Pure spin-current generated by reflection at a normal metal/2DEG interface

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by  Jacob Linder

Purespin-currentgeneratedbyreflectionatanormalmetal|2DEGinterface
Jacob Linder,1 Takehito Yokoyama,2 and Asle Sudbø1
1DepartmentofPhysics,NorwegianUniversityofScienceandTechnology,N-7491Trondheim,Norway
2Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
(Dated:ReceivedJanuary22,2010)
Theconceptofaspin-currentisausefultoolinunderstandingspin-transportinhybridsystems,butitsverydef-
initionisproblematicinsystemswherespin-orbitcouplingeffectsarestrong.Intheabsenceofspin-dependent
scattering,thespin-currentremainswell-defined.Wehereproposeamethodforgeneratingpurespin-currentsin
anormalmetalwherethespin-currentconsequentlydoesnotsufferfromtheaforementionedproblemspertain-
ingtoitsverydefinitionorspin-relaxationprocesses. Morespecifically,weshowhowanunpolarizedincident
0
charge-currentcaninduceapuretransversespin-currentbymeansofscatteringatanormalmetal|2DEGinter-
1
face. ThisoccursforbothRashbaandDresselhausspin-orbitcoupling. Anexperimentalsetupforobservation
0
ofthiseffectisproposed.
2

n PACSnumbers:
a
J
2 I. INTRODUCTION tothecouplingbetweenthespinofthecharge-carriersandan
2 electric field. On the other hand, it is disadvantageous since
it breaks conservation of spin and renders it a poor quantum
The study of spin-transport in hybrid systems with mag-
] number. It would be highlydesirable to find a wayof utiliz-
l netic elements is of crucial importance both in order to un-
l ingthefirstaspectofspin-orbitcouplingandatthesametime
a derstand the basic physics of spin-transport and to find new
h functionaldevicesemergingfromfundamentalresearch1–8.In circumventthedifficultyassociatedwiththelatter. Here, we
- thiscontext,theideaofaspin-currentisanaturalextensionof proposeawaytoachievepreciselythis.
s
e thetraditionalcharge-current,andisaheavilyemployedtool The experimental setup we have in mind is shown in Fig.
m in the characterization of spin transport. The most straight-
. forward definition of a spin-current is, in analogy with the
t Polarizing
a charge-current,simplythespincarriedbyaparticletimesits
ferromagnet
m velocity. However, there are subtleties associated with this
- definition,inparticularwhenspin-orbitcouplingispresentin
d the system9,10. To illustrate this point, consider the general
Normal
n continuityequationforspindensitySSS: metal 2DEG
o
c
[ ∂tSSS+∇·jjjS=T. (1)

2 Here, jjj is the spin-current whereas T represents a spin-
v S z
sink/sourcetermthatcauses jjj tobenon-conserved. Forin- y
1 S
stance,theeffectofspin-transfertorque,whereaspin-current
8
5 is absorbed by a magnetic order parameter, may be incorpo- x
1 ratedintoT.ThetermT willingeneralbepresentinsystems
. wherethespinoperatordoesnotcommutewiththeHamilto- Spin-current
6
nian. Now, the problem with the above equation is that one Magnetization Charge/spin
0
z reflection
9 may absorb a portion, or in fact the entirety, of T into the y
0 definition of the spin-current by writing T =−∇·PPP, which
φ
v: holdsforsystemswheretheaveragespintorquedensityvan- Charge-current θ
i ishesinthebulk. Thecontinuityequationthentakestheform M x
X ∂SSS+∇· jjj(cid:48) =0, renderingthespin-current jjj(cid:48) = jjj +PPPtobe y
∂t S S S Spin-orbitcoupling
r aconservedquantity,asopposedto jjj . Therefore,thereisan
a S
inherentambiguityinthespin-currentsinceonemaydefineit
FIG. 1: (Color online) A current bias is applied to a trilayer con-
in an arbitrary way by combining elements of jjj and T. At
S sisting of a polarizing ferromagnet, a normal metal, and finally a
thesametime,itisclearthatinanormalmetallicregionwith-
materialwithspin-orbitcoupling,e.g. a2DEG.Duetoreflectionat
out any spin-sink/source term, the conventional definition of
the second interface bordering on the material with spin-orbit cou-
the spin-current serves well and is conserved. This fact will pling,transversecharge-andspin-currentsareinducedinthenormal
featureprominentlybelow. metal region. The magnetization in the polarizing ferromagnet is
Theinfluenceofspin-orbitcouplingonaspin-currentisac- misalignedanangleφfromthezzz-axis,whiletheangleofincidence
companied by welcomed as well as troublesome effects. On isdenotedθ. Iftheincidentcurrentisunpolarized,i.e. withoutthe
theonehand,theinfluenceisbeneficialinthesensethatitof- polarizing ferromagnet, a pure transverse spin-current is generated
bymeansofspin-reflectionoffthebarrier.
