A microscopic approach to spin dynamics about the meaning of

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

Amicroscopicapproachtospindynamics:aboutthemeaningofspinrelaxationtimes

C.LechnerandU.R¨ossler

Institutf¨urTheoretischePhysik,Universit¨atRegensburg,D-93040Regensburg(Dated:February2,2008)

arXiv:cond-mat/0412370v1 [cond-mat.other] 14 Dec 2004

WepresentanapproachtospindynamicsbyextendingtheopticalBlochequationsforthedriventwo-levelsystemtoderivemicroscopicexpressionsforthetransverseandlongitudinalspinrelaxationtimes.Thisisdoneforthe6-levelsystemofelectronandholesubbandstatesinasemiconductororasemiconductorquantumstructuretoaccountforthedegrees-of-freedomofthecarrierspinandthepolarizationoftheexcitinglightandincludesthescatteringbetweencarriersandlatticevibrationsonamicroscopiclevel.Forthesubsystemofthespin-splitelectronsubbandswetreattheelectron-phononinteractioninsecondorderandderiveasetofequationsofmotionforthe2×2spin-densitymatrixwhichdescribestheelectronspindynamicsandcontainsmicroscopicexpressionsforthelongitudinal(T1)andthetransverse(T2)spinrelaxationtimes.Theirmeaningwillbediscussedinrelationtoexperimentalinvestigationsofthesequantities.

PACSnumbers:72.25.Fe,72.25.Rb,78.47.+p

I.INTRODUCTION

TheBlochequations,originallyformulatedasequa-tionsofmotion(EOM)formagneticmomentshaveturnedouttoapplyingeneraltothedynamicsofquan-tummechanicaltwo-levelsystems[2].Oneprominentex-amplearetheopticalBlochequations(OBE)inatomicorsemiconductorphysicswiththecomponentsoftheBlochvectorcomposedoftheentriesofthedensitymatrixforadriventwo-levelsystemunderexcitationbyascalarlight eld(seee.g.Ref.3).Usuallycarrierscatteringisac-countedforbyaddingphenomenologicaldampingtermsconnectedwithalongitudinal(T1)andatransverse(T2)relaxationtime.InthecontextofOBE,T1character-izesthedecaytimeofthepopulationinversionortherelaxationintoanequilibriumdistribution,whileT2isthetimescaleonwhichthecoherencebetweenexcitinglightandopticalpolarizationgetslost.Afurtherevolu-tionoftheOBEarethesemiconductorBlochequations(SBE),whichwereformulatedtodescribeopticalphe-nomenainsemiconductorsunderintenseexcitationbyin-cludingmany-particletermsduetoCoulombinteractionbetweenthecarriers[4].Theseequationsyieldamicro-scopicformulationofT1andT2causedbycarrier-carrier[5]orcarrier-phononscattering[6,7].Inspiteoftheirsuccessfulapplicationtocarrierdynamics,theOBEandSBE,intheiroriginalform,arenotcapabletocontributetothecurrenttopicofspindynamicsinsemiconduc-tors.Recently,thisshortcomingwaspartiallyovercomebyextendingtheSBEwithrespecttothespindegree-of-freedomofthecarriers(includingspin-orbitcoupling)andthepolarizationdegree-of-freedomoftheexcitinglight[8],necessarytocreateanon-equilibriumspindis-tributionbyopticalorientationTheseextendedSBE(derivedbyapplyingtheHartree-Focktruncation)arerestricted,however,tothecoherentregimeandhencefallshortofdescribingscatteringasoriginofspinrelax-ationandspindecoherence,whicharekeyissuesofspin-tronicsandquantumcomputation[10].Ontheotherhand,thestructureoftheseequationsresemblesthose

usedinthephenomenologicalapproachofspindynamics11,12],thusindicatingthepossibilityofarrivingatamicroscopicapproachtospinrelaxationinthelanguageofBlochequations.

Itistheaimofthispapertoprovideamicroscopicformulationofspindynamicsinsemiconductorsandsemiconductorheterostructures.Wedothisbystart-ingfromtheextendedOBEforthe6-levelsystemofelectronandholesubbandstates,containingthespinofthecarriersandthepolarizationoftheexcitinglight(thiscorrespondstotakingintoaccountonlythesingle-particlecontributionstotheSBEofRef.8)andincludetheelectron-phononinteractionasapossiblescatteringmechanism.Forthesubsystemoftheconductionbandstates(spin-splitbyspin-orbitcoupling)weformulatethefulldynamicsasasetofEOMforthe2×2spin-densitymatrixandthephononassisteddensitymatrices.Byusingacorrelation-basedtruncationschemeinsec-ondorderBornapproximation,wederivethescatteringrates(intheBoltzmannlimitandbeyond)toarriveatequationsdescribingtheelectronspindynamicsinclud-ingrelaxation(duetoelectron-phononinteraction)onastrictlymicroscopiclevel,whileexistingtheories(seee.g.Ref.areamixtureofmicroscopicsandphenomenol-ogy.Wewanttostressalsothat,regardingthecreationofanon-equilibriumspinpopulation,ourtheoreticalcon-ceptdi ersfromsomeexperimentalsituations:inourOBEanon-equilibriumspinpolarizationisduetoopti-calorientation,whileinspintronicdevicesitisusuallycreatedbyspininjection[13].However,thisdi erencewillnotbecomerelevantinthecontextofthispapercon-centratingonthespinrelaxationduetocarrier-phononinteraction.

Thispaperisorganizedasfollows:InSec.IIwein-troducethesystemHamiltonian,formulatethefulldy-namicsofthesystemwithouttruncationandderivetheEOMfortheelectronsubsystem.InSecs.IIIandIVwepresentthecorrelation-basedtruncationschemeusedtoachieveaclosedsetofequationsfortheentriesinthe2×2densitymatrixrelatedtothespin-splitconduction

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

bandstates.ItrepresentsanextensionofthecoherentOBEforthespin-densitymatrixbycontributionsdue

to

electron-phonon

scattering.

In

Sec.

V

we

relate

the

dy-

namics

ofthedensitymatrixwiththoseofexperimentalobservablesanddiscussthemeaningofthecorrespondingspinrelaxationtimes.Finally,wedrawtheconclusionsofourresultsandgiveanoutlook.

discussiontothecaseofaquantumwellstructure(QW),buttheequationscanbeformulatedinthesamewayforabulksemiconductor.Weconsiderasix-levelsystemconsistingofstatesfromthespin-splitlowestelectronsubband(withangularmomentumorpseudospinindicesmc=±12)andlighthole(mv=±1

Thse

2

II.

SPIN-DEPENDENTOBEINCLUDINGCARRIER-PHONONINTERACTION

TheHamiltonianofthesystemisformulatedinsecondquantizationusingthenotationofRef.8.Werestrictour

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

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3

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

tionPmcmv(k)= c mc(k)vmv(k) .Withoutelectron-phononinteractiontheEOMofallentriesofthe6×6densitymatrixwouldformaclosedsetofequationsrep-resentingthecoherentspin-dependentOBEforthesys-tem.Adetailedtheoreticalstudyoftheopticalcoher-enceandpolarizationdynamics,yetwithoutaddressingthespin/pseudospin,canbefoundinRef.14.

Thepseudospindynamics,inparticulartherelaxationanddecoherence,iscontainedinthetimeevolutionofthediagonalblocks,whichshallbeexempli edherefortheelectronsystem.Thesamestepsofcalculationwouldleadtothecorrespondingequationsfortheholesystem,

whichhoweveraremorecomplicatedduetotheaddi-tionalorbitaldegrees-of-freedom.The2×2pseudospin-densitymatrixfortheelectronsis

mcmc(k) mc mc(k)(mcm¯c)

.(6) (k)=

mcmc(k) mc mc(k)Thesingleentriesareexpectationvaluesofproductsofacreationandanannihilationoperator mcm¯c(k)=

cmc(k)cm¯c(k) .WeevaluatethecommutatorsofthesystemHamiltonian,Eq.(1),withc ¯c(k)andmc(k)cm

takethethermalexpectationvalueto ndtheirEOM

i t mcm¯c(k)¯c(k)) mcm¯c(k)=( mc(k) m

cv cv

+E(t)·dm¯cmv(k)¯cmv(k)Pmcmv(k) E(t)·dmcmv(k)Pm

mvqm′c

ee

(k q) (q) c gm+′m(q) c′(k+q)b(q)cm¯c(k) gmmc(k)b(q)cm′¯cm′mccccc

e e

+gm(k)b(q)cm′(k+q) .(q) c′(q) cm′(k q)b(q)cm¯c(k) gm′mm¯cmcccccc

(7)

The rsttwolinesarethesingle-particlecontributions

oftheSBEinRef.[8]:theydescribethedynamicscausedbythespin-splitenergylevelsandbytheex-citationoftheelectronsofeitherpseudospinfromthevalencesubbandsdependingonthepolarizationofthedrivinglight eld.Thethree-operatortermsspecifythescatteringofanelectron(inoneofthespin-splitsub-bands)fromonektoanotherone(inthesameorthe

otherspin-splitsubband)therebyabsorbingoremittingaphonon,asvisualizedinFig.3.Thethree-operatorterms(ortheirthermalexpectationvalues)establishthephonon-assisteddensitymatrix[6],whoseentriesobeyEOMsofwhichwepresentasanexampletheonefor

sm′(k+q)b(q)c(km¯c(k+q,q)= cm′¯c) cmc

i tsm′¯c(k+q,q)=cm

(k) ω(q)sm′ m′(k+q) m¯¯c(k+q,q)ccmc

e′′′+gm(k+q+q)b(q)b(q)cm′(q) cm¯c(k) ′ ′cmcc

k′q′

m cm ′c

m vm ′v

′ ′e

(q′) cm+gm′m¯c(k) c(k+q q)b(q)b(q)cmc c ′e

(k+q′) (k+q)b(q)b (q′)cm gm ′¯c(q) cm′ ′ccmc

′e′′

gm(k+q)b(q)b(q)cm c(k q) ¯cm c(q) cm′c

′ e

(k′+q)cm(k+q)cm+gm¯c(k) ′ c(q) cm ′ c(k)cm′ccmc h′

gm(q) v(k)cm′(k ′m m vvvc

′

′

.+q)vv¯c(k) ′(k+q)cm

(8)

AscanbeseenfromEqs.(7)and(8),werunintoa

hierarchyproblemwithEOMscontainingtermswithan

increasingnumberofoperators,whichistypicalforsys-

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

temswithinteractions.Thishierarchyproblemcanbeovercomebyapropertruncation.Thestandardproce-dureistoneglecttheexistenceofcoherentphononscor-respondingtotheexpectationvalueofasinglebosonicoperator( rstorderfactorization)andtotakeintoac-countonlytheexpectationvalueswhichleadtoaphononoccupation

number

[6,7].

III.

THEBOLTZMANNLIMIT

wherex(t)standsforthethree-operatortermandy(t)correspondstoproductsbetweenphononoccupationfunctionsβ(q)= b (q)b(q) andentriesoftheelectron

′

densitymatrix mcm¯c(k).AspresentedinRef.6,equa-tionsofthistypecanformallybeintegratedtoyield

x(t)=x(t0)e

iω(t t0)

t t0

+

e iωty(t t′)dt′.(10)

′

Thegoalofthetruncationistogradually lteroutthe

scatteringtermsuptoacertainorderintheinteractionrelevantfortheinvestigateddynamics.ToexpressthescatteringcontributionsintheBoltzmannlimitcausedbyelectron-phononinteraction,weformulatethefollowingrulesforthetruncation:

1.Afterfactorizationofthefour-operatortermsonlyexpressionscontainingamacroscopicexpectationvaluearetakenintoaccount.2.ScatteringtermscontributingintheBoltzmannlimitarethoseproportionaltothesquaredabsolutevalueoftheinteractionmatrixelementinEq.(7).Thismeansthatweneglecttheso-called“polariza-tionscattering”duetointer-andintrabandpro-cesses[22]forwhichwerefertothenextsection.Applyingtheserulesmodi esEq.(8)andleadstoanequationwiththefollowingcharacteristicstructure

tx(t)= iωx(t)+y(t),

(9)

