Abstract. We consider a model of the Hall effect when a quantum well (QW)
with a parabolic potential V (z) = mωz2z2/2 (where m and ωz are the effective
mass of electron and the confinement frequency of QW, respectively) is subjected
to a crossed dc electric field (EF) E⃗1 = (E1, 0, 0) and magnetic field B⃗ =
(0, 0, B) in the presence of a strong electromagnetic wave (EMW) characterized
by electric field E⃗ = (0, E0 sin (Ωt) , 0) (where E0 and Ω are the amplitude
and the frequency of the EMW, respectively). By using the quantum kinetic
equation for electrons and considering the electro-optical phonon interaction, we
obtain analytical expressions for the conductivity as well as the Hall coefficient
(HC) with a dependence on B, E1, E0, Ω, the temperature T of the system and
the characteristic parameters of QW. The analytical results are computationally
evaluated and graphically plotted for a specific quantum well, GaAs/AlGaAs.
Numerical results for the conductivity componentσxx show the resonant peaks
which can be explained by the magnetophonon resonance and optically detected
magnetophonon resonance conditions. Also, the HC reaches saturation as the
magnetic field or the EMW frequency increases and weakly depends on the
amplitude of the EMW. Furthermore, the HC in this study is always negative while
it has both negative and positive values in the case of in-plane magnetic field.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 154-166
This paper is available online at
THE HALL COEFFICIENT IN PARABOLIC QUANTUMWELLS
WITH A PERPENDICULAR MAGNETIC FIELD UNDER THE INFLUENCE
OF LASER RADIATION
Bui Dinh Hoi, Do Tuan Long, Pham Thi Trang,
Le Thi Thiem and Nguyen Quang Bau
Faculty of Physics, College of Natural Science,
Vietnam National University, Hanoi
Abstract. We consider a model of the Hall effect when a quantum well (QW)
with a parabolic potential V (z) = mω2zz
2/2 (where m and ωz are the effective
mass of electron and the confinement frequency of QW, respectively) is subjected
to a crossed dc electric field (EF) E⃗1 = (E1, 0, 0) and magnetic field B⃗ =
(0, 0, B) in the presence of a strong electromagnetic wave (EMW) characterized
by electric field E⃗ = (0, E0 sin (Ωt) , 0) (where E0 and Ω are the amplitude
and the frequency of the EMW, respectively). By using the quantum kinetic
equation for electrons and considering the electro-optical phonon interaction, we
obtain analytical expressions for the conductivity as well as the Hall coefficient
(HC) with a dependence on B, E1, E0, Ω, the temperature T of the system and
the characteristic parameters of QW. The analytical results are computationally
evaluated and graphically plotted for a specific quantum well, GaAs/AlGaAs.
Numerical results for the conductivity componentσxx show the resonant peaks
which can be explained by the magnetophonon resonance and optically detected
magnetophonon resonance conditions. Also, the HC reaches saturation as the
magnetic field or the EMW frequency increases and weakly depends on the
amplitude of the EMW. Furthermore, the HC in this study is always negative while
it has both negative and positive values in the case of in-plane magnetic field.
Keywords: Hall effect, quantum kinetic equation, parabolic quantumwells, electron
- phonon interaction.
Received November 14, 2012. Accepted October 8, 2013.
Contact Bui Dinh Hoi, e-mail address: hoibd@nuce.edu.vn
154
The Hall coefficient in parabolic quantum wells with a perpendicular magnetic field...
1. Introduction
The propagation of an electromagnetic wave (EMW) in materials leads to changes
in probability of scattering of carriers, and thus, leads to their unusual properties in
comparison to the case of absence of the EMW. Under the influence of the EMW,
selection rules satisfying the law of energy conservation in the scattering processes
of electrons with carriers are changed. In the past few years, there have been many
papers dealing with problems related to the incidence of electromagnetic wave (EMW)
in low-dimensional semiconductor systems (see Ref. [1] for some examples). Also, the
Hall effect in bulk semiconductors in the presence of an EMW has been studied in
much detail using the quantum kinetic equation method. The odd magnetoresistance was
calculated when the nonlinear semiconductors were subjected to a magnetic field and
an EMW with low frequency [4], the nonlinearity resulted from the nonparabolicity of
distribution functions of carriers. This problem was also studied in the presence of both
low frequency and high frequency EMW [5]. Moreover, in these works, the dependence of
magnetoresistance as well as magnetoconductivity on the relative angle of applied fields
has been considered carefully. The behaviors of this effect are much more interesting in
low-dimensional systems, especially a two-dimensional electron gas (2DEG) system.