fers a way of manipulating the spin-current of a system due
2
1. We assume that a charge-current, which may or may not II. THEORY
be spin-polarized, flows into a normal metallic region that is
sandwiched between the polarizing ferromagnet and a mate-
Toaddresstheabovefindingsinaquantitativeway,weem-
rial with strong spin-orbit coupling, e.g. a two-dimensional
ploy a scattering matrix approach and calculate the resulting
electron gas (2DEG). It should be noted that real ferromag-
charge- and spin-currents in the system when a current bias
netsdonotactasperfectspin-polarizers,butitisnevertheless isappliedinthexxx-direction. Thespin-currentisingenerala
instructive to consider how the transport of charge and spin
tensorsinceithasadirectionofflowinreal-spaceandapo-
is influenced by a polarization of the incident current, simi-
larization in spin-space. In the normal metal (N) region, we
larly to Ref.25 in the context of spin-transfer torque. As we
maywrite
shallsee,themostinterestingeffectsoccurwhentheincident
current is unpolarized, rendering the spin-polarizer obsolete. jjj =Im{ψ†∇⊗σσσˆψ}/(2m ), (2)
S N
Thechiefmotivationforincludingthepolarizeristhussimply
togainaphysicalunderstandingofhowthespin-polarization
where ⊗ is the tensor product between the gradient operator
interactswiththespin-orbitcouplingpresentinthe2DEGre-
andthespinoperator.Inordertoevaluatethespin-current,we
gion.
need to construct the scattering states partaking in the trans-
portprocesses. Thequasiparticlestatesareobtainedbysolv-
ing the matrix equation which diagonalizes the Hamiltonian,
namely:
Although this setup is certainly simple, it offers some [Hˆ0(x)−h(x)σˆz+α(x)(kyσˆx−kxσˆy)]ψ=εψ, (3)
highlyinterestingpossibilitieswithregardtothespin-currents
flowinginthesystem. Thecrucialaspectisthescatteringtak- with Hˆ0 = [k2/(2m(x))−µ]1ˆ. The effective electron mass
ing place at the interface between the normal region and the m(x)isassumedtobedifferentinthenormalmetaland2DEG
region with spin-orbit coupling. We will show how the scat- regions. Wehaveheretakenintoaccountthepossibilityofa
teringstemmingfromthespin-orbitcouplinggeneratestrans- magnetization in the region with spin-orbit coupling, assum-
versecurrentsflowinginthenormalregion. Sincethesecur- ingthatitpointsalongthezzz-direction. Thefollowingderiva-
rents flow in the normal region, they are not subject to the tionsaremadeundertheassumptionofaspin-orbitcoupling
difficulties associated with either the definition of the spin- of the Rashba-type [as employed in Eq. (3)], but the proce-
current or the spin relaxation length hampered by spin-orbit dureisidenticalforaDresselhaus-typeHamiltonianwherethe
coupling. Moreover, we find that these currents are highly spin-orbitcouplingtermreadsα(kyσˆy−kxσˆx).SimilarHamil-
sensitive to the spin orientation of the incoming current, i.e. tonians were considered also in Refs.11,12. In order to gain
the polarization direction φ in the ferromagnet. In fact, the somebasicunderstandingoftheroleofspin-orbitcouplingin
transversecharge-currentvanishescompletelywhenφ=π/2 our setup, we consider a N|2DEG junction with an incident
foraRashba-typespin-orbitcoupling,regardlessoftheother spin-current at Fermi level from the N side. The interface is
parametersintheproblem. Thetransversespin-current,how- locatedatx=0,andhenceh(x)=hΘ(x),α(x)=αΘ(x),with
