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.

JOURNAL OF SCIENCE OF HNUE 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) = mωz2 z /2 (where m and ωz are the effective mass of electron and the confinement frequency of QW, respectively) is subjected ⃗ = (E1 , 0, 0) and magnetic field B ⃗ = to a crossed dc electric field (EF) E (0, 0, B) in the presence of a strong electromagnetic wave (EMW) characterized ⃗ = (0, E0 sin (Ωt) , 0) (where E0 and Ω are the amplitude by electric field E 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 quantum wells, 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 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 ⃗ = (E1 , 0, 0) and V (z) = mωz2 z /2, subjected to a crossed dc electric field (EF) E ⃗ ⃗ 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 ⃗ = (0, E0 sin (Ωt) , 0) We only consider the case of high temperatures when field E 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 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 ⃗ = (E1 , 0, 0) and assumed to be V (z) = mωz2 z /2, subjected to a crossed dc EF E ⃗ magnetic field B = (0, 0, B) If a strong EMW (laser radiation) is applied along the z ⃗ = (0, E0 sin (Ωt) , 0), the Hamiltonian of the direction with the electric field vector E electron-optical phonon system in the above mentioned PQW in the second quantization representation can be written as H0 = ∑ N,n,⃗ky U= H = H0 + U, ( ∑ e ⃗ ) + εN,n ⃗k y − A (t) aN,n,⃗k aN,n,⃗ky + ωq⃗b+ q, q⃗ b⃗ y c n,n′ (2.2) q⃗ ∑ ∑∑ N,N ′ (2.1) DN,n,N ′ ,n′ (⃗q) a+ N ′ ,n′⃗k ( qy y +⃗ ) aN,n,ky b⃗q + b+ −⃗ q , (2.3) q⃗,⃗ky where ⃗ky = (0, ky , 0); ωq⃗ is the energy of an optical phonon with the wave vector ⃗q = (qx , qy , qz ); a+ and aN,n,⃗ky (b+ q⃗ and bq⃗ ) are the creation and annihilation N,n,⃗ky ⃗ (t) is the vector potential of the EMW operators of electron (phonon), respectively; A 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 ⟩ = ΦN (x − x0 ) eiky y Φn (z) , (2.4) Ly ) ( ) ( 1 εN,n ⃗ky = N + ωc + εn − vd ky + mvd2 N, n = 0, 1, (2.5) 2 where N 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 = −ℓB (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 √ ( ) ( ) z2 z √ Φn (z) = exp − Hn , (2.6) n 2ℓz ℓz n! πℓz 156 The Hall coefficient in parabolic quantum wells with a perpendicular magnetic field √ ( ) ( ) z2 z √ Φn (z) = exp − Hn (2.7) n 2ℓz ℓz n! πℓz √ with Hn (z) being the Hermite polynomial of nth order and ℓz = / (mωz ) The matrix element of interaction, DN,n,N ′ ,n′ (⃗q) is given by [10, 13] |DN,n,N ′ ,n′ (⃗q)|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 = ) −1 2πe2 ω0 χ−1 / (κ0 V0 q ) where κ0 is the electric constant (vacuum permittivity), ∞ − χ0 V0 the normalization volume of specimen, χ0 and χ0 are the static and high-frequency dielectric constants, respectively, In,n′ (±qz ) = ⟨n| e±iqz z |n′ ⟩ is the form factor of [ ′ ]2 ′ −N with LN (u) electron, and |JN,N ′ (u)|2 = (N ′ !