InsertingthisresultintotheEOMof mcm¯c(k)leadstoanon-Markovianintegro-di erentialequation,whichcanbesolvedanalyticallybyapplyingtheMarkovandadiabaticlimit[22,23].ThiscorrespondstouseinsteadofEq.(10)thefollowingexpression

P

x(t)= i

q,m′c

′′′(k+q)β(q)(k) ω(q)1 (k+q) δ (q)||gm′mcmcmcmccmc

2

′(k q)(1+β(q)).(k q) mc(k)+ ω(q)1 m′+δ m′

cmcc

(13)

Ithasthecharacteristicformofexpressionsobtained

fromFermi’sGoldenRule:alltermsareproportionaltotheabsolutesquaredvalueoftheinteractionmatrixelementsandtotheδ-functiontowarrantenergyconser-

vationinthescatteringprocess.Γinmcmc(k)hasthesameformbutwithchangedphononandelectronoccupationfactors.

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

Fortheo -diagonalentrieswemaywritethescatteringcontributionsas

e p

(k) mc mc(k),(14) t mc mc(k)|scatt1= Γmc mc

with

Γeπ

m p

c mc

(k)=T=Γ1,k inm′out

cm′c(k)+Γm′cm′c

(k)m′c

1

(16)

i

Σ¯e m′c m′c

(q) m′c m′c(k+q).q

pm′c

(18)

IncontrasttoEq.

(14)onehastosumhereoverthepseu-dospinindexandtheintheself-energyΣ¯e wavepvectorwhichentersdi erently

m′′(q)andin m′c m′c(k+q).Acorrespondingscatteringc mc

contributionwasfoundinRef.[5,6]fortheinterbandpolarization,i.e.fortheo -diagonalentryofthe2×2densitymatrixconsideredthere.Inordertopresentthestructureoftheself-energyweextractallcontributionscontaining(accordingtoEq.(11))aδ-functionbywriting

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

πp¯e Γ(q)=′′mc mc

i

mv

E(t)·dcvmcmv(k)Pmcmv(k) h.c.

(20)

in Γoutmcmc(k) mcmc(k)+Γmc mc(k)(1 mcmc(k))

t mc mc(k)=

1

cv cv

E(t)·d mcmv(k)Pmcmv(k) E(t)·dmcmv(k)P mcmv(k)

mv

qm′c

i

e p

(k) mc mc(k)+ Γmc mc

p¯e Σ′(q) m′ m′(k+q).m′ccc mc

(21)

Itcontainsinamicroscopicformulationthepseu-dospindynamicsinelectronsubbandsduetospin-orbit

coupling,spin-selectiveopticalexcitationandelectron-phononinteraction(forcarrier-carrierinteractionseetheremarkattheendof

thispaper).Byproperlyde ningaBlochvectorasinSec.IIIandlookingatthedampingtermsinthecorrespondingBlochequationswecanagainspecifythelongitudinalandtransversepseudospinrelax-ationtimes.Asitturnedout,onlyT2,kismodi edbyadditionalterms(beyondtheBoltzmannlimit)discussedinthissection,whileT1,kremainsunchanged.

V.CHANGINGTHESPINBASIS

WhendescribingexperimentsdesignedtomeasurethespinrelaxationtimeτSRandthespindecoherencetimeτSDofasystem(seee.g.Ref.13andreferencestherein),abasisisusedwithspinstatesorientedrelativetoa xeddirection,e.g.thegrowthdirectionoftheQWstructure.Accordingtothischoice,spinsarespin-up(↑)orspin-down(↓)whenalignedparallelorantiparalleltothisdi-rection,butinthepresenceofspin-orbitinteractionspinisnotagoodquantumnumber.Consequently,thekineticpartoftheHamiltonian(includingspin-orbitterms)forageneralwavevectorkisnotdiagonal.Inordertobe

