantum wells, Electron-phonon interaction
DOI: 10.3938/jkps.60.59

I. INTRODUCTION

magnetic field [9,10] and the effect of the presence of an
additional (high frequency) EMW [11] have been studied in much detail in bulk semiconductors by using the
quantum kinetic equation method.
Quantum well with parabolic potential (QWPP) is a
2DEG system in which electrons are free to move in

two directions, but are confined in the third due to the
parabolic potential. Beside other 2DEG systems, in recent years many physicists have been interested in investigating the quantum Hall effect in a QWPP from many
different aspects [12–19]. These works, however, only
considered the case when the EMW was absent and when
the temperature so that electron-impurity and electronacoustic phonon interactions were dominant (condition
for the quantum Hall effect). To our knowledge, the Hall
effect in a QWPP in the presence of an EMW remains
a problem to study. The aim of this our work is to apply the quantum kinetic equation method to study the
Hall effect in a QWPP subjected to a crossed dc electric
field E1 = (0, 0, E1 ) and magnetic field B = (0, B, 0) in
the presence of an EMW characterized by electric field
E = (E0 sin Ωt, 0, 0), the confinement potential being assumed to be V (z) = mωz z 2 /2. We only consider the
case in which the electron - optical phonon interaction
is assumed to be dominant and electron gas to be nondegenerate. We derive the analytical expressions for the
conductivity tensor and the Hall coefficient (HC). The

Recently, there has been considerable interest in the
behavior of low-dimensional systems, in particular, twodimensional electron gas (2DEG) systems, such as quantum wells and compositional and doped superlattices.
The confinement of electrons in these systems considerably enhances the electron mobility and leads to unusual behaviors under external stimuli. As a result, the
properties of low-dimensional systems, especially electrical and optical properties, are very different in comparison from those of normal semiconductors [1, 2]. There
have been many papers dealing with problems related
to the incidence of electromagnetic wave (EMW) in lowdimensional systems. The linear absorption of a weak
electromagnetic wave caused by confined electrons in
low-dimensional systems has been investigated by using
the Kubo - Mori method [3,4]. Calculations of the nonlinear absorption coefficients of a strong electromagnetic
wave by using the quantum kinetic equation for electrons
in bulk semiconductors [5,6], in quantum wires [7] and in
compositional semiconductor superlattices [8] have also
been reported. Also, the Hall effect, where a sample is
subjected to a crossed time-dependent electric field and

∗ E-mail:

; Tel: +84-913-348-020

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Journal of the Korean Physical Society, Vol. 60, No. 1, January 2012

paper is organized as follows. In the next section, we
describe the simple model of the parabolic quantum well
and present briefly the basic formulae for the calculation. Numerical results and discussion are given in Sec.
III. Finally, remarks and conclusions are shown briefly
in Sec. IV.

resentation can be written as
H = H0 + U,
H0 =

εN
N,kx

+
q

II. HALL EFFECT IN A QUANTUM WELL
WITH A PARABOLIC POTENTIAL IN
THE PRESENCE OF LASER RADIATION

1. Electronic Structure in a Parabolic Quantum
Well

Consider a perfect infinitely high QWPP structure
subjected to a crossed electric field E1 = (0, 0, E1 ) and
magnetic field B = (0, B, 0) and choose a vector potential A = (zB, 0, 0) to describe the applied DC magnetic
field. If the confinement potential is assumed to take
the form V (z) = mωz z 2 /2, then the single-particle wave
function and its eigenenergy are given by [20]
1 ik⊥ r
e
φN (z − z0 ) ,
2π
1
εN (kx ) = ωp N +
2

(1)

Ψ (r) =

+

1
2m

2 2
kx

−

kx ωc + eE1
ωp

2

,

N = 0, 1, 2, . . . ,

(2)
(3)

where m and e are the effective mass and the charge of
a conduction electron, respectively, k⊥ = (kx , ky ) is its
wave vector in the (x, y) plan; z0 = ( kx ωc + eE1 )/mωp2 ;
ωp2 = ωz2 + ωc2 , ωz and ωc = eB/m are the confinement
and the cyclotron frequencies, respectively, and

