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Impact of confined LO-phonons on the Hall effect in doped semiconductor superlattices
Journal of Science: Advanced Materials and Devices 1 (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 3 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
1. 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 do 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: [email protected] (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.
2. 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 xy 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 1 ¼ ðE1 ; 0; 0Þ; a ! ! magnetic field B ¼ ð0; 0; BÞ and laser radiation E 0 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 1 (2016) 209e213
confinement potential and these external fields, the single-wave function of an electron and its discrete energy now becomes [8,9]:
1
Jð! r Þ ¼ pffiffiffi FN ðx x0 Þeiky y fn ðzÞ;
(1)
! 1 1 Zuc þ n þ Zup Zyd ky εN;n k y ¼ N þ 2 2 1 þ me y2d ; N; n ¼ 0; 1; 2…; 2
(2)
Ly
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 2 is the normalization length along the y direction; up ¼ 4εp0emne D plasma frequency; ε0 is the electric constant; FN is the harmonic oscillator wave function, here x0 ¼ [2B ðky me yd =ZÞ with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [B ¼ Z=me uc is the radius of the Landau orbit in the xy plane; fn ðzÞ being the electron subband wave functions due to the ma! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 H z terial confinement potential: fn ðzÞ ¼ 2n n!1pffiffipffi[ exp 2[ n [z 2 z
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]:
mp ! ! ; q ¼ ð q ⊥ ; qm Þ; qm ¼ d
m ¼ 1; 2; 3:::;
2 q ⊥ þ q2m ; u2 ! ¼ 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 0 J 0 ðuÞ now becomes [7,10]: ! ¼ Cm;! q ⊥ n;n N;N N;n;N 0 ;n0 ; q ⊥
2 2pe2 Zu 1 1 0 C ! ¼ m; q ⊥ ε0 V0 c∞ c0
1 ; !2 q ⊥ þ q2m
h i2 0 JN;N0 ðuÞ2 ¼ N !eu uN0 N LN0 N ðuÞ ; N N!
m ¼ In;n0
Nd X k¼1
rffiffiffi Zd 2 mpz hðmÞcos d d 0
þ hðm þ 1Þsin
mpz fn0 ðz kdÞfn ðz 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 1 if m is even, h(m) ¼ 0 if m is odd; fn (z) and fn0 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
the DSSL. This leads the quantum kinetic equation for electron distribution to now become:
! f ! f0 h! !i vf ! ! N;n; k y N;n; k y E 1 þ Zuc k y ∧ h ; ¼ ! t Zv k y ! ! 2 þ∞ X 2 X eE ! m 2 2 0 J þ ðuÞ J ; q C ! In;n0 N;N y s m; q ⊥ me U2 s¼∞ 0 0 N ;n ;
! m; q ⊥ f
! ! N ! þ1 f ! N ! m; q ⊥ N 0 ;n0 ; k y þ q y N;n; k y m; q ⊥ ! ! ! d εN0 ;n0 k y þ q y εN;n k y Zu ! sZU m; q ⊥
þ f ! ! N ! f ! N ! þ1 m; q ⊥ N 0 ;n0 ; k y q y m; q ⊥ N;n; k y ! ! ! d εN0 ;n0 k y q y εN;n k y þ Zu ! sZU ; m; q ⊥ (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 expfb½εF εN;n ð k y Þg; where εF 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 1,0,1, meaning the processes with more than one photon are ignored. After some manipulation, the expression for the conductivity tensor is obtained:
sip ¼
8 > > > < >
t dik uc tεijk hj þ u2c t2 hi hk dkp yd A 2 2 > 1 þ uc t > > > :
X
exp b εF εN;n þ dkp uc tεklp hl þ u2c t2 hk hp
N;n
X N0 ;n0 ;
t pe2 Zu0 A 1 1 2 2 me ε0 c∞ c0 1 þ uc t
N;n;m
exp b εF εN;n m 2 I 0 ðb1 þ b2 þ b3 þ b4 þ b5 þ b6 expðbZu0 Þ 1 n;n 9 > > > > = þ 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 1 (2016) 209e213
" # [2B q2m N0 þ N þ 1 eB[ N 0 ! 2 1 1 b1 ¼ xðMÞ dðX1 Þ; Z N! M Mþ1 2 b7 ¼
b2 ¼
b3 ¼
e2 E 2 2me U4
e2 E2
