DSpace at VNU: Transport properties of a quasi-two-dimensional electron gas in a SiGe Si SiGe quantum well including temperature and magnetic field effects

Superlattices and Microstructures 64 (2013) 245–250 Contents lists available at ScienceDirect Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices Transport properties of a quasi-two-dimensional electron gas in a SiGe/Si/SiGe quantum well including temperature and magnetic field effects Nguyen Quoc Khanh ⇑, Nguyen Minh Quan Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Street, 5th District, Ho Chi Minh City, Viet Nam a r t i c l e i n f o Article history: Received 14 July 2013 Accepted 27 September 2013 Available online October 2013 Keywords: Scattering mechanisms Magnetoresistance Quantum well a b s t r a c t We investigate the mobility and resistivity of a quasi-twodimensional electron gas in a SiGe/Si/SiGe quantum well at arbitrary temperatures for two cases: with and without in-plane magnetic field We consider two scattering mechanisms: remote charged-impurity and interface-roughness scattering We study the dependence of transport properties on the carrier density, layer thickness, magnetic field and temperature Our results can be used to obtain information about the scattering mechanisms in the SiGe/Si/SiGe quantum well Ó 2013 Elsevier Ltd All rights reserved Introduction During the last few decades, much attention has been devoted to the transport properties of modulation-doped Si/SiGe heterostructures because of their high mobility and perfectives for applications [1–7] Recently, Gold has calculated the zero temperature mobility of the nonpolarized quasi-two-dimensional electron gas (Q2DEG) in a SiGe/Si/SiGe quantum well (QW), taking into account many-body effects, beyond the random-phase approximation via a local-field correction (LFC) [8], and obtained good agreement with recent experimental results [4,5] The scattering mechanism, which is responsible for limiting the mobility, can be determined by comparing experimental results with those of theoretical calculations [1–9] Recent measurements and calculations of transport properties of a 2DEG in a magnetic field give additional tool for determining the main scattering mechanism [10–16] To the author’s knowledge, there is no calculation of transport properties of the ⇑ Corresponding author Fax: +84 38350096 E-mail address: nqkhanh@phys.hcmuns.edu.vn (N.Q Khanh) 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.spmi.2013.09.036 246 N.Q Khanh, N.M Quan / Superlattices and Microstructures 64 (2013) 245–250 spin-polarized Q2DEG in a SiGe/Si/SiGe QW at finite temperatures Therefore, we calculate, in this paper, the finite temperature mobility of Q2DEG in a SiGe/Si/SiGe QW for charged-impurity scattering and study the effects of the LFC, magnetic field and layer thickness on transport properties We also discuss the importance of interface-roughness scattering (IRS) Theory We consider a 2DEG with parabolic dispersion determined by the effective mass mà We assume that the electron gas is in the xy plane with infinite confinement for z < and z > L For z L, the electron pffiffiffiffiffiffiffiffi gas in the lowest subband is described by the wave function wð0 z LÞ ¼ 2=L sin ðpz=LÞ [1] When the in-plane magnetic field B is applied to the system, the carrier densities n± for spin up/down are not equal [11,17] At T = we have n± = n(1 ± B/Bs)/2 for B < Bs with n+ = n and nÀ = for B P Bs Here n = n+ + nÀ is the total density and Bs is the so-called saturation field given by glBBs = 2EF where g is the electron spin g-factor, lB is the Bohr magneton and EF is the Fermi energy For T > 0, n± is determined using the Fermi distribution function in the standard manner [11,17] The energy averaged transport relaxation time for the (±) components are given in the Boltzmann theory by: h i Æ desðeÞe À @f @eðeÞ h i h sÆ i ẳ R ặ de e @f @ eeị R 1ị where [1,11] 1 ẳ skị 2phe qị ẳ ỵ Z 2k hjUqịj2 i ẵ2 qị q2 dq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4k À q2 ð2Þ 2pe2 F C qịẵ1 GqịPq; Tị; 2L q 3ị Pq; Tị ẳ Pỵ q; Tị ỵ P q; Tị; Pặ q; Tị ẳ ặ q; EFặ ị P F C qị ẳ b Z P0ặ q; l0 ị dl0 0 ặ qị P 2b lặ l0 ị cosh ẳ g v mà 2ph  ð4Þ ð5Þ ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2kF Ỉ 1À 1À Hðq À 2kF Æ Þ5; q ð6Þ   8p2 32p4 eaq 3aq ỵ 4p2 ỵ a2 q2 aq a2 q2 4p2 ỵ a2 q2 7ị 2 with f ặ eị ẳ 1=f1 ỵ expbẵe lặ Tịịg; b ẳ kB Tị , lặ ẳ lnẵ1 ỵ expbEFặ ị=b, EFặ ẳ  h kFặ =2m ị and 2 e ẳ h k =2m ị Here, m is the effective mass in xy-plane, gv is the valley degeneracy, G(q) is the LFC à describing the exchange–correlation effects [8,18–21] and hjUðqÞj2 i is the random potential which depends on the scattering mechanism [1] For charged-impurities of density Ni located on the plane with z = zi we have: hjU R qịj2 i ẳ Ni  2 2pe2 F R ðq; zi Þ2 2L q ð8Þ with the form factor FR(q, zi) for the electron–impurity interaction as given in Ref [1] Here eL is the background static dielectric constant For the interface-roughness scattering the random potential is given by [1]: N.Q Khanh, N.M Quan / Superlattices and Microstructures 64 (2013) 245250 hjU S qịj2 i ẳ   à 2  4 4p m p 2 ðeF DKÞ2 eÀq K =4 a2 mz kF a 247 ð9Þ where D represents the average height of the roughness perpendicular to the 2DEG and K represents the correlation length parameter of the roughness in the plane of the 2DEG and mz is the effective mass perpendicular to the xy-plane The mobility of the nonpolarized and fully polarized 2DEG is given by l0 ¼ e < s > =mà The resistivity is defined by q = 1/r where r = r+ + rÀ is the total conductivity and r± is the conductivity of the (±) spin subband given by [11]: rặ ẳ nặ e2 hsặ i m 10ị The authors of Ref [5] have found the strong decrease of the mobility at low electron densities This behavior is likely to be a precursor of localization which may lead to a metal–insulator transition (MIT) It was shown that multiple-scattering effects (MSE) can account for this MIT at low electron density where the mobility is determined by impurity scattering [22–23] We use the symbol l for the mobility when MSE are taken into account For n > nMIT the mobility can be written as l = lo(1 À A) with A The parameter A describes the MSE and depends on the random potential, the screening function and compressibility of the electron gas, and is given by [8,23]: A¼ 2pn2 Z hjUqịj2 iẵPo qị qdq ẵ2 qị : 11ị For n < nMIT, where A > 1, the mobility vanishes: l = Numerical results In this section, we present our numerical calculations for the mobility and resistivity of a Q2DEG in a SiGe/Si/SiGe QW using the following parameters [8]: eL = 12.5, gv = 2, mà = 0.19mo and mz = 0.916mo, where mo is the free electron mass We study the dependence of the mobility and resistivity on the LFC, impurity position, layer-thickness, magnetic field and temperature 3.1 The many-body effects To treat the many-body effects we use the LFC which is very important for low electron densities In the Hubbard approximation, only exchange effects are taken into account and the LFC has the form [5,24]: GH qị ẳ q q gv gs q2 ỵ kF 12ị where gs is the spin degeneracy We also use numerical results for the LFC G(q) as reported in Ref [21] where both exchange and correlation effects are taken into account The may-body effects on the mobility lo of the nonpolarized Q2DEG in silicon QW of width a = 100 Å for remote impurity scattering (RIS) are displayed in Fig of Ref [8] Similar results for the mobility lo of the fully polarized Q2DEG and the resistance ratio q(Bs)/q(B = 0) for RIS with Ni =  1012 cmÀ2 and zi = À100 Å are plotted in Fig We see that the use of a LFC is very important and the Hubbard approximation is not sufficient at very low densities We note that Dolgopolov and co-workers have analyzed the mobility as a function of electron density and concluded that the LFCs are approximately double the Hubbard form [5] 3.2 The impurity position and layer-thickness effects We have calculated the critical density nMIT as a function of the impurity density Ni and the well width a for a nonpolarized 2DEG at zero temperature The results shown in Fig indicate that nMIT 248 N.Q Khanh, N.M Quan / Superlattices and Microstructures 64 (2013) 245–250 10 105 G GH G=0 G=0 ρ(Βs)/ρ(0) mobility μ0(cm2/Vs) 10 G GH 103 102 10 0.1 0.1 10 11 10 11 -2 electron density n(10 cm ) -2 electron density n(10 cm ) Fig The mobility lo of the fully polarized Q2DEG (left) and the resistance ratio qBs ị=qB ẳ 0ị (right) for RIS in different approximations for G(q) -2 11 electron density nMIT(10 cm ) -2 11 electron density nMIT(10 cm ) 10 a = 100 Å z i = +a/2 zi = z i = -a/2 11 z i = +a/2 zi = z i = -a/2 z i = -a z i = -a 0.1 0.1 -2 N i = 10 cm 0.1 10 11 -2 impurity density N i (10 cm ) 40 80 120 