Physica E 67 (2015) 84–88 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe Transport properties of the two-dimensional electron gas in wide AlP quantum wells including temperature and correlation effects Vo Van Tai, Nguyen Quoc Khanh n Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Street, 5th District, Ho Chi Minh, Vietnam H I G H L I G H T S The difference between the results of GH and GGA model is remarkable The temperature effects are notable at very low temperature T $ 0.3TF The correlation effects increase the critical density for a MIT considerably art ic l e i nf o a b s t r a c t Article history: Received 15 August 2014 Received in revised form November 2014 Accepted 24 November 2014 Available online 26 November 2014 We investigate the mobility, magnetoresistance and scattering time of a quasi-two-dimensional electron gas in a GaP/AlP/GaP quantum well of width L4Lc ¼45.7 Å at zero and finite temperatures We consider the interface-roughness and impurity scattering, and study the dependence of the mobility, the resistance and scattering time ratio on the carrier density and quantum well width for different values of the impurity position and temperature using different approximations for the local-field correction In the case of zero temperature and Hubbard local-field correction our results reduce to those of Gold and Marty (Phys Rev B 76 (2007) 165309) [3] We also study the correlation and multiple scattering effects on the total mobility and the critical density for a metal–insulator transition & 2014 Elsevier B.V All rights reserved Keywords: AlP quantum wells 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 L oLc ¼45.7 Å, and valley degeneracy gv ¼2 for well width L4 Lc [3,4] Recently, we have calculated the mobility, scattering time and magnetoresistance for a GaP/AlP/GaP QW with L oLc including the temperature and exchange-correlations effects [5] In this paper, we present our calculation for the case of wide GaP/AlP/GaP QW with L 4Lc We consider interface-roughness and randomly distributed charged impurities as source of disorder We investigate the dependence of the mobility, the resistance and scattering time ratio on the carrier density and QW width for different values of the impurity position n Corresponding author Fax: ỵ84 38350096 E-mail address: nqkhanh@phys.hcmuns.edu.vn (N.Q Khanh) http://dx.doi.org/10.1016/j.physe.2014.11.015 1386-9477/& 2014 Elsevier B.V All rights reserved and temperature We also study the correlation and multiple scattering effects (MSE) [6] on the total mobility and the critical density for a metal–insulator transition (MIT) [6,7] Theory We assume that the electron gas (EG), with parabolic dispersion determined by the effective mass mn, is in the xy plane with infinite confinement for z o0 and z 4L For rz rL, the EG in the lowest subband is described by the wave function ψ (0 rzr L) ¼ 2/L sin(πz/L) [5,8] When the in-plane magnetic field B is applied to the system, the carrier densities n± for spin up/down are not equal [5,9,10] At T¼ we have n± = n⎛ B⎞ ⎜1 ± ⎟, B < Bs Bs ⎠ 2⎝ n+ = n, n− = 0, B ≥ Bs (1) Here n = n+ + n− is the total density and Bs is the so-called saturation field given by gμB Bs = 2EF where g is the electron spin g- V.V Tai, N.Q Khanh / Physica E 67 (2015) 84–88 factor, μB is the Bohr magneton and EF is the Fermi energy For T 40, n± is determined using the Fermi distribution function and given by [5,9] − e2x / t + n t ln n− = n − n+ n+ = (e2x / t − 1)2 + 4e (2 + 2x)/ t (2) where x = B /Bs and t = T /TF with TF is the Fermi temperature The energy averaged transport relaxation time for the (7 ) components are given in the Boltzmann theory by [5,8,9] τ± ∫ dε τ (ε) ε [ − (∂f ± (ε)/∂ε)] = ∫ dε ε [ − (∂f ± (ε)/∂ε)] (3) where 1 = 2π ℏε τ (k) ∈ (q) = + ∫0 2k U (q) q2dq [∈ (q)]2 4k − q , (4) 2π e FC (q)[1 − G (q)] Π (q, T ), ∈L q (5) Π (q, T ) = Π+ (q, T ) + Π − (q, T ) Π± (q, T ) = β ∫0 ∞ dμ ′ Π±0 (q, μ′) c osh2 (β/2) Π±0 (q, EF ± ) ≡ Π±0 (q) = FC (q) = 4π (6) (μ± − μ′) ⎡ g v m⁎ ⎢ 1− 2π ℏ2 ⎢⎣ , (7) ⎤ ⎛ 2k F ± ⎞2 ⎥ ± Θ 1−⎜ ( q k ) − ⎟ F ⎥, ⎝ q ⎠ ⎦ ⎛ 8π 32π − e−aq ⎞ ⎜3aq + ⎟, − 2 aq a2q2 4π + a2q2 ⎠ +aq ⎝ (8) (9) with β = (kB T )−1, ε = ℏ2k 2/(2m⁎) and ∈L denotes the backg round static dielectric constant Here kF ± = (4πn± /gv )1/2, EF ± = ℏ2k 2F ±/(2m⁎), μ± = ln [ − + exp (βEF ± )]/β , and Π± (q , T ) is the 2D Fermi wave vector, Fermi energy, chemical potential, Fermi distribution function and polarizability for the up/down spin state, respectively G(q) is the local-field correction (LFC) describing the exchange-correlation effects [8,11] and U (q) is the random potential which depends on the scattering mechanism [8] For interface-roughness scattering (IRS) the random potential is given by [8] UIRS (q) ⎛ 4π ⎞ ⎛ m⁎ ⎞2 ⎛ π ⎞4 2 = 2⎜ ⎟⎜ ⎟ ⎜ ⎟ (ε F ΔΛ)2e−q Λ /4 ⎝ a2 ⎠ ⎝ m z ⎠ ⎝ k F a ⎠ 85 where σ = σ+ + σ − is the total conductivity and σ± is the conductivity of the (7 ) spin subband given by σ± = n± e2 τ± /m⁎ [9] It was shown that multiple-scattering effects can account for the MIT at low electron density where interaction effects become inefficient to screen the random potential created by the disorder [6,7] 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 [3,6,7] A= 4πn2 ∫0 ∞ U (q) [Π o (q)]2 qdq [∈ (q)]2 (12) For n4 nMIT, where A o1, the 2DEG is in a metallic phase and for n onMIT, where A 41, the 2DEG is in an insulating phase and the mobility vanishes Numerical results For the case L 4Lc we use the following parameters [1–3,12]: ∈L ¼9.8, gv ¼ 2, mn ¼ 0.52mo and mz ¼0.3mo, where mo is the free electron mass The LFC is very important at low electron densities In the Hubbard approximation, only exchange effects are taken into account and the LFC has the form GH (q) = q /[gv gs q2 + kF2 ] where gs is the spin degeneracy [3–5] We also use analytical expressions of the LFC (GGA) according to the numerical results obtained in Ref [11] where both exchange and correlation effects are taken into account In Fig 1, we show the mobility μ versus electron density n for a QW of width L ¼60 Å for IRS with Δ ¼3 Å and Λ ¼ 50 Å for different temperatures in two G(q) models It is seen that 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, and hence reduces the mobility The temperature effect is remarkable for T $ 0.3TF ( $ 1.6 K for n ¼1011 cm À ) We have chosen Δ ¼3 Å and Λ ¼50 Å because, using these values, Gold and Marty [4] have calculated the mobility for thin AlP QW of width L¼40 Å and obtained good agreement with experimental results Furthermore, although there is not very much known about the parameters Δ and Λ, Δ values around Å and Λ values between 60 and 10 Å seem to be most realistic [4,13] Our results can be helpful for experimenters in determining the interface-roughness parameters Δ and Λ for GaP/AlP/GaP QW structures The minimum in the (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 remote charged impurity scattering (CIS) the random potential has the form UCIS (q) ⎛ 2πe2 ⎞2 ⎟ FCIS (q, zi )2 = Ni ⎜ ⎝ ∈L q ⎠ (11) where Ni is 2D impurity density, zi is the distance between remote impurities and 2DEG, and FCIS (q , zi ) is the form factor for the electron–impurity interaction given in Ref [8] The mobility of the nonpolarized and fully polarized 2DEG is given by μ = e < τ > /m⁎ The resistivity is defined by ρ = 1/σ Fig Mobility μ versus electron density n for a QW of width L ¼60 Å for IRS with Δ ¼3 Å and Λ ¼50 Å for different temperatures