Coherent control of optical injection of spin and currents in topological insulators

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by  Rodrigo A. Muniz

Coherentcontrolofopticalinjectionofspinandcurrentsintopologicalinsulators
Rodrigo A. Muniz and J. E. Sipe
Department of Physics and Institute for Optical Sciences,
University of Toronto, Toronto ON, M5S 1A7, Canada
(Dated:January8,2014)
Topologicalinsulatorshavesurfacestateswitharemarkablehelicalspinstructure,withpromisingprospects
forapplicationsinspintronics. Strategiesforgeneratingspinpolarizedcurrents, suchastheuseofmagnetic
contacts and photoinjection, have been the focus of extensive research. While several optical methods for
injectingcurrentshavebeenexplored,theyhaveallfocusedonone-photonabsorption.
Here we consider the use of both a fundamental optical field and its second harmonic, which allows the
4 injectionofspinpolarizedcarriersandcurrentbyanonlinearprocessinvolvingquantuminterferencebetween
1 one-andtwo-photonabsorption. Generalexpressionsarederivedfortheinjectionratesinagenerictwo-band
0 system, including those for one- and two-photon absorption processes as well as their interference. Results
2 aregivenforcarrier,spindensityandcurrentinjectionratesonthesurfaceoftopologicalinsulators,forboth
linearlyandcircularlypolarizedlight. Weidentifytheconditionsthatwouldbenecessaryforexperimentally
n
verifyingthesepredictions.
a
J

6
I. INTRODUCTION semiconductors20,21,andcurrentsingraphene22–24.Ithaseven

been proposed that it could be used to inject a macroscopic
]
l Berrycurvatureinsemiconductorquantumwells25. Herewe
l Three-dimensional topological insulators are fascinating
a present predictions of the optical injection of carrier density,
materials, with a band gap in the bulk and protected midgap
h
- states on their surfaces1,2. The surface electronic bands are spin polarization, charge current, and spin current at the sur-
s face of a topological insulator. In order to identify the fun-
described by a single Dirac cone with a helical spin struc-
e
damentalpropertiesofcoherentcontrolintopologicalinsula-
m ture,whichistheequivalentofadominantRashbaspin-orbit
tors, weuseaHamiltonianwithaperfectlysymmetricDirac
coupling term in the Hamiltonian. This property leads to a
t. number of interesting features, including non-magnetic scat- cone, and restrict the analysis to light at normal incidence.
a
m tering,themagnetoelectriceffect3,4,andtheformationofMa- Thisalsohelpstocontrasttheresultsofcoherentcontrolwith
joranafermionsintheproximityofsuperconductors5. Dueto thoseobtainedbyothermeans.Wekeepaσzmassterminthe
d- the effective spin-orbit coupling, the spin and current of the HamiltonianinordertoanalyzethedependenceontheBerry
n surface states are closely related6, providing an exciting op- phase,whichhasinterestingeffectsontheinjectionrates.
o portunity for technological applications using spin polarized In Sec. II we present the calculation of optical injection
c currents. There have already been several studies using the ratesforanarbitraryquantityusingFermi’sgoldenrule,con-
[
proximityofamagneticmetalforinjectingspinpolarization sideringoneandtwophotonsabsorptionprocessesaswellas

1 andcurrent7–10. theirinterference. InSec. IIIweprovidegeneralexpressions
v fortheinjectionratesofagenerictwo-bandsystem,especially
Anotherfruitfulapproachformanipulatingcurrentsinma-
1 for carrier density, spin density, charge current and spin cur-
terials involves optical excitation. The optical properties of
4
rentoperators. Sincetwo-bandmodelscanbeused,asafirst
2 topologicalinsulatorsurfacestatesareveryinterestingthem-
approximation,tocomputeopticalpropertiesofalargenum-
1 selves, with features such as the injected current depending
. explicitly on the Berry phase11,12. The injection of spin and ber of materials, the expressions derived there should be of
1
useevenbeyondtheirapplicationtotopologicalinsulators. In
0 currentbyonephotonabsorptionprocesseshasbeenstudied
4 indifferentcircumstances12–14.Inordertobreaktherotational Sec. IVweapplytheresultsofSec. IIItotopologicalinsula-
1 symmetrystemmingfromtheDiraccone-anecessarystepfor tors. In Sec. V we present the results for linearly and circu-
: larlypolarizedlight,referringtotheAppendicesAandBfor
v generating a current - the use of an in-plane magnetic field,
details.InSec.VIweendwithadiscussionofinterestingfea-
i the application of strain, and an oblique angle of incidence
X turesinourresultsandthepossibilitiesfortheirexperimental
haveallbeenconsidered. Correctionsduetosnowflakewarp-
r verification,includingestimatesfortheexpectedexperimental
ing have been included; even a surprisingly relevant contri-
a
results.Sincetheexperimentaltechniquesrequiredtoconfirm
bution from the Zeeman coupling of the light field has been
identified15. Nonlinear effects due to the second harmonic ourresultsarewellestablished,wecanexpectthatsuchexper-
havealsobeenconsidered16–19,especiallyinthetreatmentof iments will help advance the understanding and applications
oftopologicalinsulators.
pulses. Howeverthefocusofeventhesestudieshasbeenon
onephotonabsorptionprocesses.
Oneofthemostinterestingtechniquesforopticalinjection
iscoherentcontrol, anexampleofwhichinvolvestuningthe II. RESPONSETOLIGHTFIELDS
interference of one and two photon absorption processes to
achieveatargetresponse. Thishasbeenemployedforinject- Thereareseveralmethodsforcomputingtheresponseofa
ing carriers, spin polarization, currents, and spin currents in systemtoexternalperturbations;oneofthesimplestandmost
2
standardmethodsisFermi’sgoldenrule. Itisespeciallysuit- FromEq. (2)wecanwritec (t)as
cv,k
ableforcoherentcontrolcalculationsbecauseitmakesevident
allthecontributionsstemmingfromone-andtwo-photonpro- c (t)=(cid:88)4 K(n) (ω) e−i(nω+i(cid:15))t , (6)
cesses and their interference. This is a feature not shared by cv,k cv,k nω−ω +i(cid:15)
n=1 cv
theKuboformalism,forinstance.
where
The calculation for the injection rates of operators using
Fermi’s golden rule has been already well explained in pre- K(1) (ω)= −ev ·A(ω)+
vious studies20. However, it has been typically assumed that cv,k +(cid:126)ec2 c(cid:80)v,kvcc(cid:48)·A(−ω)vc(cid:48)v·A(2ω) − vv(cid:48)v·A(−ω)vcv(cid:48)·A(2ω)
thefundamentalphotonenergyisbelowthebandgap,asisthe (cid:126)2c2v(cid:48)c(cid:48) 2ω−ωc(cid:48)v+i(cid:15) 2ω−ωcv(cid:48)+i(cid:15)
case for most studies of semiconductors. Since we will deal + e2 (cid:80)vcc(cid:48)·A(2ω)vc(cid:48)v·A(−ω) − vv(cid:48)v·A(2ω)vcv(cid:48)·A(−ω),
with systems that are gapless, there will be an additional in- (cid:126)2c2v(cid:48)c(cid:48) −ω−ωc(cid:48)v+i(cid:15) −ω−ωcv(cid:48)+i(cid:15)
(7)
terference term. And in order to make the notation clear we
and
willpresentthemainstepsofthefullcalculation.
