Polarized Electric Current in Semiclassical Transport with Spin-Orbit Interaction

8/13/2019 Polarized Electric Current in Semiclassical Transport with Spin-Orbit Interaction

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Polarized Electric Current in Semiclassical Transport with Spin-Orbit Interaction

P. G. Silvestrov1,2 and E. G. Mishchenko3

1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands2Theoretische Physik III, Ruhr-Universitat Bochum, 44780 Bochum, Germany

3Department of Physics, University of Utah, Salt Lake City, UT 84112

Semiclassical solutions of two-dimensional Schrodinger equation with spin-orbit interaction andsmooth potential are considered. In the leading order, spin polarization is in-plane and follows theevolution of the electron momentum for a given subband. Out-of-plane spin polarization appearsas a quantum correction, for which an explicit expression is obtained. We demonstrate how spin-polarized currents can be achieved with the help of a barrier or quantum point contact open fortransmission only in the lower subband.

PACS numbers: 72.25.-b, 73.23.-b, 72.25.Hg, 03.65.Sq

I. INTRODUCTION

Achieving spin manipulation in nanodevices by meansof electric fields (without using less selective magneticfields) represents the ultimate goal of spintronics. Spin-

orbit interaction, which couples electron momentum toits spin, is one of the most promising tools for realiz-ing spin-polarized transport[1,2]. Several schemes lead-ing either to spin accumulation or to polarization of thetransmitted current induced by the spin-orbit interac-tion have been put forward. Predictions of electric fieldinduced spin-accumulation at the boundaries of a sam-ple, which originates from asymmetric scattering fromimpurities[3,4] (extrinsic spin-Hall effect) or from spin-orbit split band structure[5,6] (intrinsic effect), has re-cently reached a stage of experimental realization[7]. In-plane bulk spin polarization appears in the electric fieldin two-dimensional systems with broken inversion sym-metry [8]. Spin polarization in quantum wires with lowcarrier density has been shown to occur due to the in-terfaces of spin-degenerate and spin-split regions[9]. In-terfaces between two-dimensional regions with differentspin-orbit splitting have also been used for that purpose,in the case of a sharp[10,11] or an arbitrary [12] inter-face, as was the scattering from a sample edge [13, 14].Other proposals include polarization due to tunnelingthrough a double-barrier structure [15, 16] and tunnel-ing between two quantum wires[17]. Reference[18] sug-gested a three-terminal device with a spin-orbit split cen-tral region as a spin filter, which was numerically testedby Refs. [19] and [20]. Reference [21] pointed to a pos-sibility of generating spin-polarized currents by utilizing

crossings of spin-orbit-split subbands belonging to differ-ent transverse channels. These proposals are still lackingexperimental realization.

In the present paper we suggest a way to polarize elec-tric currents by passing them through a region where, byincreasing the external electrostatic potential, the upperspin-orbit-split subband is locally positioned above theFermi level. The proposed method utilizes electric gat-ing whose effect is two fold: (i) it completely suppressestransmission via the upper spin-orbit-split subband, and

(ii) it allows transmission only in a narrow interval of in-cident angles in the lower subband. In contrast to theproposals which advocate strong variations of the spin-orbit coupling and, thus, rely on strong gate voltages,our method requires only weak potentials of the orderof a few millivolts (which is a typical scale of the Fermi

energy). In addition, we predict a specific pinch-off be-havior of the conductance, which would allow to detectpolarized currents without actual measurement of spin.

We consider ballistic electron transport in gated two-dimensional electron gas with the Hamiltonian

H= p2

2m+ (pyx pxy) + m

2

2 + V(x, y). (1)

For the sake of simplicity we concentrate on the case ofthe Rashba spin-orbit interaction (the same method,however, can be used for more complicated interactions).

Construction of semiclassical solutions of the Schrodingerequation with the Hamiltonian(1) follows the reasoningof the conventional WKB approach[22,23,24,25], whichis valid for a smooth potential, h|V| min(p3/m,p2).The advantages of semiclassics are twofold. First, it al-lows us to obtain approximate analytical solutions forotherwise complicated problems. Second, as we will see,it turns out to be especially simple to achieve strongpolarization of electron transmission in the semiclassicalregime.

The Mexican hat shape of the effective kinetic energyin the case of spin-orbit interaction leads to a variety ofunusual classical trajectories (see Fig. 2 below), whichhave never been investigated before. Our approach em-ploys strong spin-orbit interaction (or smooth externalpotential) sufficient to affect individual electron trajec-tories, in contrast to previous semiclassical treatments[26,27] which consider spin-orbit interaction as a pertur-bation. Still we do not require the spin-orbit interactionto be comparable with the bulk value of the Fermi energy.To produce spin-polarized current, it will be sufficient tomake spin-orbit interaction comparable with the kineticenergy at some particular area of the system, for exam-ple, near the pinch-off of a quantum point contact.
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II. SEMICLASSICAL WAVE FUNCTION

Without the external potential V, the electron spec-trum consists of the two subbands, E(px, py) = (p m)2/2m. The subbands meet at only one point, p = 0,and the spin in each subband is always aligned with oneof the in-plane directions perpendicular to the momen-tump. The semiclassical electron dynamics[22] naturally

captures the essential features of this translationally in-variant limit. The classical motion in each subband isdetermined by the equations of motion which follow fromthe effective Hamiltonian:

Heff=(p m)2

2m + V(x, y). (2)

Despite the fact that spin does not appear in this equa-tion, one can easily construct semiclassical wave func-tions, which have spin pointed within the xy plane per-pendicular to the momentum:

0= ueiS/h, u=

2p

py+ ipx

py ipx

. (3)

Here the action S is related to the momentum by p =S, and = uu is the classical density for a familyof classical trajectories corresponding to a given energyE. The action S obeys the classical Hamilton-Jacobiequation, (|S| m)2/2m+ V = E. Application ofthe Hamiltonian (1) to the approximate wave function0 gives, after some algebra,

H0 = E0 ih2

( v) 0+ hFz0, (4)

where (summation over repeating indices is assumed)

F = p m2mp3 (pypiipxpxpiipy)pyx pxy2p . (5)

The second term in the rhs of Eq.(4) vanishes due to thecontinuity equation

v= 0, v= p/m p/p. (6)The last ( z) term in (4) indicates that the spin ofan accelerated electron cannot exactly stay in the planeof propagation and acquires a small hV projectiononto the z axis. To take into account this out-of-plainspin precession one has to go beyond the approximationof Eq. (3), which is done by

= (1 + hfz)0. (7)

Since (H E)f z0 =2pfz0, to the lowest orderin h, one can relate the functions F and f

f= F/2p, (8)and find the out-of-plane spin density [F is found fromEq.(5)]

z= h2p

F. (9)

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5

0

0.5

1

1.5

2

G[e

2/h]

2 /m2

G

FIG. 1: Conductance (in units ofe2/h), and spin polarizationof the current vs gate voltage (in units ofm2/2). Dashedlines show the smoothed curves (14,15), solid lines show thequantized values for mL/h = 10.5. Dotted line shows theconductance without spin-orbit interaction.

Note that Eq. (9) does not describe the nonadiabatictransitions between subbands. After the electron leaves

the region with nonzero potential gradient, V= 0, thein-plain spin orientation is restored.

The out-of-plane polarization of the electron flow inthe external potential is a subject of the rapidly develop-ing field of the spin-Hall effect [3,4,5,6,7]. Our result,Eqs. (5) and (9), incorporates previous calculations ofRef.[6] which were restricted to the one-dimensional formof the potential,V(x), with py being the integral of mo-tion. The validity of Eq. (9), however, is not restricted toa simple one-dimensional case and describes the out-of-plain polarization foranysmooth two-dimensional poten-tial (including confining potentials which create quantumwires, quantum dots, etc.). In particular, Eq. (9) mayserve as a good starting point for an analytical calcula-tion of the edge spin accumulation in ballistic quantumwires [28, 29]. We leave further investigation of theseinteresting effects for subsequent research.

Solutions of the form, Eq. (3), have clear and im-portant consequences. During its motion, an electronchanges the momentumpbut always remains in the samespin-subband. To change the subband the electron tra-

jectory should pass through the degeneracy point whereboth components of momentum vanish simultaneously,p = 0, which is generically impossible. Moreover, withthe proper use of potential barriers, one may realize asituation where electrons of only one subbandare trans-mitted and the others are totally reflected. This leads to

strong polarization of the transmitted electron flow.

III. SHARVIN CONDUCTANCE

To give an example of such a spin-polarized current letus consider transmission through a barrier,V(x), varyingalong the direction of a current propagation. We assumeperiodic boundary conditions in the perpendicular direc-tion (y +L y). As such a condition makes py the


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integral of motion, mixing of orbital channels, which isstrongly suppressed for generic smooth potential (2), isnow absent exactly. For a smooth potentialV(x) the con-duction channels may either be perfectly transmitting orcompletely closed. The conserved transverse momentumtakes the quantized values, pny = 2hn/L. Consider thefunctions

En(px) =(pn

m)2

2m , pn =

p2x+pny 2 (10)

For n= 0 the function En(px) splits into two distinctbranches. At any point x the equation

En(px) = EF V(x) (11)

yields solutions pLx and pRx , corresponding to left- and

right-moving electrons. Application of a small bias im-plies, e.g., the excess of right movers over left movers farto the left from the barrier. Particles are transmittedfreely above the barrier if Eq. (11) has a solution, pRx , foranyx. Let = EF Vmax be the difference between theFermi energy and the maximum of the potential. Thenth channel in the upper branch opens when

= (2h|n| + mL)2 /2mL2. (12)For the lower branch En(px) Eq. (11)has four solutions(two for right and two for left movers) for |n| < mL/2hand x close to the top of the barrier. However, far fromthe barrier (where the excess of right-movers is created)there are still only two crossings described by Eq. (11),one for right and one for left movers. As a result, all theextra electrons injected atx= follow the evolutionof a solution of Eq. (11) with the largest positivepx. Forall |n| < mL/2hsuch a solution does exist for any pos-itive. Thus, at = 0 as many as n0 = mL/h chan-nels open up simultaneously. The channels with highervalues|n| > mL/2h in the lower subband En openwhen

= (2h|n| mL)2 /2mL2. (13)According to the Landauer formula, ballistic conductanceis given by the total number of open channels multipliedby the conductance quantum G0= e

2/h

G= G0L

h

2m + m, 0< < m2/2

2

2m, > m2/2. (14)

This dependence G() is shown in Fig. 1. The strik-ing evidence of the presence of spin-orbit interaction isthe huge jump of the conductance at the pinch-off point,as opposed to the conventional square-root increase inthe absence of spin-orbit coupling. This jump is a con-sequence of the Mexican-hat shape of the spectrumE(px, py). Accuracy of Eqs. (12) and (13) is sufficientto resolve the steps in the conductance due to the discretevalues of|n| = 0, 1, 2,..., (conductance quantization), asshown in Fig. 1. The steps in G() are abrupt in thelimit dV/dx 0.

Close to the pinch-off, at


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IV. QUANTUM POINT CONTACT

Let us consider probably the most experimentally rel-evant example of a quantum point contact, described bythe potential

V(x, y) = m2x2

2 +

m2y2

2 . (16)

We will see that even in this simple model the electronflow in the presence of spin-orbit interaction acquires anumber of interesting and peculiar features. Classicalequations of motion follow in the usual manner from theeffective Hamiltonian (2): r= Heff/p, p= Heff/r.We consider quantum point contact (QPC) close to theopening with only the lower E subband contributing tothe conductance. A crucial property of the HamiltonianHeff, Eq. (2), is the existence of a circle of minima of thekinetic energy at |p| =m. Expanding around a point onthis circle, px0 = m cos , py0 = m sin , one readilyfinds the equations of motion for P=px cos +pysin m

m,

P+ (2 cos 2 + 2 sin 2)P= 0, = 0. (17)The trajectory is found from the relations, x =Pcos /m , y =Psin /m. We observe from Eq. (17)that only the trajectories within the angle

tan || 0, and | | < 0,P< 0.The appearance of the latter range of integration is highly

non-trivial. A simple reasoning shows that the particleswith the velocity antiparallel to the momentum (vx> 0,

px < 0) should not contribute to the conduction in thecase of a transition through a one-dimensional barrierV = V(x), see Eq. (14). Despite corresponding to theright-moving electrons, these states do not originate inthe left lead. Indeed, they exist only in the vicinity ofx = 0, but disappear as x and, thus, cannotbe populated by the excess electrons (except due to thetunneling transitions which are irrelevant in the semi-classical regime). Such trajectories, however,do exist inQPC, Eq. (16), as demonstrated in Fig. 2. After passingthrough QPC the trajectory bounces at the wall reversingits velocity. This kind of classical turning points, whereboth components of the velocity vanish simultaneously,are specific for the effective Hamiltonian (2). The exis-tence of transmitting trajectories with || < 0, < 0results in the doubling of the conductance. Simple cal-culation yields

