The Hall effect is studied theoretically in a doped  semiconductor superlattice (DSSL) subjected to a crossed dc electric field  and magnetic field in the presence of an intense electromagnetic wave (EMW).  By using the quantum kinetic equation for electrons interacting with  acoustic phonons at low temperature, we obtain expressions for the  magnetoresistance as well as the Hall coefficient in dependence on  the   external fields and characteristic parameters of the DSSL. Analytical  results are numerically evaluated for the GaAs:Si/GaAs:Be DSSL. The  dependence of the magnetoresistance on the magnetic field is consistent with  the result obtained for some two-dimensional electron systems. The Hall  coefficient depends weakly on the magnetic field and its value in the  presence of the EMW is smaller than  that of the case without EMW

Bau, Nguyen Quang

Hoi, Bui Dinh

Phong, Tran Cong

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 24, No. 3S1 (2014), pp. 45-50
DOI:10.15625/0868-3166/24/3S1/5135
HALL EFFECT IN THE DOPED SEMICONDUCTOR SUPERLATTICE WITH
AN IN-PLANE MAGNETIC FIELD UNDER INFLUENCE OF AN INTENSE
ELECTROMAGNETIC WAVE
NGUYEN QUANG BAU
Faculty of Physics, Hanoi University of Science, Vietnam National University, Hanoi, Vietnam
BUI DINH HOI
Department of Physics, National University of Civil Engineering, Hanoi, Vietnam
TRAN CONG PHONG
Center for Theoretical and Computational Physics, Hue University’s College of Education,
Hue city, Vietnam
E-mail: hoibd@nuce.edu.vn
Received 04 April 2014
Accepted for publication 24 August 2014
Abstract. The Hall effect is studied theoretically in a doped semiconductor superlattice (DSSL) subjected to a crossed
dc electric field and magnetic field in the presence of an intense electromagnetic wave (EMW). By using the quantum
kinetic equation for electrons interacting with acoustic phonons at low temperature, we obtain expressions for the mag-
netoresistance as well as the Hall coefficient in dependence on the external fields and characteristic parameters of
the DSSL. Analytical results are numerically evaluated for the GaAs:Si/GaAs:Be DSSL. The dependence of the magne-
toresistance on the magnetic field is consistent with the result obtained for some two-dimensional electron systems. The
Hall coefficient depends weakly on the magnetic field and its value in the presence of the EMW is smaller than that of
the case without EMW.
Keywords: Hall effect, magnetoresistance, electron-phonon interaction, doping superlattice.
I. INTRODUCTION
A doped semiconductor superlattice (DSSL) is formed by layers of n-type and p-type semi-
conductors arranged alternatively between an intrinsic semiconductor (so called n-i-p-i superlat-
tice). The most important advantage of DSSLs is the possibility of adjustment of their parameters
to get long lifetimes, high carrier mobility. The electrical and optical properties of the DSSL
structure may be modulated within a wide range of values. This kind of low-dimensional system,
has attracted much attention, both theoretically and experimentally.
In two-dimensional (2D) electron systems, the Hall effect has been studied in many aspects
(see [1] for recent reviews). However, most of previous works only considered the case when
an electromagnetic wave (EMW) was absent and at the temperature that electron - electron and
electron - impurity interactions to be dominant (conditions for the integer and fractional quan-
tum Hall effect). The propagation of an EMW in materials leads to the change in the scattering
c©2014 Vietnam Academy of Science and Technology
46 HALL EFFECT ON THE DOPED SEMICONDUCTOR SUPERLATTICE WITH AN IN-PLANE MAGNETIC FIELD...
probability of carriers, and thus, leads to their unusual behaviors in comparison to the case of the
absence of the EMW. In some recent works, we have used the quantum kinetic equation method
to study the influence of a high-frequency EMW on the Hall effect in parabolic quantum wells [2],
doping semiconductor superlattices [3] at high temperatures, and in rectangular quantum wells at
low temperature [4]. We also have used this method to investigate the acoustoelectric and mag-
netoacoustoelectric effects [5, 6]. The Hall effect in low-dimensional systems, especially in 2D
semiconductor systems, under the influence of a high-frequency EMW still remains problems to
study, especially by analytical and computational methods.
In this work, by using the quantum kinetic equation method, we study the Hall effect in the
DSSL subjected to an in-plane magnetic field in the presence of an intense EMW. We only con-
sider the case of low temperatures when the electron - acoustic phonon interaction is assumed to be
dominant and electron gas is weakly degenerate. We derive analytical expressions for the magne-
toresistance (MR) and the Hall coefficient (HC) taking account of arbitrary transitions between the
energy levels. The analytical result is numerically evaluated and graphed for the GaAs:Si/GaAs:Be
DSSL. In the next section, we write out the electronic structure and the Hamiltonian of the system.
