DSpace at VNU: Transport properties of a spin-polarized quasi-two-dimensional electron gas in an InP In1-xGaxAs InP quantum well including temperature effects

Physica E 43 (2011) 1712–1716 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe Transport properties of a spin-polarized quasi-two-dimensional electron gas in an InP/In1 À xGaxAs/InP quantum well including temperature effects 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 City, Vietnam a r t i c l e i n f o a b s t r a c t Article history: Received 29 March 2011 Received in revised form 18 May 2011 Accepted 25 May 2011 Available online June 2011 We investigate the mobility and resistivity of a quasi-two-dimensional electron gas in an InP/In1 À xGaxAs/ InP quantum well at arbitrary temperatures and spin polarizations caused by an applied in-plane magnetic field We consider the carrier density, impurity concentration and layer thickness parameters such that the ionized impurity and alloy disorder scattering are the main mechanisms We investigate the dependence of the mobility and resistivity on the carrier density, layer thickness and magnetic field & 2011 Elsevier B.V All rights reserved Introduction The transport properties of a quasi-two-dimensional electron gas (Q2DEG) in the lattice matched InP/In0.53Ga0.47As/InP quantum well (QW) have been studied by several authors [1–5] It is an attractive system for high-speed electronic device applications due to the negligible concentration of DX centers and discolations on the InP donor layers [1] The scattering mechanism, which is responsible for limiting the mobility, can be determined by comparing experimental results with those of theoretical calculations [6–9] Recent measurements and calculations of transport properties of a 2DEG in a magnetic field give additional tool for determining the main scattering mechanism [10–14] To the author’s knowledge, there is no calculation of transport properties of the spin-polarized 2DEG in an InP/In1À xGaxAs/InP quantum well at finite temperatures Therefore, we decide to investigate here in this paper the magnetic field and temperature effects on the mobility and resistivity of a 2DEG in an InP/In1À xGaxAs/InP quantum well Theory We consider a single InP/In1 À xGaxAs/InP QW of width a with infinite confinement We assume that the electrons are free to move in x–y plane with the effective mass mn and confined in the z-direction We neglect the subband structure and include only the lowest subband in our calculation The wave function for the z-direction is given via [6] r pz czị ẳ sin , oz o a ð1Þ a a and is zero for all other z n Fax: ỵ848 8350096 E-mail address: nqkhanh@phys.hcmuns.edu.vn 1386-9477/$ - see front matter & 2011 Elsevier B.V All rights reserved doi:10.1016/j.physe.2011.05.028 When the in-plane magnetic field is applied to the system, the carrier densities n for spin up/down are not equal At T¼0 we have [11]   < n ¼ n 17 B , B o Bs Bs ð2Þ : n ỵ ẳ n, n- ẳ 0, B Z Bs Here nẳn ỵ ỵn is the total density and Bs is the so-called saturation field given by gmBBs ¼2EF where g is the electron spin g-factor and mB is the Bohr magnetron For T40, n is determined using the Fermi distribution function and given by [11,15] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b=t ỵ 2bị=t < n ỵ ẳ n2 t ln1e2b=t ỵ e 2ị ỵ 4e 3ị : n ẳ nn ỵ where bẳ B/Bs and t ¼T/TF with TF is the Fermi temperature The energy averaged transport relaxation time for the ( 7) components are given in the Boltzmann theory by [7,11] R deteịeẵ@f eị=@e 4ị /t 7S ẳ R deeẵ@f eị=@e Here t(e) is the energy dependent relaxation time, and f (e) is the Fermi distribution function f eị ẳ 1ỵ ebẵe-m Tị 5ị where b ẳ(kBT) and m7 ẳ b ln ỵ expðbEF Þ ð6Þ is the chemical potential for the up/down spin state (with the Fermi energy EF ) The energy dependent relaxation time t(e) depends on the scattering mechanism and given by [7–9] Z 