ever, remains non-zero. Interestingly, we find that for both Θ(x) the Heaviside step function. The incident spin-current
Rashba and Dresselhaus spin-orbit coupling, a pure trans- isassumedtobepolarizedintheyyy−zzz-planewithanangleφ
verse spin-current is generated in the normal metal when relative to the zzz-axis. Solving for the eigenvalues and eigen-
the incident current is completely unpolarized, i.e. without vectorsofEq. (3),weobtainthefollowingwavefunctions:
the polarizing ferromagnet in Fig. 1. This suggests that
thecharge-andspin-reflectedcurrentsinducedinthenormal (cid:104)(cid:18)c(cid:19) (cid:110) (cid:18)c(cid:19) (cid:18)ıs(cid:19)(cid:111) (cid:105)
metalregionbenefitfromthreemajoradvantages:(i)thespin- ψN= ıs eıkθx+ r↑ ıs +r↓ c e−ıkθx eıkyy (4)
currentisconservedanditsdefinitionisunambiguous,(ii)the
spinrelaxationlengthisnotinfluencedbyspin-orbitcoupling on the N side. It should be noted that considering both inci-
since it is absent in the normal region, and (iii) the charge- dent waves with φ=0 and φ=π effectively gives an unpo-
and spin-currents may be controlled in a well-defined way larizedincidentcurrent,whichweshallcommentonlater. On
simplybyadjustingthemagnetizationdirectionofthepolar- the2DEGside,wehave:
izing ferromagnet. Thus, the environment where the charge-
andspin-currentsofinterestpropagate(normalmetalregion) ψ =(cid:104)t N (cid:18)1(cid:19)eıkx↑x+t N (cid:18)u↓(cid:19)eıkx↓x(cid:105)eıkyy (5)
isnon-hostiletowardsspin,whilethecontrolparameterstun- 2DEG ↑ ↑ u ↓ ↓ 1
↑
ingthesecurrentsarelocatedinadifferentpartofthesystem
(the ferromagnet and the material with spin-orbit coupling) Above,wehavedefinedc=cos(φ/2),s=sin(φ/2),andk =
√ θ
thanwheretheactualcurrentspropagate. Thisgreatlyfacili- k cosθ,wherek = 2m µ istheFermiwavevectoronthe
F F N N
tatestheopportunitytoexertcontroloverthespin-current. It N side and θ is the angle of incidence. We have introduced
should be mentioned that in the context of mesoscopic spin- thequantities
tronics, itisingeneraldesirableandthusroutinetoconsider
leadswithoutspin-orbitcouplinginordertocharacterizespin (cid:113)
kσ= (kσ)2−k2sin2θ, (6)
transportinanunproblematicway. x F
3
x 10−5 Transversecharge-current x 10−4 Transversespin-current(x-polarization)
1 4
0.8 3
0.6 2
0.4 1
0.2 0
F F
/v 0 /v−1
jQ xjS
−0.2 −2 φ/π=0.0
−0.4 −3 0.25
0.5
−0.6 −4 1
0.75
−0.8 −5 1.0
−1 −6
−3 −2.5 −2 −1.5 −1 −0.5 0 −3 −2.5 −2 −1.5 −1 −0.5 0
x/λF x/λF
x 10−3 Transversespin-current(y-polarization) x 10−4 Transversespin-current(z-polarization)
1.5 12
10
1
8
0.5
6
/vF 0 /vF
yjS zjS 4
−0.5
2
−1 0
−1.5 −2
−3 −2.5 −2 −1.5 −1 −0.5 0 −3 −2.5 −2 −1.5 −1 −0.5 0
x/λF x/λF
FIG.2: (Coloronline)Plotoftheinducedtransversecharge-andspin-currentsflowingparalleltothebarrierinthenormalmetalregion. We
havesetα=1×10−4andassumedaRashba-typespin-orbitcoupling.Forφ/π=0.5,thetransversecharge-currentvanishes,whichmakesit
possibletoobtainapurespin-currentsignalbycontrollingthemagnetizationdirectionofthepolarizingferromagnet. Thetransversecharge-
currentalsovanisheswhentheincidentcurrentisunpolarized,i.e. asuperpositionofφ=0andφ=π. Thiscanbeverifieddirectlyfromthe
figure.Notethatthecurvesforanglesφand(π−φ)aredegenerateforthex-andz-polarizationsofthespin-current.
inadditiontoN =(1+|u2|)−1/2and conditionsattheinterface:
σ σ
[r ∂ ψ (x,y)−∂ ψ (x,y)]| =ηˆψ (0,y),
m x 2DEG x N x=0 N
ψ (0,y)=ψ (0,y), ηˆ =(2m V 1ˆ+m αıσˆ ).