The Hall effect in 2DEGs has attracted a great deal of interest in recent years.
However, most previous works considered only the case when the EMW was absent
and the temperature so that electron-electron and electron-impurity interactions were
dominant (conditions for the integral and fractional quantum Hall effect) (see Ref. [8]
for a recent review). To our knowledge, the Hall effect in the PQWs at relatively high
temperatures, especially in the presence of laser radiation (strong EMW) continues to be
a subject of study. Therefore, in a recent work [1] we studied this effect in a PQW when
a magnetic field is oriented in the plane of free motion of electrons (the x − y plane).
The influence of a strong EMW was considered in detail. To show the differences of
the effect when changing the directions of external fields, in this work, using the quantum
kinetic equation method we study the Hall effect in a PQWwith the confinement potential
V (z) = mω2zz
2/2, subjected to a crossed dc electric field (EF) E⃗1 = (E1, 0, 0) and
magnetic field B⃗ = (0, 0, B) (B⃗ is applied perpendicularly to the plane of free motion
of electrons - the x− y plane) in the presence of a strong EMW characterized by electric
field E⃗ = (0, E0 sin (Ωt) , 0). We only consider the case of high temperatures when
the electron-optical phonon interaction is assumed to be dominant and electron gas is
nondegenerate. We derive analytical expressions for the conductivity tensor and the Hall
coefficient (HC) taking into account the arbitrary transitions between the Landau levels
and between the subbands. The analytical result is numerically evaluated and graphed
for a specific quantum well, GaAs/AlGaAs, to show clearly the dependence of the Hall
conductivity and the HC on above parameters.
155
Bui Dinh Hoi, Do Tuan Long, Pham Thi Trang, Le Thi Thiem and Nguyen Quang Bau
2. Content
2.1. The Hall effect in a parabolic quantum well under the influence of
laser radiation
2.1.1. Quantum kinetic equations for electrons
We consider a perfect infinitely high PQW structure with the confinement potential
assumed to be V (z) = mω2zz
2/2, subjected to a crossed dc EF E⃗1 = (E1, 0, 0) and
magnetic field B⃗ = (0, 0, B). If a strong EMW (laser radiation) is applied along the z
direction with the electric field vector E⃗ = (0, E0 sin (Ωt) , 0), the Hamiltonian of the
electron-optical phonon system in the above mentioned PQW in the second quantization
representation can be written as
H = H0 +U, (2.1)
H0 =
∑
N,n,⃗ky
εN,n
(
k⃗y − e~cA⃗ (t)
)
a+
N,n,⃗ky
aN,n,⃗ky +
∑
q⃗
~ωq⃗b+q⃗ bq⃗, (2.2)
U =
∑
N,N ′
∑
n,n′
∑
q⃗,⃗ky
DN,n,N ′,n′ (q⃗) a
+
N ′,n′k⃗y+q⃗y
aN,n,ky
(
bq⃗ + b
+
−q⃗
)
, (2.3)
where k⃗y = (0, ky, 0); ~ωq⃗ is the energy of an optical phonon with the wave vector
q⃗ = (qx, qy, qz); a+N,n,⃗ky and aN,n,⃗ky (b
+
q⃗ and bq⃗) are the creation and annihilation
operators of electron (phonon), respectively; A⃗ (t) is the vector potential of the EMW.
This Hamiltonian has the same number of terms as in the case of an in-plane magnetic
field [1], however, due to the change in directions of the external field, the single-particle
wave function and its total eigenenergy are now totally modified and are given by [10, 13]
Ψ(r⃗) ≡ |N,n, ky⟩ =
√
1
Ly
ΦN (x− x0) eikyyΦn (z) , (2.4)
εN,n
(
k⃗y
)
=
(
N +
1
2
)
~ωc + εn − ~vdky + 1
2
mv2d N,n = 0, 1, 2 . . . (2.5)
whereN is the Landau level index, n is the subband index, Ly is the normalization length
in the y direction, ωc = eB/m is the cyclotron frequency and vd = E1/B is the drift
velocity of electron. Also, ΦN represents harmonic oscillator wave functions centered at
x0 = −ℓ2B (ky −mvd/~) where ℓB =
√
~/ (mωc) is the radius of the Landau orbit in the
x− y plane. and Φn(z) and εn are the wave functions and the subband energy values due
to the parabolic confinement potential in the z direction, respectively, given by
Φn (z) =
√
1
2nn!
