The hall coefficient in parabolic quantum wells with a perpendicular magnetic field under the influence of laser radiation

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.

Mathematical and Physical Sci., 2013, Vol 58, No 7, pp 154-166

This paper is available online at http://stdb.hnue.edu.vn

THE HALL COEFFICIENT IN PARABOLIC QUANTUM WELLS 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) = mω2z z2/2 (where m and ω z are the effective mass of electron and the confinement frequency of QW, respectively) is subjected

to a crossed dc electric field (EF) ⃗ E1 = (E1, 0, 0) and magnetic field ⃗ B = (0, 0, B) in the presence of a strong electromagnetic wave (EMW) characterized

by electric field ⃗ E = (0, E0sin (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 componentσ xx 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 quantum wells, electron

- phonon interaction

Received November 14, 2012 Accepted October 8, 2013.

Contact Bui Dinh Hoi, e-mail address: hoibd@nuce.edu.vn

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 PQW with the confinement potential

V (z) = mω2

z z2/2, subjected to a crossed dc electric field (EF) ⃗ E1 = (E1, 0, 0) and

magnetic field ⃗ B = (0, 0, B) ( ⃗ B 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 ⃗ E = (0, E0sin (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

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) = mω2

z z2/2, subjected to a crossed dc EF ⃗ E1 = (E1, 0, 0) and

magnetic field ⃗ B = (0, 0, B) If a strong EMW (laser radiation) is applied along the z

direction with the electric field vector ⃗ E = (0, E0sin (Ωt) , 0), the Hamiltonian of the

electron-optical phonon system in the above mentioned PQW in the second quantization representation can be written as

N,n,⃗ k y

εN,n

(

⃗ky − ~c e A (t) ⃗ )

a+ N,n,⃗ k y a N,n,⃗ k

y +∑

⃗

~ω ⃗b+⃗ b⃗, (2.2)

N,N ′

∑

n,n ′

∑

⃗ q,⃗ k y

DN,n,N ′ ,n ′ (⃗ q) a+

N ′ ,n ′ ⃗ k y +⃗ q y aN,n,k y

(

b⃗ + b+−⃗q

)

where ⃗k y = (0, k y , 0); ~ω ⃗ is the energy of an optical phonon with the wave vector

⃗

q = (q x , q y , q z ); a+

N,n,⃗ k y and a N,n,⃗ k

y (b+⃗ and b ⃗) are the creation and annihilation

operators of electron (phonon), respectively; ⃗ A (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 (⃗ r) ≡ |N, n, ky⟩ =

√ 1

LyΦN (x − x0) e ik y yΦn (z) , (2.4)

ε N,n

(

⃗ky)

=

(

N + 1

2

)

~ω c + ε n − ~vd k y+ 1

2mv

2

d N, n = 0, 1, 2 (2.5)

where N is the Landau level index, n is the subband index, L yis the normalization length

in the y direction, ω c = eB/m is the cyclotron frequency and v d = E1/B is the drift

velocity of electron Also, ΦN represents harmonic oscillator wave functions centered at

x0 =−ℓ2

B (k y − mvd / ~) where ℓ B =√

~/ (mω c) is the radius of the Landau orbit in the

x − y plane and Φn (z) and ε nare the wave functions and the subband energy values due

to the parabolic confinement potential in the z direction, respectively, given by

Φn (z) =

√

1

2n n! √

πℓ z exp

(

− z2

2ℓ2

z

)

H n

(

z

ℓ z

)

Φn (z) =

√

1

2n n! √

πℓ z exp

(

− z2

2ℓ2

)

Hn

(

z

ℓ z

)

(2.7)

with H n (z) being the Hermite polynomial of n th order and ℓ z =√

~/ (mω z) The matrix

element of interaction, D N,n,N ′ ,n ′ (⃗ q) is given by [10, 13]

|DN,n,N ′ ,n ′ (⃗ q) |2

=|C⃗|2|In,n ′(±qz)|2|JN,N ′ (u) |2

(2.8)

where C ⃗ is the electron-phonon interaction constant which depends on a scattering mechanism for electron-optical phonon interaction [5, 13] |C⃗|2