2k /9UðqÞ9 S q2 dq 1 p ẳ 7ị tkị 2p_e ½ A ðqފ2 4k2 Àq2 N.Q Khanh / Physica E 43 (2011) 17121716 where e ẳ _2 k2 =2mn ị, U(q) is the random potential for wave number q and [1618] A qị ẳ ỵ 2pe2 FC qịẵ1GqịPq,Tị AL q ð8Þ is the finite wave vector dielectric screening function Here G(q) is the local field correction (LFC), FC(q) is the Coulomb form factor arising from the subband wave functions c(z), AL is the background static dielectric constant and P(q,T) is the 2D irreducible finite-temperature polarizability function given by P(q,T) ẳ P ỵ (q,T)ỵ P (q,T) with P (q,T) are the polarizabilities of the polarized up/down spin states At finite temperature we have [11,19] Z P07 ðq, m0 ị b 9ị P q,Tị ẳ dm cosh b2 ðm Àm0 Þ where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 m 2kF 0 P ðq,EF Þ  P qị ẳ 1 Yq2kF ị5 q p_ n ð10Þ À1 and for our infinite quantum well model, we have [6]   8p2 32p4 1ÀeÀaq À 3aq ỵ FC qị ẳ aq a2 q2 4p2 ỵ a2 q2 4p2 ỵa2 q2 We will use the Hubbard approximation GH qị ẳ gs 12ị q p for the 2 q ỵ kF LFC [20] where gs is the spin degeneracy For the unpolarized electron gas, we apply gs ¼2 and for the fully polarized electron gas, we use gs ¼ In this paper we will consider four scattering mechanisms: surface-roughness (S), alloy disorder (A), remote (R) and homogenous background (B) doping The random potentials for these scattering mechanisms are given as follows [4,6]    n 2  4p m p 2 /9US qị9 S ẳ 2 eF DLị2 eq L =4 ð13Þ mz kF a a /9UR ðqÞ9 S ¼ ni FR ðq,zi Þ ¼     A3 ðdVÞ2 4a  In this section, we present our numerical calculations for the mobility and resistivity of a Q2DEG in an InP/In1 À xGaxAs/InP QW using the following parameters [4]: NB ¼1016 cm À 3, ˚ L ¼50 A, ˚ eL ¼13.3, x ¼0.47, dV ¼0.6 eV, ni ¼1011 cm À 2, D ¼1 A, A¼5.9 A˚ and mn ¼mz ¼0.041mo, where mo is the vacuum mass of the electron The mobility of the unpolarized and fully polarized 2DEG is given by m ¼e/mn o t The mobility m limited by different scattering mechanisms versus electron density n at T¼0, B¼0 for the well width a¼100 A˚ (thin lines) and a ¼150 A˚ (thick lines) is plotted in Fig It is seen from the figure that the contribution of surface-roughness scattering to the mobility can be neglected for a $ 150 A˚ and no1012 cm À We note that our results are similar to those given earlier by Gold [4] In Fig we show the mobility limited by the alloy disorder, remote and background impurity scattering versus electron density n at T¼0 for B ¼0 (thin lines) and B¼Bs (thick lines) and ˚ We observe that the alloy disorder scattering depends a¼150 A strongly on the magnetic field at low densities This dependence stems from the dependence of the screening function on the spinpolarization caused by the magnetic field At higher densities n $ 1012 cm À the alloy disorder scattering shows a weak dependence on the magnetic field and becomes the main scattering mechanism in mobility limitation For a comparison, we now discuss the scattering mechanisms in an AlxGa1 À xAs/GaAs/AlxGa1 À xAs QW In the case of large aluminum concentration, the band edge discontinuity increases leading to increasing confinement of the electron wave function in the GaAs layer and correspondingly decreasing degree of wave function penetration ð14Þ 2 2pe2 FR q,zi ị2 AL q 8p2 aq 4p2 ỵ a2 q2 qzi Àaq Þ, zi o > > e ð1Àe > > < À Á 2 qz qazi ị ỵ a2pq2 sin2 pazi ,  1À e i À e > > > Àqðzi ÀaÞ > ð1ÀeÀaq Þ, zi a : 2e Numerical results 107 ð15Þ 106 (cm2/Vs) /9UA qị9 S ẳ x1xị z-direction, L is the correlation length parameter of the roughness in the xy direction, A3 is the alloy unit cell, dV is the spatial average of the fluctuating alloy potential over the alloy unit cell, ni is the 2D impurity density, zi is the distance between remote impurities and 2DEG and NB is the density of homogenous background impurities 3.1 The mobility pffiffiffiffiffiffiffiffiffiffiffiffiffiffi with kF ¼ 4pn is the 2D Fermi wave vector for the spin up/down carriers The Coulomb form factor is given by Z ỵ1 Z ỵ1 2 dz9cðzÞ9 dz0 9cðz0 Þ9 eÀq9zÀz ð11Þ FC qị ẳ 1713 rzi 105 NB = 1016cm-3 ni = 1011cm-2, zi = -a/2 δV = 0.6 eV, x = 0.47 Λ = 50Å, Δ = 1Å ð16Þ  2 2pe2 /9UB ðqÞ9 S ¼ NB a FB ðqÞ AL q FB ðqÞ ¼  2 4p2 aq e ỵ 2 eaq 1ị aq 4p2 ỵ a2 q2 aq a q  ! 