Effect of Laser Radiation on the Energy. Spectrum of 2D Electrons with Rashba Interaction

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1 Advanced Studies in Theoretical Physics Vol. 8, 14, no. 17, HIKAI Ltd, Effect of Laser adiation on the Energy Spectrum of D Electrons with ashba Interaction S.V. Kryuchkov (a,b), E.I. Kukhar (a) and V.I. Konchenkov (b) (a) Physical Laboratory of Low Dimensional Systems Volgograd State Socio-Pedagogical University 466, V.I. Lenina av., 7, Volgograd, ussia (b) Volgograd State Technical University 45, V.I. Lenina av., 8, Volgograd, ussia Copyright 14 S.V. Kryuchkov, E.I. Kukhar and V.I. Konchenkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Quasienergy spectrum of the D electron gas with ashba interaction under the influence of high-frequency electromagnetic wave polarized elliptically was obtained. Splitting of energy subbands in the material at the presence of high-frequency laser radiation was revealed. Magnetoconductivity as function of wave intensity was calculated. Conductivity of D electron gas was shown to have minimum at the certain values of amplitude of high-frequency electric field. Keywords: Quasienergy, ashba Hamiltonian, spin-orbit interaction, magnetoconductivity 1 Introduction Spin states of electrons in semiconductor structures and problems of spin dynamics attract a substantial attention of researchers due to the intensive development of spintronics and heterostructures technology [4,11]. Effect of spin-orbit (SO) interaction on the states of D electrons in magnetic field was investigated theoretically in [3], where energy spectrum describing Landau levels

2 73 S.V. Kryuchkov, E.I. Kukhar and V.I. Konchenkov modified by this interaction was obtained. Taking into account SO interaction a magneto-phonon resonance in quantum wells and so-called spin Hall ect were studied in [] and [1] correspondingly. Optical properties of low-dimensional structures in conditions of SO interaction were investigated in [5,8]. A study of spin-dependent phenomena in low-dimensional systems takes a significant part in developing of information storage and processing devices [7]. The advantage of low-dimensional systems is the dependence of intensity of SO interaction on structure parameters [11] (geometry dimensions, confinement intensity, ective mass, etc.). In [6] a usage of circularly polarized optical radiation for controlling the electron spins in quantum dots was proposed. High-frequency (HF) electromagnetic (EM) radiation is known to modify the electron spectrum of semiconductor structures significantly [1]. Effect of HF EM radiation on electron dynamics in graphene, described by the Dirac equation, was investigated in [1,9], where the energy gap was shown to be induced dynamically by HF field in originally gapless graphene. Below we study the modification of electron spectrum of D electron gas (DEG) with SO interaction by the action of HF EM radiation. Besides we discuss the influence of HF radiation on the magnetoconductivity of DEG. Modification of electron spectrum of D-electrons with ashba interaction in HF field DEG is considered to be place in a xy plane. EM radiation with frequency ω and amplitude of electric field intensity E propagates along the Oz so that A t = E ω sinωt, sin ωt + ϕ, in the plane xy its vector potential is: ( ) ( ){ ( )} ( c = 1). Dimensional quantization of motion of D electrons along the axis Oz leads to the appearance of linear in quasimomentum p term, described SO interaction [11]. This term has a form Hˆ ( p ) = α ( pyσx pxσ y) (ashba model). Here σ are the Pauli matrixes, constant α is defined by the confinement potential, α ~ ev m. We take into account the SO interaction by adding into the Hamiltonian a term Ĥ. So spinor ψ describing the motion of D electron in the field of EM wave satisfies the equation ( h = 1) : i t ψ = Hˆ p + ea ψ + Hˆ p + ea. (1) ˆ p ( ) ( )ψ Here H ( p ) = m is Hamiltonian of system without SO interaction, m is an ective mass of electron. The solution of (1) satisfies the Floquet theorem [9]: ψ t = u t exp iε t, () where () t ( ) ( ) ( ) u is a spinor, which components u ( t) and ( t) u are periodical func-

3 Effect of laser radiation on the energy spectrum 733 tions with the period into (1) we obtain: π ω, ε is a quasienergy [9]. After the substitution () u + Hˆ i t ( + ea) u + H ( p + ea) u = ε u p, (3) ˆ Figure 1: Dispersion law of DEG with ashba interaction. Dashed line corresponds to + sign in (5), solid line to sign in (5). The components of spinor u ( t) are expended into Fourier series: () t = u + u () t + u () t... u is a constant term of the spinor u () t. u 1 +, where Frequency of the EM radiation is supposed to satisfy the next condition: ω >> ( αee ) 1. These inequality allows us to take only the first two terms of the Fourier series for the components of the spinor u ( t) : u( t) = u + u1( t). After substitution the last equality into equation (3) and averaging over the small oscillations we obtain: ˆ ( ) ˆ pe α pe sinϕ H p u + H ( p) u + u + σ zu = ε u, (4) m ω where p E = ee ω. From (4) we derive the quasienergy: 4 p + pe α pe sin ϕ ε = ± α p +. (5) m ω In the absence of the HF ( E = ) the dispersion law of DEG with ashba interaction contains two branches which coincide at the point of the Brillouin zone corresponding to the momentum p = [11]. At the presence HF EM radiation ( E ) these branches become separate in the point p = (Fig. 1). A value of 3 the energy splitting in this point is Δ E = α e E sinϕ ω. The splitting is seen ϕ = π and to be to be maximum for EM wave with circular polarization ( ) equal to zero for radiation polarized linearly ( ϕ = ). 3 Influence of HF field on magnetoconductivity of D electrons with ashba interaction Modification of electronic spectrum of DEG with ashba interaction in HF EM field leads to changing of DEG conductivity. Calculations based on

