DSpace at VNU: Transport properties of the two-dimensional electron gas in AlP quantum wells at finite temperature including magnetic field and exchange-correlation effects

Physica E 58 (2014) 84–87 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe Transport properties of the two-dimensional electron gas in AlP quantum wells at finite temperature including magnetic field and exchange–correlation effects Nguyen Quoc Khanh n, Vo Van Tai Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Street, 5th District, Ho Chi Minh City, Vietnam H I G H L I G H T S      The effects of LFC and temperature on μ are remarkable for n o1012 cm À or T $ 0.3TF The effects of LFC and temperature on τt =τs are nearly canceled in the ratio for IRS For CIS, the temperature effects on τt =τs are notable at T $ 0.3TF only for GH At low n, the dependence of the resistance ratio on LFC decreases when T increases With decreasing L, nMIT increases and becomes nearly independent of LFC and zi art ic l e i nf o a b s t r a c t Article history: Received 22 September 2013 Received in revised form 19 November 2013 Accepted 25 November 2013 Available online December 2013 We investigate the transport scattering time, the single-particle relaxation time and the magnetoresistance of a quasi-two-dimensional electron gas in a GaP/AlP/GaP quantum well at zero and finite temperatures We consider the interface-roughness and impurity scattering, and study the dependence of the mobility, scattering time and magnetoresistance on the carrier density, temperature and local-field correction In the case of zero temperature and Hubbard local-field correction our results reduce to those of Gold and Marty (Physica E 40 (2008) 2028; Phys Rev B 76 (2007) 165309) We also discuss the possibility of a metal–insulator transition which might happen at low density & 2013 Elsevier B.V All rights reserved Keywords: AlP quantum well Magnetoresistance Scattering time Temperature effect Introduction GaP/AlP/GaP quantum well (QW) structures, where the electron gas is located in the AlP, have been studied recently at low temperatures via cyclotron resonance, quantum Hall effect, Shubnikov de Haas oscillations [1] and intersubband spectroscopy [2] In this structure, due to biaxial strain in the AlP and confinement effects in the quantum well of width L, the electron gas has valley degeneracy gv ¼1 for well width LoLc ¼45 Å, and valley degeneracy gv ¼2 for well width L4Lc [3] Recently, Gold and Marty have calculated the transport scattering time, single-particle relaxation time and the magnetoresistance for GaP/AlP/GaP QW with LoLc [4] In such thin QW, interface-roughness scattering (IRS) is the dominant scattering mechanism [5] The scattering mechanism, which is responsible for limiting the mobility, can be determined by n Corresponding author Fax: +848 38350096 E-mail address: nqkhanh@phys.hcmuns.edu.vn (N.Q Khanh) 1386-9477/$ - see front matter & 2013 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.physe.2013.11.021 comparing experimental results with those of theoretical calculations [6–8] Recent measurements and calculations of transport properties of a quasi-two-dimensional electron gas (Q2DEG) in a magnetic field give additional tool for determining the main scattering mechanism and scattering parameters [9–13] To the author's knowledge, there is no calculation of transport properties of the spin-polarized Q2DEG in a GaP/AlP/GaP QW at finite temperatures Therefore, in this paper, we calculate the mobility, the scattering time and the magnetoresistance of the 2DEG realized in AlP for IRS and charged impurity scattering (CIS) at zero and finite temperature, taking into account exchange– correlation effects We also discuss the possibility of a metal– insulator transition (MIT) which might happen at low density and calculate the critical electron density nMIT for the MIT Theory We consider a Q2DEG with parabolic dispersion determined by the effective mass mn We assume that the electron gas is in the xy N.Q Khanh, V.V Tai / Physica E 58 (2014) 84–87 plane with infinite confinement for z o0 and z 4L For 0r zrL, the electron gas in theplowest ffiffiffiffiffiffiffiffi subband is described by the wave function ψ(0 r zrL)¼ 2=L sin(πz/L) [14] When the in-plane magnetic field B is