Journal of Science: Advanced Materials and Devices (2016) 209e213 Contents lists available at ScienceDirect Journal of Science: Advanced Materials and Devices journal homepage: www.elsevier.com/locate/jsamd Original article Impact of confined LO-phonons on the Hall effect in doped semiconductor superlattices Nguyen Quang Bau*, Do Tuan Long Faculty of Physics, Hanoi University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam a r t i c l e i n f o a b s t r a c t Article history: Received June 2016 Received in revised form 10 June 2016 Accepted 10 June 2016 Available online 18 June 2016 Based on the quantum kinetic equation method, the Hall effect in doped semiconductor superlattices (DSSL) has been theoretically studied under the influence of confined LO-phonons and the laser radiation The analytical expression of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient of a GaAs:Si/GaAs:Be DSSL is obtained in terms of the external fields, lattice period and doping concentration The quantum numbers N, n, m were varied in order to characterize the effect of electron and LO-phonon confinement Numerical evaluations showed that LO-phonon confinement enhanced the probability of electron scattering, thus increasing the number of resonance peaks in the Hall conductivity tensor and decreasing the magnitude of the magnetoresistance as well as the Hall coefficient when compared to the case of bulk phonons The nearly linear increase of the magnetoresistance with temperature was found to be in good agreement with experiment © 2016 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Keywords: Confined LO-phonons The Hall effect The magnetoresistance Doped superlattices Semiconductor Introduction It is wellknown that the effect of phonon confinement in lowdimensional semiconductor systems leads to a change in the probability of carrier scattering, thus creating new behaviours of materials in comparison to the case of unconfined phonons [1,2] Consequently, there have been many published works dealing with the influence of confined phonons on the optical, electrical, and magnetic properties of low-dimensional semiconductor systems such as the influence of confined phonons on the absorption coefficient of strong electromagnetic waves [3] and carrier capture processes [4], as well as the resonant quasiconfined optical phonons in semiconductor superlattices [5] In semiconductors systems, the optical phonons branches not overlap and it can be considered to be confined, the wave vector of confined optical phonon contained the quantized component and the in-plane one [1,6] The different boundary conditions placed on the electrostatic potential or vibrational amplitude of the phonons, lead to be distinct confined phonon models such as the guided mode model, the slab mode model and the Huang-Zhu model [7] In the previous work [8], we have studied the Hall effect in doped semiconductor superlattices (DSSL) with bulk phonons Through the works of [2,3,6], the contribution of phonon confinement is shown to be * Corresponding author E-mail address: nguyenquangbau54@gmail.com (N.Q Bau) Peer review under responsibility of Vietnam National University, Hanoi important in the properties of low-dimensional semiconductor systems and should not be neglected Thus, in this work, we continue studying the impact of the confined LO-phonons on the Hall effect in DSSLs subjected to a dc electric field, a perpendicular magnetic field and varying laser