Josephson current noise above T c in superconducting tunnel junctions

Transcription

1 PHYSICAL REVIEW B 78, Josphson currnt nois bov T c in suprconducting tunnl junctions Alx Lvchnko School of Physics nd Astronomy, Univrsity of Minnsot, Minnpolis, Minnsot 55455, USA Rcivd 19 Mrch 008; rvisd mnuscript rcivd 6 July 008; publishd 11 Sptmbr 008 Tunnl junction btwn two suprconductors is considrd in th vicinity of th criti tmprtur. Suprconductiv fluctutions bov T c giv ris to th nois of th c Josphson currnt lthough th currnt itslf is zro in vrg. As rsult of fluctutions, currnt nois spctrum is pkd t th Josphson frquncy, which my b considrd s prcursor of suprconductivity in th norml stt. Tmprtur dpndnc nd shp of th Josphson currnt nois rsonnc lin is cultd for vrious junction configurtions. DOI: /PhysRvB PACS numbrs: k, r I. INTRODUCTION In th vicinity of th criti tmprtur T c trnsport proprtis of mtls r strongly ffctd by suprconductiv fluctutions. For xmpl, in th tmprtur rgion T T c T c, whr fluctutions r th most pronouncd, Drud conductivity cquirs noticbl Aslmzov-Lrkin, Mki- Thompson, nd dnsity-of-stts DOS corrctions. Mny othr kintic nd thrmodynmic cofficints such s mgntic suscptibility, ht conductivity, Hll cofficint, nd ultrsonic ttnution r lso modifid by fluctutions. On my consult rcnt book Rf. 1 for xhustiv ovrviw of rsults nd litrtur in this fild. Mostly immditly ftr th pionring works on suprconductiv fluctutions,,3 it ws noticd tht nlog of th c Josphson ffct my surviv in th norml stt bov th criti tmprtur. 4,5 Th lttr is lso ttributd to th formtion of fluctuting Coopr pirs. Indd, considr wk trnsprncy tunnl junction btwn two suprconductors. In this cs Josphson currnt is givn by I J t=i c sin J t, whr J = V is th Josphson frquncy nd currnt mplitud I c is proportionl to th product of suprconductiv ordr prmtrs LR, tkn from th lft L nd right R to th contct r. Abov th criti tmprtur Josphson currnt vnishs I J L R =0 sinc ordr prmtr is zro in vrg LR 0. Howvr, currnt squrd I J L R L R, which givs nois of th Josphson currnt, is pprntly not zro du to nontrivil vrg LR LR of spc nd tim fluctuting ordr prmtrs. As consqunc, nois powr spctrum S J, dfind s th Fourir trnsform of Josphson currnt-currnt corrltion function, shows distinctiv pk t th Josphson frquncy J, which is xprimntlly n ccssibl ffct. Th pk hight S mx =S J = J is strong function of T T c, usully som powr lw, which mks it possibl to dtct nois signl in th immdit vicinity of th criti tmprtur T T c T c. Although this obsrvtion ws thr for long tim, th intrst to it ws rcntly rvivd. It ws strssd 6 8 tht msurmnts of th Josphson currnt nois my b spcilly fruitful in studis of th high-tmprtur suprconductivity. Indd, whthr suprconductiv piring fluctutions xist in th psudogp rgim of th high-t c mtrils my b probd by th Josphson tunnling. Thus, xistnc of th Josphson ffct bov T c my b thought s th prcursor of suprconductivity. So fr fluctutions of th Josphson currnt bov th criti tmprtur wr studid ithr for th nrrow contcts, 4,7,8 tking into ccount only tmporl fluctutions of th ordr prmtr, or for th msoscopic rings. 