Superconductivity vs bound state formation in a two-band superconductor with small Fermi energy -- applications to Fe-pnictides/chalcogenides and doped SrTiO3

1.6 MB·

by  Andrey V Chubukov

Superconductivity vs bound state formation in a two-band superconductor with small
Fermi energy – applications to Fe-pnictides/chalcogenides and doped SrTiO
3
Andrey V. Chubukov,1 Ilya Eremin,2 and Dmitri V. Efremov3
1Department of Physics, University of Minnesota, Minneapolis, Minnesota 55455, USA
2Institut fur Theoretische Physik III, Ruhr-Universitat Bochum, D-44801 Bochum, Germany
3Leibniz-Institut fur Festkorper- und Werkstoffforschung, D-01069 Dresden, Germany
(Dated: January 11, 2016)
Weanalyzetheinterplaybetweensuperconductivityandtheformationofboundpairsoffermions
(BCS-BECcrossover)ina2DmodelofinteractingfermionswithsmallFermienergyE andweak
F
6 attractive interaction, which extends to energies well above EF. The 2D case is special because
1 two-particleboundstateformsatarbitraryweakinteraction,andalreadyatweakcouplingonehas
0 todistinguishbetweenboundstateformationandsuperconductivity. Webrieflyreviewthesituation
2 in the one-band model and then consider two different two-band models – one with one hole band
andoneelectronbandandanotherwithtwoholeortwoelectronbands. Ineachcaseweobtainthe
n
boundstateenergy2E fortwofermionsinavacuumandsolvethesetofcoupledequationsforthe
a 0
pairinggapsandthechemicalpotentialstoobtaintheonsettemperatureofthepairing,T andthe
J ins
7 quasiparticle dispersion at T = 0. We then compute the superfluid stiffness ρs(T = 0) and obtain
the actual T . For definiteness, we set E in one band to be near zero and consider different ratios
c F
ofE andE intheotherband. Weshowthat,atE (cid:29)E ,thebehaviorofbothtwo-bandmodels
] 0 F F 0
n is BCS-like in the sense that Tc ≈ Tins (cid:28) EF and ∆ ∼ Tc. At EF < E0, the two models behave
o differently: inthemodelwithtwohole/twoelectronbands,Tins ∼E0/logEEF0,∆∼(E0EF)1/2,and
c T ∼ E , like in the one-band model. In between T and T the system displays preformed pair
c F ins c
- behavior. In the model with one hole and one electron band, T remains of order T , and both
r c ins
p remainfiniteatEF =0andoforderE0. Thepreformedpairbehaviorstilldoesexistinthismodel
u becauseT isnumericallysmallerthanT . Forbothmodelswere-expressT intermsofthefully
c ins ins
s renormalizedtwo-particlescatteringamplitudebyextendingtotwo-bandcasethemethodpioneered
.
t by Gorkov and Melik-Barkhudarov back in 1961. We apply our results for the model with a hole
a
and an electron band to Fe-pnictides and Fe-chalcogenides in which superconducting gap has been
m
detected on the bands which do not cross the Fermi level, and to FeSe, in which superconducting
- gapiscomparabletotheFermienergy. Weapplytheresultsforthemodelwithtwoelectronbands
d
toNb-dopedSrTiO andarguethatourtheoryexplainsrapidincreaseofT whenbothbandsstart
3 c
n
crossing the Fermi level.
o
c
PACSnumbers:
[

1
v I. INTRODUCTION therbarelycrosstheFermilevelorarefullylocatedbelow
8 or above it5. The ”extreme” case in this respect is FeSe.
7 In this material Fermi energies on all hole and electron
The discovery of superconductivity in Fe-pnictides
6 pockets are small and are comparable to the magnitudes
1 and later in Fe-chalcogenides opened up several new di- of the superconducting gaps (the reported E on differ-
F
0 rections in the study of non-phononic mechanisms of
ent bands vary between 4 and 10 meV, while the gaps
1. electronic pairing in multi-band correlated electron sys- are 3-5meV6,7).
0 tems1,2. Two issues were brought about by recent angle-
6 resolved photoemission and other experiments in Fe- The observation of a sizable superconducting gap on
1 based superconductors (FeSCs). First, in recent exper- a band which does not cross the Fermi level was orig-
: iments on LiFe Co As, Miao et al. observed3 a fi- inally interpreted3 as the indication that the pairing in
v 1−x x
i nite superconductive gap of 4−5meV on the hole band, theFeSCsisastrongcouplingphenomenonforwhichthe
X
which is located below the Fermi level, with the top of pairinggapisnotconfinedtotheFermisurfaceanddevel-
r the band at 4-8meV away from E . Moreover, the gap opsatallmomentaintheBrillouinzone. Later,however,
a F
on this hole band is larger than the gaps on electron the experiments were re-interpreted8 in a more conven-
bands, which cross the Fermi level. A similar observa- tional weak/moderate coupling scenario, as the conse-
tion has been reported for FeTe Se 4, where super- quence of the fact that in FsSCs the pairing interaction
0.6 0.4
conductivity with the gap ∆ = 1.3meV has been ob- primarily”hopes”apairoffermionswithmomentakand
served on an electron band which lies above the Fermi −k from one band to the other9,10. In this situation, the
level,withthebottomofthebandat0.7meVawayfrom gap on the band, which does not cross the Fermi level is
E . Second, recent photoemission measurements have determined by the density of states at E of the band
F F
demonstrated that in almost all Fe-based supperconduc- which does cross the Fermi level. A solution of the cou-
tors either electron or hole pockets are more tiny than pled set of BCS gap equations for fermions in the two
previously thought andthe corresponding dispersionsei- bandsthenshowsthatonecrossingissufficienttoobtain
2
consider the situation when U remains energy indepen-
dentuptoanenergyΛ,whichwellexceedsE ,seeFig.1.