4 734 S.V. Kryuchkov, E.I. Kukhar and V.I. Konchenkov Boltzmann equation written in a relaxation time τ approximation give for the components of magnetoconductivity tensor the next expression: e τ f ( p) β μ σxx + iσxy = μ+, (6) p 1 iωcτ 1 iωτ c where ωc = ehμ, H is the intensity of the magnetic field which is applied perpendicularly to the DEG plane, μ = pε p, β = pε, f ( p) is an equilibrium state function. Further we consider the case θ << Δ E (θ is the temperature expressed in energy units). In such situation the initial momenta of electrons are located near the minima of the low dispersion line (Fig. 1, solid line) and the electron transitions to the top dispersion line can be neglected. Therefore to calculate the magnetoconductivity we take the sign in the formula (5). Figure : The dependence of the components of the magnetoconductivity tensor of DEG with SO interaction on the magnitude of HF EM radiation. a) H=3 T; b) H=5 T; c) H=1 T. The dependences of the components of the conductivity tensor on the amplitude of HF EM radiation plotted by the formula (6) are shown in Fig. 1 ( 1 3 E1 = m ω e, θ ~ 4. K). Dependence of conductivity on the amplitude of HF radiation is seen from the Fig. to have the non-monotonic character. Dependence of the diagonal conductivity σ xx on the amplitude of HF radiation is shown in Fig. 3 for H = and ϕ = π. At θ = the conductivity is: 4 1 q, q < 1, σ xx = σ, (7) 1 1, q > 1. q where q = E E1, σ = e nτ m, n is a surface concentration of charge carriers.

5 Effect of laser radiation on the energy spectrum 735 Figure 3: Dependence of the diagonal conductivity of DEG with ashba interaction on the amplitude of HF electric field. 4 Conclusions The dependence of conductivity σ xx on the amplitude E is seen from Fig. 3 to have minimum at E = E1. This fact is related with increasing of ective mass ( ) 1 m = ε p when value of E approximates to E 1. The value of the energy splitting in this case is Δ E = meα. Such result can be used for experimental measurements of splitting of dispersion lines in electron spectrum of DEG with ashba interaction. Note that the second formula of (7) is applicable if the next condition is performed: α << ω m. Nevertheless we can t put α is zero because of the inequality Δ E >> θ must be performed. The value α ~ ev m [4,11] is suitable for the formula (7). Acknowledgements. The work was supported by the FB Project r povolzhye a and with the funding of the Ministry of Education and Science of the ussian Federation within the base part of the State task 14/411 (Project Code: 5). eferences [1] D.S.L. Abergel, T. Chakraborty, Irradiated bilayer graphene, Nanotechnology, (11) 153. [] D.S.L. Abergel, V.I. Fal ko, Spin-orbit-assisted electron-phonon interaction and the magnetophonon resonance in semiconductor quantum wells, Phys. ev. B, 77 (8),

6 736 S.V. Kryuchkov, E.I. Kukhar and V.I. Konchenkov [3] P.S. Alekseev, M.V. Yakunin, I.N. Yassievich, Effect of spin-orbit coupling on the spectrum of two-dimensional electrons in a magnetic field, Semiconductors, 41 (7), [4] I. Appelbaum, B. Huang, D.J. Monsma, Electronic measurement and control of spin transport in silicon, Nature, 447 (7) [5] V.V. Bel kov, S.D. Ganichev, P. Schneider, C. Back, M. Oestreich, J. udolph, D. Hagele, L.E. Golub, W. Wegscheider, W. Prettl, Circular photogalvanic ect at inter-band excitation in semiconductor quantum wells, Solid State Commun., 18 (3), [6] J. Berezovsky, M.H. Mikkelsen, N.G. Stoltz, L.A. Coldren, D.D. Awschalom, Picosecond coherent optical manipulation of a single electron spin in a quantum dot, Science, 3 (8), [7] S. Datta, B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett., 56 (199), 665. [8] V.D. Dymnikov, O.V. Konstantinov, Spin-orbit mixing in III V semiconductors at the Γ point, Physics of the Solid State, 51 (9), [9] T. Oka, H. Aoki, Photovoltaic Hall ect in graphene, Phys. ev. B, 79 (9) 8146(). [1] V.I. Perel, S.A. Tarasenko, I.N. Yassievich, S.D. Ganichev, V.V. Bel kov, W. Prettl, Spin-dependent tunneling through a symmetric semiconductor barrier, Phys. ev. B, 67 (3), 134(). [11] C.S. Tang, A.G. Mal shukov, K.A. Chao, Generation of spin current and polarization under dynamic gate control of spin-orbit interaction in low-dimensional semiconductor systems, Phys. ev. B, 71 (5), eceived: June 17, 14

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