=

2πe2~ω0

(

χ −1 ∞ − χ −1

0

)

/ (κ0V0q2) 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, I n,n ′(±qz) = ⟨n| e ±iq z z |n ′ ⟩ is the form factor of

electron, and |JN,N ′ (u) |2

= (N ′ !/N !) e −u u N ′ −N[

L N N ′ −N (u)

]2

with L N

M (x) is the associated Laguerre polynomial, u = ℓ2B q2

⊥ , q2⊥ = q x2 + q2

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

−(e ⃗ E1+ ω c

[

⃗k y ∧ ⃗h ]) ∂f N,n,⃗ k

y

~∂⃗k y

+⃗ky m

∂f N,n,⃗ k

y

∂⃗ r

=− f N,n,⃗ k y − f0

2π

~

∑

N ′ ,n ′

∑

⃗

|DN,n,N ′ ,n ′ (⃗ q) |2

+∞

∑

s=−∞

J s2

(

λ

Ω )

×{[¯

N ′ ,n ′ ,⃗ k y +⃗ q y (N ⃗+ 1)− ¯ f N,n,⃗ k

y N ⃗

]

δ

(

ε N ′ ,n ′

(

⃗k y + ⃗ q y

)

− εN,n(⃗k y)

− ~ω⃗ − s~Ω)

+

[

¯

N ′ ,n ′ ,⃗ k y −⃗q y N ⃗ − ¯ f N,n,⃗ k

y (N ⃗+ 1)

]

δ

(

ε N ′ ,n ′

(

⃗k y − ⃗qy)− εN,n(⃗k y)

+~ω ⃗ − s~Ω)},

(2.9)

where ⃗h = ⃗ B/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), f N,n,⃗ k

yis 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,⃗ k

y (N ⃗) is the time-independent component of the distribution function

of electrons (phonons), J s (x) is the s th -order Bessel function of argument x; δ ( ) being the Dirac’s delta function, and λ = eE0q y /(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

of equation (2.9) by m e ⃗ky δ

(

ε − εN,n(⃗ky))

and carry out the summation over N and ⃗k y,

we have the equation for the partial current density ⃗j N,n,N ′ ,n ′ (ε) (the current caused by electrons that have energy of ε):

⃗j N,n,N ′ ,n ′ (ε)

[

⃗h ∧ ⃗jN,n,N ′ ,n ′ (ε)

]

= ⃗ Q N,n (ε) + ⃗ S N,n,N ′ ,n ′ (ε) , (2.10) where

⃗

Q N,n (ε) = − e

m

∑

N,n,⃗ k y

⃗ky

(

⃗

F ∂f N,n,⃗ k y

~∂⃗k y

)

δ(ε − εN,n (⃗k y )), F = e ⃗ ⃗ E1 (2.11)

and

⃗

SN,n,N ′ ,n ′ (ε) =

= 2πe

m~

∑

⃗

k y ,⃗ q

∑

N ′ ,n ′

∑

N,n

|DN,n,N ′ ,n ′ (⃗ q) |2N ⃗ ⃗k y{[

¯

N ′ ,n ′ ,⃗ k y +⃗ q y − ¯ f N,n,⃗ k

y

] [(

1− λ2

2Ω2

)

× δ(ε N ′ ,n ′

(

⃗k y + ⃗ q y

)

− εN,n(⃗k y)

− ~ω⃗)

2

4Ω2δ

(

ε N ′ ,n ′

(

⃗k y + ⃗ q y

)

− εN,n(⃗k y)

− ~ω⃗+~Ω)+ λ

2

4Ω2

×δ(ε N ′ ,n ′

(

⃗k y + ⃗ q y

)

− εN,n(⃗k y)

− ~ω⃗ − ~Ω)]

+

[

¯

N ′ ,n ′ ,⃗ k y −⃗q y − ¯ f N,n,⃗ k

y

] [(

1− λ2

2Ω2

)

δ

(

ε N ′ ,n ′

(

⃗k y − ⃗qy)− εN,n(⃗k y)

+~ω ⃗

)

