Multichannel effects in Rashba quantum wires M. M. Gelabert,1 Lloren¸c Serra,1,2 David Sa´nchez,1 and Rosa Lo´pez1 1Departament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain 2 Institut de F´ısica Interdisciplinar i de Sistemes Complexos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain. (Dated: January 26, 2010) We investigate intersubband mixing effects in multichannel quantum wires in the presence of Rashba spin-orbit coupling and attached to two terminals. When the contacts are ferromagnetic 0 and their magnetization direction is perpendicular to the Rashba field, the spin-transistor current 1 is expected to depend in a oscillatory way on the Rashba coupling strength due to spin coherent 0 oscillations ofthetravellingelectrons. Nevertheless,wefindthatthepresenceofmanypropagating 2 modes strongly influences the spin precession effect, leading to (i) a quenching of the oscillations and (ii) strongly irregular curves for high values of the Rashba coupling. We also observe that n in the case of leads’ magnetization parallel to the Rashba field, the conductance departs from a a uniform value as the Rashba strength increases. We also discuss the Rashba interaction induced J currentpolarization effectswhenthecontactsarenotmagneticandinvestigatehowthismechanism 9 is affected bythe presenceof several propagating channels. 2 PACSnumbers: 71.70.Ej,72.25.Dc,73.63.Nm ] l l a h I. INTRODUCTION nels. Confinement in the transversal direction is accom- - plished with potentials leading to subband spacings of- s ten smaller than a few meV, the order of magnitude of e Since the discovery of the giant magnetoresistance m effect,1,2 researchinspintronicshas beendevelopingata the Fermi energy in low-dimensional systems. As a con- sequence, multiple subbands are populated and channel . fast pace. An important requirement for practical appli- t mixing effects become relevant in many situations. In a cationsofthisnoveltechnologyisthegeneration,control m and manipulation of spin-polarized currents preferably fact, the Rashba interaction itself includes an intersub- - usingelectricfieldsonly.3 Spin-orbitinteractionsinsemi- bandmixingtermwhichcouplesadjacentsubbandswith d conductor materials are promising tools to achieve that opposite spins. This coupling has been recently demon- n goal. In particular, the Rashba interaction,4 a type of strated to give rise to strongly modulated conductance o curves,12–15 especiallyclosetotheonsetofhigher-energy spin-orbit coupling that originates from a lack of inver- c plateaus,duetoFanointerference16betweenpropagating [ sion symmetry in semiconductor heterostructures (such wavesandRashbainducedlocalizedlevels.14 Inthepres- asInAsorGaAs),hasbeenexperimentallyshowntopos- 1 sess a high degree of tunability using gate contacts.5 enceofin-planemagneticfields,Rashbacouplinginduced v intersubband mixing effects are shown17 to reduce the 3 Since the spin-orbit interaction couples the electron visibility of anomalous conductance steps,18 and to pro- 9 momentum and its spin, the Rashba field behaves as an duce transmission asymmetric lineshapes even in purely 3 effectivemagneticfieldthatisresponsibleforspincoher- one-dimensional systems.19 5 ent oscillations, which can be exploited in spintronics. 1. Based on this property, Datta and Das suggested a spin In this paper, we analyze the role of intersubband 0 field-effect transistor.6 It consists of a one-dimensional coupling effects in multichannel quantum wires. Our 0 ballistic channel sandwiched by two ferromagnetic con- modelconsistsofaquantumwirewithalocalizedRashba 1 tacts. Their proposal relies on the control of the cur- spin-orbit interaction coupled to ferromagnetic leads v: rent along the channel using the Rashba interaction via with magnetizationperpendicular to the directionof the i athirdterminal(thegate)andtherelativeorientationof Rashbafield. WefindthattheRashbaintersubbandcou- X theleads’magnetizations. Thelengthofthechanneland pling term modifies the spin precession effect in a dra- r the intensity of the Rashba strength determine the flow matic way. Typically, one finds a few oscillation cycles a ofthecurrent. Realizationofthespintransistorwashin- in the conductance curves before arriving at a strongly deredbysomelimitations,suchasthemismatchproblem