ThefullHamiltonianinthepresenceoftheexternalpertur- Kc(2v,)k(ω)= −(cid:126)ecvcv,k·A(2ω)+
bnaiatinonwVithexotu(tt)tihseHpe(tr)tu=rbHat0io+nV.eTxth(te),wwahveerfeunHc0tiiosnthineHthaempirletos-- +(cid:126)e22c2v(cid:80)(cid:48)c(cid:48)vcc(cid:48)·Aω−(ωω)cv(cid:48)vc+(cid:48)vi·(cid:15)A(ω) − vv(cid:48)v·Aω−(ωω)cvv(cid:48)c+v(cid:48)i·(cid:15)A(ω),
(8)
enceoftheexternalperturbationcanhaveacontributionfrom
with similar expressions for K(3) and K(4) ; the first only
an excitation of a valence v band electron to a conduction c cv,k cv,k
has terms with products of field amplitudes A(±ω)A(±2ω),
band
andthesecondonlyhastermswithproductsA(±2ω)A(±2ω).
|Ψ(t)(cid:105)= c0|0(cid:105)+ccv,k(t)|cv,k(cid:105)+..., Thenwehave
(cid:12) (cid:12) (1)
ρcv,k = (cid:12)(cid:12)ccv,k(cid:12)(cid:12)2, d (cid:16)c∗ c (cid:17) = π(cid:80)K(n)∗ (ω)K(n) (ω)
w|cvh,ekre(cid:105)|0=(cid:105) ias†c,tkhaev,gkr|o0u(cid:105)nidssttahteesotfatHe0wwitihthafinlleeldecvtraolenn-hcoelbeapnadisr;, dt c(cid:48)v(cid:48),k cv,k t=0 ·(cid:2)δn(cid:0)ncω(cid:48)v(cid:48),−kωcv,k(cid:1)cv+,kδ(cid:0)nω−ωc(cid:48)v(cid:48),k(cid:1)(cid:3).(9)
witha† denotingtheelectroncreationoperator,and WeassumethattheamplitudeA(±2ω)ofthesecondharmonic
c,k
fieldismuchsmallerthattheamplitude A(±ω)ofthefunda-
ccv,k(t)= +(cid:82)−t(cid:82)∞−t∞di(cid:126)t1die(cid:126)t1−i(cid:82)ω−ctv∞1tϑdi(c(cid:126)t1v2),ke−(itω1c)vtϑ(c2v),k(t1,t2), (2) mtthheaennsteaoclnofien-edpldhh.oatorSmninopcnerioctcweasroes-eptshh,oettnwonon-eppgrhloeoccteotesndse,psaronacrdeesomsnelusycihKnvw(1o)elva(ikωnegr)
cv,k
where ωcv = ωc − ωv; here ϑ(c1v),k(t1) = (cid:104)cv,k|Vext(t1)|0(cid:105) and Kc(2v,)k(ω) remain. Since Kc(3v,)k(ω) and Kc(4v,)k(ω) do not
and ϑ(2) (t ,t ) = (cid:104)cv,k|V (t )V (t )|0(cid:105). This allows have linear terms in the field amplitude they do not support
cv,k 1 2 ext 1 ext 2
us to compute the injection rate for the density (cid:104)M(cid:105) of anyinterferenceprocess,butonlytwo-photonabsorption.
a quantity associated with a single-particle operator M = SincealltheK(n) (ω)aremultipliedbyaδfunctioninEq.
cv,k
(cid:80) a† M a , where α and β are band indices. The ex- (9),wecanwrite
k α,k αβ,k(cid:68) β(cid:69),k
pressionfor M˙ is K(1) (ω)= ievb Eb(ω)
cv,k (cid:126)ω cv,k
−Wbc (2ω,−ω)Eb(−ω)Ec(2ω),
ddt (cid:104)M(cid:105)= L1D (cid:88) (cid:0)Mc(cid:48)c,kδv(cid:48)v−Mv(cid:48)v,kδc(cid:48)c(cid:1)ddt (cid:16)c∗c(cid:48)v(cid:48),kccv,k(cid:17), Kc(2v,)k(ω)= 2i(cid:126)eωvcbcvv,,kkEb(2ω) (10)
cv,c(cid:48)v(cid:48),k +Wbc (ω,ω)Eb(ω)Ec(ω),
(3) cv,k
where Listheunidimensionallengthand Disthespatialdi- where
mensionofthesystem.
Ωbc (ω )= −e2 (cid:80) vbcnvcnv ,
Specifyingtheperturbationtobeanincidentlaserfield,us- (cid:16)cv,k α(cid:17) (cid:126)2ω2 n ωα−ωnv (cid:16) (cid:17) (11)
ingtheminimalcouplingHamiltonianwehave Wbc ω ,ω = Ωbc (ω )+Ωcb ω ,
cv,k α β cv,k α cv,k β
ϑ(c2vϑ),kcv(,tk1,(tt12))== −ce22ec(cid:80)vvcvc,ck(cid:48)··AA((tt11))eeiiωωccvc(cid:48)t1t1,vc(cid:48)v·A(t2)eiωc(cid:48)vt2 a−nicd(nEω)(−t1)E(=nω).−cT−h1e∂tiAnje(ct)tiownarsateufsoerda,nsooperAato(nrωM) ca=n
c(cid:48)
−ce22(cid:80)v(cid:48)vv(cid:48)v·A(t1)eiωv(cid:48)vt1vcv(cid:48) ·A(t2)eiωcv(cid:48)t2, tthoennsabbesdoercpotimonpopsreodceisnsteoscwonitthriabnutaiodndsitiforonmaloinnteerafnedretnwcoeptehrom-
(4) (cid:68)M˙(cid:69)=(cid:68)M˙ (cid:69)+(cid:68)M˙ (cid:69)+(cid:68)M˙ (cid:69)where
wheree = −|e|isthechargeoftheelectron,v isthevelocity 1 2 i
operator,andthevectorpotentialis (cid:68)M˙ (cid:69)= (cid:80) Λbc(nω)Eb(−nω)Ec(nω),
1 1
A(t)=(cid:88)A(ωα)e−i(ωα+i(cid:15))t, (5) (cid:68)M˙ (cid:69)= Λn=b1c,2de(ω)Eb(−ω)Ec(−ω)Ed(ω)Ee(ω), (12)
ωα (cid:68)M˙2(cid:69)= (cid:80)2 Λbcd(ω)Eb(−ω)Ec(−ω)Ed(2ω)+cc,
withω = ±ω,±2ω;here(cid:15) → 0+ describestheturningonof i n=1,2 i(n)
α
thefieldfromt=−∞. with
3
Λbc(nω)= π (cid:80) (cid:0)M δ −M δ (cid:1)Γbc (k,ω)(cid:2)δ(cid:0)nω−ω (cid:1)+δ(cid:0)nω−ω (cid:1)(cid:3),
1 LD c(cid:48)c,k v(cid:48)v v(cid:48)v,k c(cid:48)c 1,c(cid:48)v(cid:48),cv cv,k c(cid:48)v(cid:48),k
cv,c(cid:48)v(cid:48),k
Λbcde(ω)= π (cid:80) (cid:0)M δ −M δ (cid:1)Γbcde (k,ω)(cid:2)δ(cid:0)2ω−ω (cid:1)+δ(cid:0)2ω−ω (cid:1)(cid:3),
2 LD c(cid:48)c,k v(cid:48)v v(cid:48)v,k c(cid:48)c 2,c(cid:48)v(cid:48),cv cv,k c(cid:48)v(cid:48),k (13)
cv,c(cid:48)v(cid:48),k
Λbcd(ω)= π (cid:80) (cid:0)M δ −M δ (cid:1)Γbcd (k,ω)(cid:2)δ(cid:0)nω−ω (cid:1)+δ(cid:0)nω−ω (cid:1)(cid:3),
i(n) LD c(cid:48)c,k v(cid:48)v v(cid:48)v,k c(cid:48)c i(n),c(cid:48)v(cid:48),cv cv,k c(cid:48)v(cid:48),k
cv,c(cid:48)v(cid:48),k
and satisfy dˆ · σψ = ±ψ , so when dˆ · σ is diagonal-
