Transverse Spin-Orbit Force in the Spin Hall Effect in Ballistic Semiconductor Wires
a r X i v :c o n d -m a t /0503415v 2 [c o n d -m a t .m e s -h a l l ] 23 J u n 2005
Transverse Spin-Orbit Force in the Spin Hall E?ect in Ballistic Semiconductor Wires
Branislav K.Nikoli′c ,Liviu P.Z?a rbo,and Sven Welack ?
Department of Physics and Astronomy,University of Delaware,Newark,DE 19716-2570,USA We introduce the spin and momentum dependent force operator which is de?ned by the Hamil-tonian of a clean semiconductor quantum wire with homogeneous Rashba spin-orbit (SO)coupling
attached to two ideal (i.e.,free of spin and charge interactions)leads.Its expectation value in the spin-polarized electronic wave packet injected through the leads explains why the center of the packet gets de?ected in the transverse direction.Moreover,the corresponding spin density will be dragged along the transverse direction to generate an out-of-plane spin accumulation of opposite signs on the lateral edges of the wire,as expected in the phenomenology of the spin Hall e?ect,when spin-↑and spin-↓polarized packets (mimicking the injection of conventional unpolarized charge cur-rent)propagate simultaneously through the wire.We also demonstrate that spin coherence of the injected spin-polarized wave packet will gradually diminish (thereby diminishing the “force”)along the SO coupled wire due to the entanglement of spin and orbital degrees of freedom of a single electron,even in the absence of any impurity scattering.
PACS numbers:72.25.Dc,71.70.Ej,03.65.Sq
The classical Hall e?ect 1is one of the most widely known phenomena of condensed matter physics because it represents manifestation of the fundamental concepts of classical electrodynamics—such as the Lorentz force—in a complicated solid state environment.A perpendicu-lar magnetic ?eld B exerts the Lorentz force F =q v ×B on current I ?owing longitudinally through metallic or semiconductor wire,thereby separating charges in the transverse direction.The charges then accumulate on the lateral edges of the wire to produce a transverse “Hall voltage”in the direction q I ×B .Thus,Hall-e?ect mea-surements reveal the nature of the current carriers.
Recent optical detection 2,3of the accumulation of spin-↑and spin-↓electrons on the opposite lateral edges of current carrying semiconductor wires opens new realm of the spin Hall e?ect .This phenomenon occurs in the ab-sence of any external magnetic ?elds.Instead,it requires the presence of SO couplings,which are tiny relativistic corrections that can,nevertheless,be much stronger in semiconductors than in vacuum.4Besides deepening our fundamental understanding of the role of SO couplings in solids,4,5the spin Hall e?ect o?ers new opportunities in the design of all-electrical semiconductor spintronic devices that do not require ferromagnetic elements or cumbersome-to-control external magnetic ?elds.5
While experimental detection of the strong signatures of the spin Hall e?ect brings to an end decades of theoret-ical speculation for its existence,it is still unclear what spin-dependent forces are responsible for the observed spin separation in di?erent semiconductor systems.One potential mechanism—asymmetric scattering of spin-↑and spin-↓electrons o?impurities with SO interaction—was invoked in the 1970s to predict the emergence of pure (i.e.,not accompanied by charge transport)spin current,in the transverse direction to the ?ow of longi-tudinal unpolarized charge current,which would deposit spins of opposite signs on the two lateral edges of the sample.6However,it has been argued 7that in systems with weak SO coupling and,therefore,no spin-splitting
of the energy bands such spin Hall e?ect of the extrinsic type (which vanishes in the absence of impurities)is too small to be observed in present experiments 2(unless it is enhanced by particular mechanisms involving intrinsic SO coupling of the bulk crystal 8).