2. Expressions for the Hall Conductivity and
the Hall Coefficient

DN,N (q)a+

a
(b
N ,kx +qx N,kx q

+ b+
−q ), (7)

where |N, kx > and |N , kx + q⊥ > are electron states
before and after scattering; ωq is the energy of an optiand
cal phonon with the wave vector q = (q⊥ , qz ); a+
N,kx

aN,kx (b+
q and bq ) are the creation and the annihilation
operators of electron (phonon), respectively; A(t) is the
vector potential of laser field; DN,N (q) = Cq IN,N (qz ),
where Cq is the electron-phonon interaction constant;
and IN,N (qz ) = < N |eiqz z |N > is the form factor of
electron. The quantum kinetic equation for electrons in
the single (constant) scattering time approximation takes
the form
∂fN,kx
∂fN,kx
− eE1 + ωc kx × h
∂t
∂ kx
fN,kx − f0
kx ∂fN,kx
=−
, (8)
+
m ∂r
τ
where kx = (kx , 0, 0), h = B/B is the unit vector in the
direction of magnetic field, the notation ‘×’ represents
the cross product, f0 is the equilibrium electron distribution function (Fermi - Dirac distribution), fN,kx is an

unknown distribution function perturbed due to the external fields, and τ is the electron momentum relaxation
time, which is assumed to be constant.
In order to find fN,kx , we use the general quantum
equation for the particle number operator [5–8] or the
electron distribution function fN,kx = a+ aN,kx :
N,kx

∂
f
=
∂t N,kx

a+

a
,H
N,kx N,kx

t

.

t

(9)

From Eqs. (8) and (9), using the Hamiltonian in Eq. (5),
we find
∂fN,kx
− eE1 + ωc kx × h

∂ kx
fN,kx − f0
2π
kx ∂fN,kx
=−
+
+
|DN,N (q)|2
m ∂r
τ
N ,q

∞

×
In the presence of an EMW with electric field vector
E = (E0 sin Ωt, 0, 0) (where E0 and Ω are the amplitude and the frequency of the EMW, respectively), the
Hamiltonian of the electron-optical phonon system in the
above-mentioned QWPP in the second quantization rep-

(6)

N,N q,kx

2

with HN (z) being the Hermite polynomial of N th order.

ω q b+
q bq ,

U =

i
φN (z − z0 ) = HN (z − z0 ) exp − (z − z0 ) /2 , (4)

(5)
e
kx − A(t) a+ aN,kx
N,kx
c

J2

l=−∞

Λ
Ω

[f¯N

,kx +qx

(Nq + 1)

− f¯N,kx Nq ]δ (εN (kx + qx ) − εN (kx ) − ω0 − Ω)
+ [f¯
Nq − f¯
(Nq + 1)]δ (εN (kx − qx )
N ,kx −qx

N,kx

−εN (kx ) + ω0 −

Ω) ,

(10)


Influence of a Strong Electromagnetic Wave (Laser Radiation) · · · – Nguyen Quang Bau and Bui Dinh Hoi

in which f¯N,kx (Nq ) is the time-independent component of the distribution function of electrons (phonons),
J (x) is the th -order Bessel function of argument x, and
Λ = (eE0 qx /mΩ2 )(1 − ωc2 /ωp2 ). Equation (10) is fairly
general and can be applied for any mechanism of interaction. In the limit of ωz → 0, i.e., the confinement
vanishes, it gives the same results as those obtained in

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bulk semiconductor [9–11].
For simplicity, we limit the problem to the case of
= −1, 0, 1. If we multiply both sides of Eq. (10) by
(e/m)kx δ (ε − εN (kx )) and carry out the summation over
N and kx , we have the equation for the partial current
density jN,N (ε) (the current caused by electrons that
have energy of ε):

jN,N (ε)
+ ωc [h × jN,N (ε)] = QN (ε) + SN,N (ε),

τ

(11)

where
QN (ε) = −

e
m

kx F
N,kx

∂fN,kx
∂ kx

δ (ε − εN (kx )) ,

F = eE1

(12)

and
SN,N (ε) =

2πe
m

2

f¯N

|DN,N (q)| Nq kx
N ,q N,kx

,kx +qx

− f¯N,kx

1−

Λ2
2Ω2

Λ2
δ (εN (kx + qx ) − εN (kx ) − ωq + Ω)
4Ω2
Λ2
Λ2
+ 2 δ (εN (kx + qx ) − εN (kx ) − ωq − Ω) + f¯N ,kx −qx − f¯N,kx 1 −
4Ω
2Ω2
2
Λ
×δ (εN (kx − qx ) − εN (kx ) + ωq ) +
δ (εN (kx − qx ) − εN (kx ) + ωq + Ω)
4Ω2
2
Λ
+ 2 δ (εN (kx − qx ) − εN (kx ) + ωq − Ω) δ (ε − εN (kx )) .