Analytical results of the calculation are presented in Sec. III. Numerical results, discussion, and
remarks are given in Sec. IV.
II. HAMILTONIAN OF ELECTRON – PHONON SYSTEM
We consider a simple model of the DSSL in which electron gas is confined by the su-
perlattice potential along the z-direction and free in the (x− y) plane. The motion of an electron
is limited in each layer of the system and its energy spectrum is quantized into discrete levels in
the z-direction. According to K. Ploog and G. H. Dohler [7], in an ideal model, the DSSL can
be treated as a multiquantum well structure in which the confinement potential in each well is
parabolic. In most cases of interest it is justified to neglect the interaction between neighboring
potential wells. Therefore, the single-particle wave function and corresponding eigen-energy in
the DSSL has the same mathematical forms with those obtained for a harmonic oscillator. If the
DSSL is subjected to a crossed dc electric field ~E1 = (0,0,E1) and magnetic field ~B= (0,B,0), the
single-particle wave function and corresponding eigenenergy of an electron in a single potential
well are given by [8]
|N,kx〉= ψ0ei~k⊥~r⊥ΦN (z− z0) , (1)
εN
(
~kx
)
=
(
N+
1
2
)
h¯ω˜+
1
2me
[
h¯2k2x −
(
h¯kxωc+ eE1
ω˜
)2]
, N = 0,1,2, ..., (2)
where ψ0 is a normalization factor, me and e are the effective mass and the charge of a con-
duction electron, respectively, ~k⊥ = (kx,ky) is its wave vector in the (x− y) plane, ~r⊥ = (x,y),
~kx = (kx,0,0), z0 = (h¯kxωc+ eE1)/meω˜2, ω˜2 = ω2c +ω2p, ωc is the cyclotron frequency and
ωp =
(
e2nD/κ0me
)1/2is the plasma frequency characterizing for the DSSL confinement in the
z-direction, where κ0 is the electronic constant (vacuum permittivity) and nD is the doping con-
centration, and [8, 9]
ΦN (z− z0) =
√
1
2NN!
√
pi`z
exp
(
−(z− z0)
2
2`2z
)
HN
(
z− z0
`z
)
(3)
NGUYEN QUANG BAU AND BUI DINH HOI 47
Fig. 1. The real part (solid line) and the imagine part (dashed
line) of the form factor (dimensionless) in the region −2pi/d ≤
qz ≤ 2pi/d. Here, d = 100nm, s0 = 50, and nD = 1023m−3.
with HN (z) being the Hermite poly-
nomials of Nth order and `z =
(h¯/(meωp))1/2. In the limit of nD→
0, the energy spectrum (2) becomes
the one obtained in normal semicon-
ductors by Kahn and Frederikse [10].
In the presence of an intense
EMW with electric field vector ~E =
(E0 sin(Ωt) ,0,0) (E0 and Ω are the
amplitude and the frequency of the
EMW, respectively), the Hamilton-
ian of the electron-acoustic phonon
system in the DSSL, in the second
quantization representation, can be
written as
H = H0+U, (4)
H0 =∑N,~kx εN
(
~kx− eh¯c
~A(t)
)
a+
N,~kx
aN,~kx +∑~q h¯ω~qb~q+b~q, (5)
U = ∑
N,N′
∑~q,~kx DN,N′ (~q)a+N′,~kx+~qxaN,~kx
(
b~q+b+−~q
)
, (6)
where
∣∣∣N,~kx〉 and ∣∣∣N′,~kx+~qx〉 are electron states before and after scattering, h¯ω~q is the energy
of a phonon with the wave vector ~q = (qx,qy,qz), a+N,~kx
and aN,~kx(b
+
~q and b~q) are the creation and
annihilation operators of electron (phonon), respectively, ~A(t) is the vector potential of the EMW,
and DN,N′ (~q) is the matrix element of interaction, given by [9, 11]∣∣DN,N′ (~q)∣∣2 = ∣∣C~q∣∣2 ∣∣IN,N′ (±qz)∣∣2 , (7)
where C~q is the electron-phonon interaction constant which depends on the interacting mechanism,
IN,N′ (±qz) is the form factor of electron, given by
IN,N′ (±qz) =
s0
∑
℘=1
d∫
0
e±iqzdϕN (z−℘d)ϕN′ (z−℘d)dz, (8)
with d is the period and s0 is the number of periods of the DSSL. In Fig. 1, we show the real and
the imagine parts of the form factor versus the component qz in two first mini Brillouin zones.
III. ANALYTICAL RESULTS
By using Hamiltonian (1) and procedures as in the previous works [2–6], we obtain an
equation for the partial current density in the single (constant) relaxation time approximation.