F
Elementary quantum mechanics shows that in 2D, two
fermions with dispersion k2/(2m) form a bound state at
arbitrarysmallattractionU,andtheboundstateenergy
is 2E , where E ∼ Λe−2/λ, and λ = mU/(2π) is a di-
0 0
mensionless coupling. We analyze the evolution of the
system behavior and the interplay between T and T
c ins
by varying the ratio E /E while keeping both E and
F 0 F
E well below Λ.
0
WebrieflyreviewBCS-BECcrossoverintheone-band
two-dimensional (2D) model and consider two different
two-band models.
FIG.1: Characteristicenergyscalesrelevanttotheinterplay
The first model, which we apply to FeSCs, consists of
between the formation of bound pairs of fermions and true
one hole and one electron band. We follow usual path
superconductivityin2DFermionicsystemswithweakattrac-
and consider the case when the dominant pairing inter-
tive pairing interaction. Λ is the upper energy cutoff, E is
F
actionisweakinter-bandpairhoppinginteractionU >0,
the Fermi energy, and E is the energy of a bound state of
0
two fermions in a vacuum, i.e., at µ = 0. At weak coupling, in which case superconducting state has s+− symmetry.
E (cid:28)Λ. We assume that E is also small and can be tuned Weusethesamecomputationalprocedureasinthestud-
0 F
by doping to be either larger or smaller than E . We show ies of one-band model.11,13,15,20,21,27,28 Namely, we first
0
thatinone-bandmodelandintwobandmodelwithtwohole obtain the bound state energy 2E for two fermions in a
0
or two electron bands, the system displays BCS-like behav- vacuum. Thenweconsidertheactualsystemwithanon-
ior at EF (cid:29) E0 and BEC-like behavior at EF (cid:28) E0. In zero density of carriers in one of the bands n = 2N E ,
the latter case, bound pais develop at T ∼E /log E0 but 0 F
ins 0 EF where N0 is a 2D density of states at low energies, and
the true superconductivity with full phase coherence devel-
solvethesetofcoupledequationsforthechemicalpoten-
ops at T ∼E . In between T and T the system displays
c F ins c tialµ(T)andthepairinggapsatafiniteT. Thesolution
preformed pair behavior and the spectral function displays
of the linearized gap equations yields the onset temper-
pseudogap behavior. In the two-band model with one hole
ature of the pairing T . The solution of non-linear gap
andoneelectronpocket,T andT alsosplitwhenE gets ins
ins c F
equationsatT =0yieldsthepairinggaps∆ . Wenext
smaller thanE0, but bothremain of order E0 even when EF h,e
vanishes. Still, superconducting Tc in this limit is several usethevaluesof∆h,e andµatT =0asinputsandcom-
timessmallerthanTins,sothereisawidetemperaturerange pute superfluid stiffness ρs(T = 0). For definiteness, we
of preformed pair behavior. considerthecasewhenthechemicalpotentialatT =0in
thewouldbenormalstateisatthebottomoftheelectron
band but crosses the hole band i.e., the Fermi energy is
BCS instability already at weak coupling8. This reason- zero for the electron band but finite for the hole band.
ing also naturally explains why the gap is larger on a We present the results in the two limits when E is
F
band which does not cross the Fermi level. either larger or smaller than E and in the case when
0
The observation of superconductivity with ∆ ∼ EF EF =E0. We show that in all cases the pairing gap de-
brought FeSCs into the orbit of long-standing discussion velops on both bands and is larger on the electron band
about the interplay between superconductivity and the (the one which does not cross the Fermi level). This
formation of the bound pair of two fermions. This issue agrees qualitatively but not quantitatively with the re-
hasbeendiscussedinthecondensedmattercontext11–28. sults obtained previously within the conventional BCS
and also for optical lattices of ultracold atoms.29. The theory8,30,neglectingtherenormalizationofthechemical
phenomenon in which bound pairs of fermions form at potential. We argue, however, that the renormalization
a higher T and condense at a smaller T is often of the chemical potential by both a finite temperature
ins c
termed Bose-Einstein condensation (BEC) because the and a finite gap is not a small perturbation when the
condensationofpreformedpairs(i.e.,thedevelopmentof bare chemical potential touches the bottom of the elec-
a macroscopic condensate) bears a direct analogy with tron band.
BEC of bosons in a Bose gas. When ∆ and T are much We argue that the onset temperature of the pairing,
c
smallerthanEF,boundpairsandtruesuperconductivity Tins (the one obtained by solving the set of equations
develop at almost the same temperature, i.e., T ≈T . for the pairing gaps and the chemical potentials) evolves
c ins
However, when EF gets smaller, superconducting Tc is when EF/E0 changes and is of order EF1/3E02/3 when
generally smaller than the onset temperature for bound E (cid:29) E and of order E when E (cid:28) E . We fur-
F 0 0 F 0
state formation. ther ague that superconducting T is of order T for
c ins
In the present communication, we discuss supercon- arbitrary E /E , but is numerically smaller than T .
