BCS superconductivity near the band edge: Exact results for one and several bands

1.4 MB·

by  D. Valentinis

BCS superconductivity near the band edge: Exact results for one and several bands
D. Valentinis, D. van der Marel, and C. Berthod
∗
Department of Quantum Matter Physics (DQMP), University of Geneva,
24 quai Ernest-Ansermet, 1211 Geneva 4, Switzerland
(Dated:January19,2016)
WerevisittheproblemofaBCSsuperconductorintheregimewheretheFermienergyissmallerthanthe
Debyeenergy.Thisregimeisrelevantforlow-densitysuperconductorssuchasSrTiO thatarenotintheBEC
3
limit,aswellasintheproblemof“shaperesonances”associatedwiththeconfinementofathree-dimensional
superconductor.Whiletheproblemisnotnew,exactresultswerelackinginthelow-densitylimit.Intwo
dimensions,wefindthattheinitialriseofthepairingtemperatureT atlowdensitynisnonanalyticand
c
fasterthananypowerofn. Inthreedimensions,wealsofindthat T isnonanalytic,butstartswithzero
c
slopeatweakcouplingandinfiniteslopeatstrongcoupling. Self-consistenttreatmentofthechemical
potentialandenergydependenceofthedensityofstatesarecrucialingredientstoobtaintheseresults.
Wealsopresentexactresultsformultibandsystemsandconfirmouranalyticalexpressionsbynumerical
simulations.
6
1
PACSnumbers:74.20.Fg,74.62.Yb,74.70.Ad
0
2

l I. INTRODUCTION per we focus on the band-edge effect on T in the bulk,
u c
emphasizingthegenericbehaviorsinthesimplecaseofan
J
electrongaswithparabolicdispersionandalocalattraction.
5 TheBardeen–Cooper–Schrieffer(BCS)theory1 remains
WerecovertheexpressionsofEagles2 intheweak-coupling
1 theonlystrongmicroscopicfoundationtosupportourunder-
limit. Inthelow-densityregime,weprovideexactrelations
standingofthefascinatingphenomenonofsuperconductiv-
] as a function of the density, which are valid at arbitrary
n ity. Amongmanyotherinsights,thetheoryprovidesasimple coupling. We also give exact numerical results in 2D and
o expressionforthecriticaltemperature T ,whichcontinues
c 3D,forone-bandandmultibandsystems. Theimplications
c to inspire the search for materials with improved perfor-
fortheproblemofquasi-2Dshaperesonancesandthecase
-
mances. Inparticular,itisexpectedthatsuperconductivity
r ofLaAlO /SrTiO ,willbereportedinseparatepublications.
p isfavoredbyalowdimensionalityduetoenhanceddensity 3 3
u ofstates(DOS)attheFermilevel.2 Forthree-dimensional Our starting point is the mean-field theory for a
s momentum-independentpairinginteractionactinginalim-
(3D)materials,anearlyproposaltousequantumconfine-
at. mentinathinfilm3 hasreceivedsustainedattentionuntil ited energy range around the Fermi surface. This theory
yields a pairing temperature which is in general higher
m recently.4 Theconfinement-inducedtwo-dimensional(2D)
thanthetemperatureofsuperconductingcoherence,espe-
subbands produce discontinuities in the DOS and abrupt
- cially when the dimensionality and/or the density is low
d changes of T as a function of film thickness have been
c and superconducting fluctuations become important.14–16
n routinelypredicted.
o We ignore these fluctuations and focus on the mean-field
c Thepurposeofthisstudyistoexploresomeconsequences equations,refrainingfrommakinganyapproximationwhen
[ of an aspect of the problem, considered by Eagles half a solving them for T . This approach is similar to previous
centuryago,2,5 butoftenoverlookedinrecentcalculations c
2
basedontheBCSgapequation. AstheFermienergycrosses
v
1 theedgeofaband,thereisaregimewherethedynamical 2ħhωD
2 cutoff of the pairing interaction is controlled by the band
5 edge(Fig.1). Thisregimeisrealizedinlow-densityelectron N0(E)
4 gases,whentheFermienergyissmallerthanthedynamical
0 rangeoftheinteraction. IndopedSrTiO ,forinstance,the (a)
1. carrierconcentrationistypically1019 cm3 3 andthecarrier EF E
−
0 massisintherange2–4electronicmasses,6 corresponding
6 toaFermitemperatureof50–100K,whiletheDebyetem- 2ħhωD N0(E)
1 peratureis513K.7Inthissituation,thecommonapproxima-
:
v tionoftakingaconstantDOSoverthefulldynamicalrange
Xi fails to give a good estimate for Tc. The near-band edge (b)
E
regime is also relevant in the quasi-2D problem of shape E
r F
a resonances,sinceeachresonanceisduetotheFermienergy
crossingasubbandedge.8–10 Thepairinginthatsubband, FIG.1. Schematicrepresentationof(a)thehigh-densityregime
aswellastheinter-subbandpairinginvolvingthatsubband, and(b)thelow-densityregimeforsuperconductingpairing. In
aredominatedbythebandedge. Asynthesisofthesetwo theformer,EF ħhωDandthedensityofstatesN0(E)canbetaken
casesis realizedin thequasi-2Dandlow-densityelectron constant.Inth(cid:29)elatter,EF(cid:174)ħhωD,theinteractioniscutbytheband
gasattheLaAlO /SrTiO interface.11–13 Inthepresentpa- edge,andthedetailsofN0(E)matter.
3 3
2
mean-field studies of the BCS-BEC crossover where the toanylocalinteractionwithadynamicalcutoff. Inalattice
renormalizedchemicalpotentialissolvedself-consistently version, for instance, the cutoff could be the bandwidth.
togetherwiththe Tc equationorthegapequationatzero N0β(E) is the DOS per spin and per unit volume for the
temperature.17,18 band β. It is defined on an absolute energy scale, such
Anexactsolutionofthegapequationrequiresonetotake thatN0β(µ)istheDOSatthechemicalpotentialµ,which
intoaccounttheenergydependenceoftheDOS,mostimpor- is common to all bands. The chemical potential must be
tantlythecutoffatthebandbottom,andthetemperature adjustedtofixthedensityaccordingto
dependenceofthechemicalpotentialµ,whichiscrucialat
low-densityn. Becausen, T ,andµallapproachzerosimul- ∞
c n=2 dE f(E)N(E). (1b)
taneously, it is essential to use the exact relation µ(n,Tc)
(cid:90)
in order to capture the correct behavior of Tc for n 0. −∞
Furthermore,oneshouldnotassumeweakcouplingan→d/or Here, f(E) = [e(E µ)/kBT +1] 1 is the Fermi distribution
− −
assume that Tc is small with respect to the Fermi energy function and N(E) is the total BCS density of states (per
and the cutoff for pairing. As a matter of fact, analytical spin) resulting from the opening of the superconducting
resultsinthisproblemarerare. InRef.19,rigorousbounds gapsatthechemicalpotentialineachband.