Much of the recent revival of interest in the spin Hall e?ect has been ignited by the predictions 9,10for substan-tially larger transverse pure spin Hall current as a re-sponse to the longitudinal electric ?eld in semiconductors with strong SO coupling which spin-splits energy bands and induces Berry phase correction to the group velocity of Bloch wave packets.11However,unusual properties of such intrinsic spin Hall current in in?nite homogeneous systems,which depends on the whole Fermi sea (i.e.,it is determined solely by the equilibrium Fermi-Dirac dis-tribution function and spin-split Bloch band structure)and it is not conserved in the bulk due to the presence of SO coupling,9,10have led to arguments that its non-zero value does not correspond to any real transport of spins 12,13so that no spin accumulation near the bound-aries and interfaces could be induced by any intrinsic mechanism (i.e.,in the absence of impurities 13).
On the other hand,quantum transport analysis of spin-charge spatial propagation through clean semiconductor wires,which is formulated in terms of genuine nonequi-librium and Fermi surface quantities (i.e.,conserved spin currents 14,15,16and spin densities 17),predicts that spin Hall accumulation 2,3of opposite signs on its lateral edges will emerge due to strong SO coupling within the wire re-gion.17Such mesoscopic spin Hall e?ect is determined by the processes on the mesoscale set by the spin precession length,15,17and depends on the whole measuring geom-etry (i.e.,boundaries,interfaces,and the attached elec-trodes)due to the e?ects of con?nement on the dynamics of transported spin in the presence of SO couplings in ?nite-size semiconductor structures.18,19
Thus,to resolve the discrepancy between di?erent the-oretical answers to such fundamental question as—Are SO interaction terms in the e?ective Hamiltonian of a
2 clean spin-split semiconductor wire capable of generating
the spin Hall like accumulation on its edges?—it is highly
desirable to develop a picture of the transverse motion of
spin density that would be as transparent as the famil-
iar picture of the transverse drift of charges due to the
Lorentz force in the classical Hall e?ect.Here we o?er
such a picture by analyzing the spin-dependent“force”,
which can be associated with any SO coupled quantum
Hamiltonian,and its e?ect on the semiclassical dynamics
of spin density of individual electrons that are injected as
spin-polarized wave packets into the Rashba SO coupled
clean semiconductor quantum wire attached to two ideal
(i.e.,interaction and disorder free)leads.
The e?ective mass Hamiltonian of the ballistic Rashba
quantum wire is given by
?H=?p2
(?σ×?p)·z+V conf(y),(1)
where?p is the momentum operator in2D space,?σ=
(?σx,?σy,?σz)is the vector of the Pauli spin operators,and
V conf(y)is the transverse potential con?ning electrons to
a wire of?nite width.We assume that the wire of dimen-
sions L x×L y is realized using the two-dimensional elec-
tron gas(2DEG),with z being the unit vector orthogonal
to its plane.Within the2DEG,carriers are subjected to
the Rashba SO coupling of strengthα,which arises due
to the structure inversion asymmetry4(of the con?ning
potential and di?ering band discontinuities at the het-
erostructure quantum well interface20).
This Hamiltonian generates a spin-dependent force op-
erator which can be extracted21,22within the Heisenberg
picture23as
?F H =m?
d r2H
2
[?H,[?r H,?H]](2) =
2α2m?
d?y H
y.
Here the Heisenberg picture operators carry the time dependence of quantum evolution,i.e.,?p H(t)= e i?Ht/ ?p e?i?Ht/ ,?σz H(t)=e i?Ht/ ?σz e?i?Ht/ ,and?y H(t)= e i?Ht/ ?y e?i?Ht/ ,where?σz,?p,and?y are in the Schr¨o dinger picture and,therefore,time-independent.