160 200 well width ( Å) Fig The electron density nMIT as a function of the impurity density (left) and the well width (right) for charged-impurity scattering decreases with increase in the well width and the distance of the impurities from the 2DEG The dependence of the critical density nMIT on the well width for zi = Àa, is much stronger compared to the case with zi = a/2 3.3 The magnetic field and temperature effects In order to describe the temperature and magnetic field effects we display in Fig the mobility lo as a function of electron density for several temperatures in two cases B = and B = 2Bs It is seen from the figure that the temperature effect is considerable for T $ 0.5TF Besides, we have found that the mobility decreases with increasing temperature and differs remarkably from its zero-temperature value for T > 0.15TF (%11 K for n =  1011 cmÀ2) The figure also indicates that the mobility of nonpolarized 2DEG limited by remote impurities is higher compared to that of the fully polarized case This effect is due to spin-splitting in the parallel magnetic field leading to reduced screening in a spin-polarized electron gas We note that for B = 2Bs the 2DEG is almost fully polarized at low temperature N.Q Khanh, N.M Quan / Superlattices and Microstructures 64 (2013) 245–250 L=200 Å z i=-125 Å L=200 Å z i=-125 Å 11 -2 N i=9.2x10 cm 106 11 B=2BS mobility μ (cm /Vs) 2 -2 N i=9.2x10 cm 106 B=0 mobility μ0 (cm /Vs) 249 T=0K T=0.5TF T=0K T=0.5TF 105 T=TF T=TF 11 105 10 -2 11 electron density n (10 cm ) 10 -2 electron density n (10 cm ) Fig The mobility lo versus electron density at B = (left) and B = 2Bs (right) versus electron density for charged-impurity scattering G GH G=0 G=0 10 ρ(Bs )/ρ(0) Δ=6 Å , Λ=30 Å a=100 Å mobility μ0 (cm /Vs) 20 G GH 0.1 10 11 -2 electron density n (10 cm ) Δ=6 Å , Λ=30 Å a=100 Å 10 0.1 10 11 -2 electron density n (10 cm ) Fig The mobility lo (left) and the resistance ratio qBs ị=qB ẳ 0ị (right) versus electron density for IRS with = Å and = 30 Å in different approximations for G(q) 3.4 The interface-roughness scattering Up to now, we have considered only RIS To evaluate the importance of other scattering mechanisms we have calculated the mobility lo and the resistance ratio q(Bs)/q(B = 0) for IRS The results for the case of a = 100 Å, D = Å and K = 30 Å [1] with different approximations for G(q) are plotted in Fig We observe that the LFC is very important at low densities and the mobility limited by IRS is much higher than that of RIS and can be neglected The figure again shows that the mobility of a nonpolarized 2DEG limited by IRS is higher compared to that of the fully polarized case Conclusions In this paper, we have investigated the effects of the LFC, impurity position, layer-thickness and spin-polarization on the mobility and resistivity of the Q2DEG in a SiGe/Si/SiGe QW at arbitrary temperature At zero temperature, our results reduced to those given in Ref [8] We have shown that the 250 N.Q Khanh, N.M Quan / Superlattices and Microstructures 64 (2013) 245–250 LFC is very important at low densities, and the critical electron density nMIT decreases with increase in the well width and the distance of the impurity layer from the Si/SiGe interface We have found that the temperature effect is remarkable for T > 0.15TF and the mobility of the fully polarized 2DEG is lower than that of the nonpolarized 2DEG We have also shown that the mobility limited by IRS is much higher than that limited by RIS Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 103.02-2011.25 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] A Gold, Phys Rev B 35 (1987) 723 F Schäffler et al, Semicond Sci Technol (1992) 260 J Tutor et al, Phys Rev B 47 (1993) 3690 Z Wilamowski et al, Phys Rev Lett 87 (2001) 026401 V.T Dolgopolov et al, Superlattices Microstruct 33 (2003) 271 T.M Lu et al, Appl Phys Lett 90 (2007) 182114 A Gold, Semicond Sci Technol 26 (2011) 045017 A Gold, J Appl Phys 108 (2010) 063710 T Ando, A.B Fowler, F Stern, Rev Mod Phys 54 (1982) 437 A Gold, V.T Dolgopolov, 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We also discuss the importance of interface-roughness... Khanh, N.M Quan / Superlattices and Microstructures 64 (2013) 245–250 spin-polarized Q2DEG in a SiGe/ Si /SiGe QW at finite temperatures Therefore, we calculate, in this paper, the finite temperature