for T wereobtainedforageneralinteraction. Theseresults For the calculation of T , it is sufficient to consider the
c c
are limited to weak coupling and to a positive chemical two equations in the limit of vanishing order parameters.
potential. We will see that the chemical potential at Tc is For T =Tc wehave
negativeinthelow-densitylimitin2Dforanycouplingand
in3Dforcouplingslargerthanacriticalvalue. Exactresults ħhωD tanh E
2hDav.2e0aHlsoowbeeveenr,rseinpcoerttehdefuonritvheerszaelrBoC-tSemgappertaotuTrcergaatipoiins ∆α=(cid:88)β Vαβ∆β(cid:90)−ħhωDdEN0β(µ+E) (cid:16)2E2kBTc(cid:17) (2a)
notobeyedinthelow-densitylimit,theseresultscannotbe
∞
usedtodeduce Tc. n=2 dE f(E) N0β(E). (2b)
Thispaperisthefirstinaseriesanditprovidesthemath- (cid:90) β
−∞ (cid:88)
ematical foundations for subsequent studies dedicated to
Wenowinsertexplicitformulasfortheenergy-dependent
shape resonances in thin films and to the LaAlO /SrTiO
3 3 densitiesofstatesandthedensityandwerewritetheequa-
interface. It is organized as follows. In Sec. II we recall
tions(2)inadimensionlessform,whichismoreconvenient
thebasiccoupledequationsgiving nand T andwewrite
c for analytical and numerical treatments. The densities of
theminadimensionlessform,foroneandseveralparabolic
statesforaparabolicbandindimensionsd=2andd=3
bands. InSec.IIIwepresentouranalyticalandnumerical
aregivenby
resultsforonebandin2Dand3D,andinSec.IVwebriefly
discussmultibandeffects. mβ d2 d 1
N0β(E)=(d−1)π 2π2ħh2 θ(E−E0β) E−E0β 2− ,
(cid:18) (cid:19) (3)
II. BCST EQUATIONFORMULTIBANDSYSTEMS (cid:128) (cid:138)
c where m isthebandmass, E istheenergyoftheband
β 0β
minimum,andθ istheHeavisidefunction. Thisdefinition
A. Dimensionlessequationsforthepairingtemperature ensuresthat N0β(µ)istheDOSevaluatedatthechemical
potentialµcommontoallbands,consistentlywithEq.(1a).
We consider a multiband metal with a local BCS pair- Therelationbetweendensity,chemicalpotential,andtem-
inginteraction V actingbetweenelectronsofopposite peratureforaparabolicbandinarbitrarydimensiond is
αβ
momenta and s−pins in bands α and β.21 We assume that
d
Cbaonodp,elreapdaiinrigngbeolocwcutrhseopnalyirifnogr ttewmopeelreactturroenTscintothaneosardmeer n=−2 m2πkBħhT2 2 Lid2 −eµk−BET0 , (4)
parameter∆α ineachband. Thisincludesthepossibilityof (cid:18) (cid:19) (cid:129) (cid:139)
a“proximity”inducedgap∆β inabandthatotherwisefeels where Lip(x) is the polylogarithm given by the series ex-
nopairingpotential(Vββ =0),viathenonzerointerband pansion Lip(x) = ∞q=1xq/qp. This function has the sign
interactionsVαβ. Themean-fieldgapequationfor∆α is of its argument and reduces to a usual logarithm in two
dimensions (p = 1(cid:80)): Li1(x) = ln(1 x). We provide a
ħhωD tanh (cid:198)2ξk2B+T∆2β breraiedfedr.erivationofEq.(4)inAp−pendix−Afortheinterested
∆α=(cid:88)β Vαβ∆β(cid:90)−ħhωDdξN0β(µ+ξ) 2 (cid:18)ξ2+∆2β(cid:19). (1a) denWseitymineausnuirtesoafll2[emneωrgDi/e(s2πinħh)u]nd/i2tswohferħheωmD,iseaxprerefesrsenthcee
mass,andwedistinguishthedimensionlessvariableswith
(cid:198)
The pairing interaction acts in a range ħhωD around the tildes,e.g.,
chemical potential µ. Although the not±ationħhωD is used
hreesruel,tswdeoennovtisrieoqnuitrheephporonbolne-mmeindiaittsedgepnaeirrianlgit,ybauntdapopulyr T˜c = ħkhBωTDc, µ˜= ħhωµD, n˜= 2[mωD/n(2πħh)]d/2, etc.
3
Thecoupledequations(2)for T become B. Propertiesofthefunctionsψ (a,b)
c d
∆α= ∆βλ¯αβψd 1+µ˜ E˜0β,T˜c (5a)
−
β
(cid:88) (cid:128) (cid:138)
n˜=−T˜cd2 (cid:88)β (cid:129)mmβ(cid:139)d2 Lid2 (cid:18)−eµ˜−T˜E˜c0β(cid:19). (5b) strTohnegefsutnscttriuocntsurψedd(eav,ebl)opasreardoisupnldayaed=221in, wFihgi.ch2.coTrhree-
sponds physically to having the chemical potential at the
Wehaveintroducedthedimensionlessfunction,
bottomofoneband. Wearemostlyinterestedinthebehav-
1 tanh x iorfor b 1,whichisexploredintheregimekBTc ħhωD
ψd(a,b)=θ(a)(cid:90)1 mind(ax,2()x+a−1)d2−1 2(cid:128)x2b(cid:138), (6) athneddpeanrstii(cid:28)ctyulaaprlpyroinacthheeslizmerito.bI→fa0<,w1,hψichd(ias,rbe)leivsafinn(cid:28)tiwtehfeonr
−
aswellasthecouplingconstants,
m d
λ¯αβ =Vαβ(d−1)π 2π2βħh2 2 ħhωD d2−1. (7)
(cid:18) (cid:19)
(cid:0) (cid:1)
Weuseabartorecallthatthesecouplingconstantsarenot
evaluatedattheFermienergylikeinthecommonpractice,
butatanenergyħhωDabovethebottomofeachband: λ¯αβ =
VαβN0β(E0β+ħhωD). Thischoiceisnaturalandleadstothe
simplestequations. Theusualdefinitionλ=VN0(µ)poses
problems when µ lies below the band bottom and more
generallybecauseµisafunctionofinteractionstrengthand
temperature.