radiation The analytical expressions of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient (HC) in DSSLs under the influence of confined LO-phonons are obtained by using the quantum kinetic equation method [3,8] This article is organized as follows: we outline the effects of confined electrons and confined LO-phonons in doped semiconductor superlattices and present the basic formulae for the calculations in Sec.2 Numerical results and discussion for the GaAs:Si/GaAs:Be doped semiconductor superlattices are given in the Sec.3 Finally, Sec.4 shows remarks and conclusions The Hall effect in DSSLs under the impact of confined LOphonons Consider a simple model for doped semiconductor superlattices in which the motion of the electrons is restricted along the z axis due to the DSSL confinement potential and free in the xÀy plane The thicknesses and concentrations of the n-doping and p-doping layer of the DSSL are assumed to be equal: da ¼ dp ¼ d/2 and na ¼ np ¼ nD, here d, nD are the period and the doping concentra! tions of the DSSL, respectively A dc electric field E ¼ ðE1 ; 0; 0Þ; a ! ! magnetic field B ¼ ð0; 0; BÞ and laser radiation E ¼ ð0; E sin Ut; 0Þ was applied to the DSSL Under the influence of the material http://dx.doi.org/10.1016/j.jsamd.2016.06.010 2468-2179/© 2016 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) 210 N.Q Bau, D.T Long / Journal of Science: Advanced Materials and Devices (2016) 209e213 confinement potential and these external fields, the single-wave function of an electron and its discrete energy now becomes [8,9]: J! r ị ẳ p FN x x0 Þeiky y fn ðzÞ; (1) ! 1 Zuc ỵ n ỵ Zup Zyd ky N;n k y ẳ N ỵ 2 þ me y2d ; N; n ¼ 0; 1; 2…; (2) Ly ! z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with [z ¼ Z=me up and Hn(z) is the Hermite polynomial of n-th order; uc ¼ eB=me is the cyclotron frequency; yd ¼ E1 =B being the drift velocity of the electron The quantized frequency of confined LO-phonon and its wave vector are given by [7]: m ¼ 1; 2; 3:::; q ỵ q2m ; u2 ! ẳ u20 n2 ! m; q (3) (4) ⊥ where n is velocity parameter and m being the quantum number characterizing the LO-phonon confinement Also, the matrix element for confined electron e confined LOphonon interaction in doped semiconductor superlattices Dm I m J uị now becomes [7,10]: ! ẳ Cm;! q ⊥ n;n N;N N;n;N ;n0 ; q ⊥ 2 2pe2 Zu 1 À C ! ¼ m; q ⊥ ε0 V0 c c0 ; !2 q ỵ q2m h i2 JN;N0 uị2 ẳ N !eu uN0 N LN0 N uị ; N N! m ẳ In;n0 Nd X kẳ1 r Zd mpz hmịcos d d ỵ hm ỵ 1ịsin N ;n ; m; q ⊥ plasma frequency; ε0 is the electric constant; FN is the harmonic oscillator wave function, here x0 ¼ [2B ky me yd =Zị with p [B ẳ Z=me uc is the radius of the Landau orbit in the xÀy plane; fn ðzÞ being the electron subband wave functions due to the ma! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 H z terial connement potential: fn zị ẳ 2n n!1pp[ exp À 2[ n [z mp ! ! ; q ¼ ð q ⊥ ; qm Þ; qm ¼ d ! f ! À f0 h! !i vf ! ! N;n; k y N;n; k y E ỵ Zuc k y ∧ h ; ¼ ! t Zv k y ! ! 2 ỵ X 2 X eE ! m 2 J ỵ uị J ; q C ! In;n0 N;N y s m; q ⊥ me U2 s¼À∞ 0 ! ! ! N ! ỵ1 f ! N ! m; q N ;n0 ; k y ỵ q y N;n; k y m; q ⊥ ! ! ! d N0 ;n0 k y ỵ q y À εN;n k y À Zu ! À sZU m; q ! ỵ f ! ! N ! f ! N ! ỵ1 m; q N ;n0 ; k y À q y m; q ⊥ N;n; k y ! ! ' !  