F 0 ins
ductivity vs bound state formation in 2D systems with Thenumericalsmallnessimpliesthatthereexistsafinite
weak attractive interaction U in the proper symmetry range of temperatures between T and T , where pairs
ins c
channel. ( s+− for the two-band model for FeSCs). We arealreadyformedbuttheirphasesarerandomandthere
3
single-band 2 electron (hole) bands 1 electron and 1 hole bands
√
E (cid:29)E T ∼ E E , µ(T )≈E T ∼E1/3E2/3, µ (T )≈E T ∼E1/3E2/3, µ (T )≈−0.5T
F 0 ins F 0 √ins F ins F 0 1 ins F ins F 0 e ins ins
T ≈T , ∆=2 E E µ (T )≈−0.5T , T ≈T µ (T )≈E , T ≈T
c ins F 0 2 ins ins c ins h ins F c ins
∆≈1.78E1/3E2/3 ∆≈1.78E1/3E2/3,
F 0 F 0
(cid:16) (cid:17)
E (cid:28)E T ≈E /log E0, µ(T )≈−E T ∼4.5E /log E0, µ (T )≈−2.3E T ∼1.13E 1+0.22EF ,
F 0 ins 0 EF ins√ 0 ins 0 EF 1 ins 0 ins 0 E0
T ∼E /8(cid:28)T , ∆=2 E E µ (T )≈−2.3E , T ∼E /8(cid:28)T , µ (T )≈3E /2, µ (T )≈−EF
c F ins F 0 2 ins √0 c F ins h ins F e ins 2
∆≈ 2E E T ∼0.22T , ∆=1.76T
F 0 c ins ins
E =E T =1.09E , µ(T )=−0.09E T ∼0.9E , µ (T )=0.1E T =1.35E , µ (T )=−0.35E
F 0 ins F ins F ins F 1 ins F ins F e ins F
µ (T )=−0.9E , ∆=1.4E µ (T )=1.35E , T ∼0.4E
2 ins F F h ins F c F
∆=2.4E
F
TABLE I: The summary of the results for the onset temperature of the pairing, T , the actual superconducting transition
ins
temperature, T , the gap magnitude at T =0, ∆, and the chemical potentials, µ , for different ratios of the Fermi energy, E ,
c i F
andtheboundstateenergyoftwofermionsinavacuum,E . Fortheonebandmodel,superconductivityisanordinarys−wave.
0
For the two-band models, superconducting state has s±- symmetry (different signs of the gaps on the two bands). The gap
magnitudes on different bands are approximately the same in the two-band models, up to small corrections, but nevertheless
thegaponthebandwhichdoesn’tcrossthechemicalpotentialatT ≥T islargerthanthatontheotherband,whichcrosses
ins
thechemicalpotential. Thechemicalpotentialssatisfyµ +µ =E forthetwobandmodelwithaholeandanelectronband,
h e F
and µ −µ =E for the model with two electron bands.
1 2 F
is no superconductivity (the ”preformed pairs” regime). at some finite E . We show that, at E (cid:29) E , the
F F 0
TheemergenceoftheboundpairsatT >T andthe behavior of this model is nearly identical to that in the
ins c
existence of the preformed pairs regime is often associ- model with a hole and an electron band. However, in
atedwiththecrossoverfromBCStoBECbehavior. Such the opposite limit E <<E , the behavior of the model
F 0
crossover has been studied in detail in the finite T anal- withtwoelectronbandsdiffersqualitativelyfromthatof
ysis of 3D one-band model (Ref.15). The temperatures the model with a hole and an electron band and is quite
T and T were found to differ strongly at E (cid:28) E : similar to the behavior of the 2D one-band model in the
ins c F 0
Tins ∼ E0/logEEF0 (cid:29) EF, while Tc ∼ EF, i.e., the ratio BEC limit. Namely, Tins ∼ E0/logEEF0 and Tc ∼ EF,
T /T vanishesatE →0. Thebehaviorinthe2Dcase such that the ratio T /T vanishes at E = 0. The
c ins F c ins F
is quite similar (see below). In our two-band model the reason is that for the two bands with the same sign of
behavior is similar to the one-band model in that T dispersion, there are no free carriers at E → 0, hence
ins F
becomes parametrically larger than E at E (cid:28) E , the pairing cannot create images of the original bands.
F F 0
but differs in that T remains finite and of the same or- Indeed, we show that in the model with two electron
c
der as order T even at E → 0. The reason, as we bands, the pairing gap ∆, which is responsible for the
ins F √
argue below, is that the development of the pairing gap Fermi surface reconstruction, scales at T =0 as E E
0 F
belowT reconstructsthefermionicdispersionandcre- and vanishes at E = 0. The same behavior holds at
ins F
ates images, with opposite dispersion, of original hole T =0 in the 2D one-band model.11,13
and electron bands. This in turn gives rise to the shift Superconductivityinasystemwithtwoelectronbands
of fermionic density from the filled hole band and empty is realized experimentally in Nb-doped SrTiO and, pos-
3
electronbandintotheseimagebands. Asaconsequence, sibly, in heterostructures of LaAlO and SrTiO (see
3 3
there appear new hole-like and electron-like bands with Refs.23,24,31 and references therein). The Fermi energy
a finite density of carriers in each band, proportional to in one of the bands is finite already at zero doping and
T . Superconducting T scales with this density and E is likely much larger than E . The other electron
ins c F 0
is a fraction of T . We show that preformed pair be- band is above the chemical potential at zero doping, but
ins
havior still exists at E (cid:28) E , but only because T , set the chemical potential µ moves up with doping and en-
F 0 c
by superconducting stiffness, is numerically smaller than ters this band once it exceeds the critical value µ∗. The
T . data indicate32,33 that, when this happens, T rapidly
ins c
Wenextconsiderthetwo-bandmodelconsistingoftwo increases. To analyze this behavior we compute T for
c
electronbands. Weagainassumethatthedominantpair- µ (cid:54)= µ∗. We show (see Fig. 12) that T indeed in-
c
inginteractionisinter-bandpairhoppingU >0andthat creases when µ exceeds µ∗, and the rate of the increase
the chemical potential is at or near the bottom of one of is (1−µ∗/µ)(E /E )2/3, i.e., it is enhanced by a large
F 0
the bands, but crosses the dispersion of the other band ratio of E /E .