9,10 W find, howvr, tht in th plnr gomtry of th tunnl junction, whr sptil vritions of th suprconductiv ordr prmtr hv to b ccountd for, pk in th currnt nois spctrum is mor pronouncd, spcilly, for th nonsymmtric junction configurtions. Motivtd by th ongoing xprimnts 11 nd possibl pplictions in probing psudogp rgim of high-t c mtrils, w rvisit problm of th Josphson currnt nois bov T c nd study nois in th plnr gomtry of tunnl junction. Within this work w focus on th tmprtur rng GiT T c /T c 1, whr Gi is th Ginzburg numbr. In this rgim fluctutions cn b considrd s smll nd cn b trtd in prturbtion thory. Th nturl xpnsion prmtr, which msurs strngth of th suprconductiv fluctutions, is Gi1. Th min rsults of th prsnt work my b summrizd s follows: i For symmtric wid junctions, whn both lctrods r in th fluctuting rgim, nd contct r A is lrg s comprd to th squr of th suprconductiv cohrnc lngth, A o, Josphson currnt nois spctrum S J hs Lorntzin-lik shp. Th pk hight ss in tmprtur s S mx T c T T c nd dpnds qudrtily on both tunnl conductnc of th junction g T nd th Ginzburg numbr Gi. For th lowst tmprtur T T c =GiT c, which is llowd by th pplicbility of th prturbtion thory, strngth of th nois is givn by S mx =/64g T T c / o /A. Of cours, xprimntlly, nois is mximl right t th trnsition T=T c ; howvr, in this cs it is vry difficult to mk ny quntittiv prdictions thortily. Thus, S mx givs n ordr of mgnitud stimt. ii For th nrrow, o A, symmtric junctions w find lso Lorntzin-lik shp of S J, which is gin qudrtic in both g T nd Gi; howvr, tmprtur dpndnc of th pk hight is diffrnt S mx T c T T c. Th stimt for th nois powr t th most vicinity of th trnsition is S mx =Gi/8g T T c /. iii In th cs of nonsymmtric junctions, whn on lctrod is lrdy suprconducting whil nothr is fluctuting, nois hs Lorntzin form. Th tmprtur dpndnc for th pk hight in this cs is th sm s for wid symmtric junction, which, howvr, pprs lrdy in th first ordr of th Ginzburg numbr nd contins lrg prfctor ln S /T c whr S is th supr /008/7810/ Th Amricn Physi Socity

2 ALEX LEVCHENKO conductiv gp. iv Corrctions to th currnt nois bov T c r not xhustd by th Josphson currnt contribution only. In ddition, suprconductiv fluctutions dplt norml-mtl DOS t th Frmi nrgy, which chngs tunnl conductnc. Th lttr trnslts into th currnt nois corrction S DOS vi fluctution dissiption thorm FDT. This ffct is linr in g T nd Gi, logrithmic in tmprtur S DOS ln T c T T c, nd hs n opposit sign s comprd to th Josphson currnt contribution. Th rst is orgnizd s follows: In th nxt sction Sc. II w prsnt in concis form our tchni mthod, Kldysh nonlinr -modl, which will b usd through out th ppr in cultion of th currnt nois powr. This formlism ws lbortd in Rfs. 1 nd 13, nd found to b vry usful nd powrful in mny pplictions. In th Sc. III w cult dnsity-of-stts nd Josphson currnt contributions to th nois spctrum bov T c. Th rsults of th work togthr with furthr discussions r summrizd in th Sc. IV. Numbr of tchni points r dlgtd to th Appndixs A1 A3. II. FORMALISM Considr voltg bisd tunnl junction of two suprconductors bov th criti tmprtur. Within -modl formlism tunnling btwn L nd R rsrvoirs of junction is dscribd by th ction, is T V = g T ˇ VˇQˇ 4 Tri L iˇ Vˇ Qˇ R, whr g T is th junction tunnl conductnc nd Qˇ LR r th Grn s functions dscribing lctron systm in th lctrods hrftr =k B =1. Both Qˇ LR r 44 mtrics in th four-dimnsionl Kldysh Nmbu spc. Mtrix ˇ = 0 z, whr i, i for i=0,x,y,z, r th sts of Puli mtrics cting in th Kldysh nd Nmbu subspcs corrspondingly, nd symbol stnds for th dirct product. Mtrix Vˇ is th sourc trm hving stndrd structur in th Kldysh spc, Vˇ t = V t V q V q t V t 0. Digonl lmnts of Vˇ r dirctly rltd to th ssily pplid voltg V t=vt, whil V q t is just its quntum componnt. This trminology stms from th Kldysh contour trms ssi nd quntum imply th symmtric nd ntisymmtric linr combintions of th fild componnts rsiding on th forwrd nd bckwrd prts of th Kldysh contour, rspctivly. 