Quasiparticle Band Structures and the GW Approximation

Transcription

1 Quapacle Ba Sucue a he W Appoxmao Ao Schlmay Iu fü Feöpefochug Fochugzeum Jülch 545 Jülch emay

2 Ba-Sucue Meaueme Agle-Reolve (Ivee) Phooemo Specocopy Specum of a ω pl :plamoeegy Meaue bg eegy (ba ucue): hν ( ) ( ) (): ou ae wh eleco ( ): xce ae wh eleco F. Ayaeawa e al. Phy. Rev. Le (996).

3 Defo of he Ba Sucue The elecoc ba ucue ma he complee e of eleco-emoval a eleco-ao eege acceble fom he gou ae.e. oal-eegy ffeece bewee wo yem wh ffee pacle umbe: () () fo occupe valece ae () () fo uoccupe couco ae I oe o oba elable umecal eul we ee a way o calculae he ba ucue ecly o fom oal-eegy ffeece becaue «(). xample: oeacg eleco [ () ] () () [ ] () ( ) ( ) () ex ( ) ( ) ex 5 4

4 Haee-Foc Theoy () () ( ) ( ) ( ) () () ( ) ( ) ( ) ( ) ( ) F H ex ( ) () ( ) () () () ( ) ( ) ( ) F H Koopma heoem allow u o efy he Haee-Foc egevalue wh he phycal ba ucue (eleco ao a emoval eege) bu he eglec of coelao goly oveemae he fuameal ba gap. ( ) () ( ) () () () ( ) ( ) ( ) ( ) F H Koopma heoem:.7 xp. 6. HF g S S. Maa e al. Phy. Rev. B (99).

5 Dey-Fucoal Theoy ( ) H xc[] () () () () () ( ) () ex () () () δ Ha fucoal: 44 ( ) δ H xc ( ( ) ) ( ()() ) ()() [] xc xc CB CBM I geeal: Jaa heoem: BM ( ) ( ) ( ) µ B I ey-fucoal heoy hee o mple elaohp bewee oal-eegy ffeece a Koh-Sham egevalue. Hece he lae cao be epee a excao eege (excep fo he hghe occupe ae).

6 Calculae Ba ap of Slco Haee-Foc (HF) [] xac xchage (XX) [] W Appoxmao (W) [4] xac xchage plu RPA coelao (XXRPA) [] xac May-Boy Peubao Theoy (exac MBPT) xac Dey-Fucoal Theoy (exac DFT) Local-Dey Appoxmao (LDA) [4] [] S. Maa e al. Phy. Rev. B (99). [] T. Koa J. Phy: Coe. Mae 94 (998). [] R. W. oby e al. Phy. Rev. B 7 59 (988). [4] W. Ku e al. Phy. Rev. Le ().

7 The ee-fuco Appoach oeacg pacle: () ( ) ( ) ( ) () () ex h h δ δ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) > > (hole) fo (eleco) fo occ occ uocc uocc e e The ee fuco a popagao ha ae he fluece of he evome fully o accou. The excao eege coepo o he pole of o aleavely o he pea of he pecal fuco A o he fequecy ax. ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) A e ω δ ω π µ ω η µ ω τ τ ω η ωτ Im g g lm Foue aaly:

8 Dyo quao h () () ( ) ( δ ) δ ( ) wh a aoal poeal () Soluo: ( ) ( ) ( ) ( ) ( ) Feyma agam: ( ) ( ) ( ) ( ) ( )... Rcha P. Feyma obel Pze 965 ( ) ( ) ( ) ( ) Fo paccal applcao Dyo equao ν ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( )] ( ω ) ω ω ω ω ω ω ω ω K ω ν ( ω ) elae he ee fuco o he ow oluo of a mple poblem. I ha a ec epeao em of popagao a caeg eve.

9 The Self-egy: Quapacle ( ) ( ) ( ) ( ) ( ) Σ ω ω ω ω ω... Σ Σ Σ () () [ ] () ( ) ( ) () ψ ψ ψ Σ H ex ( ) () ( ) ( ) η µ ω ψ ψ ω η g lm ( ) ( ) ( ) ( ) µ µ > < fo fo I eacg yem eleco-eleco caeg ca be goouly ecbe by he elf-eegy Σ whch geeal complex olocal a eegy-epee.