k k± k± k
izeditisrepresentedbyσ , andthereisaunitarymatrixU
ΓΓb1bcc,cd(cid:48)ev(cid:48),cv((kk,,ωω))== W(cid:126)2eω2b2cvbv(cid:48)c(cid:48)(vωccv,,ω)∗Wde (ω,ω), tchaautsepeSrUfor(m2)satnhdeSchOa(n3g)ehozavfebtahseiss,admˆke·aσlge=braU,kwσezcUak†n.wBrietke-
2,c(cid:48)v(cid:48),cv c(cid:48)v(cid:48),k cv,k (14)
Γbcd (k,ω)= ievb Wcd (2ω,−ω), U σ U† = (R zˆ)·σ,whereR representsarotationaround
i(1),c(cid:48)v(cid:48),cv (cid:126)ω v(cid:48)c(cid:48) cv,k k z k k k (cid:16) (cid:17)
Γbi(c2d),c(cid:48)v(cid:48),cv(k,ω)= 2i(cid:126)eωWcbvc,k(ω,ω)∗vdc(cid:48)v(cid:48). the axis nˆk by an a(cid:12)ngle φk(cid:12), so Uk = exp −iφ2knˆk·σ ; we
TheΛbc(ω)andΛbcd(ω)termshaveusuallybeenignoredin p(cid:16)ut nˆk = zˆ ×dˆk(cid:17)/(cid:12)(cid:12)zˆ×dˆk(cid:12)(cid:12) and cosφk = zˆ ·dˆk. The triad
1 i(1) nˆ ,dˆ ,nˆ ×dˆ formsanorthonormalbasis,soanarbitrary
theliterature,sincetheyvanishforsystemswithagapwhere k k k k
operatorcanbewrittenas
thefirstharmonicfallsbelowthebandgap. Thetwointerfer-
enceprocessesareshowninFig. 1. Thequantitiesforwhich wˆ ·σ = (nˆ ·wˆ)nˆ ·σ+(cid:16)dˆ ·wˆ(cid:17)dˆ ·σ
k k k k
the injection rates will be computed are the densities associ- (cid:104)(cid:16) (cid:17) (cid:105)(cid:16) (cid:17) (16)
+ nˆ ×dˆ ·wˆ nˆ ×dˆ ·σ,
ated with the carriers (cid:104)n(cid:105), spin (cid:104)S(cid:105), charge current (cid:104)J (cid:105), and k k k k
c
spin current (cid:104)J (cid:105). We denote the response coefficients asso-
S (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) andinthebasisofeigenvectors
ciatedwiththequantities(cid:104)n˙(cid:105), S˙ , J˙ , J˙ respectivelyby
c S (cid:16) (cid:17)
ξ,ζ,η,µ. U† (wˆ ·σ)U = (nˆ ·wˆ)nˆ ·σ+ dˆ ·wˆ zˆ·σ
k k (cid:104)k(cid:16) k(cid:17) (cid:105) k (17)
+ nˆ ×dˆ ·wˆ (nˆ ×zˆ)·σ,
k k k
whichallowsanyoperatortobeexpressedsimply.
−ω
Operators
2ω ω
ω 2ω
ω The quantities of interest are the densities of injected car-
riers(cid:104)n(cid:105),spin(cid:104)S(cid:105),chargecurrent(cid:104)J (cid:105)andspincurrent(cid:104)J (cid:105),
c S
whicharecomputedbelow.
Wekeeptrackoftheinjectedcarriersbycomputingtheden-
sity of electrons injected into the conduction band. The cor-
respondingnumberoperatorhasmatrixelementsn = 1and
cc
FIG.1. (Coloronline)One-andtwo-photoninterferenceprocesses n =0.
vv
illustratesonthehelicalDiraccone. Theoneinthelefthasenergy We suppose that the components of the spin operator are
(cid:126)ωcorrespondsto(i)1,whiletheotherontherighthasenergy2(cid:126)ω givenbySa = (cid:126)aˆ ·σ,anddecomposeaˆ ·σaccordingtoEq.
correspondsto(i)2. 2
(17),so
Sa = (cid:126)Tr(cid:2)Sa(I+σ )(cid:3)= (cid:126)dˆ ·aˆ,
Scac = 2(cid:126)Tr(cid:2)Sa(I−σz)(cid:3)=−2(cid:126)kdˆ ·aˆ (18)
vv 2 z 2 k
III. TWO-BANDSYSTEMS
are the matrix elements needed. Note that even though Sa
cc
and Sa are matrix elements of the spin operator in the basis
AnyHermitian2×2matrixcanbewrittenasalinearcom- vv
of eigenstates, they are being expressed in terms of the pa-
binationofPaulimatricesσandtheidentityσ . Soageneric
Hamiltonian for two bands is H = (cid:80) a† H0 a , where rameters of the Hamiltonian in its non-diagonal form of Eq.
k α,k αβ,k β,k (15).
α,β=1,2arebandindices,and
Thematrixassociatedwiththevelocityoperatoris?
(cid:34) (cid:35)
Hk =(cid:126)(cid:36)kσ0+(cid:126)dk·σ =(cid:126) (cid:36)dkxk++iddkkyz (cid:36)dkxk−−iddkkyz (15) vak = 1(cid:126)∂kaHk =∂ka(cid:36)kσ0+∂kadk·σ, (19)
denotes the Hamiltonian at each lattice momentum k. The anddecomposingitaccordingtoEq. (17)gives
e(+ig)en=encerganieds(a−re)E=k±v=re(cid:126)pr(e(cid:36)sekn±tindgk)thwehceorenddukct=ion|dakn|,dwvitah- vak = ∂ka(cid:36)(cid:104)k(cid:16)σ0+∂kad(cid:17)kσz+d(cid:105)k(cid:16)nˆk·∂kadˆk(cid:17)nˆk·σ (20)
lence bands respectively, so ωcv,k = 2dk. The eigenstates +dk nˆk×dˆk ·∂kadˆk (nˆk×zˆ)·σ.
4
Thevelocitymatrixelementsare whereeachD−1dimensionalintegralisovertheregionspec-
ifiedbytheenergymatchingconditionω =nω.
va = ∂ (cid:36) +∂ d , cv
cc ka k ka k
va = ∂ (cid:36) −∂ d ,
vvav = dka(cid:16)nˆk +iknˆa k×dˆ (cid:17)·(cid:16)∂ dˆ (cid:17)nˆ ·(xˆ −iyˆ), (21)
cv k(cid:16) k k k(cid:17) (cid:16) ka k(cid:17) k IV. TOPOLOGICALINSULATORS
va = d nˆ −inˆ ×dˆ · ∂ dˆ nˆ ·(xˆ +iyˆ).
vc k k k k ka k k
Itisalsonecessarytocomputeproductsoftwovelocitymatrix Whenthephotonenergyissmallerthanthebulkbandgap
elements, ofatopologicalinsulators,onlytheprotectedstateslocalized
(cid:104) (cid:16) (cid:17)(cid:105) ontheirsurfaceswillcontributetotheopticalabsorptionand
va vb =d2 ∂ dˆ ·∂ dˆ +idˆ · ∂ dˆ ×∂ dˆ . (22)
cv vc k ka k kb k k ka k kb k injection.