N 2DEG N 0 N y
kσ=[2m2DEGµ2DEG+2m22DEGα2 (8)
(cid:113)
+2m2DEGσ h2+m22DEGα4+2m2DEGµ2DEGα2]1/2, Above, rm = mN/m2DEG denotes the ratio of the electron
masses in the N and 2DEG regions. Employing the notation
σα(k −ıσkσ)
u =− y x , ofBlonder-Tinkham-Klapwijktheory13,wedefinethedimen-
σ (cid:112)
h+ α2(kσ)2+h2 sionless parameter Z =2m V /k to characterize the inter-
N 0 F
(7) face transparency. The higher the value of Z, the stronger
the interface barrier potential. An ideal interface is charac-
terizedbyZ=0. Theactualbarrierpotentialismodeledasa
delta-functionandisproportionaltoV . Notethatthebound-
0
Allwave-functionsarenormalizedtounity. Foranglesofin- aryconditionsproperlytakeintoaccounttheoff-diagonalel-
cidence that satisfy sinθ>kσ/k , the transmitted electrons ementsinthevelocityoperator,asdemandedinthepresence
F
become evanescent. To ensure their decay in the 2DEG re- ofspin-orbitcoupling14.Thetransversecharge-current j and
Q
gion,oneshouldthensetkσ=Re{kσ}+ı|Im{kσ}|. Thescat- spin-current jjj =(jx,jy,jz) are finally obtained by integrat-
x x x S S S S
teringcoefficients{r ,t }areobtainedbyusingtheboundary ing over all angles of incidence. Introducing a generalized
σ σ
4
x 10−5 x 10−3 useµN=5eVandtheelectronmassmN=0.51MeV.Inthe
4 1 2DEGregion,wesetµ =50meVwithaneffectiveelec-
φ/π 0.0 2DEG
2 0.8 0.5 tronmassm2DEG=0.1mN,i.e. rm=10. Wesetthespin-orbit
1.0 couplingparametertoα=1×10−4 tomodelatypicalvalue
/vF 0 /vF0.6 forasemiconductor23. Wehaveinvestigatednumericallythe
jQ xjS0.4 influenceoftheexchangefieldinthe2DEGregion,andfound
−2 0.2 onlyminorquantitativechangesintheresultsforvaluesupto
h/µ =0.5. Therefore,weshallherefocusonthecaseof
−4 0 2DEG
0 0.5 1 0 0.5 1 a 2DEG without magnetization, i.e. set h=0 and consider
α x 10−3 α x 10−3 solely the effect of spin-orbit coupling. To model interface
x 10−5 resistance, we set Z =2mNV0/kF =3 as a reasonable mea-
0.01 1
0.01 sure for a rather low transmissivity interface. The angularly
0 resolvedtransmissioncoefficientT(θ)isrelatedtothebarrier
0.005
0 −0.01 parameter Z as T(θ)=4cos2θ/(4cos2θ+Z2). The Fermi-
F F 0 0.5 1
yj/vS 0 zj/vS−1 x 10−3 vaercetnoorrmmiaslmizaetdcbhaarbrioevrepomteanytiianlpZri→nciZp(cid:48)lewbitehiZn(cid:48)c>orpZo,rtahtuesdlionwto-
−0.005
eringthetransmissivity24.
−0.01 −2
0 0.5 1 0 0.5 1
α x 10−3 α x 10−3
A. Rashbaspin-orbitcoupling
FIG.3: (Coloronline)Thetransversecurrentsevaluatedatx/λF =
−1.0asafunctionofthespin-orbitcouplingstrengthαforRashba- Wefirstinvestigatethetransversecharge-andspin-currents
typespin-orbitcoupling. Inallcases, themagnitudeofthecurrent flowing in the yyy-direction when a current-bias is applied to
increaseswithα. thejunction,usingEq. (9)withaRashba-typespin-orbitcou-
pling. The result is shown in Fig. 2 for several misorienta-
tionanglesφoftheincidentcurrent. Asseen,thepresenceof
current-vector jjj =(j ,jjj ) and τττˆ =(1ˆ,σσσˆ/2), we may write
Q S spin-orbit coupling induces non-zero transverse charge- and
thetransversecurrentas
spin-currents in the normal region. The charge-current van-
(cid:90) π/2 ishes at φ/π = 0.5 or if the incident current is completely
jjj= dθIm{ψ†∂yτττˆψ}/mN. (9) unpolarized (i.e. a superposition of φ=0 and φ=π). This
−π/2
suggestsaremarkableeffect: simplybyrotatingthemagneti-
Let us underline here that Eq. (9) naturally accounts for the zationinthepolarizingferromagnetrelativetothespin-orbit
contribution from different angles of incidence to the trans- coupling vector that resides in the xy-plane, it is possible to
verse current. In Eq. (9), the derivation operator ∂ brings tunethecharge-andspin-currentsinthenormalmetalregion,
y
a factor sinθ to the integrand which thus ensures that angles andinparticularonecanobtainapurespin-currentsignalfor
of incidence close to ±π/2 contribute strongly to the trans- φ/π=0.5. Both the charge- and spin-currents display oscil-
versecurrent,astheyshould. Ifwehadbeenconcernedwith lations and decay to a constant, in general non-zero value in
the spin-current flowing perpendicular to the barrier, the re- the bulk of the normal metal region. The oscillations appear
placement∂y→∂xwouldhavebeenmade,leadingtoafactor asaresultofinterferencetermsofthetypeRe{r↑e−2ıkθx}and
of cosθ as usual in that case. In principle, one could also Im{r↓e−2ıkθx}generatedwheninsertingthewavefunctionEq.