# 3 0 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 xðMÞ dðX2 Þ; Z N! M Mþ1 2
# 3 0 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 xðMÞ dðX3 Þ; Z N! M Mþ1 2
4me U4
e2 E2
b1 ;
4me U4
" # [2B q2m N0 þ N þ 1 eB[ N! 2 1 1 xðMÞ dðX4 Þ; b4 ¼ Z N0 ! M Mþ1 2 b8 ¼
b5 ¼
b6 ¼
e2 E 2 2me U4
e2 E2
# 3 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 x ðMÞ dðX5 Þ; Z N0 ! M Mþ1 2
# 3 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 xðMÞ dðX6 Þ; Z N0 ! M Mþ1 2
4me U4
e2 E2 4me U4
b4 ;
where
X1 ¼ ðN0 NÞZuc þ ðn0 nÞZup eE1 [ Zum ; X2 ¼ X1 ZU; X3 ¼ X1 þ ZU; X4 ¼ ðN N 0 ÞZuc þ ðn0 nÞZup þ eE1 [ þ Zum ; X5 ¼ X4 ZU; εN;n ¼
Nþ
X6 ¼ X4 þ ZU;
n e2 ZbLy I 1 1 1 Zuc þ n þ Zup þ me y2d ; A ¼ 0 ; 2 2 2 2pme
M ¼ jN N0 j; xð1Þ ¼ 0 and xðMÞ ¼ 1=ðM 1Þ if M > 1;
a ¼ Lx =2[2B ;
I¼
a 1 ½expðabZyd Þ þ expðabZyd Þ ½expðabZyd Þ 2 bZyd ðbZyd Þ expðabZyd Þ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here the appearance of [ ¼ ð N þ 1=2 þ N þ 1 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 dðXÞ ¼ p1 2 G 2 where G ¼ Z=t is the damping factor, to X þG 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 1 RH ¼ : 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. 3. 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 1012 s, n ¼ 8:73 104 ms1 , Zu0 ¼ 36:6meV, U ¼ 4:1012 s1 , 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 0 NÞZuc ¼ Z u20 n2 m2 p2 me d2 þ eE1 [±ðn0 nÞZup ±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 ¼ Zu0 ðn0 nÞZup ZU; Zu0 ðn0 nÞZup ; Zu0 ðn0 nÞZup þ ZU; Zu0 ZU; Zu0 ; Zu0 þ ZU; Zu0 þ ðn0 nÞZup ZU; Zu0 þ ðn0 nÞZup ; Zu0 þ ðn0 nÞZup þZU; here eE1 [≪Zu0 and it should be neglected for simplicity. When phonons are confined, in this case we have m ¼ 1 / 2, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LO-phonon frequency is now modified to u1 ¼ u20 n2 p2 =me d2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and u2 ¼ u20 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. 2 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
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N.Q. Bau, D.T. Long / Journal of Science: Advanced Materials and Devices 1 (2016) 209e213
(a)
−4
x 10
bulk phonon confined LO−phonon
0 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 0
5
10 15 20 25 Cyclotron energy (meV)
30
35
−18
Conductivity tensor σ (arb.units) xx
2 bulk phonon confined LO−phonon
(b)
450 400 350 300
n =1018 m−3
250
d=15 nm
D
200 150 100 50 0 0
5
10 15 20 25 Cyclotron energy (meV)
30
35
Fig. 1. 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 m3, d ¼ 12 nm (Fig. 1a) and nD ¼ 1018 m3, d ¼ 15 nm(Fig. 1b).
4 6 8 The laser amplitude (V/m)
10 5 x 10
Fig. 3. The dependence of the Hall coefficient on the laser amplitude for bulk phonon m ¼ 0 (dotted curve) and confined phonon m ¼ 1 (dashed curve), m ¼ 1 / 2 (solid curve), here B ¼ 2.5 T, nD ¼ 3.1020 m3 and d ¼ 12 nm.
in LO-phonon confinement. The current density rises with electron scattering, thus, the magnetoresistance decreases as a result. Fig. 3 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.
4. 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 0 50
100
150 200 Temperature (K)
250
300
Fig. 2. The dependence of the magnetoresistance rxx on the temperature T for bulk phonon m ¼ 0 (dotted curve) and confined phonon m ¼ 1 (dashed curve), m ¼ 1 / 2 (solid curve), here B ¼ 2.5 T, nD ¼ 3.1020 m3 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. 2 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 1 (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. 8 (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.
