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AbInitioCalculationsinaUniformMagneticFieldUsingPeriodicSupercellsWeiCaiandGiuliaGalliLawrenceLivermoreNationalLaboratory,UniversityofCalifornia,Livermore,California94550,USA(Received22September2003;published5May2004)Wepresentaformulationofabinitioelectronicstructurecalculationsinanitemagneticeld,whichretainsthesimplicityandefciencyoftechniqueswidelyusedinrstprinciplesmoleculardynamicssimulations,basedonplane-wavebasissetsandFouriertransforms.
Inadditionwediscussresultsobtainedwiththismethodfortheenergyspectrumofinteractingelectronsinquantumwells,andfortheelectronicpropertiesofdenseuiddeuteriuminauniformmagneticeld.
DOI:10.
1103/PhysRevLett.
92.
186402PACSnumbers:71.
15.
–m,71.
10.
–w,71.
70.
DiInthelasttwodecades,abinitioelectronicstructuremethodsbasedondensityfunctionaltheory(DFT)havebecomematuretechniques,whicharenowwidelyusedtoinvestigatethestructuralandelectronicpropertiesofbothcondensedandnitesystems,e.
g.
,moleculesandclusters[1].
Inparticular,theformulationofabinitiomoleculardynamics(MD)[2]haspermittedkeyprogressinthepredictionofnitetemperaturepropertiesofma-terialsentirelyfromrstprinciples.
ThemostwidelyusedimplementationofabinitioMDandofelectronicstruc-turecalculationsforcondensedsystemsisbasedonpseu-dopotentialsandplane-wave(PW)basissets.
TheuseofPWhasseveraladvantages.
Theconvergenceoftotalenergyandforcecalculationscanbecontrolledbyasingleparameter(kineticenergycutoff)andimprovedtoarbitraryaccuracy.
Atomicforcescanbeeasilycom-putedwithoutevaluatingtheso-calledPulaycontribu-tions[3]andefcientfastFouriertransform(FFT)techniquescanbeapplied.
PWbasissetscallfortheuseofperiodicboundaryconditions(PBC),whichcon-venientlyeliminatesurfaceandinterfaceeffectsandallowforasmallsimulationcelltomimicthebulkbehaviorofmaterials.
Todate,mostabinitioinvestigationshavefocusedongroundstatepropertiesintheabsenceofexternalelectro-magneticelds.
DuetotechnicaldifcultiesindescribingniteeldswithinPWformulationsandusingPBC,al-mostallstudieswithelectromagneticeldshavebeencarriedoutperturbatively.
Withinthisapproach,simula-tionsareperformedatzeroeldandelectricpolarizabil-ityandmagneticsusceptibilityarecomputedbasedonlinearresponsetheory[4].
Whilethistechniquecanbeusedwhentheappliedeldissufcientlysmall,therearemanysituations,e.
g.
,condensedsystems—notablyhy-drogen—instarsandplanets[5],wheretheeffectofaniteeldcannotbetreatedinaperturbativefashion.
Recentlytherehasbeenprogressinexplicitlyincorpo-ratinganiteelectriceldincondensed-phaseabinitiosimulations[6].
AnonpeturbativeBlochsolutionoftheSchro¨dinger'sequationinanitemagneticeldwasalsoproposed[7].
Yetnoattempthasbeenmadetoformulateelectronicstructurecalculationsincludingnitemag-neticeldsinthecontextofcondensed-phaseabinitioMDsimulations.
InthisLetter,wedescribeaformulationofself-consistentabinitiocalculationswithinDFTwheretheeffectofanite,uniformmagneticeldistreatedinanonperturbativemanner,usingalgorithmsbasedonPWbasissetsandFFT.
ThesealgorithmshavebeenkeyinthedevelopmentofsimpleandefcientrstprinciplesMDtechniques.
Wepresentapplicationsofthisnewmethodtointeractingelectronsinaquantumwellanddenseliquiddeuteriuminauniformmagneticeld.
Magneticperiodicboundaryconditions.
—TheHamil-tonianofanelectroninaperiodicpotentialV~rrandauniformmagneticeld~BBisH12m~ppe~AA~rr2V~rr;(1)where~AA~rristhevectorpotential(~BBr~AA).
