Interplay between quantum dissipation and an in-plane magnetic field in the spin ratchet effect

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by  Sergey Smirnov

Interplay between quantum dissipationand anin-plane magneticfieldinthe spinratchet effect
Sergey Smirnov,1 Dario Bercioux,2 Milena Grifoni,1 and Klaus Richter1
1Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany
2PhysikalischesInstitutandFreiburgInstituteforAdvancedStudies,Universita¨tFreiburg,D-79104Freiburg,Germany
(Dated:January14,2009)
Weinvestigatetheexistenceofthepurespinratcheteffectinadissipativequasi-one-dimensionalsystemwith
Rashbaspin-orbitinteraction. Thesystemisadditionallyplacedintoatransverseuniformstationaryin-plane
magneticfield.Itisshownthattheeffectexistsatlowtemperaturesandpurespincurrentscanbegeneratedby
9
applyinganunbiasedacdrivingtothesystem. Ananalyticalexpressionfortheratchetspincurrentisderived.
0
Fromthisexpressionitfollowsthatthespinratcheteffectappearsasaresultofthesimultaneouspresenceof
0
thespin-orbitinteraction,couplingbetweentheorbitaldegreesoffreedomandspatialasymmetry.Inthispaper
2
weconsiderthecaseofabrokenspatialsymmetrybyvirtueofasymmetricperiodicpotentials. Itturnsoutthat

n anexternalmagneticfielddoesnothaveanyimpactontheexistenceofthespinratcheteffect,butitinfluences
a itsefficiencyenhancingorreducingthemagnitudeofthespincurrent.
J

4 PACSnumbers:72.25.Dc,03.65.Yz,73.23.-b,05.60.Gg
1

] I. INTRODUCTION aparticleinsteadofonlythechargeone. Oneessentialdiffer-
l
l encebetweenspinandchargeisthataparticlecanhavemore
a
thanonespin state while ithasonlyonechargestate. In the
h Itiswellknownthatadirectedstationaryflowofparticles
- in a system can be created by unbiased external forces. In contextoftransportitisimportantthatthespinstateofaparti-
s
clecanstronglydependonthetransportconditions,inparticu-
e generalthispossibilityariseswhenthesystemisnotinvariant
m under reflections in real space. This fact is mainly indepen- laronthetransportdirection,asithappensforexampleinsys-
temswithspin-orbitinteraction. Thisfacthasfoundedanew
. dent of the mechanics which underpins the particle motion,
t arena for different spin devices used to store, transform and
a classicalorquantum.However,themicroscopicoriginofthis
m effect, conventionallycalled the ratcheteffect, is differentin transfermiscellaneousinformation.Thepossibilitytotransfer
thespinseparatelyfromchargeplaysanimportantrole. This
- the classicalandquantumcase. One principlesourceof that
d differenceis quantummechanicaltunnellingwhich does not can be implemented by so-called pure spin currents, that is
n spincurrentswhicharenotaccompaniedbychargecurrents.
haveanalogsintheclassicalmechanics.Correspondingly,one
o Thus the generation of such currents has been extensively
usually distinguishes between classical and quantum ratchet
c discussed. Among different mechanisms of spin-orbit inter-
[ effects. Inthispaperweconcentrateonthelatteroneinadis-
sipativesystem. SuchdissipativeratchetsystemsactasBrow- action Rashba spin-orbit interaction (RSOI)8 plays a distin-
2 nian motors1,2 turning Brownian into directed motion. The guishedrolebecauseitprovidesanopportunitytocontrolthe
v spin-orbitcouplingstrengthbyanexternalelectricfield. The
existence of the ratchet effect in a quantum dissipative one-
6 changeinthebandstructurespawnedbythespin-orbitinter-
dimensional (1D) system which lacks the spatial symmetry
9
actionleadstooneofthemostremarkableeffectsinspintron-
2 hasbeenfirsttheoreticallypredictedinRef.3. Later,withina
ics,theintrinsicspin-Halleffect,firstpredictedbyMurakami
1 tight-bindingmodelwherethelowestbandsarenarrow,ithas
et al.9 for hole-dopedsemiconductorswith the spin-orbitin-
. beendisclosedthataratchetstateoftheparticletransportcan
9
teraction of the effective Luttingermodel for holes and later
0 only be achieved when at least the two lowest Bloch bands
8 contribute to transport4. To obtain the ratchet effect in sys- by Sinova et al.10 in a high-mobility two-dimensional elec-
tron gas (2DEG) with RSOI. The spin current which results
0 temswithweakperiodicpotentialsatleasttwoharmonicsof
: thepotentialshouldenterthedynamicalequations5.Rectifica- fromtheintrinsicspin-Halleffectispureanditsexperimental
v detectionwasdiscussed,e.g.,byWunderlichetal.11 Another
i tioncanalsotakeplaceina single-bandtight-bindingmodel
X kindofspin-Halleffect,theextrinsicspin-Halleffect,isare-
where the spatial asymmetry is concealed from the electron
sult of the spin-orbit interaction as well. The spin currents
r dynamics.Onewaytoachievethisistouseunbiasedexternal
a forceswithharmonicmixing6. related to the extrinsic spin-Hall effect are also pure. Such
pure spin currentswere experimentallydetected throughop-
Coherentchargeratchetsbasedonmolecularwireswithan
ticalmeasuringofelectronspinaccumulationattheedgesof
asymmetric level structure of the orbital energies were pro-
thesamples12 andthroughthereciprocalspin-Halleffect13,14
posedinRef.7.Inthiscaseweakdissipationoriginatesfroma
inRef.15. Anotherapproachtocreatepurespincurrentsisto
weakcouplingbetweenthewireedgesandleadswhichrepre-
usepolarizedlight.Forexampleinnoncentrosymmetricsemi-
sentfermionicreservoirs.Incontrasttothesystemsdescribed
conductorsone-photonabsorption of linearly polarized light
abovein this system there is no dissipation in the wire. The
inducespurespincurrents16. Thepurespin currentresponse
ratcheteffectisaresultofthedissipativecouplingofthewire
to linearly and circularly polarized light irradiation, exciting
tofermionicbaths.
electronsfromvalencebandsintotheconductionbands,was
In a different branch of condensed matter a new research
studied by Li et al.17 and by Zhou et al.18 for 2DEGs with
fieldhasemergedduringthelastdecade,namelyspintronics,
RSOI. An alternativetechniqueof getting purespin currents
whereonetriestomakeuseofthespindegreeoffreedomof
2
isquantumspinpumping.Theideaofquantumspinpumping where Hˆ is the Hamiltonianof the isolated periodic system,
comesfromthegeneralideaofelectronpumping19. Electron Hˆ (t)describesanexternaldrivingandHˆ representsthe
ext bath
pumping assumes that in a given system any voltage bias is termresponsiblefordissipativeprocesses.
