Ab initio theory of superconductivity. I. Density functional formalism and approximate functionals

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1 PHYSICAL REVIEW B 72, Ab nto thory of suprconductvty. I. Dnsty functonal formalsm and approxmat functonals M. Lüdrs, 1,2 M. A. L. Marqus, 2,3 N. N. Lathotaks, 2,3 A. Flors, 3,4 G. Profta, 5 L. Fast, 2,6 A. Contnnza, 5 S. Massdda, 4, * and E. K. U. Gross 2,3 1 Darsbury Laboratory, Warrngton WA4 4AD, Untd Kngdom 2 Insttut für Thortsch Physk, Unvrstät Würzburg, Am Hubland, D Würzburg, Grmany 3 Insttut für Thortsch Physk, Fr Unvrstät Brln, Arnmall 14, D Brln, Grmany 4 INFM SLACS, Sardnan Laboratory for Computatonal Matrals Scnc and Dpartmnto d Scnz Fsch, Unvrstá dgl Stud d Caglar, S. P. Monsrrato-Sstu km 0.700, I Monsrrato (Caglar), Italy 5 CASTI Isttuto Nazonal Fsca dlla Matra (INFM) and Dpartmnto d Fsca, Unvrstá dgl stud dll Aqula, I Coppto (L Aqula) Italy 6 SP Swdsh Natonal Tstng and Rsarch Insttut, P. O. Box 857, S Borås, Swdn Rcvd 1 Jun 2004; rvsd manuscrpt rcvd 25 Fbruary 2005; publshd 29 July 2005 An approach to th dscrpton of suprconductors n thrmal qulbrum s dvlopd wthn a formally xact dnsty functonal framwork. Th thory s formulatd n trms of thr dnsts: th ordnary lctron dnsty, th suprconductng ordr paramtr, and th dagonal of th nuclar N-body dnsty matrx. Th lctron dnsty and th ordr paramtr ar dtrmnd by Kohn-Sham quatons that rsmbl th Bogolubov d Gnns quatons. Th nuclar dnsty matrx follows from a Schrödngr quaton wth an ffctv N-body ntracton. Ths quatons ar coupld to ach othr va xchang-corrlaton potntals whch ar unvrsal functonals of th thr dnsts. Approxmatons of ths xchang-corrlaton functonals ar drvd usng th dagrammatc tchnqus of many-body prturbaton thory. Th bar Coulomb rpulson btwn th lctrons and th lctron-phonon ntracton ntr ths prturbatv tratmnt on th sam footng. In ths way, a truly ab nto dscrpton s achvd whch dos not contan any mprcal paramtrs. DOI: /PhysRvB PACS numbrs: Jb, Kc, z, Ad I. INTRODUCTION On of th grat challngs of modrn condnsd-mattr thory s th prdcton of matral spcfc proprts of suprconductors, such as th crtcal tmpratur T c or th gap at zro tmpratur 0. Th modl of Bardn, Coopr, and Schrffr 1 BCS succssfully dscrbs th unvrsal faturs of suprconductors,.., thos faturs that all convntonal, wak-couplng suprconductors hav n common, lk th unvrsal valu of th rato 2 0 /k B T c. Th grat achvmnt of BCS thory was th mcroscopc dntfcaton of th suprconductng ordr paramtr whch ld, aftr mor than 50 yars of strugglng, to a mcroscopc undrstandng of th phnomnon of suprconductvty. BCS thory, howvr, cannot b consdrd a prdctv thory n th sns that t would allow th computaton of matral-spcfc proprts. Morovr, matrals wth strong lctron-phonon couplng, such as nobum or lad, ar poorly dscrbd by BCS thory. In ths strong-couplng matrals, phonon rtardaton ffcts play a vry mportant rol. A propr tratmnt of thos ffcts was dvlopd by Elashbrg. 2,3 Hs thory can b vwd as a GW approxmaton 4 n trms of th Nambu-Gorkov 5 Grn s functons. Elashbrg s thory not only achvs a succssful dscrpton of th strong-couplng smpl mtals lk Nb and Pb, t also provds a convncng xplanaton of th suprconductng faturs of mor complx matrals such as MgB 2. 6 In spt of ts trmndous succss, Elashbrg thory, n ts practcal mplmntaton, has to b consdrd a smphnomnologcal thory. Whl th lctron-phonon ntracton s prfctly accountd for, corrlaton ffcts du to th lctron-lctron Coulomb rpulson ar dffcult to handl n ths thory. Thos ffcts ar condnsd n a sngl paramtr *, whch rprsnts a masur of th ffctv lctronc rpulson. Although * could, n prncpl, b calculatd by dagrammatc tchnqus, 3 frst-prncpls stmats of * ar xtrmly hard to mak, and n practc, * s tratd as an adjustabl paramtr, usually chosn such that th xprmntal T c s rproducd. Th goal of ths work s to dvlop a tru ab nto thory for suprconductvty whch dos not contan any adjustabl paramtrs. Th crucal pont s to trat th lctron-phonon ntracton and th Coulombc lctron-lctron rpulson on th sam footng. Ths s achvd wthn a dnsty functonal framwork. Dnsty functonal thory 7 9 DFT njoys normous popularty as an lctronc-structur mthod n sold-stat physcs, quantum chmstry, and matrals scnc. DFT combns good accuracy wth modrat numrcal ffort and s oftn th mthod of choc spcally for larg molculs and solds wth a bg unt cll. DFT s basd on th Hohnbrg-Kohn 7 thorm whch nsurs a rgorous 1:1 corrspondnc btwn th ground-stat dnsty and th xtrnal potntal. At fnt tmpratur, th corrspondnc holds 10 btwn th dnsty n thrmal qulbrum and th xtrnal potntal. As a consqunc, all physcal obsrvabls of an ntractng lctron systm bcom functonals of th dnsty. Th practcal mplmntaton of DFT rsts on th /2005/722/ /$ Th Amrcan Physcal Socty

2 LÜDERS t al. Kohn-Sham 8 schm whch maps th ntractng systm of ntrst on an auxlary nonntractng systm wth th sam ground-stat thrmal dnsty. Tradtonal DFT, by ts vry natur, nvtably nvolvs th Born-Oppnhmr approxmaton: On s supposd to calculat th lctronc ground-stat thrmal dnsty that corrsponds 1:1 to th lctrostatc potntal of clampd nucl. To ovrcom ths lmtaton Krbch and Gross 11 KG rcntly prsntd a multcomponnt DFT whch trats both lctrons and nucl quantum mchancally on th sam footng. Th KG thory nvolvs two dnsts: th lctronc dnsty nr rfrrng to a body-fxd coordnat fram, and th dagonal R of th nuclar N-body dnsty matrx. Th xchang-corrlaton functonal apparng n th rsultng Kohn-Sham quatons dpnds on both dnsts and contans, formally, all nonadabatc couplngs btwn th lctrons and th nucl. Hnc, n prncpl, th KG framwork should b abl to dscrb convntonal suprconductvty. In practc, howvr, t s not advsabl to attmpt such a dscrpton. Th stuaton s qut smlar to th DFT tratmnt of magntc ffcts: By vrtu of th Hohnbrg-Kohn thorm, magntc ffcts can b dscrbd on th bass of th dnsty alon. In partcular, th ordr paramtr of spn magntsm, th spn magntzaton mr, s a functonal of th dnsty m=mn. Howvr, ths functonal has to b hghly non-local and ts xplct form s unknown. Thrfor, t s advsabl to nclud th ordr paramtr mr as an addtonal dnsty n th DFT formulaton. 12 Ths vrson of DFT s known as spn DFT. It s th standard form of DFT that s mployd n all practcal applcatons. A smlar da was suggstd n 1988 by Olvra, Gross, and Kohn 13 OGK to trat suprconductors n a DFT framwork. OGK proposd th ncluson of th ordr paramtr r,r that charactrzs th suprconductng phas as a basc dnsty n th DFT formulaton. OGK dalt wth snglt ordr paramtrs. A gnralzaton to trplt ordr paramtrs was gvn latr. 14 Th complxts of th many-body problm wr cast nto an xchang and corrlaton trm, but n contrast to ordnary dnsty functonal thory whr a varty of functonals has appard ovr th past 30 yars, vry fw xchang-corrlaton functonals hav bn proposd for th suprconductng stat. To our knowldg, only a local dnsty approxmaton dscrbng th purly lctronc ntractons has bn prsntd. 15 Howvr, th usfulnss of th OGK approach was dmonstratd by Györffy and coworkrs n thr study of nobum and YBa 2 Cu 3 O 7 usng a smphnomnologcal paramtrzaton of th xchangcorrlaton functonal. 17,18 Th OGK formulaton was trggrd by th dscovry of hgh-t c suprconductors 19 whr an lctronc parng mchansm was blvd to b domnant. Hnc, th OGK dscrpton trats th Coulomb rpulson btwn lctrons formally xactly whl th lctron-phonon couplng only ntrs through a gvn, nonrtardd BCS-typ lctronlctron ntracton,.., strong lctron-phonon couplng cannot b dalt wth. In ths work w dvlop a DFT for suprconductors basd on th thr dnsts nr,r,r and R. Th formalsm can thus b vwd as a suprconductng gnralzaton of KG thory or as a strong-couplng gnralzaton of OGK formalsm. It lads to a st of formally xact Kohn-Sham quatons for lctrons and nucl. Ths quatons contan xchang-corrlaton potntals whch ar unvrsal functonals of th thr dnsts n,, and. For th tm bng, w do not study ffcts found n th prsnc of magntc flds. Thos can b accommodatd by gnralzng th framwork to nclud th currnt dnsty as an addtonal varabl. 20,21 Th succss of any dnsty functonal thory crucally dpnds on th avalablty of accurat approxmatons for th xchang-corrlaton functonals. Th man body of ths papr s dvotd to th constructon of such approxmat xchang-corrlaton functonals. Dagrammatc many-body prturbaton thory s usd for ths purpos. In a scond papr hncforth rfrrd to as II, ths approxmat functonals ar mployd to calculat suprconductng proprts of lmntal mtals. Th prsnt papr s organzd as follows. In Sc. II w drv a multcomponnt DFT for th suprconductng stat. Ths thory lads to a st of Kohn-Sham quatons that ar dscrbd n th followng scton. Scton IV s dvotd to th dvlopmnt of Kohn-Sham prturbaton thory. Th rsultng xchang-corrlaton potntals ar dscussd n Sc. V. A smpl BCS-lk modl s dscrbd n Sc. VI. Ths modl s usd, n Sc. VII, to analyz th approxmat xchang-corrlaton krnls ntrng th lnarzd DFT gap quaton. Fnally, n Sc. VIII, th xchang-corrlaton contrbutons to th nonlnar gap quaton ar dscussd. II. MULTICOMPONENT DFT FOR SUPERCONDUCTORS It s clar that a balancd tratmnt of th lctron-phonon and Coulomb ntractons has to start from th many-body lctron-nuclar Hamltonan atomc unts ar usd throughout ths papr Ĥ = Tˆ + Û + Tˆ n + Û nn + Û n, whr Tˆ rprsnts th lctronc kntc nrgy, Û th lctron-lctron ntracton, Tˆ n th nuclar kntc nrgy, and Û nn th Coulomb rpulson btwn th nucl. Th ntracton btwn th lctrons and th nucl s dscrbd by th trm Û n = PHYSICAL REVIEW B 72, d 3 r d 3 R ˆ rˆ R Z ˆ Rˆ r, r R whr ˆ r and ˆ R ar, rspctvly, lctron and nuclar craton oprators. For smplcty w assum th nucl to b dntcal, and w nglct th nuclar spn dgrs of frdom. Th xtnson of ths framwork to a mor gnral cas s straghtforward. Not that thr s no xtrnal potntal n th Hamltonan. To dvlop a multcomponnt DFT for th lctronnuclar systm w hav to procd wth car. Th Hamltonan 1 dscrbs a translatonally nvarant and sotropc systm. Thus, both th lctronc and nuclar on-partcl

3 Ab nto THEORY OF SUPERCONDUCTIVITY. I. dnsts ar constant and thrfor not usful to charactrz th systm. Ths problm can b solvd by adoptng a bodyfxd rfrnc fram. 11,22 In ths papr w ar ntrstd n nfnt solds whr th nucl prform small oscllatons around th qulbrum postons. Furthrmor, w assum that th sold s not rotatng as a whol. Fortunatly, n ths cas, th body-fxd rfrnc fram concds wth th normal Cartsan systm commonly usd to dscrb solds. Th stuaton s vry dffrnt for fnt systms, whch hav to b handld by usng approprat ntrnal coordnats. In ordr to formulat a Hohnbrg-Kohn typ statmnt, th Hamltonan 1 s gnralzd to Ĥ = Tˆ + Tˆ n + Û n + Û + Vˆ xt + Vˆ xt n + ˆ xt Nˆ. 3 Th xtrnal potntal for th lctrons s dfnd as Vˆ xt = d 3 r ˆ rv xt rˆ r. 4 Snc, at ths lvl, th nucl ar takn nto account xplctly, th lattc potntal s not tratd as an xtrnal fld, but s ncludd va th ntracton trm Û n. Th trm Vˆ xt s ntroducd as a mathmatcal trck to prov th Hohnbrg- Kohn thorm, and wll b takn to zro at th nd of th n drvaton. Vˆ xt s a multplcatv N-body oprator wth rspct to th nuclar coordnats n Vˆ xt = d 3 R v n xt R ˆ R, 5 whr w hav dfnd R =R 1,R 2,,R N,d 3 R =d 3 R 1 d 3 R 2 d 3 R N, and ˆ R = ˆ R 1 ˆ R N ˆ R N ˆ R 1 s th dagonal part of th nuclar N-partcl dnsty matrx n oprator. Not that th trm Vˆ xt ncluds th ntracton btwn th nucl Û nn to whch t rducs f no othr xtrnal nuclar potntals ar prsnt. Th trm 6 PHYSICAL REVIEW B 72, nr = ˆ rˆ r 8 s dfnd n th usual way. Th brackt dnots th thrmal avrag Â=Trˆ 0Â, wth th grand canoncal statstcal dnsty oprator ˆ 0= Ĥ /Tr Ĥ n th suprconductng stat. W furthrmor dfn th nvrs tmpratur =1/T. Th anomalous dnsty r,r = ˆ rˆ r s th ordr paramtr charactrzng th snglt suprconductng stat. Ths quantty s fnt for suprconductors blow th transton tmpratur and zro abov ths tmpratur. To dscrb th nuclar dgrs of frdom, w us th dagonal part of th nuclar N-partcl dnsty matrx R =ˆ R. Altrnatvly, on could dfn a multcomponnt DFT usng th on-partcl dnsty for th nucl n n R=ˆ Rˆ R. Howvr, n th followng t wll b convnnt to transform th nuclar dgrs of frdom to collctv phonon coordnats. Usng n n R would lad to a on-body quaton for nonntractng nucl. Thus, th nuclar Kohn-Sham quaton would not lad to ralstc phonons wth a propr dsprson rlaton. Only Enstn phonons could b prsnt n ths systm. Ths s also clar from th fact that a systm of nonntractng partcls dos not xhbt collctv mods. Wth our choc of R, th nucl oby an N-partcl quaton, vry smlar to th Born- Oppnhmr quaton, and whr phonon coordnats can b asly ntroducd. As usual, th Hohnbrg-Kohn thorm guarants a onto-on mappng btwn th st of th dnsts nr,r,r,r n thrmal qulbrum and th st of thr conjugat potntals v xt r, xt r,r,v n xt R. Asa consqunc all obsrvabls ar functonals of th st of dnsts. Fnally t assurs that th grand-canoncal potntal n,, = Fn,, + d 3 rnrv xt r 9 ˆ xt = d r 3 d 3 r * xt r,rˆ rˆ r + H.c. 7 dscrbs an xtrnal parng fld, and usually vanshs unlss our systm s n th proxmty of an adjacnt suprconductor. Howvr, ths trm s rqurd to brak th gaug nvaranc of th Hamltonan, and th lmt xt 0 can only b takn at th nd of th drvaton. Not that th trm 7 dscrbs a snglt parng fld. Th xtnson to trplt suprconductors s straghtforward. 14 Fnally, stands for th chmcal potntal, and Nˆ s th numbr oprator for th lctrons w trat th lctronc dgrs of frdom n a grand-canoncal nsmbl. Our multcomponnt formulaton s basd on thr dnsts. Th lctronc dnsty d r 3 d 3 rr,r * xt r,r + H.c. + d 3 R R v n xt R 10 s mnmzd by th qulbrum dnsts. W us th notaton Af to dnot that A s a functonal of f. Th functonal Fn,, s unvrsal, n th sns that t dos not dpnd on th xtrnal potntals, and s dfnd by Fn,, = T n,, + T n n,, + U n n,, + U n,, 1 Sn,,, 11 whr S stands for th ntropy of th systm

4 LÜDERS t al. Sn,, = Trˆ 0n,,lnˆ 0n,,. 12 Th proof of th thorm follows closly th proof of th Hohnbrg-Kohn thorm at fnt tmpraturs 10 and wll not b prsntd hr. In standard DFT on normally dfns a Kohn Sham systm,.., a nonntractng systm chosn such that t has th sam ground-stat dnsty as th ntractng on. In our formulaton, th Kohn-Sham systm conssts of nonntractng suprconductng lctrons, and ntractng nucl. It s dscrbd by th thrmodynamc potntal cf. Eq. 10 s n,, = F s n,, + d 3 rnrv s r s d r 3 d 3 rr,r * s r,r + H.c. + d 3 R R v n s R, 13 whr F s s th countrpart of Eq. 11 for th Kohn-Sham systm,.., F s n,, = T s n,, + T s n n,, 1 S sn,,. 14 Hr T s n,,,t n s n,,, and S s n,, ar th lctronc and nuclar kntc nrgs and th ntropy of th Kohn-Sham systm, rspctvly. From Eq. 13 t s clar that th Kohn-Sham nucl ntract wth ach othr through th N-body potntal v n s R whl thy do not ntract wth th lctrons. By applyng th Hohnbrg-Kohn thorm to both th ntractng and th nonntractng systms, and rqurng th dnsts of th Kohn-Sham systm to rproduc th dnsts of th fully ntractng on, w can dntfy th xprssons for th ffctv Kohn-Sham potntals. As usual, ths nclud contrbutons from xtrnal flds, Hartr, and xchang-corrlaton trms. Th lattr account for all th many-body ffcts stmmng from th lctron-lctron and lctron-nuclar ntractons. To smplfy th xprssons, w now st th auxlary xtrnal potntals to zro. Th Kohn-Sham potntal for th lctrons v s r rads as d 3 R R v s n,,r = Z r R + d 3 r nr r r + v xc n,,r. 15 Th frst trm, th lctron-nuclar Hartr potntal, rducs to th usual nuclar attracton potntal f w assum that th nucl ar classcal and prfctly localzd at thr qulbrum postons. Ths trm s usually tratd as th xtrnal potntal n standard DFT. Th last two contrbutons to v s r ar, rspctvly, th Hartr potntal, whch accounts for th classcal rpulson btwn th lctrons, and th xchang-corrlaton trm. Th anomalous Kohn-Sham potntal s s gvn by s n,,r,r = r,r r r + xcn,,r,r. 16 Not that th frst trm, th so-calld anomalous Hartr potntal, gvs rs to a postv contrbuton to th nrgy. Fnally, th nuclar potntal s Z 2 v n s n,,r = R R Z + v n xc n,,r. d 3 nr r r R 17 Th frst trm stms from Û nn, and dscrbs th bar nuclar-nuclar rpulson. Th scond s th nuclar-lctron Hartr trm and s th countrpart of th frst trm n Eq. 15. As n standard DFT, th xchang-corrlaton potntals ar dfnd as functonal drvatvs, n,,r = F xcn,,, 18a nr v xc xc n,,r,r = F xcn,, *, 18b r,r v n xc n,,r = F xcn,,. 18c R Th xchang-corrlaton fr nrgy s dfnd through th quaton Fn,, = F s n,, + F xc n,, + U nn + E H n, + E n H n,. 19 Thr ar two contrbutons to E H, on stmmng from th lctronc Hartr potntal, and th othr from th anomalous Hartr potntal, E H n, = 2 1 d r 3 d 3 r nrnr r r + d r 3 d 3 r r,r2. 20 r r Fnally, E H n dnots th lctron-nuclar Hartr nrgy E H n n, = Z PHYSICAL REVIEW B 72, d 3 r d 3 R III. THE KOHN-SHAM EQUATIONS nrr. 21 r R Th problm of mnmzng th Kohn-Sham grand canoncal potntal 13 can b transformd nto a st of thr dffrntal quatons that hav to b solvd slf-consstntly. Two of thm ar coupld and dscrb th lctronc dgrs of frdom. Thr algbrac structur s smlar to th Bogolubov d Gnns quatons. Th thrd s an quaton for th nucl rsmblng th famlar nuclar Born- Oppnhmr quaton

5 Ab nto THEORY OF SUPERCONDUCTIVITY. I. PHYSICAL REVIEW B 72, A. Elctronc quatons Th Kohn-Sham Bogolubov d Gnns KS-BdG quatons 16 rad v s r u r + d 3 r s r,rv r = Ẽ u r, 22a v s r v r + d 3 r s * r,ru r = Ẽ v r, 22b whr u r and v r ar th partcl and hol ampltuds. Ths quaton s vry smlar to th Kohn-Sham quatons n th OGK formalsm. 13 Howvr, n our formulaton th lattc potntal s not consdrd as an xtrnal potntal but ntrs va th lctron-on Hartr trm. Furthrmor, our xchang-corrlaton potntals dpnd paramtrcally on th nuclar dnsty matrx, and thrfor on th phonons. Although ths quatons hav th structur of statc quatons, thy contan, n prncpl, all rtardaton ffcts through th xchang-corrlaton potntals. A drct soluton of th Kohn-Sham Bogolubov d Gnns quatons 17 s facd wth th problm that on nds xtrmly hgh accuracy to rsolv th suprconductng nrgy gap, whch s about thr ordrs of magntud smallr than typcal lctronc nrgs. At th sam tm, on has to covr th whol nrgy rang of th lctronc band structur. Th so-calld dcouplng approxmaton rlvs th problm by sparatng ths two nrgy scals. Th partcl and hol ampltuds can b xpandd n th complt st of wav functons of th normal-stat Kohn-Sham quaton v s n,,r r = r 23 whch can b solvd by standard band-structur mthods. Th dcouplng approxmaton thn mpls th followng form for th partcl and hol ampltuds, whr only on trm of th xpanson s rtand: u ru r, v rv r. 24 In ths way th gnvalus n Eq. 22 bcom Ẽ =±E, whr E = 2 + 2, 25 and =. Ths form of th gnnrgs allows th ntrprtaton of th matrx lmnts of th par potntal as th gap functon of th suprconductor. Th coffcnts u and v also hav smpl xprssons wthn ths approxmaton, u = 1 2 sgnẽ 1+ Ẽ, 26a v = Ẽ Fnally, th matrx lmnts ar dfnd as = d 3 r d 3 r * r s r,r r, 26b 27 and s th phas = /. Wthn th dcouplng approxmaton, th normal and th anomalous dnsts can b asly obtand from th partcl and hol ampltuds, nr = 1 E tanh 2 E r 2, r,r = 1 2 E tanh 2 E r * r. 28a 28b Th valdty and lmtatons of th dcouplng approxmaton wll b dscussd n dtal n II. B. Nuclar quaton Th Kohn-Sham quaton for th nucl has th form 2 2M + v s n R l R = E l l R. 29 Ths quaton has th sam structur as th usual nuclar Born-Oppnhmr quaton. W mphasz that th Kohn- Sham quaton 29 dos not rly on any approxmaton and s, n prncpl, xact. As alrady mntond, w ar ntrstd n solds at rlatvly low tmpratur, whr th nucl prform small-ampltud oscllatons around thr qulbrum postons. In ths cas, w can xpand th v n s n,, n a Taylor srs around th qulbrum postons, and transform th nuclar dgrs of frdom nto collctv phonon coordnats. In harmonc ordr, th nuclar Kohn-Sham Hamltonan thn rads Ĥ ph s =,qbˆ,qbˆ,q ,q 2, whr,q ar th phonon gnfrquncs, and bˆ,q dstroys a phonon from th branch and wav vctor q. Not that th phonon gnfrquncs ar functonals of th st of dnsts n,,, and can thrfor b affctd by th suprconductng ordr paramtr. Wthn th harmonc approxmaton, th nuclar dnsty matrx R s gvn by R = n,q h,q Q 2, 31,q whr n dnot th Bos occupaton numbrs and h,q Q ar harmonc oscllator wav functons rfrrng to th collctv coordnats Q. C. Gap quaton In Fg. 1 w sktch th Kohn-Sham slf-consstnt procdur wthn th dcouplng approxmaton. W start by fnd-

6 LÜDERS t al. PHYSICAL REVIEW B 72, s rathr dmandng. Furthrmor, w ar rqurd to provd good approxmatons for th thr functonals v s n,,, s n,,, and v s n n,,. As our objctv s to study suprconductvty, w wll mak two smplfyng assumptons. v s n,, can b wll approxmatd by th ground-stat functonal usd n standard dnsty functonal thory v s GS n. As th nrgy scal of th phonons and of th suprconductng gap s thr ordrs of magntud smallr than lctronc nrgy scals lk th Frm nrgy ths s a vry rasonabl assumpton. Th nuclar functonal v s n n,, s approxmatd by th ground-stat Born- Oppnhmr nrgy surfac. It s wll known that calculatons prformd wthn th Born-Oppnhmr approxmaton yld phonon frquncs that ar n xcllnt agrmnt wth xprmntal dsprsons. 26 W thrfor xpct ths to b an xcllnt approxmaton to th Kohn-Sham phonons. Howvr, w ar nglctng th ffct of th suprconductng par potntal on th phonon dsprson. Ths ffct has bn masurd xprmntally, 27 and t turns out to b qut small. Not that ths two approxmatons ar quvalnt to fxng v s n,, and v s n n,, to v s,0 n,, and v s n,0 n,,. By nsrtng Eqs. 28 n Eq. 16 and by subsquntly nsrtng Eq. 16 on th rght-hand sd of Eq. 27, w obtan th DFT gap quaton FIG. 1. Schmatc flow chart for th tratv Kohn-Sham schm wthn th dcouplng approxmaton. ng sutabl approxmatons for th Kohn-Sham potntals to start th cycl: for v s,0 n,, w can us th Kohn-Sham potntal stmmng from a standard DFT calculaton for th nonsuprconductng ground stat,.., v s GS n. Ths s a vry good approxmaton as th dpndnc of v s n,, on and s crtanly vry wak for th usual suprconductors at low tmpratur; th startng par potntal s 0 n,, can b approxmatd by a squar wll cntrd at th Frm nrgy wth wdth of th ordr of th Dby frquncy and hght computd from a BCS modl; fnally, for v s n,0 n,, w can us th ground-stat Born-Oppnhmr potntal. Th nxt two stps of th slf-consstnt cycl can b prformd n paralll. 1 Equaton 23 s solvd to obtan th wav functons s and th gnnrgs s. Ths can thn b usd wthn th dcouplng approxmaton, Eq. 24, to calculat th normal and anomalous dnsts through Eqs. 28. W not that th chmcal potntal ntrng Eq. 23 has to b adjustd at vry traton, such that th dnsty nr ntgrats to th corrct partcl numbr N. 2 Wth v s n n,, w solv th nuclar quaton 29 by xpandng th nuclar potntal to harmonc ordr to obtan th phonon gnfrquncs and gnmods. Th nuclar dnsty matrx thn follows from Eq. 31. Fnally, th st of dnsts n,, s usd to valuat th nw Kohn-Sham potntals v s, s,v s n from th dfntons 15,16,17. At ths pont, f slf-consstncy s rachd, th cycl s stoppd. Othrws, th nw potntals ar usd to rstart th cycl. It s clar that, vn wthn th dcouplng approxmaton, th task of solvng th slf-consstnt cycl dpctd n Fg. 1 = Hxc,, 32 whr Hxc stands for th sum of th Hartr and xchangcorrlaton contrbutons to th functonal. Not that through th xchang-corrlaton functonal th rght-hand sd of Eq. 32 bcoms a hghly complcatd functonal of and. Th dpndnc on th gap functon s totally dffrnt from th usual man-fld gap quaton cf. Sc. VIII. Aftr ths smplfyng approxmatons, a Kohn-Sham calculaton procds as follows. Usng a standard bandstructur cod, th ground-stat wav functons and band structur ar obtand. Th Born-Oppnhmr phonon frquncs and gnmods ar obtand from lnar-rspons 26 calculatons, agan usng standard tools wdly avalabl to th communty. Th gap quaton 32 s tratd untl slf-consstncy s achvd. W can now s how th dffrnt nrgy scals ar sparatd: Th normal dnsty, th anomalous dnsty, and th phonon proprts ar obtand from thr sparat quatons. In th vcnty of th transton tmpratur, th anomalous dnsty wll b vanshngly small. In ths rgm, th gap quaton can b lnarzd n, ladng to = 1 2 j F Hxc,j tanh/2 j j j, 33 whr th anomalous Hartr xchang-corrlaton krnl of th homognous ntgral quaton rads F Hxc,j = Hxc j =0 = 2 E H + F xc * j. =0 34 W mphasz that th lnarzd gap quaton can only b usd to calculat th suprconductng transton tmpratur

7 Ab nto THEORY OF SUPERCONDUCTIVITY. I. In partcular, th functon that stms from th soluton of ths quaton dos not hav any physcal ntrprtaton. W can gan furthr nsght nto Eq. 33 f w sparat th krnl F Hxc,j nto a purly dagonal trm Z and a nondagonal part K,j = Z 1 2 j K,j tanh/2 j j j. 35 Ths quaton has th sam structur as th BCS gap quaton wth th krnl K,j rplacng th modl ntracton of BCS thory. On th othr hand, Z plays a smlar rol as th rnormalzaton trm n th Elashbrg quatons. Ths smlarty allows on to ntrprt th krnl K,j as an ffctv ntracton rsponsbl for th bndng of th Coopr pars. Th functon K,j s a quantty of cntral mportanc n th dnsty functonal thory for suprconductors. By studyng K,j for a spcfc matral as a functon of and j on can larn whch orbtals ar rsponsbl for suprconductvty and, ultmatly, by dntfyng thos parts of th xchangcorrlaton functonal phononc and/or Coulombc that lad to an ffctv attracton, on can trac th mchansm rsponsbl for th suprconductng stat. Not that Eq. 35 s consdrably smplr than th Elashbrg quatons as t dos not dpndnt xplctly on th frquncy. Howvr, phonon rtardaton ffcts ar ncludd through th xchang-corrlaton trms. Furthrmor, Eq. 35 s not a man-fld quaton as n BCS thory but contans corrlaton ffcts. A lnarzd gap quaton can also b drvd wthout th dcouplng approxmaton, 25,28 ladng to a smlar quaton, but wth a four-pont krnl. From ths pont of vw, th dcouplng approxmaton can b vwd as a dagonal approxmaton to ths four-pont krnl, nglctng th corrspondng off-dagonal lmnts. IV. KOHN-SHAM PERTURBATION THEORY In th prvous sctons w showd how to dvlop a dnsty functonal thory for th suprconductng stat. Th man quaton of ths thory, th gap quaton 32, allows, n prncpl, th calculaton of th suprconductng gap for any systm. Howvr, to solv ths quaton on nds approxmatons for xc, th xchang-corrlaton contrbuton to th Kohn-Sham par potntal. In th followng, w wll dvlop such approxmatons by applyng Kohn-Sham prturbaton thory, as dscrbd by Görlng and Lvy, 29 to suprconductng systms. 25,30 Ths prturbaton thory, that wll trat both th lctron-lctron and lctron-phonon ntractons on th sam footng, s a gnralzaton of th mthod usd by Kurth t al. to calculat th xchangcorrlaton nrgy of th unform suprconductng lctron gas. 15 Our startng pont s th Hamltonan of th lctronnuclar systm as dfnd n Eq. 3. Ths Hamltonan s thn splt nto a sutably chosn rfrnc Hamltonan Ĥ 0 and th rmandr, whch s tratd as a prturbaton. Th most approprat rfrnc systm for ths formalsm s as follows. Th nuclar Kohn-Sham Hamltonan 29 rgorously dfns th nuclar qulbrum postons R 0. Whn xpandd to harmonc ordr around ths postons t can b wrttn as th phonon Hamltonan Ĥ ph 30. Nxt w dfn a Born-Oppnhmr Kohn-Sham Hamltonan va a rgd-lattc potntal gvn by th qulbrum coordnats R 0, whr Vˆ Hxc = Ĥ BO = Tˆ + Vˆ latt,r 0 + Vˆ Hxc + ˆ Hxc, 36 d 3 r ˆ rˆ r d 3 r nr r r + v xc r, 37 whl ˆ Hxc ncluds th anomalous Hartr and xchangcorrlaton contrbutons ˆ Hxc = d r 3 d 3 rˆ rˆ r * r,r + * r r xc r,r + H.c.. 38 Wth th choc Ĥ 0 =Ĥ ph +Ĥ BO, th ntracton Hamltonan rads Ĥ 1 = Û + Û ph n BO Vˆ Hxc Vˆ Hxc ˆ Hxc. 39 Th Born-Oppnhmr lctron-phonon couplng oprator -ph s gvn by Û BO PHYSICAL REVIEW B 72, Û -ph BO = d 3 rv BO,q rˆ rˆ rˆ,q,,q 40 whr V BO,q r s bascally th gradnt of th lctronc Kohn-Sham potntal wth rspct to th nuclar coordnats and th phononc fld oprator s ˆ,q=bˆ,q+bˆ, q. Not that now w hav st th auxlary xtrnal potntals Vˆ xt and n ˆ xt to zro, and Vˆ xt to th bar ntrnuclar ntracton. W blv that ths s th most physcal way to splt th Hamltonan, snc th lctronc-structur calculaton for nr, n practc, s usually prformd for fxd nuclar postons; th nuclar dynamcs s absorbd n th xchang corrlaton functonals. Furthrmor, standard calculatons for th lctron-phonon couplng, whch ar basd on lnar rspons thory, nvolv th abov couplng potntals V BO,q r. Not that, bsds th ntracton trms Û and Û -ph BO, th prturbaton ncluds th Hartr and xchang-corrlaton potntals. In th Appndx w gv som mor dtals of ths constructon. W now dvlop a many-body prturbatonal approach, whch wll ultmatly lad to xplct xprssons for th xchang-corrlaton functonals. Th constructon of ths approach follows closly th standard many-body prturbaton thory dscrbd n many txtbooks. 31 Our objctv s to xpand th dffrnc = s n a powr srs. From ths dffrnc and wth th dfntons 10 and 13 t s straghtforward to drv an xprsson for th functonal F xc

8 LÜDERS t al. In standard prturbaton thory s wrttn as an xpanson n powrs of 2 and g 2, whr th lctron charg and g th lctron-phonon couplng constant masur, rspctvly, th strngth of th Coulomb and of th lctronphonon ntractons. In our approach, howvr, vry ordr n prturbaton thory contans all ordrs n 2 and g 2. Ths s du to th spcal form of th prturbaton Ĥ 1 that nvolvs th xchang-corrlaton potntals whch contan mplctly all ordrs n th ntractons. Thrfor, for book-kpng purposs, w multply th prturbaton Ĥ 1 by a dmnsonlss paramtr. In ths way, th trms apparng n th prturbatv xpanson can b labld by powrs of. Th grand-canoncal potntal can b wrttn as = 1 lnz, 41 PHYSICAL REVIEW B 72, F s r,r = Tˆ ˆ rˆ r s. 45b In Fynman dagrams ths propagators appar as lns wth two arrows pontng outward for F s and as lns wth two arrows pontng nward for F s. Th Grn s functon 44 and th anomalous avrags 45 can b convnntly assmbld nto a matrx Grn s functon n Nambu-Gorkov spac. 5 Fnally, as th prturbaton ncluds th lctronphonon ntracton trm Ĥ -ph, som dagrams contan th phonon propagator rprsntd as a curly ln s D,q, = Tˆ ˆ,qˆ,q s. 46 As usual n fnt-tmpratur many-body thory, t s convnnt to work n magnary frquncy spac. Th tm Fourr transform of th Grn s functon 45 s dfnd as whr th partton functon has th dfnton Z =Trxp Ĥ. From ths xprsson t follows drctly that s G r,r = 1 n G r,r; n, n s 47 s = 1 ln Z Z s, 42 whr Z s s th partton functon of th Kohn-Sham systm. It s thn straghtforward, usng th standard machnry of many-body prturbaton thory, to wrt th rato Z/Z s as a srs xpanson n whch can b valuatd wth th hlp of Wck s thorm. Morovr, th numbr of trms n th srs can b rducd by usng a gnralzaton of th lnkdclustr thorm. 32 Th fnal rsult rads whr n =2n+1/ ar th odd Matsubara frquncs. Th frquncy-dpndnt anomalous propagators hav a smlar dfnton. In Matsubara spac th Kohn-Sham propagators rad, n trms of th Kohn-Sham partcl and hol ampltuds and of th Kohn-Sham gnnrgs, s G, r,r; n =, u ru * r n E + v rv * r, n + E 48a s = 1 all connctd dagrams. 43 s F, r,r; n =, sgn In ths xprsson th sum runs ovr all connctd Fynman dagrams that ar topologcally dstnct. Altrnatvly, on can xpand dagrammatcally th propagators and us th Galtsk-Mgdal formula to valuat th nrgy. 33 Th two approachs ar quvalnt. Thr ar svral Kohn-Sham propagators that ntr th Fynman dagrams. Frst w hav th contracton that rducs to th usual Grn s functon for systms that ar not suprconductng s G r,r = Tˆ ˆ rˆ r s, 44 whr Tˆ s th tm-ordrng oprator, whch ordrs th oprators from rght to lft n ascndng magnary tm ordr, and th avrag s s don wth rspct to th Kohn-Sham statstcal dnsty oprator ˆ s. Ths Grn s functon s rprsntd n th Fynman dagrams by a ln wth two arrows pontng n th sam drcton. Furthrmor, du to th prsnc of parng flds n th Kohn-Sham systm 13, th followng anomalous avrags ar nonvanshng for suprconductng systms: s F r,r = Tˆ ˆ rˆ r s, 45a v * ru r n + E F s, r,r; n =, sgn u * rv r n + E u rv * r, n E 48b v ru * r. n E 48c On th othr hand, th phonon propagator dpnds on th vn Matsubara frquncs n =2n/, s D,q n = 2,q n +,q In frst ordr n thr s only on dagram contrbutng to F xc. Ths dagram, dpctd n Fg. 2a, s of purly lctronc orgn and has th sam form as th standard xchang dagram. Th wavy ln n Fg. 2a rprsnts th Coulomb ntracton. Ths trm can b wrttn as for smplcty w wrt all contrbutons to F xc wthn th dcouplng approxmaton

9 Ab nto THEORY OF SUPERCONDUCTIVITY. I. FIG. 2. Lowst-ordr lctronc a and phononc b, c contrbutons to F xc. F a xc = 1 v,j1 4,j E tanh 2 E 1 j E j tanh 2 E j, 50 whr th matrx lmnts of th Coulomb ntracton ar dfnd as v,j = d r 3 d 3 r * 1 r r r r jr * j r. 51 As th xpctaton valu,q =0, th lctron-phonon ntracton dos not contrbut to F xc n frst-ordr prturbaton thory n. Th frst nonvanshng trms appar n scond ordr n frst ordr n g 2 and ar dpctd n Fgs. 2b and 2c. Th frst of ths trms Fg. 2b s th countrpart of th anomalous trm that contrbuts to th lctronc Hartr nrgy 20. Its analytc form can b wrttn as b = 1 2,q F xc g,q * 2 j E E j IE,E j,,q IE, E j,,q, 52 whr w dfnd th matrx lmnts of th lctron-phonon couplng constant g,q whl th functon I s = d 3 r * rv BO,q r j r, 53 IE,E, = E 2 E Th thr fractons n th sum com from th dnomnators of th two Grn s functons G s and from th phonon propagator D s. It s possbl to prform th frquncy sums usng standard complx contour ntgraton mthods. Th fnal rsult s IE,E j, = f E f E j n E E j + E E j Ej E+. 55 E E j + A dagram analogous to th on dpctd n Fg. 2b but wth th phonon propagator rplacd by th bar Coulomb ntracton xsts as wll. Ths dagram s th anomalous Hartr trm whch appars as th scond trm on th rghthand sd of Eq. 20. Snc ths trm s tratd xplctly n th lctronc Kohn-Sham quatons t s not part of th xchang-corrlaton functonal. Not that xprsson 52 s so much mor complcatd than th anomalous Hartr trm smply bcaus th phonon propagator dscrbs a rtardd ntracton. If w assumd a rtardd lctronc ntracton nstad of th nstantanous Coulomb potntal 1/r r th anomalous Hartr contrbuton would look vry smlar to 52. Th scond phononc trm that contrbuts to F xc s dpctd n Fg. 2c. Ths Fynman dagram has th sam structur as th lctronc xchang trm Fg. 2a. Howvr, and agan du to rtardaton ffcts, ts analytc structur s mor complcatd than Eq. 50, namly, c = 1 2,q F xc g,q 21+ j,e j,,q E E jie +1 E j E jie, E j,,q. b Th slf-nrgy dagrams contrbutng to F xc c and F xc 56 rsmbl th slf-nrgy dagrams that appar n Elashbrg thory. 2,3 By vrtu of Mgdal s thorm, 34 vrtx corrctons should b small and ar thrfor nglctd, both n Elashbrg thory and n our tratmnt. Thr s, howvr, an mportant dffrnc: n Elashbrg thory th propagators that ntr th slf-nrgy dagrams ar drssd propagators, whl n our cas w hav bar Kohn-Sham propagators. By usng th bar propagators w nglct mor dagrams than thos contanng vrtx corrctons. W postpon a mor dtald dscusson of ths problm to th scton on th xchang-corrlaton krnls. V. FUNCTIONAL DERIVATIVES AND THE CHAIN RULE W hav sn n Eq. 18b how th anomalous xchangcorrlaton potntal s dfnd as th functonal drvatv of th xchang-corrlaton fr-nrgy functonal wth rspct to th anomalous dnsty. Howvr, th contrbutons to th xchang-corrlaton fr-nrgy functonal that stm from Kohn-Sham prturbaton thory ar only known as xplct functonals of th Kohn-Sham orbtals r, th Kohn- Sham sngl-partcl nrgs, and th par potntal. Of cours, th Hohnbrg-Kohn thorm tlls us that ths quantts ar thmslvs functonals of th dnsts, so th fr nrgy s an mplct functonal of th dnsts. Furthrmor, f on maks th addtonal approxmaton that v xc PHYSICAL REVIEW B 72, dos not dpnd on thn th Kohn-Sham orbtals r and th gnnrgs ar also ndpndnt of th anomalous dnsty. F xc s thn a functon of th chmcal potntal and a functonal of th complx par potntal F xc = F xc, 2,. 57 For convnnc, nstad of workng wth and *,w prfr usng th modulus squard of th par potntal and ts phas as ndpndnt varabls. Th anomalous xchangcorrlaton potntal can thus b valuatd usng th chan rul for functonal drvatvs

10 LÜDERS t al. xc = F xc * F xc j 2 j j 2 * + F xc j *. j 58 Th partal drvatvs of F xc can now b calculatd drctly. Th rmanng functonal drvatvs ar somhow hardr to obtan, but can b drvd from th dfntons of th dnsts, Eqs. 28, and from th fact that th partcl dnsty and th anomalous dnsty ar ndpndnt varabls. Ths lattr condton can b xprssd through th rlaton nx * r,r =0. Morovr, w mak us of th two trval condtons * 59 * =,j, * j =0. 60 j From th abov condtons, and aftr som algbra, w arrv at th xprssons for th rmanng functonal drvatvs, j 2 * = 2 Y j 0E j 2 j,j Y j 1 j 2 *, E j *,=,j * tanh/2e, 1 * = Z Z 0 j. j Th functons Z 0 and Z 1 hav th dfntons Z 0 = E Y 0 /2tanh/2E, Z 1 = Y 1 0 Y, cosh 2 /2E and, fnally, th two functons Y 0 and Y 1 rad Y 0 = 2 E tanh 2 E + Y 1 = E tanh 2 E /2 2 cosh 2 /2E, /2 cosh 2 /2E. 61a 61b 61c 62 63a 63b Thr xsts anothr mthod to obtan xchangcorrlaton functonals usng Kohn-Sham prturbaton thory wthout rsortng to functonal drvatvs. Ths mthod follows th das of Sham and Schlütr, 35 and provds a vry drct conncton btwn many-body prturbaton thory and th normal and anomalous xchang-corrlaton functonals. In th followng w wll gv a brf account of th Sham-Schlütr mthod for suprconductors. Thr s a smpl conncton btwn th lctron dnsty nr and th ntractng many-body Grn s functon 1 nr = lm ng r,r; n n Th dfnton of th ntractng Grn s functon s smlar to Eq. 44, but wth th thrmal avrag wghtd by th ntractng statstcal oprator ˆ 0. Th nfntsmal s usd to nsur th corrct ordrng of th fld oprators n Eq. 44 so that Eq. 64 s satsfd. In th sam way t s smpl to prov that th anomalous dnsty obys th rlaton 1 r,r = lm nf r,r; n, n whr F s th ntractng anomalous propagator. On th othr hand, w dfnd th Kohn-Sham systm as th nonntractng systm whos both normal and anomalous dnsts ar qual to th dnsts of th ntractng systm. Thrfor, on can qually calculat th ntractng dnsts from th Kohn-Sham Grn s functons 1 nr = lm 0 + PHYSICAL REVIEW B 72, n s ng r,r; n, 66 wth an quaton smlar to Eq. 65 rlatng and F s.w thn xpand prturbatvly th ntractng Grn s functons n Eqs. 64 and 65, and quat th two quatons for nr, and th two quatons for r,r. As th prturbaton 39 ncluds th trms Vˆ xc and ˆ xc, th rsultng quatons form a systm of two coupld ntgral quatons that allow th dtrmnaton of v xc and xc. Thos ntgral quatons ar th so-calld Sham-Schlütr quatons. Th two mthods to obtan th xchang-corrlaton potntals ar concptually qut dffrnt. Th frst uss th dfntons 18 to drv th xchang-corrlaton potntals usng a srs of chan ruls. Th Sham-Schlütr approach s closr to many-body prturbaton thory, and provds a natural rlatonshp btwn th Grn s functon and th xchang-corrlaton potntals of DFT. Howvr, both approachs lad to th sam rsult f th fr nrgy n th frst mthod s xpandd up to th sam ordr n prturbaton thory as th Grn s functons n th scond mthod. VI. A SIMPLE BCS-LIKE MODEL W now ntroduc a smpl modl that wll allow us to study n dtal th functonals dvlopd n ths artcl. For smplcty, w assum that th par potntal has s-wav symmtry and bhavs approxmatly lk =. Ths assumpton s satsfd by all tradtonal suprconductors. In ths modl, w can transform th gap quaton nto a ondmnsonal quaton n nrgy spac = Z 1 dnk, 2 tanh/2, 67 whr N s th dnsty of stats at th nrgy. Its possbl to furthr smplfy ths quaton by assumng a BCS-lk modl for both K and Z. If w assum that th krnl K and th rnormalzaton trm Z ar constant n a shll of wdth c around th Frm nrgy and zro outsd ths rgon, Eq. 67 can b solvd analytcally for th transton tmpratur T c

11 Ab nto THEORY OF SUPERCONDUCTIVITY. I. PHYSICAL REVIEW B 72, T c xp 1+Z0 N0K0, 68 whr K0 and Z0 ar th valus of K, and Z at th Frm surfac. Equaton 68 has xactly th sam structur as McMllan s formula, 36,37 whch s an approxmat soluton of th Elashbrg quatons. Ths lattr formula rads T c = ln 1.20 xp * Th numbr *, th Coulomb psudopotntal of Elashbrg thory, masurs th strngth of th lctron-lctron ntracton. Ths paramtr s qut hard to calculat and s oftn fttd to xprmntal data. As * s normally postv, t tnds to dcras th suprconductng transton tmpratur. On th othr hand, s a masur of th lctron-phonon couplng strngth =2 d 2 F. 70 Th bhavor of T c wth s vry nonlnar. For small valus of,t c grows xponntally; Howvr, as ncrass, th suprconductng transton tmpratur saturats. Th paramtr ln s a wghtd avrag of th phonon frquncs ln = xp 2 d ln 2 F 71 and s of th ordr of th Dby frquncy of th matral. Fnally, th Elashbrg spctral functon s th lctronphonon couplng constant avragd on th Frm surfac, 2 F = 1 N0,q g,q j,q. 72 It s wdly accptd that McMllan s formula gvs a qut accurat dscrpton of th transton tmpratur for smpl, BCS-lk, suprconductors. Thrfor, by comparng xprssons 68 and 69 for th phonon-only cas,.., * =0, w obtan that for BCS-lk suprconductors N0K0, Z0. 73 Ths s an xtrmly mportant proprty of th xchangcorrlaton krnl, whch should b satsfd by any approxmat functonal. VII. APPROXIMATIONS TO THE ANOMALOUS HARTREE EXCHANGE-CORRELATION KERNEL From th prturbatv xpanson of th xchangcorrlaton fr nrgy t s clar that w can splt th fr nrgy nto thr parts. Th frst contans th purly lctronc trms,.., th trms that do not contan xplctly th lctron-phonon couplng constant; th scond, trms only nvolvng th lctron-phonon couplng constant; and th last, whch w dfn as th total fr nrgy mnus th two frst parts, wll hav mxd contrbutons from th Coulomb and lctron-phonon ntractons. Th xchang-corrlaton FIG. 3. Th functon N0K ph, for Al, Nb, and Pb, calculatd at T=0 K. potntals and th xchang-corrlaton krnls can b splt n th sam way. In ths scton w dvlop xchang-corrlaton krnls to b usd n th lnarzd gap quaton 35. Functonals that can b usd n th nonlnar gap quaton 32 ar dscussd latr. Ths scton s dvdd nto two parts. Frst w look at th purly lctron-phonon contrbutons to th xchangcorrlaton krnl. Such functonals ar dvlopd usng th machnry of Kohn-Sham prturbaton thory togthr wth th chan rul ntroducd arlr. In th scond part, w turn our attnton to th purly lctronc part of th krnl. Two functonals wll b prsntd: th frst has th form of a local dnsty approxmaton LDA, whl th scond s a functonal that avods th drct computaton of th scrnd Coulomb matrx lmnts. Th mxd contrbutons apparng n th prturbatonal xpanson of th fr nrgy ar nglctd n th currnt tratmnt. A. Elctron-phonon contrbutons In frst ordr n g 2 thr ar two trms stmmng from th lctron-phonon ntracton that contrbut to th xchangcorrlaton fr nrgy: F b xc gvn by Eq. 52, and F c xc gvn by Eq. 56. Th xchang-corrlaton krnl drvd from F b xc s nondagonal and has th form K ph = 2 tanh/2 tanh/2 j,q g,q 2 I, j,,q I, j,,q. 74 To gan furthr nsght nto ths trm, w us a smplfd modl: w approxmat th lctron-phonon couplng constants by thr avrag valu at th Frm surfac and th lctronc nrgy dsprson s rplacd by th fr-lctron modl. In Fg. 3 w dpct th dagonal K ph, for alumnum, nobum, and lad at zro tmpratur for ths smplfd modl. As ths contrbuton to th xchang-corrlaton krnl xhbts partcl-hol symmtry w only plot th rgon 0. Ths trm s sharply pakd at th Frm nrgy not th logarthmc scal on th axs. Furthrmor, th wdth of th curvs for ach matral s of th ordr of th

12 LÜDERS t al. PHYSICAL REVIEW B 72, corrspondng Dby frquncy. Th valu of th krnl at th Frm nrgy can b calculatd analytcally, N0K ph 0,0 = d 2 F coth At zro tmpratur, th valu of N0K ph 0,0 rducs to, whch s th valu xpctd from th comparson to Mc- Mllan s formula cf. Eq. 73. Howvr, at hghr tmpratur N0K ph 0,0 dcrass monotoncally. Th scond phononc contrbuton to th xchangcorrlaton krnl comng from th Kohn-Sham prturbaton thory PT orgnats from th dagram F c xc. It s a dagonal trm, whch rads Z ph,pt 2 = /2/cosh 2 /2 j j 1 /2 snh/2 cosh/2 jl jl g,q 2 I j, l,,q,q 1 + tanh/2 j,q 2 g,q 1 I, j,,q I, j,,q 2I, j,,q, whr th functon I s dfnd as 76 I, j,,q = I, j,,q. 77 If w try to apply th smplfd modl prsntd arlr w fnd that Z ph,pt dvrgs logarthmcally. Ths dvrgnc can b tracd back to th substtuton of g,q by ts valu at th Frm surfac. Ths problm can b solvd by rtanng th full dpndnc of th lctron-phonon couplng constant on th ndcs and j:g,q thn dcays as a functon of nrgy thrby makng th ntgrals prsnt n Eq. 76 convrgnt. A closr analyss of th xprssons also rvals that th dvrgnt part of th ntgrands s antsymmtrc around th Frm surfac. Thrfor, th dvrgnt ntgrals would vansh n th cas of partcl-hol symmtry. It sms thn rasonabl to nglct th antsymmtrc part of th ntgrands, rtanng only th symmtrc part. Th nw functonal rads Z ph,sym 1 = tanh/2 j g,j,q 2,q I, j, + I, j,. 78 In Fg. 4 ths trm s plottd for nobum for svral tmpraturs. It turns out that th functon Z ph,sym s a smooth functon of th nrgy, and ts valu at th Frm surfac FIG. 4. Th dpndnc of Z ph,sym on tmpratur for nobum. k =0 s approxmatly 2. Ths s twc th valu xpctd from th comparson to McMllan s formula cf. Eq. 73. Furthrmor, a carful analyss of Fg. 4 suggsts that Z ph,sym can b wrttn as th sum of two trms: on broadr and vry wakly tmpratur dpndnt; a scond contrbuton whos wdth dcrass sgnfcantly wth th tmpratur. Both trms contrbut wth approxmatly to th valu of Z ph,sym at th Frm surfac. As th rnormalzaton trm Z ph,sym appars to b too larg, on can xpct that transton tmpraturs calculatd wth ths functonal wll b too small. Th stuaton should b worst for th strong-couplng suprconductors lk nobum or lad, whr th rnormalzaton s larg. Ths s confrmd by Tabl I whr w lst th transton tmpraturs obtand wth th phononc part of th functonal. Ths numbrs ar compard to solutons of Elashbrg s quaton whr w nglctd th lctron-lctron rpulson. W blv that th shortcomngs of ths functonal can b tracd back to th followng: Mgdal s thorm tlls us that, to a vry good approxmaton, w can nglct n th prturbatv xpanson dagrams ncludng vrtx corrctons du to th lctron-phonon ntracton. Howvr, dagrams ncludng slf-nrgy nsrtons of phononc orgn should b ncludd to hav a consstnt dscrpton of th lctronphonon ntracton. Thrfor th bar Grn s functons ntrng n th dagrams dpctd n Fgs. 2b and 2c should b rplacd by drssd propagators. In a frst stp to mprov our functonals w drssd th propagators wth a subst of phonon slf-nrgy nsrtons. W found that th nondagonal trm K ph s bascally nsnstv, whl th trm Z ph,sym s rducd by roughly 20%. Ths s almost half th corrcton ncssary to satsfy Eq. 73. W xpct that th othr 30% s accountd for by th rmanng slf-nrgy nsrtons. Howvr, ths approach s qut nvolvd numrcally, so w choos a dffrnt path to mprov our functonal. W know that th phonon rnormalzaton trm should hav th valu at th Frm surfac. Furthrmor, ths trm should hav a wdth comparabl to th Dby frquncy. It s clar that th broadr contrbuton to Z phsym obys ths rqurmnts. W thrfor propos to sparat th two parts of Z ph,sym and us th part as our rnormalzaton trm

13 Ab nto THEORY OF SUPERCONDUCTIVITY. I. TABLE I. Transton tmpraturs from numrcal solutons of th phonon-only DFT and Elashbrg quatons. All tmpraturs ar n klvn. W blv that ths procdur s at last partally justfd by th rsults obtand by drssng th Grn s functons. Th functonal corrctd n ths way rads Z ph 1 = tanh/2 g,q 2 J, j,,q j,q + J, j,,q, 79 whr th functon J s dfnd by Al Nb Mo Ta V Pb K ph +Z ph,sym K ph +Z ph Elashbrg PHYSICAL REVIEW B 72, a xchang trm F xc dpctd n Fg. 2a. Th ntracton that ntrs ths xprssons s th bar Coulomb ntracton 1/r r. Howvr, lctrons n a mtal do not fl th bar Coulomb ntracton, but a much wakr ntracton, scrnd by th sa of lctrons. To tak ths nto account, w tak a stp back, and propos an altrnatv form to th nrgy functonal basd on th suprconductng vrson of th local dnsty approxmaton. 15 In ths approach th xchang-corrlaton nrgy of th nhomognous systm s wrttn n trms of th xchang-corrlaton nrgy dnsty of th homognous suprconductng lctron gas, F SCLDA xc nr,r,k = d 3 R f hom xc n,k n=nr, = W R, k 82 whr W R,k s th Wgnr transform of th anomalous dnsty of th nhomognous systm, gvn by Fnally w hav J,, = J,, J,,. 80 W R,k = d 3 s ks R + s 2, R s f + n J,, = f f f f Th functonal Z ph s smooth both as a functon of th nrgy and as a functon of th tmpratur. Furthrmor, t has approxmatly th valu at th Frm surfac. Th functonal 79, togthr wth th phononc krnl 74, s a cntral rsult of our work. It s th functonal that wll b usd n th calculatons of II. In Tabl I, w prsnt th phonon-only transton tmpraturs calculatd wth ths functonal. All T c s ar n qut good agrmnt wth transton tmpraturs calculatd from Elashbrg s quaton. W mphasz that th transton tmpraturs n Tabl I ar gvn xclusvly for th purpos of tstng and/or calbratng th approxmatons mad for th phononc part of th xchang-corrlaton functonal. T c s rsultng from sttng * =0 n th Elashbrg quatons and sttng th Coulomb trms to zro n th DFT contxt hav, of cours, nothng to do wth th T c s obsrvd n natur. For rsults ncludng th Coulomb trms, w rfr th radr to II. B. Elctron-lctron contrbutons W now dvlop functonals that tak nto account th Coulombc part of th ntracton. Thr ar two trms n th nrgy functonal that gv contrbutons to th lnarzd gap quaton 35. Th frst s th anomalous contrbuton to th Hartr nrgy, gvn by Eq. 20, and th scond s th It s asy to s that ths dfnton rducs to th usual LDA for nonsuprconductng systms n th lmt 0. Morovr, t s possbl to prov that ths s th only consstnt dfnton of a LDA for th suprconductng stat. 38 As an approxmaton to th xchang-corrlaton nrgy of th lctron-gas on could tak th random-phas approxmaton functonal proposd n Rf. 15. Howvr, ths functonal has only bn valuatd for a vry smpl class of par potntals, namly, Gaussans cntrd at th Frm surfac. W thrfor propos an altrnatv and smplr form to th Coulombc contrbuton to F xc. For convnnc, w approxmat togthr th anomalous Hartr and th xchang-corrlaton contrbutons. Our approxmaton rads f hom Hxc n f hom,ns xc n = d 3 r rr r 2 v TF r r, 84 whr v TF r r s th Coulomb ntracton scrnd by a Thomas-Frm modl. In coordnat spac th Thomas- Frm ntracton rads v TF r r = k TFr r, 85 r r wth th Thomas-Frm scrnng lngth k TF gvn by

14 LÜDERS t al. k 2 TF =4N0. 86 By nsrtng xprsson 84 n th dfnton of th LDA, Eq. 82, w can dntfy ths approxmaton as a Thomas- Frm scrnd anomalous Hartr contrbuton to th fr nrgy. Th anomalous Hartr xchang-corrlaton krnl stmmng from ths trm s smply K TF = v TF, 87 whr th matrx lmnts of th Thomas-Frm ntracton ar dfnd by PHYSICAL REVIEW B 72, wth k=2 and k=2. Usng th BCS-lk two-wll modl on can xtract th countrpart of th Coulomb psudopotntal * from Elashbrg thory. A crud stmat for r s =2 gvs a valu around 0.1, whch compars wll wth th typcal valus of * for smpl mtals * = It should b strssd agan at ths pont that th prsnt mthod dos not rqur *. Th stmats gvn hr ar usd to dmonstrat to whch valus of * our ab nto Coulomb trms corrspond. Whl th rplacmnt of th Kohn-Sham orbtals n Eq. 88 by plan wavs may b accptabl for smpl mtals, t wll b too crud for mor complx matrals. In thos cass t s stll possbl to avod th drct computaton of th scrnd Coulomb matrx lmnts 88, by gong along th lns dscrbd by Sham and Kohn. 39 W brfly outln hr th man ponts of ths classcal papr, whch dals wth an approxmat way of gttng an lctron slf-nrgy for th normal stat. W assum, as usual wthn th LDA, that our systm can b dscrbd around th pont r by a homognous lctron gas of dnsty nr. Th wav functons of ths lctron gas can b locally xprssd as plan wavs of momntum pr whos valu s dtrmnd, n a smclasscal way, from th lctron nrgy of th ral systm. In th smplst form, th mappng can b obtand from Eqs. 4.5 and 4.13 of Rf. 39 as p 2 2 = + h nr, 90 whr h n s th chmcal potntal of a nonntractng homognous lctron gas wth dnsty n. Furthrmor, w approxmat h (nr) by th constant h n, whr n s th avrag dnsty of th matral. W suggst hr to approxmat th Coulomb ntracton krnl btwn lctrons at nrgs k and k by th corrspondng quantts n th fr-lctron gas. W thn rplac p 2 /2 + h =, and rwrt th ntracton 89 as v TF = d r 3 d 3 r * r rv TF r r j r * j r 88 whr th s ar th Kohn-Sham orbtals of th nhomognous systm at hand. In II w wll compar th rsults obtand wth th abov approxmaton wth furthr smplfd xprssons. In th smplst modl, th Kohn-Sham orbtals ar takn to b plan wavs wth a parabolc dsprson. In ths cas, th krnl can b wrttn n nrgy spac aftr avragng ovr th angls as K TF, = kk ln k + k2 2 + k TF k k 2 2, 89 + k TF K SK = 2 / j 2 j + k 2 TF /2. 2 j ln + j +2 j + k TF In prncpl, on could consdr not only p but also k TF as locally dpndnt on th dnsty nr. In our smplfd approach, howvr, w fx th Thomas-Frm scrnng lngth to a constant valu. Equaton 90 s concvd n trms of wav packts, and s vald f nr dos not vary too much on th scal of th Frm lngth, xactly as n th normal stat LDA. On can spculat, howvr, that whn appld to th suprconductng stat th rlvant lngth scal bcoms th cohrnc lngth, normally much largr than th atomc scal. Thrfor, w may assum that local varatons of th dnsty on th atomc scal wll not affct th fnal suprconductng proprts. It should b notd that ths approxmaton, although drvd n th sprt of a LDA, s not a local dnsty approxmaton, snc t dos not dpnd xplctly on th dnsts, but mplctly va th sngl-partcl nrgs. VIII. FUNCTIONALS FOR THE NONLINEAR GAP EQUATION In ths scton w provd approxmatons to th xchang-corrlaton krnl that can b usd n th nonlnar gap quaton 32. Ths functonals wll oby on constrant, namly, that upon lnarzaton thy wll rduc to th functonals prsntd n th prvous scton. Ths assurs that th gap functons obtand from Eq. 32 and th transton tmpraturs calculatd from Eq. 32 ar consstnt. Furthrmor, w rqur ths functonals to b wll bhavd,.., wthout dscontnuts or any othr knd of pathologcal bhavors. Th smplst way to drv an xchang-corrlaton functonal s to us th xprssons drvd through Kohn-Sham prturbaton thory n th dfnton 18b. For xampl, th frst phononc contrbuton F b xc Fg. 2b ylds th contrbuton