d εN0 ;n0 k y À q y À N;n k y ỵ Zu ! sZU ; m; q ⊥ & where N, n are the Landau level index and the subband index, respectively; Z is the Planck constant; me is the effective mass of an electron; ky, Ly being the wave vector of the electron and the 1=2 is the normalization length along the y direction; up ¼ 4εp0emne D z the DSSL This leads the quantum kinetic equation for electron distribution to now become: mpz à fn0 ðz À kdÞfn ðz À kdÞ; d where ε0 is the electric constant; V0 is the normalization volume of specimen; c0 and c∞ are the static and the high frequency dielectric constants; LM N ðuÞ is the associated Laguerre polynomial, !2 u ¼ [2B q ⊥ =2; Nd being the the number of periods of the DSSL; h(m) ¼ if m is even, h(m) ¼ if m is odd; fn (z) and fÃn0 ðzÞ are the electron sub-band wave functions in the initial and final states The effect of LO-phonon confinement and these external fields change the probability of electron scattering, thus modifying the Hamiltonian of the confined electron e confined phonon system in  f (5) ! ! where h ¼ B is the unit vector along the magnetic field; the noB tation “∧” represents the vector product; f0 is the equilibrium electron distribution function, t is the momentum relaxation time of electron, which is assumed to be a constant, Js(x) is the sth-order Bessel function of argument x and d(X) being the Dirac delta function The electron distribution function is now non-equilibrium and the current density is nonlinear as a result Let us consider that the ! electron gas is non-degenerate, f0 ¼ n0 expfbẵF N;n k y ịg; where F is the Fermi level, b ¼ 1/kBT and kB is the Boltzmann constant For simplicity, we limit the problem to the cases of s ¼ À1,0,1, meaning the processes with more than one photon are ignored After some manipulation, the expression for the conductivity tensor is obtained: sip ¼ > > > < > t dik À uc tijk hj ỵ u2c t2 hi hk dkp yd A 2 > ỵ uc t > > > :  X È À ÁÉ exp b F N;n ỵ dkp uc tklp hl þ u2c t2 hk hp N;n  X N0 ;n0 ; t pe2 Zu0 A 1 À 2 me c c0 ỵ uc t N;n;m È À ÁÉ exp b εF À εN;n m 2 I b1 ỵ b2 ỵ b3 ỵ b4 ỵ b5 ỵ b6 expbZu0 ị n;n > > > > = ỵ b7 þ b8 Þ ; > > > > ; (6) where symbols i, j, k, l, p correspond the components x, y, z of the Cartesian coordinates, dik is the Kronecker delta and εijk being the antisymmetric Levi e Civita tensor The terms b1, b2, …, b8 are given below: N.Q Bau, D.T Long / Journal of Science: Advanced Materials and Devices (2016) 209e213 " # [2B q2m N0 ỵ N ỵ eB[ N ! 1 b1 ẳ xMị dX1 ị; Z N! M Mỵ1 b7 ẳ b2 ẳ b3 ẳ e2 E 2me U4 e2 E2 # 3 2 " [2B q2m N0 ỵ N ỵ eB[ N! 1À xðMÞ dðX2 Þ; Z N! M Mỵ1 # 3 2 " [2B q2m N0 ỵ N ỵ eB[ N! 1 xMị dX3 ị; Z N! M Mỵ1 4me U4 e2 E2 b1 ; 4me U4 " # [2B q2m N0 ỵ N ỵ eB[ N! 1 xMị dX4 ị; b4 ẳ Z N0 ! M Mỵ1 b8 ẳ b5 ẳ b6 ¼ e2 E 2me U4 e2 E2 # 3 2 " [2B q2m N0 ỵ N ỵ eB[ N! 1 À x ðMÞ dðX5 Þ; Z N0 ! M Mỵ1 # 3 2 " [2B q2m N0 ỵ N ỵ eB[ N! 1À xðMÞ dðX6 Þ; Z N0 ! M Mỵ1 4me U4 e2 E2 4me U4 b4 ; where X1 ẳ N0 NịZuc ỵ n0 nịZup À eE1 [ À Zum ; X2 ¼ X1 À ZU; X3 ẳ X1 ỵ ZU; X4 ẳ N N ịZuc ỵ n0 nịZup ỵ eE1 [ þ Zum ; X5 ¼ X4 À ZU; εN;n ẳ Nỵ X6 ẳ X4 ỵ ZU; n e2 ZbLy I 1 Zuc ỵ n þ Zup þ me y2d ; A ¼ ; 2 2pme M ¼ jN À N0 j; x1ị ẳ and xMị ẳ 1=M 1ị if M > 1; a ¼ Lx =2[2B ; I¼ a ẵexpabZyd ị ỵ expabZyd ị ẵexpabZyd ị bZyd ðbZyd Þ À expðÀabZyd Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here the appearance of [ ẳ N ỵ 1=2 ỵ N þ þ 1=2 Þ[B =2 assumes an effective phonon momentum eyd qy zeE1 [, which sim! plifies the summation of q ⊥ [9] The delta functions are also replaced by dXị ẳ p1 G where G ẳ Z=t is the damping factor, to X ỵG avoid divergence [9,11] The component rxx of the magnetoresistance and the Hall coefficient are given by [9]: rxx ¼ sxx : s2xx þ s2yx syx RH ¼ À : B s2xx þ s2yx 211 (7) (8) where syx and sxx are derived by formula (6) Through equations (6)e(8), the impact of confined LO-phonons on the Hall effect is interpreted by the dependence of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient on the quantum number m characterizing the LO-phonon confinement and the other parameters of the external fields as well as the DSSL The different form of the confined LO-phonon wave vector and frequency lead to considerable changes of the theoretical results in comparison with the bulk phonons from the previous study [8] When m goes to zero, we obtain results as the case of bulk phonon in doped semiconductor superlattices Numerical results and discussion To clarify the obtained theoretical results, in this section, we present in detail the numerical evaluation of the Hall conductivity, the magnetoresistance and the Hall coefficient for the GaAs:Si/ GaAs:Be doped semiconductor superlattices Parameters used in this calculation are as follows: me ¼ 0.067 m0, (m0 is the free mass of an electron), c∞ ¼ 10:9, c0 ¼ 12:9, εF ¼ 50meV, t ¼ 10À12 s, n ¼ 8:73  104 msÀ1 , Zu0 ¼ 36:6meV, U ¼ 4:1012 sÀ1 , T ¼ 290K, E1 ¼ 2:102 V=m, E ¼ 105 V=m, Lx ¼ Ly ¼ 100nm, Nd ¼ 3, N ¼ 0, N0 ¼ 2, n ¼ 0, n0 ¼ 0/1 (the transition between the lowest and the first excited level of an electron) As we can see in Fig 1, there are multiple resonance peaks of the Hall conductivity tensor sxx These peaks correspond to the condition: ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN NịZuc ẳ Z u20 n2 m2 p2 me d2 ỵ eE1 [n0 nịZup ZU; which is called the intersubband magnetophonon resonance (MPR) condition [12e15] From the left to the right, in Fig 1a, resonance peaks of the conductivity tensor in case of bulk phonons correspond to the conditions 2Zuc ẳ Zu0 n0 nịZup ZU; Zu0 À ðn0 À nÞZup ; Zu0 À ðn0 nịZup ỵ ZU; Zu0 ZU; Zu0 ; Zu0 þ ZU; Zu0 þ ðn0 À nÞZup À ZU; Zu0 þ ðn0 À nÞZup ; Zu0 þ ðn0 À nÞZup þZU; here eE1 [≪Zu0 and it should be neglected for simplicity When phonons are confined, in this case we have m ¼ / 2, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LO-phonon frequency is now modified to u1 ¼ u20 À n2 p2 =me d2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and u2 ¼ u20 À 4n2 p2 =me d2 , thus, giving the additional resonance peaks of the conductivity tensor It is easy to see that the parameters of the superlattices have important roles on the MPR condition Indeed, the small value of the doping concentration nD leads to a weak material confinement effect Thus, in Fig 1b, the resonance peaks, which are associated with the effect of confined electrons, have mostly disappeared with only the center peaks being observed With increasing period d of a DSSL, the contribution from LO-phonon confinement decreases thus corresponding resonance peaks will also be difficult to detect Therefore, the confinement of LO-phonons, as well as doped superlattice parameters, make a remarkable impact on the magneto-phonon resonance condition Fig shows the dependence of the magnetoresistance on the temperature at different values of quantum number m which characterizes the LO-phonon confinement It can be seen that the 212 N.Q Bau, D.T Long / Journal of Science: Advanced Materials and Devices (2016) 209e213 (a) −4 x 10 bulk phonon confined LO−phonon m=0 m=1 m=2 −2 500 Hall coefficient (arb.units) Conductivity tensor σ (arb.units) xx 600 n =3.5 × 1020 m−3 400 D d=12 nm 300 200 100 −4 −6 −8 −10 −12 −14 −16 0 10 15 20 25 Cyclotron