absentandtheparticleflowisaresultofacyclicvariationof Theisolatedquasi-1Dperiodicsystemisformedina2DEG
atleasttwoparametersofthissystem. Whentheelectronspin (x−zplane)withRSOIusingaperiodicpotentialalongthex-
is involved due to some mechanisms, various quantum spin axisandaharmonicconfinementalongthez-axis.Thewhole
pumpsemerge. Forexamplespinpumpsbasedonelectronic system isin a uniformstationarymagneticfield alongthez-
interactions20, magnetic barriers21, carbon nanotubes22 have axis:
been discussed. A spin pump based on a quantum dot was
experimentallyimplementedbyWatsonetal.23Thepurespin Hˆ = ~2kˆ2 + mω02zˆ2 − ~2kso σˆ kˆ −σˆ kˆ +
x z z x
currentgenerationusingthespinratcheteffectincoherentand 2m 2 m (2)
dissipative systems with RSOI was investigated in Refs. 24 zˆ2 (cid:0) (cid:1)
+U(xˆ) 1+γ −gµ σˆ H ,
and25,respectively.Thespinratcheteffectinthepresenceof L2 B z 0
(cid:18) (cid:19)
a non-uniform static magnetic field without spin-orbit inter-
where H is the z-component of the magnetic field H =
action, the Zeemanratcheteffect, was studied in Ref. 26 for 0 0
(0,0,H ), andwehaveusedthegaugeinwhichthecompo-
coherentquantumwiresformedina2DEG.However,thespin 0
nentsofthevectorpotentialareA = −H y,A = A = 0
ratcheteffectin a dissipative system in an externalmagnetic x 0 y z
(Landaugauge).Additionally,wehavetakenintoaccountthe
fieldhasnotbeenconsidereduptonow.
factthatina2DEGy =0. InEq. (2)theoperatorkˆisrelated
In this paper we extendthe results of Ref. 25 to includea
tothemomentumoperatorpˆ aspˆ = ~kˆ, ω istheharmonic
transversein-planeuniformstationarymagneticfield.Specifi- 0
cally,weconsidernon-interactingelectronsinaquantumwire confinementstrength, kso the spin-orbit interaction strength,
γ thestrengthofthecouplingbetweentheorbitaldegreesof
formedbyaharmonictransverseconfinementina2DEGwith
RSOI. Theelectronsarealsosubjecttoa1D periodicpoten- freedom x and z, g the electron spin g-factor, µB the Bohr
magneton, and U(xˆ) denotes the periodic potential with pe-
tial along the wire direction and the in-plane magnetic field
perpendicular to the wire. An orbital coupling between this riodL,
originallyisolatedsystemandanexternalenvironmentcauses
U(x+L)=U(x). (3)
dissipative processes affecting indirectly the spin dynamics
throughRSOI. In the following we assume that the periodic structure is
An externalac drivingoriginatesin our work from an ap- subject to an external homogeneous time-dependent electric
pliedacelectricfield. Weshowthatforsuchadrivingthenet field.Onlythex-componentoftheelectricfieldvectorisnon-
stationary charge current is strongly suppressed if the trans- zero,thatistheelectricfieldisparalleloranti-paralleltothe
portisgovernedonlybyelectronsoftheBlochsub-bandsre- x-axis. Experimentallythiscanbeimplementedusingforex-
lated to the same Bloch band which would result from the amplelinearlypolarizedlight.Theexternalforcethuscouples
correspondingtruly 1D problem without RSOI. However, at onlytothex-componentoftheelectroncoordinateoperator:
thesametimeandunderthesameconditionsanetstationary
Hˆ =−F(t)xˆ, (4)
spincurrentturnsouttobeactivatedina spatiallyasymmet- ext
ric situation and for finite values of the spin-orbit coupling where the force F(t) is unbiased. In this work we use the
strengthandthecouplingstrengthbetweentheorbitaldegrees time-dependence
offreedom. Themagneticfielddoesnotdestroythispicture,
F(t)=Fcos(Ω(t−t )). (5)
butitcanpartlyreduceoronthecontraryenhancetheratchet 0
effect.
The term ”unbiased external force” should not be confused
Thepaperisorganizedasfollows.InSectionIIwedescribe with voltage bias. An external force is called unbiased if it
amodelwithinwhicharatchetlikebehaviorofthespintrans- isperiodicintimeanditsmeanvalue,thatisitsaverageover
portcanbeachievedandpresentamasterequationintermsof one period, is equal to zero. It is obviously our case as one
populationsandtransitionratesbetweenthebasisstatesused canseefromEq.(5).
to calculate the charge and spin currents. These basis states The system is also coupled to an external bath. In the
arethenthoroughlydiscussedinSectionIII. Atight-binding present work we assume the transverse confinement to be
modelis formulatedin Section IV. In Section V we present strongenoughso thatthe probabilitiesof directbath-excited
the transition rates and their properties. Finally, in Section transitionsbetweenthetransversemodesarenegligiblysmall.
VI we derive analytical expressions for the charge and spin In other words, the wire is truly 1D from the point of view
currentsandexplorethespinratcheteffectinthesystem. of the bath which directly changes only the dynamics along
the wire. Thus in our model the external environment cou-
plestotheelectronicdegreesoffreedomonlythroughxˆ. The
II. FORMULATIONOFTHEPROBLEM bathitselfaswellasitsinteractionwiththequantumwireare
describedwithintheCaldeira-Leggettmodel27,28,
ThefullHamiltonianofourproblemis
1 NO pˆ2 c 2
Hˆ = α +m ω2 xˆ − α xˆ . (6)
Hˆfull(t)=Hˆ +Hˆext(t)+Hˆbath, (1) bath 2α=1(cid:20)mα α α(cid:18) α mαωα2 (cid:19) (cid:21)
X
3
Thebathisfullycharacterizedbyitsspectraldensitydefined that is when the system turns into an insulator, the spin cur-
as rentgivenbyEq. (11)goestozero. Belowwewillcalculate
only the spin current polarized along the z-axis and denote
π NO c2 this current as J , i.e., J (t) ≡ Jz(t). The components of
J(ω)≡ α δ(ω−ω ). (7) S S S
2 m ω α thespin currentpolarizedalongthe xandy axesare zeroas
α α
αX=1 shown in Appendix B. The discussion of the difference be-
Itisimportanttoemphasizethat,duetothespin-orbitinterac- tween the conventionaldefinitionof the spin currentand the
tionandorbit-orbitcoupling,thedirectdissipativeinteraction spincurrentdefinitionusedinourworkcanalsobefoundin
betweenthelongitudinaldynamicsinthewireandtheexter- AppendixB.
nalenvironmenthasanindirectimpactonthetransitionrates It is convenientto calculate the traces in (9) and (11) us-
betweendifferenttransversemodes.Thetransversedynamics ingthebasiswhichdiagonalizesbothxˆandσˆz, becausethis
in the wire indirectly feels the presence of the external bath requires to determine only the diagonal elements of the re-
throughthespin-orbitinteractionandorbit-orbitcoupling. duced density matrix. In a quasi-1D periodic system with
Thedynamicalquantitiesofinterestarethechargeandspin RSOI the energy spectrum can be related to the one of the
currents. Specifically, the longitudinal charge current J (t) correspondingtruly 1D problemwithoutRSOI34. This links
C
is given (see for example Ref. 4) as a statistical average of the Bloch bands of that truly 1D problem to the Bloch sub-
thelongitudinalchargecurrentoperatorJˆ (t), i.e. the prod- bandsofthe quasi-1Dproblem. The generalstructureof the
C
uctofthevelocityoperatorvˆ(t)andtheelementaryelectronic results obtained in Ref. 34 is retained in the presence of the
charge−e, orbit-orbit coupling and a uniform stationary magnetic field
along the z-axis. A slight change of the theory is given in
Jˆ (t)=−evˆ(t), (8) AppendixC. Weshallconsiderelectrontransportundersuch
C
conditionswhenonlyafinitenumberoftheBlochsub-bands
isinvolved. Thebasiswhichdiagonalizesxˆandσˆ becomes
z
d
inthiscasediscrete. ThetotalnumberoftheBlochsub-bands
J (t)=−e Tr[xˆρˆ(t)], (9)
C
dt isequaltotheproductofthenumber,N ,oftheBlochbands
B
from the corresponding truly 1D problem without magnetic
where ρˆ(t) = Tr Wˆ(t) is the reducedstatistical operator
bath fieldandwithoutspin-orbitcoupling,thenumber,N , ofthe
ofthesystem,thatisthefulloneWˆ(t)withthebathdegrees t
transverse modes and the number of the spin states. In this
offreedomtracedout.