+ λ

2

4Ω2δ

(

ε N ′ ,n ′

(

⃗k y − ⃗qy)− εN,n(⃗k y)

+~ω ⃗+~Ω)

2

4Ω2δ

(

ε N ′ ,n ′

(

⃗k y − ⃗qy)− εN,n(⃗k y)

− ~ω⃗ − ~Ω)]}×δ(ε − εN,n(⃗k y))

(2.12)

Solving (2.10) we have the expression for ⃗j N,n,N ′ ,n ′ (ε) as follows:

⃗j N,n,N ′ ,n ′ (ε) =

τ

1 + ω2τ2

{(

⃗

Q N,n (ε) + ⃗ S N,n,N ′ ,n ′ (ε)

)

− ωc τ

(

[⃗h ∧ ⃗QN,n (ε)] + [⃗h ∧ ⃗SN,n,N ′ ,n ′ (ε)]

)

+ω c2τ2

(

⃗

Q N,n (ε)⃗h + ⃗ S N,n,N ′ ,n ′ (ε)⃗h

)

⃗h}

(2.13) The total current density is given by

⃗

J =

∞

∫

0

⃗j N,n,N ′ ,n ′ (ε)dε or J i = σ im E 1m (2.14)

Inserting (2.13) into (2.14) we obtain the expressions for the current J ias well as the

conductivity σ imafter 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, ω ⃗ ≈ ω0, N ⃗ ≈ N0 =

k B T /( ~ω0), where ω0 is the frequency of the longitudinal optical phonon, assumed to

be constant, and k B is the Boltzmann constant Otherwise, the summations over ⃗k y and ⃗ q

are transformed into the integrals as follows [10]

∑

⃗ k y

( ) → L y

2

L x /2ℓ2

B

∫

−L x /2ℓ2

B

∑

⃗

( ) → V0

4π2

+∞

∫

0

( )q ⊥ dq ⊥

+∞

∫

−∞

dq z = V0

4π2ℓ2

B

+∞

∫

0

( )du

+∞

∫

−∞

dq z , (2.16)

here, L x is the normalization length in the x direction After some mathematical

manipulation, we find the expression for the conductivity tensor:

σ im= e

2τ

~

(

1 + ω c2τ2)−1(

δ ij − ωc τ ε ijk h k + ω2c τ2h i h j)

×{aδ jm + bδ jℓ[

δ ℓm − ωc τ ε ℓmp h p + ω c2τ2h ℓ h m]}

where δ ij is the Kronecker delta, ε ijk 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 = − ~βv d L y I

2πm

∑

N,n

with εF being the Fermi level, and

b = βAN0LyI

8π2m2

τ

1 + ω2

c τ2

∑

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] ,

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 ′)~ω c + (n ′ − n)~ωz + eE1ξ + ~ω0] ,

b6 =− θ

2M

(

eBξ

~

)3

exp [β(εF− εN,n)]

[

N !

(N + M )!

]2

× δ [(N − N ′)~ω c + (n ′ − n)~ωz + eE1ξ + ~ω0] ,

b7 = θ

4M

(

eBξ

~

)3

exp [β(εF− εN,n)]

[

N !

(N + M )!

]2

× δ [(N − N ′)~ω c + (n ′ − n)~ωz + eE1ξ + ~ω0+~Ω] ,

b8 = θ

4M

(

eBξ

~

)3

exp [β(εF− εN,n)]

[

N !

(N + M )!

]2

× δ [(N − N ′)~ω c + (n ′ − n)~ωz + eE1ξ + ~ω0− ~Ω] ,

M = |N −N ′ | = 1, 2, 3, , α = ~vd , θ = e2E02/ (m2Ω4), A = 2πe2~ω0

(

χ −1 ∞ − χ −1

0

)

/κ,

ξ =(√

N + 1/2 +√

N + 1 + 1/2

)

ℓ B /2, β = 1/(k B T ),

ε N,n =(

N + 1

2

)

~ω c+(

n + 1 2

)

~ω z+1

2mv2

d,

I = a1(αβ) −1 [exp (αβa1) + exp (−αβa1)]− (αβ) −2 [exp (αβa1)− exp (−αβa1)],

a1 = L x /2ℓ2

B, and we have set

I(n, n ′) =

+∞

∫

−∞

|In,n ′(±qz)|2

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)