Foruniform~BB,~AA~rrisnotperiodicandtheelectronwavefunction~rrcannotsatisfyPBC.
However,physicalob-servablesmaystillretaintranslationalinvarianceproper-tiesinsuchconditions.
Forexample,inaclassicalpictureamagneticelddoesnotdoanyworkwhenanelectronmovesfromonepointtoanotherinspace(Lorentzforceisalwaysperpendiculartoelectronvelocity),sothattheelectronickineticenergyistranslationallyinvariant.
ThissuggeststhatPBCmaybegeneralizedtodescribeelec-tronsinauniformmagneticeld.
Let~ccbetheperiodicityofthepotentialV~rr,i.
e.
,V~rr~ccV~rr.
If~rrisaneigenfunctionofH,then~rr~ccisaneigenfunctionofH0,whichdiffersfromHonlybyitsvectorpotential~AA0~rr~AA~rr~cc.
When~BBisuniform,~AA~rrislinear,i.
e.
,~AA~rr~cc~AA~rr~AA~cc.
Therefore,wecanregardtheabovetranslationofHasagaugetransformationfor~AA:~AA0~rr~AA~rrr~rr,with~rr~AA~cc~rr.
Gaugeinvarianceinsuresthat0~rrexpieh~rr~rrisalsoaneigenfunctionofH0.
InthespiritoftheBlochtheorem,wecanrequire~rr~cctoequal0~rr,uptoaphasefactorexpi~kk~cc:~rr~ccexpieh~AA~cc~rri~kk~cc~rr;(2)PHYSICALREVIEWLETTERSweekending7MAY2004VOLUME92,NUMBER18186402-10031-9007=04=92(18)=186402(4)$22.
502004TheAmericanPhysicalSociety186402-1Equation(2)expressestheso-calledmagneticperiodicboundarycondition(MPBC),whichwasrstsuggestedinRef.
[8].
ItisstraightforwardtoseethatifthewavefunctionsatisesMPBC,thenthechargedensity~rrj~rrj2—ameasurablequantity,isaperiodicfunction.
Inthezero-eldcase,Eq.
(2)simplyreducestotheBlochtheorem.
Forsimplicityinthefollowingwewillonlydiscussthecaseof~kk0(point).
Considerarectangularsimulationcell[forwhichV~rrisperiodic]ofdimensionaandbalongxandy,respec-tively.
LetauniformmagneticeldbeparalleltothezaxisandadopttheLandaugauge:~AA~rr0;Bx;0.
Asthewavefunctionisperiodicalongthezaxis,inthefollowingwewillnotdiscussexplicitlythezdependenceof.
IntheLandaugauge,MPBCcanbeexpressedasxa;yexpieBahyx;y;x;ybx;y:(3)AninterestingpropertyofMPBCcanbeobtainedbyconsideringthephasechangeofasonemovesalongtheedgeofthesimulationcell.
OneaccumulatesatotalphaseofeBab=haftercompletingoneloop,whichmustequal2n(ninteger)forself-consistency.
Therefore,theenforcementofMPBCrequiresthetotalmagneticuxthroughthesimulationcelltobeanintegermultipleofthefundamentalquanta0h=e:Babnh=en0.
Dependingonthesizeofthesupercell(aandb),thisquantizationimposesaconstraintonthemagnitudeofmagneticeldsthatonecanconsiderusingMPBC.
ImplementingMPBCwithinaplane-wave-likeformu-lation.
—WerstreviewthebasicsofabinitiocalculationsusingPBCandPWbasis.
Inthezero-eldcase,therealspacewavefunctionx;ycanbeexpressedbyitsFouriercomponentsckx;ky,wherekxnxGxandkynyGy,Gx2=a,Gy2=b(nx;nyintegers).
Thewavefunctionisthentruncatedinreciprocalspace,nx2Nx=21;Nx=2,ny2Ny=21;Ny=2,sothatitisrepresentedbyanNxNyarrayofcomplexnumbers.
FFTtechniquescanbeusedtogoefcientlyfromrealtoreciprocalspacerepresentations.
Ofcoursethetransfor-mationfromx;ytockx;kycanbedonenumericallyinonestepbyatwo-dimensionalFFT.
However,letusconsideratwo-stepprocess,wherex;yisFouriertransformedalongtheyaxisrst,leadingtofx;ky,whichisthentransformedalongthexaxis,resultinginckx;ky.
Wedenefx;kyasthewavefunctioninthe''intermediate''space,sincekyisareciprocalspacevariablewhilexisarealspaceone.
fx;kycanberegardedasasetofone-dimensionalperiodicfunctionsofx,eachcorrespondingtoadifferentky(thereareNyofthemintotal).
ThereforetheFouriertransformoffintoreciprocalspacecanbeconsideredasNyindividualone-dimensionalFFTsalongthexaxis.
WhenB0,x;ysatisesMPBCasinEq.
(3).
Forsimplicityconsiderthesmallesteldvaluepermitted,Bh=eab,i.
e.
,n1.
Becausex;yisperiodiciny,itcanbeFouriertransformedintotheintermediatespace,x;y!
FFTyfx;ky.
Inthisspace,MPBCbecomes:fxa;kyfx;kyGy:(4)Thustheintermediatewavefunctionfx;kycannolongerberegardedasasetofindependent,periodicfunctionsofx.
Instead,alloftheNyfunctionsarenowinterconnected.
Infact,ifwedeneanewvariable^xxxaky=Gy,Eq.
(4)canbeautomaticallysatisedbylettingfbeaone-dimensionalfunctionof^xx,f^xxfx;kyfxa;kyGy(5)f^xxcannowbeFouriertransformedintothereciprocalspace,f^xx!
FFT^xxck^xx,wheretheck^xxaretheFouriercoefcientsofthewavefunctionintoplane-wave-like,orthonormalbasisfunctionssatisfyingMPBC.
Theeffec-tivereductionofdimensionality(fromtwotoone)ofwavefunctionsduetothepresenceofamagneticeldhasbeennoticedpreviously[9].
ThetopologychangeoftheintermediatespaceisillustratedinFig.
1.
Whilefx;kyatB0canberegardedasasetofindependentrings(eachrepresentingaperiodicfunctionofxfordifferentky),itbecomesalongspiralwhenBh=eab.
Thissituationisanalogoustothatofacrystallatticecontainingascrewdislocation[10].
EvaluationoftotalenergiesusingMPBC.
—AkeystepinabinitiosimulationsisthecalculationofH,givenanarbitrarywavefunction.
OnceHiscomputed,itera-tivealgorithmsandMDtechniquescanbeappliedtoykFFTYRealspace),(yxψxy),(yxkkcReciprocalspacexkykFFTXIntermediatespace),(ykxxykfykFFTYRealspace),(yxψxyyGxyk),(ykxfyGyyGkaxx/+=FFTk)=(f),(ykxfxxxx)(ckxReciprocalspaceIntermediatespaceUnfold(a)(b)FIG.
1(coloronline).
Thereal-spacewavefunctionx;ycanbeFouriertransformedintoreciprocalspaceckx;kyintwosteps,viaanintermediate-spacewavefunctionfx;ky(seetext).
(a)AtB0,fx;kycanberegardedasasetofone-dimensionalperiodicfunctions,orrings.
(b)AtBh=eab,MPBCrequiresfx;kytobealongspiral.
Theresultingwavefunctioninintermediateandreciprocalspaceiseffectivelyonedimensional.
PHYSICALREVIEWLETTERSweekending7MAY2004VOLUME92,NUMBER18186402-2186402-2computetotalenergiesandforces.
WhenB0,thetwocomponentsofH—kineticenergy^TTandpotentialenergy^VV,arediagonalinreciprocalandrealspace,respectively:^TTckx;kyh2=2mk2xk2yckx;ky,^VVx;yVx;yx;y.
Therefore,^TTand^VVcanbeeasilycomputedinthesetwospacesseparately,and^HHisobtainedbyas-semblingthemtogetherviaFFT.
WhenBh=eab,theHamiltoniancanbeseparatedintothreecomponents,eachdiagonalinadifferentspace,H12m~ppe~AAx;y2Vx;y12mih@x2ih@yeBx2Vx;yh22m@2xh2G2y2ma2xi@yaGy2Vx;y^TTx^TTy^VV:(6)The''xcomponent''ofthekineticenergyisdiagonalinreciprocalspace,^TTxck^xxhk2^xx=2mck^xx.
The''ycomponent''becomesaharmonicpotentialintheinter-mediatespace,^TTyf^xxh2G2y=2ma2^xx2f^xx.
Thepo-tentialenergyisdiagonalinrealspaceasusual.
ThereforeHcanbeobtainedefcientlybycalculatingthesethreecomponentsseparatelyinthreespaces,followedbyanassemblyviaFFT,i.
e.
,),(yxψFFTx)(ckx)(fxFFTy)(fxTxyT)(ckxV),(yxψ+FFTx+FFTyFFTxH)(ckxThereal-tointermediate-toreciprocal-spaceFouriertransformsillustratedaboveapplytowavefunctionsonly.
DFTcalculationsalsorequireFouriertransformingthechargedensityjj2.
Sinceisasimpleperiodicfunctioninbothxandydirections,itsFouriertransformcanbeperformedbyordinarytwo-dimensionalFFTasinthezero-eldcase.
Forsimplicitywehaveonlyconsideredthecaseofthelowestpermittedmagneticeld.
Ingeneral,n=0canbelargerthan1.
Inthiscase,theintermediatespacecanbevisualizedasnspiralsinterlacedwitheachother(seeFig.
1).
TheFouriertransformbetweenintermediateandreciprocalspaceshouldthenbecarriedoutbynindependentone-dimensionalFFTs.
Moretechnicalde-tailswillbepresentedinaforthcomingLetter.
Results.
—Wehaveimplementedtheformalismdis-cussedabovetostudydifferentsystemsinamagneticeldwithincreasinglevelsofcomplexity.
First,asaproofofprinciplewesolvedthewellknownproblemofasingleelectroninauniformmagneticeldandcorrectlyrepro-ducedtheenergyspectrumofequallyspacedLandaulevels[11].
Wethencomputedtheenergyspectrumofasingleelectronandthatoftwointeractingelectronsinatwo-dimensionalquantumdot.
InRef.
[12],thisquantumdotwasmodeledasasquarepotentialwellwithenergyzeroinsideandinniteoutsidethewell.
Becausetheelectronwavefunctionisentirelylocalizedwithinthedot,thisproblemcanbesolvedwithoutusingasupercellandplane-wave-likebasisfunctions.
Asabenchmarkforourmethod,wesolvethisproblemusingasupercellenclosingthedot[seeinsetofFig.
2(a)].
Thepotentialenergyoutsidethedotissetto0.
1eV.
Foreachmagneticeld,weobtainthelowest64singleelectronlevels.
Theenergyspectrumofthetwo-electronsystemisthenob-tainedbydiagonalizingtheHamiltonianmatrixinthespacespannedbythesesingleelectronwavefunctions.
AsshowninFig.
2,theagreementwithpreviousresultsisverygood.
Thesmalldiscrepancyisattributedtothefactthatinourstudythepotentialenergyoutsidethequantumdotisnotstrictlyinnite.
Ourmethodisreadilyappli-cabletothemorechallengingproblemofaperiodicarrayofquantumdots,withelectronwavefunctionsnotcom-pletelylocalizedwithineachdot.
InthiscasethemethodofRef.
[12]isnolongerapplicable.
Wealsocalculatedthelowestthreelevelsofahydrogenatominmagneticelds[Fig.
3(a)].
Again,supercelltech-niquesarenotrequiredforthisproblem,sothatprevious00.
020.
040.
060.
020.
040.
060.
08B(T)E(meV)1600nm800nm00.
010.
020.
030.
040.
050.
060.
320.
340.
360.
380.
4B(T)E(meV)(a)(b)FIG.
2.
(a)Energyspectrumofsingleelectroninquantumdotasafunctionofmagneticeld.
Theinsetshowsthegeometryofquantumdot(shadedarea)andsimulationcell(outersquare).
(b)Energyspectrumoftwointeractingelec-tronsinquantumdotasafunctionofmagneticeld.
Filledandopencirclesindicatespinsingletandtripletstatesfromthiswork.
ThickandthinlinesareforsingletandtripletstatesfromRef.
[12].
02468x10420151050B(T)E(eV)00.
511.
522.
53x10521.