work we shall use the model with N = 1, N = 2. Since
B t
For the longitudinalspin currentoperatorwe use the defi-
there are only two spin states, the totalnumberof the Bloch
nitionsuggestedbyShietal.29,
sub-bandsinourproblemisequaltofour. Therepresentation
in terms of the eigen-states of the coordinate operator for a
d
Jˆi(t)= σˆ xˆ , (10) model with discrete x-values is called discrete variable rep-
S dt i
resentation (DVR)35. Let us call σ-DVR the representation
(cid:0) (cid:1)
which was further developed and applied to a two- inwhichboththecoordinateandspinoperatorsarediagonal.
dimensionalholegasbyZhangetal.30 Theadvantageofthis Denotingtheσ-DVRbasisstatesas{|αi}andeigen-valuesof
definitionovertheconventionalone(JˆSi = σˆivˆ)isthatusing xˆ and σˆz in a state |αi throughxα and σα, respectively, the
thecorrespondingspincurrent, chargeandspincurrents(9)and(11)arerewrittenas
d
JSi(t)= ddtTr σˆixˆρˆ(t) , (11) JC(t)=−eXα xαdtPα(t), (12)
(cid:0) (cid:1) d
thecontinuityequationforthespindensitycanoftenbewrit- J (t)= σ x P (t),
S α α α
tenwithouta sourceterm,whichmeansthatthespincurrent dt
α
X
definedinthiswayisconserved. Thisconservedspincurrent
where P (t) ≡ hα|ρˆ(t)|αi is the population of the σ-DVR
canbeuniquelyrelatedto thespinaccumulationata sample α
state|αiattimet.
boundary. Theout-of-planepolarizedspin accumulationcan
We are interested in the long-time limit of the currents
experimentallybemeasuredwithKerrrotationmicroscopy31
J¯ (t)andJ¯ (t)averagedoverthedrivingperiodT = 2π/Ω
ortheFaradayrotationtechnique32. Thein-planespinpolar- C S
with the time average of a time dependentfunctionf(t) de-
izationisnotdirectlymeasuredbyKerrrotationmicroscopy,
butitcanstillbescannedbyamagneto-opticKerrmicroscope finedasf¯(t)≡(1/T) t+T dt′f(t′). From(12)itfollows
t
using, e.g.,the cleavededgetechnologyasdiscussed byKo-
R d
tissek et al.33 Even when the continuity equation contains a J¯ (t)=−e x P¯ (t),
C α α
dt
source term, there is still one advantage of the spin current Xα (13)
operator definition (10). This definition leads to a very rea- d
J¯ (t)= σ x P¯ (t).
sonablephysicalresult:thecorrespondingspincurrentin(11) S α αdt α
vanishes in insulators. In Section VI we will return to this Xα
pointand analyticallyprovethat when the periodicpotential Theadvantageofworkingintheσ-DVRbasisisthatreal-
gets stronger and as a result the energy bands get narrower, timepathintegraltechniquescanbeusedtoexactlytraceout
4
the bath degrees of freedom36,37. Moreover, at driving fre- for example, the system is divided into the elementary cells
quencies larger than the ones characterizing the internal dy- in such a way that the origin of coordinates is at the center
namicsofthequasi-1Dsystemcoupledtothebath,theaver- of an elementary cell, then −L/2 < d 6 L/2. In Eq.
γ;ζ,j
aged populationsP¯ (t) can be found from the master equa- (18) we have taken into account that the periodic potential
α
tion, U (x),introducedinAppendixC,dependsonγ andj,and
γ,j
thus the eigen-values distributed within one elementary cell
d
P¯ (t)= Γ¯ P¯ (t)− Γ¯ P¯ (t), (14) alsoacquireadependenceonγ andj.
α αβ β βα α
dt From(16)and(18)itfollowsthatonecanlabeltheeigen-
β β
X X
(β6=α) (β6=α) states of xˆ with the quantum numbers ζ, m, j, σ, that is
as |ζ,m,j,σi , and in the {|l,k ,j,σi } representation
valid at long times. In Eq. (14) Γ¯ is an averaged transi- γ,j B γ,j
αβ theseeigen-stateshavetheform:
tionratefromthestate|βitothestate|αi. Inordertoobtain
concreteexpressionsfortheaveragedcurrentstheσ-DVRba- γ,j′hl,kB,j′,σ′|ζ,m,j,σiγ,j =
(19)
sis must be found explicitly. This is the subject of the next =δj′,jδσ′,σ γ,jhl,kB,j,σ|ζ,m,j,σiγ,j.
section.
The corresponding eigen-values are x = x .
γ;ζ,m,j,σ γ;ζ,m,j
Fromtheeigen-valueequation
III. DIAGONALIZATIONOFσˆz ANDxˆ:THEσ-DVR xˆ|ζ,m,j,σiγ,j =xγ;ζ,m,j|ζ,m,j,σiγ,j (20)
BASIS
writteninthe{|l,k ,j,σi }representationthroughtheuse
B γ,j
of(16),
The eigen-statesof the σˆ operatorwere foundin Ref. 34
z
(see Eq. (12) therein) for a model without coupling be- hl,k +σk |xˆ|l′,k′ +σk i ×
γ,j B so B so γ,j
tweentheorbitaldegreesoffreedomandmagneticfield. The lX′,kB′ (21)
changesnecessaryto includethose two effectsare discussed × hl′,k′ ,j,σ|ζ,m,j,σi =
inAppendixC. Theeigen-valueequationfortheσˆ operator γ,j B γ,j
z
=x hl,k ,j,σ|ζ,m,j,σi ,
is γ;ζ,m,j γ,j B γ,j
itfollowsthat
σˆ |l,k ,j,σi =σ|l,k ,j,σi . (15)
z B γ,j B γ,j hl,k ,j,σ =1|ζ,m,j,σ =1i =
γ,j B γ,j
InEq. (15)l,kB,j,σ standfortheBlochbandindex,quasi- = γ,jhl,kB+kso|ζ,miγ,j,
(22)
momentum, transverse mode index and z-projection of the
hl,k ,j,σ =−1|ζ,m,j,σ =−1i =
γ,j B γ,j
spin,respectively.Sinceinthepresenceoftheorbit-orbitcou-
= hl,k −k |ζ,mi .
plingtheperiodicpotentialU (x)(seeAppendixC)depends γ,j B so γ,j
γ,j
onγandj,wehavelabeledtheket-symbolwiththesubscript Since |ζ,m,j,σi is also the eigen-state of σˆ corre-
γ,j z
γ,j. Intheensuinganalysiswe followthesameruleandla- sponding to the eigen-value σ = σ, we infer that the
ζ,m,j,σ
belallthebra-andket-symbolswiththesubscriptγ,j,thatis σ-DVRbasisstates|αifromtheprevioussectionarejustthe
γ,jh···|and|···iγ,j. |ζ,m,j,σiγ,j states,thatis{|αi}≡{|ζ,m,j,σiγ,j}.