In an N-band system, the relations (5) provide N +1
equationsfortheN+1unknowns,whichare T˜c,µ˜,andthe
N 1ratios∆β/∆1. Wecanassumethat∆1=0without
los−sofgenerality,becausethereisatleastonen(cid:54) onzerogap
parameterat T andwearefreetonumberthebandssuch
c
that∆1 isthisone. WenoweliminatetheN 1gapratios
andreducetheproblemtoapairofequation−sfor T˜ andµ˜.
c
Withthenewdefinitions rβ =∆β/∆1 and
Λαβ(µ˜,T˜c)=λ¯αβψd(1+µ˜ E˜0β,T˜c), (8)
−
thesetofN equations(5a)becomestheeigenvalueproblem
Λr = r with r = (1,r2,...,rN). This means that, when
evaluatedatavalueof T˜c solvingEq.(5a),thematrixΛhas
atleastoneuniteigenvalue. Inotherwords, T˜ corresponds
c
to the largest temperature that satisfies the characteristic
equationdet(11 Λ)=0. Thetwocoupleddimensionless
equationsgiving−nand T forN bandsaretherefore
c
0=det[11 Λ(µ˜,T˜c)] (9a)
−
n˜= T˜cd2 N mα d2 Lid eµ˜−T˜E˜c0α . (9b)
− (cid:88)α=1(cid:129) m (cid:139) 2 (cid:18)− (cid:19)
The existence of a nontrivial solution to Eq. (5a) clearly
impliesEq.(9a). Theconverseisalsotrue: Thevanishingof FIG.2. Representationofthefunctionsψd(a,b)definedinEq.(6)
fordimensionsd=2(top)andd=3(bottom).Physically,thea
thedeterminantinEq.(9a)issufficienttoenforcethatthe axiscorrespondstovaryingµaroundthebandbottom(a=1)and
matrixΛhasoneuniteigenvalue,whichprovidesasolution the baxisisproportionaltoT.Thebluelinesshowthebehavior
c
toEq.(5a). Theequations(9)havethesamestructurein for b=0and a<1. Thetworedlinesineachgraphshowthe
2D and 3D, the quantitative differences stemming mostly asymptotic bdependenciesfor1<a<2anda>2,respectively.
fromdifferentfunctionsψd(a,b). Inthenextparagraphwe The green lines show cuts at the value ψd(a,b) = 2.5, which
discussthepropertiesofthesefunctions,whichweshalluse correspondtothepathfollowedinthe(a,b)planebythesolution
inthefollowingsectionstoderiveanalyticalresults. oftheBCSequations(9)foronebandandforλ¯=0.4.
4
b=0. Thelimitingvalueisgivenby andintheregimeµ>ħhωD:
ψd(0<a<1,b 0)= 1b dx (bx+a2x−1)d2−1 ψ3(a>2,b→0)=(cid:112)a+ a−2+ a−1
→ (cid:90)1−ba ln a−1 (cid:112)a(cid:112)−1−(cid:112)a(cid:112)−28eγ−2 . (14)
ln(cid:112)1 a (d=2) × (cid:112)a+(cid:112)a 1(cid:200)(cid:112)a 1+(cid:112)a 2 πb 
=− − (10)  − − − 
(cid:112)a−(cid:112)1−asin−1((cid:112)a) (d=3). Tasheresedalisnyems.pLtoatsitclyb,einhatvhieorhsigahr-edienndsiictay,tehdigihn-TFigs.e2ct(obrotato>m2)
c
These limiting behaviors are indicated on the graphs as and b , the function reduces simply to ψd(a,b) =
bblue0li,nbesu.tIifnadi>ffe1r,enψtdw(aa,ybs)indivtheergtewsolorgaanrgitehsm1ic<alaly<fo2r (aF−ig1u)rd→e/22−∞1a/ls(o2bsh).owsaparticularcutatthevalueψd =2.5.
an→d a > 2. The former range corresponds physically to SincetheBCSequation(9a)foronebandissimplyψd =
0fo<rthµe<paħhiωrinDg,siunctehrathctaitonth,ewbhailnedtheedglaettseertsratnhgeeloiswtehrecuustuoaffl 1(1/λ¯+,µ˜th,Te˜cs)efcourtλ¯s =sho0w.4.thNeotleocthuastotfhethaeppsoroluxtiimonastio(an,sb()11=)
to(14)showninredunderestimatethefunctionψ atlow
regime, where the Fermi energy is larger than the Debye d
b;usingtheminsteadoftheexactfunctionsthusleadsto
energy. Intwodimensions,theasymptoticbehaviorisquite
underestimating T .
simple: Ifa>2,wehavethewell-knownresult, c
1
b tanh(x/2) 2eγ
ψ2(a>2,b 0)= dx 2x =ln πb , (11) III. ONEPARABOLICBANDIN2DAND3D
→ 1
(cid:90)−b (cid:18) (cid:19)
A. Analyticalresults
with γ 0.577 the Euler constant. The function is inde-
penden≈tof a,becausetheDOSisconstantovertherange
Forasingleband,weplacetheoriginofenergyatthebot-
of integration when µ >ħhωD. If 1 < a < 2, we evaluate tomofthebandandweusethebandmassasthereference
thefunctionbyextendingtheintegraltoreproducethecase
mass. Thecoupledequations(9)for T becomesimply:
a>2andsubtractingthedifference: c
d
1b tanh(x/2) 1−ba 1 1=λ¯ψd 1+µ˜,T˜c , n˜=−T˜c2 Lid2 −eµ˜/T˜c . (15)
ψ2(1<a<2,b→0)= 1 dx 2x − 1 dx −2x In2Dtherela(cid:128)tionbetw(cid:138)eenn˜andµ˜ canb(cid:16)etriviall(cid:17)yinverted
(cid:90)−b 2e(cid:90)γ−b andthetwoequationsreducetoasingleimplicitrelation
=ln a 1 πb . (12) for T˜c asafunctionofn˜andλ¯:
−
(cid:18) (cid:19)
Equations(11)and(12)arerepresen(cid:112)tedinFig.2(top)as 1=λ¯ψ2 1+T˜cln en˜/T˜c 1 ,T˜c . (16)
−
redlines. Inordertoobtaintheexactasymptoticbehavior
(cid:16) (cid:17)
for b 0inthreedimensions, weintroducethefunction Atnottoolowdensity,weseef(cid:0)romthea(cid:1)symptoticexpres-
t(x) a→s a piece-wise linear approximation of tanh(x/2)— sionsindicatedinFig.2thatthepairingtemperaturecrosses
namely, 1for x < 2, x/2for x <2,and+1for x >2— overbetweentworegimesatµ˜=1. In2Dwehave
and we−calculate an−alytically th|e|integral with tanh(x/2)
rmeaptlaiocnedanbdy tth(xe)e.xTahcetrdeisffuelrteinscebetweenthelatterapproxi- 2eγ 1 µ˜ µ˜(cid:174)1
T˜c ≈ π exp −λ¯ ×1(cid:112) µ˜>1 (d=2). (17)
1 (cid:18) (cid:19)
