Charge and spin current in a quasi-one-dimensional quantum wire with spin-orbit coupling
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Charge and spin current in a quasi-one-dimensional quantum wire with spin-orbitcoupling
K. E. Nagaev
1, 2 and A. S. Goremykina
1, 2 Kotelnikov Institute of Radioengineering and Electronics, Mokhovaya 11-7, Moscow, 125009 Russia Moscow Institute of Physics and Technology, Institutsky per. 9, Dolgoprudny, 141700 Russia (Dated: April 26, 2018)We show that Rashba spin-orbit coupling may result in an energy gap in the spectrum of electronsin a two-mode quantum wire if a suitable confining potential is chosen. This leads to a dip inthe conductance and a spike in the spin current at the corresponding position of the Fermi level.Therefore one may control the charge and spin currents by means of electrostatic gates withoutusing magnetic field or magnetic materials.
PACS numbers: 72.25.-b, 73.23.-b, 73.63.Rt
I. INTRODUCTION
Spintronics, or spin electronics, involves the study ofactive control and manipulation of spin degrees of free-dom in solid-state systems and is a rapidly growing fieldof science. The key purpose of these studies is the gen-eration, control, and manipulation of spin-polarized cur-rents. A useful tool for achieving this goal is the spin-orbit interaction, which couples the spin of an electronwith its spatial motion in a presence of a certain asym-metry of the conductor. For example, Rashba spin-orbitinteraction is due to a lack of inversion symmetry in semi-conductor heterostructures such as InAs or GaAs. Theadvantage of this type of interaction is that it can betuned by means of electrostatic gates.
In truly single-mode quantum channels, spin-orbit in-teraction alone neither changes the electric current norresults in a spin current if no magnetic field or magneticmaterials are involved. In this case, spin-orbit interac-tion does not change the energy-band topology and canbe simply eliminated from the Hamiltonian by means ofa unitary transform. A prototypical scheme of a spin field-effect transistorbased on Rashba interaction and single-mode ballisticchannel with ferromagnetic electrodes was proposed byDatta and Das more than two decades ago. Recently,such a device was experimentally realized. The current in a single-mode quantum channel also de-pends on the spin-orbit interaction if a magnetic field isapplied parallel to the channel or normally to the planeof the heterostructure (i.e. in the direction of Rashbafield). The interplay of the spin-orbit interaction withmagnetic field significantly modifies the band structureand produces an energy gap in the spectrum togetherwith additional subband extrema. This results in a de-crease in the charge current and a net spin current as theFermi level passes through the gap. These effects wererecently experimentally observed by Quay et al. Many authors studied spin and charge transport inmultimode quantum channels in the absence of magneticfield or magnetic ordering. Governale and Z¨ulicke con-sidered a long channel with parabolic confinement po- tential and took into account the mixing of differenttransverse-quantization subbands by the spin-orbit inter-action. This mixing results in an asymmetric distortionof the dispersion curves but does not open any gaps in thespectrum. As a consequence, the spin-orbit interactionin a presence of voltage drop across the channel resultsin a spin accumulation inside the channel but does notlead to a spin current or deviations from the standardconductance quantization. There is also a number of nu-merical calculations of the spin current, but thesepapers deal with stepwise constrictions and the resultsare obscured by the interference effects. A more realisticgeometry of a saddle-point contact in two-dimensionalpotential landscape was considered in Ref. 15, but theRashba interaction was taken into account there as a per-turbation. In Refs. 16 and 17, a quasi-one-dimensionalwire with localized region of Rashba interaction was con-sidered and nonzero spin current was predicted for suf-ficiently sharp boundaries of the region. Unusual tra-jectories were revealed by Silvestrov and Mishchenko within the quasiclassical approach to exist near these re-gions. However in all the above papers, the spin currentand the deviations from perfect conductance quantiza-tion are related with the mixing of subbands in the tran-sition areas between the quantum contact and reservoirsby spin-orbit interaction and crucially depend on the ge-ometry and properties of these regions. It is hard to seeany general regularities concerning the magnitude of theeffect.In this paper, we propose a mechanism of spin currentgeneration that relies on the energy band structure deepin the wire rather than on the reflection effects in thetransition areas and leads to 100% spin-polarized currentat definite positions of the Fermi level. This mechanism isreminiscent of the one in Ref. 8 but requires no magneticfield. II. THE MODEL
Consider a quasi-one-dimensional conducting channelformed in two-dimensional electron gas by means of elec-
FIG. 1. Dispersion curves for the two lowest subbands of aquantum wire with spin-orbit interaction and parabolic trans-verse confinement. The dispersion curves are distorted bylevel crossing but exhibit no local maxima.The color of thecurve designates the dominant spin projection. trostatic gates. The transition between the reservoirs andthe channel is assumed to be adiabatic, and the length ofthe channel is much larger than that of the transition re-gions. We assume that the Rashba spin-orbit interactionis present in the channel but absent in the reservoirs, sothe spin current through the system is well-defined. TheHamiltonian of the system is of the formˆ H = ˆ p x m + ˆ p z m + U ( x, z )+ α ( x ) ~ (ˆ p x ˆ σ z − ˆ p z ˆ σ x ) − i ∂α∂x ˆ σ z , (1)where U ( x, z ) is the confining potential and α ( x ) is theparameter of spin-orbit coupling. Both quantities aresmooth functions of the longitudinal coordinate x thatare constant almost throughout the whole length of chan-nel and vanish in the reservoirs. It is now straightforwardto make use of the adiabatic approximation and introducea complete set of of eigenfunctions ϕ n ( x, z ) and eigenen-ergies ε n corresponding to the transverse motion of elec-trons in the z direction. This leads to a set of coupledequations of the form (cid:20) ˆ p x m + α ( x ) ~ ˆ σ z ˆ p x − i dαdx ˆ σ z + ε m (cid:21) ¯ ψ m ( x ) − α ( x ) ~ ˆ σ x X n h m | p z | n i ¯ ψ n ( x ) = ε ¯ ψ m ( x ) (2)for the spinors ¯ ψ n = ( u n , v n ) T that describe the longi-tudinal dependence of the spin-up and spin-down ampli-tudes of wave-function in the n -th transverse quantummode.If the matrix elements of transverse momentum be-tween different modes ϕ m and ϕ n were zero, the twofoldspin degeneracy of these modes would be lifted by thespin-orbit interaction and one should see two sets of FIG. 2. The system under consideration. The current flowsin the x direction, and the negative voltage at the additionalmiddle gate changes the confining potential from one-well to adouble-well shape. As the negative voltage increases, a max-imum appears in the lower dispersion curve. parabolic dispersion curves shifted along the k x axis thatwould correspond to the two possible projections of spinon the z axis. The curves of each set would have minimaat k x = ± mα/ ~ and intersect without affecting eachother.Nonzero matrix elements h m | p z | n i result in anticross-ing of the dispersion curves with different n and spinprojection and lead to an asymmetric distortion of them(see Fig. 1). However this does not give rise to new max-ima and minima in these curves for the case of standardparabolic confining potential. The reason is that thelevels of transverse quantization are evenly spaced andit is impossible to isolate a pair of them with a smallseparation. In other words, the vertical separation ofanticrossing curves is too large as compared with theirhorizontal shifts.The failure of the approximate two-band model thatpredicts a nonmonotonic behavior of the curves may beunderstood as follows. In the absence of band mixing, thetwo curves corresponding to two subsequent transverse-quantization levels and different spin projection wouldcross at k x = ∆ ε/ α , where ∆ ε = ε n +1 − ε n . To form amaximum, the crossing branches of these curves shouldhave different signs of slope k x + 2 mα/ ~ > k x − mα/ ~ < ε < mα / ~ . On the other hand, the bandmixing term α h n + 1 | p z | n i / ~ would lead to the splittingof the curves at the crossing point of the order of Ω ∼√ m ∆ ε α/ ~ . The two-band model is justified only if Ω ≪ ∆ ε , i.e. ∆ ε ≫ mα / ~ , which is incompatible with theprevious condition. Exact calculations show that allthe dispersion curves have only one minimum and hencethe dependence of the conductance of the channel on theFermi energy exhibits only the conventional 2 e /h steps,while the spin current is absent. FIG. 3. Dispersion curves for a pair of closely spaced energylevels with small matrix element of transverse momentum in aquantum wire with spin-orbit interaction. The level crossingresults in appearance of local maxima in the lower curves.