√
πℓz
exp
(
− z
2
2ℓ2z
)
Hn
(
z
ℓz
)
, (2.6)
156
The Hall coefficient in parabolic quantum wells with a perpendicular magnetic field...
Φn (z) =
√
1
2nn!
√
πℓz
exp
(
− z
2
2ℓ2z
)
Hn
(
z
ℓz
)
(2.7)
withHn (z) being the Hermite polynomial of nth order and ℓz =
√
~/ (mωz). The matrix
element of interaction, DN,n,N ′,n′ (q⃗) is given by [10, 13]
|DN,n,N ′,n′ (q⃗)|2 = |Cq⃗|2|In,n′ (±qz)|2|JN,N ′ (u)|2 (2.8)
where Cq⃗ is the electron-phonon interaction constant which depends on a
scattering mechanism for electron-optical phonon interaction [5, 13] |Cq⃗|2 =
2πe2~ω0
(
χ−1∞ − χ−10
)
/ (κ0V0q
2) where κ0 is the electric constant (vacuum permittivity),
V0 the normalization volume of specimen, χ0 and χ0 are the static and high-frequency
dielectric constants, respectively, In,n′ (±qz) = ⟨n| e±iqzz |n′⟩ is the form factor of
electron, and |JN,N ′(u)|2 = (N ′!/N !) e−uuN ′−N
[
LN
′−N
N (u)
]2
with LNM(x) is the
associated Laguerre polynomial, u = ℓ2Bq
2
⊥, q
2
⊥ = q
2
x + q
2
y . By using Hamiltonian (2.1)
and procedures performed in the previous works [1, 4, 5], we obtain the quantum kinetic
equation for electrons in the single (constant) scattering time approximation
−
(
eE⃗1 + ωc
[
k⃗y ∧ h⃗
]) ∂fN,n,⃗ky
~∂k⃗y
+
k⃗y
m
∂fN,n,⃗ky
∂r⃗
= −
fN,n,⃗ky − f0
τ
+
2π
~
∑
N ′,n′
∑
q⃗
|DN,n,N ′,n′ (q⃗)|2
+∞∑
s=−∞
J2s
(
λ
Ω
)
×
{[
f¯N ′,n′ ,⃗ky+q⃗y (Nq⃗ + 1)− f¯N,n,⃗kyNq⃗
]
δ
(
εN ′,n′
(
k⃗y + q⃗y
)
− εN,n
(
k⃗y
)
− ~ωq⃗ − s~Ω
)
+
[
f¯N ′,n′ ,⃗ky−q⃗yNq⃗ − f¯N,n,⃗ky (Nq⃗ + 1)
]
δ
(
εN ′,n′
(
k⃗y − q⃗y
)
− εN,n
(
k⃗y
)
+ ~ωq⃗ − s~Ω
)}
,
(2.9)
where h⃗ = B⃗/B is the unit vector along the magnetic field, the notation ∧ represents
the cross product (or vector product), f0 is the equilibrium electron distribution function
(Fermi-Dirac distribution), fN,n,⃗ky is an unknown electron distribution function perturbed
due to the external fields, τ is the electron momentum relaxation time which is assumed to
be constant, f¯N,n,⃗ky (Nq⃗) is the time-independent component of the distribution function
of electrons (phonons), Js(x) is the sth-order Bessel function of argument x; δ (...) being
the Dirac’s delta function, and λ = eE0qy/(mΩ). Equation (2.9) is fairly general and can
be applied for any mechanism of interaction. In the following, we will use it to derive the
conductivity tensor as well as the HC.