b tanh(x/2) t(x) 
lim dx bx+a 1 −
b→0(cid:90)1−mibn(a,2) (cid:112) − 2x T˜c is independent of µ˜ (hence of n˜) for µ˜ > 1, due to
4eγ 1 theconstantDOSandtheconventionalBCSexpressionis
= a 1ln − . recovered. In3Dwefind
π
−
(cid:130) (cid:140)
(cid:112)
Expandingthenonsingulartermstoleadingorderin b,we 8eγ 2 1 e(cid:112)1+1/µ˜
T˜ − µ˜ exp
finallygetintheregime0<µ<ħhωD: c ≈ π −λ¯ µ˜1+ 1+1/µ˜
(cid:112)
ψ3(1<a<2,b 0) 1  (cid:112) µ˜(cid:174)1(cid:112)
→
a 1 8eγ 2  (d=3). (18)
=(cid:112)a+ a 1ln − − , (13) × e(cid:112)1 1/µ˜ 1 (cid:112)1 1/µ˜ µ˜>1
− (cid:130)(cid:112)a+(cid:112)a 1 πb (cid:140)  − 1−+(cid:112)1−1/µ˜
(cid:112) − (cid:199) −

5
The product λ¯ µ˜ is the coupling evaluated at the chem- toleadingorderin1 a. Thisyields
ical potential, which enters the exponential as expected. −
ERreqegfu.iam2tieiofnowsfev(1aa7ldi)dm(cid:112)iatintydtoh(fa1tt8hk)eBsaTercee=xidp(ereneγst/isπciao)ln∆tso,,wbEuhqtsic.nh(o2its)itanrnutdehe(in3l)othwoef- T˜c ≈n˜32 exp(cid:150)W(cid:130)8(13/πλ¯2−n˜231)2(cid:140)(cid:153)
density and/or strong-coupling regimes (see below). We (d=3,λ¯>1,n 0). (20)
emphasizethattheseapproximationsresultfromexpanding →
tthheerfeufonrcetiaocncuψrda(tea,obn)lyininththeelilmimititbT→00foartafin>it1eapnodsitaivree Linikcreeainse2sDf,asTtcersttahratns wanityhpaonwienrfinoiftensilfoλp¯e>at1n. =If λ¯0 a<nd1
c
µ. These equations are not accurate→in the high-density there is no finite solution a to the equation ψ3(a,0) =
regimewhere T islarge. Ournumericalresultsshowthat 1/λ¯, meaning that µ = 0 at zero density. As can be seen
c
Eqs.(17)and(18)providearatherpoorapproximationas in Fig. 2, the curvature along the cut for ψ3 > 1 is such
soonas Tc reachesafewtenthsofħhωD. that µ˜ > T˜c. In the limit T˜c 0 we can use the large-x
expansion Li3/2( ex) 4/→(3(cid:112)π)x3/2 and recover µ˜ =
Wenowturntothelow-densityregion. In2Danycutof (3(cid:112)πn˜/4)2/−3,whic−histh→ezero-temperaturenoninteracting
thefunctionψ2(a,b)atthevalue1/λ¯ convergesat b=0 result. Usingtheasymptoticform(13)wefinallyobtain
to a value a<1, given by the relation ln(cid:112)1 a=1/λ¯
(fipsrneoietaecFhingee.sg2az)te.irvoHe.evnTachlueisethiµ˜semacinhce=omn−jiuceag−la2tp/eλo¯dtweenhftf−eieanclttchooefntvd−heeerngpseiastyirtioanpga- T˜c = 8eπγ−2 (cid:130)3(cid:112)4πn˜(cid:140)23 exp−(cid:18)λ1¯ −1(cid:19)(cid:18)3(cid:112)4πn˜(cid:19)13
interaction and the DOS discontinuity: At any finite cou- (d=3,λ¯<1,n 0). (21)
plingthemomentumdistributionisspreadandanegative
→
chemical potential leads to a finite density even at zero
Thisfunctionstartswithzeroslopeatn=0andincreases
temperature. Thechemicalpotentialatzerodensityisre-
slower than any power of n. It is exactly equivalent to
lµ˜amteind=toE˜tbh/e(2en+erE˜gby).EbEqoufathtieontw(o1-6p)arftoircleT˜cboun0dbsetcaotembeys wthaeveresscualttt1e7riknBgTlcen=gt(h8aeγ−is2/cπom)EpFuetxepd[wπ/it(h2koFuarsi)n]teifratchteions-
1 = λ¯ln T˜cln(en˜/T˜c 1), which can be→solved for n˜ potential, namely 4πħh2ass/m = V/(λ¯ 1). This potential
Tasheal−fautntecrtie(cid:112)oxnp−roefssT˜ico:nn˜sh=ow−T˜scltnh{a1t+n˜iesxspm[−alelexrpt(h−a2n/Tλ˜¯c)/wT˜hce]n}. whahsynµob=ou0ndasttazteerofordetwnsoitpyaritnicltehsisifcλ¯a−s<e.1,Twhheicchheaxnpgleainosf
bothapproachzero,suchthatinthislimitwecanreplace behavioratλ¯ =1isdiscontinuousaccordingtoEqs.(20)
ln(en˜/T˜c 1)byln(n˜/T˜c). Wethusfindthesolution, and (21), both functions giving T˜c n˜2/3 with different
− pre-factors. ∝
Theanalyticalexpressions(17)–(21)arecomparedbelow
withthenumericalresults. Notethatifthereferencemassis
T˜c =n˜exp W e−n˜2/λ¯ (d=2,n 0). (19) ninotthtehseebeaqnudatmioansss.mα,onemustreplacen˜byn˜(m/mα)d/2
 (cid:32) (cid:33) →
AstheBCSmean-fieldtheoryisnotbelievedtobeauseful
  modelin1D,wehavenotdiscussedthiscase. Forcomplete-
ness, and because it has been argued that the singularity
W(x) is the Lambert function (or “product logarithm”), of the 1D DOS could induce large enhancements of Tc in
whichgivestheprincipalsolutionoftheequation x =WeW. stripedquasi-1Dsuperconductors,8 weshowinAppendixB
Equation(19)isnonanalyticinbothλ¯ andn˜. Itgivesa T thatthepairingtemperatureisalsocontinuousandnonana-
c
startingwithaninfiniteslopeatn=0andincreasingfaster lyticatthebottomofa1Dband.
thananypowerofn(inthesensethattherunningexponent Beforeclosingthissection,wepointoutthatthesolution
given by the logarithmic derivative approaches zero for ofthegapequationatT =0doesnotgenerallyallowoneto
n 0). An approximation of (19) valid to logarithmic deduce T . Althoughthefocusofthepresentpaperison T ,
c c
acc→uracywasgivenearlier.24 wegiveinAppendixCexactresultsforthezero-temperature
gapin 2Dat lowdensity, forthe purposeof showingthat
In 3D the function ψ3(a < 1,0) approaches 1 for theusualBCSgapto Tc ratioisnotobeyedinthislimit.