insert a weight-factor f(θ) inside the integrand of Eq. (9) (4) into Eq. (9). From these expressions, it is seen that as
which models a statistical distribution of the incoming parti- |x|grows,theexponentvariesmorerapidlywithθ,suchthat
cles. If the experimental geometry dictates that the incident the angular averaging in Eq. (9) eventually completely can-
quasiparticles are collimated near θ=0, one could use e.g. celsoutthex-dependenttermsgivingrisetotheoscillations.
f(θ)=cosθ. Wehavecheckedexplicitlythatourresultsun- The magnitude of the oscillations are therefore the strongest
dergo only a minor quantitative change when including such closesttothebarrier(x/λ →0−). Animportantobservation
F
aweight-factor,andwehererestrictourattentiontothecase is that these transverse currents are not subject to the inher-
withoutanysuchstatisticaldistribution f(θ). entproblemwithspin-orbitcouplingrelatedtothedefinition
of the spin-current or spin-relaxation processes. We proceed
toshowthatincreasingspin-orbitcouplinginducesastronger
III. RESULTS spin-reflectedcurrent, asdepictedinFig. 3. Inallcases, the
magnitudeofthecurrentincreaseswithα.
Let us now discuss our choice of parameters for the phys- FromFig.2,weseethatthechargecurrentsatisfies j (φ)=
Q
ical quantities entering the model. Unless specifically stated −j (π−φ). The exact expressions for the reflection coeffi-
Q
otherwise,thefiguresareobtainedusingtheparametervalues cientsaretoounwieldytopermitananalyticalexpressionfor
below. We have distinguished between the electron masses j (φ) through solving Eq. (9) by hand, but numerically we
Q
and Fermi levels in the N and 2DEG region, as these differ findthat j (φ)∝cos(φ). Asaresult,itfollowsthatforanin-
Q
greatly in realistic samples. In the normal metal region, we cidentcurrentwhichisunpolarized,i.e. byremovingthepo-
5
x 10−4 Transversecharge-current x 10−3 Transversespin-current(x-polarization)
8 1.5
6
1
4
2
0.5
0
/vQF −2 x/vFS 0
j φ/π=0.0 j
−4
0.25 −0.5
−6
0.5 1
−8 0.75 −1
−10 1.0
−12 −1.5
−3 −2.5 −2 −1.5 −1 −0.5 0 −3 −2.5 −2 −1.5 −1 −0.5 0
x/λF x/λF
x 10−4 Transversespin-current(y-polarization) x 10−6 Transversespin-current(z-polarization)
4 5
3 4
2
3
1
2
0
/vF −1 /vF 1
yjS zjS
−2 0
−3
−1
−4
−2
−5
−6 −3
−3 −2.5 −2 −1.5 −1 −0.5 0 −3 −2.5 −2 −1.5 −1 −0.5 0
x/λF x/λF
FIG.4: (Coloronline)Plotoftheinducedtransversecharge-andspin-currentsflowingparalleltothebarrierinthenormalmetalregionfora
Dresselhaus-typespin-orbitcoupling.Thetransversecharge-currentsnolongervanishesatφ/π=0.5,butforanunpolarizedincidentcurrent
(superpositionofφ=0andφ=π),thetransversecharge-currentisabsentjustasintheRashba-case.
larizingferromagnet,apuretransversespin-currentmayagain two contributions are scattered in opposite directions due to
begenerated. Weremindthereaderthatanunpolarizedinci- thespin-orbitcoupling,15andthusthenetcharge-currentvan-
dent current can be thought of as a superposition of an inci- isheswhereasthespin-currentisnon-zero.
dent φ=0 and φ=π wave which leads to a total transverse
charge-current:
jQ(0)+jQ(π)= jQ(0)−jQ(π−π)=0. (10) B. Dresselhausspin-orbitcoupling
The result is therefore a pure transverse spin-current. Effec-
tively,thisamountstoaconversionfromapurecharge-current Let us also briefly investigate how the transverse charge-
flowinginthexxxˆ-directiontoapurespin-currentflowinginthe andspin-currentsflowingintheyyy-directionareinfluencedby
yyyˆ-direction. Whereas such a scenario is also found inside a a Dresselhaus-type spin-orbit coupling in the 2DEG region,
2DEGsubjecttothespin-Halleffect,animportantdifference incontrasttotheRashba-casetreatedintheprevioussection.
fromourresultsisthatinthatcase, thespin-currentflowsin In Fig. 4, we plot the transverse charge- and spin-currents
theregionwherespin-orbitcouplingeffectsarestrong.There- flowingalongthebarrier. Duetothedifferentstructureofthe
fore, boththedefinitionofthespin-currentanditsrelaxation Dresselhausspin-orbitcouplingcomparedtotheRashba-type,
lengthbecomeproblematic. Inourcase,bothoftheseissues the x- and y-polarizations of the spin-current change roles.
are eluded since the spin-current flows in a normal metal re- FortheDresselhaus-type,boththey-andz-polarizationofthe
gion by means of reflection off a barrier in the presence of spin-currentareinsensitivetoavariationinφ.Similarlytothe
spin-orbit coupling. The vanishing of the transverse charge- Rashba case, however, the magnitude of the transverse cur-
current is understood by realizing that an injected unpolar- rents all increase with α as shown in Fig. 5. It should be
ized charge-current may be viewed as a coherent superposi- notedthatthex-polarizationofthespin-currentdoesnotvan-
tion of spin-↑ and spin-↓ electrons with equal weight. The ishcompletelyatφ/π=0.5: itissimplystronglysuppressed
6
currents are plotted in Fig. 7 as a function of angle of in-
15x 10−4 0.01 cidence for a Rashba-type spin-orbit coupling, being evalu-
ated at x/λ =−1.0. We here focus on the most interest-
F
10 0.005 ingcaseφ/π=0.5. Asseen,boththecharge-currentandthe
y-polarization of the spin-current are antisymmetric around
F F
/v 5 /v 0 θ=0,leadingtoavanishingnetcurrentuponperformingthe
Q xS
j j angular integration. In contrast, the x- and z-polarizations of
0 −0.005
the spin-current are symmetric around θ=0, yielding a net
contributiontothetotalcurrent.Thespecificformofthespin-
−5 −0.01
0 0.5 1 0 0.5 1 orbit coupling potential should also influence the symmetry
α x 10−3 α x 10−3 properties. In Fig. 7, we have used a standard Rashba-form
8x 10−4 20x 10−6 with HˆSOC =α(kyσˆx−kxσˆy). In the case of a Dresselhaus-
φ/π 0.0 form HˆSOC =α(kyσˆy−kxσˆx), one would expect that the x-
15
6 0.5 andy-polarizationofthespin-currentwouldinterchangetheir
F F10 1.0 symmetry properties since the two Hamiltonians are related
/v 4 /v bythesubstitutionσˆ ↔σˆ . Thispictureisverifiedbycom-
yjS zjS 5 x y
paringthex-andy-polarizationsofthespin-currentinFig. 2
2
0 with Fig. 4. The transverse charge-current nevertheless re-
mainszeroinbothcasesforanincidentunpolarizedcurrent.
0 −5
0 0.5 1 0 0.5 1 WecommentmoreontheroleofaddingaDresselhaustermto
α x 10−3 α x 10−3 Hˆ laterinthissection. Finally,wenotethattheoscillationsof
thecurrentsinFig. 7increaseinrapidityas|x|increases,i.e.
FIG.5: (Coloronline)Thetransversecurrentsevaluatedatx/λF = farther inside the N region. The reason for this is the previ-
−1.0 as a function of the spin-orbit coupling strength α for ouslymentionedinterferencetermsofthetypeRe{r↑e−2ıkθx}
Dresselhaus-typespin-orbitcoupling. Inallcases,themagnitudeof andIm{r↓e−2ıkθx}intheexpressionforthecurrent.
thecurrentincreaseswithα.
The assumption of ballistic transport and a sharp inter-
face at the N|2DEG region is certainly an approximation to
real systems where the 2DEG is often not characterized by
comparedtotheothervaluesofφ.
the ballistic regime due to impurity scattering, e.g. in InAs.
Increased impurity scattering randomizes the momentum of
scatteredparticles,whichisdetrimentaltowavefunctionsthat
IV. DISCUSSION
The results presented in this paper suggest a method for x 10−3
2
obtaining a conversion from charge- to spin-currents. The
method we propose exploits the possibility of manipulating
1
thecurrentbymeansofspin-orbitcoupling,whereasitatthe
F
sametimerendersthespin-currentimmunetowardsthecom- /v 0
xS
plicating and adverse effects of spin-orbit coupling with re- j
−1
gardtorelaxationprocessesandtheverydefinitionofaspin-
current. Inordertoobservethetransversecharge-spincurrent
−2
separation,onewouldneedtofindawaytoprobethepresence −1 −0.8 −0.6 −0.4 −0.2 0
x/λ
F
ofaspin-flowinthetransversedirection.Thiscouldinprinci-
plebeachievedbymeasuringforinstancespinaccumulation x 10−3 Z=0 1 2 3
attheedgeofthenormalmetalwirewithopticaltechnique7,8. 4
Above,wekepttheinterfacebarrierpotentialfixedatZ=3, 3
corresponding toa transmissioncoefficient of about T (cid:39)0.3
F
fornormalincidence. Todemonstratethatourresultsremain /v 2
zS
qualitativelyunaltereduponvaringthebarrierpotentialZ,we j
1
plotinFig. 6thetransversespin-currentsforanincidentcur-
rentwithφ=π/2.ForaRashba-typespin-orbitcoupling,this
0
isequivalenttothevanishingoftransversechargecurrent,as −1 −0.8 −0.6 −0.4 −0.2 0
x/λ
seenine.g. Fig. 2. TheroleofthebarrierpotentialZ isseen F
inFig. 6tosimplyreducethemagnitudeofthespin-current,
anddoesnotinfluencetheresultsqualitatively. FIG. 6: (Color online) The transverse spin-currents Ix and Iz as a
S S
Itisalsoinstructivetoconsidertheangularlyresolvedtrans- functionofthespatialdistancex/λFpenetratedintothenormalmetal
versecurrentsinordertounderstandthescatteringprocesses regionforRashba-typespin-orbitcoupling. Wehaveheresetφ/π=
onamoremicroscopiclevel.Thetransversecharge-andspin- 0.5andconsideredseveralbarrierstrengthsZ.
7
x 10−4 x 10−4
4 4 2 8
6
2 2 1
j/vQF 0 xj/vFS 0 yj/vFS 0 zj/vFS24
−2 −2 −1 0
−−40.5 0 0.5 −−40.5 0 0.5 −−20.5 0 0.5 −−20.5 0 0.5
θ/π θ/π θ/π θ/π
FIG.7: (Coloronline)Theangularlyresolvedtransversechargeandspin-currentsevaluatedatx/λF =−1.0inthenormalmetalregionfora
Rashba-typespin-orbitcoupling.Wehaveheresetφ/π=0.5.
arehighlysensitivetotheorientationofthemomentumonthe Some previous works have also investigated spin-
Fermisurface. Rashbaspin-orbitcouplingcanbeinterpreted dependent scattering in hybrid structures in the presence of
as a wave vector-dependent Zeeman field that is altered dra- spin-orbitcoupling18–21 asapossiblemeanofobtainingcon-
maticallywhenanelectronscattersfromonemomentumori- trollable spin-currents. In contrast to our results, however,
entation to another, even if the magnitude of the momentum the spin-currents obtained in these works suffer from all the
remainsthesame. Thistypeofscatteringeffectivelyrandom- problems related to spin-currents that we have elaborated on
izestheelectronspin. previously, since the spin-current flows in the 2DEG-region.
To show that the possibility of obtaining pure trans- We emphasize that we have demonstrated the possibility of
versespin-currentspersistseveninthenonballistictransport- having a transverse, dissipationless spin-current26 in the ab-
regime,weconsiderasituationwherebothRashbaandDres- sence of any accompanying charge-current. The charge-spin
selhaus spin-orbit coupling are present and tuned to be of currentseparationcouldalsofindpotentialuseasaspin-filter.
equal magnitude by means of proper gating, as discussed in Spin-filtering effects in spin-orbit coupled systems by selec-
Refs.16,17. In this case, the scattering eigenvectors are inde- tive angular beam injection have been discussed previously
pendent of momentum, and thus survive angular averaging in Ref.22. However, in previous discussions it has transpired
over the Fermi surface in the presence of non-magnetic im- that any net spin-current vanishes when taking into account
purities. Toseethis,considertheHamiltonian allpossibleanglesofincidence. Inourcase,thespin-current
survivestheaveragingandisthuseasiertoaccessexperimen-
Hˆ =Hˆ0+αR(kyσˆx−kxσˆy)+αD(kyσˆy−kxσˆx), (11) tally since angular filtering is a much more difficult task in
quantumelectronicsthanin,forinstance,optics.
whereα andα representthespin-orbitcouplinginteraction
R D
parameter of Rashba and Dresselhaus type, respectively. In
thecasewheretheseareequal,αR=αD≡α,oneobtainsthe V. SUMMARY
eigenvalues
√ In summary, we have investigated the transport of charge
ε±=k2/(2m)−µ± 2α(ky−kx), (12) and spin in a normal metal|2DEG junction, taking into ac-
count a spin-polarization of the incident current and a mag-
withbelongingeigenvectors
netic exchange energy in the 2DEG region. We find that it
1 (cid:18)1√−i(cid:19) 1 (cid:32) 1 (cid:33) itsopaopsusirbelseption-ocbutrarienntascimonpvleyrsbiyonrefflroecmtioanpoufrfetchheabrgarer-iceurrsreepn-t
ψ+= √2 12 , ψ−= √2 −(1√+ı) . (13) arating the normal and 2DEG region. More specifically, an
2
incident unpolarized charge-current flowing towards the bar-
Theabovewavefunctionsψ arenotsensitivetothedirection rier is converted into a pure transverse spin-current flowing
±
ofmomentum.Incontrast,thewavefunctionsinEq. (5)where parallel to the barrier due to spin-dependent scattering off
a pure Rashba spin-orbit coupling was used are strongly de- the barrier induced by the spin-orbit coupling. We empha-
pendentonthemomentumorientationduetotheu factors.In size that the spin-current flowing in the normal metal region
σ
thepresentcaseofcombinedRashba+Dresselhausspin-orbit is unambiguously defined and also rendered insensitive to
coupling, the transverse currents scattered off the barrier are the adverse spin-relaxation effects accompanying spin-orbit
obtained using a similar framework as described previously. coupling. The method we propose to generate a pure spin-
Consideringaninjectedunpolarizedcurrent, wefindthatthe current in fact utilizes the desirable properties of spin-orbit
transversecharge-currentagainvanisheswhereasthex-andy- coupling for facilitated control over spin-transport while si-
componentsofthespin-currentremain. Thus, themethodof multaneously avoiding the complicating effects of spin-orbit
generating pure spin-currents suggested here should display coupling pertaining to the definition of the spin-current and
robustness against impurity effects, although a more careful spin-relaxation. Moreover, we have studied how the trans-
investigationofthismattercertainlyiswarranted. verse charge- and spin-currents can be controlled by spin-
8
polarizingtheincidentcurrent. Itisfoundthatitispossibleto wheresssˆ=σσσˆ/2,weobtain
tune the transverse charge-current to zero simply by rotating
themagnetizationofthepolarizingferromagnet,thusleaving
apurespin-currentflowingparalleltothebarrier. Ourresults ı∂SSS=ı[ı(Hˆψ)†sssˆψ−ıψ†sssˆHˆψ]=2ıIm{ψ†sssˆHˆψ} (A3)
t
mayopenupnewperspectivesforthegenerationandcontrol
overpurespin-currents.
by means of the Schro¨dinger equation ı∂ψ=Hˆψ. Making
t
useoftheaboveequations,weobtain
Acknowledgments
J.L. and A.S. were supported by the Research Council of ∂tSSS+∇·jjjS=TSOC+TFM, (A4)
Norway, Grant No. 167498/V30 (STORFORSK). T.Y. ac-
knowledgessupportbyJSPS.
where jjj istheconventionalspin-current
S
AppendixA:Spin-densitycontinuityequation
jjj =Im{ψ†∇⊗σσσˆψ}/(2m). (A5)
S
We here briefly outline the derivation of the continuity
equation of the spin-density, setting h¯ =c=1. We consider
Moreover,wehavedefined
thegenericHamiltonian
ppp2
Hˆ = +HˆSOC+HˆFM+V(x), TSOC=Im{ψ†σσσˆHˆSOCψ}, TFM=Re{ψ†(σσσˆ×hhh)ψ}. (A6)
2m
Hˆ =A(ppp)σˆ −B(ppp)σˆ , Hˆ =−σσσˆ·hhh, (A1)
SOC x y FM
In particular, the term T may be interpreted as a spin-
FM
withV(x)containingallpotentialenergyterms. Definingthe transfer torque to the magnetic order parameter, leading to a
spin-densityas non-conserved spin-current even in the absence of spin-orbit
coupling. TheabovetreatmentisvalidbothforRasbha-and
SSS=ψ†sssˆψ (A2)
Dresselhaus-typespin-orbitcoupling.
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