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 q y by eBξ/~

where ξ is a constant of the order of ℓ B The purpose is to impose simplicity in performing

the integral over q y This has been done in Ref 10 and is equivalent to assuming an

effective phonon momentum ev dqy ≈ eE1ξ The HC is given by the formula [7]

R H =−1

B

σ yx

σ2

xx + σ2

yx

where σ yx and σ xx 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 −12 s,L x = L y = 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

σ xx on the cyclotron energy ~ω c 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~ω z Therefore, from left to right, the peaks correspond to the values of cyclotron energy, respectively, satisfy the conditions

~ω c =~ω0+ eE1ξ − ~ωz,~ω c=~ω0+ eE1ξ and ~ω c=~ω0+ eE1ξ + ~ω z 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~ω c= 34.59

meV), then eE1ξ ≈ 0.0277 meV ≪ ~ω0 So, the condition for the second peak can be written as approximately ~ω c = ~ω0, as we can see in the figure at ~ω c = 36.6meV.

This is precisely the MPR condition The conditions for the first and the third peaks also become ~ω c = ~ω0 − ~ωz and ~ω c = ~ω0 +~ω z, 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 σ xx as a function of the cyclotron energy ~ω 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 σ xxon 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

~ω c =~ω0± ~ωz ± ~Ω Concretely, from left to right the peaks of this curve correspond

to the conditions:~ω c =~ω0 − ~ωz − ~Ω, ~ωc = ~ω0− ~ωz,~ω c = ~ω0− ~ωz +~Ω,

~ω c =~ω0− ~Ω,~ωc=~ω0,~ω c=~ω0+~Ω,~ω c =~ω0+~ω z − ~Ω, ~ωc =~ω0+~ω z,

~ω c = ~ω0 +~ω z +~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/emission Moreover,

the conductivity in the case of E0 = 105V.m −1 is not significantly different considerably

compared to the case of E0 = 0, and this shows that the influence of EMW amplitude

on the conductivity is not respectable The presence of a strong EMW only leads to the appearance of additional resonant peaks satisfying the selection rules for the transition

of electrons between the states The above conditions with the presence of are actually

the optically detected MPR conditions They were obtained in both bulk semiconductors [2, 6, 12, 13] and recently in quantum wires [11] by other methods in which the authors presented many applications of this behavior in detail, especially the determination of the electronic spectrum of systems The main purpose of our present work is to make a comparison between the results in this study and the case of a in-plane magnetic field [1]

To do this, in the following we numerically evaluate the HC as functions of the external fields, the temperature and the confinement frequency of the PQW

Figure 2 The HC as a function of the magnetic field at the different values

of the temperature: T = 200K (solid curve),

T = 250K (dashed curve), T = 300K (dotted curve)

Here, ω z = 0.5 × ω0, E1 = 5× 103V.m −1 , E0 = 105V.m −1 , Ω = 5 × 1013s −1

In Figure 2 we consider the dependence of the HC on the magnetic field at different values of temperature For the chosen parameters, it is seen that the HC increases strongly

as the magnetic field increases in the region of approximately 5T to 10T As the magnetic field increases continuously, the HC changes slightly and reaches saturation This behavior

is similar to the results obtained previously in the case of the in-plane magnetic field [1] and in some works at low temperature (see Ref [3] and references therein) Furthermore,

in the region of strong magnetic field the HC depends very weakly on the temperature because when the magnetic field increases, radii of the Landau orbits decrease so the electron density (and followed by the HC) reaches saturation and almost does not vary with the temperature

Figure 3 shows the HC as a function of the EMW frequency at different values of

the temperature for B = 4T All the curves have resonant peaks which can be explained similar to that in Figure 1 In the region Ω <1013s −1 the HC decreases quickly with the frequency, then reaches saturation as the frequency increases continuously (approximately

Ω > 2 × 1013s −1) This behavior is similar to the case of the in-plane magnetic field [1] Moreover, at this (small) magnetic field the HC depends strongly on the temperature and

is always negative