Quantum and Transport Mobilities of a Two-Dimensional Electron Gas in the Presence of the Rashba Spi

第27卷第2期半导体学报Vol.27 No.22006年2月CHINESE JOURNAL OF SEMICONDUCTORSFeb. ,2006Quantum and Transport Mobilities of a Two-Dimensional ElectronGas in the Presence of the Rashba Spin-Orbit InteractionXu Wen-.2,↑(1 Deparlment of Theoretical Physics, Research School of Physical Sciences and Engineering ，Australian National Universiy, Canberra ACT 0200， Australia)(2 Institute of Solid State Physis, Chinese Academy of Sciences, Hefei 230031, China)Abstract: A systematic theoretical approach is developed to study the electronic and transport properties of a two-dimensional electron gas (2DEG) in the presence of spin-orbit interactions induced by the Rashba effect. Thestandard random- phase approximation is employed to calculate the screening length caused by electron-electron in-teraction in different transition channels. The quantum and transport mobilities in different spin branches are eval-uated using the momentum- balance equation derived from the Boltzmann equation,in which the electron interac-tions with both the remote and background impurities are taken into account in an InAlAs/ InGaAs heterojunctionat low-temperatures.Key words: InGaAs/ InAlAs HEMT; 2D modeling and simulation; polarization charges; quantum effectsPACC: 7200; 7110CCLC number: 047Document code: AArticle D: 0253-4177(2006)02-0204-14magnetic-field, such as InAs- and InGaAs-based1 Introductiontwo-dimensional electron gases ( 2DEGs)[1，has .led to recent proposals dealing with advancedIn recent years ，the investigation of spin po-electronic devices such as spin-transistorsl，spin-larized electronic systems has become a fast-grow-filters[3] ,and spin-waveguides(4]. It is known thatng research field in condensed matter physics andin narrow-gap semiconductor nanostructures suchsemiconductor electronics, owing to the interest-as quantum wells, the zero-magnetic-field spinng physics behind them and to important devicesplitting (or spontaneous spin spltting) of theapplications. It has been realized that spin-elec-carriers can be achieved by the inversion asymme-tronic ( or spintronic) systems and devices can betry of the microscopic confining potential due torealized on the basis of diluted magnetic semicon-the presence of the heterojunction5J . This corre-ductors and narrow-gap semiconductor nanostruc-sponds to an inhomogeneous surface electric fieldtures. In the former case , the spin-orbit interactionand , hence , is electrically equivalent to the Rashba(SOI) is achieved in the presence of an externalspin-spltting or Rashba effectf6]. The publishedmagnetic field, and devices can be operated evenexperimental results-7,8] have indicated that inat room-temperature. In the latter case,the SOI isInAs- and InGaAs-based 2DEG systems, the spon-introduced due to the innate features of the mate-taneous spin splitting is mainly induced by therial systems, and a strong spin polarization caRashba effect ( with an SU(2) symmetry) whichnormally be achieved at relatively low-tempera-can be enhanced further with increasing the gatetures. At present ,one important aspect in the fieldvoltage applied. Other contributions such as theof spintronics is the study of electronic systemsDresselhaus term (with an SU(1, 1) symmetry)with a finite spin-splitting realized from narrow-are relatively weak, because they come mainlygap semiconductor nanostructures in the absencefrom the bulk- inversion asymmetry of the materi-of an external magnetic field. The progress madealL9中国煤化工ength of the Rash-in realizing spin-split electron gas systems at zero-baTresponding SOI inHHCNMHG↑Corresponding author. Email: wen105@ rsphysse. anu. edu. auReceived 17 October 2005⑥2006 Chinese Institute of Electronics第2期Xu Wen: Quantum and Transport Mobilities of a Two-Dimensional Electron Gas ...205these systems can be controlled artificially by ap-tion[1,5] . Including the lowest order of the SOI in-plying a gate voltagel7.8], changing the sampleduced by the Rashba effect, the single-electrongrowth parameters[1o] ,etc.Hamiltonian is given, in the absence of electronicIn order to apply spintronic systems as elec-scattering centers,bytronic devices such as spin-transistors,it is of fun-damental importance to study the effect of SOI onHo=P+°(σXp)z+U(z)(1)the electronic and transport properties of thesenovel systems. At present,one of the most power-where p =(px,p,) is the momentum operator inful and most popularly used experimental tech-which px= - iha/ax,m' is the electron effectiveniques to identify the Rashba spin splitting ismass, U (z) is the confining potential energymagneto-transport measurement carried out inalong the growth-direction, and a is the Rashbaquantized magnetic fields and low-temperatures atparameter which measures the strength of thewhich the Shubnikov-de Hass (SdH) oscillationsspin-orbit coupling. Due to the Pauli spin matricesσ= (σx ,, ,σz) ,this Hamiltonian is a 2x2 matrix.are observable1.8.10~12) . From the periodicity, am-plitude, and profile of the SdH oscillations, theFurthermore, this Hamiltonian suggests that thedensity and quantum mobility (or lifetime) in dif-SOI induced by the Rashba effect does not affectferent spin branches ，together with the Rashbathe electronic states along the growth direction.parameter, can be determined experimentally.The solutions of the corresponding SchrodingerThese techniques are akin to those employed inequation are readily obtained4] as .the investigation of spin-degenerate 2DEGs in theWan(R)=|σ,k,n)= ()e"ψ,(z) (2) .2\ isepresence of more than one occupied electronicsubbandC13,14] . It has been observed experimentallywhere k= (kx,k,) is the electron wavevector alongthat in InAs- and' InGaAs-based spintronic sys-the 2D plane,R= (r,z)= (x,y,z),中is the angletems, although the electron densities can differbetween k and the x-axis, and σ=土1 refers tosignificantly in different spin branches[10~12] ，thedifferent spin branches in k-space. The corre-quantum mobilities, determined from the ampli-sponding energy spectrum is given bytudes of the SdH oscillations at relatively low andEon(k) = E。(k)十en =h2k2+ oak +εn (3)2 m"intermediate magnetic fields, depend very weaklywith k = (k}; + k})1/2.In Eqs. (2) and (3), theon the SOI strength[10] . This result is in sharp con-wavefunction ψn(z) and energy En for an electrontrast to what has been seen in spin-degeneratein the n th electronic subband are determined by a(e. g. , GaAs-based) 2DEGs with more than onespin-independent Schrodinger equation along theoccupied subband, where both quantum and trans-growth-direction, because SOI does not affect theport mobilities differ significantly in different e-electron states along the z -direction.lectronic subbands. In order to understand this im-From the electron energy spectrum given byportant and interesting experimental finding andEq. (3) ,one can immediately see that in the pres-to achieve an in-depth understanding of how SOIence of the Rashba spin-splitting: (1) the elec-affects the electronic and transport properties of atronic states are split into two spin branches in k-2DEG，we develop a tractable theoretical apspace and electrons are oriented perpendicular toproach to examine quantum and transport mobili-the electronic momentum in the 2D-plane;(2) theties pertinent to a spin-split 2DEG in this paper.energy dispersion of a 2DEG is not parabolic dueto the inclusion of the SOI term;and (3) the ener-2 One-particle aspectsgy levels for the土spin branches depend stronglyon electron wavevector ( or momentum ), whereFor a typical 2DEG formed in the xy-planethe energy separation between the two spin bran-(taken as the 2D-plane) in narrow-gap semicon-ches中国煤化工ese features are inductor nanostructures, such as InGaAs/InAlAssharCNMH Gspin-degenerateheterojunctions in which the growth-direction isLen's function for ataken along the z-axis, the effect of SOI can bespin-split 2DEG can be obtained,in the (E,k) orobtained from a k● p band-structure calcula-(spectrum，momentum) representation ,as206半导体学报第27卷Gonm(E) = [E- Eon(k)+ i8]-. (4)2.0-Thus , the density-of-states (DoS) for a spin-split2DEG can be determined from the imaginary partof G如n (E). In this paper , we consider an InGaAs/D(EVD。InAlAs heterojunction in which only the lowest e-1.0lectronic subband is present (i.e.,n'= n = 0),D(目D。0.5-D(E)/D。and we measure the energy from Eo. In such acase , the electron DoS in different spin branches is01 o234obtained asElectron energy.E/EaD.(E)= 2&[E- E,(k)](5)Fig.1 Density-of-states in different spin brancheswhere for spin-up(D: (E)) as a function of electron energy ED. (E) =°B(E)(1-VE+E.)E(6)why electron density in the spin-down channel isalways larger than that in the spin-up channel. Afor spin-downdirect and important application of the DoS for a[@(E)(1+VE+E,E。spin-split 2DEG is to determine the Fermi energyof the system and the electron density in different20(- E)@(E+E.)VE+E.」(7)spin branches. Applying the DoS for a 2DEG withSOI to the condition of total electron number con-and the total DoS isservation and the definition n。= 2f(E,(k)) withf(x) being the Fermi-Dirac function, the FermiD(E)=D.(E)+D_(E)=energy Ep and electron density n。in the σ spinDo0(E)+日(- E)O(E+ E.)NE+E. |(8)branch are obtained, respectively , for low temper-atures T-→0,asHere,@(x) is the unit-step function,E is the elec-tron energy, D。= m" /nh,and E。= c?m" /2h'.Ep = F(πn。一k2)(9)n'These theoretical results indicate that in contrastandto a spin-degenerate 2DEG whose DoS is givenn。=∩°_σke√2rn.- R(10)simply by D(E) = D,@(E),the DoS for a spin-split 2DEG depends strongly on SOI, which re-for nc≥k:/π(or a≤h2√mne/m° ).Here,n.= n-sults in the following:(1) spin-up and spin-down+ n. is the total electron density and k。= m' a/electrons have different DoS; (2)the DoS dependsh. For= n,≤k/π we have Ep≤0, and only thenot only on those step-functions but also on E。=lower-energy‘-’spin branch is occupied by e-(E + E,), because of a nonparabolic energy spec-lectrons (see Fig. 1), with n+ = 0 and n- = n。trum given by Eq. (3);and (3) for the spin-downprecisely at ne = k:/π; in this case the electronschannel the DoS can exit even in the negative en-are entirely in the‘-' spin branch. However,itergy regime, whereas D. (E) only exits when Eshould be noted that the condition n。< < k2/π can> 0. Furthermore, we note that D. (E)/D。=only be satisfied in a system with very low elec-(∞,0.5] when E=[E.,∞),whereas D. (E)/D。tron density and very large Rashba parameter,=[0,0.5] when E= [0,∞). This implies that D.which has not yet been realized experimentally.(E) is always larger than D+ (E) and a larger D.Therefore ,in this paper , we only consider the situ-(E) can be observed at a larger E.These interest-ation where both土spin branches are occupied bying features can be more clearly seen from Fig. 1electrons, namely the situation where nc> k2/π.where the DoS in different spin-orbits is shown asFrom Eq. (10), we obtain the relationa function of electron energy.中国煤化工in.- k:(11)The theoretical results obtained and shownwhicCNMHGpaper.above indicate that in the low energy regime,CHAnouner simpie way 1o understand why thewhich is most probably occupied by electrons , thepresence of SOI can lead to different electronDoS for the‘-’branch is always larger than thatdensities in different spin branches is to look intofor the‘+，branch,and this is the main reason第2期Xu Wen: Quantum and Transport Mobilities of a Two-Dimeasional Electron Gas”207the dispersion relation E,(k) versus k shown in1.00Fig.2. In Fig. 2, the solid parabolic curve corre-n.