F 0
4
FIG. 2: The bare dispersion for the models considered in the present manuscript: (a) a one-band model of 2D fermions with
the parabolic dispersion and a positive bare chemical potential (i.e., a non-zero E ); (b) a two-band model with one hole and
F
one electron band separated in the momentum space. For definiteness, we set the bare chemical potential such that it touches
the bottom of the electron band and crosses the hole band at a finite distance from its top; (c) a two-band model with two
electron bands separated in the momentum space. For definiteness we set the bare chemical potential to touch the bottom of
one band and cross the other.
Another goal of our work is to compare the analysis when E = 0. We, however, show that GMB approach
F
of BCS/BEC crossover with the approach put forward remains valid even when E < E , and T can be ex-
F 0 ins
by Gorkov and Melik-Barkhudarov (GMB) back in 1961 plicitly expressed via the exact 2D scattering amplitude,
(Refs.34,35). GMB considered a one-band model with with E in the prefactor.
F
attractive Hubbard interaction U at weak coupling in
The paper is organized as follows. In Sec.II we
D=3. They argued that superconductivity comes from
review one-band 2D Fermi system with small E . We
F
fermionswithenergiesnotexceedingEF,whileallcontri- reproduceearlierresults11,13,28 fortheonsettemperature
butions to the pairing susceptibility from fermions with
for the pairing, T , the pairing gap, the renormalized
ins
higher energies can be absorbed into the renormaliza-
chemical potential, and the spin stiffness. We argue that
tion of the original 4-fermion interaction into quantum-
superconducting T scales with E and vanishes when
c F
mechanical scattering amplitude. GMB explicitly sepa-
E = 0. In Sec. III we consider in detail the case of
F
ratedtheCooperlogarithm(associatedwiththepresence
one hole and one electron bands, relevant to FeSCs. We
of a sharp Fermi surface at E (cid:54)= 0) from the renormal-
F show that in this model both T and T remain finite
ins c
ization of the interaction into the scattering amplitude
even when neither bands crosses the Fermi level. The
and obtained Tc = 0.277EFe−π/(2|a|kF), where a is the superconducting T is smaller than T in this case,
c ins
s-wave scattering length. They argued that this is the
but the smallness is only numerical. In Sec. IV we
rightformula forcomparisonwiththeexperimentaldata
consider the case of two hole/two electron pockets and
becausethescatteringlengthisthephysicallyobservable
show that that T remains non-zero as long as one of
c
parameter, while the interaction U is not.
the band crosses Fermi level but vanishes when E = 0
F
TheGMBanalysisdoesnotincludephasefluctuations,
for both bands. In Sec. V we review GMB formalism
hence their instability temperature is the same as T ,
ins and then apply it first to 2D one-band model and then
re-expressed in terms of scattering amplitude. In the
to 2D model with a hole and an electron band. In both
original GMB analysis (which we review in Sec. V be-
cases we show that the instability temperature T
GMB
low) |a|k is assumed to be small, and no bound state
F is precisely T , even when E → 0. We present our
ins F
develops. WeextendGMBanalysistoone-bandandtwo-
conclusions in Sec. VI. Discussion of some technical
bandmodelsin2Dandwillspecificallyconsiderthelimit
details is moved into the Appendix.
E <E . Inthislimit,thescatteringamplitudediverges
F 0
at the onset of bound state development at T ∼ E
0
and changes sign at a smaller T. It is then a–priori
unclear whether the onset temperature of the pairing,
T , can be expressed via the 2D scattering amplitude
ins
a (dimensionless in 2D) with E in the prefactor, par- II. ONE-BAND MODEL
2 F
ticularly given that the ratio T /E tends to infinity
ins F
To set the stage for the analysis of the two-band model we first review pairing and superconductivity in
5
the 2D one-band model. Consider a set of 2D fermions holds in powers of T /E , this variation generally can-
ins F
with the parabolic dispersion ε = k2 and chemical not be neglected, i.e., the equation for the pairing vertex
k 2m
potential µ = E , see Fig.2(a). We assume that at T = T has to be combined with the equation for
0 F ins
fermions get paired by a weak attractive pairing inter- the chemical potential µ(T ). The latter follows from
ins
action U(q,Ω), which for simplicity we approximate as the condition that the total number of fermions is con-
momentum and frequency independent U up to upper served.13,15 Thetwocoupledequationsare(µ=µ(T ))
ins
momentumcutoffq andcorrespondingfrequencycut-
offΛ=qm2ax/(2m).mFaoxrelectron-phononinteractionΛis 1 = λ(cid:90) Λdεtanh2εT−inµs
of the order of Debye frequency. The actual dispersion 2 ε−µ
0
does not have to be parabolic, however at weak coupling λ(cid:32)(cid:90) µ tanh x (cid:90) Λ tanh x (cid:33)
energiesrelevantforthepairingaremuchsmallerthanΛ = dx 2Tins + dx 2Tins
andε = k2 canbejustviewedastheleadingterminthe 2 0 x 0 x
k 2m
expansion of the lattice dispersion in small momentum. (cid:90) Λ 1
We introduce the dimensionless parameter EF = 0 dεe(ε−µ)/Tins +1 =Tinslog(1+eµ/Tins)
m|U| (3)
λ=N |U|= , (1)
0 2π
At E (cid:29)E , the solution of these equations yields
F 0
wthhaetreλNis0s=ma2mlπl nisumthbeedr.enTshiteycoofnsvteantteisoninal2wDe.aWkecoausspulimnge Tins = 1.13(ΛEF)1/2e−λ1 ∼(cid:112)EFE0
BCS analysis is valid when the attraction is confined to µ(Tins) ≈ EF (4)
energies much smaller than E , i.e., when Λ(cid:28)E . We
F F In the opposite limit E (cid:28)E we obtain28
considertheoppositesituationwhenE ismuchsmaller F 0
F
than the cutoff energy Λ. T = E0
For two fermions with 2D k2 dispersion in a vacuum ins log E0
(i.e., at E = 0), an arbitrary small attraction U gives EF
F µ(T ) ≈ −E (5)
risetotheformationofaboundstate36. Theboundstate ins 0
energy at T =0 is 2E0, where This behavior is rather similar to that in three-
dimensional case.15 The behavior of T and µ at in-
E0 =Λe−λ2 (2) termediate EF ∼E0 can be easily obtaiinnesd numerically.
