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

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/