a 1. Therefore we have the same situation as in
2D→if λ¯ > 1. In this case the chemical potential ap-
proaches a finite negative value given by the solution of
B. Numericalresults
µ˜min+1 µ˜minsin−1 µ˜min+1 =1/λ¯ astheden-
sity approa−ches−zero. Since µ˜ is finite and negative in ThenumericalsolutionofEq.(16)isshowninFig.3. T˜
(cid:112)the limit T˜ (cid:112)0, we can (cid:0)u(cid:112)se the asy(cid:1)mptotic expression c
c reachesaplateauathighdensityduetotheconstantDOSof
Li3/2( ex) →ex for x andgetthechemicalpoten- theband. Forλ¯ oforderone,thevalue T ontheplateau
−tialµ˜=−T˜cln(→n˜/T˜c3/2). E→qua−ti∞on(15)canthenbesolvedfor departs significantly from the approximca,∞te solution (17),
T˜ intherelevantregime µ˜ 1,bymakinguseofEq.(10) whichbecomesworsewithincreasingλ¯,whilethesimple
c
− (cid:28)
6
0.5 0.8
(a) λ¯=1 (a) ¯λ = 1.5
0.4
0.6
n˜∞ T˜c, ¯λ = 1
T˜c 0.3 λ¯=0.75 ∞ T˜c 0.4 λ¯ = 0.75
0.2 λ¯=0.5 λ¯ =0.5
0.2
0.1
λ¯=0.3 λ¯ =0.3
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
n˜ n˜
100 100
(b) (b)
λ¯ =1.5
10 2
−
10 1
− )
˜T/c,∞ λ¯=1 1 ˜T1/)(c10−4
˜Tc 10 2 T˜c,∞ ˜T˜n(c ¯λ = 1
−
10 6
−
λ¯=0.3 00 1 λ¯ 2 λ¯=0.3
10 3 10 8
−10−8 10−6 10−4 10−2 100 −10−8 10−6 10−4 10−2 100
n˜/n˜ n˜
∞
FIG.3. (a)Pairingtemperatureasafunctionofelectrondensity FIG.4. (a)Pairingtemperatureasafunctionofelectrondensity
foroneparabolicbandintwodimensions. T isexpressedinunits for one parabolic band in three dimensions. T is expressed in
c c
ofħhωD/kBandninunitsofmωD/(πħh).Thethinhorizontallines unitsofħhωD/kBandninunitsof2[mωD/(2πħh)]3/2.Thethinand
showtheapproximatesolution(17)foreachλ¯.Theverticalbars dashedlinesshowEq.(18),evaluatedusingµ˜0=(3(cid:112)πn˜/4)2/3for
indicaten˜ [Eq.(22)]. Thedashedlinesshowtheapproximate µ˜. Theverticalbarsindicateµ˜0=1. (b)Samedataonalog-log
scaling T˜c∞= T˜c, (n˜/n˜ )1/2. (b) Same data normalized. The scale. ThedashedlinesshowEqs.(20)and(21). Equation(21)
whitedashedlin∞esare∞thepredictionofEq.(19)andtheblack wasusedforλ¯ =1. Theshort-dashedwhitelineforλ¯ =1.5is
dashedlineindicatesthesquare-rootbehaviorforn˜(cid:174)n˜ .(Inset) obtainedwithoutexpandingEq.(10)arounda=1(seetext).
Maximum pairing temperature as a function of λ¯ (so∞lid line),
comparedwithEq.(17)(dotted)andλ¯/2(dashed).
Eq.(19)vanishescontinuouslyforn˜ 0,thecorrectpicture
isthatofa T tendingasymptotical→lytoadiscontinuityof
large-Tc result T˜c, = λ¯/2 becomes increasingly reliable sizezerowithc decreasingn.
[insetofFig.3(b)]∞. Thedensityn˜ atwhichtheplateauis The numerical results for the 3D case are displayed in
reachedcorrespondstoµ ħhωD co∞incidingwiththebottom Fig.4. Alsoshownisthehigh-densityapproximation(18),
oftheband,whichmeans−: evaluated with µ˜ replaced by its zero-temperature nonin-
teracting value µ˜ . The approximation falls on top of the
n˜ =T˜c, ln e1/T˜c,∞+1 . (22) numerical data fo0r small λ¯, but deviates significantly for
∞ ∞ largercoupling. Thegoodagreementatweakcouplingis
(cid:16) (cid:17)
For n˜ (cid:174) n˜ , Eq. (17) gives T˜c/T˜c, µ˜1/2. Since µ˜ is due to a cancellation of errors: the agreement worsens if
very close∞to a linear function of n˜∞a≈t intermediate and µ˜ rather than µ˜ is used in Eq. (18). The reason is that
0
highdensities(seebelow),weexpecttohavetheuniversal µ(Tc)<µ0andtheuseofµ0alwaysleadstooverestimating
scaling T˜c/T˜c, (n˜/n˜ )1/2. This is well obeyed by the Tc. Thishappenstocompensatetheunderestimationof Tc
data. ∞ ≈ ∞ duetotheuseofEqs.(13)and(14).
Closeton˜=0thebehaviorisnonuniversal,inthesense At low density, the change of behavior from a convex
thatthecurvesdonotcollapseif n˜and T˜ arerescaledby increaseforλ¯<1toaconcaveincreaseforλ¯>1isvisible
c
n˜ and T˜ [Fig.3(b)]. Thenumericaldataareinperfect onthelog-logplotinFig.4(b)—whereaconvexfunction
c,
ag∞reement∞withthelimitingbehavior(19)atallcouplings. hasaslopelargerthanunity. Thelow-density,low-coupling
Theflatteningofthecurvesinthelog-logplotshowsthat limit(21)describesthenumericaldataperfectly. Thelow-
the running exponent η(n) in Tc nη(n) approaches zero density,high-couplingexpression(20)deviatesslightlydue
for n 0. Thisissuggestiveofa∝discontinuityin Tc(n)at totheuseofEq.(10)atlowestorderin1 a. Thissmall
n=0,→reminiscentoftheDOSdiscontinuity. However,since discrepancydisappearsifµ˜ isevaluatedwit−houtexpanding
7
0.3 0.3 momentum distribution above µ , than there are states
2D 3D 0
00..12 ¯λ =0.2¯λ5=0.5¯λ =1 ¯λ =1.5 0.2 ¯λ=0.5¯λ =1¯λ =1.5¯λ =2 vrtahealsmelouroeewvf,oeibtrdyhebraieennlmcoarwaemianµosibu0n.englTtonhww.ehteihcqehuziilenirbcorr-ieutaemsmecspheweraimtthuicriaenlcnprooenateisnnintteigaralλ¯cmtaiunnsdgt
Tc 0.1
@ 0
0.0005
˜µ λ¯=0.25 0.005
−0.1 0 0 0.5 ¯λ=1 IV. MULTIBANDEFFECTS
−0.2 e 2/−λ¯0.00050 0.0005 0.1 0 Theinterestraisedbymultibandsuperconductors,inpar-
0.3 − − − 0 0.0005 ticularMgB2 andtheiron-basedfamily,hastriggeredmany
− 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 studiesovertheyears.25 Herewediscussmultibandeffects
n˜ n˜ thatoccurnearabandedgeandareassociatedwiththelow
10−12 0 densityinoneofthebands.
