Journal 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) = mωz z /2 (where m and ωz 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 = (E0 sin Ωt, 0, 0) (where E0 and Ω are the amplitude and the frequency of EMW, respectively) We obtain analytic expressions for the components σzz and σxz of the Hall conductivity as well as a Hall coefficient with a dependence on B, E1 , E0 , Ω, temperature T 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 amplitude E0 and the frequency Ω of EMW and by using the dependences on the magnetic field B 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 magnetic field [9,10] and the effect of the presence of an additional (high frequency) EMW [11] have been studied 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 recent years many physicists have been interested in investigating 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 electron-impurity and electronacoustic 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 apply 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 = (E0 sin Ωt, 0, 0), the confinement potential being assumed to be V (z) = mωz z /2 We only consider the case in which the electron - optical phonon interaction is assumed to be dominant and electron gas to be nondegenerate We derive the analytical expressions for the conductivity tensor and the Hall coefficient (HC) The Recently, there has been considerable interest in the behavior of low-dimensional systems, in particular, twodimensional electron gas (2DEG) systems, such as quantum wells and compositional and doped superlattices The confinement of electrons in these systems considerably enhances the electron mobility and leads to unusual behaviors under external stimuli As a result, the properties of low-dimensional systems, especially electrical and optical properties, are very different in comparison from those of normal semiconductors [1, 2] There have been many papers dealing with problems related to the incidence of electromagnetic wave (EMW) in lowdimensional 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 nonlinear 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 -59- -60- Journal of the Korean Physical Society, Vol 60, No 1, January 2012 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 calculation Numerical results and discussion are given in Sec III Finally, remarks and conclusions are shown briefly in Sec IV resentation can be written as H = H0 + U, H0 = εN N,kx + q II HALL EFFECT IN A QUANTUM WELL WITH A PARABOLIC POTENTIAL IN THE PRESENCE OF LASER RADIATION 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 potential A = (zB, 0, 0) to describe the applied DC magnetic field If the confinement potential is assumed to take the form V (z) = mωz z /2, then the single-particle wave function and its eigenenergy are given by [20] ik⊥ r e φN (z − z0 ) , 2π εN (kx ) = ωp N + (1) Ψ (r) = + 2m 2 kx − kx ωc + eE1 ωp , N = 0, 1, 2, , (2) (3) where m and e are the effective mass and the charge of a conduction electron, respectively, k⊥ = (kx , ky ) is its wave vector in the (x, y) plan; z0 = ( kx ωc + eE1 )/mωp2 ; ωp2 = ωz2 + ωc2 , ωz and ωc = eB/m are the confinement and the cyclotron frequencies, respectively, and Expressions for the Hall Conductivity and the Hall Coefficient DN,N (q)a+ a (b N ,kx +qx N,kx q + b+ −q ), (7) where |N, kx > and |N , kx + q⊥ > are electron states before and after scattering; ωq is the energy of an optiand cal phonon with the wave vector q = (q⊥ , qz ); a+ N,kx aN,kx (b+ q and bq ) are the creation and the annihilation operators of electron (phonon), respectively; A(t) is the vector potential of laser field; DN,N (q) = Cq IN,N (qz ), where Cq is the electron-phonon interaction constant; and IN,N (qz ) = < N |eiqz z |N > is the form factor of electron The quantum kinetic equation for electrons in the single (constant) scattering time approximation takes the form ∂fN,kx ∂fN,kx − eE1 + ωc kx × h ∂t ∂ kx fN,kx − f0 kx ∂fN,kx =− , (8) + m ∂r τ where kx = (kx , 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 distribution function (Fermi - Dirac distribution), fN,kx is an unknown distribution function perturbed due to the external fields, and τ is the electron momentum relaxation time, which is assumed to be constant In order to find fN,kx , we use the general quantum equation for the particle number operator [5–8] or the electron distribution function fN,kx = a+ aN,kx : N,kx ∂ f = ∂t N,kx a+ a ,H N,kx N,kx t t (9) From Eqs (8) and (9), using the Hamiltonian in Eq (5), we find ∂fN,kx − eE1 + ωc kx × h ∂ kx fN,kx − f0 2π kx ∂fN,kx =− + + |DN,N (q)|2 m ∂r τ N ,q ∞ × In the presence of an EMW with electric field vector E = (E0 sin Ωt, 0, 0) (where E0 and Ω are the amplitude 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- (6) N,N q,kx with HN (z) being the Hermite polynomial of N th order ω q b+ q bq , U = i φN (z − z0 ) = HN (z − z0 ) exp − (z − z0 ) /2 , (4) (5) e kx − A(t) a+ aN,kx N,kx c J2 l=−∞ Λ Ω [f¯N ,kx +qx (Nq + 1) − f¯N,kx Nq ]δ (εN (kx + qx ) − εN (kx ) − ω0 − Ω) + [f¯ Nq − f¯ (Nq + 1)]δ (εN (kx − qx ) N ,kx −qx N,kx −εN (kx ) + ω0 − Ω) , (10) Influence of a Strong Electromagnetic Wave (Laser Radiation) · · · – Nguyen Quang Bau and Bui Dinh Hoi in which f¯N,kx (Nq ) is the time-independent component of the distribution function of electrons (phonons), J (x) is the th -order Bessel function of argument x, and Λ = (eE0 qx /mΩ2 )(1 − ωc2 /ωp2 ) Equation (10) is fairly general and can be applied for any mechanism of interaction In the limit of ωz → 0, i.e., the confinement vanishes, it gives the same results as those obtained in -61- bulk semiconductor [9–11] For simplicity, we limit the problem to the case of = −1, 0, If we multiply both sides of Eq (10) by (e/m)kx δ (ε − εN (kx )) and carry out the summation over N and kx , we have the equation for the partial current density jN,N (ε) (the current caused by electrons that have energy of ε): jN,N (ε) + ωc [h × jN,N (ε)] = QN (ε) + SN,N (ε), τ (11) where QN (ε) = − e m kx F N,kx ∂fN,kx ∂ kx δ (ε − εN (kx )) , F = eE1 (12) and SN,N (ε) = 2πe m f¯N |DN,N (q)| Nq kx N ,q N,kx ,kx +qx − f¯N,kx 1− Λ2 2Ω2 Λ2 δ (εN (kx + qx ) − εN (kx ) − ωq + Ω) 4Ω2 Λ2 Λ2 + δ (εN (kx + qx ) − εN (kx ) − ωq − Ω) + f¯N ,kx −qx − f¯N,kx − 4Ω 2Ω2 Λ ×δ (εN (kx − qx ) − εN (kx ) + ωq ) + δ (εN (kx − qx ) − εN (kx ) + ωq + Ω) 4Ω2 Λ + δ (εN (kx − qx ) − εN (kx ) + ωq − Ω) δ (ε − εN (kx )) 4Ω ×δ (εN (kx + qx ) − εN (kx ) − ωq ) + ∞ dielectric constants, respectively After some calculation, we find the expression for conductivity tensor: The total current density is given by J = jN,N (ε)dε or Ji = σim E1m We now consider only the electronoptical phonon interaction We also consider the electron gas to be nondegenerate (the Fermi-Dirac distribution becomes a Boltzmann distribution) In this case, ωq ω0 is taken, and Cq is [5,6] |Cq |2 = 2πe2 ω0 0q 1 − χ∞ χ0 , σim = (14) where is the electric constant (vacuum permittivity), and χ0 and χ∞ are the static and the high-frequency a = b = b1 = e2 Lx 2πm 2πeN0 m π αβ exp β εF − N + N τ δij − ωc τ εijk hk + ωc2 τ hi hj + ωc2 τ τ be ×{aδjm + δjl m + ωc2 τ × δlm − ωc τ εlmp hp + ωc2 τ hl hm }, (15) , (16) where ωp + e2 E12 γ2 + 2mωp 4α {b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8 } , (17) N,N −βALx I (N, N ) exp β εF − N + 64π α2 × α (13) C12 α2 K 12 β |C1 | − γK0 β |C1 | ωp + γ2 e2 E12 C1 + − 2mωp 4α + C1 C12 α2 − 14 K− 12 β |C1 | , (18) -62- Journal of the Korean Physical Society, Vol 60, No 1, January 2012 b2 = −βθALx I (N, N ) exp β εF − N + 64π α2 × α b3 = K 32 β |C1 | −γ C12 α2 C22 α2 K 32 β |C2 | −γ C22 α2 ωp + × α C32 α2 K 32 β |C3 | −γ C32 α2 ωp + K1 + C1 C12 α2 K 12 β |C1 | , (19) K 12 β |C2 | , (20) K 12 β |C3 | , (21) γ2 e2 E12 C2 + − 2mωp2 4α β |C2 | K1 −βθALx I (N, N ) exp β εF − N + 128π α2 γ2 e2 E12 C1 + − 2mωp2 4α β |C1 | K1 −βθALx I (N, N ) exp β εF − N + 128π α2 × α b4 = C12 α2 ωp + + C2 C22 α2 γ2 e2 E12 C3 + − 2mωp2 4α β |C3 | + C3 C32 α2 b5 = b1 (C1 → D1 ), b6 = b2 (C1 → D1 ), b7 = b3 (C2 → D2 ), b8 = b4 (C3 → D3 ), β = 1/(kB T ), α = ( /2m)(1 − ωc2 /ωp2 ), γ = eE1 ωc /mωp2 , 2πe2 ω0 −1 e2 E02 (1 − ωc2 /ωp2 ), A = χ∞ − χ−1 , m Ω = (N − N ) ωp − ω0 , C2 = C1 + Ω, C3 = C1 − Ω, = (N − N ) ωp + ω0 , D2 = D1 + Ω, D3 = D1 − Ω, θ = C1 D1 (22) (23) (24) (25) and ∞ I(N, N ) = −∞ |IN,N (qz )|2 dqz (26) The HC is given by the formula [21] RH = σxz ρxz =− , + σ2 B B σxz zz (27) where σxz and σxx are 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 Fig 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, ωz = 0.5 × ω0 , E = × 105 V/m, E0 = 105 V/m, and T = 270 K III NUMERICAL RESULTS AND DISCUSSION In this section, we present detailed numerical calculations 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 following 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 Lx = 10−9 m The HC is plotted as function of the EMW frequency at different values of the magnetic field in Fig 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 frequency 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 and Fig 3, we show the dependence of the