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[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 AlxGa1xAs/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.
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 3 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
1. 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 do 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: [email protected] (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.
2. 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 xy 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 1 ¼ ðE1 ; 0; 0Þ; a ! ! magnetic field B ¼ ð0; 0; BÞ and laser radiation E 0 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 1 (2016) 209e213
confinement potential and these external fields, the single-wave function of an electron and its discrete energy now becomes [8,9]:
1
Jð! r Þ ¼ pffiffiffi FN ðx x0 Þeiky y fn ðzÞ;
(1)
! 1 1 Zuc þ n þ Zup Zyd ky εN;n k y ¼ N þ 2 2 1 þ me y2d ; N; n ¼ 0; 1; 2…; 2
(2)
Ly
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 2 is the normalization length along the y direction; up ¼ 4εp0emne D plasma frequency; ε0 is the electric constant; FN is the harmonic oscillator wave function, here x0 ¼ [2B ðky me yd =ZÞ with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [B ¼ Z=me uc is the radius of the Landau orbit in the xy plane; fn ðzÞ being the electron subband wave functions due to the ma! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 H z terial confinement potential: fn ðzÞ ¼ 2n n!1pffiffipffi[ exp 2[ n [z 2 z
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]:
mp ! ! ; q ¼ ð q ⊥ ; qm Þ; qm ¼ d
m ¼ 1; 2; 3:::;
2 q ⊥ þ q2m ; u2 ! ¼ 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 0 J 0 ðuÞ now becomes [7,10]: ! ¼ Cm;! q ⊥ n;n N;N N;n;N 0 ;n0 ; q ⊥
2 2pe2 Zu 1 1 0 C ! ¼ m; q ⊥ ε0 V0 c∞ c0
1 ; !2 q ⊥ þ q2m
h i2 0 JN;N0 ðuÞ2 ¼ N !eu uN0 N LN0 N ðuÞ ; N N!
m ¼ In;n0
Nd X k¼1
rffiffiffi Zd 2 mpz hðmÞcos d d 0
þ hðm þ 1Þsin
mpz fn0 ðz kdÞfn ðz 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 1 if m is even, h(m) ¼ 0 if m is odd; fn (z) and fn0 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
the DSSL. This leads the quantum kinetic equation for electron distribution to now become:
! f ! f0 h! !i vf ! ! N;n; k y N;n; k y E 1 þ Zuc k y ∧ h ; ¼ ! t Zv k y ! ! 2 þ∞ X 2 X eE ! m 2 2 0 J þ ðuÞ J ; q C ! In;n0 N;N y s m; q ⊥ me U2 s¼∞ 0 0 N ;n ;
! m; q ⊥ f
! ! N ! þ1 f ! N ! m; q ⊥ N 0 ;n0 ; k y þ q y N;n; k y m; q ⊥ ! ! ! d εN0 ;n0 k y þ q y εN;n k y Zu ! sZU m; q ⊥
þ f ! ! N ! f ! N ! þ1 m; q ⊥ N 0 ;n0 ; k y q y m; q ⊥ N;n; k y ! ! ! d εN0 ;n0 k y q y εN;n k y þ Zu ! sZU ; m; q ⊥ (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 expfb½εF εN;n ð k y Þg; where εF 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 1,0,1, meaning the processes with more than one photon are ignored. After some manipulation, the expression for the conductivity tensor is obtained:
sip ¼
8 > > > < >
t dik uc tεijk hj þ u2c t2 hi hk dkp yd A 2 2 > 1 þ uc t > > > :
X
exp b εF εN;n þ dkp uc tεklp hl þ u2c t2 hk hp
N;n
X N0 ;n0 ;
t pe2 Zu0 A 1 1 2 2 me ε0 c∞ c0 1 þ uc t
N;n;m
exp b εF εN;n m 2 I 0 ðb1 þ b2 þ b3 þ b4 þ b5 þ b6 expðbZu0 Þ 1 n;n 9 > > > > = þ 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 1 (2016) 209e213
" # [2B q2m N0 þ N þ 1 eB[ N 0 ! 2 1 1 b1 ¼ xðMÞ dðX1 Þ; Z N! M Mþ1 2 b7 ¼
b2 ¼
b3 ¼
e2 E 2 2me U4
e2 E2
