59∼64Influence 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, Col
Trang 1Journal 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) = mω z z2/2 (where m and ω z 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 = (E0sin Ωt, 0, 0)
(whereE0 and Ω are the amplitude and the frequency of EMW, respectively) We obtain analytic
expressions for the componentsσ zzandσ xzof the Hall conductivity as well as a Hall coefficient with
a dependence onB, E1,E0, Ω, temperatureT 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 amplitudeE0 and the frequency Ω of EMW and by using the dependences
on the magnetic fieldB 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
Recently, there has been considerable interest in the
behavior of low-dimensional systems, in particular,
two-dimensional electron gas (2DEG) systems, such as
quan-tum wells and compositional and doped superlattices
The confinement of electrons in these systems
consid-erably enhances the electron mobility and leads to
un-usual behaviors under external stimuli As a result, the
properties of low-dimensional systems, especially
electri-cal and optielectri-cal properties, are very different in
compar-ison from those of normal semiconductors [1, 2] There
have been many papers dealing with problems related
to the incidence of electromagnetic wave (EMW) in
low-dimensional 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
non-linear 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: nguyenquangbau54@gmail.com; Tel: +84-913-348-020
magnetic field [9,10] and the effect of the presence of an additional (high frequency) EMW [11] have been stud-ied 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 re-cent years many physicists have been interested in inves-tigating 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 impurity and electron-acoustic 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 ap-ply 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 = (E0sin Ωt, 0, 0), the confinement potential being as-sumed to be V (z) = mω z z2/2 We only consider the
case in which the electron - optical phonon interaction
is assumed to be dominant and electron gas to be non-degenerate We derive the analytical expressions for the conductivity tensor and the Hall coefficient (HC) The
Trang 2-59-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
calcula-tion Numerical results and discussion are given in Sec
III Finally, remarks and conclusions are shown briefly
in Sec IV
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
poten-tial A = (zB, 0, 0) to describe the applied DC magnetic
field If the confinement potential is assumed to take
the form V (z) = mω z z2/2, then the single-particle wave
function and its eigenenergy are given by [20]
Ψ (r) = 1
2π e
ik ⊥ r φ N (z − z0) , (1)
ε N (k x) = ω p
N +1
2
+ 1
2m
2k2x −
k x ω c + eE1
ω p
2
, (2)
where m and e are the effective mass and the charge of
a conduction electron, respectively, k ⊥ = (k x , k y) is its
wave vector in the (x, y) plan; z0= (k x ω c + eE1)/mω2;
ω2p = ω z2+ ω c2, ω z and ω c = eB/m are the confinement
and the cyclotron frequencies, respectively, and
φ N (z − z0) = H N (z − z0) exp
− (z − z0)2
/2
, (4) with H N (z) being the Hermite polynomial of N thorder
2 Expressions for the Hall Conductivity and
the Hall Coefficient
In the presence of an EMW with electric field vector
E = (E0sin Ωt, 0, 0) (where E0 and Ω are the
ampli-tude 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-resentation can be written as
H0 =
N,k x
ε N
k x − c e A(t) a+
N,k x a N,k x
q
N,N
q,k x
D N,N (q)a+N ,k
x +q x a N,k x (b q + b+−q ), (7)
where |N,k x > and |N , k x + q ⊥ > are electron states
before and after scattering;ω qis the energy of an