14 Finlly, trc oprtion Tr... in Eq. 1 ssums summtion ovr th mtrix structur s wll s tim nd sptil intgrtions. Th origin of phs fctors xpiˇ Vˇ in Eq. 1 is from gug trnsformtion, which movs diffrnt lctrochmi potntils of lctrons in th lds from th Grn s functions to th tunnling trm. Dynmics of th Grn s functions is govrnd by th -modl ction, 1,13 1 i is Q L,Q R = Trˇ ˇ ˇ =L,R =L,R 4 TrD Qˇ 4ˇ t Qˇ +4iˇ Qˇ, 3 whr is th br norml-mtl dnsity of stts t th Frmi nrgy, D is th diffusion cofficint, is th suprconductiv coupling constnt, nd ˇ = x 0. Th mtrix suprconductiv ordr prmtr ˇ r,t is ˇ = ˆ ˆ q PHYSICAL REVIEW B 78, ˆ q = ˆ ˆ, Action 3 is subjct to th nonlinr constrint Qˇ =1. Physi quntitis of intrst r obtind from th ction vi its functionl diffrntition with rspct to th pproprit quntum sourc. For xmpl, tunnl currnt is found from th qution, It = i ZV V q, ZV = DQ isq L,Q R, 5 tv q =0 whr SQ L,Q R =S +S T. Corrsponding nois powr spctrum is dfind s + S dt t ZV = V q tv q it t. 6 tv q =0 Th procdur of xtrcting physi obsrvbls, outlind bov, is rthr gnrl within Kldysh tchniqu. Howvr, for th problm t hnd, informtion ncodd in ctions 1 nd 3 is xcssiv. Indd, S dscribs not only dynmics of th ordr prmtr ˇ but lso contins xplicitly lctronic dgrs of frdom in th form of th Qˇ mtrics, which complicts furthr nlysis. Simplifiction is possibl rlizing tht dynmics of Qˇ is fst s comprd to tht of ˇ. Th lttr is govrnd by th tim s Q 1/T, whil th formr by 1/T T c, nd noticbly Q whn TT c. Undr this condition, on my intgrt out fst lctronic dgrs of frdom from ction 3 nd find n ffctiv thory, which dscribs spc nd tim fluctutions of th suprconductiv ordr prmtr only. This progrm ws rlizd for Eq. 3 in th rcnt work 15 nd w will follow hr th sm rout in dling with th tunnl trm S T V. Lt us outlin ssntil lmnts of th mthod. Hving intrst in th ffcts of suprconductiv fluctutions, it is rsonbl to strt from th norml-mtl stt with th Grn s functions Qˇ LR=Qˇ N givn by Qˇ N = 1 R F 0 1 A z, F = tnh T, 7 which minimizs ction 3 for ˇ =0. On trts thn ˇ in prturbtion thory on top of Qˇ N. Tchnily this progrm is

3 JOSEPHSON CURRENT NOISE ABOVE T c IN rlizd in svrl stps. At th first stg on projcts Q mtrics s Qˇ = iwˇ / Qˇ N iwˇ /, whr Wˇ r crris informtion bout fst lctronic dgrs of frdom. Mtrix Wˇ is prmtrizd by th two complx filds c q nd c q Coopr mods, which will b intgrtd out vntully. It is convnint to choos with Wˇ = c + + c Wˇ = Ř Wˇ Ř 1, c + + c, 10 whr = x i y /, = 0 z /, nd Ř = Ř 1 = 1 F On brings thn Eq. 8 into ction 3 nd xpnds S Q S W, to th scond ordr in th Coopr mods W =c,c dtils of this procdur r providd in th Appndix A1. On finds thn tht to th lding ordr in th coupling TrQˇ ˇ, Coopr mods r connctd to th suprconductiv ordr prmtr ccording to th rltions, c R q = C q q, c c q = C A c q q, 1 whr w hv introducd rtrdd dvncd Coopron propgtor, C RA 1 q = D q i +, nd th form fctors, c q = q + F q, q 13 c q q = q F q. 14 Knowing rltions 1 Gussin intgrtion ovr th Coopr mods is strightforwrd, DW xpis W, = xpis ff. 15 Th corrsponding qudrtic form S W, should b tkn from Eq. 46 nd on finds s rsult, S ff = Tr Lˆ 1, T =, q. 16 =L,R Th propgtor Lˆ 1 q, govrns suprconductiv ordrprmtr dynmics. It hs typi bosonic structur in th Kldysh spc, with Lˆ 1 q, = 0 L R 1 L 1 A L 1, K L 1 RA q, = D q + 1 8T GL i, c L K 1 q, = B L R 1 q, L A 1 q,, nd GL =/8T T c nd B =coth/t. Noticbly, ffctiv ction 16 is much simplr thn th originl on Eq. 3. Howvr, wht is