# 3 0 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 xðMÞ dðX2 Þ; Z N! M Mþ1 2
# 3 0 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 xðMÞ dðX3 Þ; Z N! M Mþ1 2
4me U4
e2 E2
b1 ;
4me U4
" # [2B q2m N0 þ N þ 1 eB[ N! 2 1 1 xðMÞ dðX4 Þ; b4 ¼ Z N0 ! M Mþ1 2 b8 ¼
b5 ¼
b6 ¼
e2 E 2 2me U4
e2 E2
# 3 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 x ðMÞ dðX5 Þ; Z N0 ! M Mþ1 2
# 3 2 " [2B q2m N0 þ N þ 1 eB[ N! 1 1 xðMÞ dðX6 Þ; Z N0 ! M Mþ1 2
4me U4
e2 E2 4me U4
b4 ;
where
X1 ¼ ðN0 NÞZuc þ ðn0 nÞZup eE1 [ Zum ; X2 ¼ X1 ZU; X3 ¼ X1 þ ZU; X4 ¼ ðN N 0 ÞZuc þ ðn0 nÞZup þ eE1 [ þ Zum ; X5 ¼ X4 ZU; εN;n ¼
Nþ
X6 ¼ X4 þ ZU;
n e2 ZbLy I 1 1 1 Zuc þ n þ Zup þ me y2d ; A ¼ 0 ; 2 2 2 2pme
M ¼ jN N0 j; xð1Þ ¼ 0 and xðMÞ ¼ 1=ðM 1Þ if M > 1;
a ¼ Lx =2[2B ;
I¼
a 1 ½expðabZyd Þ þ expðabZyd Þ ½expðabZyd Þ 2 bZyd ðbZyd Þ expðabZyd Þ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here the appearance of [ ¼ ð N þ 1=2 þ N þ 1 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 dðXÞ ¼ p1 2 G 2 where G ¼ Z=t is the damping factor, to X þG 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 1 RH ¼ : 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. 3. 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 1012 s, n ¼ 8:73 104 ms1 , Zu0 ¼ 36:6meV, U ¼ 4:1012 s1 , 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 0 NÞZuc ¼ Z u20 n2 m2 p2 me d2 þ eE1 [±ðn0 nÞZup ±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 ¼ Zu0 ðn0 nÞZup ZU; Zu0 ðn0 nÞZup ; Zu0 ðn0 nÞZup þ ZU; Zu0 ZU; Zu0 ; Zu0 þ ZU; Zu0 þ ðn0 nÞZup ZU; Zu0 þ ðn0 nÞZup ; Zu0 þ ðn0 nÞZup þZU; here eE1 [≪Zu0 and it should be neglected for simplicity. When phonons are confined, in this case we have m ¼ 1 / 2, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LO-phonon frequency is now modified to u1 ¼ u20 n2 p2 =me d2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and u2 ¼ u20 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. 2 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 1 (2016) 209e213
(a)
−4
x 10
bulk phonon confined LO−phonon
0 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 0
5
10 15 20 25 Cyclotron energy (meV)
30
35
−18
Conductivity tensor σ (arb.units) xx
2 bulk phonon confined LO−phonon
(b)
450 400 350 300
n =1018 m−3
250
d=15 nm
D
200 150 100 50 0 0
5
10 15 20 25 Cyclotron energy (meV)
30
35
Fig. 1. 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 m3, d ¼ 12 nm (Fig. 1a) and nD ¼ 1018 m3, d ¼ 15 nm(Fig. 1b).
4 6 8 The laser amplitude (V/m)
10 5 x 10
Fig. 3. The dependence of the Hall coefficient on the laser amplitude for bulk phonon m ¼ 0 (dotted curve) and confined phonon m ¼ 1 (dashed curve), m ¼ 1 / 2 (solid curve), here B ¼ 2.5 T, nD ¼ 3.1020 m3 and d ¼ 12 nm.
in LO-phonon confinement. The current density rises with electron scattering, thus, the magnetoresistance decreases as a result. Fig. 3 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.
4. 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 0 50
100
150 200 Temperature (K)
250
300
Fig. 2. The dependence of the magnetoresistance rxx on the temperature T for bulk phonon m ¼ 0 (dotted curve) and confined phonon m ¼ 1 (dashed curve), m ¼ 1 / 2 (solid curve), here B ¼ 2.5 T, nD ¼ 3.1020 m3 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. 2 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 1 (2016) 209e213
Vietnam (Nafosted 103.01e2015.22) and Vietnam International Education Development (Project 911).
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