irregulardomainathighvaluesofthe Rashbaparameter (which results in poor injection of spin-polarizedcurrent in which case the intersubband coupling produces an ef- between a ferromagnet and a semiconductor)7 and the fective randomization of the injected spins independent idealization of ballistic transport.8 However, recent ex- of the relative orientation of the leads’ magnetization. periments on quasi-two dimensional structures9, already Therefore, our results point out a serious limitation of discussed in Refs. 10,11, have overcome these obstacles the spin transistor performance, even in the ideal cases and have obtained a behavior which looks similar to the of perfect spin injection and fully ballistic propagation. spin transistor effect. On the other hand, Rashba interaction has lately de- In reality, strictly one-dimensional channels are hard servedmuchattentionasagenerationprocedureofspin- to fabricate and one must deal mostly with quasi-one polarized currents. Several methods have been proposed dimensionalsystems containing many propagatingchan- in different setups (see Refs. 20–41, although the list is 2 by no means exhaustive). We here consider a simple system: a Rashba quantum wire attached to two non- magneticleads. We findthatthe Rashbainteractioncan produce a highly polarized electric current and that the effect is purely due to interchannel coupling. For quan- tum waveguides supporting a single propagating mode, the polarization effect vanishes.42–44 Since the Rashba interaction is localized, we calculate the generated po- larization as a function of the interface smoothness and show that the highest values of the polarization are ob- tainedwhen the transitionbetweenthe regionswith and FIG. 1: (Color online) Sketch of thephysical system (a) and without spin-orbit interaction is abrupt. of the spatial variation of Rashba intensity α(x) and gate In Sec.II we discuss the physicalsystemand establish potential Vg(x) (b). thetheoreticalmodeltocalculatethelinearconductance. Section III is devoted to the numerical results when the contacts are ferromagnetic. The spin polarization effect α(x) α(x) i ′ = p σ + − p + α(x) σ , (2) in the case of normal contacts is analyzed in Sec. IV. y x x y ¯h ¯h 2 Finally, Sec. V contains our conclusions. (cid:18) (cid:19) where,asusual,spinisrepresentedbythevectorofPauli matrices~σwhilep andp aretheCartesiancomponents x y II. PHYSICAL SYSTEM AND MODEL of the electron’s linear momentum. The Rashba inten- sity α(x) varies smoothly taking a constant value α in- 0 side the Rashba dot and vanishing elsewhere. The term We consider a quasi-one dimensional system (a quan- proportional to p is responsible for spin precession of tum wire) with a localized Rashba interaction (the x an injected electron.6 The intersubband coupling term Rashba dot) coupled to semi-infinite leads. Figure 1 proportionaltop couplesadjacentsubbandswithoppo- shows a sketch of the physical system. Transport oc- y site spins. Finally, the term with the derivative α′(x) is curs along the x direction. We characterize the Rashba dot as a small region of length ℓ with strong spin-orbit addedinEq.(2)toensuretheHermitiancharacterofthe Hamiltonian. coupling with strength α . The spin polarization in the 0 leadsisdescribedusingtheStonermodelforitinerantfer- As mentioned above,the Stoner field ∆(x) is constant in the left and right asymptotic regions (∆ ) and it romagnets. Due to exchange interactionamong the elec- ℓ,r smoothly vanishes at distances d towards the left and trons, the electronic bands in the asymptotic regionsbe- ℓ,r rightoftheRashbadot. Theseareassumedlargeenough comespinsplitwithasplittingphenomenologicallygiven such that all evanescent states at the interface vanish by an effective field ∆ , which we take as a parameter. 0 before reaching the leads. The gate potential aligning This approximation is good at low temperatures (lower the band bottom of the different regions is taken as than the Curie temperature) and for electron densities V (x)=|∆(x)|. Anequivalentchoicebutlocalizedtothe large enough so that strong correlations can be safely g neglected.45 Denoting the Stoner field in left and right Rashba dot would be Vg(x) = |∆(x)|−∆0. All spatial transitions in α(x) and ∆(x) are described using Fermi- regions by ∆ and ∆ , respectively, the parallel configu- ℓ r like type functions characterized by a small diffusivity ration is described by ∆ = ∆ = ∆ while the antipar- ℓ r 0 a.46 In general, a is assumed to be small enough, al- allel corresponds to ∆ = −∆ = ∆ . In addition, we ℓ r 0 though we shall also discuss below the dependence with assume that a local gate potential V (x) is aligning the g this parameter in some cases. potential bottom of the successive regions. This way we ForagivenenergyE theelectronwavefunctionfulfills remove unwanted conductance modifications due to the potential mismatches,7 thus focussing on the properties Schr¨odinger’s equation induced purely by the spin-orbit coupling. (H−E)Ψ=0, (3) The system Hamiltonian reads with the appropriate boundary conditions. Our method ¯h2 d2 d2 1 H = − + + mω2y2 of solution combines discretization of the longitudinal 2m dx2 dy2 2 0 variable x in a uniform grid with a basis expansion in (cid:18) (cid:19) + Vg(x)+∆(x)nˆ·~σ+HR . (1) transverseeigenfunctionsφn(y)andineigenspinorsχs(η) along a direction given by a unitary vector nˆ, The confinement along the direction y, perpendicular ∞ to the current, is taken as parabolic with oscillator fre- Ψ= ψ (x)φ (y)χ (η), (4) quency ω , which defines the length ℓ = ¯h/mω . The ns n s 0 0 0 s=±n=0 inhomogeneous Rashba coupling H is given by XX R p where s = ± is the spin quantum number while η =↑,↓ H ≡ H(1)+H(2) denotes the twofold spin discrete variable. In terms of R R R 3 the polar and azimuthal angles (θ,φ) corresponding to the spin quantization axis nˆ we can write cos θ sin θ 2 2 χ+ ≡ ; χ− ≡ . (5) sin θ(cid:0) (cid:1)eiφ ! −cos (cid:0)θ(cid:1)eiφ ! 2 2 The transvers(cid:0)e (cid:1)eigenfunctions are the s(cid:0)olu(cid:1)tions of the harmonic 1D oscillator ¯h2 d2 1 − + mω2y2 φ (y)=ε φ (y), (6) 2mdy2 2 0 n n n (cid:18) (cid:19) with 1 ε = n+ ¯hω ; n=0,1,... . (7) n 0 2 (cid:18) (cid:19) Projecting Eq. (3) onto the basis we obtain the equa- tions for the unknown channel amplitudes ψ (x) ns ¯h2 ′′ − ψ (x) + V (x)+s∆(x)+ε −E ψ (x) 2m ns g n ns (cid:18) (cid:19) ′ ′ + hns|HR|nsiψn′s′(x)=0. (8) FIG. 2: (Color online) Conductance as a function of Rashba n′s′ couplingintensity. BlackcorrespondstothecompleteRashba X interaction while grey (red color) to the neglect of H(1). R Notice that the Rashba interaction is the only source of The leads are spin-polarized along x. Upper, intermediate interchannel coupling since, in general, the matrix ele- and lower panel correspond to Np = 1, 5 and 10 propagat- ment hns|HR|n′s′i will be non diagonal. Using the sepa- ing modes, respectively. We take the parameters ℓ = 8ℓ0, ration in two spin-orbit contributions introduced in Eq. E =Np¯hω0, ∆ℓ =∆r =10h¯ω0, dℓ =dr =10ℓ0, a=0.1ℓ0. (2) we can write hns|HR(1)|n′s′i = α(¯hx)hn|py|n′ihs|σx|s′i, (9) sgceenneaitryioinwαhe(xn)HisR(1a)lsiostiankcelundiendtoaancdcowuhnetn. TsphaeceAipnpheonmdoix- hns|HR(2)|n′s′i = −α(¯hx)px+ 2iα′(x) δnn′hs|σy|s′i. ctoonctoaminpsuttheetdheetatrilasnosmf tishseioenmtpnl′os′y,neds,ni.uem.,etrhicealprmobetahboild- (cid:18) (cid:19) ity amplitude from a given left incident mode ns to the (10) rightmoden′s′. Then,usingthescatteringapproachthe Equations (9) and (10) clearly show that, in general, linear-response conductance is given by, both H(1) and H(2) couple channels with opposite spins throughRthe matrRix elements hs|σx|s′i and hs|σy|s′i. Of G=G0 |tn′s′,ns|2 , (11) course, if the spin quantization axis nˆ is chosen along ns,n′s′ X the x or y axis then either hs|σ |s′i or hs|σ |s′i become x y diagonal. Regarding the coupling between transverse where G =e2/h is the conductance quantum. For later 0 modes,wenoticethatHR(2)isalwaysdiagonal(δnn′)while discussionon the polarizationof the transmittedcurrent H(1) is connecting modes differing in one subband in- we also define the polarized conductance Gp, R dex (n′ = n±1) through the oscillator matrix element hn|py|n′i. Gp =G0 s′|tn′s′,ns| , (12) IfweneglectH(1) asinstrictone-dimensionalsystems, ns,n′s′ R X Eq.(8)involvesasinglemode n. If,inaddition,thespin axis is chosen along y then the two spin modes uncou- and the relative polarization p (−1≤p≤1), ple and no spin oscillation is allowed; in other directions (x or z) a rigid spin precession should be expected if all G p p= . (13) the contribution between parenthesis in Eq. (10) is as- G sumed constant. This precessionis the underlying work- ing mechanism of the Datta-Das spin transistor.6 Below We shall pay special attention to the multichannel case we investigate the solution of Eq. (8) in the general case consideringenergiesE inEq.(8)suchthatupto10prop- in order to analyze the robustness of the spin precession agating modes are active in the leads. 4 FIG. 3: (Color online) Same as Fig. 2 for polarized leads FIG. 4: (Color online) Same as Fig. 2 for parallel polarized along xbut in antiparallel orientations, i.e., ∆ =10h¯ω and leads along y. ℓ 0 ∆r =−10h¯ω0. with n=1,2,... (red symbols). III. RESULTS FOR SPIN POLARIZED LEADS Figure3containstheresultsforpolarizedleadsalongx but in antiparallel directions. In this case, when α ≈ 0 0 Figure 2 shows the results for polarized leads oriented the conductance vanishes due to the spin valve effect. along x. When H(1) is neglected the conductance for 5 As α0 increases, however, the conductance rises and the R and 10 propagating modes displays an almost sinusoidal spin valve effect is effectively destroyed by the presence behavior with only minor distortions. These deviations, ofthe Rashbadot. For big enoughvalues the systembe- which are enhanced in the single mode case, can be at- havessimilarlytothecaseofparallelpolarizedleads(Fig. tributed to the quantum interference with the Rashba 2), displaying irregular oscillations around a mean value dot.14 The presentresults confirm, therefore,the preces- ≈ Np/2. For strong spin-orbit couplings and high num- sionscenariomentionedabovebutonlywhenthenumber ber of modes no clear distinction between parallel and of modes is large enough and interband coupling is ne- antiparallel orientations is then to be expected. This is glected. Quite remarkably, however, this scenario is not a consequence of the strong subband mixing. In fact, if robust with the inclusion of HR(1). When the full Rashba HR(1) is neglected (red symbols) there is a full correspon- interaction is considered only for small values of α the dence between the conductance nodes of the parallel ge- 0 conductance behaves in a regular way. Very rapidly as ometrywiththemaximaoftheantiparallelone;ascould α increases G fluctuates in a staggeredway that resem- expected from the simplified rigid precession scenario. 0 bles the conductance fluctuations of disordered systems. The above results are not modified if other values of The mean value, in units of G , is ≈0.5N , with N the ∆ are used, provided they are large enough to ensure 0 p p ℓ,r number of active channels, while the amplitude of the full polarization of the leads. The same is true for dis- fluctuation decreases when N increases. tancesd . Theyshouldbelargeenoughtoallowthede- p ℓ,r The existence of the first conductance minimum has cayofevanescentstatesattheinterfaceswiththeRashba been clearly seen in the experiments of Ref. 9. Our re- dotandatthepointswhereStonerfieldsareswitchedon. sultsareinagreementwiththisexperiment,buttheyalso We consider next polarized leads along y and z; that predict that successive maxima and minima are heavily is, in directions that are perpendicular to the quantum distortedorevenfullywashedout. Itisalsoworthnotic- wire. For z polarizations the results are very similar to ingthatthefirstconductanceminimumfortheblackdots the x ones alreadydiscussedandthus willnotbe shown. occurs at a slightly lower value of α than that of the Figures4 and5 containthe results for y-polarizedparal- 0 grey (red color) data, indicating that the minima α lel and antiparallel leads. A first conspicuous difference min are somewhatcontractedwith respect to the simple pre- with the results ofFigs. 2 and3 is thatthe greysymbols diction from the Rashba dot length: 2mℓα = nπ¯h2, (red color) do not display wide sinusoidal oscillations. min 5 confirmed.9 Our results reproduce that behavior (Fig. 4) and they also suggest the antiparallel y orientation (Fig. 5) as an interesting configuration for a spin-orbit- controlled device. Indeed, the initial rise of conductance inthe multichannelcase,interpretedaboveas a Rashba- induced destruction of the spin valve, could be used as the conducting (ON) state of the device. One should check, however, that the evolution of G(α ) from zero 0 to the higher values remains smooth for increasing num- bers ofpropagatingchannels. The presentresultsdo not elucidate this point but they seem to indicate that for N =10 propagating modes the initial rise of G(α ) oc- p 0 curs more rapidly than for N =5. In a future work we p shalltreatthecontinuumcase,havinganinfinitenumber of transverse states, using a different approach from the present one. The results shown above are not much modified if the interfaces with the Stoner fields at distaces d and d to ℓ r the left and right of the Rashba dot, respectively (See Fig. 1), are smoothed by increasing the corresponding Fermi-function parameter.46 This confirms that the con- ductance modifications are an effect of the Rashba dot, and not of the Stoner field interfaces. Indeed, the more FIG.5: (Coloronline)SameasFig.2forpolarizedleadsalong diffuse the interface, the more reflectionless and thus y in antiparallel orientation. more ideal is the description of the contact. In the next section we shall discuss the case of nonpolarized leads (∆ =0), but we have also calculatedsome casesof par- 0 (1) TheconductancewhenH isneglectedisactuallymax- tial polarizationby decreasing ∆ when both s=+ and R 0 imal for the parallel case and stays rather constant with − transverse states are active, although their number is some small oscillations at large α’s that disappear when not perfectly balanced. We have found that the conduc- the number of channels increases. On the other hand, tance is qualitatively similar to the fully polarized case, G vanishes for the antiparallel orientation. We under- with irregular behaviour at large values of α . 0 stand this spin-valve behavior as a complete absence of spin precession, resulting from the fact that H is spin R diagonal in this approximation [cf. Eq. (10)]. IV. RASHBA POLARIZERS IncludingH(1)inthey-polarizedgeometryagainyields R qualitativemodificationsofthelinearconductance(black Ithasbeenrecentlypointedout32,40 thataRashbadot symbols in Figs. 4 and 5). Except for the antiparallel can act as a current polarizer in such a way that when one-channel case, G shows staggering behavior at large a non polarized current enters the dot from the left, the α0’s, quite similarly to the x-polarized results. On av- transmitted current to the right may attain an impor- erage, the conductance is somewhat reduced from the tant degree of spin polarization in y direction. For this maximal value in the parallel case (Fig. 4) and, remark- to occur, it has been shown that at least two propagat- ably, takes a finite value in the antiparallel distribution ing modes of opposite spin must interfere.32,40 In wires (Fig. 5). Forα0 ∼0.2h¯ω0ℓ0 the antiparallelconductance with parabolic transverse confinement this means that has already reached a value close to Np/2 and to the the energy should at least exceed 1.5h¯ω0 such that the eventual saturation value. The Rashba coupling is thus four modes {0+,0−,1+,1−} are active and the inter- quite effective in allowing transmission by flipping spins ference occurs in subsets {0+,1−} and {0−,1+}. The of the polarizedincoming electrons towardsthe opposite resulting spin polarizationis very sensitive to the energy spinorientationoftheoutgoingones. Thesinglechannel (see Fig. 3 of Ref. 40) and a large enhancement of the limit (upper panel of Fig. 5) is obviously an exception polarization p, Eq. (13) is obtained when the energy is since even the black symbols vanish in this case. This suchthataFano-typeresonancewithaquasiboundstate is easily understood noticing that the incident ns = 0+ from a higher evanescent band is formed. This type of mode couples in the Rashba dot with modes 1−,2+,..., resonances which lead to the Fano-Rashba effect was in- but not with 0−, which is the only propagating mode vestigated in Ref. 14. The polarization of the transmit- in the right lead. Therefore, no conduction is possible tedcurrentis zeroif, insteadofy, otherdirectionfor the under this conditions. quantization axis are chosen. Experimentally, the absence of conductance oscilla- The preference for the transverse y direction in polar- tion in the parallel y-oriented configuration has been ization is an example of chirality induced by the Rashba 6 fect connected with the formation of quasibound states that tend to block the current for a given spin direction. When the number of channels is increased (lower panels of Fig. 6) both G and p show reduced staggering oscilla- tions with increasing α, as in Figs. 2-5. There is also an overalltendency to smoothly reduce G and increase p in a linear way with α. With increasing number of chan- nels the slopesof these straightlines are reducedandfor α ≈ 2h¯ω ℓ the polarization reaches the values ≈ 0.2 0 0 0 and ≈ 0.1 for 10 and 20 propagating channels, respec- tively. In almost all cases the polarization is positive, indicating that the transmitted current is preferentially polarized along +y. A. Smooth interfaces In this subsection we discuss how the results are af- fected by the way in which the Rashba field is switched on spatially. For this, we vary the parameter a in the Fermi functions describing the transitions shown in Fig. FIG. 6: (Color online) Conductance G, black symbols with 1.46 For largevalues ofa the edges arequite smoothand left scale, and polarization of transmitted current, grey sym- correspond to an adiabatic turn-on or turn-of in space. bols(redincolor)withrightscale,asafunctionoftheRashba On the contrary, abrupt changes are given by the limit intensity. We have used the same parameters as in Fig. 2, a → 0. Our method is based on a grid discretization of except for the Stoner fields which are here taken to vanish. the variable x and its only requirement is that the grid Upper, intermediate and lower panel correspond to Np = 4, should be fine enough to describe the spatial variations. 10 and 20 propagating modes, respectively. The results discussed above have been obtained using a = 0.1ℓ , a rather small value describing abrupt tran- 0 sitions in space. We have checked that either using a interaction. This is possible even with a time-reversal smaller value a=0.05ℓ or a larger value a=ℓ the be- 0 0 invariantHamiltonianlikeEq.(2)becauseourboundary haviors of the conductance in the presence of polarized condition (left incidence) is not time reversal invariant. leads discussedinSec. II,namely the staggeringfor high Indeed,if weconsiderthe time reversedboundarycondi- values of α and the modification due to intersubband 0 tion, i.e., incidence from the right, the current transmit- coupling, are not qualitatively changed. Of course, it ted to the left is polarizedinthe opposite direction. The should be fulfilled that the Rashba dot length ℓ is much superposition of both solutions completely restores the greater than a in order to still allow the transition to symmetry without any preferred spin direction. The re- reach to the saturation value α . More delicate is the 0 versalofthepolarizationfortheright-to-lefttransmission polarization p discussed in the preceding subsection and can be seen as a peculiar behavior of Rashba polarizers Fig 6. In Fig 7 we show the evolution with a of G and p that makesthem fragile inthe presence ofmagnetic bar- when N =5 channels are propagating in the wire. The p rierslikethoseofSec. III.Indeed,onecouldnaivelythink polarization vanishes when a increases, indicating that thatwhenthe Rashbadotactsasacurrentpolarizerthe smooth edges do not favor the appearance of polarized left-to-righttransmissionwithy-magnetizedleadsshould currents. Inthisdiffuse-edgelimittheconductancetakes beveryhighinparallelconfigurationandverylowinan- themaximalvalueG=N G asinapurelyballisticwire p 0 tiparallel configuration. This is not the case, however, without any Rashba dot. The evolution for α = ¯hω ℓ 0 0 0 because of multiple backwards and forwards reflections (upperpanel)isquitesmoothbutforα =2h¯ω ℓ (lower 0 0 0 with their associated inversions of p (see lower panels of panel) superimposed to the overall behavior we find ir- Figs. 4 and 5). regular maxima and minima as in previous results. In this section we assume nonmagnetic leads by tak- ing ∆ = 0, i.e., vanishing Stoner fields in Fig. 1, and ℓ,r analyze the evolution of the polarization and the con- V. CONCLUSIONS ductance when the number of active channels increases. As shown in Fig. 6 upper panel, high polarizations p are Recent experiments have proved the feasibility of the obtained for the minimal number of channels N = 4 spin transistor proposed by Datta and Das some years p and strong spin-orbit intensities α . The clear correla- ago.6,9 This device, usually presented as a paradigm of 0 tion between G and p, conductance minima correspond spintronics, is expected to open new ways to overcome to maxima in polarization, indicate that this is an ef- present limitations of electronics. In this paper we have 7 (Spain) Grant FIS2008-00781. Appendix A: Resolution method This appendix gives some details of the practical method to solve Eq. (8) and the corresponding bound- ary conditions. We use a method based on the quantum transmitting boundary algorithm.47,48 A fictitious parti- tioning of the system in central and asymptotic regions (contacts)is introduced. The boundariesforthe leftand right contacts are at x and x , respectively. In the con- ℓ r tacts the band amplitudes take the form ψ (x)=a eisckc,ns(x−xc)+b e−isckc,ns(x−xc) , ns c,ns c,ns (A1) where c = ℓ,r is a label referring to left (ℓ) and right FIG. 7: (Color online) Conductance G, black symbols with (r) contacts, respectively, and we defined s = 1 and ℓ left scale, and polarization of transmitted current, grey sym- s = −1. The incident and reflected amplitudes for a r bols (redin color) with right scale, asafunction of thediffu- givenmodensandcontactcaregivenbya andb , sivity a in the Fermi functions describing the spatial transi- c,ns c,ns respectively. Thisexpressionisforapropagatingchannel tionsin Fig. 1. Wehaveused thesame parameters asin Fig. in contact c, for which ε + |∆ | + s∆ < E and its 6,andavalueoftheRashbaintensityα =h¯ω ℓ and2¯hω ℓ n c c 0 0 0 0 0 for the upperand lower panels, respectively. corresponding wavenumber k = 2m∗(E−ε −|∆ |−s∆ )/¯h, (A2) c,ns n c c discussedsome specific aspects relatedto the Rashbain- teraction, including the so-called intersubband coupling, p is a real number. Equation (A1) also applies to evanes- relevantforabetterunderstandingofthephysicalmech- cent modes, ε +|∆ |+s∆ > E, if we assume in this anisms behind the spin transistors and spin polarizers. n c c case a =0 and a purely imaginary wavenumber Taking the wire containing the Rashba dot oriented c,ns along x we have analyzed the transmission in the pres- ence of polarizedleads along x, y or z,and with increas- k =i 2m∗(ε +|∆ |+s∆ −E)/¯h. (A3) c,ns n c c ing number of propagating channels. The cases of par- p allel and antiparallel polarized leads along x and y have Notice that the output amplitudes can be obtained from been explicitly shown. The evolution with Rashba in- the wave function right at the interface, tensity shows dramatic modifications when the Rashba intersubband coupling is included. These modifications b =ψ (x )−a . (A4) are specially relevant at strong values of α , where stag- c,ns ns c c,ns 0 geringoscillationsofGhavebeenfound. Ingeneral,only a first smoothoscillationof G(α ) remains when the full Substituting Eq. (A4) in Eq. (A1) we obtain 0 Rashba interaction is considered, while successive ones areheavilydistortedorevenfully washedout. The spin- ψ (x)−ψ (x )e−isckc,ns(x−xc) = ns ns c valvebehavioriseffectively destroyedby the Rashbadot 2ia sin(s k (x−x )), (A5) and the conductance for both parallel and antiparallel c,ns c c,ns c leads is relatively high. TheroleofRashbadotsasspinpolarizershasbeendis- that is the quantum-transmitting-boundaryequation for cussedandexplicitly calculatedassumingthe leadstobe the contacts. nonpolarized. AsmoothlinearincreaseinpwithRashba Equations(8)and(A5),forthecentralandcontactre- intensity has been observed in the multichannel case. In gions,respectively,formaclosedsetthatdoesnotinvoke the limit of adiabatic transitions the polarization van- the wave function at any external point. Of course, this ishes. These overall smooth behaviors are superimposed is not true for any of these two subsets separately, since by irregular changes for high values of α0. central and contact regions are connected through the derivativeinEq.(8)andofψ (x )inEq.(A5). Inprac- ns c tice,weuseauniformgridinxwithn-pointformulaefor Acknowledgments the derivatives (n ≈5−11) and truncate the expansion in transverse bands, Eq. (4), to include typically 30-60 Useful discussions with M.-S. Choi are gratefully ac- terms. 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