510.
500.
5B(T)E(eV)(a)(b)FIG.
3.
(a)Lowestlevelsofahydrogenatomasafunctionofmagneticeld:forthiswork,solidlineforpreviousresults[5].
(b)ChangeofbindingenergyofanH2moleculeinthe1gstate(twospinsantiparallel)asafunctionofmagneticeld:forourresultusingHartree-Fockapproximationwithinter-protondistancexedat0:74A,andsolidlinefromRef.
[13].
PHYSICALREVIEWLETTERSweekending7MAY2004VOLUME92,NUMBER18186402-3186402-3data[5]existforcomparison.
Weusedacubicsupercellof14Awithanenergycutoffof103eV,andourresultsagreeverywellwithearlierreports.
Theconstantdiffer-enceinthegroundstateisduetothewellknownproblemforPWtoresolvetheCoulombsingularityatthenucleus.
WealsocomputedthechangeofbindingenergyofahydrogenmoleculeasafunctionofmagneticeldusingtheHartree-Fockapproximation[Fig.
3(b)],againincloseagreementwithearlierresults[13].
NoticethatthemagneticeldhereisaboutsevenordersofmagnitudeshigherthanthatinFig.
2.
Thisdemonstratesthecorrect-nessandaccuracyofourmethodregardlessofthemag-nitudeofthemagneticeld.
Finally,weimplementedourformalismwithinthelocaldensityapproximation(LDA)ofDFTandcarriedoutself-consistentcalculationsoftheelectronicproper-tiesofdenseuiddeuterium[Fig.
4(a)].
Thepositionsof128deuteriumionsareobtainedfromasnapshotofanearlierabinitioMDsimulation[14]attemperature5000Kanddensity5105mol=m3underzeromagneticeld.
Thesimulationcellisacubewithlength7:52Aandanenergycutoffof2:7103eVisused.
WehavefoundthattheinstantaneousbandgapEgisstronglyinuencedbythemagneticeld:Eg0:176eVatB0withEg0:272eVatB104T.
Whilenoappreciabledifferenceisobservedinthetotalchargedensityatthesetwomagneticelds(whichissomewhatsurprising),theden-sityofindividualelectroniclevelschangesdramatically.
Fig.
4(b)plotsthechargedensityofthehighestoccupiedmolecularorbital(HOMO)forbothB0(blue)andB104T(red).
Weseethatthisstateisassociatedwithdiffer-entatomsatthesetwomagneticeldvalues.
Similarconsiderationsapplytothelowestunoccupiedmolecularorbital(LUMO).
Thereforeweexpectastrongmagneticeldtohaveasignicantinuenceontheelectromagneticandopticalresponseofcompresseduiddeuterium.
Insummary,wedevelopedamethodforabinitiocalculationsinthepresenceofauniformmagneticeld.
OurapproachretainsthesimplicityandefciencyofelectronicstructurecalculationsbasedonPWandFFT,andcanbeappliedtobothniteandcondensedsystems.
OurformulationopensthewaytoperformingabinitioMDsimulationsinanitemagneticeld.
Calculationsofvelocityindependentcomponentofionicforcesisex-pectedtobestraightforward,atleastforlocalpseudo-potentials.
Inaddition,ageometricalcomponentcomingfromtheBerryphasecontributestotheLorenzforceonthenuclei[15].
Generalizationofthisapproachtoincludeacouplingofthemagneticeldwithspindegreesoffreedomisunderway.
WethankE.
PollockforusefuldiscussionsandL.
Kraussforhelponvisualization.
ThisworkwasperformedundertheauspicesofU.
S.
DepartmentofEnergybyUniversityofCaliforniaLawrenceLivermoreNationalLaboratoryunderContractNo.
W-7405-Eng-48.
W.
C.
issupportedbytheUniversityRelationshipProgramatLLNL.
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FIG.
4(color).
(a)Totalchargedensityofadensedeuteriumuid(seetext),whichremainsessentiallythesameasBgoesfrom0to104T.
(b)ThechargedensitiesoftheHOMOstateforB0(blue)andB104T(red)aredistributedondiffer-entatoms.
PHYSICALREVIEWLETTERSweekending7MAY2004VOLUME92,NUMBER18186402-4186402-4