importnt to mphsiz, is tht S ff cpturs corrctly ll th rlvnt low-nrgy xcittions of r,t. Aftr ths tchni prliminris w turn now to th pplictions of th gnrl formlism bsd on th ffctiv ction S ff. III. CURRENT NOISE ABOVE T c A. Tunnl currnt nois Th first pprnt ffct of suprconductiv fluctutions is modifiction of th norml-mtl dnsity of stts. Bing flt in th norml stt, cquirs strong nrgy dpndnc in th vicinity of T c with dip round Frmi nrgy. 16 Th lttr supprsss tunnl conductnc of th junction, which influncs tunnl currnt nd s th rsult its nois. Suprconductiv fluctutions corrction to th tunnl currnt ws studid in Rf. 17. Hr w cult corrsponding corrction to th nois. Although th rsult of this cultion follows immditly from th fluctution-dissiption rltion it is still usful to s how it pprs within th -modl pproch. To this nd, ssum nonsymmtric tunnl junction: lt us sy tht lft lctrod is in its norml stt, whil th right on is in th fluctuting rgim. To cult nois powr, on uss gnrl dfinition Eq. 6 nd insrts Qˇ L =Qˇ N nd Qˇ RQˇ N1+iWˇ Wˇ / Rf. 18 into th tunnling prt of ction 1. Aftr th diffrntition, which is don with th hlp of th formul, xpiˇ Vˇ V q = it tˇ xpivtˇ, t V q =0 19 whr ˇ = x z, on finds for th nois Hr PHYSICAL REVIEW B 78, S = S S + S DOS. S S =g T T u T cothu T, 0 1 with u =V, is just th Schottky formul for th nois in th norml tunnl junction, whil th corrsponding fluctutions corrction is

4 ALEX LEVCHENKO PHYSICAL REVIEW B 78, S DOS = g + T dt ts + t,t + S t,t 8 it t, whr S t,t =TrQˇ NŘ Wˇ qwˇ q Ř ˇ Qˇ Nˇ ivt tˇ i t t. 3 Quntum vrging in Eq. 3, dnotd by th ngulr brckts..., should b prformd with ffctiv ction 16, nmly,...=d...xpis ff. Rl tht fluctution mtrix Wˇ is xprssd through th Coopr mods c nd c, which r functionlly dpndnt on th ordr prmtr vi Eq. 1. Th nottion S DOS in Eq. nd its ctul rltion to th dnsity-of-stts supprssion r motivtd in Appndix A. Th linr in Wˇ trm in Eq. 3 is not writtn xplicitly sinc it dos not contribut to th finl rsult. Th finl commnt in ordr of Eq. is tht trcs of S functions llow rthr simpl nd convnint digrmmtic rprsnttion shown in Fig. 1. At this point on cults th product of Wˇ mtrics in Eq. 3 nd prforms Gussin functionl intgrtion ovr th fluctuting ordr prmtr using Eqs. 1 nd 16. Th rsulting vrgs r c qc q = i/ L Kq, + F L R q, + F L A q, Dq i +, 4 c qc q = i/ L Kq, F L A q, F L R q, Dq + i +. 5 Nxt fw stps r concptully simpl. i On trcs Eq. 3 ovr its mtrix structur first nd thn prforms tim Fourir trnsforms in Eq. dt t i Vt t = V, which rmovs intgrtion. ii Obsrv tht for th intgrtion, trm contining F L A q, in th vrg cc nd trm contining F L R q, in th vrg c c do not contribut to S s bing intgrls of purly dvncd nd rtrdd functions, rspctivly. As rsult, on tks cc +c c =i Imcc. Finlly on chngs momntum sum into th intgrl q d q/4, ssuming tht th lctrods r qusi-twodimnsionl films, nd introducs dimnsionlss vribls x =Dq /T, y= /T, nd z=+/4t. Aftr ths stps Eq. bcoms function. Clos look on Eq. 7 llows us to rwrit it in th form, S DOS = I DOS u coth u T, 8 whr I DOS is th tunnl currnt corrction cultd in Rf. 17, which is priori xpctd rsult from FDT. S DOS = 16Gi 3 g TT coth u + + T dydz 0 dx F z+u /T F z u /T R x + + y x + iy 4iz. 6 Hr =1/T c GL, nd w introducd Ginzburg numbr Gi =1/D. Aftr th rmining intgrtions s Appndix A3 for dtils on finds s rsult, S DOS = 4Gi g TT ln T c T T c Im 1 1 iu T, coth u T 7 whr 1 z is th first-ordr drivtiv of th digmm FIG. 1. Suprconductiv fluctution contributions to th currnt nois. Digrms nd b corrspond to th ffcts coming from th fluctutions in th dnsity of stts for th nonsymmtric nd symmtric junctions. Digrms c nd d r th fluctuting Josphson currnt contribution for th symmtric nd suprconductorfluctuting mtl junctions corrspondingly. Lddrs rprsnt Cooprons, Eq. 13, wvy lins stnd for th fluctutions propgtor, Eq. 18, nd crossd boxs dpict tunnl conductnc g T