4Ω

×δ (εN (kx + qx ) − εN (kx ) − ωq ) +

∞

dielectric constants, respectively.
After some calculation, we find the expression for conductivity tensor:

The total current density is given by J = 0 jN,N (ε)dε
or Ji = σim E1m . We now consider only the electronoptical phonon interaction. We also consider the electron
gas to be nondegenerate (the Fermi-Dirac distribution
becomes a Boltzmann distribution). In this case, ωq
ω0 is taken, and Cq is [5,6]
|Cq |2 =

2πe2 ω0
2
0q

1
1
−
χ∞
χ0

,

σim =

(14)

where 0 is the electric constant (vacuum permittivity),
and χ0 and χ∞ are the static and the high-frequency

a =
b =
b1 =

e2 Lx
2πm
2πeN0
m

π
αβ

exp β εF − N +
N

1
2

τ
δij − ωc τ εijk hk + ωc2 τ 2 hi hj
1 + ωc2 τ 2
τ
be
×{aδjm +
δjl

m 1 + ωc2 τ 2

× δlm − ωc τ εlmp hp + ωc2 τ 2 hl hm },

(15)

,

(16)

where

ωp +

e2 E12
γ2
+
2
2mωp
4α

{b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8 } ,

(17)

N,N

−βALx
1
I (N, N ) exp β εF − N +

64π 3 α2
2
× α

(13)

C12
α2

1
4

K 12

β |C1 |
2

− γK0

β |C1 |
2

ωp +

γ2
e2 E12
C1
+
−
2

2mωp
2
4α

+ C1

C12
α2

− 14

K− 12

β |C1 |
2

,

(18)


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Journal of the Korean Physical Society, Vol. 60, No. 1, January 2012

b2 =

−βθALx
1
I (N, N ) exp β εF − N +

64π 3 α2
2
× α

b3 =

3
4

K 32

β |C1 |
2

−γ

C12
α2

1
2

C22
α2

3
4

K 32

β |C2 |
2

−γ

C22
α2

ωp +

1
2

× α

C32
α2

3
4

K 32

β |C3 |
2

−γ

C32
α2

ωp +

1
2

K1

+ C1

C12
α2

1
4

K 12

β |C1 |
2

,

(19)

K 12

β |C2 |
2

,

(20)

K 12

β |C3 |
2

,

(21)

γ2
e2 E12
C2
+
−
2mωp2
2
4α

β |C2 |
2

K1

−βθALx
1
I (N, N ) exp β εF − N +

128π 3 α2
2

γ2
e2 E12
C1
+
−
2mωp2
2
4α

β |C1 |
2

K1

−βθALx
1
I (N, N ) exp β εF − N +
128π 3 α2
2
× α

b4 =

C12
α2

ωp +

+ C2

C22
α2

1
4

γ2
e2 E12
C3
+
−
2mωp2
2
4α

β |C3 |
2

+ C3

C32
α2

1
4

b5 = b1 (C1 → D1 ), b6 = b2 (C1 → D1 ), b7 = b3 (C2 → D2 ), b8 = b4 (C3 → D3 ),

β = 1/(kB T ), α = ( 2 /2m)(1 − ωc2 /ωp2 ), γ = eE1 ωc /mωp2 ,
2πe2 ω0 −1
e2 E02
(1 − ωc2 /ωp2 ), A =
χ∞ − χ−1
,
0
2
4
m Ω
0
= (N − N ) ωp − ω0 , C2 = C1 + Ω, C3 = C1 − Ω,
= (N − N ) ωp + ω0 , D2 = D1 + Ω, D3 = D1 − Ω,

θ =
C1
D1

(22)
(23)
(24)

(25)

and
∞

I(N, N ) =

−∞

|IN,N (qz )|2 dqz .

(26)

The HC is given by the formula [21]
RH =

1
σxz
ρxz
=−
,
2 + σ2
B
B σxz
zz

(27)

where σxz and σxx are given by Eq. (15). Equation (27)
shows the dependence of the HC on the external fields,
including the EMW. It is obtained for arbitrary values
of the indices N and N . In the next section, we will
give a deeper insight into this dependence by carrying
out a numerical evaluation with the help of computer
programm.

Fig. 1. Hall coefficients (arb. units) as functions of the
EMW frequency Ω at B = 4.00 T (solid line), B = 4.05 T

(dashed line), and B = 4.10 T (dotted line). Here, ωz =
0.5 × ω0 , E = 5 × 105 V/m, E0 = 105 V/m, and T = 270 K.

III. NUMERICAL RESULTS AND
DISCUSSION
In this section, we present detailed numerical calculations of the HC in a QWPP subjected to uniform
crossed magnetic and electric fields in the presence of
an EMW. For the numerical evaluation, we consider the
model of a QWPP of GaAs/AlGaAs with the following parameters: [20, 22] εF = 50 meV, χ∞ = 10.9,
χ0 = 12.9, ω0 = 36.25 meV (optical phonon frequency),
and m = 0.067 × m0 (m0 is the mass of a free electron).
For the sake of simplicity, we also choose N = 0, N = 1,
τ = 10−12 s, and Lx = 10−9 m.