To do this, we assume that the electrons system is degenerate and the distribution function has
the form of Heaviside step function. For the electron – acoustic phonon interaction, h¯ω~q = h¯vsq
and [11] ∣∣C~q∣∣2 = ξ 2q2ρvsV0 , (9)
48 HALL EFFECT ON THE DOPED SEMICONDUCTOR SUPERLATTICE WITH AN IN-PLANE MAGNETIC FIELD...
where vs, ξ , ρ , and V0 are, respectively, the sound velocity, the acoustic deformation potential, the
mass density, and the normalization volume of the specimen. After some manipulation, we obtain
the expression for the conductivity tensor:
σim =
eτ
me
bτ
1+ω2c τ2
(
δi j−ωcτεi jkhk +ω2c τ2hih j
)
δ jl
[
δ`m−ωcτε`mphp+ω2c τ2h`hm
]
, (10)
where τ is the momentum relaxation time, δi j is the Kronecker delta, εi jk being the antisymmetrical
Levi - Civita tensor, the Latin symbols i, j,k,m, l, p stand for the components x,y,z of the Cartesian
coordinates,
b =
2eLxA√
2pi2meh¯
∑N,N′ I
(
N,N′
)
[b1+b2+b3+b4], (11)
b1 =
1√
∆k∆
(1)
q
{
k+0
{(
q+1(+)
)3 [
exp
(√
2β h¯vsq+1(+)
)
−1
]−1
+
(
q−1(+)
)3 [
exp
(√
2β h¯vsq−1(+)
)
−1
]−1}
+k−0
{(
q+1(−)
)3 [
exp
(√
2β h¯vsq+1(−)
)
−1
]−1
+
(
q−1(−)
)3 [
exp
(√
2β h¯vsq−1(−)
)
−1
]−1}}
,
(12)
b2 =− θ
2
√
∆k∆
(1)
q
{
k+0
{(
q+1(+)
)5 [
exp
(√
2β h¯vsq+1(+)
)
−1
]−1
+
(
q−1(+)
)5 [
exp
(√
2β h¯vsq−1(+)
)
−1
]−1}
+ k−0
{(
q+1(−)
)5 [
exp
(√
2β h¯vsq+1(−)
)
−1
]−1
+
(
q−1(−)
)5 [
exp
(√
2β h¯vsq−1(−)
)
−1
]−1}}
,
(13)
b3 =
θ
4
√
∆k∆
(2)
q
{
k+0
{(
q+2(+)
)5 [
exp
(√
2β h¯vsq+2(+)
)
−1
]−1
+
(
q−2(+)
)5 [
exp
(√
2β h¯vsq−2(+)
)
−1
]−1}
+k−0
{(
q+2(−)
)5 [
exp
(√
2β h¯vsq+2(−)
)
−1
]−1
+
(
q−2(−)
)5 [
exp
(√
2β h¯vsq−2(−)
)
−1
]−1}}
,
(14)
b4 =
θ
4
√
∆k∆
(3)
q
{
k+0
{(
q+3(+)
)5 [
exp
(√
2β h¯vsq+3(+)
)
−1
]−1
+
(
q−3(+)
)5 [
exp
(√
2β h¯vsq−3(+)
)
−1
]−1}
+ k−0
{(
q+3(−)
)5 [
exp
(√
2β h¯vsq+3(−)
)
−1
]−1
+
(
q−3(−)
)5 [
exp
(√
2β h¯vsq−3(−)
)
−1
]−1}}
,
(15)
β = 1/(kBT ) , θ = e2E20
(
1−ω2c /ω˜2
)
/m2eΩ4, ∆k = γ2−4αδ , α =
(
h¯2/2me
)(
1−ω2c /ω˜2
)
, γ = eE1h¯ωc/meω˜2,
δ = (N+1/2) h¯ω˜−e2E21/
(
2meω˜2
)−εF , ∆(`)q = ∆k−4αC`(`= 1,2,3), C1 = (N′−N) h¯ω˜ , C2 =C1+ h¯Ω,
C3 =C1− h¯Ω, k±0 =
(
γ±√∆k
)
/(2α), q±
`(+)
=
(
−√∆k±
√
∆(`)q
)
/(2α), q±
`(−) =
(√
∆k±
√
∆(`)q
)
/(2α),
A = ξ 2/(2ρvs), εF is the Fermi level, kB being the Boltzmann constant, T is the temperature,Lx is
the normalization length in the x-direction, and
I(N,N′) =
+pi/d∫
−pi/d
∣∣IN,N′ (±qz)∣∣2 dqz. (16)
The MR and the HC are, respectively, given by
ρzz(B) =
σzz
σ2zz+σ2xz
; RH =− 1B
σxz
σ2zz+σ2xz
. (17)
NGUYEN QUANG BAU AND BUI DINH HOI 49
IV. NUMERICAL RESULTS
In the following, we will give a deeper insight into analytical results by carrying out a
numerical evaluation. For the computation, we consider the n-i-p-i DSSL of GaAs:Si/GaAs:Be
with the following parameters [5–8]: ξ = 13.5eV , ρ = 5320 kg ·m−3, vs = 5378 m · s−1, εF = 50
meV, me = 0.067×m0 (m0 is the mass of a free electron), d = 100 nm, s0 = 50. For simplicity,
we consider only the ground and the first-excited states: N = 0,N′ = 1.
Unfortunately, we are not aware of theoretical and experimental works on this problem
other than [12–14] in which the MR and the HC are reported in bulk semiconductors [12], par-
abolic quantum wells [13], and compositional superlattices [14]. Therefore, we cannot compare
our calculation with the experiment. However, we can compare our results with those obtained
in [12–14] in some limits such as low and high doping concentration, weak and strong magnetic
field.
Fig. 2. The relative MR change as func-
tion of the magnetic field in the case of the
absence of the EMW. Here, nD = 1020m−3,
τ = 10−12s, and T = 4.2K.
Fig. 3. The MR as function of the mag-
netic field at high doping concentration for
two cases: the absence and the presence
of the EMW. Here, nD = 7×1024m−3, the
other parameters are the same as in Fig. 1.
Fig. 2 shows the relative change of the MR
∆ρzz/ρzz (0) = ρ (ρzz (B)−ρzz (0))/ρzz (0) with mag-
netic field at T = 4.2K and low doping concentration
(nD = 1020m−3). It is seen that there occurs the nega-
tive MR. In the range of weak magnetic field, the resis-
tance monotonically decreases with increasing the mag-
netic field. This behavior is similar to the one classically
observed in [12] for bulk semiconductors in which the
parameter γ = kBTτ/h¯ is small and in the InSb parabolic
quantum well with weakly degenerate electron gas [13].
The relative change is largest at Bm≈ 1.35 T and is about
45% . In the range B > Bm there is a monotonically in-
creasing function of the magnetic field as observed in
InSb quantum well. However, the most noticeable dif-
ference in this calculation is that the relative MR at low
doping concentration is always negative at any magnetic
field, i.e., the MR can not change its sign as the mag-
netic field increases. This can be explained as follow.
In the calculation for parabolic quantum well [13], the
authors have assumed a model of an infinite potential as
we have done in this calculation. However, at low dop-
ing concentration, this approximation is maybe invalid,
i.e, the confinement potential is not high enough for this
assumption. From these results, we can conclude that at
low doping concentration, the DSSL structure behaves
as both bulk and 2D semiconductors.
In the range of high concentration, the assump-
tion of infinite confinement potential with parabolic law
is well done. In order to illustrate for this conclusion, in
Fig. 3 we plot the MR (in arbitrary units) versus the mag-
netic field at the doping concentration of 7× 1024m−3.
50 HALL EFFECT ON THE DOPED SEMICONDUCTOR SUPERLATTICE WITH AN IN-PLANE MAGNETIC FIELD...
The figure shows that there exists a value of the magnetic field at that the MR changes sign. As the
magnetic field increases, the MR strongly increases. This behavior has been well observed in the
InSb parabolic quantum well [13] and in GaAs/AlGaAs compositional semiconductor superlat-
tices [14] at high magnetic field. It is also seen from Fig. 3 that the presence of the EMW changes
the MR considerably. The MR is positive in the absence of the EMW, whereas it is negative if the
EMW is swiched on.
Fig. 4. The HC as function of the mag-
netic field for two cases: the absence and
the presence of the EMW. Here, nD =
7×1020m−3, and τ = 5.7×10−11s.
Fig. 4 shows the dependence of the HC on the
magnetic field. The HC is seen to weakly decrease with
increasing the field. In the range of weak field, the HC
is nearly independent of the magnetic field. This depen-
dence is consistent with the result obtained for a degen-
erate electrons system [12] where γ is small (γ = 5). It is
usually observed in experiments. It is also seen that the
presence of the intense EMW does not change the be-
havior of the HC but changes its value. The EMW leads
to a decrease of the HC at a fixed magnetic field.
In conclusion, the Hall effect in DSSLs subjected
to a crossed dc electric field and magnetic field in the
presence of an intense EMW has been theoretically stud-
ied. In general, the results without the EMW are in ac-
cordance with those obtained in some 2D electron sys-
tems. To our knowledge, the results for the presence of
the EMW are new and experimental works are needed to test their validity.
ACKNOWLEDGMENT
This research is funded by the Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 103.01-2011.18.
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Hall Effect on the Doped Semiconductor Superlattice with an In-plane Magnetic Field Under Influence of an Intense Electromagnetic Wave

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Hall Effect on the Doped Semiconductor Superlattice with an In-plane Magnetic Field Under Influence of an Intense Electromagnetic Wave

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