5 JOSEPHSON CURRENT NOISE ABOVE T c IN In complt nlogy on cn cult corrsponding corrction to th nois for th symmtric junction whn both lctrods r in th fluctuting rgim. In this cs Grn sfunction mtrix Qˇ L hs to b xpndd in fluctutions Wˇ lso nd on fcs digrm shown in Fig. 1b. Th rsult of th cultion cn gin b cst in th form of Eq. 8, whr I DOS should b rplcd by th pproprit scond-ordr fluctution corrction known from Rf. 17. Furthrmor, if on is bl to cult I DOS compltly, mning to ll ordrs of prturbtion thory, thn for th nois of th tunnl currnt Eq. 8 cn b considrd s th xct rsult, which is gin consqunc of FDT. B. Josphson currnt nois Contribution to th nois spctrum coming from th Josphson ffcts is vry much diffrnt thn tht of dnsity of stts. First of ll thr is no simpl FDT rltion similr to Eq. 8. Scondly, th physi mchnism, which lds to th nois, is diffrnt. Probbly th simplst wy to s this is to strt from th dfinition of th currnt in Eq. 5. Assuming symmtric junction configurtion, on xpnds thn ch Grn s-function mtrix to th linr ordr in fluctutions Qˇ LR iqˇ NWˇ LR in th tunnl prt of ction 1, which givs for th currnt, I J t = ig T ˇ VˇQˇ NWˇ 4 V q t Tri L iˇ Vˇ Qˇ NWˇ R. 9 To procd furthr, w will simplify Eq. 9, xploring sprtion of th tim ss btwn lctronic nd ordrprmtr dgrs of frdom. Indd, on should notic tht s it follows from Eq. 18 rlvnt nrgis nd momnt for th ordr-prmtr vritions r Dq 1 GL, whil th rlvnt frmionic nrgis ntring th Coopron in Eq. 13 r 1/T. As rsult, nonlo rltions btwn Coopr mods nd ordr prmtr in Eqs. 10 nd 1 cn b pproximtd s 19 Wˇ tt r ˆ tt ˆ tt r, whr t RA ˆ R tt = t t 0 0 A t t,, ˆ tt r = r, t + t + + r, t + t, 30 r th rtrdd dvncd stp functions. Physily Eq. 30 implis tht Coopron is short rngd, hving chrctristic lngth s o = D/Tc, s comprd to th long-rngd fluctutions of th ordr prmtr, which propgts to th distncs of th ordr of GL = DGL o. Thus, rltions 1 r ffctivly lo, which simplifis furthr nlysis considrbly. Equtions 30 llow us to trc Kldysh subspc in Eq. 9 xplicitly to rriv t I J t = g T Tr t t 1 F t1 t t t ˆ tt L z ˆ t R t ivt+t z, 1 31 whr w hv usd Eq. 19 nd wrot trc in th rlspc rprsnttion not tht Tr... hr dos not imply tim t intgrtion. Chnging intgrtion vribls t 1 =t nd t 3 =t, nd rsing, in th units of tmprtur T,T, on finds for Eq. 31 n quivlnt rprsnttion, I J t = ig + T dd T sinh L Tr N ˆ t,t T R z ˆ t T,t iv t T, T 3 whr w usd quilibrium frmionic distribution function in th tim domin F t = it/sinhtt. Th most significnt contribution to th bov intgrls coms from 1. At this rng rtios,/t chng on th s of invrs tmprtur, whil s w lrdy discussd, ordr-prmtr vritions r st by t GL 1/T. Thus, prforming nd intgrtions on my nglct,/t dpndnc of th ordr prmtrs. As th rsult w find I J t = ig T 4T d r A Rr,t L r,t i J t c.c.. 33 Finlly w r rdy to cult corrsponding contribution to th currnt nois. On brings two currnts from Eq. 33 into Eq. 6 nd pirs fluctuting ordr prmtrs using corrltion function, r,t b r,t = i bl K r r,t t, 34 which follows from Eqs. 16 nd 18. As rsult, Josphson currnt corrction to th nois of wid symmtric junction is S J = 1 g T 4 4T c d + r dtl A K r,t i t, 35 whr = J. Corrsponding digrmmtic rprsnttion of Eq. 35 is shown in Fig. 1c. Rmining intgrtions in Eq. 35 cn b don in th osd form s Appndix A3 for dtils, providing S J = Gi 64T c g TT c A T o c T T c N GL, Nz = 4 z ln 1+z /4. 36 Anlogous cultion in th cs of th nrrow symmtric junction, which is obtind from Eq. 35 by rplcing L K r,t L K 0,t nd rmoving sptil intgrtion, givs for th nois spctrum s dtils in Appndix A3, S J = PHYSICAL REVIEW B 78, Gi 8T c g TT c Mz =1 + T c T T cm GL, lnxdx 1+x + z