HC on the magnetic field at different values of the tem- Influence of a Strong Electromagnetic Wave (Laser Radiation) · · · – Nguyen Quang Bau and Bui Dinh Hoi Fig 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, ωz = 0.5 × ω0 , E = × 105 V/m, E0 = 105 V/m, and Ω = × 1013 s−1 Fig Hall coefficients (arb units) as functions of the dc electric field at different values of confinement frequency: ωz = 0.3 × ω0 (solid line), ωz = 0.4 × ω0 (dashed line), and ωz = 0.5 × ω0 (dotted line) Here, T = 270 K, B = T, E0 = 105 V/m, and Ω = × 1013 s−1 perature T and on the the dc electric field E1 at different values of the confinement frequency ωz , respectively; the necessary parameters involved in the computation are the same as those in Fig We can describe the behavior of the HC in Fig as follows: Each curve has one maximum and one minimum 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) Particularly, 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 maxima and the minima to the right Also, the values of the HC at maxima (minima) at different temperatures are very different; for instance, the maximum at tempera- -63- Fig 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, ωz = 0.5 × ω0 , B = T, E = × 105 V/m, and Ω = × 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 sensitive to the temperature The dependences of the HC on the dc electric field E1 and the confinement frequency in Fig can be analyzed similarly Figure shows the dependence of the HC on the amplitude E0 of the EMW at different values of the temperature 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 evidently 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 electron 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 dependences 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 parameters of the system From the numerical results, we can summarize the main points as follows: The HC depends nonlinearly on the amplitude E0 of the EMW, and -64- Journal of the Korean Physical Society, Vol 60, No 1, January 2012 it increases strongly with increasing EMW frequency for the small values of the EMW frequency and reaches saturation 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 temperatures are very different; for instance, the maximum at a temperature of 260 K is approximately twice that at 270 K, as shown in Fig 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 Development of Vietnam (NAFOSTED) and the Project of Basic Research in Natural Science, Vietnam National University in Hanoi (project code: QG TD 10 02) REFERENCES [1] N Nishiguchi, Phys Rev B 52, 5279 (1995) [2] P Zhao, Phys Rev B 49, 13589 (1994) [3] T C Phong and N Q Bau, J Korean Phys Soc 42, 647 (2003) [4] N Q Bau and T C Phong, J Phys Soc Jpn 67, 3875 (1998) [5] V V Pavlovich and E M Epshtein, Sov Phys Semicond 11, 809 (1977) [6] G M Shmelev, L A Chaikovskii and N Q Bau, Sov Phys Semicond 12, 1932 (1978) [7] N Q Bau, D M Hung and N B Ngoc, J Korean Phys Soc 54, 765 (2009) [8] N Q Bau and H D Trien, J Korean Phys Soc 56, 120 (2010) [9] E M Epshtein, Sov Phys Semicond 10, 1414 (1976) [10] E M Epshtein, Zh Tekh Fiz., Pisma 2, 234 (1976) [11] G M Shmelev, G I Xurcan and N H Shon, Sov Phys Semicond 15, 156 (1981) [12] K Ensslin, M Sundaram, A Wixforth, J H English and A C Gosssard, Phys Rev B 43, 9988 (1991) [13] E G Gwin, R M Westervelt, P F Hopkins, A J Rimberg, M Sundaram and A C Gossard, Phys Rev B 39, 6260 (1989) [14] V Halonen, Phys Rev B 47, 4003 (1993) [15] V Halonen, Phys Rev B 47, 1001 (1993) [16] S Fujita and Y Okamura, Phys Rev B 69, 155313 (2004) [17] G M Gusev, A A Quivy, T E Lamas and J R Leite, Phys Status Solidi C 1, S181 (2004) [18] C Ellenberger, B Simoviˇc, R Lecturcq, T Ihn, S E Ulloa, K Ensslin, D C Driscoll and A C Gossard, Phys Rev B 74, 195313 (2006) [19] A M O de Zevallos, N C Mamani, G M Gusev, A A Quivy and T E Lamas, Phys Rev B 75, 205324 (2007) [20] X Chen, J Phys Condens Matter 9, 8249 (1997) [21] M Charbonneau, K M van Vliet and P Vasilopoulos, J Math Phys 23, 318 (1982) [22] S C Lee, J Korean Phys Soc 51, 1979 (2007)  ... 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,... minimum with a negative value When the magnetic field is increased further, the HC increases continuously (with negative values) Particularly, the values of HC at the maxima are much larger and at the. .. confirms once again that the HC is quite sensitive to the change in the temperature IV CONCLUSIONS In this work, we have studied the in uence of laser radiation on the Hall effect in quantum wells