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SurvivalofFloquet−BlochStatesinthePresenceofScattering
SvenAeschlimann,,RazvanKrause,MarianaChávez-Cervantes,
UmbertoDeGiovannini,HannesHübener,StivenForti,CamillaColetti,KerstinHanff,KaiRossnagel,
AngelRubio,andIsabellaGierz*
CiteThis:,21,5028−5035ReadOnline
ACCESSMetrics&MoreArticleRecommendations*sıSupportingInformation
ABSTRACT:Floquettheoryhasspawnedmanyexciting
possibilitiesforelectronicstructurecontrolwithlight,with
demonstrationinsolids,however,
particular,theinfluenceofscatteringontheformationofFloquet−
-
andangle-resolvedphotoemissionspectroscopywithtime-depend-
entdensityfunctionaltheoryandatwo-levelmodelwithrelaxation
toinvestigatethesurvivalofFloquet−Blochstatesinthepresence
−Blochstateswillbedestroyed
ifscatteringactivatedbyelectronicexcitationspreventsthe
two-levelmodelalsoshowsthatFloquet−Blochstatesreappearat
indicatetheimportanceoflongscatteringtimescombinedwithstrongdrivingfieldsforthesuccessfulrealizationofvariousFloquet
phenomena.
KEYWORDS:Floquet−Blochstates,dissipation,time-andangle-resolvedphotoemissionspectroscopy,
time-dependentdensityfunctionaltheory,driventwo-levelsystemwithdissipation
iththerecentdevelopmentofstrong-fieldterahertzandbetweenTDDFTsimulationsandtr-ARPESmeasurementsfor
Wmid-infraredlasersources,Floquetengineering,
thecoherentinteractionofstronglightfieldswithBlochofgraphene,however,TDDFTsimulationspredictthe
,2021at03:36:42(UTC).electronsinsideasolidisusedtomanipulatethebandopeningofvariousbandgapsintheelectronicdispersion
structureofthematerial,becomesaviableapproachforthatarenotobservedinthetr-
−
numberoffascinatingphenomena,includingdynamicalaresonantlydriventwo-
1
localizationofchargecarriersandlight-inducedtopologicalThismodelshowsthatFloquet−Blochstateswillbedestroyed
See,3
activatedbyelectronicexcitationspreventsthe
4−6
,however,theBlochelectronsfromfollowingthedrivingfieldcoherently.
abilityoftheBlochelectronstocoherentlyfollowthedrivingThemodelalsoshowsthatFloquet−Blochstatesreappearat
fi
withextremelylongscatteringtimessuchasthetopological
7,8abouttheexperimentalandtheoreticalmethodsandthedata
insulatorBi2Se3andexfoliatedgrapheneatlowtemper-analysisareprovidedintheSupportingInformation.
−
10TheexperimentalobservationofFloquetBlochstateswith
presenceofdissipationisahotlydebatedtopicbecauseofitstr-ARPESisimpededbythefactthatinthepresenceofa
relevanceforfutureapplicationsandthepossibilitytouse
dissipationtostabilizenovelFloquetphasessuchasthetime
−
:February25,2021
Here,weusetime-andangle-resolvedphotoemissionRevised:May10,2021
spectroscopy(tr-ARPES)combinedwithtime-dependentPublished:June4,2021
densityfunctionaltheory(TDDFT)andatwo-levelmodel
withrelaxationtoinvestigatethesurvivalofFloquet−Bloch
©
AmericanChemicalSociety/
,21,5028−5035
ℏωfi
-ARPESdataforWSe2forppandspdrivingpulses,respectively,atdrive=280meVwithapeakdrivingeld
−
ofEvac==0fs,
markthemomentumrangefortheenergydistributioncurves(EDCs)-inducedchangesofthephotocurrent
att=
(EDCs)
(dark)linesarethedata(fit).(dark)redareasshowtheindividual
Lorentziansofthesidebands(mainbands).Column5showstheintensityoftheuppermostreplicabandobtainedbyintegratingthetransient
EDCsincolumn4overtheenergyrangeindicatedbytheblackarrowasafunctionofpump−probedelaytogetherwithaGaussianfit.
strongdrivinglaserfieldphotoemissionoccursfromFloquet−states,whilethereplicabandsappearingforppdrivingpulses
Blochstates(photon-dressedinitialstates)toVolkovstatescontaincontributionsfrombothFloquet−BlochandVolkov
(photon-dressedfree-electronfinalstates).8,20,21Photon-(Supporting
dressingofboththeinitialandthefinalstateresultsintheInformation)inFigure1(b1−b4)weobtainreplicaband
formationofreplicabandsinthephotoelectronspectrumthatintensitiesof(±)%and(±)%forppandsp
areseparatedfromtheoriginalbandstructurebyintegerdrivingpulses,
geometryemployedinthepresentstudy,theVolkovwithintheerrorbars.
fi
contributiontotherst-orderreplicabandsiszerofors-Todeterminethelifetimeofthedrive-inducedreplicabands
fi
polarized(sp)drivingpulseswheretheelectriceldvectorliesweintegratethetime-dependentEDCspresentedinFigure
intheplaneofthesamplesurfaceandisperpendiculartothe1(a4andb4)overtheenergyrangeindicatedbytheblack
directionalongwhichthetransientbandstructureismeasured.
Inthiscase,theexperimentalobservationofrst-orderreplicafittedwithaGaussian[seeFigure1(a5andb5)],indicating
bandswithtr-ARPESprovidesdirectevidenceforthethatthelifetimeofthedrive-inducedreplicabandsis
formationofFloquet−Blochstates21(SupportingInforma-
determinedbythepump−
tion).
durationoftheMIRdrivewas280fsinpanela5and390fs
InFigure1wepresenttr-ARPESsnapshotsofthevalence
Γinpanelb5determinedbythedispersionofthewaveplates
bandofbulkWSe2measuredalongtheK-directioninthe
vicinityoftheK-.
ofFigure1showthemeasuredbandstructurewithoutandInFigure2wepresentTDDFTsimulationsfordirect
withp-polarized(pp)mid-infrared(MIR)drivingfield,comparisonwiththetr-(a1)
-inducedchangesoftheshowsthesimulatedspectrumwithoutmid-infrareddrive.
photocurrentobtainedbysubtractingFigure1(a1)fromPanelsa2anda3ofFigure2showthecorrespondingspectrain
fi
Figure1(a2)areshowninFigure1(a3).Dashedwhitelinesthepresenceofppandspdrivingpulses,respectively,witheld
inFigure1(a1anda2)indicatetheareaofintegrationforthestrengthanddrivingfrequencymatchingtheexperimental
−
energydistributioncurves(EDCs)presentedinFigure1(a4).(b1b3)weshow
ThecorrespondingdataforspdrivingpulsesarepresentedinEDCsextractedalongthedashedredlinesinFigure2(a1−a3).
Figure1(b1−b4).Ingoodagreementwiththetr-ARPESexperimentsinFigure1
Weobserveclearindicationsfortheformationofreplicawefindthat,inthepresenceofbothppandspdrivingpulses,
(SupportingInformation)theobservationofreplicabandsforcomparisonofthetheoreticalandexperimentalEDCsreveals
spdrivingpulsesindicatestheformationofFloquet−Blochthatthetheoreticalreplicabandintensityissimilartothe
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,21,5028−5035
(a1),inthepresenceofappdrivingpulse(a2),andinthepresenceofanspdriving
pulse(a3).Dashedredlinesina1−a3indicatethepositionsfortheenergydistributioncurves(EDCs)inpanelsb1−-point
atequilibrium(b1),inthepresenceofappdrivingpulse(b2),andinthepresenceofanspdrivingpulse(b3).
experimentalvalueforppdrivingpulsesandsignificantlyaccordingtothemodelfromref20,containcontributionsfrom
−
Havingdemonstratedthecapabilityofourtr-ARPESsetup[Figure3(a3)]thereplicabandsarefoundtobemuchweaker
togenerateandresolveFloquet−BlochstateswenowturntobecauseoftheirpureFloquet−Blochcharacter(Supporting
thefascinatingscenarioofalight-inducedtopologicalphaseInformation).Also,thesimulationsclearlyshowthepredicted
,
circularlypolarized(cp)lightfieldsispredictedtoopenagapinthepresenceofacircularlypolarized(cp)drivingpulseis
attheDiracpointandturngrapheneintoatopologicalshowninFigure3(a4).AsidefromreplicabandsandRabigaps
(Rabigapsinthefollowing)thespectrumshowsapronouncedbandgapattheDiracpoint.
arepredictedtoopenawayfromtheDiracpointwheretheInordertogetabetterimpressionoftheintensityofthe
unperturbedbandstructurecrossesthenth-orderFloquetreplicabandsaswellasthesizeofthedynamicalbandgaps,
,22BecausethematrixelementforinterbandFigure3(b1−b4)presentsEDCsextractedalongthedashed
−
transitionsingrapheneishighlyanisotropic2325withnodesverticallinesinFigure3(a1−a4).FromLorentzianfitstothe
(maxima)inthedirectionparallel(perpendicular)totheEDCinFigure3(b3)weextractaRabigapof200meVforthe
polarizationofthedrivingfield,,LorentzianfitsoftheEDCsinFigure
alongthedirectioninmomentumspacethatisperpendicular3(b4)yield100meVfortheRabigapand60meVforthegap
.
−probedelay
howmid-,onlyoneof
,thetwobranchesoftheDiracconeisvisibleinthespectrum.
drivingfrequencyaswellasfieldstrengthwerechosentoIncontrasttothesimulationsthatwereperformedforaneutral
(a1)showsthegraphenelayer,theepitaxialgraphenesamplesusedforthetr-
-
oftheDiracconeisinvisiblebecauseofphotoemissionmatrixbelowtheFermilevel(SupportingInformation).Thesecond
,27InthepresenceofappdrivingfieldthecolumnofFigure4showsthephotocurrentatzeropump−
spectruminFigure3(a2)showsstrongreplicabandsthat,,sp,andcpdrivingpulsesare
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,21,5028−5035
(greenlines)atequilibrium(a1),inthepresenceofappdriving
pulse(a2),inthepresenceofanspdrivingpulse(a3),andinthepresenceofacpdrivingpulse(a4).Blueandreddashedlinesindicatethe
positionsfortheenergydistributioncurves(EDCs)inpanelsb1-(b1),andinthepresenceofapp
drivingpulse(b2).EDCthroughtheDiracpoint(blue)andatthepositionwheretheRabisplittingismostpronounced(red)inthepresenceofan
spdrivingpulse(b3)andinthepresenceofacpdrivingpulse(b4).
fi
showninr