/N !) e−u uN −N LN M (x) is the N 2 associated Laguerre polynomial, u = ℓ2B q⊥ , q⊥ = qx2 + qy2 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 ( [ ]) ∂f ⃗ ⃗ky ∂fN,n,⃗ky N,n,ky ⃗ + ωc ⃗ky ∧ ⃗h − eE + m ∂⃗r ∂⃗ky ( ) +∞ ∑ fN,n,⃗ky − f0 2π ∑ ∑ λ 2 =− + |DN,n,N ′ ,n′ (⃗q)| Js τ Ω s=−∞ N ′ ,n′ q⃗ {[ ] ( ( ) ( ) ) × f¯N ′ ,n′ ,⃗ky +⃗qy (N⃗q + 1) − f¯N,n,⃗ky Nq⃗ δ εN ′ ,n′ ⃗ky + ⃗qy − εN,n ⃗ky − ωq⃗ − s Ω [ ] ( ( ) ( ) )} ¯ ¯ ⃗ ⃗ ′ ′ + fN ′ ,n′ ,⃗ky −⃗qy Nq⃗ − fN,n,⃗ky (N⃗q + 1) δ εN ,n ky − ⃗qy − εN,n ky + ωq⃗ − 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), 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 = eE0 qy /(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, 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 ε − εN,n ⃗ky and carry out the summation over N and ⃗ky , we have the equation for the partial current density ⃗jN,n,N ′ ,n′ (ε) (the current caused by electrons that have energy of ε): e⃗ k δ m y ( [ ] ⃗jN,n,N ′ ,n′ (ε) ⃗ N,n (ε) + S ⃗N,n,N ′ ,n′ (ε) , + ωc ⃗h ∧ ⃗jN,n,N ′ ,n′ (ε) = Q τ where ∑ ⃗ky ⃗ N,n (ε) = − e Q m N,n,⃗ky ( ) ∂fN,n,⃗ky F⃗ δ(ε − εN,n (⃗ky )), ∂⃗ky ⃗1 F⃗ = eE (2.10) (2.11) and ⃗N,n,N ′ ,n′ (ε) = S ) {[ ] [( 2 λ 2πe ∑ ∑ ∑ |DN,n,N ′ ,n′ (⃗q)| N⃗q⃗ky f¯N ′ ,n′ ,⃗ky +⃗qy − f¯N,n,⃗ky 1− = m 2Ω2 ′ ,n′ N,n N ⃗ky ,⃗ q ( ( ) ( ) ) × δ εN ′ ,n′ ⃗ky + ⃗qy − εN,n ⃗ky − ωq⃗ ( ) ( ) ) λ2 ( λ2 ⃗ ⃗ ′ ′ + δ ε k + ⃗ q − ε k − ω + Ω + N ,n y y N,n y q⃗ 4Ω 4Ω2 ( ( ) ( ) )] ×δ εN ′ ,n′ ⃗ky + ⃗qy − εN,n ⃗ky − ωq⃗ − Ω ) ( ( ) ( ) ) ] [( [ λ2 ⃗ky − ⃗qy − εN,n ⃗ky + ωq⃗ ¯ ¯ ′ ,n′ δ ε + fN ′ ,n′ ,⃗ky −⃗qy − fN,n,⃗ky 1− N 2Ω2 ( ) ( ) ) λ2 ( + δ εN ′ ,n′ ⃗ky − ⃗qy − εN,n ⃗ky + ωq⃗ + Ω 4Ω ( ) ( ) )]} ( ( )) λ2 ( ⃗ ⃗ δ εN ′ ,n′ ky − ⃗qy − εN,n ky − ωq⃗ − Ω + ×δ ε − εN,n ⃗ky (2.12) 4Ω2 Solving (2.10) we have the expression for ⃗jN,n,N ′ ,n′ (ε) as follows: ⃗jN,n,N ′ ,n′ (ε) = {( ) ( ) τ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ QN,n (ε) + SN,n,N ′ ,n′ (ε) − ωc τ [h ∧ QN,n (ε)] + [h ∧ SN,n,N ′ ,n′ (ε)] + ωc2 τ ( ) } ⃗ N,n (ε)⃗h + S ⃗N,n,N ′ ,n′ (ε)⃗h ⃗h (2.13) +ω τ Q c The total current density is given by ∫∞ J⃗ = ⃗jN,n,N ′ ,n′ (ε)dε 158 or Ji = σim E1m (2.14) 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 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 = kB T /( 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 ⃗ky and ⃗q are transformed into the integrals as follows [10] ∑ ⃗ky ∑ q⃗ V0 ( ) → 4π ∫ Lx /2ℓ2B Ly ( ) → ( )dky , (2.15) −Lx /2ℓ2B ∫+∞ ∫+∞ ∫+∞ ∫+∞ V0 ( )q⊥ dq⊥ dqz = 2 ( )du dqz , 4π ℓB −∞ (2.16) −∞ here, Lx is the normalization length in the x direction After some mathematical manipulation, we find the expression for the conductivity tensor: )−1 ( ) e2 τ ( + ωc2 τ δij − ωc τ εijk hk + ωc2 τ hi hj { [ ]} × aδjm + bδjℓ δℓm − ωc τ εℓmp hp + ωc2 τ hℓ hm , σim = (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 βvd Ly I ∑ β (εF −εN,n ) a=− e , (2.18) 2πm N,n with εF being the Fermi level, and ∑∑ βAN0 Ly I τ b= I (n, n′ ) {b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8 }, 8π m2 + ωc2 τ N,n N ′ ,n′ (2.19) ( ) [ ]2 (N + M )! eBξ exp [β(εF − εN,n )] b1 = M N! ′ ′ × δ [(N − N ) ωc + (n − n) ωz − eE1 ξ − ω0 ] , [ ( )3 ]2 θ eBξ (N + M )! b2 = − exp [β(εF − εN,n )] 2M N! ′ ′ × δ [(N − N ) ωc + (n − n) ωz − eE1 ξ − ω0 ] , 159 Bui Dinh Hoi, Do Tuan Long, Pham Thi Trang, Le Thi Thiem and Nguyen Quang Bau [ ( )3 ]2 θ eBξ (N + M )! b3 = exp [β(εF − εN,n )] 4M N! ′ ′ × δ [(N − N ) ωc + (n − n) ωz − eE1 ξ − ω0 + Ω] , ( )3 [ ]2 θ eBξ (N + M )! b4 = exp [β(εF − εN,n )] 4M N! ′ ′ × δ [(N − N ) ωc + (n − n) ωz − eE1 ξ − ω0 − Ω] , ( ) [ ]2 eBξ N! b5 = exp [β(εF − εN,n )] M (N + M )! ′ ′ × δ [(N − N ) ωc + (n − n) ωz + eE1 ξ + ω0 ] , ( )3 [ ]2 θ eBξ N! b6 = − exp [β(εF − εN,n )] 2M (N + M )! ′ ′ × δ [(N − N ) ωc + (n − n) ωz + eE1 ξ + ω0 ] , ( )3 [ ]2 N! θ eBξ exp [β(εF − εN,n )] b7 = 4M (N + M )! ′ ′ × δ [(N − N ) ωc + (n − n) ωz + eE1 ξ + ω0 + Ω] , ( )3 [ ]2 θ eBξ N! b8 = exp [β(εF − εN,n )] 4M (N + M )! ′ ′ × δ [(N − N ) ωc + (n − n) ωz + eE1 ξ + ω0 − Ω] , ( ) −1 M = |N −N ′ | = 1, 2, 3, , α = vd , θ = e2 E02 / (m2 Ω4 ), A = 2πe2 ω0 χ−1 /κ, ∞ − χ0 ) (√ √ N + 1/2 + N + + 1/2 ℓB /2, β = 1/(kB T ), ξ= ( ) ( ) εN,n = N + 12 ωc + n + 12 ωz + 12 mvd2 , I = a1 (αβ)−1 [exp (αβa1 ) + exp (−αβa1 )] − (αβ)−2 [exp (αβa1 ) − exp (−αβa1 )], a1 = Lx /2ℓ2B , and we have set ∫+∞ I(n, n′ ) = |In,n′ (±qz )|2 dqz (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] ) ( Γ δ(X) = (2.21) π X + Γ2 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 evd qy ≈ eE1 ξ The HC is given by the formula [7] RH = − σyx , 2 B σxx + σyx (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 = 10−12 s,Lx = Ly = 10−9 m and only consider the transitions N = 0, N ′ = 1, n = 0, n′ = ÷ (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 + eE1 ξ ± ∆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′ = 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 + eE1 ξ − ωz , ωc = ω0 + eE1 ξ and ωc = ω0 + eE1 ξ + ωz However, the values of the term eE1 ξ 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 eE1 ξ ≈ 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 The mangetoconductivity σxx as a function of the cyclotron energy ωc for E0 = (solid curve) and E0 = 105 V.m−1 Here, ωz = 0.5 × ω0 , Ω = × 1013 s−1 , E1 = × 103 V.m−1 , and T = 270 K The dashed curve in Figure shows the dependence of σxx on the cyclotron energy in the presence of a strong EMW with amplitude E0 = 105 V.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/emission Moreover, the conductivity in the case of E0 = 105 V.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 162 The Hall coefficient in parabolic quantum wells with a perpendicular magnetic field 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 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 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, ωz = 0.5 × ω0 , E1 = × 103 V.m−1 , E0 = 105 V.m−1 , Ω = × 1013 s−1 In Figure 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 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 In the region Ω × 1013 s−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 163 Bui Dinh Hoi, Do Tuan Long, Pham Thi Trang, Le Thi Thiem and Nguyen Quang Bau Figure The HC as a function of the