Itisconvenienttostartthediagonalizationofthecoordinate
operatorwritingitsmatrixinthe{|l,k ,j,σi }representa-
B γ,j
tion: IV. σ-DVRREPRESENTATIONANDITSTIGHT-BINDING
MODEL
γ,j′hl′,kB′ ,j′,σ′|xˆ|l,kB,j,σiγ,j = (16)
=δj′,jδσ′,σ γ,jhl′,kB′ +σkso|xˆ|l,kB+σksoiγ,j. Let us represent the Hamiltonian Hˆ in the σ-DVR basis
obtainedintheprevioussectioninordertoderiveaneffective
Thediagonalblocks,
tight-bindingmodel.
γ,jhl′,kB′ ,j,σ =1|xˆ|l,kB,j,σ =1iγ,j = Usingthe{|ζ,m,j,σiγ,j}basistheHamiltonianHˆ canbe
= hl′,k′ +k |xˆ|l,k +k i , ∀j, writtenas
γ,jhγl′,,jkB′ ,j,Bσ =s−o1|xˆ|l,kBB,j,sσo=γ,j−1iγ,j = (17) Hˆ = γ,j′hζ′,m′,j′,σ′|Hˆ|ζ,m,j,σiγ,j×
= γ,jhl′,kB′ −kso|xˆ|l,kB−ksoiγ,j, ∀j, ζ′ζ,m,Xm′,,jj,′σ,σ′ (23)
areunitaryequivalentforagivenvalueoftheindexjandthus ×|ζ′,m′,j′,σ′iγ,j′ γ,jhζ,m,j,σ|,
theeigen-valuesofxˆdonotdependonσ. withthematrix
AsitisshowninAppendixA,theeigen-valuesofthema-
trix hl′,k′ |xˆ|l,k i are γ,j′hζ′,m′,j′,σ′|Hˆ|ζ,m,j,σiγ,j = εγ;l,η(kB)×
γ,j B B γ,j
l,XkB,η
xγ;ζ,m,j =mL+dγ;ζ,j, (18) × γ,j′hζ′,m′|l,kB+σ′ksoiγ,j′× (24)
where m = 0,±1,±2..., ζ = 1,2,...,NB and the eigen- × γ,jhl,kB+σkso|ζ,miγ,j θγ;l,kB,η(j′,σ′)×
values d are distributed within one elementary cell. If, ×θ∗ (j,σ).
γ;ζ,j γ;l,kB,η
5
Thetight-bindingapproximationof(23)isobtainedifone wefinallyhave
assumesthatthematrixelements(24)with|m′−m|>1are
4
negligiblysmall.
Hˆ = ε |m,ξi hm,ξ|+
We consider temperatures low enough and assume that γ;ξ γ,ξ γ,ξ
m (cid:20)ξ=1
electronspopulateonlythelowestBlochsub-bandswithl=1 X X
(i.e.,N =1).Underthisconditiontheperiodicpotentialcan 4
beofarBbitraryshapeandtheonlylimitationonitisthevalid- + ∆iγn;tξr′a,ξ(m)|m,ξ′iγ,ξ′ γ,ξhm,ξ|+
ityofthetight-bindingapproximation. ξ6=Xξ′=1 (31)
Below we thoroughly study the case where the four low- 4
est Bloch sub-bands are the ones with l = 1, η = 1,2,3,4 + ∆iγn;tξe′r,ξ,b(m)|m,ξ′iγ,ξ′ γ,ξhm+1,ξ|+
and the only ones which are populated with electrons. For ξ,ξ′=1(cid:18)
X
simplicity we consider weak orbit-orbit coupling and calcu-
late the corresponding eigen-energies ε (k ) and eigen- +∆iγn;tξe′r,ξ,f(m)|m+1,ξ′iγ,ξ′ γ,ξhm,ξ| .
γ;l,η B
spinors θ (j,σ) retaining only the first two transverse (cid:19)(cid:21)
modes,thγa;tl,iksBj,η=0,1. InthiscaseHˆ hastheform Equation(31)representsatight-bindingmodelwhichcannow
beusedtoperformactualcalculationsofquantumtransportin
adissipativesystem.
Hˆ = εγ;j,σ|m,j,σiγ,j γ,jhm,j,σ|+ Toconcludethissection,wewouldliketonotethatbecause
m (cid:20)j,σ ofthesimultaneouspresenceoftheharmonicconfinementand
X X
+ ∆iγn;tjr,σa′;j,σ(m)|m,j,σ′iγ,j γ,jhm,j,σ|+ RsySsOteImthise csyhsatreamctesrpizlietds ibnytoξtw=o1s,u4bsaynsdtemthse.seTchoendfirostnesubby-
j,σ′6=σ
X ξ = 2,3. Thesesubsystemsaretotallydecoupled:thereisno
+ ∆iγn;tjr′a,σ′;j,σ(m)|m,j′,σ′iγ,j′ γ,jhm,j,σ|+ (25) electronexchangebetweenthem. Sucha state ofaffairsper-
j′6=j,σ′,σ sistsifoneconsidersmorethantwotransversemodes. Inthis
X
work,forsimplicity,weonlyconsideronesubsystem,namely
+ ∆iγn;tje′r,σ,b′;j,σ(m)|m,j′,σ′iγ,j′ γ,jhm+1,j,σ|+ theonewithξ =1,4.Suchuncoupledsubsystemsalsoappear
j′,j,σ′,(cid:18)σ withinthehardwallmodelofthetransverseconfinement38.
X
+∆iγn;tje′r,σ,f′;j,σ(m)|m+1,j′,σ′iγ,j′ γ,jhm,j,σ| ,
(cid:19)(cid:21) V. TRANSITIONRATES
where
The tight-binding model introduced in Section IV relies
|m,j,σi ≡|ζ =1,m,j,σi , (26) uponthefactthatthehoppingmatrixelements(27)aresmall.
γ,j γ,j
Inthiscasethesecond-orderapproximationfortheaveraged
and we have defined the on-site energies ε and hop- transitionratesinEq. (14)canbeusedgiving4,39
γ;j,σ
∆piniγng;tje′r,mσ,f′a;jt,rσix(me)leamsefonltlsow∆siγn;tjr′a,σ′;j,σ(m), ∆iγn;tje′r,σ,b′;j,σ(m) and Γ¯mγ;ξ′,′m,ξ = |∆mγ~;ξ′2,′m,ξ|2×
∞
εγ;j,σ ≡ γ,jhm,j,σ|Hˆ|m,j,σiγ,j, × dτe−[(xγ;m,ξ−xγ;m′,ξ′)2/~]Q(τ)+i[(εγ;ξ−εγ;ξ′)/~]τ× (32)
∆iγn;tjr′a,σ′;j,σ(m)(j′,σ′≡)6=(j,σ)γ,j′hm,j′,σ′|Hˆ|m,j,σiγ,j, (27) Z−∞ ×J 2F(xγ;m,ξ−xγ;m′,ξ′)sin Ωτ ,
∆iγn;tje′r,σ,b′;j,σ(m)≡ γ,j′hm,j′,σ′|Hˆ|m+1,j,σiγ,j, 0(cid:20) ~Ω (cid:18) 2 (cid:19)(cid:21)
∆iγn;tje′r,σ,f′;j,σ(m)≡ γ,j′hm+1,j′,σ′|Hˆ|m,j,σiγ,j. w∆hmγe;ξ′r,′em,ξx≡γ;mγ,,ξξ′≡hmx′γ,;ξζ′=|H1ˆ,m|m,ξ,=ξiγm,ξLth+ehdoγp;ξpwinigthmdaγt;rξix≡eldeγm;1e,njt,
Notethat betweenthestates |m′,ξ′iγ,ξ′ and|m,ξiγ,ξ, J0(x) the zero-
orderBesselfunctionandQ(τ)thetwiceintegratedbathcor-
[∆iγn;tjr′a,σ′;j,σ(m)]∗ =∆iγn;tjr,σa;j′,σ′(m), (28) relationfunction37:
1 ∞ J(ω) ~ωβ
Q(τ)= dω coth ×
π ω2 2
[∆inter,b (m)]∗ =∆inter,f (m). (29) Z0 (cid:20) (cid:18) (cid:19) (33)
γ;j′,σ′;j,σ γ;j,σ;j′,σ′
×[1−cos(ωτ)]+isin(ωτ) ,
Introducingthenotations
(cid:21)
whereJ(ω)isgivenbyEq. (7)andβ istheinversetempera-
{ξ}≡{(j,σ)},
ture.
