Spin densities in parabolic quantum wires with Rashba spin-orbit interaction

0.1 MB·English

by  S. I. Erlingsson

pssheaderwillbeprovidedbythepublisher
7
0 Spin densities in parabolic quantum wires with Rashba spin-
0
2 orbit interaction

n
a SigurdurI.Erlingsson 1,2,J.CarlosEgues1,3,andDanielLoss1
J ∗
1 DepartmentofPhysicsandAstronomy,UniversityofBasel,Klingelbergstrasse82,CH-4056,Switzerland
3
2 2 ScienceInstitute,UniversityofIceland,Dunhagi3,IS-107Reykjavik,Iceland
3 DepartmentofPhysicsandInformatics,UniversityofSa˜oPauloatSa˜oCarlos,13560-970Sa˜oCarlos/SP,

] Brazil
l
l
a
h
- Keywords Rashbainteraction,quantumwires,spindensity
s
e Using canonical transformations we diagonalize approximately the Hamiltonian of a gaussian wire with
m Rashbaspin-orbitinteraction. Thisproceedureallowsustoobtaintheenergydispersionrelationsandthe
wavefunctions with good accuracy, even in systems with relatively strong Rashba coupling. With these
.
t eigenstatesonecancalculatethenon-equilibriumspindensitiesinducedbyapplyingbiasvoltagesacross
a
thesample.Wefocusonthez-componentofthespindensity,whichisrelatedtothespinHalleffect.
m
-
d Copyrightlinewillbeprovidedbythepublisher
n
o Thespin-orbitinteractionopensupthepossibilitytomanipulatetheelectronspinusingelectricalmeans,
c either with appliedbias orgate voltage[1]. One manifestationof such electricalspin controlis the spin
[
Halleffect[2,3,4]wherespincurrentsarecreatedviaaninterplaybetweenspin-orbitinteractionandthe

1 applied electric field. When the spin Hall effect is considered in systems with finite transverse size the
v edgesplaysanimportantrole. Indisorderedsystemsonewouldexpectabuildingupofspindensity(spin
4
accumulation)withoppositesignatthetwoedges.Thispictureisnotnecessarilyvalidinballisticsystems
6
andthewavefunctions(orwavepackets)themselvesbecomeimportantquantities[5]. Thewavefunctions
5
1 canthenbeusedtocalculatethebiasvoltageinducedspindensity[6,7,8,9,10,11,12,13,14,15].
0 Inthis paperwe calculateanalyticallyan approximateeigenspectrumof the RashbaHamiltonianin a
7 gaussianquantumwire. Thisallowsustocalculateanalyticallytherelevantmatrixelementsanddensity
0
of states for each transverse mode in the wire with good accuracy, even for relatively strong spin-orbit
/
t coupling. TheHamiltonianforthequantumwiremadeina2DEGwithRashbaspin-orbitinteraction,for
a
m agivenwavevectorkalongthewire,is
- ~2k2 1 α
d H = + mω2y2+ (~kσ p σ ) (1)
n 2m 2 ~ y − y x
o ~2k2 1 α α
= +~ω(a a+ )+τ αk (a τ +aτ )+ (a τ +aτ ). (2)
c 2m † 2 z − √2ℓ † + − √2ℓ † − +
:
v
In Eq. (2) we introduced for convenience new spin matrices where τ = σ and τ raises and lowers
i z y ±
X betweentheσ eigenstates.Forpositivemomentaonlystates k,n, and k,n+1, crossinenergy.All
y
r otherstates areseparatedbyatleast~ω, whichallowsusto|treatt↑hieterm| proporti↓oinalto (a†τ+ +aτ )
a perturbativelyink ℓ α/ℓ~ω.NotethatourHamiltonianistimereversalsymmetric(thereisnoextern−al
so
≡
sourceofmagneticfield).HencewecanfocusontheposivekstatesandusetheKramesrelationsǫ (k)=
n,s
ǫ ( k)andψ (r)=ψ (r)toobtainthenegativekstates.
