Ab initio many-body calculation of excitons in solid Ne and Ar

arXiv:cond

-mat/041

258

4v1 [co

n

d-mat.other] 21 Dec 2004Abinitiomany-bodycalculationofexcitonsinsolidNeandAr

S.Galami´c-Mulaomerovi´candC.H.Patterson

DepartmentofPhysicsandCentreforScientificComputation,

UniversityofDublin,TrinityCollege,Dublin2,Ireland

(Dated:February2,2008)

Absorptionspectra,excitonenergylevelsandwavefunctionsforsolidNeandArhavebeencalculatedfrom

firstprinciplesusingmany-bodytechniques.ElectronicbandstructuresofNeandArwerecalculatedusing

theGWapproximation.ExcitonstateswerecalculatedbydiagonalizinganexcitonHamiltonianderivedfrom

theparticle-holeGreenfunction,whoseequationofmotionistheBethe-Salpeterequation.Singletandtriplet

excitonseriesupton=5forNeandn=3forArwereobtained.Bindingenergiesandlongitudinal-transverse

splittingsofn=1excitonsareinexcellentagreementwithexperiment.Plotsofcorrelatedelectron-holewave

functionsshowthattheelectron-holecomplexisdelocalisedoverroughly7a.u.insolidAr.

PACSnumbers:71.10.Li,71.35.-y,71.35.Aa,78.40.-q

I.INTRODUCTION

Electronicandopticalpropertiesofraregassolids(RGS)

NeandArhavebeenstudiedexperimentally1,2,3,4,5,6,7,8,9,10and

theoretically11,12,13,14,15,16,17,18andhavebeenthesubjectof

severalreviewarticles19,20,21.Opticalabsorptionspectraare

characterizedbysharpexcitonpeaksatenergiesuptosev-

eraleVbelowthefundamentalbandgap.ExcitonsinRGS

consistofaholeinap-typevalencebandandanelectron

inans-typeconductionband.Themomentumofthehole

canbeeitherj=3/2(spintriplet)orj=1/2(spinsin-

glet).Spin-orbitcouplingmixesthesestatesandsopairsof

transverseexcitonlines,labelledbyprincipalexcitonquan-

tumnumbersnandn′,areobservedinopticalabsorptionex-

periments.Whenspin-orbitcouplingisweak,asinNe,ener-

giesoftheseexcitonsoughttobeclosetothespinsingletand

tripletenergies.Creationoflongitudinalexcitonsinelectron

energylossexperimentsproducesalongitudinalpolarisation

fieldwhichcouplestotheassociatedlong-rangemacroscopic

electricfieldandleadstoenergysplittingoflongitudinal(L)

andtransverse(T)excitonsofthesameprincipalquantum

number.

Anintegralequationapproachhasbeenappliedtoexcitons

ininsulatorsforover40years22.Thisincorporatesscreened

electron-holeattractionandexcitonexchangetermsinthe

Hamiltonian.EarlyapplicationsofthisapproachtoRGS12,13

usedanapproximationinwhichtheexcitonwavefunction

wasrestrictedtoasingleunitcell(onesiteapproximation).

Followingthis,Rescaandcoworkersdevelopedanapproach

whichtookintoaccountthedelocalisednatureoftheexci-

tonwavefunctionbysolvinganeffectivemassequation14,15,18.

Inthisapproachtheelectron-holeattractiontermwasun-

screenedwhenbothelectronandholewereonasinglesite

andscreenedbyamacroscopicdielectricconstantfactorwhen

theywereondifferentsites.Modificationoftheelectron-

holeattractiontermasafunctionofelectron-holeseparation

inthiswayleadsnaturallytoaquantumdefectcorrection,

En=Eg−Bex/(n+δn)2,totheWannierformulaforex-

citonenergies,En=Eg−Bex/n2.Egisthefundamen-

talbandgapandBexistheexcitonbindingenergy.Resca

andResta15showedthattheformerexpressioncouldpredict

excitonenergiesratherwellwithaweakdependenceofthequantumdefect,δn,onn.However,subsequentdirectmea-

surementsofthefundamentalbandgapsinRGS23showed

thatthefundamentalgapderivedfromafittotheWannier

formula(excludingthen′=1exciton)yieldedavaluefor

Egingoodagreementwiththeexperimentalvalueswhilethe

quantumdefectmodelyieldedquitedifferentvalues.Bern-

storffetal.24concludedthatRGSexcitons’donotpossess

atomicparentage’.ExcitonwavefunctionsforRGSarethere-

forewellknowntohavestrongbutincompletelocalization

oftheelectronandholeonthesamesite.Thisintermedi-

atebindingcharacterofexcitonsinRGSmeansthattheyare

farfromthewell-separatedelectron-holepairlimit,which

appliestosemiconductorsandiswelldescribedbytheef-

fectivemassapproximationtheory25.Arecentcalculation11,

whichusedascreenedCoulombelectron-holeinteractionand

aSlater-Kosterparametrizationofthebandstructure,found

goodagreementwithexperimentalexcitonenergiesanddelo-

calizationoftheelectron-holewavefunctionoverthreenearest

neighbordistances.

Theintegralequationapproachwasappliedtodiamond26,27

andsilicon28byHankeandShaminthe1970’susingatight-

bindingparameterizationofthebandstructure.Strinati29

showedhowtheequationofmotionfortheparticle-hole

Greenfunction,theBethe-Salpeterequation30,forcore-hole

excitonscouldbereducedtoaneffectiveeigenvalueprob-

lem.Recentabinitiocalculationsofvalenceexcitonsincrys-

tallinesolids31,32,33havebeenbasedonthiseffectiveeigen-

valueproblemandtheBethe-Salpeterformalismhasbeen

reviewed34.Inthispaperwepresentabinitiocalculations

ofexcitonicabsorptionspectraandcorrelatedelectron-hole

wavefunctionsforexcitonsinNeandAr.Quasiparticle

bandenergiesarecalculatedusingtheGWapproximation35

andtheBethe-Salpeterformalism34,whichincludesstatically

screenedelectron-holeattractionandexcitonexchangeterms,

isusedtocalculatetheopticalspectrum.Thisisgeneratedus-

ingmatrixelementsbetweenthegroundstateandcorrelated

electron-holeexcitedstatesandthelongitudinal-transverse

(LT)splittingofexcitonsisinvestigated.

Theremainderofthepaperisarrangedasfollows:InSec-

tionIIthetheoreticalformalismispresented,inSectionIIIre-

sultsofcalculationsofopticalspectraandcorrelatedelectron-

holewavefunctionsinsolidNeandAraregivenandfinally