6 ALEX LEVCHENKO PHYSICAL REVIEW B 78, In similr fshion on my considr nonsymmtric tunnl junction. Assum tht on of th lctrods is in th dp suprconducting stt, with wll dfind gp in th xcittion spctrum S, whil th othr is in th fluctuting rgim. W st thn on of th Qˇ mtrics to b suprconductiv Grn s function Qˇ L=Qˇ S, whr Qˇ S = Qˆ S R Qˆ S K 0 Qˆ S A, Qˆ S K = Qˆ S R Fˆ Fˆ Qˆ S A, 38 Fˆ =F z, nd Qˆ S RA = 1 i0 S S S, 39 whil xpnding th othr on in Coopr mods Qˇ R iqˇ NWˇ R. Th rsulting xprssion for th currnt rds I J t = g T 4 ˇ VˇQˇ V q t Tri S iˇ Vˇ Qˇ NWˇ R. 40 Following th sm stps s in th cs of th symmtric junction, crrying out diffrntition with th hlp of Eq. 19 nd trcing consquntly Kldysh nd Nmbu subspcs nd prforming tim intgrls, on finds for th currnt, ln I J t = ig T S d r 4 T A Rr,t i J t c.c., 41 whr w ssumd tht S T c. Squring Eq. 41 nd vrging ovr th ordr-prmtr fluctutions with th hlp of Eq. 34, wgt S J = i g T 4 ln S d + r dtl K r,t T A i t. 4 Prforming th rmining intgrtions, on finds nois spctrum of th nonsymmtric junction s corrsponding digrm in Fig. 1d, S J = 3 Gi 64T c g TT c T c T T c ln S o T A L GL, Lz = 1 1+z. 43 Spctrl lin shps for Eqs. 36, 37, nd 43 r plottd in Fig.. IV. DISCUSSIONS W hv considrd ffcts of suprconductiv fluctutions on th currnt nois in tunnl junctions bov th criti tmprtur. Svrl contributions wr idntifid. Th simplst on origints from th fluctution supprssion of FIG.. Color onlin Shp of th Josphson currnt nois spctrl lins in th vicinity of th rsonncs z= GL. th dnsity of stts. This ffct givs ngtiv contribution to th currnt nois, which is only logrithmic in tmprtur S DOS lnt T c, whrs dip in th dnsity of stts t th Frmi nrgy hs much strongr tmprtur dpndnc 0T T c. Somhow currnt nd its nois gt supprssd, wkr thn th dnsity of stts itslf. Anothr intrsting point is tht currnt nois is strongly modifid only t th chrctristic voltgs VT c, whil corrsponding ftur in th dnsity of stts pprs t nrgis T T c, s Eq. 50. It turns out tht highr-ordr fluctution ffcts, similr to tht shown in Fig. 1b, rstor dditionl structur of th nois signl t VT T c. Corrction S DOS is linr in Gi nd in tunnl conductnc g T. This is in contrst to th Josphson currnt contribution to th nois. Th lttr is qudrtic in fluctutions nd in tunnling, nd nhncs nois t th frquncis in th vicinity of th Josphson frquncy J. Th pk t = J is wll dfind nd is strongly tmprtur dpndnt, which mks it possibl to dtct it xprimntlly. W hv found tht dpnding on th junction configurtion: symmtric or nonsymmtric nd nrrow or wid, nois rsonnc lin hs diffrnt shps in th frquncy domin Fig. nd diffrnt tmprtur dpndncis. Closing this sction w should mntion tht in th fild of fluctuting suprconductivity on usully idntifis thr typs of fluctution corrctions. Aprt from dnsity of stts, thr r lso so ld Aslmzov-Lrkin AL nd Mki- Thompson MT trms mntiond in Sc. I. It is quit nturl to sk how AL nd MT procsss modify currnt nois nd how thy cn b idntifid within -modl formlism. As n ttmpt to nswr, on should rl tht in ddition to th simpl tunnling trm S T V, considrd in this work, on my hv yt nothr on is A V=g A /16 Tr iˇ VˇQˇ L iˇ VˇQˇ R, which ws nglctd. It corrsponds to Andrv procsss, nd g A is Andrv conductnc. Using S A V, instd of S T V, on my follow th sm routing xpnding Qˇ -mtrics in fluctutions Wˇ to obtin dditionl contribution to th nois. Howvr, mong ll th trms mrging in prturbtiv xpnsion, sprtion on AL nd MT contributions bcoms mbiguous. Nvrthlss, th problm is vry intrsting nd rquirs furthr studis