=10'"cm20.75-sponds to the absence of the SOI (a= 0),i.e.,ton/n.ki =kμ = kp. The intersections of the curves for0.50a≠0(E. (k) and E. (k)) with the Fermi levelEp ,projected onto the k-axis,give the Fermi wav-n./m."9 0.25evectors ki and k# .The difference ki -ki ata≠0 leads to a difference in k-space area:π(ki )210-21011010≠π(k$ )2. Accordingly, the densities n- and n+Rashba parameter a/(eV.m)are different when the SOI is present. Because kiFig.3 Electron density n. in different spin chan-is always larger than k; ,n- is always larger thannels as a function of the Rashba parameter a at an+。fixed total electron density n,=n. + n-E.(k)E.(A)/E.(K)E, Fa-1.6x10"'eV.mn/m。0.50-E 0.25-Fig.2 Dispersion relation E,(k) versus k for 2DEGsThe solid parabolic curve,for a= 0，corresponds tootal ectron densit/cmithe absence of sol where k; =ki = kp. The inter-sections of the dashed curves (E.(k)) fora≠0 withFig. 4 Dependence of electron distribution in thethe Fermi level Er(dotted line) ,rojected onto the k-土spin branches on total electron density at a fixedaxis,give the Fermi wavevectors ki and ki for dif-Rashba parameterferent spin branches.of electrons in semiconductor-based 2DEG sys-The dependence of electron distribution intems. For a modulation-doped InGaAs/InAIAsheterojunction , the main sources of electron-impu-different spin branches on the Rashba parameterrity scattering come from the ionized remote-im.a and total electron density n。 is shown respec-purities in the InAlAs layer and from the chargedtively in Figs. 3 and 4. These results are obtainedbackground impurities in the InGaAs layer. Thefor an InGaAs-based 2DEG structure.It should beCoulomb potential induced by electron interactionnoted that with increasing a in Fig.3 and/or de-with charged impurities takes the formcreasing ne in Fig. 4, Fermi energy decreases (seeV(R-R,)= Ze(12)Eq. (9)) and,consequently, more electrons are inKTR-R。Tthe spin-down orbit because it has a lower energywhere the impurity is located at R。= (r。,zo)=and higher DoS. This is in line with experimental(x.,yo,za),Z is its charge number,andx is thefindingsf8.10-12]) . The results shown in Figs.3 andstatic dielectric constant of the material. In the4,together with those given by Eqs. (6) and (7)，absence of electron- electron (c-e) screening, thissuggest that in a 2DEG, spin polarization increa-potential results in an electron-impurity interac-ses with increasing Rashba parameter and/or withtion Hamiltonian in momentum representation af-decreasing total electron density.ter the Fourier transformation中国煤化工(13)3 Electron-impurity scatteringYHCNMHGwhereq! 4x1y,, s Lic iactor of the FourierAt low temperatures, electron-impurity scat-transform, which corresponds to the change oftering is the principal channel for the relaxationelectron wavevector (or momentum) during an208半导体学报第27卷electron-impurity scattering event. For electronσ' branch to the σ branch , the bare e-e interactioninteractions with charged impurities in an elec-in the presence of SOI becomestronic system,we may assume that the system un-Vp(k,q) = V,G。(q) xder study can be separated into the electron of in-[4Aw8n + iB(1-80)7(17)terest and the impurities, i.e. ,lo,k,n;c)= |σ,k,n)|c> where |c) represents the state of the im-where Go(q)= Jdz1 Jdzz |4(z;)|2 |46(z)|e-913-2|purity system. Thus, the matrix element for elec-with 4o(z) being the ground-state electron wave-tron-impurity interaction is obtained asfunction along the growth-direction, Am = (k +qcosψ)/|k+q|,Bw = qsinψ/|k+q|, and ψ is the .Unmnd(q,R.) =2rZe2 Jn(z.Txangle between k and q. It should be noted that inFnn(q,z.)hs. (0)e" ir'e81. .(14)contrast to a spin-degenerate 2DEG for which theHere,n;(za)= |习[。dk xupper (lower) case refers to j=1 or 4 for intra-πh2 rqSO transition (j=2 or 3 for inter-SO transition).fLE.(k)]k(k + q)____H'(k,q) .(2k + q + 2dk.)(k+ q+|k-qT)Moreover , the inverse dielectric function matrix(27)for a spin-split 2DEG iswhere[1-ai-a01- a2- a;H转(k,q) = 11K(A) + n(AB+,A) +∈-'(2,q) =2-ai1-a;ai1- ai9(9+ 2d<)[n(AC; ,A)- I(AB+ ,A)](28)4k(k+ dk,)(21)K(x) is a complete elliptic integral of the firstwhere ai = a1/(1+ a1+ as),ai = a2/(1 + a2kind,I(n,x)= I(π/2,n,x) is a complete ellipticaz),ai = a3/(1+ a2 + ag),and a; = as/(1 + a1integral of the third kind,A=(k+ q-|k- q|)/+ as).With the inverse dielectric function matrix,(k+q+|k-q|),B: =[(2k+ q)/q]*', and C+=[(q- 2ok.)= (2k + q+ 2dk.)]*1. For the casewe can calculate the matrix element for the elec-of a low-temperature limit (i.e, T→0), we ob-tron-impurity interaction in the presence of e-etainscreening , through the definition U ( q ,R。) =2∈;'(q)U9(q,R。). Here∈y(q)=∈y($→0,K, (q)= 16e?m"-G。(q)2 dkxπh'xq !q) is the element of the static dielectric functionk(k+ q)。下H(k,q)matrix. Using Eq. (15), the square of the matrix(2k + q+ 2ok.)(k + q+|k-q|element for the electron-impurity interaction in29)the presence of e-e screening becomesThese results imply that in the presence of SOI,| U,(q,0)|2 =| U.(q,6)|2 =| U, (q,0) |8v.othe intra- and inter-SO transitions have different(22)screening lengths under the RPA approach. Infor intra-SO scattering ,andparticular , we note that in a long wavelength limit| U2(q,B)|2 =| U3(q,0)|2 =| U_ (q,B) 1|8r.Hw (i.e,q→0 which means that the e-e interaction(23)does not change the electron wavevector or mo-for inter-SO scattering. Here，mentum),K +(q)→∞whereas K. (q)→0./2rZe2 2h; (0)|U2(q.0)p=(kC) [q+k;(q)FX5 Spin-dependent quantum and trans-|dzgn;(z。)FE(q,zo)(24)port mobilitiesin which the contribution from the impurity dis-From the above presented results, we obtaintribution along the growth direction has beenfor the electronic transition rate given by Fermi'ssummed over, h: (0) = (1土cos0)/2,and the in-golden ruleverse screening lengths for intra- and inter-SO e-W..(k' ,k) =lectronic transitions are, respectively,K. (q) = q[a:(q)+ a,(q)]=2n I U。(q,0) I8v.+r8[E。(k')- E,(k)]-号V,Go(q)2(1+ Au)x(30)which is the probability for an electron to be scat-f[E.(k+q)]- f[E。(k)](25)tered from a state |σ,k> to a state |o' ,k" > dueE.(k+q)- E。(k)to screened electron-impurity scattering. We nowandK. (q) = q[a2(q) + a3(q)] = .consid中国煤化工d Fs applied alongthe 2fYHCNMHGx direction) ofa- YV,Go(q)2(1+ Am)X2DEG.1ir ule stcauy siale Liic corresponding semi-of[E。(k+q)]- f[E_。(k)](26)classical Boltzmann equation, for nondegenerateE.(k+q)- E_。(k)statistics, reads210半导体学报第27卷eFxxaf.(k) =4=(- - ) correspond to different channels for e-akxlectronic transition. From the quantum and trans-2[f.(K)W.(k,K)- f.(k) W..(K ,k)]port mobilities, the quantum and transport life-(31)times for an electron in different spin brancheswhere f。(k) is the momentum-distribution func-are given respectively by塌= m*喝/e and τ{ =m *pi/e . Furthermore, the average transport mob-tion for an electron at a state |σ,k> . We assumeility ,measured in a conventional transport experi-that f,(k) can be described by the drifted energyment,is given asdistribution function f。(k)= f(E,(kx- m* v。/h,n+μ+n-k,)),where v。 is the average drift-velocity of anμ=n(39)electron in the σ spin branch along the x directionUsing Eq. (30) we obtain, for electron-impu.due to the presence of Fx.Then the momentum-rity scattering,balance equation[17,18 can be derived by mutipl-- m*kying kx with both sides of the Boltzmann equation”)= 4n"heJ。d)。dkicke.cosa)xand by summing over k. In doing so , we have| U。(q,0) |'f(E) |E=E,(k) .(40)二n。=Here,q = Vk3。+ k°- 2kk..cos0 and k。。 = k -(σ' -σ) k. For low temperatures,i.e., for T- →0,[k2f。(k) Wi。(k' ,k) - ksf.(k) W。(k ,k)]we have f"(E)=af(E)/aE≈-δ(Ep- E) with Ep(32)being the Fermi energy. Then we obtainFor weak driving fields Fx we assume v。 《kx/(B.。\m*2kk”。m' and obtain(c.)= 4,4πieex.(d(e.cos)xf。(k)≈U。(q,0) |2 lk= 7m,41)f(E。(k))- k;v,(1 +ak./k)f"(E) Ig=E.k)After using Eq. (11 ), for different transition(33)channels we havewhere f (E) = af(E)/aE. Thus the momentum-= 2co(n. /n-)l2(n+ /n.) Xbalance equation givesn。= 2C[xB.-piC.] (34)a0(。( cosa)U, (q,0) |2(42)Here出= v。/Fx is the transport mobility for an e-with co=m°'n。√4πn-/(2πh e/2rπn.- k百) andlectron in spin branch σ andq1 = 4/πn+ sin(0/2),(..)=- t(k,)xU_ (q-,B) |2k,(1 + ok。/k) Wi,(k' ,k)f(E) |ε=E,() (35)(43)It can be seen that the term B。。is induced bysmall-angle scattering between k' and k. Hence,bywith q- = 2/rn。一(πn。- k2)cos0,definition, the quantum mobility of electrons in√n47n)U_ (q-,0) |2spin branch σ，喝，is given bycos0(36)(44)andThe transport mobility ,μ: ,in different spin bran-ches can be determined by solving Eq. (34) , which(&)=2co(n. /nod)*d() u. (q.O)|2reads(45)(B、- C。+ B2)n.+ Czn_μr=(B-C1+ B3)(B,- C.+ B2)- CzC3with q4 = 4√πn- sin (0/2). These results indicate(37)that corresponding to different electronic transi-tion中国煤化工impurity scattering,theFvevector or momen-C3n+ (B,- C1+ B3)n_YHr =;tumC N M H Gso sattering,k;=(B-C1+B3)(B,一Cs+B2)-C2C3(38)√4πn+ and q=q1=[0,4Vπn+] for a transitionHere,again,1=(+ +),2=(+ -),3=(- +) and within the‘+ ' spin branch,and ks= V4πn- and第2期Xu Wen: Quantum and Transport Mobilities of a Two-Dimensional Electron Gas ...211q= qs = [0,4√πn- ] for a transition within thematrix element induced by scattering with remotebranch. Whereas for inter-SO scattering,q =and background impurities is given respectively byq- =[2k,2√2πn。- k幻] is the same for both a|U:(q,0)|= N,(2rZe* ))xktransition from the‘+，spin branch to the‘-’h: (0)e-2qbranch and a transition from the‘ -，branch to[q+ K:(q)]^ (x+1)6(48)the‘+’branch. Furthermore,q-→0 can only be aand .case for intra-sO scattering. For inter-SO scatter-/2rZe2、h+ (0)ing, q≠0, which implies that an inter-sO transi-| U? (q.0)|°= Nok)[q+K:(q)]xtion can only be achieved via varying the3x5+18x4+43x3+48x2+24x+2.(49)wavevector ( or momentum) of an electron, be4x(x + 1)6cause the spin-pltting depends explicitly on elec-where x=q= b,Nb= Nop:/db and b= [(48rm"tron wavevector.Xe2/kh2](Ndqpl + 11n,/32)]/3 defines the thick-ness (≈3/b) of the triangular well. These results6 InGaAs/InAlAs heterojunctionindicate that similar to a spin-degenerate 2DEG,electrons in a heterojunction interact more strong-In an InGaAs/InAlAs heterojunction , the im-ly with background impurities than with remotepurity scattering comes mainly from: (1) ionizedimpurities, especially when q→0. This is mainlyremote impurities within a narrow space chargedue to the fact that background impurities are lo-layer in the InAlAs region with a concentrationcated in the layer where the majority of conduc-N, at a spacer distance s from the interface ，dueting electrons are. The form factor for the e-e in-to modulation-doping; and (2) charged back-teraction isground impurities with a depletion charge density3x2 +9x+8Go(q) =(50)Naepnl and a depletion length d in the InGaAs lay-8(x+ 1)8er,due to the effect of depletion. In general, theseimpurity concentrations and their distributions a-7 Numerical results and discussionlong the growth-direction are not well known, be-cause the ionization of modulation-doped impuri-The results of this section pertain to InGaAs/ties ( including deep-centers) and the depletionInAlAs heterojunctions at low temperatures. Thelength and charge density of the background immaterial parameters corresponding to InGaAs arepurities are not easily determined experimentally .taken as follows:(1) electron effective mass m 'In conjunction with a typical spintronic device re-=0.042m。with m。being the rest-mass of an e-alized from an InGaAs/InAlAs heterojunctionT] ,lectron; (2) static dielectric constant k = 12.9;andin this paper we model the remote and back-(3) the typical depletion charge density Naepnl = 2ground impurity distributions respectively asX 1010 cm~2. Furthermore，the electron distribu-n,(z。) = N,8(z。+ s)(46)tion n+ in different spin branches is determinedandusing Eq. (10) (also see Figs.3 and 4).In the cal- .no(za) = (Nep/ d)@(z。)(47)culations, the charge number of an impurity isThese assumptions are mainly based on the facttaken to be Z=1.that the width of the charge layer for modulation-7.1 Screening lengthdoping is relatively narrow and the depletionlength in the InGaAs layer is much longer thanFrom the results presented in Section 5, wethe effective thickness of the confining potentialknow that for electron-impurity scattering at low-for electrons.temperatures , the change of electron wavevectorIn the present work , we apply the usual trian-or momentum (i.e. ,a depends on electron densi-gular well approximation to model the confiningty n中国煤化工dfferent transition .potential normal to the interface of the hetero-chanYHCN MH Gy the effect of c-ejunction and use the corresponding variationalinteraction on transport coetficients induced bywave function for the ground subband[15] . Thus,electron-impurity scattering, it is convenient andthe square of the electron-impurity interactionuseful to look into the angular dependence of the212半导体学报第27卷respectively, for transitions within the‘+，andn.-2xI0'cm2‘-’spin channels. These results indicate that for。10.0a=2xI0~"eV.mintra-SO transitions, K, (q) increases with in-creasing a or with decreasing n.. Together withthose obtained for electron distribution (or spin1.0K.(q)polarization) shown in Figs.3 and 4,an important-K(qJK@Jconclusion we can draw is that the inclusion of0.1室3(50 90 120 150 1801()q=(m")9sin(62)n.-=2x10"cm2Fig.5 Inverse screening length K.(q),s=士,fortransition within the‘土’spin branch as a function ofangle θ at a fixed total electron density ne and a fixedRashba parametera Here, θ is the angle between thex 2x01initial electronic wavevector ( or momentum) k and5x1012the final wavevector k' ,q: = 4√πn. sin(/2) for atransition within the‘+，branch,q+ = 4√πn- sin(0/a-2x10"LeV.m2)foratransitionwithinthe‘-’branch，q-=言102√πne- (πn。- k)coso for inter-So transition, andnx is the electron density in the ‘土’spin branch.n.=8x10*cm2Note that K. (q-) is negative.2x10'8x101screen length for different transition channels. In0306090 120 150180Fig. 5, the inverse screening length Kt (q) is(°)shown as a function of θ (the angle between theFig.6 Angular dependence of inverse screeninginitial wavevector k and the final wavevector k'length K. (q) for transition within the ‘+’spiduring a scattering event) at a fixed total electronbranch The results are shown at a fixed total elec-density n。and a fixed Rashba parameter a. Fromtron density n。for different Rashba parameters a (up-these results,we note that:(1) | K: (q) | decrea-per panel) and at a fixed a for different n. (lowerses with increasing 0, which implies that a strongpanel).effect of e-e screening can be achieved at smallscattering angles; (2) for intra-SO transition with-SOI can enhance the effect of e-e screening in ain the‘+’or‘-’spin branch,K+ (q)→+∞2DEG for intra-SO transitions. From Figs. 6 andwhen θ→0(i.e. ,q→0); (3) K_ (q) for inter-SO7,we note that K+ (q1) (induced by a transitiontransition is negative and finite when θ= [0,π];within the‘+’branch) depends more strongly on(4) at a fixed 0,K. (q1) for a transition withina and n。