Things become different if the confinement is non-parabolic. Consider, e. g., the system in which U ( z ) hasthe shape of a double potential well. Such a potentialmay be formed by means of a negatively biased centralgate on top of the quantum wire (see Fig. 2). In the caseof a high impenetrable barrier between the wells, each ofthem would possess the same set of energy levels, so thelevels of the whole system would be doubly degenerate. Afinite tunneling through the barrier lifts the degeneracy,and therefore one gets a set of pairs of levels with verysmall spacings inside the pair. In this case, the two-bandmodel is justified and taking into account only the twolowest levels, one obtains the following expression for theresulting dispersion curves ε = ~ k x m + ε + ε ± p ( ε − ε ± αk x ) + 4 | p | α / ~ , (3)where p = h | p z | i . The upper sign at 2 αk x underthe square root corresponds to the mixture of | ↑i and | ↓i states, and the lower sign corresponds to the mix-ture of | ↓i and | ↑i states. The two pairs of the re-sulting curves are symmetric with respect to k x = 0.The lower dispersion curve may have one or two minimaas a function of k x depending on the relations between∆ ε = ε − ε , p , and α (see Fig. 3). The upper min-imum disappears by merging with the local maximum,i. e. when the points where dε/dk x = 0 and d ε/dk x = 0coincide. Therefore it follows from Eq. (3) that the sec-ond minimum exists if (cid:12)(cid:12)(cid:12)(cid:12) ~ p mα (cid:12)(cid:12)(cid:12)(cid:12) / + (cid:18) ~ ∆ ε mα (cid:19) / < . (4)Apparently one can meet this condition by making theoverlap of the wave functions in the two wells sufficiently small. For example, if a square quantum well with in-finitely high external walls is symmetrically cut by a δ -like barrier in the middle, both ∆ ε and | p | are inverselyproportional to the effective strength of the barrier k . III. THE CONDUCTANCE
The existence of a local maximum in the lower pair ofthe dispersion curves leads to significant changes in theconductance of the wire. If the Fermi level lies betweenthe lower and upper minima µ and µ in the disper-sion curves (see Fig. 3), it intersects two branches withpositive (negative) group velocity that correspond to thetwo different spin projections in the z direction, and theconductance is 2 e /h , while the spin current is absent. Ifthe Fermi level lies between the upper minimum µ andthe local maximum µ in the lower curves or above theminimum in the two upper dispersion curves µ , it inter-sect two branches with positive (negative) group velocityand one spin projection and two branches with positive(negative) group velocity with the other spin projection.This results in the 4 e /h conductance and yields no spincurrent. However if the Fermi level falls within the gapbetween the local maximum µ in the lower curves andthe minimum µ in the upper curves, it intersect twobranches with positive group velocities and one spin pro-jection and two branches with negative velocities and theother spin projection. Therefore in the case of a suffi-ciently long wire the conductance exhibits a dip to 2 e /h where the current is 100% spin polarized.To calculate the current through the wire, we use theLandauer - B¨uttiker formula for the zero-temperaturetotal electric conductance G = e h X n L ,n R ,σ L ,σ R | t n R σ R ,n L σ L | (5)where t n R σ R ,n L σ L are the transmission amplitudes fromthe state in transverse mode n L with spin projection σ L in the left lead to the state in the mode n R with spinprojection σ R in the right lead. The spin conductance G sz = I s /V with respect to the z axis is given by G sz = − e π X n L ,n R ,σ L (cid:0) t ∗ n R ↑ ,n L σ L t n R ↑ ,n L σ L − t ∗ n R ↓ ,n L σ L t n R ↓ ,n L σ L (cid:1) . (6)In general, the transmission amplitudes t n R σ R ,n L σ L canbe calculated only numerically. Analytical results may beobtained for the particular case of strong and nearly con-stant spin-orbit interaction if one neglects the reflectionfrom the boundary regions where the interaction and theconfining potential vanish. This is possible if both quan-tities go to zero in the leads sufficiently smoothly. Tomake this evident, we perform a unitary transformationof the Hamiltonian with matrix ˆ S ( x ) = exp[ − i ˆ σ z ξ ( x ) / ,ξ ( x ) = 2 m ~ Z x −∞ dx ′ α ( x ′ ) , (7)which eliminates the term linear in ˆ p x in it and bringsEqs. (2) to the form d ¯ ψ dx + ( m α ~ − + ∆ k ) ¯ ψ = − mα ( x ) p ~ − (ˆ σ x cos ξ − ˆ σ y sin ξ ) ¯ ψ , (8a) d ¯ ψ dx + ( m α ~ − + ∆ k ) ¯ ψ = − mα ( x ) p ∗ ~ − (ˆ σ x cos ξ − ˆ σ y sin ξ ) ¯ ψ , (8b)where ∆ k , = 2 m ( ε − ε , ) / ~ . Even though α and U are smooth functions of x , the right-hand sides of equa-tions (8) contains rapidly oscillating functions cos ξ andsin ξ that lead to interband scattering. These equationsare similar to those of mechanical parametric resonance and can be solved in a similar way. If the detuning inboth bands is small and the interband coupling is weak,i. e. ∆ k , ≪ m α / ~ and | p | ≪ mα/ ~ , the coupledcomponents of the wave function may be presented in theform u = A ( x ) e iξ/ + B e − iξ/ , (9a) v = C ( x ) e iξ/ + D e − iξ/ , (9b)where A , B , C , and D are amplitudes that slowlyvary on the scale of ~ / ( mα ). Substituting Eqs. (9b)into (8), neglecting the second derivatives of slowly vary-ing quantities and collecting the terms proportional toexp( ± iξ/
2) leads to a system of first-order equations2 imα ~ dA dx = − ∆ k A − mαp ~ D , (10a)2 imα ~ dB dx = ∆ k B , (10b)2 imα ~ dC dx = − ∆ k C , (10c)2 imα ~ dD dx = ∆ k D + 2 mαp ∗ ~ A . (10d)While the standalone Eqs. (10b) and (10c) have purelyoscillating solutions for any choice of parameters, the so-lutions of coupled equations (10a) and (10d) may expo-nentially grow or decay. If we assume them to be propor-tional to e sx , one easily finds the roots of characteristicequations of system (10a) - (10d) s , = i ~ ∆ k − ∆ k mα ± κ,κ = s(cid:12)(cid:12)(cid:12) p ~ (cid:12)(cid:12)(cid:12) − ~ (cid:18) ∆ k + ∆ k mα (cid:19) . (11) FIG. 4. The dependence of normalized conductance andspin current on the Fermi energy. The upper and lowersolid curves present G ( ε ) and I s numerically calculated for | p | = 0 . mα/ ~ , ∆ ε = 0 . mα / ~ , and L = 20 ~ /mα .The dashed curve shows G ( ε ) calculated by means of Eq. (12). Solving Eqs. (10a) - (10d) and similar equations for u and v results in the transmission amplitudes from theleft to the right | t ↑ , ↑ | = | t ↑ , ↑ | = 16 m α | p | − ~ (∆ k + ∆ k ) m α | p | cosh ( κL ) − ~ (∆ k + ∆ k ) (12)with | t ↓ , ↓ | = | t ↓ , ↓ | = 1 and zero spin-mixing orband-mixing transmission amplitudes. A substitution ofthese amplitudes into Eqs. (5) and (6) suggests thatthe electric conductance as a function of the Fermi en-ergy has a dip at the second quantization plateau, whichcorresponds to a spike in the spin current. This is dueto blocking of the current from the left to the right forspin-up electrons inside the gap in the spectrum. In thestrong-interaction approximation, the dip is centered at ε = ( ε + ε ) / α | p | / ~ .Figure 4 shows the calculated electric conductance andspin current for | p | = 0 . mα/ ~ , ∆ ε = 0 . mα / ~ ,and L = 20 ~ /mα . Solid lines show the values obtainedby a numerical solution of Eqs. (8) with account takenof spatial variations of ε , and α , and the dashed lineshows analytical results calculated by means of Eq. (12).Both the numerically calculated conductance and spincurrent exhibit an oscillatory behavior as the Fermi levelapproaches the spectrum gap from below. This behavioris explained by quantum interference effects that arisedue to the reflections of electrons from the ceiling of theallowed band at the edges of the wire where it goes down(see Fig. 5). The amplitude of the oscillations increasesas the gap is approached because the reflection amplitudeincreases. FIG. 5. Spatial dependence of the gap in the spectrum. Thelower curve shows the position of the maximum in the lowerdispersion curve, and the upper curve shows the position ofthe minimum in the upper one. Electrons experience partialreflections at the points where the Fermi level crosses the gapin the spectrum.
IV. CONCLUSION
We have shown that Rashba spin-orbit interaction mayopen additional gaps in the spectrum of a multichan-nel quantum wire if the transverse confining potential ischosen appropriately. This happens if the energy lev- els of transverse quantization come in pairs and the ma-trix elements of transverse momentum between the cor-responding states is sufficiently small. In this case, theconductance of the wire exhibits a dip and the spin cur-rent exhibits a spike inside the gap. If the contact issufficiently long, the conductance in the dip drops from4 e /h to 2 e /h and the current is fully spin-polarized inthe transverse in-plane direction. This effect may be usedfor designing an all-electrical spin transistor. By apply-ing a negative voltage to the middle longitudinal gate,one may increase the degree of spin polarization of thecurrent from zero to 100% if the Fermi level is adjustedappropriately.One of the main advantages of using electric bias forspin control is the ability to make it time-dependent.This can lead to non-trivial effects in the transport. In the future, it would be of interest to study the effectsof time-periodic bias in our model.
ACKNOWLEDGMENTS
This work was supported by Russian Foundation forBasic Research, grant 13-02-01238-a, and by the programof Russian Academy of Sciences. I. ˘Zuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323 (2004). E. I. Rashba, Fiz. Tverd. Tela (Leningrad) , 1224 (1960)[Sov. Phys. Solid State , 1109 (1960)]. J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.Rev. Lett. , 1335 (1997). G. Engels, J. Lange, Th. Sch¨apers, and H. L¨uth, Phys.Rev. B , R1958 (1997). L. S. Levitov and E. I. Rashba, Phys. Rev. B S. Datta and B. Das, Appl. Phys. Lett. , 665 (1990). H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, andM. Johnson, Science , 1515 (2009). Y. V. Pershin, J. A. Nesteroff, and V. Privman, Phys. Rev.B , 121306 (2004). C. H. L. Quay, T. L. Hughes, J. A. Sulpizio, L. N. Pfeiffer,K. W. Baldwin, K. W. West, D. Goldhaber-Gordon andR. de Picciotto, Nature Physics , 336 (2010). M. Governale and U. Z¨ulicke, Phys. Rev. B ,073311(2002); Solid State Commun. , 581 (2004). M. Eto, T. Hayashi, and Y. Kurotani, J. Phys. Soc. Jpn. , 1934 (2005). F. Zhai and H. Q. Xu, Phys. Rev. B , 035306 (2007). J.-F. Liu, Zh.-Ch. Zhong, L. Chen, D. Li, Ch. Zhang, andZh. Ma, Phys. Rev. B , 195304 (2007). F. Zhai, K. Chang, and H. Q. Xu, Appl. Phys. Lett. ,102111 (2008). V. A. Sablikov, Phys. Rev. B , 115301 (2010). D. S´anchez and L. Serra, Phys. Rev. B , 153313 (2006). M. M. Gelabert, L. Serra, D. S´anchez, and R. L´opez, Phys.Rev. B , 165317 (2010). P. G. Silvestrov and E. G. Mishchenko, Phys. Rev. B The existence of maxima in the dispersion curves was pre-dicted by A. V. Moroz and C. H. W. Barnes, Phys. Rev.B , 14272 (1999). However subsequent papers of otherauthors found no maxima in them. M. B¨uttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys.Rev. B , 6207 (1985). F. Zhai and H. Q. Xu, Phys. Rev. Lett. , 246601 (2005). L.D. Landau, E.M. Lifshitz Mechanics. Vol. 1 (3rd ed.).Butterworth-Heinemann (1976). This corresponds to L = 220 nm in AlGaAs/GaAs het-erostructures with α = 4 × − eV · cm and to L = 18 nmin InAs quantum wells with α = 3 × − eV · cm. Almas F. Sadreev and E. Ya. Sherman, Phys. Rev. B88