The second term above is the Berry curvature; we can track Thestandardeffectivemodelobtainedfromak·papprox-
thecontributionstoopticalproperties thatdependonit. The imationforthematerialsinthefamilyofBiSeTeresultsina
charge current is expressed in terms of the velocity operator four-bandHamiltonian26
byJ =ev.  
m A k 0 A k
21(cid:16)WSaevbkd+efivbkneSa(cid:17)thsoefosrpainsysctuemrrewntheroepSeraat=or(cid:126)2aˆa·sσJwSaeb,khave= Hk =ε0kI4×4+(cid:126) A10kkz A−2m1kk+z Am2kk− −A201k−z , (29)
U†Jab U = (cid:126)(cid:104)aˆ ·dˆ (∂ d )+d (cid:16)aˆ ·∂ dˆ (cid:17)(cid:105)+ A2k+ 0 −A1kz −mk
k S,k k 2 k kb k k kb k (23)
+(∂kb(cid:36)k)Uk†SaUk, where k± = kx ±iky, ε0k = C +D1kz2 +D2k⊥2 and mk = m−
andthecomponentsare B1kz2−B2k⊥2 withtheconstantsdependingonthematerial.
In order to consider interfaces along the zˆ direction one
Jab = (cid:126)aˆ·dˆk(∂kbdk)+dkaˆ·∂kbdˆk+(∂kb(cid:36)k)aˆ·dˆk, canwritetheHamiltonianinaseparatedformH = H⊥+Hz,
JSab,cc = (cid:126)aˆ·dˆk(∂kbdk)+dkaˆ·∂2kbdˆk−(∂kb(cid:36)k)aˆ·dˆk, (24) and in the limit of k⊥ → 0 the transverse pakrt Hk⊥kcan bke
S,vv 2 neglected. Also, Hz isblockdiagonalandseparatesintotwo
k
whichcompletesthelistofnecessarymatrixelements. sectors according to the spin of the electrons. The boundary
conditionsforsurfacestatesthenleadtoonlyonesolutionfor
each sector, giving two independent states. Next, the 4×4
Opticalinjectioncoefficients bulk Hamiltonian with lattice momentum near the Γ point is
projected on the subspace spanned by the two independent
The quantities necessary for computing the injection rates surface states, and an effective Hamiltonian is obtained for
arereadilyobtainedfromtheequationsabove,giving thesurfacestates,
(cid:18) (cid:19)
WbcΩ(cid:16)bcωvc,k,(ωωα(cid:17))== (cid:126)−2eeω222 (cid:20)vωbcvαvbc∂−cvkωcccvdcvk++vvbcωvccvvα∂cvvkbd,k(cid:21), (25) Hk⊥ =(cid:126)(cid:16)C0+D0k2(cid:17)σ0−(cid:126)A0(zˆ×k)·σ, (30)
cv,k α β (cid:126)2ω2 ωα ωβ whichisvalidinthelimitoflargeslabthickness27andinclud-
wherewehaveusedthefactthatωα+ωβ =ωcv. Therefore ing terms up to the second order in k. The parameters C0,
D andA canbedeterminedintermsofthebulkparameters
(cid:104) (cid:105) 0 0
Γbc (k,ω)= e2dk2 ∂kbdˆk·∂kcdˆk−idˆk·(∂kbdˆk×∂kcdˆk) , appearinginEq. (29).
Γb1c,dcve(k,ω)= e4 (cid:20)vdcv∂kedk+vec(cid:126)v2∂ωkd2dk(cid:21)(cid:20)vbvc∂kcdk+vcvc∂kbdk(cid:21), Inordertoidentifythemostbasicfeaturesofcoherentcon-
2,cv (cid:126)4ω4 ω ω trolopticalinjectionintopologicalinsulatorswewillcompute
(26) theinjectionratesstartingfromtheHamiltonianofEq. (30).
and However,inordertokeeptrackofhowtheBerrycurvatureef-
(cid:20) (cid:21)
Γbcd (k,ω)= ie3 vccvvbvc∂kddk − vdcvvbvc∂kcdk , fectstheopticalresponsewekeepaσz masstermthatwould
i(1),cv (cid:126)3ω3 2ω ω correspond, for instance, to an external magnetic field along
(cid:20) (cid:21) (27)
Γbcd (k,ω)= ie3 vdcvvbvc∂kcdk+vdcvvcvc∂kbdk . thezˆ direction. The2DHamiltonianweconsideristhen
i(2),cv 2(cid:126)3ω3 ω
(cid:16) (cid:17)
Foratwo-bandsystemthereisonlyonevalenceandonecon- H =(cid:126) C +D k2 σ −(cid:126)A (zˆ×k)·σ+(cid:126)∆ σ , (31)
k⊥ 0 0 0 0 m z
ductionband,c(cid:48) =candv(cid:48) =v,soEqs. (13)and(14)become
simpler; in the continuum limit the momentum sums are ex- whichcorrespondstoEq. (15)with(cid:36) =C −Dk2andd =
k 0 k
pressedby A (zˆ×k)+∆ zˆ sonˆ =−kˆ and
0 m k
ΛΛb2b1ccde(n(ωω))== ωωccvv(cid:82)(cid:82)==n2ωω((((22ddππkk))))DDDD−−−−1111((MMcccc,,kk−−MM||∇∇vvvvkk,,kkωω))cΓΓcvvb1b2||cc,,ccdvve((kk,,nωω)),, (28) ∂∂kkbbddˆkk == −Ad20Akk0bd(,zkˆ×bˆ) − A20kdbk2dˆk. (32)
Λbcd(ω)= (cid:82) (dk)D−1 (Mcc,k−Mvv,k)Γbi,ccdv,(n)(k,ω), This allows us to compute the optical injection coefficients.
i(n) ωcv=nω(2π)D−1 |∇kωcv| SinceinthebasisofEq. (30)thespinoperatorisrepresented
5
bySa = (cid:126)aˆ ·σ,fromEq. (18),Eq. (21)andEq. (24)wecan The expressions for the various coefficients that follow from
2
identifythematrixelementsoftheoperatorsofinterest these expressions are the main results of this paper, and are
detailedinAppendixB.