2aq 3a3 q3 81eaq ị ỵ ỵ 4p2 ỵ a2 q2 p2 8p4 104 17ị 103 1010 1011 1012 B R A S 1013 -2 n (cm ) ð18Þ In above expressions mz is the mass perpendicular to the interface, D is the average height of the roughness in the Fig Mobility m versus electron density n at T¼ and B¼ The lines refer to the mobility limited by: surface-roughness (S), alloy disorder (A), remote (R) and background (B) impurity scattering for the well width a ¼100 A˚ (thin lines) and a¼ 150 A˚ (thick lines) 1714 N.Q Khanh / Physica E 43 (2011) 1712–1716 107 16 NB = 10 cm ni = 1011cm-2, zi = -a/2 δV = 0.6 eV, x = 0.47 105 B R A 104 103 1010 NB = 1016cm-3 ni = 1011cm-2, zi = -a/2 δV = 0.6 eV, x = 0.47 ρ (Bs)/ρ (B = 0) μ (cm2/Vs) 106 -3 B R A 1011 n (cm-2) 1010 1012 Fig Mobility m limited by alloy disorder, remote and background impurity scattering versus electron density n at T¼ for B¼0 (thin lines) and B ¼Bs (thick ˚ lines) and a¼ 150 A 1011 n (cm-2) 1012 Fig Resistivity ratio r(Bs)/r(B¼ 0) versus electron density n at T¼0 for ˚ The thin and thick lines correspond to the cases of G(q) ¼0 and a¼ 150 A G(q) ¼ GH(q), respectively 107 2000 105 103 1010 1011 ni = 1011cm-2, zi = -a/2 1200 δV = 0.6 eV, x = 0.47 B R A 800 B R A 104 n = 1011 cm-2 NB = 1016cm-3 1600 ρ (Ω) μ (cm2/Vs) 106 NB = 1016cm-3 ni = 1011cm-2, zi = -a/2 δV = 0.6 eV, x = 0.47 400 100 1012 n (cm-2) Fig Mobility m limited by alloy disorder, remote and background impurity ˚ The thin and scattering versus electron density n at T¼0 and B¼ for a¼150 A thick lines correspond to the cases of G(q) ¼ GH(q) and G(q) ¼0, respectively 120 140 160 180 200 a (Å) Fig Resistivity r due to alloy disorder, remote and background impurity scattering versus the well width at T¼ for B ¼0 and B¼ Bs (thick lines) of the ( 7) spin subband given by into the AlxGa1 À xAs barrier layer Thus, our infinite confining potential well model is reasonable and the alloy disorder scattering can be neglected [5] Furthermore, interfaces extremely flat are obtainable by the state-of-art molecular-beam epitaxy technology and interface roughness scattering is still excluded from our calculations [21] We have also found that, for scattering parameters used in this paper, the mobility limited by ionized impurities is about two times lower than that in an InP/In1 À x GaxAs/InP QW due to the higher electron effective mass in GaAs (mn ¼0.067mo) We now discuss the effect of the LFC G(q) appeared in the screening function (8) on the mobility We use the Hubbard approximation GH(q) for the LFC The results for T ¼0, B ¼0 and a ¼150 A˚ plotted in Fig indicate that the effect of the LFC is remarkable at low densities 3.2 The resistivity The resistivity of the polarized 2DEG is given by r ¼ 1/s where s ¼ s ỵ ỵ s is the total conductivity and s is the conductivity s7 ¼ n e2 /t S mn ð19Þ Results for the resistivity ratio r(Bs)/r(B¼0) versus electron density n at T¼0 for a ¼150 A˚ are shown in Fig We observe again that the effect of the LFC is remarkable at low densities We note that our results are similar to those given in earlier works [20,22] for other structures The dependence of the resistivity on the well width at T¼0 for two cases of B¼ and B ¼Bs is depicted in Fig It is seen that the resistivity shows a weak dependence on the well width for homogeneous background doping In the case of remote doping and alloy disorder scattering the resistivity decreases with increase in the well width In Fig we plot the temperature dependence of the resistivity ˚ As seen from the figure, the resistivity due to the for a ¼150 A alloy disorder scattering