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

consistentwiththisconvention,wetranslatetheresultsofSects.IIIandIV,formulatedintheeigenstatesofH0,tothespin-up/downbasis.Theunitarytransformationconnectingthetwobasissystemsdependsonthewavevectorkandthetypeofspin-orbitinteractiontobecon-sidered.Tokeepthediscussionasgeneralaspossible,wetakeintoaccountthetwomostfrequentlydiscussedmechanismsofspin-orbitcoupling,namelythelinearizedDresselhaustermandtheRashbaspin-orbitinteraction[24,25].Accordingly,wehaveinsteadofH√

0theHamil-tonian

H↑↓=Hkin+HR+HD,

(22)

withthekineticenergyHkin=

2

d+(k)+2 {A dk mc mc(k)}

d (k)+2i {A 2

(k) 2i {A

k m k mc mc(k)}c mc(k)}d+(k) 2 {Ak mc mc(k)}

,

withd±(k)= mcmc(k)± mc mc(k).Foraparticulartransformationtoexpress

choiceofthespin-orbitinteraction(RashbaorDressel-haus)thecorrespondingunitarytransformationcanbeS=c(k)}

derivedonthebasisofthisresult.

4 {A k mc mSpin-dynamicsexperiments,suchastime-resolvedpho-toluminescenceorFaradayrotation(foranoverviewofC=

recentexperimentsusingthesetechniquesseeRef.10) k

k

orphotogalvanice ect[26]donotaimatthedynamicsofthedensitymatrixofaninpidualkbutatquantitiessuchasthespinpolarization

S=

( ↑↑(k) ↓↓(k))(26)

k

andthespincoherence[21]

C=

|,

(27)

k

| ↑↓(k)de nedforthewholepopulationofthetwo-levelsystem.

TheirdecayischaracterizedbythespinrelaxationtimeτSRandthespindecoherencetimeτSD.Withthere-sultsofSects.IIIandIVwearenowinthestatetoformulatetherelationbetweenthesequantitiesandtherelaxationtimesT1,kandT2,kbyapplyingtheunitary

8

(25)

(28)

We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin relaxation times. This is done for the 6-level system of electron an

9

VI.

CONCLUSIONS

Inthispaperwehavepresentedamicroscopicformula-tionofspindynamicsinsemiconductorheterostructures.Itisbasedonthedensitymatrixapproachanditspar-ticularform,theopticalBlochequations.Startingfromthe6-levelsystemofconductionandvalencebandstatesdrivenbyopticalexcitationandincludingcarrier-phononinteractionwederiveexplicitlytheEOMforthe2×2den-sitymatrixoftheelectronsubsystemwhoseenergylev-elsarespin-splitduetospin-orbitcoupling.Weemployatruncationschemetoincludeelectron-phononinterac-tioninsecondorder.Inthislimitwederivemicroscopicexpressionsforthelongitudinalandtransverse(pseudo-)spinrelaxationtimesfortheinpidualspin-splittwo-levelsystemata xedk.Finallyaconnectionbetweentheseresultsandspinrelaxationtimescharacterizingthedynamicsofawholepopulationandaccessiblebyexper-imentsisestablished.Ittakesintoaccountthedi er-entsetsofeigenstatesusedinourmicroscopicderivation

(whichdiagonalizesthespin-orbitcoupling)andinthein-terpretationofthemeasurabletimes(witha xedaxisforspinquantizationandnondiagonalspin-orbitcoupling).Thusweprovideatthesametimeamicroscopicformu-lationofspindynamicsanditsrelationtoexperiments.Wewouldliketoemphasizethattheconceptpresentedherecanbeextendedtoincludealsocarrier-carrierinter-actionthusarrivingatanextensionofthecoherentSBEofRef.8.ForpreliminaryresultswerefertoRef.27.Furtherstepswillbenumericalevaluationsofthemicro-scopicexpressionsforrealisticquantumstructuresandtheexplicittreatmentofthespindynamicsfortheholesystem.

VII.

ACKNOWLEDGMENT

Wethankfullyacknowledge nancialsupportfromtheDFGviaForschergruppe320/2-1“Ferromagnet-Halbleiter-Nanostrukturen”.

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