10 The W Appoxmao W: HF : Σ ( ) ( ) W ( ) ( ) v( ) δ ( ) W () ( ) F ( ) The W appoxmao fo he elf-eegy a exee Haee-Foc meho wh a ceee Coulomb eaco. Bee exac exchage ecbe yamc ceeg a yem of ea eleco whch he oma coelao effec may ol. Hocally moel ceeg fuco wee fequely employe. Moe calculao ue he aom-phae appoxmao (RPA): W P RPA ( ω ) v( ) v( ) P( ω) W ( ω) ( ) ( ) ( ) RPA

11 Peubave W Teame Quapacle equao: [ () () ] ψ () ( ) ψ ( ex H Σ ) ψ () [ () () () ] ψ () [ Σ( ) () δ ( )] ψ ( ) ψ ( ) ex H xc h KS () xc Σ ( ) Koh-Sham equao: h KS KS KS KS () () () F-oe peubao heoy: KS () Σ( ) ( ) KS KS I pacce he W elf-eegy coeco uually evaluae o-elf-coely wh f-oe peubao heoy. The calculae ae he acually pope ey fucoal!

12 The W Space-Tme Meho Σ occ KS KS KS KS τ τ ( τ ) () KS KS ( ) e Θ() τ () ( ) e Θ( τ ) ( τ ) ( τ ) W ( τ ) uocc FFT P ( τ ) ( τ ) ( τ ) FFT Max veo W ω v ω ( ) ( ) ( ) FFT ( ω ) δ P ( ω) v ( ) KS ( ) KS KS Σ xc Aalyc couao o eal eege M. M. Rege L. Sebec I. D. Whe H.. Roa a R. W. oby Comp. Phy. Commu. 7 (999). L. Sebec A. Rubo L. Reg M. Toe I. D. Whe a R. W. oby Comp. Phy. Commu. 5 5 ().

13 The Ba ap of Semcouco heoecal ba gap S e W LDA O a CS expemeal ba gap C MgO The W appoxmao yel vey goo excao peca fo maeal wh wea o meum coelao clug all ypcal emcouco. Sogly coelae maeal uch a O a Mo-Hubba ulao ae le accuaely ecbe becaue he W elf-eegy me mpoa cobuo.

14 The Ba Wh of Meal Ba wh of alal meal L a K LDA W xp J.. ohup e al. Phy. Rev. B (989). M. P. Suh e al. Phy. Rev. B (988). Homogeeou eleco ga xac DFT: Haee-Foc: F F F W: [ Σ( F ) Σ( )] F > π F < F F The W appoxmao coecly ecbe he coelao-uce ba aowg ue o he heave quapacle effecve ma compae o he bae eleco ma. a Ba wh J.. ohup e al. Phy. Rev. Le (987).

15 Quapacle Lfeme Lfeme of exce eleco he couco ba of lve τ Im Σ( ) h R. Keylg e al. Phy. Rev. B 6 67 (). : xp. (Tme-Reolve Two-Phoo Phooemo Specocopy)

16 Defec o Semcouco Suface A vacacy A o p-aa() a I a II A a quvale: P vacacy P o p-ip(). Legel e al. Phy. Rev. Le (994).

17 Chage Tao Level Defec fomao eegy: lab ( q µ ) ( q) µ q( µ ) fom e q I BM e fom geomey chage ae q / q fom q / q ( q ) ( q ) Sepaao of ffee eegy cobuo: IP() LDA xp. / (e).47.75±. / (e).54 xpeme (STM a Phooeleco Specocopy): Ph. be e al. Phy. Rev. Le (). / / ( ) ( ) ( ) ( ) BM elaxao eegy eleco affy ( ) ( ) ( ) ( ) BM elaxao eegy ozao poeal

18 aa() Suface Ba Sucue Suface ba Defec level (chage ae ) W poece W bul ba LDA X M W LDA Γ X xp:.6±.4 e S. Alo e al. J. Chem. Phy ().

19 Impove Chage Tao Level IP() / (e) / (e) aa() / (e) / (e) LDA W xp ± LDA W / / ( ) ( ) ( ) ( ) BM elaxao eegy eleco affy ( ) ( ) ( ) ( ) BM elaxao eegy ozao poeal

20 Cocluo Ule he Koh-Sham egevalue he ee-fuco appoach popely ecbe eleco emoval a ao eege a meaue phooemo pecocopy. The W appoxmao fo he elf-eegy combe exac exchage wh he oma coelao cobuo (RPA) fo maeal wh elocale eleco. I coecly ecbe he ba gap of emcouco he ba wh of meal a he fe lfeme of quapacle excao. The W appoxmao uually apple peubavely combao wh a aa ey-fucoal calculao. Leaue:. F. Ayaeawa a O. uao Rep. Pog. Phy. 6 7 (998).. W.. Aulbu L. Jöo a J. W. Wl Sol Sae Phyc ee by H. heech a F. Spaepe (Acaemc Sa Dego ) ol. 54 p..