− 2D λ¯=0.5 Itisclearfromtheprevioussectionthataknowledgeof
10−10 λ¯=0.25 theself-consistentchemicalpotentialisrequiredtounder-
˜n)−10 8 −0.2 λ¯=1 stand the behavior of Tc close to a band minimum. This
˜µ(0 − − raisesthequestionoftheroleplayedbyperturbationsthat
@T)c−−10−6 ¯λ =0.5 −0.4 λ¯=1.5 asuffpeecrtctohnedcuhcetminigcablapnodte(nNtiBa)l,bsuencheaatshththeepsruepseenrcceonodfuacntionng-
˜µ( −10−4 ¯λ = 1 0.6 bcaanndon(lSyBa)l.teInrtthheesaubpseernccoendoufcitnintegrbparnopdecrotiuepsloinfgth,ethSeBNbBy
−10−2 λ¯ =1.5 − 3D λ¯=2 cthheankgeiyngobthseervchateimonicwalapsottheanttiµali.sInfin2iDteaannddinne3gDatfivoerλa¯t>th1e,
100 0.8
− 0 1 2 3 − 0 1 2 3 band bottom, such that the nonanalytic behavior of Tc is
n˜ n˜ notcontrolledbyµ. AnNBisthereforenotexpectedingen-
eraltochangethisnonanalyticbehaviorqualitatively. An
FIG.5. (Toprow)ChemicalpotentialatT inthelow-densitylimit.
c exception—confirmingtherule—occurswhenthebottomof
Thedotsshowthesolutionofψd(1+µ˜,0)=1/λ¯,withψd(a,0) theNBcoincidespreciselywiththeenergyatwhichtheSB
givenbyEq.(10).Theinsetsshowthatµ(n=0)<0in2Dforall
λ¯,whileµ(n=0)=0in3Dforλ¯(cid:182)1. (Bottomrow)Difference beginstobepopulated. Forthispeculiararrangement,the
betweenthechemicalpotentialat Tc andthezero-temperature NBcontrolstherelationbetweenµandninthelimitTc 0
noninteractingvalueµ˜0.Theverticalbarsindicateµ˜=1. and Tc displaysasimpleanalyticdependenceonn,whic→his
linearin2Dand n2/3 in3D.ThisisillustratedinFig.6(a)
forthe2Dcase. S∝olvingthecoupledequations(9)fortwo
Eq. (10). The value λ¯ = 1 is somewhat peculiar: The bandsintheappropriateregime,wegetarelationbetween
numericsshowstheexpectedn˜2/3scaling,butthepre-factor n˜and T˜c whichisaccuratenearthebandminimum:
isneitherunityasimpliedbyEq.(20),nor0.742asgiven
byFEigqu.r(e215)s,hbouwts∼th0e.6c.hemicalpotentialcalculatednumeri- n˜=T˜c mm1 ln 1+e−exp(−T˜c2/λ¯11)
ccoaullpyliantgT,ca.sIdnis2cDussµedcoanbvoevreg.eIsnt3oDaµnetegnadtisvteovzaelruoeaftonra=n0y (cid:20) (cid:18) +mm2 ln 1(cid:19)+e−exp(−2/T˜λ¯c11)−E˜02 . (23)
ifλ¯(cid:182)1. Ifλ¯>1itconvergestoanegativevalue. Theden- (cid:18) (cid:19)(cid:21)
sityatwhichµ=0isgivenforλ¯(cid:166)1byn˜ 0.62(1 1/λ¯)3. This reproduces the near-band edge behavior as shown
Thiscoincideswiththecondition1/(kFas≈) 0.68.−Theef- inFig.6(a)andinparticulargivesthelineardependence
fectofincreasingthepairinginteractionis≈mainlytoshift n˜ = T˜c(m2/m)ln(2) at the transition point where E˜02 =
the µ(n) curve downwards. At n=0 this shift is entirely exp( 2/λ¯11).
duetotheinteraction-inducedspreadingofthemomentum −If E˜0−2 < exp( 2/λ¯11), the SB is not populated at low
distribution. Atfinitenpartoftheshiftisduetothethermal densityand−super−conductivityappearsatsomefiniteden-
smearing. sity. In all cases, the Tc(n) curve is “stretched” to higher
As the density increases, the behavior is qualitatively densities with respect to the one-band result due to the
differentin2Dand3D:whileµ˜ approachesµ˜0=n˜ in2D, carriers“lost”intheNB.Forinstance,ifthebandminima
this does not happen in 3D. In 2D the chemical potential are degenerate, we see from Eqs. (9) that the one-band
is µ˜ = T˜cln(en˜/T˜c 1). Since T˜c saturates for n˜ > n˜ , we Tc(n)curveissimplymodifiedbyarescalingofthedensity
have n˜ > T˜c at l−arge n˜ and µ˜ approaches exponen∞tially n n/[1+(m1/m2)d/2]. Thisimpliesthatthepairingtem-
thevaluen˜. Thisispeculiartothe2DconstantDOS,since pe→rature is necessarily reduced by a nonsuperconducting
bothinteractionandtemperatureredistributestatesinequal band,intheabsenceofinterbandcoupling.
amountsbelowandaboveµ . In3Dthesquare-rootDOS Both attractive and repulsive interband interactions in-
0
impliesthattherearemorestatesaddedinthetailofthe crease T fortwobands,26,27 asillustratedbythefactthat
c
8
0 0.05 1
0.5 0.05 λ¯12=0 (a) 2D 3D
+1 0.8 λ¯22=2 λ¯22=2
0.4
0 +0.5 0.6
0.3 0 T˜c λ¯22=0.5
T˜c 0.4 λ¯22=0.5
0.2 0.5 0.2
−
0
0.1 −e−2/λ¯11 E˜02=−1 1
mm2 e−2/λ¯11 E˜02 2D 3D
− −
0 (cid:0) (cid:1) 0.8
λ¯12=0.2
+1
0.4 0.6
+0.5 T˜
c
0 0.4
0.3
T˜ λ¯12=0.5 λ¯12=0.5
c
0.2 0.5 0.2 λ¯12=0.2 λ¯12=0.2
− λ¯12=0 λ¯12=0
0
0.1 E˜02= 1 0 1 2 3 4 0 1 2 3 4
− n˜ n˜
(b)
0
0 0.5 1 1.5 2 FIG.7. (Toppanels)ChangeofTc inducedbyasecondsupercon-
n˜ ductingbandin2Dand3D,withoutinterbandcoupling.(Dashed
blacklines)Pairingtemperatureforasinglebandwithcoupling
wFIiGth.6co.upPlainirgisngλ¯1t1em=p1e,raλ¯t2u2r=ef0o,ratnwdo(baa)nλ¯d1s2i=n0tw,o(bd)iλm¯1e2n=sio0n.2s,. λt¯h1e1t=wo1-abnanddmsayssstemm1w/mith=E˜102.(SE˜o0l1id=li0n.e7s5),Pmai2ri=ngmt1e,mapnedracotuurpelifnogr
E02istheenergyminimumofthesecondband,measuredfromthe λ¯22 =0.5(red) andλ¯22 =2−(blue). (Dash-dotted)Case of the
energyminimumofthefirst. Themassesarem1=m2=m. The secondbandalone. (Dotted)Caseofthetwo-bandsystemwith
dashedlinein(a)istheone-bandresult,shiftedhorizontallyfor λ¯11 =0. (Bottompanels)Increaseof Tc byinterbandcoupling.
easiercomparison.(Inset)Blowupofthetransitionregion.Curves (Solidlines)Nointerbandcoupling,samedataandcoloringasin
areshownforE˜02= exp( 2/λ¯11)+δE˜,withδE˜rangingfrom thetoppanels.Dottedanddashedlinescorrespondtointerband
−fo0r.δ04E˜(=blu0e.,Trihgehtw)hto−ite+d0a.0s−4he(dreldin,elesfts)h.oNwotEeqt.h(e2l3in).eaIrnb(ebh)a,vtihoer couplingλ¯12=0.2and0.5,respectively.
dottedlinesaretheresultforλ¯12=0andthedashedwhitelines
showEq.(24).
Reference6reportsanonmonotonicdependenceofthepair-
Eq.(9)involveonlyλ¯212: interbandinteractionsdonotin- ingtemperatureoncarrierconcentrationindopedSrTiO3:
duceinterbandpairinginthepresentmodel,butreinforce thiscannotbeinterpretedonthebasisofEq.(9)without
theintrabandpairingbysecond-orderprocessesinvolving invokingdensity-dependentinteractions.
theotherband. If T startsatfinitedensity,theinterband
c
coupling leads to a tail in the Tc(n) curve. In the regime Asecondbandcanneverthelessleadtoadecreaseof Tc
wherethechemicalpotentialisbelowtheSBbutwellinto atfixeddensity. Specifically,consideraone-bandsystemat
theNB,wefind,forinstance,in2D: somedensitywithcouplingλ¯ andpairingtemperatureT0;
11 c
2eγ m addasecondbandathigherenergywithcouplingλ¯22(cid:182)λ¯11
T˜c = π (cid:114)n˜m2 exp(cid:168)λ¯1212 (cid:150)λ¯11+ln −E˜022−n˜mm2 (cid:153)(cid:171). (24) athnedsnaomiendteernbsaitnydhcaosuTpcli(cid:182)ngT;c0t.heTnhitshceatnwboe-briagnodrosuyssltyemprowviethn
(cid:16) (cid:17) bymanipulatingEqs.(9).
ThisiscomparedwiththenumericalresultinFig.6(b).
We move on to the case of two superconducting bands Ifthesecondbandhasacouplingλ¯ >λ¯ , T exceeds
22 11 c
and begin with general trends. The observation that T T0 at high enough density and follows the dependence
c c
is an increasing function of density28 remains true in the thatwouldcorrespondtoanonsuperconductingfirstband.
near-bandedgeregime. Itispossibletoshowthattheprop- ThesevarioustrendsareillustratedinFig.7(toppanels).
erty dTc/dn(cid:190)0 is guaranteed by Eq. (9) for an arbitrary Figure 7 (bottom panels) shows the effect of interband
numberofbandsandanyvaluesofthecouplingconstants. interaction,whichisgenericallyanincreaseof T .
c
9
V. CONCLUSION for drawing our attention to a previous approximation of
Eq.(19)andtoP.Pieriforpointingouttheequivalenceof
Insummary,inthelow-densityregimewherethedynami- Eq. (21) with a known result in the BEC limit. This work
calrangeofthepairinginteractionissetbythebandedge, was supported by the Swiss National Science Foundation
thepairingtemperature T dependsontheelectrondensity underDivisionII.
c
ninanonanalyticway. Forparabolicbands,weprovidedex-
actasymptoticformulasdescribingthisdependency,taking
intoaccounttheenergyvariationoftheelectronicDOS,as
wellasthevariationofthechemicalpotentialwithinterac- AppendixA:SimpleproofofEq.(4)
tionstrengthandtemperature. Inoneandtwodimensions
and in three dimensions at strong enough coupling—in
Thedensityofafree-electrongasindimensiond ispro-
other words, when there is a bound solution to the two-
portionaltothevolumeofthed-dimensionalFermisphere,
particleproblem—thechemicalpotential(at T )becomes
c smearedbytheFermifunction:
negativeatlowdensity: Asaresultthe Tc(n)curvestarts
withinfiniteslopeandincreasesfasterthananypowerofn.
ddk 1
Otherwise,i.e.,inthreedimensionsatweakcoupling,the
n=2
cchuervmeicsatlarptostewnittihalzaeprporsolaocphee,saznedroitatinlocwreadseenssistlyo,wtheerTthc(ann) (cid:90) (2π)d exp ħh2k2k/B2Tm−µ +1
anypowerofn. ddk (cid:16) x (cid:17)
Our results may be relevant for low-density supercon- =2 (2π)d x exp ħh2k2 ,
ductors. InSrTiO3,oxygenreductionandniobiumdoping (cid:90) − 2mkBT
allowsonetotunethecarrierdensity6 inarangesuchthat (cid:16) (cid:17)
thedimensionlessdensityn˜variestypicallybetween10−2 with x = exp µ/kBT . Inordertoevaluatetheintegral,
adn˜nuvdcaer1di0ess.hItenyepttihcceaalLrlyraiAefrrloOdm3e/n1Ss0riTtyi1Oct3oani1n.atelIsnrofathbceee,lttouhwnee-fiddee2ln9dss-ieutyfcfheracttnhgianet- wwerituesek2th=−e exdip=(cid:0)a1nksi2i.onThx(cid:1)i/s(xle−adas)to=a−p(cid:80)ro∞qd=u1cxtqo/faqGaanudssiwane
− integrals:
ofthesedomains,ourexactformulasdifferfromtheusual
(cid:80)
formulasvalidathigherdensities. Inthenumericalillustra-
tλ¯ioonfsoorfdtehreopnree,swenhticphampearywaeppheaavrevuesreydlacroguepilnincgocmopnasrtaisnotns n= 2 ∞ xq d ∞ dki exp q ħh2ki2
2π 2mk T
tothetypicalvaluesoftheorder0.1reportedforSrTiO . We − q=1 i=1(cid:90) (cid:130)− B (cid:140)
3 (cid:88) (cid:89) −∞
efemrspfhraosmizethtehautsuoaulrddeefifinnitiitoionn,soufcthhethcaotuinpltinhgreceodnismtaenntssiodnifs- = 2 ∞ xq d mkBT = 2 mkBT d2 ∞ xq .
oursarebiggerthattheusualonesbyafactor(µ/ħhωD)1/2, − (cid:88)q=1 (cid:89)i=1(cid:200)2πħh2q − (cid:18)2πħh2 (cid:19) (cid:88)q=1qd/2
whichistypicallythreeinSrTiO .