2.1.2. Analytical expressions for the conductivity tensor and the Hall coefficient
To keep things simple, we limit the problem to the cases of s = −1, 0, 1. This
means that processes with more than one photon are ignored. If we multiply both sides
157
Bui Dinh Hoi, Do Tuan Long, Pham Thi Trang, Le Thi Thiem and Nguyen Quang Bau
of equation (2.9) by e
m
k⃗yδ
(
ε− εN,n
(
k⃗y
))
and carry out the summation over N and k⃗y,
we have the equation for the partial current density j⃗N,n,N ′,n′ (ε) (the current caused by
electrons that have energy of ε):
j⃗N,n,N ′,n′ (ε)
τ
+ ωc
[
h⃗ ∧ j⃗N,n,N ′,n′ (ε)
]
= Q⃗N,n (ε) + S⃗N,n,N ′,n′ (ε) , (2.10)
where
Q⃗N,n (ε) = − e
m
∑
N,n,⃗ky
k⃗y
(
F⃗
∂fN,n,⃗ky
~∂k⃗y
)
δ(ε− εN,n(k⃗y)), F⃗ = eE⃗1 (2.11)
and
S⃗N,n,N ′,n′ (ε) =
=
2πe
m~
∑
k⃗y,q⃗
∑
N ′,n′
∑
N,n
|DN,n,N ′,n′ (q⃗)|
2
Nq⃗k⃗y
{[
f¯N ′,n′ ,⃗ky+q⃗y − f¯N,n,⃗ky
] [(
1− λ
2
2Ω2
)
× δ
(
εN ′,n′
(
k⃗y + q⃗y
)
− εN,n
(
k⃗y
)
− ~ωq⃗
)
+
λ2
4Ω2
δ
(
εN ′,n′
(
k⃗y + q⃗y
)
− εN,n
(
k⃗y
)
− ~ωq⃗ + ~Ω
)
+
λ2
4Ω2
×δ
(
εN ′,n′
(
k⃗y + q⃗y
)
− εN,n
(
k⃗y
)
− ~ωq⃗ − ~Ω
)]
+
[
f¯N ′,n′ ,⃗ky−q⃗y − f¯N,n,⃗ky
] [(
1− λ
2
2Ω2
)
δ
(
εN ′,n′
(
k⃗y − q⃗y
)
− εN,n
(
k⃗y
)
+ ~ωq⃗
)
+
λ2
4Ω2
δ
(
εN ′,n′
(
k⃗y − q⃗y
)
− εN,n
(
k⃗y
)
+ ~ωq⃗ + ~Ω
)
+
λ2
4Ω2
δ
(
εN ′,n′
(
k⃗y − q⃗y
)
− εN,n
(
k⃗y
)
− ~ωq⃗ − ~Ω
)]}
×δ
(
ε− εN,n
(
k⃗y
))
(2.12)
Solving (2.10) we have the expression for j⃗N,n,N ′,n′ (ε) as follows:
j⃗N,n,N ′,n′ (ε) =
τ
1 + ω2cτ
2
{(
Q⃗N,n(ε) + S⃗N,n,N ′,n′(ε)
)
− ωcτ
(
[⃗h ∧ Q⃗N,n(ε)] + [⃗h ∧ S⃗N,n,N ′,n′(ε)]
)
+ω2cτ
2
(
Q⃗N,n(ε)⃗h+ S⃗N,n,N ′,n′(ε)⃗h
)
h⃗
}
(2.13)
The total current density is given by
J⃗ =
∞∫
0
j⃗N,n,N ′,n′ (ε)dε or Ji = σimE1m. (2.14)
158
The Hall coefficient in parabolic quantum wells with a perpendicular magnetic field...
Inserting (2.13) into (2.14) we obtain the expressions for the current Ji as well as the
conductivity σim after carrying out the analytical calculation. To do this, we consider only
the electron-optical phonon interaction at high temperatures, and the electron system as
being nondegenerate and assumed to obey the Boltzmann distribution function in this
case. Also, we assume that phonons are dispersionless, i.e, ωq⃗ ≈ ω0, Nq⃗ ≈ N0 =
kBT/(~ω0), where ω0 is the frequency of the longitudinal optical phonon, assumed to
be constant, and kB is the Boltzmann constant. Otherwise, the summations over k⃗y and q⃗
are transformed into the integrals as follows [10]
∑
k⃗y
(...)→ Ly
2
Lx/2ℓ2B∫
−Lx/2ℓ2B
(...)dky, (2.15)
∑
q⃗
(...)→ V0
4π2
+∞∫
0
(...)q⊥dq⊥
+∞∫
−∞
dqz =
V0
4π2ℓ2B
+∞∫
0
(...)du
+∞∫
−∞
dqz, (2.16)
here, Lx is the normalization length in the x direction. After some mathematical
manipulation, we find the expression for the conductivity tensor:
σim =
e2τ
~
(
1 + ω2cτ
2
)−1 (
δij − ωcτεijkhk + ω2cτ 2hihj
)
× {aδjm + bδjℓ [δℓm − ωcτεℓmphp + ω2cτ 2hℓhm]} , (2.17)
where δij is the Kronecker delta, εijk is the antisymmetric Levi - Civita tensor; the Latin
symbols i, j, k, l,m, p stand for the components x, y, z of the Cartesian coordinates
a = −~βvdLyI
2πm
∑
N,n
eβ(εF−εN;n), (2.18)
with εF being the Fermi level, and
b =
βAN0LyI
8π2m2
τ
1 + ω2cτ
2
∑
N,n
∑
N ′,n′
I (n, n′) {b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8},
(2.19)
b1 =
1
M
(
eBξ
~
)
exp [β(εF − εN,n)]
[
(N +M)!