opti-cal phonon with the wave vector q = (q ⊥ , q z ); a+N,k
x and
a N,k x (b+q and b q) are the creation and the annihilation
operators of electron (phonon), respectively; A(t) is the vector potential of laser field; D N,N (q) = C q I N,N (q z),
where C q is the electron-phonon interaction constant;
and I N,N (q z ) = < N |e iq z z |N > is the form factor of
electron The quantum kinetic equation for electrons in the single (constant) scattering time approximation takes the form
∂f N,k x
∂t − e E1+ ω c
k x × h N,k x
∂k x
+k x
m
∂f N,k x
∂r =− f N,k x − f0
τ , (8) where k x = (k x , 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
distri-bution function (Fermi - Dirac distridistri-bution), f N,k x is an unknown distribution function perturbed due to the
ex-ternal fields, and τ is the electron momentum relaxation
time, which is assumed to be constant
In order to find f N,k x, we use the general quantum equation for the particle number operator [5–8] or the
electron distribution function f N,k x=
a+N,k
x a N,k x
t:
i ∂
∂t f N,k x=
a+N,k
x a N,k x , H
From Eqs (8) and (9), using the Hamiltonian in Eq (5),
we find
− e E1+ ω c
k x × h N,k x
∂k x
+k x
m
∂f N,k x
∂r =− f N,k x − f0
2π
N ,q
|D N,N (q)|2
× ∞
l=−∞
J 2
Λ
Ω [ ¯f N ,k x +q x (N q+ 1)
− ¯ f N,k x N q ]δ (ε N (k x + q x)− ε N (k x)− ω0
+ [ ¯f N ,k x −q x N q − ¯ f N,k x (N q + 1)]δ (ε N (k x − q x)
−ε N (k x) +ω0 , (10)
Trang 3Influence of a Strong Electromagnetic Wave (Laser Radiation) · · · – Nguyen Quang Bau and Bui Dinh Hoi
-61-in which ¯f N,k x (N q) is the time-independent
compo-nent of the distribution function of electrons (phonons),
J th -order Bessel function of argument x, and
Λ = (eE0q x /mΩ2)(1− ω2
c /ω2) Equation (10) is fairly general and can be applied for any mechanism of
inter-action In the limit of ω z → 0, i.e., the confinement
vanishes, it gives the same results as those obtained in
bulk semiconductor [9–11]
For simplicity, we limit the problem to the case of
(e/m)k x δ (ε − ε N (k x)) and carry out the summation over
N and k x, we have the equation for the partial current
density j N,N (ε) (the current caused by electrons that have energy of ε):
j N,N (ε)
where
Q N (ε) = − e
m
N,k x
k x
F ∂f N,k x
∂k x
and
S N,N (ε) = 2πe
m
N ,q
N,k x
|D N,N (q)|2N q k x¯
f N ,k x +q x − ¯ f N,k x
1− Λ2
2Ω2
×δ (ε N (k x + q x)− ε N (k x)− ω q) + Λ
2
4Ω2δ (ε N (k x + q x)− ε N (k x)− ω q+Ω) + Λ2
4Ω2δ (ε N (k x + q x)− ε N (k x)− ω q − Ω)+¯
f N ,k x −q x − ¯ f N,k x
1− Λ2
2Ω2
×δ (ε N (k x − q x)− ε N (k x) +ω q) + Λ
2
4Ω2δ (ε N (k x − q x)− ε N (k x) +ω q+Ω) + Λ2
4Ω2δ (ε N (k x − q x)− ε N (k x) +ω q − Ω)δ (ε − ε N (k x )) (13)
The total current density is given by J =∞
0 j N,N (ε)dε
or J i = σ im E 1m We now consider only the
electron-optical phonon interaction We also consider the electron
gas to be nondegenerate (the Fermi-Dirac distribution
becomes a Boltzmann distribution) In this case, ω q
ω0is taken, and C q is [5,6]
|C q |2= 2πe2ω0
0q2
1
χ ∞ − 1
χ0
where 0 is the electric constant (vacuum permittivity),
and χ0 and χ ∞ are the static and the high-frequency
dielectric constants, respectively
After some calculation, we find the expression for con-ductivity tensor:
σ im = τ
1 + ω2c τ2
δ ij − ω c τ ε ijk h k + ω c2τ2h i h j
×{aδ jm+be
m
τ
1 + ω2c τ2δ jl
×δ lm − ω c τ ε lmp h p + ω2c τ2h l h m
}, (15) where
a = e
2L x
2πm
π αβ
N
exp
β
ε F −
N +12
ω p+ e2E12 2mω2 +
γ2 4α
b = 2πeN0
m
N,N
{b1+ b2+ b3+ b4+ b5+ b6+ b7+ b8} , (17)
b1 = −βAL x
64π3α2I (N, N
) exp
β
ε F −
N +1
2
ω p+ e2E2 2mω2 − C1
2 +
γ2 4α
×
α
C2
α2
1
K1
β |C1|
2
− γK0
β |C1|
2
+ C1
C2
α2
−1
K −1
β |C1|
2
Trang 4
b2 = −βθAL x
64π3α2 I (N, N
) exp
β
ε F −
N +12
ω p+ e2E2 2mω2 − C1
2 +
γ2 4α
×
α
C12
α2
3