Sa −Sa = (cid:126)aˆ ·dˆ = (cid:126)A0kz×a+(cid:126)∆mzˆ·aˆ,
cc vv k dk
Jabvac−c−Javbavv == (cid:126)2∂(∂kadεk =)aˆ2·Ad20kdkˆa,= 2(cid:126)D0kb[A0kz×a+∆mzˆ·aˆ]. (33)
S,cc S,vv kb k k dk V. RESULTS
Theysatisfytherelations
Sz −Sz = (cid:126)∆m = (cid:126)∆m (n −n ), Forthesystemweareconsidering,one-andtwo-photonab-
cc vv dk dk cc vv sorption processes inject scalar quantities while interference
v = −2A0z(cid:126)ˆ×S, (34) processesinjectvectorialones. Weconfirmwithinourmodel
(cid:16) (cid:17)
Jzb −Jzb = 2(cid:126)D0kb∆m = (cid:126)D0∆m vb −vb , that carriers are injected by one- and two-photon absorption
S,cc S,vv dk A20 cc vv processes,butnotfromtheinterferencebetweenthem. Con-
wherethesecondequationistheidentityexploredbyRaghu versely,chargecurrentisinjectedsolelyfromtheinterference
et al. [6]; the first states that the zˆ component of the spin processes and not from the one- and two-photon absorption
densitySz merelycorrespondstothespinpolarizationofthe processes. However, thereareadditionalpeculiaritiesforthe
injected carriers; and the third identifies the zˆ component of spindensityandspincurrentinjection.
thespincurrent JSz asentirelyduetothespinpolarizationof Duetotherelations(34),thein-planespindensityfollows
thechargecurrent.Bothspindensityandcurrentarenon-zero thechargecurrentinjection,stemmingonlyfromtheinterfer-
only in the presence of the σz mass ∆m. It should be noted enceprocesses;theout-of-planespindensityonlyhascontri-
thatthespincurrentistypicallynotaconservedquantity,and butionsfromtheone-andtwo-photonabsorptionprocesses.It
indeeditisnotconservedatthesurfaceoftopologicalinsula- simplycorrespondstothespinpolarizationoftheinjectedcar-
tors. Nevertheless, we still compute its optical injection rate riers,whichisproportionaltotheσ masstermintheHamil-
z
because depending on the experimental technique, the spin tonian.
separationtowhichitleadsmightbedetected(ortunneledto
Asimilarsituationholdsforthespincurrent. Thespincur-
anothermaterial)beforethespinsrelax28,29.
rent of the zˆ component of spin follows the charge current
Eq. (22)isthen
and simply amounts to the net spin polarization of the carri-
(cid:16) (cid:17)
vb vc = A2bˆ ·cˆ− A40kbkc +iA20∆mzˆ· bˆ ×cˆ (35) eOrns othfethoethceurrhreanntd;,itthiesionb-tpalianneedsfprionmcuthrreeninttiesrfaerreensucelttoefrmthse.
cv vc 0 d2 d
k k Diracconewithchiralspins;itdoesnotrequireanetspinpo-
whichallowsustocomputetheΓcoefficientsofEq. (26)and larization generated by a σz mass term. It is obtained from
one-andtwo-photonabsorptionandhasnocontributionfrom
Eq. (27),giving
interferenceprocesses.
Γbc (k,ω)= e2 vc vb , Below we present the injection rates for the quantities of
Γb1c,dcev(k,ω)= (cid:126)e42ωA402 (cid:20)ckvevdcvvc+kdvecv(cid:21)(cid:20)kcvbvc+kbvcvc(cid:21), (36) interest,consideringlinearandcircularpolarizations. InAp-
2,cv (cid:126)4ω4 ωdk ωdk pendixAweshowthegeneralexpressionsfortheopticalin-
jectioncoefficients,andinAppendixBwepresenttheexplicit
and
formofthecoefficientsrelatedtolinearandcircularpolariza-
(cid:20) (cid:21)
Γbcd (k,ω)= ie3A20 kdvccvvbvc−2kcvdcvvbvc , tionsoftheincidentlight,whicharereferredtobelow.
Γbi(c1d),cv(k,ω)= 2ie(cid:126)33Aω203 (cid:20)kcvdcvvbvcω+dkkbvdcvvcvc(cid:21). (37) The values of the parameters A0, D0 and ∆m = µ2B(cid:126)gBused
i(2),cv 2(cid:126)3ω3 ωdk fortheplotsorspecificestimatesaregiveninTableI;theycor-
respondtotheparametersof Bi Te foranappliedmagnetic
Theopticalinjectioncoefficientscannowbecomputed. 2 3
fieldaround10T.26
The only relevant states for our calculations are localized
onthesurface,sothemomentumintegralsarealltwodimen-
sional. Since ωcv,k does not depend on the direction of the A0 D0 (cid:126)∆m Eω E2ω
momentum,whentheintegralsareperformedinpolarcoordi- 5·105m 7·10−4m2 1.5·10−2eV 104V 72V
s s m m
nates(k,θ),thekintegralsimplydealswiththedeltafunction
setting2d = ωor2d = 2ωdependingontheterm. SoEq. TABLEI.Valuesoftheparametersusedfortheplots.
k k
(28)becomes
ΛΛbb1ccde(n(ωω))== (cid:82)(cid:82) 2ddπθθddkk((MMcccc,,kk−−MM2vvAvv,,kk20))ΓΓb1b2cc,,ccdvve((kk,,nωω))(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)dk=n2ω, , (38) daraemWienendtciacolantasivindedeorEffi2thωeeld=laarmg7e2pslmVtitfiufedolerdstihnoetfensEesωictoie=nsda1lh0lao4rwmVmeofdonwricit,thhewinhfuitchnhe-
Λ2bcd(ω)= (cid:82) 2dπθdk(Mcc,k−M2vAv,20k)Γbi,ccdv,(n)(k,ω(cid:12))d(cid:12)(cid:12)(cid:12)(cid:12)k=ω . pfoerrttuhrebaitnivjeecrteedgimcaer.riTerhedseensviatylu,essodwepeenexdpolaninthehoewxptrheesysioanres
i(n) 2π 2A20 (cid:12)dk=n2ω obtainedinSec. VI.
6
A. Linearpolarizations Perpendicularorientations
The one- and two-photon processes do not depend on the HerewehaveE(ω) = Eωeiθ1eˆω andE(2ω) = E2ωeiθ2eˆ2ω
relativeorientationofthefundamentalE(ω) = Eωeiθ1eˆω and with eˆ2ω = zˆ × eˆω. The relative phase parameter is again
second harmonic E(2ω) = E2ωeiθ2eˆ2ω fields, where Eω and ∆θ=θ2−2θ1.
E are real. Therefore we show here the results for the in- Thechargecurrentinjectionrateisgivenby
2ω
jectioncoefficientsΛ andΛ , whiletheresultsforΛ and (cid:68) (cid:69) (cid:104) (cid:105)
dΛiic(2u)laarrepdoilsaprilzaayteidonfso.r1thespec2ialcasesofparallelandpi(e1r)pen- J˙ia = −·s2ineˆ2(ω∆θ·)aˆEImω2(cid:104)Eη2ωyi(x1x)y(ω)+ηyi(x2x)y(ω)(cid:105) (44)
Thecarrierdensityinjectionrateisgivenby +2eˆω·aˆRe ηix(x1x)y(ω)+ηix(x2x)y(ω)
·cos(∆θ)E2E ,
(cid:104)n˙ (cid:105)= ξxx(ω)E2 +ξxx(2ω)E2 , ω 2ω
1 1 ω 1 2ω (39)
(cid:104)n˙ (cid:105)= ξxxxx(ω)E4, andthespindensityandcurrentfollowEq. (43).
2 2 ω
From Eqs. (B6) and (B7) in Appendix B we can identify
andthezˆcomponentofthespindensityinjectionrateisgiven twodifferentcontributionstotheinjection: onethatisrelated
by to theBerry curvature, andthus depends on ∆ , andanother
m
(cid:68)S˙z(cid:69)= 2(cid:126)∆m (cid:104)ξxx(ω)E2 + 1ξxx(2ω)E2 (cid:105), thatisindependentof∆m.
(cid:68) 1(cid:69) ω 1 ω 2 1 2ω (40) Againthedirectionofthepolarizationvectorprovidescon-
S˙z = (cid:126)∆mξxxxx(ω)E4.
2 ω 2 ω troloftheangleoftheinjectedvectorialquantities. Therela-
tivephasecanstillcontroltheirmagnitudeandorientation,but
This result simply corresponds to the net polarization of the
itcanalsoswitchbetweenthetworegimes:thefirstwherethe
injectedcarriers.
Thechargecurrentinjectionratevanishes;(cid:68)J˙a(cid:69)=(cid:68)J˙a(cid:69)=0. photoinjectionstemsfromtheBerrycurvature,andthesecond
1 2 whereitdoesnot.
Thespincurrentinjectionrateis
(cid:68)(cid:68)JJ˙˙Saabb,1(cid:69)(cid:69)== n(+z=(cid:80)ˆn1=(cid:80),×21,(2ezˆeˆˆn×ω)e··ˆnaaˆˆω((cid:16))zeˆˆ·a׈ (cid:16)e·ˆeˆbnˆnω(cid:17)ω)µ··ybxˆbˆx(cid:17)µxµx1xxyy1xx(xxωx()n(nEωω4))+EEn2ωn2ω,+ (41) −−ms21()a)1.125x 1020 hhnn˙˙12((2ωω)i)i −−ms21()b)1.25x 1024 τeτˆ===xˆ+−11
S,2 +eˆω·anˆω(zˆ×eˆωn)ω·bˆµ2xyx2xxx(ω)Eω4,ω nh˙i(0.5 nh˙i(0.15
thefirsttermineachequationgivesaspincurrentindependent 00 0.05 0.1 0.15 00 0.05 0.1 0.15
of the applied field polarization, and is due the helical spin ¯hω(eV) ¯hω(eV)
structure.
FIG.2. (Coloronline)(a)Carrierdensityinjectionratesfromone-
andtwo-photonabsorptionprocessesattotalenergy2(cid:126)ω. (b)Car-
rierdensityinjectionratesforlinear(eˆ = xˆ)andcircular(τ = ±1)
Parallelorientations ω
polarizationsoftheincidentfields.
Onlytheinterferenceprocessesdependontherelativeori-
entationoftheE(ω)andE(2ω). HerethefieldsareE(ω)= (a) x 1012 (b) x 1013
Eωeiθ1eˆω andE(2ω)= E2ωeiθ2eˆω. Therelativephaseparam- J˙xy 4
eteTrhise∆chθa=rgθe2c−ur2reθn1.tinjectionrateisgivenby −−11s) 15 DDJ˙SSyxEE −−s11) 3
m 10 m
(cid:68)J˙i(cid:69)= −·s2ineˆω(∆Imθ)(cid:104)Eηiω2x(x1Ex)x2ω(ω. )+ηix(x2x)x(ω)(cid:105) (42) ˙JeV(SDE 5 ˙JeV(SDE12 ττ==+−11
DuetoEq. (34),theinplanespindensityandthespinzˆ cur- 00 0.05 0.1 0.15 00 0.05 0.1 0.15
(cid:68) (cid:69) ¯hω(eV) ¯hω(eV)
rentinjectionratesaregivenintermsof J˙ (ω) by
i
FIG.3. (Coloronline)Planarspincurrentdensityinjectionratesfor
(cid:68) (cid:69) (cid:68) (cid:69)
S˙ = (cid:126) zˆ× J˙ , (a)linearpolarizationsalongthexˆ directionand(b)circularpolar-
(cid:68)J˙zi(cid:69)= 2(cid:126)AD00e∆m (cid:68)J˙(cid:69),i (43) izations.
S,i A2e i
0
the spin current merely corresponds to the magnetization of
thecarriersofthechargecurrent.
B. Circularpolarizations
Thedirectionofthepolarizationvectorprovidescontrolof
the angle of the injected vectorial quantities, while the rela-
tivephaseparameterofthelightbeamscancontrolonlytheir ForcircularpolarizationsE(ω) = Eωeiθ1pˆτ1 and√E(2ω) =
magnitudeandorientation. E2ωeiθ2pˆτ2 whereτ1,τ2 = ±1andpˆ± = (xˆ ±iyˆ)/ 2,sopˆτ·
7
(a) x 106 (b) x 106 (c) x 107
5 DJ˙i,(1)E 1102 eeˆˆω2ω 2.25 ττ ==+−11
−1s) 4 DJ˙i,(2)E −s1) 8 −s1)
−1m 3 J˙ −m1 −m1 1.5
A D iE A 6 A
˙J(E2 ˙J(E 4 ˙J(E 1
D D D
1 0.5
2
0 0 0
0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15
¯hω(eV) ¯hω(eV) ¯hω(eV)
FIG.4.(Coloronline)Currentdensityinjectionratesfor(a)linearpolarizationswithparallelorientations(b)linearpolarizationswithperpen-
dicularorientations,showingthecomponentsofthecurrentalongtheeˆ andeˆ directions,and(c)circularpolarizations.
ω 2ω
pˆτ = 0andpˆ+·pˆ− = 1aswellaspˆ−×pˆ+ = izˆ. Therelative VI. DISCUSSION
phaseparameterisstill∆θ=θ −2θ . Againtheoneandtwo
2 1
photonsprocessesdonotdependontherelativehelicityofthe In order to determine the validity our calculations for the
E(ω)andE(2ω)fieldsandarepresentedfirst. optical injection rates, we have to consider the fraction of
Thecarrierdensityinjectionrateisnowgivenby theinjectedcarrierpopulationrelativetothetotalnumberof
(cid:104)n˙ (cid:105)= ξ−+(ω)E2 +ξ−+(2ω)E2 , statesintherangeofenergiescoveredbythelaserpulse. The
(cid:104)n˙1(cid:105)= ξ1−−++(ω)ωE4. 1 2ω (45) durationofthepulse T setsthefrequencybroadeningof the
2 2 ω laser ∆ω = 2π, which in turn - via the dispersion relation,
T
The spin density injection is still given by Eq. (40), and for whichweassumeE =(cid:126)A khereforsimplicity-determines
k 0
thespincurrentwehave the area of the Brillouin zone that can be populated by car-
(cid:68)(cid:68)J˙Sab,1(cid:69)(cid:69)= 2i(cid:16)(cid:16)aˆ ×bˆ(cid:17)(cid:17)·zˆn=(cid:80)1,2µ−1+−+(nω)En2ω, (46) orifersstaates=a2vπaikl∆abkl,ewinhetrheiska=reaAωo0fatnhde∆Bkril=lou∆Aiω0n.zTonheeinsuam/bae1r,
J˙ab = 2i aˆ ×bˆ ·zˆµ−+−−++(ω)E4. wherea = (2π)2 istheareaoccupiedbyonestate. Themaxi-
S,2 2 ω 1 L2
mumamplitudesofthelaserfieldsarerestrictedbythecondi-
Circularlypolarizedlightdoesnotbreakrotationalsymmetry,
tionthatthenumberofinjectedcarrierswithadditionalenergy
thereforethesecondtermofEq. (41)isnotpresent.
2(cid:126)ωisatmost5%ofthetotalnumberofcarrierstatesinthe
allowedenergyrange
Equalhelicities (cid:16)ξxx(2ω)E2 +ξxxxx(ω)E4(cid:17)TL2 <0.05 a . (48)
1 2ω 2 ω a
1
The interference processes depend on the relative helicity We then estimate the amplitudes by imposing the additional
of the two fields. We first consider the fields with the same condition ξxx(2ω)E2 = ξxxxx(ω)E4, which gives optimal
1 2ω 2 ω
helicity,E(ω)= Eωeiθ1pˆτandE(2ω)= E2ωeiθ2pˆτ. interferencebetweentheabsorptionprocesses20. Finally, the
Thechargecurrentinjectionrateisgivenby fieldamplitudesarelimitedby
(cid:68) (cid:69)
J˙ia = 2·i[(cid:104)aηy+−c−o+s((∆ωθ))++ηa+x−−s+in((ω∆)θ(cid:105))E]2E . (47) (eE2ω)2 = 4A20(cid:126)(2eωE4ω)4 < 1.A62(cid:126)T2ω22. (49)
i(1) i(2) ω 2ω 0
ThespindensityandcurrentfollowsEq. (43). For pulses lasting 1ns with a frequency of 30meV, the field
The relative phase displacement between the two light amplitudes found are Eω = 104mV for the fundamental and
beamscannowcontrolthedirectionoftheinjectedquantities. E2ω = 72mV for the second harmonic, which correspond to
Especiallyforfrequenciesnearthegap, theinjectionrates laser intensities of 9.9 W and 0.65mW, respectively. We use
cm2 cm2
for different helicities τ depend strongly on the chirality of thesevaluesforall(cid:126)ωinFigs. 2,3,and4,althoughfor(cid:126)ω <
the electronic states, identified by ∆ /|∆ |. The helicity of 30meVsmalleramplitudeswouldberequiredtoguaranteeEq.
m m
theincidentlighthasnoeffectforvanishing∆ . (48),andcouldbefoundbyusingEq. (49).
m
Intheabsenceoftheσ massterm, thecarrierandcharge
z
current injection rates are very similar to the ones found for
Oppositehelicities graphene23,exceptfortheadjustmentsduetohavingonlyone
Dirac cone and a smaller Fermi velocity. However, even in
HerewehaveE(ω) = Eωeiθ1pˆτ andE(2ω) = E2ωeiθ2pˆ−τ. thiscasethereisalsoinjectionofthetransversespinfollow-
The injection rates from interference all vanish for the four ing the same form of the injected current, a signature char-
operatorsofinterest. acteristic of topological insulators. The magnitude of these
8
injectedquantitiesarealsoofthesameorderofthevaluesfor andEngineeringResearchCouncilofCanada(NSERC).
graphene,whichhasalreadybeenmeasured22.
Another distinctive trait shared with graphene is the rela-
tivelylowaveragevelocityoftheinjectedcarrierswhencom-
pared to semiconductors. This is due to the one photon ab- AppendixA:Generalexpressionsfortheopticalinjection
sorptionatthefundamentalfrequency,forbiddeninsemicon- coefficients
ductors because of the bandgap. This gives rise to the extra
interferenceprocesswithtotalenergy(cid:126)ω,whichusuallypar- InordertoevaluatethecoefficientsinEq. (38),thefollow-
tiallycancelstheinjectedcurrentstemmingfromtheinterfer-
ingintegralsarehelpful
enceprocesswithtotalenergy2(cid:126)ω.
Severalparticularfeaturesarefoundinthepresenceofthe (cid:82)
lBaerirzraytipohnasseofinthdeucoipntgic∆almfiσezldtes,rmw,heesnpaenciianlltyerefostrincgircinutlearrpplaoy- ϕaϕbacdb == (cid:82) 22ddππθθkkkaakk2kbbk4c=kdaˆ=2·bˆ,aˆ·bˆ(cˆ·dˆ)+aˆ·cˆ(8bˆ·dˆ)+aˆ·dˆ(bˆ·cˆ), (A1)
between the helicity τ of the incident fields and the chirality ϕabcdef = (cid:82) dθkakbkckdkekf = (cid:80) aˆ·bˆ(cˆ·dˆ)eˆ·fˆ,
∆ /|∆ | of the Dirac cone can greatly suppress or enhance 2π k6 48
m m pairings
opticalinjection. Inordertoobservethesefeaturesacombi-
nation of high magnetic field and low temperature is neces-
and
sary. BecausetheZeemancoupling(cid:126)∆ = µBgBneedstobe
m 2
abebneoeervgneyotuesgmhhopufelordartnuBorite2TbkeeB3Tm..uFWcoherlaearsgtpiemrroanthtoeauntnhctaheted7Z7eeKffeemacnta,dnt6hgTeapps.hhRootueoland- (cid:82) (cid:82)dθk2dπθavvcccvvvbvbc == 0A,20(dk22+d∆k22m)cˆ·bˆ +iA20∆mzdˆk·(cˆ×bˆ), (A2)
2π cv vc
sonable photon energies for the fundamental field would not also
bemuchlargerthan(cid:126)ω=30meV,whichcanbeachievedwith
quantumcascadelasers. (cid:82) dθkdkbvc ve = (dk2−∆2m)(cid:104)dk2cˆ·eˆ(dˆ·bˆ)−2(dk2−∆2m)ϕbcde(cid:105)
When lasers of similar intensity are considered, the mag- 2π cv vc 2dk2 (A3)
nitude of the injected currents obtained from coherent con- +i(dk2−∆2m)∆mdˆ·bˆ(cˆ×eˆ)·zˆ,
trol seem to be considerably larger than the values found by 2dk
otherapproaches,likeapplyinganin-planemagneticfieldor
and
oblique incidence. Therefore it can play a crucial role in the
questforharnessingtheexoticpropertiesoftopologicalinsu- (cid:82)
dθkckdkeva vb = 0,
latorsforspintronicapplications. (cid:82) dθk2πckdkekfvcavvvbc = (dk2−∆2m)2(cid:104)dk2aˆ·bˆϕcdef−(dk2−∆2m)ϕabcdef(cid:105)
2π cv vc A2d2
0 k
ACKNOWLEDGMENTS +i(dk2−∆2m)2∆m(aˆ×bˆ)·zˆϕcdef.
A20dk
(A4)
WethankJulienRiouxandJinLuoChengforhelpfuldis- The above equations combined with Eq. (33), Eq. (35), Eq.
cussions. This work was supported by the Natural Sciences (36)andEq. (37)givesthefollowingresults.
Oneandtwophotonabsorption
Thecarrierdensitycoefficientsare
(cid:20) (cid:18) (cid:19) (cid:21)
ξbc(ω)= Θ(ω−2∆m)e2 bˆ·cˆ 1+ 4∆2m −i∆mzˆ·(bˆ×cˆ) ,
1 2(cid:126)2ω 4 ω2 ω
(cid:18) (cid:19)(cid:18)(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21)(cid:19)
ξbcde(ω)= Θ(ω−∆m)e4A20 1− ∆2m (bˆ·dˆ)cˆ·eˆ+(bˆ·eˆ)cˆ·dˆ −2ϕbcde 1− ∆2m +i∆m cˆ·eˆ(dˆ×bˆ)·zˆ+bˆ·eˆ(dˆ×cˆ)·zˆ+cˆ·dˆ(eˆ×bˆ)·zˆ+dˆ·bˆ(eˆ×cˆ)·zˆ .
2 (cid:126)4ω5 ω2 2 ω2 ω 4
(A5)
Thechargecurrentcoefficientsvanish,ηabc(ω)=ηabcde(ω)=0.
1 2
Thespindensitycoefficientscanbewrittenintermsofthecarrierdensityonesas
ζabc(ω)= 2(cid:126)∆m(zˆ·aˆ)ξbc(ω),
1 ω 1 (A6)
ζabcde(ω)= (cid:126)∆m(zˆ·aˆ)ξbcde(ω),
2 ω 2
whichisaconsequenceofEq. (34).