N !
]2
× δ [(N ′ −N)~ωc + (n′ − n)~ωz − eE1ξ − ~ω0] ,
b2 = − θ
2M
(
eBξ
~
)3
exp [β(εF − εN,n)]
[
(N +M)!
N !
]2
× δ [(N ′ −N)~ωc + (n′ − n)~ωz − eE1ξ − ~ω0] ,
159
Bui Dinh Hoi, Do Tuan Long, Pham Thi Trang, Le Thi Thiem and Nguyen Quang Bau
b3 =
θ
4M
(
eBξ
~
)3
exp [β(εF − εN,n)]
[
(N +M)!
N !
]2
× δ [(N ′ −N)~ωc + (n′ − n)~ωz − eE1ξ − ~ω0 + ~Ω] ,
b4 =
θ
4M
(
eBξ
~
)3
exp [β(εF − εN,n)]
[
(N +M)!
N !
]2
× δ [(N ′ −N)~ωc + (n′ − n)~ωz − eE1ξ − ~ω0 − ~Ω] ,
b5 =
1
M
(
eBξ
~
)
exp [β(εF − εN,n)]
[
N !
(N +M)!
]2
× δ [(N −N ′)~ωc + (n′ − n)~ωz + eE1ξ + ~ω0] ,
b6 = − θ
2M
(
eBξ
~
)3
exp [β(εF − εN,n)]
[
N !
(N +M)!
]2
× δ [(N −N ′)~ωc + (n′ − n)~ωz + eE1ξ + ~ω0] ,
b7 =
θ
4M
(
eBξ
~
)3
exp [β(εF − εN,n)]
[
N !
(N +M)!
]2
× δ [(N −N ′)~ωc + (n′ − n)~ωz + eE1ξ + ~ω0 + ~Ω] ,
b8 =
θ
4M
(
eBξ
~
)3
exp [β(εF − εN,n)]
[
N !
(N +M)!
]2
× δ [(N −N ′)~ωc + (n′ − n)~ωz + eE1ξ + ~ω0 − ~Ω] ,
M = |N−N ′| = 1, 2, 3, ..., α = ~vd, θ = e2E20/ (m2Ω4),A = 2πe2~ω0
(
χ−1∞ − χ−10
)
/κ,
ξ =
(√
N + 1/2 +
√
N + 1 + 1/2
)
ℓB/2, β = 1/(kBT ),
εN,n =
(
N + 1
2
)
~ωc +
(
n+ 1
2
)
~ωz + 12mv
2
d,
I = a1(αβ)
−1 [exp (αβa1) + exp (−αβa1)]− (αβ)−2 [exp (αβa1)− exp (−αβa1)],
a1 = Lx/2ℓ
2
B, and we have set
I(n, n′) =
+∞∫
−∞
|In,n′ (±qz)|2dqz (2.20)
which will be numerically evaluated by a computational program. The divergence of delta
functions is avoided by replacing them by the Lorentzians as [9]
δ(X) =
1
π
(
Γ
X2 + Γ2
)
(2.21)
160
The Hall coefficient in parabolic quantum wells with a perpendicular magnetic field...
where Γ is the damping factor associated with the momentum relaxation time τ by
Γ = ~ /τ . The appearance of the parameter ξ is due to the replacement of qy by eBξ/~
where ξ is a constant of the order of ℓB. The purpose is to impose simplicity in performing
the integral over qy. This has been done in Ref. 10 and is equivalent to assuming an
effective phonon momentum evdqy ≈ eE1ξ. The HC is given by the formula [7]
RH = − 1
B
σyx
σ2xx + σ
2
yx
, (2.22)
where σyx and σxx are given by Eq. (2.17). Equations (2.17) and (2.22) show the
complicated dependencies of the Hall conductivity tensor and the HC on the external
fields, including the EMW. It is obtained for arbitrary values of the indices N, n,N ′ and
n′. However, it contains the term I (n, n′) for which it is diffcult to produce an exact
analytical result due to the presence of the Hermite polynomials. We will numerically
evaluate this term using the computational method. Also, it is seen that the change
in the direction of the magnetic field has modified the wave function and energy of
electrons and, consequently, the obtained results are now very different from our previous
results [1]. In the next section, we will give a deeper insight into these results by carrying
out a numerical evaluation and a graphic consideration using the computational method.