K3
β |C1|
2
− γ
C12
α2
1
K1
β |C1|
2
+ C1
C12
α2
1
K1
β |C1|
2
b3 = −βθAL x
128π3α2I (N, N
) exp
β
ε F −
N +12
ω p+ e2E12 2mω2 − C22+ γ2
4α
×
α
C2
α2
3
K3
β |C2|
2
− γ
C2
α2
1
K1
β |C2|
2
+ C2
C2
α2
1
K1
β |C2|
2
b4 = −βθAL x
128π3α2I (N, N
) exp
β
ε F −
N +1
2
ω p+ e2E2 2mω2 − C3
2 +
γ2 4α
×
α
C2
α2
3
K3
β |C3|
2
− γ
C2
α2
1
K1
β |C3|
2
+ C3
C2
α2
1
K1
β |C3|
2
b5 = b1(C1→ D1), b6= b2(C1→ D1), b7= b3(C2→ D2), b8= b4(C3→ D3), (22)
β = 1/(k B T ), α = (2/2m)(1 − ω c2/ω p2), γ = eE1ω c /mω p2, (23)
θ = e2E2
m2Ω4(1− ω2
c /ω p2), A = 2πe2ω0
0
χ −1 ∞ − χ −1
0
C1 = (N − N)ω p − ω0, C2= C1+Ω, C3= C1− Ω,
D1 = (N − N)ω p+ω0, D2= D1+Ω, D3= D1− Ω, (25)
and
I(N, N ) =
∞
−∞ |I N,N (q z)|2dq z (26) The HC is given by the formula [21]
R H =ρ xz
B =−1
B
σ xz
σ xz2 + σ2zz , (27) where σ xz and σ xxare 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
III NUMERICAL RESULTS AND
DISCUSSION
In this section, we present detailed numerical
calcu-lations 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
follow-ing 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 L x= 10−9 m.
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, ω z =
0.5 × ω0,E = 5 × 105 V/m,E0= 105 V/m, andT = 270 K.
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
fre-quency 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
Trang 5tem-Influence of a Strong Electromagnetic Wave (Laser Radiation) · · · – Nguyen Quang Bau and Bui Dinh Hoi
-63-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,ω z = 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:
ω z = 0.3 × ω0 (solid line),ω z = 0.4 × ω0 (dashed line), and
ω z = 0.5 × ω0 (dotted line) Here, T = 270 K, B = 4 T,
E0= 105 V/m, and Ω = 5× 1013s−1.
perature T and on the the dc electric field E1at different
values of the confinement frequency ω z, 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
maxi-mum and one minimaxi-mum 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)
Particu-larly, 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
max-ima and the minmax-ima to the right Also, the values of the
HC at maxima (minima) at different temperatures are
very different; for instance, the maximum at
tempera-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, ω z = 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 sensi-tive to the temperature The dependences of the HC on
the dc electric field E1and the confinement frequency in
Fig 3 can be analyzed similarly
Figure 4 shows the dependence of the HC on the
am-plitude E0of the EMW at different values of the
temper-ature 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 ev-idently 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 elec-tron 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
de-pendences 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 pa-rameters of the system From the numerical results, we can summarize the main points as follows: The HC
de-pends nonlinearly on the amplitude E0of the EMW, and
Trang 6it increases strongly with increasing EMW frequency for
the small values of the EMW frequency and reaches
sat-uration 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
tem-peratures 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
De-velopment 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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... increased further, the HCincreases continuously (with negative values)
Particu-larly, the values of HC at the maxima are much larger
and at the minima, they are much smaller... class="text_page_counter">Trang 5
tem -In? ??uence of a Strong Electromagnetic Wave (Laser Radiation) · · · – Nguyen Quang Bau and Bui Dinh Hoi... 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