shows a weak dependence on temperature The temperature dependences of the resistivity for a ¼150 A˚ in two cases of B¼0 and B ¼Bs are plotted in Figs and 8, N.Q Khanh / Physica E 43 (2011) 1712–1716 1600 n = 1011 cm-2 NB = 1016cm-3 ni = 1011cm-2, zi = -a/2 δV = 0.6 eV, x = 0.47 B R A 1400 B/Bc = B R A NB = 1016cm-3 1011cm-2, 1000 ρ (Ω) ρ (Ω) 1200 105 1715 800 10 ni = zi = -a/2 δV = 0.6 eV, x = 0.47 103 600 400 200 0.2 0.4 0.6 102 0.8 T/TF Fig Resistivity r due to alloy disorder, remote and background impurity scattering as a function of the temperature for a¼ 150 A˚ in two cases of B¼0 and B¼ Bs (thick lines) 105 B=0 NB = 1016 cm-3 ρ (Ω) 104 ni = 1011cm-2, zi = -a/2 δV = 0.6 eV, x = 0.47 B R A 103 102 12 16 20 24 T (K) 28 32 36 40 Fig Resistivity r due to alloy disorder, remote and background impurity scattering as a function of the temperature for a¼150 A˚ and B¼0 in two cases of n¼ 1010 cm À and n¼ 1011 cm À (thick lines) respectively We observe that at high temperatures the resistivity shows a weak temperature dependence 12 16 20 24 T (K) 28 32 36 40 Fig Resistivity r due to alloy disorder, remote and background impurity scattering as a function of the temperature for a ¼150 A˚ and B¼ Bs in two cases of n¼ 1010 cm À and n¼ 1011 cm À (thick lines) Hubbard LFC used in this paper is not exact We believe, however, that our results are reasonable for carrier densities larger than 1011 cm À [27] For lower densities, we have to use more exact LFCs [28–30] Third, we have excluded the phonon contribution from our calculations The phonon effects, however, are negligible for the temperature range considered in this paper [21] In conclusion, we have calculated the mobility and resistivity of a Q2DEG in InP/In1 À xGaxAs/InP QW in an applied in-plane magnetic field at arbitrary temperatures for three scattering mechanisms: alloy disorder, remote and homogenous background doping We have investigated the dependence of the mobility and resistivity on the carrier density, layer thickness and magnetic field We have shown that the contribution of surface-roughness scattering to the mobility can be neglected for a $ 150 A˚ and no1012 cm À Our results and new measurements of transport properties can be used to obtain information about the scattering mechanisms in the InP/In1 À xGaxAs/InP QWs [3] Acknowledgment The author wishes to thank the Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) for the financial support He also thanks the referees for requiring him to be more precise in preparing this manuscript References Discussion and conclusion We now discuss the validity and limitations of our results First, we note that our saturation field Bs is defined with respect to a non-interacting system In order to get better results, we have to take into account the inter-particle interactions [23,24] The authors in Ref [24] have shown that, in the density range studied in this paper, the saturation field for interacting systems Bsi is somewhat less than that for non-interacting systems and the dependence of the spin-polarization on the ratio B/Bsi is similar to that of non-interacting systems Therefore, except the saturation field value, our results for 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Quoc Khanh, Physica B 396 (2007) 187  ... the main scattering mechanism in mobility limitation For a comparison, we now discuss the scattering mechanisms in an AlxGa1 À xAs/GaAs/AlxGa1 À xAs QW In the case of large aluminum concentration,... InP /In1 À xGaxAs /InP QW in an applied in- plane magnetic field at arbitrary temperatures for three scattering mechanisms: alloy disorder, remote and homogenous background doping We have investigated... for a $ 150 A and no1012 cm À Our results and new measurements of transport properties can be used to obtain information about the scattering mechanisms in the InP /In1 À xGaxAs /InP QWs [3] Acknowledgment