For E =E , T ≈1.09E and µ(T )≈−0.09E .
The bound state develops at T = 1.13E . The 2D F 0 ins F ins F
0 0 Note that at E (cid:29) E , the instability temperature
scattering amplitude a2 ∝ 1/logTT0 diverges at T = T0 T (cid:28) E , whilFe at E 0(cid:28) E , T (cid:29) E and µ(T )
and changes sign from negative at T >T to positive at ins F F 0 ins F ins
0 is negative.
T < T . We consider the system at a non-zero E , i.e.,
0 F Theprefactor1.13inEq. (4)isinfactobtainedbygo-
at a finite density of fermions n = mE /π. We show
F ingbeyondlogarithmicalaccuracyintheparticle-particle
that the system behavior is different at E (cid:29)E and at
F 0 channel. To get the correct prefactor one also needs to
E (cid:28)E .
F 0 includefermionicself-energytoorderλandtherenormal-
ization of U by corrections from the particle-hole chan-
A. The onset temperature of the pairing, the
nel.34,35 These two renormalizations are not essential for
pairing gap and the renormalization of the chemical
our consideration and in the bulk of the text we neglect
potential
them. For completeness, however, we obtain the result
for T with the full prefactor in the Appendix.
The onset temperature of the pairing instability, T ins
ins The pairing gap ∆ and the renormalized chemical po-
(not necessary a true superconducting transition tem-
tential µ at T < T are obtained by solving simulta-
perature)isobtainedbyintroducinginfinitesimalpairing ins
neously the non-linear gap equation and the equation on
vertex and dressing it by renormalizations to obtain the
µ(T). The set looks particularly simple at T = 0 (here
pairing susceptibility. The temperature T is the one
ins µ=µ(T =0), ∆=∆(T =0)):
at which the pairing susceptibility diverges. To logarith-
mical accuracy, one needs to keep only ladder series of λ(cid:90) Λ 1
renormalizations in the particle-particle channel and ne- 1 = dε(cid:112) ,
2 (ε−µ)2+∆2
glectallrenormalizationscomingfromparticle-holechan- 0
(cid:32) (cid:33)
nel because the first contain series of λlog Λ while the 1(cid:90) ∞ ε−µ
lattercontainseriesinλ. WeassumeandthTeinnsverifythat EF = 2 dε 1− (cid:112)(ε−µ)2+∆2 . (6)
in all cases that we consider, T (cid:28)Λ, hence log Λ is 0
ins Tins
a large factor. However, as we will see, the ratio of Tins Solving these equations we obtain at T =0
EF
is small only when EF (cid:29)E0 and is actually large in the µ+(cid:112)µ2+∆2 =2E
opposite limit E (cid:28) E . Because temperature varia- F
F 0 (cid:112)
tion of the chemical potential µ(T) in the normal state µ2+∆2−µ=2E (7)
0
6

k T<Tins
E =
T~T kin 
ins k
k+q_ k+q_
2 2
(T~Tins) 2 22(T Tins) E = q q + q q
pot    
q_ q_
-k+ -k+
E
F k 2 2
FIG. 4: Diagrammatic representation of the kinetic and
potential energy of a one-band superconductor. The sum
E +E (q = 0) gives the condensation energy, and the
kin pot
prefactorfortheq2 terminE (q)determinesthesuperfluid
pot
stiffness.
FIG. 3: The dispersion in the one-band model. Red dashed
line – the bare dispersion (the one which the system would
haveatT =0intheabsenceofthepairing). Blackline–the
which the system would have at T =0 in the absence of
dispersion right above T , blue line – the dispersion below
ins the pairing).
T . Theplotisforthecasewhenthechemicalpotentialµis
ins
already negative at T =T . Observe that the minimal gap
(cid:112) ins
is ∆2+µ2 and the minimum of the dispersion is at k =0
rather than at kF. B. Superconducting Tc
The temperature T appears in the ladder approx-
ins
hence
imation as the transition temperature, but is actually
only the crossover temperature as pair formation by it-
µ=E −E
F 0
self does not break the gauge symmetry. To obtain the
(cid:112)
∆=2 EFE0 (8) actual Tc, at which the gauge symmetry is broken (i.e.,
the phases of bound pairs order), one needs to treat the
These results were first obtained in Ref.13.
phase φ(r) as fluctuating variable and compute the en-
When E (cid:29) E , the expressions for µ and ∆ are the
F 0 ergy cost of phase variation δE = (1/2)ρ (T)(cid:82) dr|∇φ|2
same as in BCS theory: s
(see Ref. [37] for a generic description of fluctuations in
µ≈E , superconductors). The prefactor ρ (T) is the superfluid
F s
∆=2(ΛEF)1/2e−λ1 =1.76Tins. (9) sotfiffBneersesz.insIkny-2KDo,stseurplietrzc-oTnhdouuclteisnsgtrTacns(itthioent)emispoefroartduerer
In the opposite limit EF (cid:28)E0, ρs(T =0) (Ref.38,39), provided that ρs(T =0) is smaller
than T . If ρ (T = 0) (cid:29) T , phase fluctuations cost
ins s ins
µ≈−E0 too much energy, and the phases of bound pairs order
(cid:18)E (cid:19)1/2 E almost immediately after the pairs develop. In this last
∆∼Tins EF logE0. (10) case Tc =Tins minus a small correction.
0 F Within our model with local interaction U, δE is the
√
Observethatwhile∆=2 E E staysthesameinboth O(q2) term in the ground state energy of an effective
F 0
limits, the ratio ∆/T changes: ∆ ∼ T at E (cid:29) E model described by the effective fermionic Hamiltonian
ins ins F 0
and ∆ (cid:28) T at E (cid:28) E . At E = 0, ∆, T , and with the anomalous term
ins F 0 F ins
∆/Tins all vanish. The vanishing of ∆ is easy to under- (cid:90)
stand – a finite gap would reconstruct fermionic disper- H = d2r∆(r)c†(r)c†(r)+h.c
anom ↑ ↓
sion and open up a hole band with a finite density of
(cid:88)
carriers proportional to ∆, what is impossible at E =0 = ∆(q)c†(k+q/2)c†(−k+q/2)+h.c
F ↑ ↓
becausethedensityoffermionsiszero. Anegativeµim- k,q
plies that the Fermi momentum kF (defined as position (11)
(cid:112)
of the minimum of E = (ε −µ)2+∆2) is zero. In
k k
fact, the Fermi momentum shifts downwards already in with∆(r)=∆eiφ(r) ≈∆ei(∇φ)r whoseFouriercomponet
the normal state at a finite T because µ(T) < E . It ∆(q)=∆δ(q−∇φ).
F
becomeszeroatT =T atE /E ≈0.882. Thedown- The ground state energy is the sum of the kinetic and
ins 0 F
ward renormalization of k has been recently obtained the potential energy. The kinetic energy depends on
F
in the study of superconductor-insulator transition.16 |∆(r)|2 = ∆2 and is not sensitive to phase fluctuations
InFig. 3weplottheactualdispersionbelowandabove (i.e., it does not have (∇φ)2 term) and is simply given
T along with the bare fermionic dispersion (the one by the convolution of the quasiparticle dispersion with a
ins
7
single fermionic Green’s function (see Fig.4). At T =0 function.
(cid:90) dε dω
E = 2N k ε G (k,ω)=
kin 0 2π k s The potential energy, on the other hand, does depend
(cid:90) dε dω iω+(ε −µ) onq. WithinthemodelofEq. (11)itisgivenbythesum
−2N k ε k (12)
0 2π kω2+(ε −µ)2+∆2 of the convolutions of two normal and two anomalous
k
Green’s functions with ∆ in the vertices18,37,40 (see Fig.
where G is the normal of the superconducting Green’s 3). In the analytic form
s
(cid:90) d2kdω
E (q)=−∆2 [G (k+q/2,ω)G (−k+q/2,−ω)+F (k+q/2,ω)F (−k+q/2,−ω)], (13)
pot (2π)3 s s s s
where F is the anomalous Green’s function. Integrating Using
s
over frequency in Eq.(13) we obtain at T =0
(cid:32) (cid:33)
∆2 d ε −µ
∆2 ∆2 (cid:90) (ε −ε )2 =− 1− k
Epot(q)=−|U| + 4 d2k((εkk+−q/µ2)2+k−∆q2/)23/2 +... ((εk−µ)2+∆2)3/2 dεk ((εk−µ)2+∆2)1(/129)
∆2 ∆2 (cid:90) ε and integrating by parts Eq.(18) we obtain
= − +q2 dε k +..(.14)
|U| 8π k((εk−µ)2+∆2)3/2 N (cid:90)
ρ (T =0)= 0 dε ×
where dots stand for the terms of higher orders in q2. s 8 k
ThedifferencebetweenEpot(q =0)+Ekin inasupercon- (cid:32)1− εk−µ (cid:33)(cid:34) d (cid:18)dεk(cid:19)2(cid:35) (20)
dthuectcoorndanendsathtieonkienneetrigcyenEergy.inTothoebtnaoinrmEal stawteegeivvaels- ((εk−µ)2+∆2)1/2 dεk dk
cond cond
uate the frequency integrals in (12) and (14) and write
The term in square brackets is simply a constant (=
the condensation energy as
2/m), and the remaining integral gives exactly the to-
tal energy density equal to 2E . As a result,
E =−N × F
cond 0
(cid:34)EF2 + ∆22 (cid:90) Λ √x2d+x ∆2 (cid:18)1− x+2(√xx+2+µ)∆2(cid:19)(cid:35) ρs = E4πF (21)
−µ
(15) Note that this result is exact for the Galilean invariant
case when the dispersion is exactly k2 , but also holds,
2m
Theintegraloverxisultra-violetconvergentandonecan up to corrections of order (E2 +E2)/Λ2, for arbitrary
0 F
safelyreplacetheupperlimitbyinfinity. Wethenobtain lattice dispersion.41
At E (cid:29) E , ρ is parametically larger than T ∼
Econd =−N0(cid:20)EF2 + ∆42 − µ2 (cid:16)µ+(cid:112)µ2+∆2(cid:17)(cid:21) (16) (cEosFtlEy0)aF1n/d2. TAs0≈thesTcon,seiq.eu.e,ncfeer,mpihoansiec flpuacitrusactioonndisnesnasree
c ins
almost immediately after they develop. In the oppo-
Using µ = E − E and ∆2 = 4E E (see Eq. (8))
F 0 F 0 site limit E (cid:28) E , ρ (T = 0) (cid:28) T , and hence
(cid:112) F 0 s ins
we immediately obtain µ + µ2+∆2 = 2EF and T ∼ ρ (T = 0) (cid:28) T . Using the criterium38 T =
(cid:16) (cid:112) (cid:17) c s ins c
µ µ+ µ2+∆2 /2 = E2 −∆2/4. Substituting into (π/2)ρ (T) and approximating ρ (T) by ρ (T = 0), we
F s s s
obtain an estimate T =E /8. (A more accurate analy-
(16) we obtain c F
sis42 yields T ∼E /log(logE /E ).)
c F 0 F
∆2 The superconducting transition temperature ap-
Econd =−N0 2 =−N0E0EF (17) proaches zero as O(EF) when EF → 0, while Tins ∼
noTmhaettperrefwachtaotrtfhoerrtahteioqE2Ft/eErm0 iisn. Eq.(14) determines sEc0a/lelsogasEEEFE0F0dlroogpEsEF0onalyndloogbavriiothumslyicvaallnyi.shTehsewrhaetinoETFc/=Tin0s.
InthetemperatureregionbetweenT andT thebound
ins c
ρ (T =0):
s pairs develop but remain incoherent. In Fig.5 we plot
T and T as functions of E /E .
ρ (T =0)=N ∆2 (cid:90) dε (cid:0)ddεkk(cid:1)2 (18) inTshesplitctingbetweenTinsa0ndTFconceEF getssmaller
s 0 8 k((εk−µ)2+∆2)3/2 than Tins (BCS-BEC crossover) and the corresponding
8
interatomic distance a ∼ 1/(mΛ)1/2 ∼ 1/q , where
0 max
q = (2mΛ)1/2. The ratio ξ /a ∼ (Λ/E )1/2 is
max 0 0 F
large, hence fermions in a bound pair are on average
located much farther away from each other than inter-
atomic spacing. Hence, the pairs cannot be viewed as
”molecules” in the real space. In this respect our result
differs from the analysis in Ref.13, where it was argued
that at E (cid:28) E , ξ becomes much smaller than the
F 0 0
interatomic spacing.
Ourresultsfortheone-bandmodeldifferfromRefs.8,30
FIG. 5: The onset temperature T for the bound state for-
ins where T was found to remain finite at E =0. The au-
mationandthesuperconductingtransitiontemperatureT in c F
c thors of8,30 solved BCS-like equations, hence their T is
theone-bandmodelasfunctionsofE /E . Thetemperatures c
0 F
in fact the onset temperature for the pairing, T . Still,
are normalized to EF (a) and to E0 (b). Observe that Tins ins
scales as E0/logE0/EF at large E0/EF. This Tins increases wefoundthateventhistemperaturevanishesatEF =0,
when plotted in units of E and decreases when plotted in once one includes into consideration temperature varia-
F
units of E . tion of the chemical potential.
0
preformed pairs behavior at Tins > T > Tc has been C. The density of states at T =0
originally studied in 3D systems.15,20,21 The physics in
2D is similar, but there is one important difference – the
In a conventional BCS superconductor with E (cid:29)T ,
distance between fermions in a bound pair (the coher- F c
µ(T =0) is positive, and the density of states (DOS) at
(cid:113)
ence length ξ ) scales as ξ ∼ |µ|/∆∼1/k , while the T =0 is, for electron dispersion
0 0 m F
N(ω)=−1Im(cid:90) d2kG (k,ω)=Im(cid:90) d2k (cid:0)u2δ(ω−E )+v2δ(ω+E )(cid:1)
π 4π2 s 4π2 k k k k
(cid:32) √ (cid:33)
= N0 √ 2ω θ(ω−∆)− ω−√ ω2−∆2θ(cid:16)ω−(cid:112)µ2+∆2(cid:17)
2 ω2−∆2 ω2−∆2
(cid:32) √ (cid:33)
+N0 √ 2|ω| θ(−ω−∆)− |ω|√+ ω2−∆2θ(cid:16)−ω−(cid:112)µ2+∆2(cid:17) , (22)
2 ω2−∆2 ω2−∆2
(cid:112)
where E = (ε −µ)2+∆2, N = m/(2π), θ(x) = 1 negative (µ(T =0)≈−E ), and the DOS is given by
k k 0 0
for x>0, µ=µ(T =0), and
1 (cid:90) d2k
N(ω)=− Im G (k,ω)
π 4π2 s
1(cid:32) ε −µ (cid:33) =Im(cid:90) d2k (cid:0)u2δ(ω−E )+v2δ(ω+E )(cid:1)
u2 = 1+ k , 4π2 k k k k
k 2 (cid:112)(εk−µ)2+∆2 (cid:32) √ (cid:33)
(cid:32) (cid:33) = N0 ω+√ ω2−∆2 θ(cid:16)ω−(cid:112)µ2+∆2(cid:17) (24)
1 ε −µ
v2 = 1− k , (23) 2 ω2−∆2
k 2 (cid:112)(εk−µ)2+∆2 (cid:32) √ (cid:33)
+N0 |ω|√− ω2−∆2 θ(cid:16)−ω−(cid:112)µ2+∆2(cid:17),
2 ω2−∆2
This N(ω) vanishes at |ω|<∆, has a square-root singu-
where µ = µ(T = 0). This N(ω) vanishes at
(cid:112)
larity 1/ |ω−∆| above the gap, and drops by a finite (cid:112)
|ω| < µ2+∆2 and jumps to a finite value at |ω| =
(cid:112)
amount at |ω| = µ2+∆2 + 0, when |ω| crosses the (cid:112)
µ2+∆2 + 0. Because µ ≈ −E , is much larger
0
edgeoftheband. TheDOSisnearlysymmetricbetween
than ∆ = 2(E E )1/2, the coherence factors u2 and
positive and negative ω, at least for ∆<ω (cid:28)E . F 0 k
F v2 are quite different: u2 ≈ 1 for all momenta, while
k k
In our case, this behavior holds for the case E (cid:29)E , v2 ≈ ∆2 is small. As the result, N(ω) in (24) is
F 0 k 4(εk+|µ|)
but not for E (cid:28) E . In the latter case, µ(T = 0) is highlyanisotropicbetweenpositiveandnegativefrequen-
F 0
9
between fermions on the two bands. The repulsive in-
teraction of this type gives rise to s+− pairing with the
nits) 1 b. units) 0.1 phase shift by π between ∆’s on the two bands.
