DSpace at VNU: Dependence of the Hall Coefficient on Doping Concentration in Doped Semiconductor Superlattices with a Perpendicular Magnetic Field under the Influence of a Laser Radiation

This article was downloaded by: [Florida State University] On: 09 October 2014, At: 04:21 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Integrated Ferroelectrics: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ginf20 Dependence of the Hall Coefficient on Doping Concentration in Doped Semiconductor Superlattices with a Perpendicular Magnetic Field under the Influence of a Laser Radiation a ab Nguyen Quang Bau & Bui Dinh Hoi a Department of Physics, College of Natural Science, Vietnam National University, Hanoi, Viet Nam b Department of Physics, National University of Civil Engineering, Hanoi, Viet Nam Published online: 23 May 2014 To cite this article: Nguyen Quang Bau & Bui Dinh Hoi (2014) Dependence of the Hall Coefficient on Doping Concentration in Doped Semiconductor Superlattices with a Perpendicular Magnetic Field under the Influence of a Laser Radiation, Integrated Ferroelectrics: An International Journal, 155:1, 39-44, DOI: 10.1080/10584587.2014.905109 To link to this article: http://dx.doi.org/10.1080/10584587.2014.905109 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, 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at 04:21 09 October 2014 NGUYEN QUANG BAU1,∗ AND BUI DINH HOI1,2 Department of Physics, College of Natural Science, Vietnam National University, Hanoi, Viet Nam Department of Physics, National University of Civil Engineering, Hanoi, Viet Nam The dependence of the Hall coefficient on doping concentration in doped semiconductor superlattices (DSSLs) under a crossed dc electric field and magnetic field in the presence of a laser radiation, is investigated by using a quantum kinetic equation for electrons Analytical results for the resistance and the Hall coefficient (HC) are computationally evaluated and graphically plotted for the GaAs:Si/GaAs:Be DSSL Numerical results for the magnetoresistance are in accordance with available theories The dependence of the HC on the doping concentration shows an oscillation whose phase is strongly affected by the laser radiation Keywords Hall coefficient, SdH oscillation, doped superlattice, quantum kinetic equation 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 There have been many papers dealing with problems related to the incidence of EMWs in semiconductor systems such as the calculations of the linear and nonlinear absorption coefficients in low-dimensional semiconductor systems [1–5], the studies of the Hall effect in bulk semiconductors in the presence of an EMW by using quantum kinetic equation [6–10] In a recent work, we have used the quantum kinetic equation method to study the influence of an intense EMW on the Hall coefficient in parabolic quantum wells with an in-plane magnetic field [11] In this work, by using this method we study the Hall effect in doped semiconductor superlattices (DSSLs), subjected to a crossed dc electric field and magnetic field (the magnetic field is applied along the DSSL axis), in the presence of a laser radiation (intense EMW) We only consider the case in which the electron-acoustic phonon interaction is assumed to be dominant and electron gas is degenerate at low temperatures We derive analytical expressions for the conductivity tensor and the Hall coefficient (HC) taking account of arbitrary transitions Received July 23, 2013; in final form January 12, 2014 ∗ Corresponding author E-mail: nguyenquangbau54@gmail.com 39 40 N Q Bau and B D Hoi between Landau levels and between subbands The paper is organized as follows In the next section, we briefly describe a regime of the problem and present basic formulae of the calculation Numerical results and discussion are also given Remarks and conclusions are shown briefly in Sec Hall Effect in a DSSL under the Influence of a Laser Radiation Downloaded by [Florida State University] at 04:21 09 October 2014 We consider a simple model of a DSSL (n-i-p-i superlattice), in which electron gas is confined by an additional potential along the z direction and free in the (x-y) plane The motion of an electron is confined in each layer of the system and its energy spectrum is quantized into discrete levels in the z direction If the DSSL is subjected to a crossed electric field E1 = (E1 , 0, 0) and magnetic field B = (0, 0, B), the single-particle wave function and its eigenenergy are given by [12, 13] (r) = φN (x − x0 ) eiky y φn (z) , Ly εN,n ky = N + ωc + n + (1) ωp − vd ky + mvd2 ; N, n = 0, 1, , (2) where m and vd = E1 /B are the effective mass and the drift velocity of a conduction electron, respectively, ky being its wave vector in the y direction, ωp and ωc = eB/m are 1/2 the plasma and the cyclotron frequencies, respectively, ωp = e2 nD /κ0 m with κ0 is the electronic constant and nD is the doping concentration, x0 = − ky /(mωc ) and Ly are the center of Landau orbits and the normalization length in the y direction, respectively, N denotes the Landau level index and n being the quantization index of energy levels in the z direction due to the DSSL potential, φN (x) and φn (z) are the harmonic wave functions By using above wave function and energy spectrum, we can write out the Hamiltonian of electrons and phonons system and obtain the quantum kinetic equation for electrons in the presence of a laser radiation with electric field vector E = (0, E0 sin ( t) , 0) (E0 and are the amplitude and the frequency of the EMW, respectively), utilizing the same procedures as in Ref 11 Then by considering the electron - acoustic phonon interaction, we obtain the expression for the conductivity tensor after