Since the force operator22depends on spin through ?σz H,which is a genuine(internal)quantum degree of free-dom,23it does not have any classical analog.Its phys-ical meaning(i.e.,measurable predictions)is contained in the quantum-mechanical expectation values,such as ?F y (t)= Ψ(t=0)|?F y H(t)|Ψ(t=0) obtained by acting with the transverse component?F y H of the vector of the
force operator(?F x H,?F y
H )on the quantum state|Ψ(t=0)
of an electron.While such“force”can always be asso-ciated with a given quantum Hamiltonian,its usefulness in understanding the evolution of quantum systems is limited—the local nature of the force equation cannot be reconciled with inherent non-locality of quantum me-chanics.For example,if the force“pushes”the volume of a wave function locally,one has to?nd a new global wave function in accord with the boundary conditions at in?n-ity(the same problem remains well-hidden in the Heisen-berg picture where time dependence is carried by the op-erators while wave functions are time-independent).Nev-ertheless,analyzing the dynamics of spin and probability densities in terms of the action of local forces can be in-sightful for particles described by wave packets(whose probability distribution is small compared to the typical length scale over which the force varies).11,23 Therefore,we examine in Fig.1the transverse SO “force” ?F y in the spin wave packet state,which at t=0 resides in the left lead as fully spin-polarized(along the z-axis)and spatially localized wave function24,25Ψ(t=0)=C sin
πy
3
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Longitudinal Packet Position
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Longitudinal Packet Position
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Longitudinal Packet Position
FIG.1:(Color online)The expectation value of the trans-
verse component of the SO force operator(upper panel)in
the quantum state of propagating spin wave packet along
the two-probe nanowire.The middle panel shows the cor-
responding transverse position of the center of the wave
packet as a function of its longitudinal coordinate.The ini-
tial state in the left lead is fully spin-polarized wave packet
Eq.(3),which is injected into the SO region of the size
L x×L y≡100a×31a(a?3nm)with strong Rashba cou-
pling t SO=α/2a=0.1t o and the corresponding spin preces-
sion length L SO=πt o a/2t SO≈15.7a weak SO coupling t SO=0.01t o and L SO≈157a>L x(lower panel). gion,its center ?y (t)= Ψ(t)|?y|Ψ(t) will be de?ected along the y-axis in the same direction as is the direc- tion of the transverse SO“force”.However,due to its inertia the packet does not follow fast oscillations of the SO“force”occurring on the scale of the spin precession length17,19L SO=πt o a/2t SO on which spin precesses by FIG.2:(Color online)The dynamics of spin density S(r)≡ [S x(r),S y(r),S z(r)]induced by simultaneous propagation of two electrons through quantum wire100a×31a with the Rashba SO coupling t so=0.1t o.Both electrons are injected at t=0from the left lead,one as spin-↑and the other one as spin-↓polarized(along the z-axis)wave packet Eq.(3).The di?erent snapshots of the sum of their spin densities are taken at the points(a),(b),(c)where the transverse SO“force”and the y-coordinate of the center of these wave packets have val- ues shown in the middle panel of Fig.1. an angleπ(note that the spin splitting generates a?nite di?erence of the Fermi momenta,which is the same for all subbands of the quantum wire in the case of parabolic energy-momentum dispersion,so that L SO is equal for all channels26).In contrast to an in?nite2DEG of the intrin- sic spin Hall e?ect,10,12,13in quantum wires electron mo- tion is con?ned in the transverse direction and the e?ec- tive momentum-dependent Rashba magnetic?eld B R(k) is,therefore,nearly parallel to this direction.19,26Thus, the change of the direction of the transverse SO“force” is due to the fact that the z-axis polarized spin will start precessing within the SO region since it is not an eigen- state of the Zeeman term?σ·B R(k)[i.e.,of the Rashba term in Eq.