ξ =1⇔(0,1), ξ =2⇔(0,−1), (30) The transition rates are functions of the orbit-orbit cou-
ξ =3⇔(1,1), ξ =4⇔(1,−1), pling strength γ because the Bloch amplitudes as well as
6
the difference ∆d ≡ d −d depend on γ. Within ∆inter,b(m) and ∆inter,f(m) only through a phase factor as
γ γ;1,0 γ;1,1 γ;ξ′,ξ γ;ξ′,ξ
thecontextofthetight-bindingmodeltheeigen-valuesd itisshowninAppendixD. From(29)and(36)itfollowsthat
γ;1,0
and d tend to zero and fulfil ∆d /l ≪ 1, where l =
γ;1,1 γ r r
min[L, ~/mω0,~Ω/F,...]. Consequently, the transition Γ¯iγn;tξe,rξ,b =Γ¯iγn;tξe,rξ,f, (38)
rates depend on γ predominantly through the Bloch ampli-
p
tudes, and in this work we pay no regard to terms of order
O(∆d /l ). Thisisalsoconsistentwithourmodeltakinginto
γ r Γ¯inter,bΓ¯inter,b =Γ¯inter,fΓ¯inter,f. (39)
account only the first two transverse modes. Keeping terms γ;ξ′,ξ γ;ξ,ξ′ γ;ξ′,ξ γ;ξ,ξ′
of order O(∆d /l ) would mean that the strength γ of the
γ r
orbit-orbitcouplingislargeenoughsothatonewouldneedto To calculate the charge and spin currents we additionally
considermorethanjustthefirsttwotransversemodesbecause needthetransitionrates
in this case the non-diagonalelementswould be comparable
withthediagonalones. Γ¯γ;ξ,ξ′ ≡Γ¯iγn;tξe,rξ,′f +Γ¯iγn;tξr,aξ′ +Γ¯iγn;tξe,rξ,′b. (40)
Usingthenotations,
As pointed out at the end of Section IV, the system is split
Γ¯mγ;ξ,m′,ξ ≡Γ¯iγn;tξr′a,ξ, ξ′ 6=ξ, intotwosubsystemsisolatedfromeachother. Sinceelectron
exchangebetweenthesubsystemsisabsentonecanwrite
Γ¯m,m+1 ≡Γ¯inter,b, (34)
γ;ξ′,ξ γ;ξ′,ξ
Γ¯m+1,m ≡Γ¯inter,f, Γ¯ =Γ¯ =Γ¯ =Γ¯ =
γ;ξ′,ξ γ;ξ′,ξ γ;1,2 γ;1,3 γ;2,1 γ;2,4 (41)
=Γ¯ =Γ¯ =Γ¯ =Γ¯ =0.
from(32)oneobtains γ;3,1 γ;3,4 γ;4,2 γ;4,3
Γ¯iγn;tξr′a,ξ =0, (35) Tsihgenilfiacsatnetqlyuasliimtipeslifayrethveeeryxpurseesfsuiolnbsefcoarustheethcheyargaleloawndusspitno
and currentswhicharederivedinthenextsection.
Γ¯iγn;tξe′r,ξ,b =|∆iγn;tξe′r,ξ,b(m)|2Jγ;ξ′,ξ,
(36) VI. CHARGEANDSPINCURRENTS
Γ¯iγn;tξe′r,ξ,f =|∆iγn;tξe′r,ξ,f(m)|2Jγ;ξ′,ξ,
where Theexpressionsforthestationaryaveragedchargeandspin
currents,
Jγ;ξ′,ξ = ~12 Z−∞∞dτe−L~2Q(τ)+i[(εγ;ξ−εγ;ξ′)/~]τ× (37) J¯C∞ ≡tl→im∞J¯C(t), J¯S∞ ≡tl→im∞J¯S(t), (42)
2FL Ωτ
×J sin .
0 ~Ω 2 can be found from the averaged master equation (14) which
(cid:20) (cid:18) (cid:19)(cid:21)
we rewrite here using the σ-DVR indices and tight-binding
Note that Γ¯inter,b and Γ¯inter,f do not depend on m due approximationintroducedinSectionIVandutilizingtheno-
γ;ξ′,ξ γ;ξ′,ξ
to the Bloch theorem which leads to an m-dependence of tationsofSectionVforthetransitionrates:
4 4
d
dtP¯γm;ξ(t)= Γ¯iγn;tξe,rξ,′fP¯γm;ξ−′1(t)+Γ¯iγn;tξr,aξ′P¯γm;ξ′(t)+Γ¯iγn;tξe,rξ,′bP¯γm;ξ+′1(t) − Γ¯iγn;tξe′r,ξ,b+Γ¯iγn;tξr′a,ξ+Γ¯iγn;tξe′r,ξ,f P¯γm;ξ(t)+
(ξξX′′=6=1ξ)(cid:2) (cid:3) (ξξX′′6==ξ1)(cid:2) (cid:3) (43)
+ Γ¯inter,fP¯m−1(t)+Γ¯inter,bP¯m+1(t) − Γ¯inter,b+Γ¯inter,f P¯m(t),
γ;ξ,ξ γ;ξ γ;ξ,ξ γ;ξ γ;ξ,ξ γ;ξ,ξ γ;ξ
(cid:2) (cid:3) (cid:2) (cid:3)
4
From(12)and(43)onefinds J¯S∞ = dγ;ξσξ −dγ;ξ′σξ′ Γ¯iγn;tξe,rξ,′f +Γ¯γin;tξe,rξ,′b +
(45)
ξ,ξ′=1
X (cid:2)(cid:0) (cid:1)(cid:0) (cid:1)
+Lσξ Γ¯iγn;tξe,rξ,′f −Γ¯iγn;tξe,rξ,′b p∞γ;ξ′,
(cid:0) (cid:1)(cid:3)
4 where we have used Eq. (18). To derive Eq. (45) we have
J¯C∞ =−eL Γ¯iγn;tξe,rξ,′f −Γ¯iγn;tξe,rξ,′b p∞γ;ξ′, (44) additionallymadeuseofEq. (35). InEq. (45)σξ ≡σζ=1,m,ξ
ξ,ξ′=1 andσ1 =σ3 =1, σ2 =σ4 =−1asitfollowsfromEq.(30).