in two G(q) models 86 V.V Tai, N.Q Khanh / Physica E 67 (2015) 84–88 Fig Resistance ratio ρ (Bs )/ρ (B = 0) versus electron density for a QW of width L ¼ 60 Å for IRS with Δ ¼ Å and Λ ¼ 50 Å at different temperatures in different approximations for the LFC mobility versus carrier density appeared in Fig has been explained by Gold [8] using an approximation derived in [14] Due to the factor e−q 2Λ2 /4 in UIRS (q) , the low temperature mobility μIRS ∼ 1/n at low density (kFΛ«1) and μIRS ∼ n3/2 at high density (kFΛ»1) The crossover of the μIRS ∼ 1/n (decreasing function of n ) to the μIRS ∼ n3/2 (increasing function of n) behavior is at n ¼gv/(2πΛ2), where the minimum of the mobility is reached In Fig we show the resistance ratio ρ (Bs )/ρ (B = 0) versus electron density for a QW of width L ¼60 Å for IRS with Δ ¼3 Å and Λ ¼50 Å at different temperatures in different approximations for the LFC We see that the resitivity of a 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 The LFC increases considerably the resistance ratio especially in the case of GGA In the Hubbard approximation, the effects of LFC are nearly canceled by the temperature effect at T $ 0.3TF for n 45 Â 1011 cm À Recall that for L¼40 Å o Lc, at low density, the Hubbard LFC GH increases and the LFC GGA decreases the resistance a ratio [5] We note that in the case of IRS, the height parameter Δ cancels out for the magnetoresistance ratio ρ (Bs )/ρ (B = 0) Without LFC, the limiting behavior for small density is ρ (Bs )/ρ (B = 0) ¼8 [15] When many-body effects described by a LFC are included, using the approximation derived in [14] for the scattering time, it can be shown that the magnetoresistance ratio might increase remarkably at low density [16] These features of the magnetoresistance ratio found in Refs [15,16] are seen also in Fig The ratio of the transport scattering time and the single-particle scattering time τt/τs can be used to determine microscopic parameters of disorder such as Λ and zi [3] In Fig 3, we show the ratio τt/τs versus electron density for a QW of width L ¼60 Å for IRS and CIS in different G(q) models In the case of IRS, only small effects due to the LFC GH are seen In the case of CIS, Ni is canceled out in the ratio τt/τs and the results for zi ¼L/2 differs strongly from that for zi ¼ À L/2 at high density For low density, the difference between the results of GH and GGA model is notable for both IRS and CIS Note that the ratio τt/τs in Fig is shown on a log-plot On a linear scale the changes due to a finite LFC are much larger The mobility versus electron density for a QW of width L¼100 Å for CIS with the impurity concentration Ni ¼n at two temperatures for different values of the distance zi of the impurity layer from the QW edge at z ¼0 is shown in Fig We use two G (q) models and observe that the correlation effects are very important at low density The dependence of the mobility on zi is more pronounced at high density The temperature effects on the mobility are remarkable at T $ 0.3TF We note that, as in the case of IRS, the minimum in the mobility μCIS versus carrier density for T¼ appeared in Fig can be explained using Gold's arguments given in [8] In Fig we show the mobility μ versus QW width L for CIS with Ni ¼n ¼1012 cm À at two temperatures for three values of the impurity position zi in two G(q) models For zi ¼L/2, we see a very weak QW width dependence of the mobility For zi r 0, the mobility increases with increasing well width because the distance between the electron gas and the impurity layer increases with increasing well width The temperature effects and the differences between the results of two G(q) models are notable for the wide range of QW width b Fig Ratio τt/τs versus electron density for a QW of width L ¼60 Å in different G(q) models (a) for IRS with Δ ¼ Å and Λ¼ 50 Å and (b) for CIS for two values of the impurity position zi V.V Tai, N.Q Khanh / Physica E 67 (2015) 84–88 Fig Mobility μ versus electron density for a QW of width L ¼ 100 Å for CIS with Ni ¼n at two temperatures for different values of impurity position zi in two G (q) models Fig Mobility μ versus QW width L for CIS at two temperatures for three values of the impurity position zi in two G(q) models a 87 Fig Total mobility versus electron density n for a QW of width L ¼ 60 Å for IRS with Δ ¼ Å and Λ ¼ 50 Å and CIS with zi ¼ À L/2 and Ni ¼ n in two G(q) models The resistance ratio ρ (Bs )/ρ (B = 0) versus QW width L for CIS with Ni ¼n at two temperatures for different values of electron density n in two G(q) models is plotted in Fig We see that the difference between the results of two G(q) models is very large for impurities both inside (zi ¼L/2 ) and outside (zi ¼ À L/2 ) the QW The temperature effects in both GH and GGA model are notable at very low temperature T $ 0.1TF ( $ 0.53 K for n ¼1011 cm À ) for the entire range of QW width considered here The total mobility versus electron density n for a QW of width L¼ 60 Å for IRS with Δ ¼3 Å and Λ ¼ 50 Å and CIS with zi ¼ ÀL/2 and Ni ¼n in two G(q) models is displayed in Fig We observe that the correlation and multiple-scattering effects are remarkable at low densities The LFC decreases the screening properties and hence increases the effective random potential and critical electron density The MSE are very strong for the case of GGA when correlation effects are taken into account At high densities (n 45 Â 1012 cm À 2) the MSE are negligible and the total mobility becomes nearly independent of the approximation used for LFC Note that the low-temperature MIT is caused by quantum fluctuations and its behavior in the critical region is very complex b Fig Resistance ratio ρ (Bs )/ρ (B = 0) versus QW width L for CIS with Ni ¼ n at two temperatures for different values of electron density n in two G(q) models (a) for zi ¼ À L/2 and (b) for zi ¼L/2 88 V.V Tai, N.Q Khanh / Physica E 67 (2015) 84–88 Therefore, an adequate description of the MIT, especially in the vicinity of the transition point, requires more sophisticated theory [17–19] Science and Technology Development (NAFOSTED) under Grant no 103.01-2014.01 References Conclusion In summary, we have calculated the mobility, scattering time and magnetoresistance of the 2DEG in a GaP/AlP/GaP QW of width L4 Lc ¼45.7 Å for interface-roughness and impurity scattering We find that the difference between the results of GH and GGA model is remarkable for n o1012 cm À The temperature effects are notable at very low temperature T $ 0.3TF ( $ 1.6 K for n ¼1011 cm À ) The MSE lead to a MIT at low density and the correlation effects increase the critical density considerably Our results can be used to obtain information about the scattering mechanism and manybody effects in GaP/AlP/GaP QW structures Acknowledgment This research is funded by Vietnam National Foundation for [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] M.P Semtsiv, et al., Phys Rev B 74 (R) (2006) 041303 M.P Semtsiv, et al., App Phys Lett 89 (2007) 184102 A Gold, R Marty, Phys Rev B 76 (2007) 165309 A Gold, R Marty, Physica E 40 (2008) 2028 Nguyen Quoc Khanh, Vo Van Tai, Physica E 58 (2014) 84 A Gold, W Götze, Phys Rev B 33 (1986) 2495 A Gold, W Götze, Solid State Commun 47 (1983) 627 A Gold, Phys Rev B 35 (1987) 723 S Das Sarma, E.H Hwang, Phys Rev B 72 (2005) 205303 E.H Hwang, S Das Sarma, 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are notable for the wide range of QW width