energy (meV) 30 35 −18 Conductivity tensor σ (arb.units) xx bulk phonon confined LO−phonon (b) 450 400 350 300 n =1018 m−3 250 d=15 nm D 200 150 100 50 0 10 15 20 25 Cyclotron energy (meV) 30 35 Fig The dependence of the conductivity tensor sxx on the cyclotron energy for confined phonon (solid curve) and bulk phonon (dashed curve), here nD ¼ 3.5  1020 mÀ3, d ¼ 12 nm (Fig 1a) and nD ¼ 1018 mÀ3, d ¼ 15 nm(Fig 1b) The laser amplitude (V/m) 10 x 10 Fig The dependence of the Hall coefficient on the laser amplitude for bulk phonon m ¼ (dotted curve) and confined phonon m ¼ (dashed curve), m ¼ / (solid curve), here B ¼ 2.5 T, nD ¼ 3.1020 mÀ3 and d ¼ 12 nm in LO-phonon confinement The current density rises with electron scattering, thus, the magnetoresistance decreases as a result Fig shows the Hall coefficient plotted as a function of laser amplitude at different values of quantum number m It can be seen that the HC decreases nonlinearly to a near-zero saturating value as the laser amplitude is increased It has been seen that the HC decreases nonlinearly to the saturation value as the raising of the laser amplitude In addition, the increasing of quantum number m leads to a faster HC decline Hence, there are new behaviours of the Hall effect in doped semiconductor superlattices due to the effect of LOphonon confinement Conclusions 0.04 0.035 ρ xx (arb.units) 0.03 m=0 m=1 m=2 0.025 0.02 0.015 0.01 0.005 50 100 150 200 Temperature (K) 250 300 Fig The dependence of the magnetoresistance rxx on the temperature T for bulk phonon m ¼ (dotted curve) and confined phonon m ¼ (dashed curve), m ¼ / (solid curve), here B ¼ 2.5 T, nD ¼ 3.1020 mÀ3 and d ¼ 12 nm magnetoresistance increases nearly linear at high temperatures This result is in accordance with that obtained in experiment [16] at the same range of the temperature Fig also shows that the increase of quantum number m leads to a decrease of the magnetoresistance The mechanism behind this decrease is likely the increase in the probability of electron scattering due to the increase So far, the influence of confined LO-phonons on the Hall effect in doped semiconductor superlattices GaAs:Si/GaAs:Be has been studied The analytical expressions for the Hall conductivity tensor, the magnetoresitance and the Hall coefficient are obtained base on quantum kinetic equation method Theoretical results are very different from previous one [8] because of the considerable contribution of the confined LO-phonon The effect of LO-phonon confinement enhances the probability of electron scattering The magnetoresistance, as well as the Hall coefficient, thus, decreases as a result In addition, the MPR condition in doped semiconductor superlattices under the influence of external fields and the effect of confined LO-phonon now contains new terms It was found that the increase of LO-phonon confinement leads to a decrease in the HC and the magnetoresistance When increasing the laser amplitude, the HC declined in magnitude to a saturation value near zero Furthermore, the near linear increase in the magnetoresistance with temperature has good agreement with experimental data [16] This study shows that confined LO-phonons create new properties and behaviours of the Hall effect in doped semiconductor superlattices Acknowledgements This paper is dedicated to the memory of Dr P.E Brommer - a founding editor of the Journal of Science: Advanced Materials and Devices This work was completed with financial