n, s k,n,s k,n, s
− − − −
Asmentionedabove,forthepositivemomentumstatestheperturbativetermis
α
V = (τ a +τ a), (3)
+ †
−2√2ℓ −
∗ Correspondingauthor:e-mail:[email protected],Phone:[phone],Fax:[phone]
pssdatawillbeprovidedbythepublisher
2 Sh.FirstAuthor,Sh.SecondAuthor,andSh.ThirdAuthor:ShortTitle
andthenon-perturbedHamiltonianis
~2k2 1 α
H0 = +~ω(a†a+ )+ατzk+ (τ+a+τ a†). (4)
2m 2 √2ℓ −
WedefineaneffectiveHamiltonianH =eSHe S whichresultsin
eff −
1 1
H = H + [S,V]+ [S,[S,V]]+o((k ℓ)4), (5)
eff 0 so
2 3!
withS chosentofulfilltheusualconditionV +[S,H ]=0. TheS whichsatisfiestheaboverequirement
0
is
S = 2√2ℓ(~ωα+2αk)(τ+a†−τ−a)+ 4ℓ2~ω(~αω2+2αk)τz(a†2−a2)+o((ksoℓ)3). (6)
ThenewtermsgeneratedbyS intheeffectiveHamiltonianare
1[S,V] = α2(τz(2a†a+1)−1) + α3 τ+aa†a+τ+a†3+h.c. +o((k ℓ)4). (7)
2 4ℓ2(~ω+2αk) √24ℓ3~ω(~ω+2αk) so
(cid:0) (cid:1)
3
The term proportionalto τ a and its hermitian conjugateare perturbative,i.e. they only coupleterms
+ †
whichareseparatedbyatleast3~ω. Furthermore,onecanshowthattheterm 1[S,[S,V]]inEq.(5)con-
3
tainsno’diagonal’terms1andthusthecorrectionstotheHamiltonian,andconsequentlyitseigenenergies,
areatmosto((k ℓ)4). TheresultingeffectiveHamiltonianisthus
so
~2k2 α2(τ (2a a+1) 1)
Heff = 2m +~ω(a†a+1/2)+αkτz + 4zℓ(~ω†+2αk)−
α α2a a α2a a
−2√2ℓ τ+a 1− 4ℓ~ω(~ω†+2αk) + 1− 4ℓ~ω(~ω†+2αk) τ−a† . (8)
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
TheaboveH canbeexactlydiagonalizedusingageneralizedrotationmatrixinspinspace
eff
cos(Θ[nˆ+1]/2) sin(Θ[nˆ+1]/2)a
U = √nˆ+1 , (9)
−a†sin(Θ√[nˆnˆ++11]/2) cos(Θ[nˆ]/2) !
wherenˆ =a aand
†
1 k ℓ2k (ksoℓ)2(nˆ+1)
cos(Θ[nˆ+1])= 2 − so − 2(1+2ksoℓ2k) . (10)
1 k ℓ2k (k(cid:16)soℓ)2(nˆ+1) 2+ (ksoℓ)2(nˆ+1)(cid:17)1 (ksoℓ)2(nˆ+1) 2
2 − so − 2(1+2ksoℓ2k) 2 − 4(1+2ksoℓ2k)
r
(cid:16) (cid:17) (cid:16) (cid:17)
TheresultingHamiltonianH = U H U isdiagonalinthespinandladderoperatorsanditseigenen-
diag † eff
ergiesforpositivekare
1 1 (k ℓ)2
ε (k) = ~ω ℓ2k2+ k ℓ2k so (11)
0,↓ (cid:18)2 2 − so − 2(1+2ksoℓ2k)(cid:19)
1 (k ℓ)2
ε (k) = ~ω ℓ2k2+n so +∆ (k) , n>0 (12)
n,↓ (cid:18)2 − 2(1+2ksoℓ2k) n (cid:19)
1 (k ℓ)2
ε (k) = ~ω ℓ2k2+(n+1) so ∆ (k) , n 0 (13)
n,↑ (cid:18)2 − 2(1+2ksoℓ2k) − n+1 (cid:19) ≥
1 (k ℓ)2n 2 (k ℓ)2 (k ℓ)2n 2
∆ (k) = k ℓ2k so + so n 1 so . (14)
n s(cid:18)2 − so − 2(1+2ksoℓ2k)(cid:19) 2 (cid:18) − 4(1+2ksoℓ2k)(cid:19)