7 JOSEPHSON CURRENT NOISE ABOVE T c IN ACKNOWLEDGMENTS Mny usful discussion with A. Kmnv, M. Rznikov, A. Vrlmov, nd spcilly G. Ctlni r kindly cknowldgd. I would lik to thnk lso Digitl Mtril Lbortory t Th Institut of Physi nd Chmi Rsrch RIKEN, whr this work ws finlizd, for thir hospitlity. This work ws supportd by th NSF Grnt No. DMR APPENDIX: Fluctutions xpnsion Within this sction w show in dtils how th trnsformtion from Eq. 3 to Eq. 16 occurs. W strt fluctutions xpnsion by tking Qˇ Qˇ N1+iWˇ Wˇ / nd bringing it into th S hr nd in wht follows subscript in Qˇ mtrix nd ll othr lmnts will b supprssd for brvity. For th trc of th grdint trm w find, TrQˇ = TrQˇ NWˇ Qˇ NWˇ = TrWˇ Wˇ, whr w mployd nticommuttivity rltion Qˇ N,Wˇ + =0 nd nonlinr constrint Qˇ N=1. Using n xplicit form of th Wˇ mtrix Eq. 10 nd trcing th product of two Wˇ ovr th Kldysh Nmbu spc, w obtin TrQˇ =TrDq c qc q + c qc q. A1 Th tim drivtiv trm in th S producs contribution Trˇ t Qˇ = i TrŘ ˇ Ř ˇ Wˇ Wˇ, whil linr in Wˇ prt trcs out to zro hr w usd Qˇ N=Ř ˇ Ř with ˇ = z z nd substitutd t i.obsrving tht Ř ˇ Ř ˇ = z 0, on finds Trˇ t Qˇ = i Tr + c qc q c qc q. A For th coupling trm btwn Cooprons nd, to th lding ordr, w hv Trˇ Qˇ =TrŘ ˇ Ř ˇ Wˇ, which trnslts into q Trˇ Qˇ = i Tr q + F qc q + q + F q qc q q q F qc q q F q qc q. A3 Combining now Eqs ll togthr nd bringing thm bck into Eq. 3, w wind for th qudrtic in Cooprons prt of ction S W,=S c c,+s c c,, whr contributions from th rtrdd c nd dvncd c Cooprons rd s PHYSICAL REVIEW B 78, is c c, = Trc Dq i + c + q + F c + + F q c, A4 c is c, = Trc Dq + i + c q F c F q c. A4b At this stg w r rdy to prform intgrtion ovr th Coopron mods. Assuming tht configurtion of th ordrprmtr fild is givn, on vris Eq. 46 with rspct to c nd c, nd obtins sttionry point qutions S c /c c =0 nd S /c =0. Th lttr r sily solvd by Eq. 1. Sinc th vlu of th Gussin intgrl is qul to tht tkn t th sddl point, on brings Eq. 1 into Eq. 46 nd ftr som strightforwrd lgbr finds Eq. 16. Furthr dtils cn b found in Rf. 15. Rltion btwn S DOS nd () Th purpos of this sction is to dmonstrt xplicitly tht S DOS indd origints from th DOS ffcts, which ws hiddn in th tchni dtils of Sc. III. To this nd w cult tmprtur dpndnc of th within Kldysh tchniqu. This illustrtion is usful for th sk of comprison with th known rsults obtind prviously from th tmprtur Mtsubr tchniqu. 16 Within -modl nrgy dpndnt dnsity of stts is xprssd in trms of Qˇ mtrix in th following wy: = 4 TrQˇ NQˇ. A5 Stting Qˇ =Qˇ N on rcovrs br norml-mtl dnsity of stts =. To ccount for th fluctutions on top of th mtllic stt, on xpnds Qˇ in Coopr mods Wˇ to th qudrtic ordr nd vrgs ovr fluctutions with th ffctiv ction from Eq. 16; = 4 Trc qc q + c qc q. A6 Obsrv tht this is prcisly th sm combintion of th Cooprons, which ntrs S DOS in th Eq., thus thy hv common origin. Furthrmor, it is sy to show tht S DOS df +u /T F u /T. Using vrgs from Eq. 4, dnsity-of-stts corrction bcoms + d L K q, + F L R q, =Im q Dq i + i. A7 whr w st =. Hr on mts th convninc of th Kldysh tchniqu, which llows us to gt physi qun

8 ALEX LEVCHENKO titis voiding nlytic continution procdur. Using xplicit form of fluctutions propgtors from Eq. 18 nd prforming frquncy nd momntum intgrtions, on finds in th qusi-two-dimnsionl cs, = Gi 16 T c whr dimnsionlss function is Fz =R0 + T T c F GL, dx 1+x1+x iz. A8 A8b In grmnt with Rf. 16 dip t th Frmi nrgy is 0 T T c, whil t lrg nrgis GL 1 dnsity-of-stts corrction rcovrs its norml vlu ccording to T c / ln GL. Intgrls for S DOS () nd S J () I Trnsformtion from Eq. 6 to Eq. 7 rquirs cultion of th intgrl, I =0 + + dx On prforms y intgrtion first, I = 0 + F z+u /T F z u /T dydz R x + + y x + iy 4iz. + dx A9 dz R F z+u /T F z u /T x + +x 4iz. A10 Sinc 1 nd rlvnt z1 on my sfly pproximt +x 4izx iz. Thn xpnding F z into th sris F z = n z/z +z n, with z n =n+1/, intrchnging ordr of summtion nd intgrtion nd rling dfinition of th nth-ordr drivtiv of th digmm function n z = 1 n+1 n! n=0 1/n+z n+1, on finds tht + dz F 1 zu /T x iz = i 1 iu T +. x A11 Rmining x intgrtion cn b tkn with logrithmic ccurcy, ignoring x dpndnc of th digmm function sinc only x1 contribut significntly, which vntully givs ln. Combining ll togthr, on finds 1 I = 1 4 ln1/im 1 iu A1 T, which in combintion with Eq. 6 rsults in Eq. 7. II Trnsition from Eq. 35 to Eq. 36 is prformd in th following wy: As th first stp on finds Kldysh componnt of th fluctution propgtor in th mixd momntum/tim rprsnttion L K q,t=l K q, it d/, which givs L K q,t = it c qt/gl, q = GL q +1. T T c q A13 On insrts thn L K r,t=l K q,t iqr dq /4 into Eq. 35, intgrts ovr r, introducs dimnsionlss tim =t/ GL, nd chngs from q to intgrtion dq =d/ GL, which givs ll togthr S J = Gi 64T c g TT c + d1 + d A T o c T T c iz, whr z = GL. Aftr intgrtion on is lft with 1 4d 4 + z, A14 A15 which dfins Nz function in Eq. 36. III Clcultion of Eq. 37 is compltly nlogous. Noticing tht L K 0,t=L K q,tdq /4 nd trnsforming to th dimnsionlss units =t/ GL nd = GL q +1, w hv S J = Gi GL 4 + g TT c d1 Aftr intgrtion on is lft with 1 PHYSICAL REVIEW B 78, dd z = z 1 + dd + iz. + d A16 1+ rccot, z A17 which ftr th intgrtion by prts rducs to Mz, with Mz function dfind by Eq A. I. Lrkin nd A. Vrlmov, Thory of Fluctutions in Suprconductors Clrndon, Oxford, 005. L. G. Aslmzov nd A. I. Lrkin, Fiz. Tvrd. Tl Lningrd 10, Sov. Phys. Solid Stt 10, K. Mki, Prog. Thor. Phys. 39, I. O. Kulik, Zh. Eksp. Tor. Fiz. Pis m Rd. 10, JETP Ltt. 10, D. J. Spino, Phys. Rv. Ltt. 4, A. A. Vrlmov, G. Blstrino, E. Milni, nd D. V. Livnov, Adv. Phys. 48,

9 JOSEPHSON CURRENT NOISE ABOVE T c IN 7 I. Mrtin nd A. Bltsky, Phys. Rv. B 6, R Xi Di, To Xing, Ti Ki Ng, nd Zho-bin Su, Phys. Rv. Ltt. 85, V. V. Dorin nd M. V. Fistul, Phys. Rv. B 46, E. Shimshoni, P. M. Goldbrt, nd N. Goldnfld, Phys. Rv. B 48, M. Rznikov privt communiction. 1 A. Kmnv nd A. Andrv, Phys. Rv. B 60, M. V. Figl mn, A. I. Lrkin, nd M. A. Skvortsov, Phys. Rv. B 61, A. Kmnv, in Nnophysics: Cohrnc nd Trnsport, ditd by H. Bouchit, Y. Gfn, S. Guron, nd G. Montmbux Elsvir, Nw York, 005, p PHYSICAL REVIEW B 78, A. Lvchnko nd A. Kmnv, Phys. Rv. B 76, E. Abrhms, M. Rdi, nd J. W. Woo, Phys. Rv. B 1, A. A. Vrlmov nd V. V. Dorin, Zh. Eksp. Tor. Fiz. 84, Sov. Phys. JETP 57, Hr dditionl subscript R in th nottions of mtrix Wˇ R ws supprssd for brvity. According to th ssumption Wˇ L =0 for th nonsymmtric configurtion. 19 Driving Eq. 30 w hv ignord quntum componnt of th ordr prmtr q r,t. Although formly prsnt it givs t th nd sublding contribution to th nois spctrum