frequency of EMW at the different values of the temperature: T = 270 K (solid curve), T = 280 K (dashed curve), T = 290 K (dotted curve) Here, ωz = 0.5 × ω0 , E1 = × 103 V.m−1 , E0 = 105 V.m−1 , B = 4T Figure The HC as a function of the amplitude of EMW at the different values of the confinement frequency: ωz = 0.48 × ω0 (solid curve), ωz = 0.50 × ω0 (dashed curve), ωz = 0.52 × ω0 (dotted curve) Here, B = 4T, Ω = × 1013 s−1 , E1 = × 103 V.m−1 , T = 270 K Figure shows the dependence of the HC on the amplitude of EMW at different values of the confinement frequency It is easy to see that the HC depends very weakly on the amplitude E0 and this dependence is only considerable when E0 is very large The HC also depends strongly on the confinement frequency and as the confinement frequency increases the absolute value of the HC decreases This is due to the raise of electron density when the confinement frequency increases From the above results we also see that the HC in this study is always negative while in the case of the in-plane magnetic field [1] it has both negative and positive values 164 The Hall coefficient in parabolic quantum wells with a perpendicular magnetic field Conclusion In this work, we have studied the Hall effect in quantum wells with parabolic potential subjected to a crossed dc electric field and a magnetic field in the presence of a strong EMW (laser radiation) We obtain the expression of the Hall conductivity and the HC when the electron-optical phonon interaction is taken into account at high temperature and the electron gas is nondegenerate The influence of the EMW is interpreted by the dependence of the Hall conductivity and the HC on the amplitude and the frequency (photon energy) of the EMW besides the dependence on the magnetic and dc EF as in the ordinary Hall effect The analytical results are numerically evaluated and plotted for a specific PQW of GaAs/AlGaAs to show clearly the dependence of the Hall conductivity and the HC on the external fields, the temperature and the parameters of the system The obtained results are very different in comparison to our previous results in the case of the in-plane magnetic field The most important difference is the appearance of the maximum peaks of conductivity satisfying the condition M ωc = ω0 + eE1 ξ ± ∆n,n′ ± Ω, M = 1, 2, 3, In the case of the absence of both the laser field and the dc EF, it becomes the usual MPR condition, M ωc = ω0 , and intersubband MPR condition, M ωc = ω0 ± ∆n,n′ (n ̸= n′ ) The HC reaches saturation as the magnetic field or the EMW frequency increases and weakly depends on the amplitude of the EMW These behaviors are similar to the case of the in-plane magnetic field Moreover, the HC in this work is always negative while it has both negative and positive values in the case of in-plane magnetic field Acknowledgments This work was completed thanks to financial support from the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) (Grant No.: 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