1 Here,diagonalreferstoa†a,τzandalsothetermscontainingτ+aandτ−a†.
(cid:13)c 2003WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
pssheaderwillbeprovidedbythepublisher 3
(a) (b)
7 7
6 6
5 5
ω ω
~ ~
/ /
k) 4 k) 4
(ns (ns
ε ε
3 3
2 2
1 1
0 0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
kℓ kℓ
Fig.1 EnergydispersionsofEqs.(11)-(13)areplotted(solidlines)forthefirst6oscillatorlevelsforksoℓ = 0.2in
(a)andksoℓ =0.4in(b). Thedashedlinesaretheresultsofnumericalcalculations. Notethatthenumericalcurves
fallontopoftheanalyticalcurvesexceptforthehighestenergycurvesin(b).
The negativek branchof the spectrumis obtainedfromthe Kramersrelations. In Fig. 1 the energydis-
persionforthefullspectrumisplotted,alongwithresultsofnumericalcalculations(dashedcurves).Even
for a relativelylarge (k ℓ = 0.4) perturbationparameter the numericaland analytical curvesare barely
so
distinguishableexceptforthehighesttransversestates,seeFig.1b.
Spindensities: Knowingthewavefunctionsandeigenenergiesallowsonetocalculatemanyphysical
quantitiesanalytically,evennon-equilibriumones. Asanexamplewe considerspindensities(transverse
tothewiredirection)andfocusonthetheaveragevalueofσ δ(y). Foranappliedbiasvoltage, σ δ(y)
z z
h i
is relatedto the spin-Halleffect[16]. Since the Hamiltonianis time reversalsymmetricthere willbe no
equilibriumspindensityinthesystem[17].ApplyingabiasvoltageliftsthissymmetrysinceKramerspairs
becomeunequallyoccupied. Forspindensities(andanyquantitywhichisoddundertimereversal)only
statesinthebiaswindowneedtobeconsidered. Startingfromthenon-equilibriumGreen’sfunctions[18]
itispossibletoshowthatforkT =0thespindensityis
τ δ(y) = ψ τ δ(y)ψ f ε (k) (µ+eV) f ε (k) µ , (15)
η eV kns η kns ns ns
h i| h | | i − − −
Xkns h (cid:0) (cid:1) (cid:0) (cid:1)i
wherethesumisoverallk whosevelocityispositive(assumingthateV > 0), i.e.thesumcontainsnot
onlyk>0statesbutcanalsoincludenegativemomentumstates. Althoughthebiasinducedspindensities
are know analytically through the wavefunctions, the equations are quite long and complicated. As an
examplewegivethefollowingmatrixelement
ψ τ δ(y)ψ = sin(Θ[n+1])φ (y)φ (y)
kn x kn n n+1
h ↑| | ↑i −
k ℓ
+ so cos(Θ[n+1]/2)2φ (y)φ (y)
1+2ksoℓ2k n n−1
n
sin(Θ[n+1])/2)2φ (y)φ (y) , (16)
n+1 n+2
−
o
where y n = φ (y) are the harmonic oscillator eigenstates. The matrix element in Eq. (16) has the
n
h | i
symmetryproperty ψ σ δ(y)ψ = ψ σ δ( y)ψ . Thisisconsistentwiththespin-Hall
kns z kns kns z kns
h | | i −h | − | i
effectphenomenologywherethetransversespincurrentleadstooppositespinpolarizationatthetwoedges
[16,5]. Thispropertyisadirectresultoftheparityoftheharmonicoscillatorstatesφ (y).