(1)]. The transverse SO“force”and the motion of the cen- 4 ter of the wave packets in Fig.1suggests that when two electrons with opposite spin-polarizations are injected si-multaneously into the SO coupled quantum wire with perfectly homogeneous25Rashba coupling,the initially unpolarized mixed spin state will evolve during propaga-tion through the wire to develop a non-zero spin density at its lateral edges.This intuitive picture is con?rmed by plotting in Fig.2the spin density, S m(t)= 2 σ,σ′c?m,σ′(t)c m,σ(t) σ′|?σ|σ ,(4) corresponding to the coherent evolution of two spin wave packets,|Ψ(t=0) =|Φ ?|↑ and|Ψ(t=0) =|Φ ?|↓ , across the wire. The mechanism underlying the decay of the transverse SO“force”intensity is explained in Fig.3,where we demonstrate that(initially coherent)spin precession is also accompanied by spin decoherence.19,30These two processes are encoded in the rotation of the spin polariza-tion vector P and the reduction of its magnitude(|P|=1 for fully coherent pure states?ρ2s=?ρs),respectively.The spin polarization vector is extracted from the density ma-trix?ρs=(1+P·?σ)/2of the spin subsystem.23The spin density matrix?ρs is obtained as the exact reduced den-sity matrix at each instant of time by tracing the pure state density matrix?ρ(t)=|Ψ(t) Ψ(t)|over the orbital degrees of freedom, ?ρs(t)=Tr o|Ψ(t) Ψ(t)|= m m|Ψ(t) Ψ(t)|m = m,σ,σ′c m,σ(t)|σ σ′|c?m,σ′(t).(5) The dynamics of the spin polarization vector and the spin density shown in Fig.3are in one-to-one correspondence 2 Tr s[?ρs(t)?σ]= m S m(t).(6) The incoming quantum state from the left lead in Fig.3 is separable|Ψ(t=0) = m,σc m,σ(t=0)|m ?|σ = |Φ ?|↑ ,and therefore fully spin coherent|P|= 1. However,in the course of propagation through SO cou-pled quantum wires it will coherently evolve into a non-separable23state where spin and orbital subsystems of the same electron appear to be entangled.19,31Note that Fig.3also shows that at the instant when the center of the wave packet enters the wire region,its quantum state is already highly entangled as quanti?ed by the non-zero von Neumann entropy(associated with the reduced den-sity matrix of either the spin?ρs or the orbital subsystem S p i n P o l a r i z a t i o n V e c t o r P Longitudinal Packet Position FIG.3:(Color online)Spin precession,as signi?ed by oscil-lations of the spin polarization vector(P x,P y,P z),and spin decoherence(as measured by decrease of the purity|P|below one)of the spin state of a single electron propagating along the Rashba quantum wire100a×31a with the SO coupling strength t SO=0.1t o(L SO≈15.7a).The electron is injected from the left lead as a spin-↑polarized wave packet,whose spin subsystem is therefore fully coherent|P|=1at t=0. The bottom panel shows the z-component of the spin density S z(r)at di?erent values of 2 log2 1+|P| 2 log2 1?|P| 5 FIG.4:(Color online)The spin accumulation(S x(r),S y(r), S z(r))induced by the ballistic?ow of unpolarized charge cur-rent,simulated by injecting one after another600pairs of spin-↑and spin-↓polarized(along the z-axis)wave packets from the left lead,through quantum wire100a×31a with the Rashba SO couplings:(a)t SO=0.1t o(L SO≈15.7a)and(b) t SO=0.01t o(L SO≈157a). ?ected at the lead/SO-region interface for strong Rashba coupling)and boundaries18,19of the con?ned structure. Thus,the decoherence mechanism revealed by Fig.3is also highly relevant for the interpretation of experiments on the transport of spin coherence in high-mobility semi-conductor32and molecular spintronic devices.33 The interplay of the oscillating and decaying(induced by spin precession and spin decoherence,respectively) transverse SO“force”and wave packet inertia leads to spin-↑electron exiting the wire with its center de?ected toward the left lateral edge and the spin-↓density appear-ing on the right edge17for strong SO coupling t SO=0.1t o in Figs.1and 2.This picture is only apparently coun-terintuitive to the na¨ive conclusion drawn from the form of the force operator itself Eq.