than K+ (qu) (by a transition within thebranch) does. The dependence of the inversethe‘+’branch is always larger than K+ (q4) fora transition with in the‘-’branch;and (5) at ascreening length induced by an inter-SO transitionfixed日, the inverse screening lengths induced byon the Rashba parameter and total electron densi-intra-SO transitions, K+ (q1) and K+ (q,), arety is shown in Fig. 8. We see that in the small 0 re-much larger than |K. (q- )| induced by inter-SOgime - K_ (q) increases with decreasing a or in-transition. Moreover,it should be noted that at acreasing ne, whereas at large 0,- K- (q) increa-fixed 0, because the transition from the‘-’spinses with increasing a or decreasing n。. Thus, forbranch to the‘+’spin branch corresponds to theinter-SO transitions, the SOI can reduce the effectsame q as for the transition from the‘+’branchof e-e screening in the small angle regime and en-to the‘-’branch, the screening length is thehance中国煤化工rge 0.~ 8 indicate thatsame for inter-SO transition channels.in theMYHC N M H! Gening length ofaThe influence of the strength of SOI and to-tal electron density on angular dependence of the2DEG differs significantly for different electronicinverse screening length is shown in Figs. 6 and 7,transition channels. In particular, the e-e screen-第2期Xu Wen: Quantum and Transport Mobilities of a Two-Dimensional Electron Gas .The main physical reason behind this importantq=4(mn)"sin(@/2)effect is that the inter-SO transition due to e-e in-n.=2x10"cm2teraction requires the change of electron wavevec-tor or momentum , because , again, the spin-split-a=Sx10+42e.Vmting depends explicitly on electron wavevector.1.02x10-1Furthermore, over a wide regime of θ or q,5x10-1|K+ (q)|≈105~10° cm-'I ,similar to the inverse0.1screening length for a spin- degenerate 2DEG.a=2x0"eV.m7.2 Quantum and transport mobilitiesg 10.0Here we study the quantum and transportmobilities in different spin branches due to elec-2x10tron interactions with remote and background im-8x10"purities in an InGaAs/InAlAs heterojunction. Al-though the concentrations N, and N, for remote-306090120150180/(°)and background impurities are normally notknown,one may assume that N,∞n。and N;》Fig.7 Inverse screening length K+ (q) for a transi-No. The former assumption is based on the facttion within the‘-’spin branch as a function ofθ Inthat the conducting electrons in the InGaAs layerupper (lower) panel, the results are shown at a fixedcome mainly from ionized donors modulation-total electron density n。for different Rashba parame-doped in the InAIAs layer. The later assumption istersa (at a fixed a for different n。).made for the case of a high quality sample inwhich the background impurity concentration inthe InGaAs layer is low. As has been shown inqc-2[mn-(m-k.)os)]nSection 6, electrons in a heterojunction interactmore strongly with background impurities than2x101Xa=Sx1012eV.mwith remote impurities. Hence, although the con-5x10centration N, is relatively low , background impu-rities can significantly affect the transport proper-.1ties of a sample.From the results presented in Sections 5 anda=2x10-"'eV.m6,we know that the square of the matrix elementfor electron-impurity scattering via inter-SO tran-sition is not divergent over the whole defined re-. n-=5xI0'*cm2gime of q or 0. Together with the fact that the e-e2x104screening relatively weakly affects the inter-SO8x101transition (see Figs. 5~8) ,in the present study weonly include the effect of e-e screening for an in-30 60 90 120 150 1808(°)tra-SO transition induced by electron-impurityscattering. Substituting n。 in Eqs. (42)~(45) theFig. 8 Angular dependence of the inverse screeningfactors B。。 and C。induced by scattering with re-length for an inter-so transition Note that K_ (q)