The HC is plotted as function of the EMW frequency
at different values of the magnetic field in Fig. 1. The
HC can be seen to increase strongly with increasing
EMW frequency for the region of small values (Ω <
2.5 × 1013 s−1 ) and reaches saturation as the EMW frequency continues to increase. Moreover, the HC is very
sensitive to the magnetic field at the chosen values of the
other parameters; concretely, the value of the HC raises
remarkably when the magnetic field increases slightly.
In Fig. 2 and Fig. 3, we show the dependence of the
HC on the magnetic field at different values of the tem-


Influence of a Strong Electromagnetic Wave (Laser Radiation) · · · – Nguyen Quang Bau and Bui Dinh Hoi

Fig. 2. Hall coefficients (arb. units) as functions of the
magnetic field at temperatures of 260 K (solid line), 270 K

(dashed line), and 280 K (dotted line). Here, ωz = 0.5 × ω0 ,
E = 5 × 105 V/m, E0 = 105 V/m, and Ω = 5 × 1013 s−1 .

Fig. 3. Hall coefficients (arb. units) as functions of the
dc electric field at different values of confinement frequency:
ωz = 0.3 × ω0 (solid line), ωz = 0.4 × ω0 (dashed line), and
ωz = 0.5 × ω0 (dotted line). Here, T = 270 K, B = 4 T,
E0 = 105 V/m, and Ω = 5 × 1013 s−1 .

perature T and on the the dc electric field E1 at different
values of the confinement frequency ωz , respectively; the
necessary parameters involved in the computation are
the same as those in Fig. 1. We can describe the behavior
of the HC in Fig. 2 as follows: Each curve has one maximum and one minimum. As the magnetic field increases,
the HC is positive, reaches the maximum value and then
decreases suddenly to a minimum with a negative value.
When the magnetic field is increased further, the HC
increases continuously (with negative values). Particularly, the values of HC at the maxima are much larger
and at the minima, they are much smaller than other
values. Moreover, the increasing temperature not only
brings down the value of the HC but also shifts the maxima and the minima to the right. Also, the values of the
HC at maxima (minima) at different temperatures are
very different; for instance, the maximum at tempera-

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Fig. 4. Hall coefficients (arb. units) as functions of the
amplitude of the electric field E0 at temperatures of 269 K
(solid line), 270 K (dashed line), and 271 K (dotted line).
Here, ωz = 0.5 × ω0 , B = 4 T, E = 5 × 105 V/m, and

Ω = 5 × 1013 s−1 .

ture of 260 K is approximately twice larger than it is at
270 K. Thus, we can conclude that the HC is very sensitive to the temperature. The dependences of the HC on
the dc electric field E1 and the confinement frequency in
Fig. 3 can be analyzed similarly.
Figure 4 shows the dependence of the HC on the amplitude E0 of the EMW at different values of the temperature. From this figure, we can see that the dependence
of the HC on the amplitude E0 is nonlinear. The HC
parabolically decreases with increasing amplitude E0 of
the EMW and strongly depends on the temperature so
that as the temperature increases, the HC decreases evidently. This confirms once again that the HC is quite
sensitive to the change in the temperature.

IV. CONCLUSIONS
In this work, we have studied the influence of laser
radiation on the Hall effect in quantum wells with a
parabolic potential subjected to crossed dc electric and
magnetic fields. The electron-optical phonon interaction
is taken into account at high temperatures, and the electron gas is nondegenerate. We obtain the expressions for
the Hall conductivity as well and the HC. The influence
of the EMW is interpreted by using the dependences of
the Hall conductivity and the HC on the amplitude E0
and the frequency Ω of the EMW and by using the dependences on the magnetic B and the dc electric field E1
as in the ordinary Hall effect. The analytical results are
numerically evaluated and plotted for a specific quantum
well, GaAs/AlGaAs, to show clearly the dependence of
the Hall conductivity on the external fields and the parameters of the system. From the numerical results, we
can summarize the main points as follows: The HC depends nonlinearly on the amplitude E0 of the EMW, and

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Journal of the Korean Physical Society, Vol. 60, No. 1, January 2012

it increases strongly with increasing EMW frequency for
the small values of the EMW frequency and reaches saturation as the EMW frequency continues to increase. As
the magnetic field increases, the HC is positive, reaches
its maximum value and then decreases suddenly to a
minimum with a negative value; also, the values of the
HC at a maxima are much larger and at the minima
are much smaller than other values. Furthermore, the
values of the HC at maxima (minima) at different temperatures are very different; for instance, the maximum
at a temperature of 260 K is approximately twice that
at 270 K, as shown in Fig. 2. This means that the HC is
very sensitive to the temperature.

ACKNOWLEDGMENTS
This work was completed with financial support from
the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) and the Project
of Basic Research in Natural Science, Vietnam National
University in Hanoi (project code: QG. TD. 10. 02).

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