3
The observation that T is a nonanalytic function of n ConsideringtheTaylorexpansionofthepolylogarithm,we
c
nearabandbottomcallsforareconsiderationoftheprob- seethattheq-suminthelastexpressionisLid/2(x),which
lem of shape resonances. These refer to oscillations of T provesEq.(4).
c
in a quasi-two-dimensional superconductor confined in a
slab, as a function of the slab thickness. The oscillations
arisewhenthechemicalpotentialcrossesthebottomofone
oftheconfinement-inducedsubbandsandwerepresented AppendixB:Resultsforoneparabolicbandin1D
intheliteratureonthesubjectasdiscontinuities.4 Ourre-
sults show that such discontinuities are artifacts, because
T vanishescontinuouslyatabandedgeinanydimension. Withtheprovisothatthefactor(d 1)πinEqs.(3)and
Thceactualdependenceof T ontheslabthicknessisthere- (7)bereplacedby1,Eqs.(2)–(9)are−validford=1. The
fore a continuous function,c which remains to be investi- asymptoticpropertiesofthefunctionψ1(a,b)are
gated. Aparticularlyinterestingsysteminthisrespectisthe
LtiacAsolOf3b/eSinrgTiaOl3owin-tdeernfascitey,swuhpiecrhcocnudmuuctliantgessythsteemch,acroancfitenreisd- ψ1(0<a<1,b 0)= sin−1((cid:112)a) π/2 1
→ (cid:112)1 a ≈ (cid:112)1 a −
inaquasi-two-dimensionalgeometry,andalsoamultiband 1 − a− 1 8eγ
system. ψ1(1<a<2,b→0)= (cid:112)a 1ln(cid:130)(cid:112)a+−(cid:112)a 1 πb(cid:140)
− −
1
ACKNOWLEDGMENTS ψ1(a>2,b 0)=
→ (cid:112)a 1
−
a 1 8eγ
WeacknowledgefruitfuldiscussionswithJ.-M.Triscone, ln − .
S.Gariglio,andM.Grilli. WearegratefultoA.V.Chubukov × (cid:130)((cid:112)a+(cid:112)a 1)((cid:112)a 1+(cid:112)a 2) πb(cid:140)
− − −
10
The corresponding weak-coupling approximations in the
(a)
regimen˜(cid:166)1are 0.6 ¯λ = 1
8eγ 1 1 0.7 5
T˜c ≈ π exp−λ¯/ µ˜1+ 1+1/µ˜ T˜c 0.4 ¯λ =
λ¯ =0.5
 (cid:112)µ˜  µ˜(cid:112)(cid:174)1 0.2 λ¯=0.3
 (d=1), (B1)
× (cid:112) 1 µ˜>1
1+(cid:112)1 1/µ˜ 0
− 0 0.5 1 1.5 2
 n˜
in agreement with the result of Ref. 2, where λ¯/ µ˜ is
100
the coupling constant evaluated at the chemical poten- (b)
tial. In the low-density limit the chemical potent(cid:112)ial ap-
proaches a finite negative value given by the solution of
1/λ¯ = sin−1 1+µ˜min / µ˜min (π/2)/ µ˜min 1.
Wecanusetheexpansion Li−1/2( e≈x) ex forl−argen−ega- 10 1
tive x andde(cid:0)d(cid:112)ucethere(cid:1)la−(cid:112)tionµ˜−=T˜cl→n(n˜/T˜c1(cid:112)/2). Wefind T˜c − λ¯ = 1
for T˜
c
T˜c ≈n˜2 exp W 12 1π+λ¯λ¯ 2 n˜12 (d=1,n→0). (B2) 10−2 ¯λ = 0.3
(cid:149) (cid:129) (cid:16) (cid:17) (cid:139)(cid:152)
The numerical results are shown in Fig. 8 and compared
10 8 10 6 10 4 10 2 100
withtheanalyticalformulas(B1)and(B2). Equation(B1) − − − −
n˜
workswellathighdensity,butseverelybreaksdownatlow
density,evenatweakcoupling. Thecancellationoferrors FIG.8. (a)Pairingtemperatureasafunctionofelectrondensity
observedinthe3Dcasealsooccursheretosomeextent,but foroneparabolicbandinonedimension. T isexpressedinunitsof
c
themainissueisthatEq.(B1)failstodescribetheregime ħhωD/kBandninunitsof2[mωD/(2πħh)]1/2.Thethinanddashed
whereµ˜<0atlowdensity,whilein3Dthisregimeisabsent linesshowEq.(B1),evaluatedusingµ˜0=(π/4)n˜2forµ˜.(b)Same
forλ¯<1. Equation(B2)isaccurateatweakcouplingwhere dataonalog-logscale.ThedashedlinesshowEq.(B2).
µ˜ 1andhasthesmallinaccuracyassociatedwiththe
−appr(cid:28)oximationmadeinsolvingforµ˜ atlargercoupling.
Thegapequationisobtainedbyreplacingtanh( )byunity
inEq.(1a). Foronebandin2Dwefind ···
AppendixC:GaptoT ratiointhelow-densitylimit
c 0 µ˜< 1
−
Wegiveherethezero-temperaturegapexplicitlyforone 1/λ¯=12ln (cid:112)1+µ˜(cid:112)2+1∆˜+2∆˜2µ˜ −1<µ˜<1 (C3)
bcaanndbeinco2mDb,inaesdawfuitnhctthioenreosfuldte(n1s9i)tyinanorddecorutoploinbgta.inTthhies ln (cid:129)1+(cid:112)∆˜1+∆˜2− (cid:139) µ˜>1.
exactgaptoT ratiointhelow-densitylimit. Theexpression  (cid:129) (cid:139)
c 
ofthedensityat T =0is For µ˜ > 1, the gap is independent of density and given
by ∆˜ = 1/sinh(1/λ¯); combined with the weak-coupling
result(17),thisyieldstheusualweak-couplingBCSratio
n=(cid:90)−∞∞dξN0(µ+ξ)1− ξ2ξ+∆2ξ, (C1) 1∆e)li/em−(ki2n/Bλa¯T.tce)S∆o˜=lvaiπmn/goenfγog.rEItnhqse.thg(aeCp2l,)owwane-ddteh(neCsn3it)ayrtrorievfigenimadte:µ˜µ=˜ <n˜+1,(n˜w−e
 (cid:198) 
whereN0(E)isthenormal-stateDOSgivenbyEq.(3)and
W∆eξ s=etθE(0ħhαω=D −0|aξn|)d∆mw=ithm∆α atsheinzeSreoc-.teIImIpaenrdatmuroevegatpo. ∆˜ = (cid:112)n˜+sin˜n(hn˜(1−/λ1¯))e−2/λ¯ (n˜<1). (C4)
dimensionlessvariables. In2Dwehaveat T =0,
Comparing Eqs. (C4) and (19) we see that T increases
c
0 µ˜< 1 fasterthan∆withincreasingdensity. Asaresultthegapto
− T ratiovanishesforn 0,asweshowinFig.9. Thisresult
n˜=12 µ˜+1+ µ˜2+∆˜2 1+∆˜2 1<µ˜<1 mcay look surprising i→n view of the fact that known two-
− −
µ˜(cid:129) (cid:112) (cid:112) (cid:139) µ˜>1. dimensionalsuperconductorstendtohaveagapto Tc ratio
(C2) largerthantheBCSvalue. ThesuppressionshowninFig.9