X (cid:2) (cid:3)
7
Thequantitiesp∞ aredefinedas couplingtoanexternalenvironment.TogetEq. (53)wehave
γ;ξ
eliminatedfromJ¯∞ theequilibriumspincurrentarisingdue
S
p (t)≡ P¯m(t), p∞ ≡ limp (t), (46) tothenon-compensation40ofthespincurrentsfromdifferent
γ;ξ γ;ξ γ;ξ γ;ξ
t→∞ bandsoftheRashba-Blochspectrumoftheisolatedsystem.It
m
X turnsoutthatthiseffectisstrongenoughtoindentureinadis-
andtheysatisfytheconstraint sipativesystem. Belowweonlyconsiderthenon-equilibrium
spincurrent,J¯∞ ,andnotthefullone,J¯∞.
p (t)+p (t)+p (t)+p (t)=1, ∀ t. (47) n−e,S S
γ;1 γ;2 γ;3 γ;4 Let us at this point also mention the dependence of the
spincurrentJ¯∞ onthemagneticfieldH . Sincethemag-
AsalreadymentionedattheendofSectionIV,weonlycon- n−e,S 0
neticfieldisappliedalongthez-axis,itcouplestothesystem
siderthesubsystemwithξ = 1,4. Thepropertiesofthesta-
throughtheσˆ operatorandthusthehoppingmatrixelements
tionaryaveragedtransportdonotdependoninitialconditions. z
∆inter,f (m)donotdependonH . Itthenfollowsthatthe
Wechoosethefollowingones: γ;1(4),4(1) 0
spin current depends on the magnetic field only through its
pγ;1(t=0)=1, pγ;4(t=0)=0. (48) dissipativeprefactor. ThedependenceonH0 comesintoplay
throughtheon-siteenergiesε . Thedifferenceε −ε
γ;1(4) γ;4 γ;1
Becauseoftheconstraint(47)pγ;2(t=0)=pγ;3(t=0)=0 whichenterstheintegralsJ andJ(0) canbe
and since there is no electron exchangebetween the subsys- γ;1(4),4(1) γ;1(4),4(1))
writtenas:
tems,thestatesofthesubsystemwithξ = 2,3remainempty
at any time, p (t) = p (t) = 0, ∀t. This leads to 1
γ;2 γ;3 ε −ε = ε(0) (k )−ε(0) (k ) +
p∞ = p∞ = 0. Then from the master equation (43) with γ;4 γ;1 N γ,1;1 B γ,0;1 B
γ;2 γ;3 (54)
theinitialconditions(48)andusing(40),(41)oneobtains XkB (cid:2) (cid:3)
+~ω +2gµ H ,
0 B 0
Γ¯ Γ¯
p∞γ;1 = Γ¯γ;1,4γ+;1,Γ¯4γ;4,1, p∞γ;4 = Γ¯γ;1,4γ+;4,Γ¯1γ;4,1. (49) whereN isthenumberoftheelementarycellsandε(γ0,)j;l(kB)
aretheeigen-valuesofthetruly1DHamiltonian
UsingEqs. (35),(38)-(40)and(49)itfollowsfrom(44)
~2kˆ2 ~ 1
J¯C∞ =0, (50) Hˆ01;Dγ,j ≡ 2mx +U(xˆ) 1+γmω L2 j+ 2 . (55)
(cid:20) 0 (cid:18) (cid:19)(cid:21)
thatistheabsenceofthestationaryaveragedchargetransport.
Therefore,inthepresenceofatransversein-planeuniform
However,usingEqs.(35),(38),(40)and(49)wegetfromEq.
stationary magnetic field the existence of the spin current is
(45)
possible under the same conditionswhich were discussed in
Ref.25. Forcompletenesswelisttheseconditionsbelow.
2L
J¯∞= Γ¯inter,fΓ¯inter,b−Γ¯inter,bΓ¯inter,f . (51) From(53)onefinds,asmentionedinSectionII,thatwhen
S Γ¯ +Γ¯ γ;1,4 γ;4,1 γ;1,4 γ;4,1
γ;1,4 γ;4,1 theelectronicstatesbecomelocalized,thestationaryaveraged
(cid:0) (cid:1)
spincurrentvanishes.Indeed,inthisinsulatinglimitthefunc-
The last expressioncan be rewritten in termsof the hopping
matrixelements∆iγn;tξe′r,ξ,f(m). MakinguseofEqs. (29),(35), tkio′nanFdγ;EkBq,.k(B′53d)oegsivneostzdeerop.endonthequasi-momentakB and
(36)and(40)wederivethestationaryaveragedspincurrent: B
When the spin-orbitinteractionis absent, that is k = 0,
so
J J wegetfrom(53)
J¯∞ =2L γ;1,4 γ;4,1 ×
S Jγ;1,4+Jγ;4,1 (52) J¯∞ =0. (56)
× |∆inter,f(m)|2−|∆inter,f(m)|2 . n−e,S kso=0
γ;1,4 γ;4,1
(cid:12)
Further, if the orbital degr(cid:12)ees of freedom x and z are not
UsingEqs. (52(cid:0))and(D2)thenon-equilibrium(cid:1)stationaryav- coupled,thatisγ =0,itfollowsfromEqs. (53)and(D6)that
eragedspincurrentcanbewrittenas
J¯∞ =0. (57)
J J J(0) J(0) n−e,S γ=0
J¯∞ =−2 γ;1,4 γ;4,1 − γ;1,4 γ;4,1 × (cid:12)
n−e,S (cid:18)Jγ;1,4+Jγ;4,1 Jγ(0;1),4+Jγ(0;4),1(cid:19) (53) amFpilnitaulldye,siafrethreeaplearnioddwicepfiont(cid:12)denftrioaml isEqsys.m(5m3e)traincd, t(hDe4)Bloch
L~3k2 ω
× mso 0 sin[(kB−kB′ )L]Im[Fγ;kB,kB′ ], J¯n∞−e,S =0, for symmetric periodic potentials. (58)
kXB,kB′
Summarizingtheresultsofthissectionweconcludethatin
whereJ(0) isgivenbyEq. (37)withF = 0andthefunc- ordertogenerateafinitestationaryaveragedspincurrentthree
γ;ξ′,ξ
tionFγ;kB,kB′ isdefinedbyEq.(D3).NotethestructureofEq. conditionsmustsimultaneouslybefulfilled:1)presenceofthe
(53). Itistheproductoftwofactorsofdifferentphysicalori- spin-orbitinteractionintheisolatedsystem;2)finitecoupling
gin.Thefactorinthesecondlinedescribestheisolatedsystem betweentheorbitaldegreesoffreedomxandz;3)absenceof
andthe factorin thefirst line ispurelydueto thedissipative therealspaceinversioncenterintheisolatedsystem.
8
Amongthesethreeconditionsthesecondconditionisper-
hapsless transparentand a simplified physicalinterpretation
] 4
isnecessary. Weproposethefollowingphysicalexplanation. ω 0 gµH = 0.0
The orbit-orbit coupling leads to the situation in which the L gµΒH0 = -0.05
ttswhrtorierenenw.ggirTtrhoehueoapnfpdvtehergleiooectpdsieitcsyritporioosdntilecganreptgrioeactrleloiinsnsteieatrqhluteovacalierttisnoetserUadalg(cxrereos)g.siisnTonththhuoeesfqctthuehenaesteeiw-lre1iorcDe-f ∞- 5J [10n-e,S 02 gggµµµΒΒΒΒHHH0000 === 0-00..03.35
anddecreasesclosertotheedges. Atthesametimetheelec- nt
e
trondistributionacrossthechanneldependsonthetransverse rr -2
u
modej. ItisgivenbytheHermitepolynomials.Forj =0the c
n
electronspopulatethecenterofthewirewhileforj =1they pi
S -4
aredistributedinregionsclosertotheedges. Hence,theelec-
tronswithj = 0arefasterthanthosewithj =1. Becauseof
themixingbetweentheconfinementandRSOIdifferenttrans- 0 1 2 3 4
Amplitude of the driving force, F [ h¯ ω/L ]
verse modes carry different spin states. Therefore, we con- 0
cludethatdifferentspinstateshavedifferentgroupvelocities
alongthewire. Thisdifferenceresultsinafinitelongitudinal FIG. 1: (Color online) Non-equilibrium spin current, J¯n∞−e,S, as a
spincurrent. functionoftheamplitude, F, ofthedrivingforcefordifferent val-
Finally,oneobservesthatatransversein-planeuniformsta- uesofthez-projectionofthemagneticfieldH0.Furtherparameters:
tionarymagneticfieldaloneisnotenoughtoproducethespin temperature kBoltz.T = 0.5, spin-orbit coupling strength kso with
k L = π/2, orbit-orbit coupling strength γ = 0.08, driving fre-
currentinadrivendissipativesystem. Themagneticfieldcan so
quencyΩ=0.2,viscositycoefficientη=0.08.