G= G04m sin 0

h

2

m

. (21)

The presence of a threshold angle 0, as well as thesquare-root dependence ofG(), are in a sharp contrastto the well-known result G = G0/h, in the absenceof spin-orbit interaction.

Equation(21) is valid in the case of many open chan-nels. Since Eq. (17) describes only the linearized elec-tron dynamics, Eq. (21) is formally valid if m2.Nevertheless, the current remains totally polarized for0< < m2/2 [similar to Eq. (15)]

y = yvx/vx = 1. (22)

With increasing the chemical potential, > m

2

/2,transmission via the upper subbandE+ kicks in and thedegree of polarization gradually decreases, similarly toEq. (15), though with different, more complicated, de-pendence of spin-polarization on. Note that transmis-sion of different orbital channels through QPC is indepen-dent as long as the confining potential (16) is smooth overa distance of the characteristic spin-orbit length h/m.It is easy to see that this requirement is equivalent to thecondition that (, ) m2/h. This is also a conditionof large conductance G G0.

V. DISCUSSION

In both analyzed systems (of ballistic Sharvin conduc-tance and of QPC) polarization of current is achievedwhen many channels are transmitting. As a consequenceof the Kramers degeneracy, transmission eigenvalues al-ways appear in pairs in the presence of time-reversal sym-metry, leading to the prohibition of the spin-current inthe lowest (n = 0) conducting channels (cf. Ref. [18]).In the case of higher channels, however, the degeneratetransmission eigenvalues belong to the same spin-orbit


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y

x

a

b c

FIG. 2: Three kinds of trajectories in the point contact. a,transmitted trajectory whose momentum is always collinearwith the velocity. b, trajectory bouncing inside the QPC.This trajectory is periodic in the linearized approximationdescribed in the text, while the exact calculation for finite am-plitude shows its slow drift. c, transmitted trajectory whosemomentum inside the contact is opposite to the velocity. Elec-trons flow from left to right. Arrows show momentum andspin orientations. Few equipotential lines are also shown.

subband and carry, respectively, the same spin polariza-tion. For example, in the case of the QPC any trans-mitted trajectoryx(t), y(t) (e.g., one of the two shown inFig. 2) is accompanied by its mirror reflectionx(t), y(t)with identical transmission.

In InAs-based heterostructures, typical value of spin-orbit coupling [36] is h = 2 1011eV m. Character-

istic spin-orbit length lR = h/m = 100 nm and en-

ergym2/2 = 0.1 meV. In order to have strongly spin-polarizing QPC, the latter should support many trans-mitting channels at chemical potential m2/2h. This condition can, equivalently, be written in termsof the width of the point contact y, see Eq. (16), asy lR. This is a realistic condition for typical ballis-tic constrictions.

To conclude, we have proposed a way to polarize cur-rents in the ballistic regime by means of using electricgates to suppress transmission in the upper spin-orbit-split subband. The polarization is stronger when thereare many transmitting channels in the lower subband.This is exactly the condition when the semiclassical ex-pansion in powers of h is applicable. An obvious advan-tage of our scheme is that we do not require the spatialmodulation of the strength of spin-orbit interaction. Nei-ther do we need a restricted angle of incident electronsin order to have a polarized current.

Acknowledgments

We have benefited from discussions withG.E.W. Bauer, C.W.J. Beenakker, and B.I. Halperin.This work was supported by the Dutch Science Foun-dation NWO/FOM, by the SFB TR 12, and bythe DOE, Office of Basic Energy Sciences, AwardNo. DEFG02-06ER46313.

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Polarized Electric Current in Semiclassical Transport with Spin-Orbit Interaction

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