9
Andthespincurrentcoefficientsare
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:21)
µabcd(ω)= Θ(ω−2∆m)e2D0 1− 4∆2m (zˆ×aˆ)·bˆ(cˆ·dˆ) −ϕ(z×a)bcd 1− 4∆2m +i∆m(zˆ×aˆ)·bˆ(dˆ×cˆ)·zˆ ,
1 4A0(cid:126) ω2 2 ω2 ω
µabcdef (ω)= 4Θ(ω−∆m)e4D0A0 (cid:18)1− ∆2m(cid:19)2(cid:20)ϕ(z×a)bdfcˆ·eˆ+ϕ(z×a)bcfdˆ·eˆ+ϕ(z×a)bdecˆ·fˆ+ϕ(z×a)bcedˆ·fˆ −ϕ(z×a)bcdef (cid:18)1− ∆2m(cid:19)(cid:21) (A7)
2 (cid:126)3ω4 ω2 4 ω2
+i4Θ(ω−∆m)e4D0A0∆m (cid:18)1− ∆2m(cid:19)2(cid:20)ϕ(z×a)bdf(eˆ×cˆ)·zˆ+ϕ(z×a)bcf(eˆ×dˆ)·zˆ+ϕ(z×a)bde(fˆ×cˆ)·zˆ+ϕ(z×a)bce(fˆ×dˆ)·zˆ(cid:21),
(cid:126)3ω4 ω ω2 4
givingspincurrentswithonlyin-planecomponentsofspin,whichisindependentofthe∆ σ massterm.
m z
Interferenceprocesses
Thecarrierdensitycoefficientsvanishξbcd(ω)=0. Thechargecurrentcoefficientsare
i(n)
(cid:18) (cid:19)(cid:18)(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21)(cid:19)
ηabcd(ω)= iΘ(ω−2∆m)e4A20 1− 4∆2m (aˆ·dˆ)bˆ·cˆ−2(aˆ·cˆ)bˆ·dˆ +ϕabcd 1− 4∆2m +i2∆m aˆ·dˆ(cˆ×bˆ)·zˆ−2aˆ·cˆ(dˆ×bˆ)·zˆ ,
i(1) 4(cid:126)3ω3 ω2 2 ω2 ω 2
(cid:18) (cid:19)(cid:18)(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21)(cid:19) (A8)
ηabcd(ω)= iΘ(ω−∆m)e4A20 1− ∆2m (aˆ·cˆ)bˆ·dˆ+(aˆ·bˆ)cˆ·dˆ −2ϕabcd 1− ∆2m +i∆m aˆ·cˆ(dˆ×bˆ)·zˆ+aˆ·bˆ(dˆ×cˆ)·zˆ ,
i(2) 2(cid:126)3ω3 ω2 2 ω2 ω 2
duetoEq. (34),thespindensitycoefficientscanbewrittenin and
termsoftheonesforthechargecurrentas
(cid:126) µ−+−+(ω)= iΘ(ω−2∆m)e2D0 (cid:18)1− 4∆2m(cid:19)(cid:16)1+τ2∆m(cid:17)2,
ζia(bnc)d(ω)= 2eA0η(i(zn×)a)bcd(ω), (A9) µ−+1−−++(ω)= iΘ(ω−16∆(cid:126)mA)e04D0A0 (cid:18)1−ω∆22m(cid:19)2(cid:16)1+τω∆m(cid:17)2, (B4)
and the spin current coefficients can also be written in terms 2 2(cid:126)3ω4 ω2 ω
oftheonesforthechargecurrentas
also µ+−−+(ω) = −µ−+−+(ω) and µ−+−−++(ω) =
µai(bnc)de(ω)= (cid:126)D0e∆Am20zˆ·aˆηbi(cnd)e(ω), (A10) −µT−2h+−e−i+1n+te(rωfe)r.encecoefficie1ntsare 2
whichfinishesthelistofopticalinjectioncoefficients.
(cid:18) (cid:19)(cid:18) (cid:19)
ηxxxx(ω)= −iΘ(ω−2∆m)e4A20 1− 4∆2m 1+ 12∆2m ,
i(1) 32(cid:126)3ω3 ω2 ω2
(cid:18) (cid:19)(cid:18) (cid:19) (B5)
AppendixB:Opticalinjectioncoefficientsforlinearand ηxxxx(ω)= iΘ(ω−∆m)e4A20 1− ∆2m 1+ 3∆2m ,
i(2) 8(cid:126)3ω3 ω2 ω2
circularpolarizations
and
The coefficients used for one and two photons absorption
processesare (cid:18) (cid:19)
ηxxxy(ω)= −Θ(ω−2∆m)e4A20 1− 4∆2m ∆m,
(cid:18) (cid:19) i(1) 2(cid:126)3ω3 ω2 ω
ξ1xx(ω)= Θ(ω8−(cid:126)22∆ωm)e2 1+ 4ω∆22m , ηxxxy(ω)= Θ(ω−∆m)e4A20 (cid:18)1− ∆2m(cid:19)∆m, (B6)
ξxxxx(ω)= Θ(ω−∆m)e4A20 (cid:18)1− ∆2m(cid:19)(cid:18)1+ 3∆2m(cid:19), (B1) i(2) 2(cid:126)3ω3 ω2 ω
2 4(cid:126)4ω5 ω2 ω2
also
and
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
µxµµyxy11xxxyxxxxxx(((ωωω)))=== −ΘΘ((Θωω(−−ω8∆2−(cid:126)∆m8A2m)(cid:126)∆0e)A4me0D2)De02A0D001(cid:18)(cid:18)11−−−4ω∆∆422m2mω∆2(cid:19)2m2(cid:19)(cid:18)34(cid:18)114+++∆ω∆2m232mω∆2(cid:19)2m,,(cid:19), (B2) ηηyiyi((xx21xx))yy((ωω))== i−ΘiΘ(ω(3ω−82(cid:126)2−(cid:126)3∆∆3ωωmm3)3)ee44AA2020 (cid:18)11−−4∆ωω∆2m222m(cid:19)2,5− 4ω∆22m , (B7)
2 4(cid:126)3ω4 ω2 ω2
µyxxxxx(ω)= −Θ(ω−∆m)e4D0A0 (cid:18)1− ∆2m(cid:19)2(cid:18)1+ 5∆2m(cid:19), and
2 4(cid:126)3ω4 ω2 ω2
forlinearpolarization. ηi+(1−)−+(ω)= iΘ(ω−4(cid:126)23∆ωm3)e4A20 (cid:18)1− 4ω∆22m(cid:19)(cid:20)14 −(cid:16)1+τ∆ωm(cid:17)2(cid:21),
Forcircularpolarizationwehave η+−−+(ω)= iΘ(ω−∆m)e4A20 (cid:18)1− ∆2m(cid:19)(cid:16)1+τ∆m(cid:17)2,
i(2) 4(cid:126)3ω3 ω2 ω
ξ−τ,+τ(ω)= Θ(ω−2∆m)e2 (cid:16)1+τ2∆m(cid:17)2, (B8)
ξ−1−++(ω)= Θ(ω8−(cid:126)∆2mω)e4A20 (cid:18)1− ∆ω2m2(cid:19)(cid:16)1+τ∆m(cid:17)2, (B3) fromwhichtheotherinjectioncoefficientsareobtained.
2 2(cid:126)4ω5 ω2 ω
10
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