(2),which would suggest that spin-↑electron is always de?ected to the right while moving along the Rashba SO region.While such situ-ation appears in wires shorter than L SO(as shown in the lower panel of Fig.1),in general,one has to take into account the ratio L x/L SO,as well as the strength of the SO force∝α2,to decipher the sign of the spin accumulation on the lateral edges and the sign of the corresponding spin currents that will be pushed into the transverse leads attached at those edges.15 When we inject pairs of spin-↑and spin-↓polarized wave packets one after another,thereby simulating the ?ow of unpolarized ballistic current through the lead–wire–lead structure(where electron does not feel any electric?eld within the clean quantum wire region),17we ?nd in Fig.4that the de?ection of the spin densities of individual electrons in the transverse direction will gen-erate non-zero spin accumulation components S z(r)and S x(r)of the opposite sign on the lateral edges of the wire. While recent experiments?nd S z(r)with such properties to be the strong signature of the spin Hall e?ect,2,3here we con?rm the conjecture of Ref.17that S x(r)can also emerge as a distinctive feature of the mesoscopic spin Hall e?ect in con?ned Rashba spin-split structures—it arises due to the precession(Fig.3)of transversally de?ected spins.Note that S x(r)=0accumulations cannot be explained by arguments based on the texturelike struc-ture26of the spin density of the eigenstates in in?nite Rashba quantum wires where26,27S x(r)≡0. In conclusion,the spin-dependent force operator,de-?ned by the SO coupling terms of the Hamiltonian of a ballistic spin-split semiconductor quantum wire,will act on the injected spin-polarized wave packets to de?ect spin-↑and spin-↓electrons in the opposite transverse di-rections.This e?ect,combined with precession and deco-herence of the de?ected spin,will lead to non-zero z-and x-components of the spin density with opposite signs on the lateral edges of the wire,which represents an example of the spin Hall e?ect phenomenology6,17that has been observed in recent experiments.2,3The intuitively ap-pealing picture of the transverse SO quantum-mechanical force operator(as a counterpart of the classical Lorentz force),which depends on spin through?σz,the strength of the Rashba SO coupling throughα2,and the momentum operator through the cross product?p×z,allows one to di?erentiate symmetry properties of the two spin Hall ac-cumulation components upon changing the Rashba elec-tric?eld(i.e.,the sign ofα)or the direction of the packet propagation:S z(r)?α=S z(r)αand S z(r)?p=?S z(r)p vs.S x(r)?α=?S x(r)α(due to opposite spin preces-sion for?α)and S x(r)?p=S x(r)p.These features are in full accord with experimentally observed behavior of the S z(r)spin Hall accumulation under the inversion of the bias voltage,3as well as with the formal quantitative quantum transport analysis17of the nonequilibrium spin accumulation induced by the?ow of unpolarized charge current through ballistic SO coupled two-probe nanos-tructures. Finally,we note thatα2dependence of the transverse SO“force”is incompatible with theα-independent(i.e.,“universal”)intrinsic spin Hall conductivityσsH=e/8π(describing the pure transverse spin Hall current j z y=σsH E x of the z-axis polarized spin in response to the longitudinally applied electric?eld E x)of an in?nite ho-mogeneous Rashba spin-split2DEG in the clean limit, which has been obtained within various bulk transport approaches.10,11,12,13,34On the other hand,it supports the picture of the SO coupling dependent spin Hall accu-mulations17S z(r),S x(r)and the corresponding spin Hall conductances15(describing the z?and the x-component of the nonequilibrium spin Hall current in the trans-verse leads attached at the lateral edges of the Rashba wire)of the mesoscopic spin Hall e?ect in con?ned struc-tures.14,15,16By the same token,the sign of the spin ac- 6 cumulation on the edges(i.e.,whether the spin current ?ows to the right or to the left in the transverse direc-tion15)cannot be determined from the properties34of σsH.Instead one has to take into account the strength of the SO couplingαand the size of the device in the units of the characteristic mesoscale L SO,as demonstrated by Figs.1and 4.This requirement stems from the os-cillatory character of the transverse SO“force”brought about by the spin precession of the de?ected spins in the e?ective magnetic?eld of the Rashba SO coupled wires of?nite width. 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