onlyaffectthemagnitudeofthespin currentwhenthe prop-
ertiesoftheisolatedsystemmeetthethreeconditionsderived
above. 0.5
]
ω 0
L 0
VII. RESULTS - 40
1
[
forInthtehitsheseocretitoicnawlmeoshdoelwdseovmeleopreedsuilntstohbetapirneevdionuusmseecrtiicoanllsy. ∞J n-e,S -0.5
Astrsucatnureex.amThpelevawlueecsoonfsitdheercaonrreInspGoanAdsin/IgnPpaqraumanettuemrs uwsierde rrent -1 ggµµΒΒHH00 == 0-0.0.05
to get the results are similar to the ones from the work of n cu ggµµΒHH0 == 0-0.0.35
~S2ckha¨p/emrs=et9a.9l.441×I1n0−p1a2rteicVu·lmar,(w~hωic0hg=ive0s.k225=m4e.V8,2×α10≡6 Spi -1.5 gµΒΒH00 = 0.3
so so
m−1), m = 0.037m (m is the free-electron mass). The
0 0
value, g = 7.5, of the electron spin g-factor (in our nota- 0 1 2
tionsg ≡ −g∗/2,whereg∗ istheeffectivegyroscopicfactor Spin-orbit coupling strength, k L/π
so
measuredexperimentally)istakenfromRef.42. Fromthese
parametersandforexamplefortheperiodofthesuper-lattice FIG. 2: (Color online) Non-equilibrium spin current, J¯n∞−e,S, as a
L = 2.5 ~/mω0 ≈ 0.24 µm, which is easily achievable functionofthespin-orbitcouplingstrength,kso,fordifferentvalues
technologicallyatpresent43,itfollowsthatksoL≈0.368π. ofthez-projectionofthemagneticfieldH0. Thedrivingamplitude
Theasypmmetricperiodicpotentialis isF =1.0~ω0/L.TheotherparametersareasinFig.1.
2πx
U(x)=~ω0 2.6 1−cos −1.9 + Topresentthe resultswe usein allthe figurestheunitsof
L
(cid:26) (cid:20) (cid:18) (cid:19)(cid:21) (59) ~ω0 and ω0 for energies and frequencies, respectively. The
4πx viscositycoefficientistakeninunitsofmω .
+1.9cos . 0
L Letusdiscusspossiblevaluesofthedrivingparameters.In
(cid:18) (cid:19)(cid:27)
adissipationlesssystem(orinasystemwithweakdissipation)
ThebathisassumedtobeOhmicwithexponentialcutoff: of size L one should restrict possible values of the driving
amplitudeandfrequency,0<FL<~ω and0<Ω<ω ,in
0 0
ω ordertostaywithinthevalidityofthemodelwiththefirsttwo
J(ω)=ηωexp − , (60)
ω transversemodesopened. Inastronglydissipativesystem,as
(cid:18) c(cid:19)
in our case, it is not necessary to fulfil the last inequalities
where η is the viscosity coefficient and ω the cutoff fre- because an electron loses a huge amount of its energy due
c
quency.Weuseω =10ω . to intensive dissipative processes. In general, our model of
c 0
9
a driven strongly dissipative system taking into account the
first four Bloch sub-bandsremains valid if at long times the 10 F = 0.5 h¯ ω / L
electronenergyaveragedoveroneperiodofthedrivingforce, ω ]0 F = 0.8 h¯ ω00 / L
ǫav(F,Ω,η)(whichisafunctionofthedrivinganddissipation 5L F = 1.0 h¯ ω0 / L
parameters), is smaller than ~ω0, ǫav(F,Ω,η) < ~ω0. This - 10
acctoannsustucamhkeedprmilvaaicnjoegretavhmeenosuitfrnoFtnogLfdt>hises~eipωlea0ctitaornondn(leΩanreg>regyvω.a0lubeescaoufsηe)ewveilnl ∞J [n-e,S 5
InFig. 1the non-equilibriumspincurrentasa functionof nt
e
r 0
the amplitude of the external driving is shown for different ur
c
valuesofthez-projectionofthemagneticfield.Forsmallval- n
ues of the driving amplitude and small magnetic fields it is Spi
seen that if the magnetic field has the same direction as the
-5
z-axis, the spin current decreases, while the opposite direc-
-0.4 -0.2 0 0.2 0.4
tionofthemagneticfieldamplifiesthespincurrent. Thisbe- Magnetic field, gµH[ h¯ ω ]
haviorcanbe physicallyunderstoodfromEq. (54). Positive Β 0 0
valuesofH canbe equivalentlyconsideredaslargervalues
of~ω , that0isofthedistancebetweenthetransversemodes. FIG. 3: (Color online) Non-equilibrium spin current, J¯n∞−e,S, as a
0 function of the magnetic field, gµBH0, for different values of the
Thisinturnleadstoadecreaseofthetransitionprobabilities
amplitude of the driving force, F. The other parameters are as in
whichsuppressesthespin current. On the contrary,negative
Fig.1.
valuesof H correspondto smaller valuesof ~ω leadingto
0 0
anincreaseofthetransitionratesandthusthespincurrentis
enhanced. Anotherphysicalexplanationisthatthemagnetic
fieldalignsthespinsalongitsdirection. Therefore,whenH
0
is positiveor negativethe spinsare forcedto pointinthe di- themagneticfield. When|H0|increasesfurther,thespincur-
rectionofthez-axisorintheoppositedirection,respectively. rentdependsnon-linearlyonH0 andacomplicatedinterplay
Thespincurrentgetsmorepolarizedinthedirectionofthez- betweenthemagneticfield,drivinganddissipativeprocesses
axisforH >0orintheoppositedirectionforH <0. Asa develops.Thisdependenceofthespincurrentonthemagnetic
0 0
consequenceitsmagnitudedecreasesforH >0orincreases fieldisdepictedinFig.3fordifferentvaluesoftheamplitude
0
forH < 0sinceitwaspolarizedinthedirectionoppositeto ofthedrivingforce.Inordertostaywithinthevalidityofour
0
theoneofthez-axisintheabsenceofthemagneticfield. model,whereonlythefirsttwotransversemodesareopened,
themagnitudeofthemagneticfieldmustsatisfythecondition:
The same dependenceof the spin currenton the magnetic
fieldwithsmallvaluesofitsmagnitude(aswellasforasmall
value of the driving amplitude FL = 1.0~ω ) is found in
0
Fig.2inviewofitsdependenceonthespin-orbitinteraction gµ |H |60.5(~ω +∆ε ), (61)
B 0 0 γ;4,1
strength k . Again for H > 0 the magnitude of the spin-
so 0
currentgetssmallerandforH < 0itgetslarger. Addition-
0
ally,onecansee thatthepresenceofthemagneticfielddoes
not change the locations of minima and maxima of the spin where∆ε ≡ ε(0) (k )−ε(0) (k ) /N. Forthe
currentas a functionof kso. Thishas the followingphysical valuesoftγh;4e,1paramekteBrsuγ,s1e;d1 toBobtainγ,t0h;1e nuBmericalresults
explanation. The minima and maxima in Fig. 2 are related wehave∆ε =P−0(cid:2).07~ω . Thusgµ |H | (cid:3)6 0.465~ω .
γ;4,1 0 B 0 0
to the periodicity of the energy spectrum in the k-space. In AsitcanbeseenfromFig.3themagnitudeofthespincurrent
terms of the band energy versus the quasi-momentumk de- decaysforlargepositivevaluesofH . Thishappensbecause
0
pendence RSOI produces a horizontal (that is the energy of thedistancebetweentheBlochsub-bandsbecomeslargeand
the bandsdoesnotchange)split of the energybandsaswell thus the transition processes are less probable. For a certain
astheirhybridization. Duetotheperiodicitythissplitcanbe negative value of H the magnitude of the spin current has
0
minimalormaximalforsomevaluesofksowhichleadstothe a maximum after which it starts to decrease and vanishes at
correspondingminimaandmaximainFig.2. Theroleofthe some point H(0) < 0. After this point and for H < H(0)
0 0 0
hybridizationisthatthesplitisneverzeroandthustheminima the spincurrentreversesits sign andits magnitudeincreases
ofthespincurrentarenotexactlyequaltozero.Incontrastto again. This behavior clearly demonstrates that the magnetic
thishorizontalsplitthemagneticfieldproducesavertical(that fieldcan,withoutchangingitsdirection,actinphase(i.e. de-
isalongtheenergyaxis)splitanditalsoproduceshybridiza- stroythespintransport)withthedissipativeprocessesaswell
tion. This verticalsplit is notcorrelatedwith the periodicity as out-of-phase (i.e. intensify the spin kinetics) with them.
of the energybands in the k-space and thus the locations of MathematicallyitcomesfromthefactthatinEq. (32)forthe
minimaandmaximaremainuntouchedbythemagneticfield. transitionratesthe magneticfield H andtheimaginarypart
0
However,thepictureexplainedaboveisonlyvalidforsmall ofthetwiceintegratedbathcorrelationfunctionIm[Q(τ)]en-
valuesofthedrivingamplitudeF andmagnitudeofthemag- ter the argumentsof the same trigonometric functions. This
neticfield|H |wherethespincurrenthasalinearresponseto isclarified byEq. (37) appropriatelyrewrittenbelowforthe
0
10
0.20 1.5x10-4
η = 0.03
ω ]0 ηη == 00..0058 0.18 1.0
L 10 η = 0.1
5
- 0 0.16 0.5 Sp
∞Jnt [1n-e,S 0 Coefficient 00..1142 -00..05 in Current
e y
n curr scosit 0.10 -1.0
pi Vi 0.08
S
-10
0.06
-0.4 -0.2 0 0.2 0.4
Magnetic field, gµH[ h¯ ω ] 0.04
Β 0 0
-0.4 -0.2 0.0 0.2 0.4
FIG. 4: (Color online) Non-equilibrium spin current, J¯n∞−e,S, as a Magnetic Field
function of the magnetic field, gµBH0, for different values of the
viscositycoefficient, η. Thedrivingamplitude isF = 1.0~ω0/L.
FIG. 5: (Color online) Contour plot of the non-equilibrium spin
TheotherparametersareasinFig.1.
current, J¯n∞−e,S [Lω0], asafunctionof themagnetic field, gµBH0
[~ω0], and viscosity coefficient, η. The other parameters areas in
Fig.1.
caseξ′ =1,ξ =4:
Jγ;1,4 = ~22 ∞dτe−L~2QR(τ)× distancebetweenthecorrespondingBlochsub-bands.
Z0
∆ε 2gµ H L2 TheminimainFigs.3and4atnegativevaluesofH0appear
×cos γ;4,1 +ω + B 0 τ − Q (τ) × (62) asaresultofacooperativeactionoftheorbit-orbitcoupling,
~ 0 ~ ~ I
(cid:20)(cid:18) (cid:19) (cid:21) confinement,magneticfield,drivinganddissipation.Itsloca-
2FL Ωτ tionchangeswhenthestrengthofthedrivinganddissipation
×J sin ,
0 ~Ω 2 varies.
(cid:20) (cid:18) (cid:19)(cid:21)
For completeness in Fig. 5 we also show the spin current
where Q (τ) ≡ Re[Q(τ)], Q (τ) ≡ Im[Q(τ)]. The phys-
R I as a contour plot using the variables H and η. The main
ical explanation of why in our system the magnetic field in- 0
effect of the interaction between the electrons and external
teracts only with the friction part of the dissipation and not
environmentis the electron dressing. The dressed electrons
with its noise part is rooted in the roles which the magnetic
areheavierandasaresultlessmobile.Sincethespindegreeof
fieldanddissipationplayforquantumcoherence. Ontheone
freedomiscarriedbythesedressedelectrons,thespincurrent
sidequantumcoherenceinadissipativesystemdiesoutdueto
decreaseswhentheviscositycoefficientgrows.
thenoisepartoftheFeynman-Vernoninfluenceweightfunc-
tional. On the other side, within the Feynman path integral
formalism,weseethatinoursystematransversein-planeuni-
formstationarymagneticfield cannotproducethe additional VIII. CONCLUSION
phase due to the integral of the vector potential along the
Feynmanpaths(seeAppendixC). Thusinoursystemquan- In conclusion, we have studied averaged stationary quan-
tumcoherenceistotallyinsensitivetothemagneticfieldand tum transport in a driven dissipative periodic quasi-one-
asaresultcannotinteractwiththenoisepartoftheFeynman- dimensional (1D) system with Rashba spin-orbit interaction
Vernoninfluenceweightfunctional. (RSOI)andplacedinatransversein-planeuniformstationary
Themutualimpactofthemagneticfieldandquantumdis- magneticfield. Forthecaseofmoderate-to-strongdissipation
sipativeprocessesonthespincurrentinthesystemisshown it has been shown that the averaged stationary charge trans-
inFig.4wherethespincurrentisplottedversusthemagnetic port is well suppressed as soon as it is restricted within the
field,H ,andtheviscositycoefficient,η,playsaroleofapa- Blochsub-bandsgrownoutofthesameBlochbandofthecor-
0
rameter.AgainforlargepositivevaluesofH thespincurrent respondingtruly 1D problemwithoutRSOI. Howeverin the
0
vanishes.Asexpected,thespincurrentgetssmallerifthedis- same situation the averaged stationary spin transport is acti-
sipationinthesystemgetsstronger.Whenthedissipationgets vated. Theanalyticalexpressionforthespincurrenthasbeen
weaker(η=0.05andη =0.03curves)theoscillationsofthe derived and its behavior as a function of the driving param-
spincurrentbecomeobservable.Theseoscillationsarerelated eters, dissipation, spin-orbit interaction strength, orbit-orbit
totheinteractionbetweenthemagneticfieldanddrivingand coupling strength and a transverse in-plane uniform station-
can be described in terms of the photon emission/absorption arymagneticfieldhasbeenanalyzed. Ourresultsonthespin
processes36 since changingH is equivalentto changingthe transport in the system have been presented and thoroughly
0