n
Thespin densities(normalizedto the wire length)for differentvaluesof the chemicalpotentialµ are
plotted in Fig. 2, assuminglinear response eV ~ω. As expectedthe spin density is oddin y andthe
≪
highesttransversemodecontributesthemostduetoitsdensityofstates beingthe highest. Notethatthe
(cid:13)c 2003WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
4 Sh.FirstAuthor,Sh.SecondAuthor,andSh.ThirdAuthor:ShortTitle
(a) (b)
0.2 0.4
0.15 µµµµµ=====11122.....57913 0.3 µµµµµ=====35344.....57913
i i
eV~2ωℓ 0.1 eV~2ωℓ 0.2
h h
y)i 0.05 y)i 0.1
σδ(z 0 σδ(z 0
h h
-0.05 -0.1
-0.1 -0.2
-0.15 -0.3
-0.2 -0.4
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
y/ℓ y/ℓ
Fig. 2 Thespin-density for different µ such that 2(a), and 4(b) harmonic oscillator levels for ksoℓ = 0.4. The
spin-densityislargestclosetoeachbandbottom,wherethedensityofstatesisgreatest.
sizeofthespindensityiscomparabletotheinducedparticledensitysincethematrixelementinEq.(15)
isorderunityduetothestrongcouplingofadjacentoscillatorstateswithoppositespin(seeEq.(8))
To summarize, we have calculated the eigenspectrumof a gaussian quantumwire with Rashba inter-
action. The spectrumallows usto calculate the non-equilibriumspin densityanalytically. We presented
results for the σ density for a few low lying oscillator states but the formalism can well handle higher
z
statesalso.
Acknowledgements ThisworkwassupportedbytheIcelandicResearchFund,theSwissNSF,theNCCRNanoscience,
EUNoEMAGMANet,DARPA,ARO,ONR,JSTICORP,CNPq,andFAPESP.
References
[1] Semiconductor SpintronicsandQuantum Computation, editedbyD.D.Awschalom, D.Loss,andN.Samarth
(Springer-Verlag,Berlin,2002).
[2] J.Sinova,D.Culcer,Q.Niu,N.A.Sinitsyn,T.Jungwirth,andA.H.MacDonald, Phys.Rev.Lett.92,126603
(2004).
[3] O.ChalaevandD.Loss,Phys.Rev.B71,245318(2005).
[4] J.Schliemann,Int.J.Mod.Phys.B20,1015(2006).
[5] G.UsajandC.Balseiro,Europhys.Lett.72,631(2005).
[6] M.LeeandC.Bruder,Phys.Rev.B72,045343(2005).
[7] A.Reynoso,G.Usaj,andC.Balseiro,Phys.Rev.B73,115342(2006).
[8] J.Wang,K.Chan,andD.Xing,Phys.Rev.B73,033316(2006).
[9] J.YaoandZ.Yang,Phys.Rev.B73,033314(2006).
[10] S.BellucciandP.Onorato,Phys.Rev.B73,045329(2006).
[11] S.DebaldandB.Kramer,Phys.Rev.B71,115322((2005)).
[12] F.MiralesandG.Kircznow,Phys.Rev.B64,024426(2001).
[13] L.Serra,D.Sa´nchez,andR.Lopez,Phys.Rev.B72,235309(2005).
[14] M.GovernaleandU.Zu¨licke,Phys.Rev.B66,073311(2002).
[15] B.Nikolic´,L.Za´rbo,andS.Welack,Phys.Rev.B72,075335(2005).
[16] B.K.Nikolic´,S.Souma,L.P.Zaˆrbo,andJ.Sinova,Phys.Rev.Lett.95,046601(2005).
[17] E.I.Rashba,Phys.Rev.B68,241315R(2003).
[18] S.Datta,ElectronicTransportinMesoscopicSystems,(CambridgeUniversityPress,Cambridge,1995).
(cid:13)c 2003WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim