KOHN LUTTINGER SUPERCONDUCTIITY IN GRAPHENE J. Gonzálz Instituto d Estructur d l Mtri, CSIC, Spin
Is it possibl to hv suprconducting instbility in grphn (by suitbl doping)? Thr hv bn lrdy svrl proposls to driv grphn towrd piring instbility: du to topologicl dfcts nd th nhncmnt of th dnsity of stts t th Frmi lvl, J. G., F. Guin nd M. A. H. ozmdino,, Phys. Rv. B 63,, 13441 (1). from th intrction with mtllic coting, B. Ucho nd A. H. Cstro Nto,, Phys. Rv. Ltt. 98,, 14681 (7). from lctronic corrltions rising from th br Coulomb rpulsion, A. M. Blck Schffr nd S. Donich,, Phys. Rv. B 75,, 13451 (7). In ll th bov proposls, significnt dnsity of stts is inducd i t th Frmi lvl. So th qustion is, how much cn th suprconducting corrltions s b nhncd by doping th lctron systm?
GRAPHENE π BAND Whn grphn is lrgly dopd, th Frmi lin dvits from n isotropic shp. Th ffct of n Hov singulrity in th spctrum is flt t bout 3. Th bnd structur cn b fittd by mns of tight binding modl with hopping up to third nighbors in th crbon lttic H = t d ik v t' ik w ik v t ik v d t' ik w ik v nd ovrlp s btwn nrst nighbors. nighbors. Th diffrnt prmtrs r dtrmind by rproducing th vlu of th Frmi vlocity, 3 vf = ( t t' + 3sd ) C -C th lvl of th sddl points rltiv to th Dirc points, t 3t' d 3d +.7 1+ s nd th curvtur of th disprsion t th sddl points, which lds to d.7 +.8s + O( s ) ()
NESTING OF THE FERMI LINE IN GRAPHENE Q As long s t, th Frmi lin shows pproximt nsting, ε ( k ) = ε ( Q + k) which lds to n nhncmnt of th suscptibility χ(q,ω). For sddl point chrctrizd by ε + ( k) α δk β δk x y th suscptibility t momntum trnsfr Q is (Q, ω) 1 1 (1 + β / 3α )(1 + 3β / α ) Λ log log 3π α + β (1 β / 3α )(1 3β / α ) ω + μ χ This hs to b comprd with th suscptibility t smll momntum trnsfr 1 1 Λ (, ω) log 4π αβ ω + μ χ Both suscptibilitis divrg whn th chmicl potntil is pinnd t th singulrity ( (μ ). In th prticulr cs of th vlnc bnd, w hv α / β 3.6 χ( Q, ω) >> χ(, ω) Whn χ(,ω) is lrg t low nrgis, w my xpct frromgntic instbility in th systm. Wht kind of instbility riss whn χ(q,ω) ) >> χ(,ω) )? (J. G., Phys.. Rv. B 78,, 5431 (8))
KOHN LUTTINGER SUPERCONDUCTIITY Th ffct rsts on th fct tht th modultion of th scttring ovr th Frmi surfc my giv ris to n ttrctiv coupling in th BCS chnnl (W. Kohn nd J. M. Luttingr,, Phys. Rv. Ltt. 15, 54 (1965)). In th D cs, th BCS vrtx obys th scling qution d ( φ, ) k k p p n( dλ Λ 1 π π d ' ( φ, ') ( ', ) If w dcompos in trms of th diffrnt rprsnttions of th point symmtry group, m, n ( φ, ) = Ψ ( φ) Ψ ( ) Λ = n( m, n m n,m,n s Th proximity to th n Hov singulrity nhncs th instbility, ty, s th dnsity of stts is n( But w must find ngtiv coupling if piring instbility is to tk plc s Λ. d 3 1 Λ ) log 4π αβ ε + μ ε dλ m, s s, n
KOHN LUTTINGER SUPERCONDUCTIITY IN GRAPHENE Q In th cs of th honycomb lttic, w hv th ignmods {cos(6nφ)}, {sin(6nφ)}, {cos((6n+3) +3)φ)}, )}, {sin((6n+3) +3)φ)}, )}, n Z, nd {cos({ cos(mφ), sin(mφ)}, m not bing multipl of 3. W cn xpnd th BCS vrtx in th form ( φ, ) =, + + + +,,4 3,3 ( cos(6φ ) + cos(6φ ')),6 ( cos(φ )cos( ) + sin(φ )sin( )) ( cos(φ )cos(4 ) sin(φ )sin(4 ) + φ ) ( cos(3φ )cos(3φ ')) + ( sin(3φ )sin(3φ ')) +... 3,3 Th vlu of (φ,φ) t th high nrgy cutoff is dtrmind by th drssing from h pirs. Th diffrnt strngth of th scttring t Q nd smll momntum trnsfr implis (,) (, π / 3) 3χ(, ω) χ( Q, ω) < (,) (, π / 3) 3( + ) <,, 4 showing tht thr is n ttrctiv coupling in th d wv chnnl.
KOHN LUTTINGER SUPERCONDUCTIITY IN GRAPHENE W kp th dominnt couplings to dl with trunction of th scling qutions: dm, Λ dλ n = n( s m, s s, n d Λ dλ, 4,,4 n(, 4,,4, 4,,4 which hv to b solvd with initil conditions (in th cs of on sit rpulsion U) U U 3( + ),, 4 nd 1 3Uχ (, ω) 1 3Uχ ( Q, ω) n( 3 1 Λ ) log 4π αβ ε + μ ε A divrgnc is obtind in th th low nrgy limit Λ t som finit vlu of Λ, tht w intrprt s th dvlopmnt of th suprconducting gp. For U 4,, w gt Th rlvnt quntity dictting th mgnitud of th suprconducting gp Δ is th chmicl potntil μ tht msurs th dvition of th Frmi lvl from th n Hov singulrity.
KOHN LUTTINGER SUPERCONDUCTIITY IN GRAPHENE W my crry out th drivtion of th ffct t tmprtur T using th propgtor G () ( q, ω) = 1 ω P iπ tnh δ ( ω ε ( k)) ω ε ( k) k T B This lds to th corrctions of th BCS vrtx in th prticl prticl chnnl Λ n( 1 π d ( φ, ) tnh dλ d ' ( φ, ') ( ', k T Λ π B Th finit tmprtur ttnuts th dnsity of stts n( ) d Λ dλ, 4,,4 Λ tnh kbt n(, 4,,4, 4,,4 For givn vlu of th chmicl potntil μ, th divrgnc in th solution of th scling qutions is lost t mximum tmprtur. Th rtio btwn th suprconducting gp nd th criticl tmprtur is vry clos to th BCS vlu, xcpt whn th Frmi lvl pprochs th n Hov singulrity. T c
KOHN LUTTINGER SUPERCONDUCTIITY IN GRAPHENE W my think of diffrnt imprfctions tht cn wkn th n Hov singulrity: cncis or voids. Thy giv ris to fturs prfrntly clos to th Dirc point. Substitutionl impuritis. Th ffcts of disordr r nhncd by th nsting of th Frmi lin, nd th n Hov singulrity is ttnutd by logrithmic corrctions, τ π () 1 1 n( Λ ) n ( 1 log +... π tτ π 4τ π ε whr is th rlxtion tim in th impurity scttring (E. P. Nkhmdov t l., Phys. Rv. Ltt. 84,, 393 ()). Howvr, in cln grphn smpls whr th mn 3 fr pth is of th ordr of th micron scl, w hv tτ π ~1. Rippls. Thy cn ct s n intrinsic sourc of scttring, modulting th hopping in th grphn lttic. This my induc splitting of th n Hov singulrity by n nrgy Δε ~ t ( / R) C C R bing th rdius of curvtur of th rippls. Howvr, R is typiclly bov th scl of 1 nm, giving ris to splitting g of th ordr of ~.1 m, which is inffctiv to ttnut th strngth of th singulrity.
In conclusion it is possibl to induc suprconducting instbility in grphn, with criticl tmprtur bov 1 K, dpnding on th bility to tun th Frmi lvl to th divrgnt dnsity of stts th suprconducting instbility will ppr upon hol doping, s th n Hov singulrity in th vlnc bnd coms with concomitnt nsting of th Frmi lin inducing th rquird ttrctiv coupling th piring instbility is bl to surviv finit tmprtur nd d disordr ffcts, providd tht th grphn smpls r sufficintly cln, with mn fr pths bov th micron scl