applied to the system, the carrier densities n for spin up/down are not equal [15–16] At T ¼0 we have n7 ẳ n BBs ị; n ỵ ẳ n; B o Bs n À ¼ 0; B ZBs 1ị Here, n ẳ n ỵ ỵn is the total density and Bs is the so-called saturation field given by g μB Bs ¼ 2EF where g is the electron spin g-factor, μB is the Bohr magneton and EF is the Fermi energy [15] For T4 0, n is determined using the Fermi distribution function and given by [16] p 2x=t 2x=t ỵ 2xị=t n n þ ¼ t ln À e þ ðe 1ị ỵ 4e 2ị n ẳ nnỵ where x ¼ B=Bs and t ¼ T=T F with TF is the Fermi temperature The energy averaged transport relaxation time for the (7 ) components is given in the Boltzmann theory by [1416] R   dịẵ f ị= ẳ R 3ị d ẵ f ị= where Z 1 ẳ kị A qị ẳ ỵ 2k jUqịj2 ẵ A qị2 q2 dq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 4k Àq2 ð4Þ 2πe2 F C qịẵ1 Gqị q; T ị; AL q 5ị q; Tị ẳ ỵ q; Tị ỵ q; Tị; q; T ị ẳ β Z ðq; E F Þ  Π ð6Þ À dμ= Á Π 07 q; μ= À Á; cosh β=2 μ À μ= n gv m 1À ðqÞ ¼ 2π ℏ ð7Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2kF 1À Θðq À2kF Þ5; q ð8Þ F C qị ẳ ỵ a2 q2  3aq þ8π 32π À e À aq À 2 aq a q ỵ a2 q2  ð9Þ where ni is the 2D impurity density, zi is the distance between remote impurities and 2DEG, and F CIS ðq; zi Þis the form factor for the electron–impurity interaction given in Ref [14] The mobility of the nonpolarized and fully polarized 2DEG is given by μ ¼ e〈τ〉=mn The resistivity is defined by ρ ¼ 1=s where s ẳ s ỵ ỵ s is the total conductivity and s is the conductivity of the (7 ) spin subband given by s ¼ n e2 〈τ 〉=mn [15] It was shown that multiple-scattering effects can account for this MIT at low electron density where interaction effects become inefficient to screen the random potential created by the disorder [21–22] The MIT is described by parameter A, which depends on the random potential, the screening function including the LFC and the compressibility of the electron gas, and is given by [3,21–22] D  E Z U qị2 o qị2 qdq Aẳ 12ị n2 ẵ A qị2 For n4 nMIT, where Ao 1, the Q2DEG is in a metallic phase and for n onMIT, where A41, the Q2DEG is in an insulating phase and the mobility vanishes Numerical results In this section, we present our numerical calculations for the transport scattering time, the single-particle relaxation time and the magnetoresistance of a Q2DEG in a GaP/AlP/GaP QW for the case L oLc using the following parameters [3–4]: A L ¼ 9.8, mn ¼0.3mo and mz ¼0.9mo, where mo is the free electron mass To treat the exchange–correlation 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 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi has the form GH ðqÞ ẳ q=ẵg v g s q2 ỵ kF where gs is the spin degeneracy We also use analytical expressions of the LFC (GGA) according to the numerical results obtained in Ref [20] where both exchange and correlation effects are taken into account 3.1 Results for interface-roughness scattering In Fig 1, we show the mobility μ versus electron density n for two different QW widths and temperatures Two approximations for the LFC, GH and GGA are used It is seen that exchange and correlation effects are very important for no 1012 cm À and the mobility depends strongly on the approximation for LFC The LFC reduces the screening, increases the effective scattering potential, 10 where Δ represents the average height of the roughness perpendicular to the 2DEG, Λ 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 For CIS the random potential has the form  2  2 2π e2 〈U CIS qị ẳ ni F CIS q; zi ị2 AL q Δ=3Å; Λ=50Å; B/Bs=0 T/TF=0 T/TF=0.3 mobility μ(104cm2/Vs) with f ị ẳ 1=f1 ỵ exp ẵ Tịịg, ẳ kB Tị , ẳ ln ẵ 2 ỵexpEF ị= , EF ẳ k F =2mn ị and ẳ k =2 mn ị Here, n m is the effective mass in xy-plane, gv is the valley degeneracy, G (q) is the local-field correction (LFC) describing the exchange– correlation effects [17–20], A L is the background static dielectric  2 constant and 〈UðqÞ 〉 is the random potential which depends on the scattering mechanism [14] For IRS the random potential is given by [14]    n 2  D  E m π 2 U IRS qị2 ẳ F ị2 e À q Λ =4 ð10Þ mz kF a a 85 L=40Å GH GGA GH 0.1 L=30Å GGA 0.1 electron density n(1012cm-2) ð11Þ Fig Mobility μ versus electron density n for IRS for two different QW widths and temperatures in two G(q) models 86 N.Q Khanh, V.V Tai / Physica E 58 (2014) 84–87 10 GH L = 40Å < LC Δ = 3Å; Λ = 50Å; B/Bs=0 G=0 G=0(T/TF=0) GH(T/TF=0) GGA(T/TF=0) GGA(T/TF=0.3) ρ(Bs)/ρ(0) GGA τt/τs T/TF=0 T/TF=0.3 1 (kFΛ)2/3 L = 40Å < LC Δ = 3Å; Λ = 50Å 0.6 0.1 2/3 0.1 10 and hence reduces the mobility The temperature effect is remarkable for T $ 0.3TF ( $2.75 K for n ¼1011 cm À 2) Note that the mobility in Fig is shown on a log-plot On a linear scale the changes due to a finite LFC and temperature are much larger The ratio of the transport scattering time and the singleparticle relaxation time τt =τs versus electron density is shown in Fig for L¼ 40 Å and different approximations for the LFC We observe that the dependence of τt =τs on the LFC and temperature is very weak because the effects of both LFC and temperature are nearly canceled in the ratio The straight lines are the analytical results for IRS given in Ref [23] Note that at high electron density τt =τs % ðkF ΛÞ2 =3 and the ratio allows us to determine kF Λ, and for a given density, the parameter Λ can be determined Results for the resistance ratio ρðBs Þ=ρðB ¼ 0Þ versus electron density for L ¼40 Å and two temperatures in different approximations for the LFC are shown in Fig We see that the resistance ratio depends strongly on approximations for the LFC The resistivity of a fully polarized 2DEG limited by IRS is higher compared to that of the nonpolarized case This effect is due to spin-splitting in the parallel magnetic field leading to reduced screening in a spin-polarized electron gas At low density, the Hubbard LFC GH increases and the LFC GGA decreases the resistance ratio This behavior may stem from the dependence of the LFC on the spinpolarization At low density, the dependence of the resistance ratio on the LFC decreases when temperature increases and the temperature effect is remarkable for T $ 0.3TF Fig Resistance ratio Bs ị=B ẳ 0ị versus electron density for IRS for L ¼ 40 Å and two temperatures in different approximations for the LFC 10 L=30Å; ni = n; B/Bs=0 zi = -L/2 T/TF = T/TF = 0.3 mobility μ(103cm2/Vs) Fig Ratio of the transport scattering time and the single-particle relaxation time τt =τs versus electron density for IRS electron density n(1012cm-2) electron density n(1012 cm-2) GGA GH zi = GH GGA zi = L/2 GH 0.1 GGA 0.1 10 electron density n(10 cm ) 12 -2 Fig Mobility μ versus electron density n for CIS for two temperatures and three values of the position of impurities zi in two approximations for the LFC 10 L=30Å; ni = n; B/Bs=0 T/TF = T/TF = 0.3 The mobility versus electron density for CIS, characterized by the impurity density ni ¼n and the distance zi of the impurity layer from the QW edge at z¼0, is shown in Fig We observe that, the LFC is very important at low density On the other hand, the dependence of the mobility on zi is more pronounced at higher density Again, the temperature effects on the mobility are remarkable at T$ 0.3TF The ratio τt =τs versus electron density for CIS for two temperatures and three values of zi in two G(q) models is displayed in Fig At low density, the finite LFC decreases τt =τs remarkably for zi ¼L/2 or G(q)¼GGA [3] At high density, the ratio τt =τs for zi ¼ L/2 differs strongly from that for zi ¼ ÀL/2 The temperature effects on τt =τs are notable for T$ 0.3TF at low density only for the Hubbard LFC The resistance ratio Bs ị=B ẳ 0ị versus electron density for impurities with density ni ¼n and zi ¼L/2 is plotted in Fig We see τt/τs 3.2 Results for charged impurity scattering GGA zi = -L/2 GH GH zi = L/2 GGA 0.1 10 electron density n (1012cm-2) Fig Ratio τt =τs versus electron density for CIS for two temperatures and zi in two approximations for the LFC N.Q Khanh, V.V Tai / Physica E 58 (2014) 84–87 nMIT is determined mainly by IRS (CIS) for small (large) QW width Therefore, when L approaches Lc, the nMIT for zi ¼ L/2 is larger than that for zi ¼ À L/2 because the CIS is stronger for impurities inside the QW in comparison with the case of impurities located outside the QW L = 30 Å zi = L/2 87 ni= n G = (T/TF = 0) ρ(Bs)/ρ(0) T/TF = T/TF = 0.3 Conclusions GH GGA 0.1 10 electron density n (1012 cm-2) Fig Resistance ratio ρðBs Þ=ρðB ¼ 0Þ versus electron density for CIS for L ¼30 Å and two temperatures in different approximations for the LFC 10 critical electron density nMIT(1012cm-2) B/Bs= 0; T/TF= Δ = 3Å;Λ = 50Å n = ni = 1x1011cm-2 conducting phase GGA insulating phase IRS IRS+RIS (zi=L/2) GH Acknowledgement IRS+RIS (zi=-L/2) This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 103.02-2011.25 0.1 10 20 We have calculated the mobility, the scattering time and magnetoresistance ratio, and the critical electron density of the 2DEG realized in AlP for interface-roughness and impurity scattering We find the remarkable effects of the LFC and temperature for n o1012 cm À and T $ 0.3TF For IRS, the effects of both the LFC and temperature on τt =τs are nearly canceled in the ratio At low density, the dependence of the resistance ratio ρðBs Þ=ρðB ¼ 0Þ on the LFC decreases when temperature increases For CIS, the temperature effects on both τt =τs and the resistance ratio are remarkable at T $ 0.3TF only for Hubbard approximation The LFC decreases the screening properties and hence increases the effective random potential and critical electron density nMIT With the decrease of the QW width, the interface-roughness scattering becomes stronger, and the critical electron density increases and becomes nearly independent of the approximations used for the LFC Near Lc, the critical density nMIT is determined mainly by CIS and the nMIT for zi ¼L/2 is larger than that for zi ¼ À L/2 We hope that our results can be used to obtain information about the scattering mechanism and many-body effects in GaP/AlP/GaP QW structures 30 40 quantum well width L(Å) Fig Critical electron density versus QW width for two values of zi and two approximations for the LFC that the effects of LFC and temperature are notable at low density only for Hubbard approximations References [1] [2] [3] [4] [5] [6] [7] 3.3 Results for the metal–insulator transition [8] The critical electron density versus QW width for two approximations for the LFC is displayed in Fig We observe that the critical electron density decreases with increasing QW width, and depends considerably on the approximation for LFC The LFC decreases the screening properties and hence increases the effective random potential and critical electron density With the decrease of the QW width, the IRS becomes stronger and the critical electron density increases For Lo15 Å, the critical electron density is high (nMIT 42 Â 1012 cm À 2) and becomes nearly independent of the approximation used for the LFC (because at n 42 Â 1012 cm À the exchange–correlation effect is small, see Fig 1) It is seen that, for scattering parameters used in Fig 7, the [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] M.P Semtsiv, et al., Phys Rev B 74 (2006) 041303 (R) M.P Semtsiv, et al., Appl Phys Lett 89 (2007) 184102 A Gold, R Marty, Phys Rev B 76 (2007) 165309 A Gold, R Marty, Physica E 40 (2008) 2028 A Gold, Solid State Commun 60 (1986) 53 T Ando, A.B Fowler, F Stern, Rev Mod Phys 54 (1982) 437 John H Davies, The Physics of Low-Dimensional Semiconductors, An Introduction, Cambridge University Press, Cambridge, 1998 Chihiro Hamaguchi, Basic Semiconductor Physics, Springer, Berlin, Heidelberg, 2001 A.A Shashkin, et al., Phys Rev B 73 (2006) 115420 V.T Dolgopolov, A Gold, JETP Lett 71 (2000) 27 T.M Lu, et al., Phys Rev B 78 (2008) 233309 E.H Hwang, S Das Sarma, Phys Rev B 87 (2013) 075306 Xiaoqing Zhou, et al., Phys Rev B 85 (2012) 041310 (R) A Gold, Phys Rev B 35 (1987) 723 S Das Sarma, E.H Hwang, Phys Rev B 72 (2005) 205303 Nguyen Quoc Khanh, Physica E 43 (2011) 1712 M Jonson, J Phys C (1976) 3055 A Gold, L Calmels, Phys Rev B 48 (1993) 11622 A Gold, Phys Rev B 50 (1994) 4297 A Gold, Z Phys B 103 (1997) 491 A Gold, Phys Rev Lett 54 (1985) 1079 A Gold, W Götze, Phys Rev B 33 (1986) 2495 A Gold, Phys Rev B 38 (1988) 10798  ... approximations for the LFC We observe that the dependence of τt =τs on the LFC and temperature is very weak because the effects of both LFC and temperature are nearly canceled in the ratio The straight... may stem from the dependence of the LFC on the spinpolarization At low density, the dependence of the resistance ratio on the LFC decreases when temperature increases and the temperature effect... calculated the mobility, the scattering time and magnetoresistance ratio, and the critical electron density of the 2DEG realized in AlP for interface-roughness and impurity scattering We find the