some manipulation: σim = τ δij − ωc τ εij k hk + ωc2 τ hi hj + ωc2 τ aδj m + τ be δj m + ωc2 τ δ m − ωc τ ε mp hp + ωc2 τ h hm , (3) where δij is the Kronecker delta, εij k being 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= eLy εN,n − εF , 2π m α (4) εF is the Fermi level, and b= 4π e m {b1 + b2 + b3 + b4 }, N,n,n (5) Dependence of the Hall Coefficient on Doping ∞ eB b1 = −γ (−1)s e−2πs 1+2 /( ωc ) 41 cos (2π s n¯ ) , s=1 n¯ = n−n Downloaded by [Florida State University] at 04:21 09 October 2014 ωp + eE1 n−n b4 = − γθ ωp + eE1 − eB cos (2π s n¯ ) , (−1)s e−2πs /( ωc ) cos (2π s n¯ ) , ∞ 1+2 /( ωc ) cos (2π s n¯ ) , / ( ωc ) , (−1)s e−2πs s=1 n−n ωp + eE1 + ,γ = N + 1/2 + +π/d ∞ 1+2 (τ is the relaxation time), I (n, n ) = /( ωc ) s=1 θ = e2 E02 / m2 mvd2 , (−1)s e−2πs / ( ωc ) , eB n¯ = = ∞ 1+2 s=1 γθ b3 = − n¯ = eB γθ b2 = s0 −π/d j =1 / ( ωc ) , ALy kB T εN,n − εF I n, n (2π )3 vs ωc vd2 N + + 1/2 B = √ B /2, B , = /τ /(mωc ), εN,n = N + φn (z − j d)| e±iqz z |φn (z − j d) ωc + n + ωp + dqz , d and s0 are the period and the number of periods, respectively, T is the temperature, kB being the Boltzmann constant,A = ξ / (2ρvs ) where vs , ξ and ρ are the sound velocity, the deformation potential constant and the mass density, respectively The magnetoresistance ρxx and the Hall coefficient RH are determined by [14]: ρxx = σyx σxx , RH = − + σ2 + σ2 σxx B σ yx xx yx (6) where σyx and σxx are given by Eq (3) In the following, we will give a deeper insight into these results by carrying out a numerical evaluation with the help of a computer program For the numerical evaluation, we consider the n-i-p-i DSSL of GaAs:Si/GaAs:Be with the following parameters [4, 5]: ξ = 13.5 eV, ρ = 5.32 gcm−3 , vs = 5378 ms−1 , εF = 50 meV, m = 0.067m0 (m0 is the mass of a free electron), τ = 10−12 s, Ly = 100 nm, d = nm We also consider transitions N = 0, N = 1, n = 0, n = Figure 1(a) shows the dependences of the magnetoresistance ρxx on the magnetic field at different values of the temperature We can see the appearance of the typical Shubnikovde Haas oscillations with the period does not depend on the temperature Our results are similar to those (for the type of oscillations) obtained experimently in a two-dimensional electron system [15] Moreover, the figure also shows that the amplitude of these oscillations at a fixed magnetic field decreases when the temperature increases Denoting A (Bn T ) and A (Bn T0 ), respectively, are amplitudes of the oscillation peaks observed at a magnetic field Downloaded by [Florida State University] at 04:21 09 October 2014 42 N Q Bau and B D Hoi Figure (a) The magnetoresistance as functions of the magnetic field at different values of the temperature in the absence of the EMW (b) The relative amplitude versus temperature Here, E1 = × 102 V/m, and nD = 1023 m−3 Bn and at temperatures T and T0 The relative amplitudes versus temperature have been shown to be [15] T sinh 2π kB mT0 / eBn A (Bn T ) = A (Bn T0 ) T0 sinh 2π kB mT / eBn (7) This relation is also plotted in Fig 1(b) for T0 = K, Bn = T, and it is seen that there is a good agreement between our calculation and Eq (7) In Fig 2(a), the magnetoresistance is plotted as a function of the magnetic field for two cases: the absence and the presence of the EMW There occurs the beat phenomenon in the case of the presence of the EMW This property has been observed in some two-dimesional electron systems (see Ref 16 and references therein) Figure 2(b) shows the dependences of the HC on the doping concentration for two cases: the absence and the presence of the EMW The HC can be seen to oscillate and decrease with increasing the doping concentration It is also seen that the presence of the Figure The dependences of the magnetoresistance on the magnetic field (a), and the dependences of the HC on the doping concentration (b) for two cases: the absence and the presence of an EMW Here, B = 3T, E1 = × 102 V/m, and T = K Dependence of the Hall Coefficient on Doping 43 EMW does not change the HC value considerably but causes the change in the phase of the oscillated HC Downloaded by [Florida State University] at 04:21 09 October 2014 Conclusion So far, we have studied the Hall effect in DSSLs subjected to crossed dc electric and magnetic fields under the influence of a laser radiation (intense EMW) The electronacoustic phonon interaction is taken into account We obtain the expressions for the magnetoresistance as well as the HC The analytical results are numerically evaluated and plotted for the GaAs:Si/GaAs:Be DSSL The dependence of the magnetoresistance ρxx on the magnetic field shows Shubnikov–de Haas oscillations with periods not depend on the temperature and amplitudes decrease with increasing the temperature The HC oscillates and decreases with increasing the doping concentration The presence of the EMW does not affect the HC value considerably but causes the change in the phase of oscillations Funding This research is funded by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) (Grant No.: 103.01-2011.18) and Vietnam National University (Grant No.: QGTD.12.01) References T C Phong, and N Q Bau, Parametric resonance of acoustic and optical phonons in a quantum well J Korean Phys Soc 42, 647–651 (2003) N Q Bau, and T C Phong, Calculations of 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Hall coefficient on doping concentration in doped semiconductor superlattices (DSSLs) under a crossed dc electric field and magnetic field in the presence of a laser radiation, is investigated... can be seen to oscillate and decrease with increasing the doping concentration It is also seen that the presence of the Figure The dependences of the magnetoresistance on the magnetic field (a) ,