2.2. Numerical results and discussion
In this section we present detailed numerical calculations of the Hall conductivity
and the HC in a PQW subjected to the uniform crossed magnetic and electric fields in
the presence of a strong EMW. For numerical evaluation, we consider the model of a
PQW of GaAs/AlGaAs with the following parameters [1, 13]: εF = 50meV, χ∞ = 10.9,
χ0 = 12.9, ~ω0 = 36.6 meV, m = 0.067 m0 (m0 is the mass of a free electron). Also, for
the sake of simplicity we choose τ = 10−12s,Lx = Ly = 10−9m and only consider the
transitions N = 0, N ′ = 1, n = 0, n′ = 0÷ 1 (the lowest and the first-excited levels).
In Figure 1, the solid curve describes the dependence of the magnetoconductivity
σxx on the cyclotron energy ~ωc in the case of absence of the EMW (E0 = 0). We can
see very clearly that this curve has three maximum peaks and the values of conductivity
at the peaks are very much larger than they are at others. Physically, the existence of the
peaks can be explained in detail as follows using the computational method to determine
their positions. All the peaks correspond to the conditions
(N ′ −N)~ωc = ~ω0 + eE1ξ ±∆n,n′ , ∆n,n′ = (n′ − n)~ωz (2.23)
This condition is generally called the intersubband magnetophonon resonance
(MPR) condition under the influence of a dc EF (all the peaks now may be called resonant
peaks). In this consideration,N ′−N = 1,∆n,n′ = 0 or ~ωz. Therefore, from left to right,
the peaks correspond to the values of cyclotron energy, respectively, satisfy the conditions
161
Bui Dinh Hoi, Do Tuan Long, Pham Thi Trang, Le Thi Thiem and Nguyen Quang Bau
~ωc = ~ω0+eE1ξ−~ωz, ~ωc = ~ω0+ eE1ξ and ~ωc = ~ω0+eE1ξ+~ωz. However, the
values of the term eE1ξ are very small in comparison to the optical phonon energy and
can be considered negligible. For instance, if we take B = 20T (approximately ~ωc = 34.59
meV), then eE1ξ ≈ 0.0277 meV ≪ ~ω0. So, the condition for the second peak can be
written as approximately ~ωc = ~ω0, as we can see in the figure at ~ωc = 36.6meV.
This is precisely the MPR condition. The conditions for the first and the third peaks
also become ~ωc = ~ω0 − ~ωz and ~ωc = ~ω0 + ~ωz, respectively. These conditions
show that they are symmetrical to the second one as we can see in the figure. At this, we
can conclude that the influence of the dc EF on the conditions for the resonant peaks is
considerable only when its value is very large.
Figure 1. The mangetoconductivity xx as a function of the cyclotron energy ~!c
for E0 = 0 (solid curve) and E0 = 105 V.m−1
Here, ωz = 0.5× ω0 , Ω = 5× 1013s−1, E1 = 5× 103V.m−1, and T = 270 K
The dashed curve in Figure 1 shows the dependence of σxx on the cyclotron energy
in the presence of a strong EMW with amplitude E0 = 105V.m−1 and the photon energy
~Ω = 6.6meV. It is seen that besides the main resonant peaks, as in the case of the absence
of the EMW, the subordinate peaks appear. The appearance of the subordinate peaks is due
to the contribution of a photon absorption/emission process that satisfies the conditions
~ωc = ~ω0 ± ~ωz ± ~Ω. Concretely, from left to right the peaks of this curve correspond
to the conditions: ~ωc = ~ω0 − ~ωz − ~Ω, ~ωc = ~ω0 − ~ωz, ~ωc = ~ω0 − ~ωz + ~Ω,
~ωc = ~ω0−~Ω,~ωc = ~ω0,~ωc = ~ω0+~Ω,~ωc = ~ω0+~ωz−~Ω, ~ωc = ~ω0+~ωz,
~ωc = ~ω0 + ~ωz + ~Ω respectively. It is also seen that the main peaks are much higher
than the subordinate peaks. This means that the possibility of a process with no photon
is much larger than it is for a process with one photon absorption/