support from the National Foundation for Science and Technology Development of N.Q Bau, D.T Long / Journal of Science: Advanced Materials and Devices (2016) 209e213 Vietnam (Nafosted 103.01e2015.22) and Vietnam International Education Development (Project 911) References [1] D.Z Mowbray, M Cardona, K Ploog, Confined LO phonons in GaAs/AlAs superlattices, Phys Rev B 43 (1991) 1598e1603 [2] J.S Bhat, B.G Mulimani, S.S Kubakaddi, Electron-confined LO phonon scattering rates in GaAs/AlAs quantum wells in the presence of a quantizing magnetic field, Semicond Sci Technol (1993) 1571e1574 [3] N.Q Bau, D.M Hung, L.T Hung, The influences of confined phonons on the nonlinear absorption coefficient of a strong electromagnetic wave by confined electrons in doping superlattices, Prog Electromagn Res Lett 15 (2010) 175e185 [4] A.M Paula, G Weber, Carrier capture processes in semiconductor superlattices due to emission of confined phonons, J Appl Phys 77 (1995) 6306e6312 [5] A Fasolino, E Molinari, J.C Maan, Resonant quasiconfined optical phonons in semiconductor superlattices, Phys Rev B 39 (1989) 3923e3926 [6] P.Y Yu, M Cardona, Fundamentals of Semiconductors, Springer Berlin, Heidelberg, 2005, pp 469e551 [7] S Rudin, T Reinecke, Electron-LO-phonon scattering rates in semiconductor quantum wells, Phys Rev B 41 (1990) 7713e7717 213 [8] N.Q Bau, B.D Hoi, Dependence of the Hall coefficient on doping concentration in doped semiconductor superlattices with a perpendicular magnetic field under the influence of a laser radiation, Integr Ferroelectr 155 (2014) 39e44 [9] M Charbonneau, K.M van Vliet, P Vasilopoulos, Linear response theory revisited III: one-body response formulas and generalized Boltzmann equations, J Math Phys 23 (1982) 318e336 [10] L Friedman, Electron-phonon scattering in superlattices, Phys Rev B 32 (1985) 955e961 [11] P Vasilopoulos, M Charbonneau, C.M Van Vliet, Linear and nonlinear electrical conduction in quasi-two-dimensional quantum wells, Phys Rev B 35 (1987) 1334e1344 [12] D.J Barnes, R.J Nicholas, F.M Peeters, X.G Wu, J.T Devreese, J Singleton, C.J.G.M Langerak, J.J Haris, C.T Foxon, Observation of optically detected magnetophonon resonance, Phys Rev Lett 66 (1991) 794e797 [13] G.Q Hai, F.M Peeters, Optically detected magnetophonon resonances in GaAs, Phys Rev B 60 (1999) 16513e16518 [14] N Mori, H Murata, K Taniguchi, C Hamaguchi, Magnetophonon-resonance theory of the two-dimensional electron gas in AlxGa1ÀxAs/GaAs single heterostructures, Phys Rev B 38 (1988) 7622e7634 [15] G.M Shmelev, G.I Tsurkan, N.H Shon, The magnetoresistance and the cyclotron resonance in semiconductors in the presence of strong electromagnetic wave, Sov Phys Semicond 15 (1981) 156e161 [16] E Waldron, J Graff, E Schubert, Influence of doping profiles on p-type AlGaN/ GaN superlattices, Phys Stat sol.(a) 188 (2001) 889e893 ... equations (6)e(8), the impact of confined LO- phonons on the Hall effect is interpreted by the dependence of the Hall conductivity tensor, the magnetoresistance and the Hall coefficient on the quantum... velocity parameter and m being the quantum number characterizing the LO- phonon confinement Also, the matrix element for confined electron e confined LOphonon interaction in doped semiconductor superlattices. .. decrease of the magnetoresistance The mechanism behind this decrease is likely the increase in the probability of electron scattering due to the increase So far, the in? ??uence of confined LO- phonons on