arb. u OS (ar0.05 In this section we consider the model with one band
DOS (0.5 D 0-3 -2.5 -2 -1.5 -1 -0.5 0 wwiitthhehleoclet-rloikne-lidkiespdeisrpsieornsioεnhkε=e =EF,kh2−−2Ekm2h .anTdhiasnmoothdeerl
/ E0 is relevant to FeSCs, at least,kat a2mquealitaFt,ieve level. The
0 Fermi energies EF,h and EF,e and the masses mh and
-3 -2 -1 0 1 2 3 m are generally not equivalent. We keep E and E
/ E e F,h F,e
0 different,butsetm =m =mtosimplifytheformulas.
h e
FIG. 6: The DOS in the single-band model at T = 0 for
EF (cid:28) E0.√We set EF = 0.1E0, in which case µ = −9EF BCSanalysisofthepairinginmulti-bandmodelswith
and ∆=2 E0EF ≈6.3EF. We added the fermionic damp- two electron bands and two or three hole bands, as in
ing γ = 0.001E . In the clean limit, the density of states
F (cid:112) FeSCs, has been presented in series of recent publica-
vanishes at |ω| < µ2+∆2 and jumps to a finite value at
tions8,30. In particular, Ref.8 considered the case of
(cid:112)
|ω| = µ2+∆2 +0 (see Eq. (24)). Due to the difference
two hole bands, only one of which crosses the Fermi
between coherence factors, the DOS is strongly particle-hole
level. A FS-constrained superconductivity in this last
anisotropicandhasmuchlargervalueatpositivefrequencies,
case emerges due to interaction between the hole band
unobservableinphotoemissionexperiments. Atlarge|ω|,the
DOStendstoafinitevalueatpositiveωandvanishesas1/ω2 with EF,h > 0 and the electron band(s). Ref.8 has
atnegativeω. Tomakethepower-lawsuppressionoftheDOS demonstrated that the presence of the additional hole
atnegativeω morevisible,weplotthenegativefrequencyre- band increases Tc, despite that this band is full located
gion separately in the inset. below the Fermi level.
cies–itisapproachesN atlargepositivefrequenciesand Weanalyzedifferentphysics–thecrossoverinthesys-
0
scales as N0∆2/(4ω2) at large negative frequencies. We tem behavior once the largest EF becomes smaller than
plot the DOS at zero temperature for EF (cid:28) E0 in Fig. thetwo-particleboundstateenergyE0. Thisphysicshas
6. Because only negative frequencies are probed in pho- not been analyzed before, to the best of our knowledge.
toemission experiments, the features associated with the We restrict to one hole and one electron band because
bound state development below T are weak and dis- the inclusion of additional bands affects the details of
ins
appearatE →0. Thislastfeaturehasbeenalsofound the analysis but does not qualitatively affect BCS-BEC
F
in the recent study of superconductor-insulator transi- crossover. Like in the previous Section, we set the upper
tion.16 energycutoffatΛ(cid:29)EF,i(i=h,e),approximateU(q,Ω)
Note in passing that within our approximate treat- by a constant below the cutoff, and set the dimension-
ment, based on the effective quadratic Hamiltonian, the less coupling coupling λ=mU/(2π) to be small. We set
DOS vanishes below |ω| = (µ2 + ∆2)1/2 already at U > 0, in which case superconducting order parameter
T < T . A more accurate treatment would require has s+− symmetry. As our goal is to analyze BCS-BEC
ins
one to compute the imaginary part of the fermionic self- crossover,weconsidertheparticularcasewhenEF,e =0,
energy at a finite temperature and analyze the feed- see Fig.2(b). The extension of the analysis to small but
back on this self-energy from the development of the finite EF,e (positive or negative) is straightforward and
boundpairs. Ongeneralgrounds,thedensityofstatesat does not bring qualitatively new physics.
|ω|<(µ2+∆2)1/2 should remain finite at temperatures
between T andT , asnosymmetryisbrokeninthisT
ins c
Theanalysisoftheboundstateenergyfortwoparticles
range. BelowT ,however,thetruegapdevelopsatthese
c
at E ≡0 does not differ from that in previous Section,
frequencies and the DOS should be as in Fig. 6. F
and the result is that the scattering amplitude diverges
at T = 1.13E , where, like before, E = Λe−2/λ. The
0 0 0
bound state energy at T =0 is 2E .
0
III. TWO-BAND MODEL WITH ONE HOLE
AND ONE ELECTRON BAND
TheonsettemperatureforthepairingatafiniteE is
F
We now extend the analysis to two-band models. We obtained by solving simultaneously the linearized equa-
consider two models – one with a hole and an electron tions for ∆ and ∆ and the equations for the chemical
e h
band, and one with two hole/two electron bands In both potentials µ (T) and µ (T), subject to µ (T)+µ (T)=
e h e h
casesweassume,tomakepresentationcompact,thatthe E . The equation for the chemical potential follows for
F
dominant pairing interaction U(q,Ω) is the pair hopping the conservation of the total number of fermions. The
10
set of equations is (µ =µ (T ),µ =µ (T )): Hence
e e ins h h ins
λ (cid:90) Λ dx x Tins =1.13D1/3EF1/3E02/3 =1.04EF1/3E02/3 (31)
∆ = − ∆ tanh
e 2 h x 2T
−µh ins This expression is valid when λlogEF/E0 (cid:28)1. At even
λ (cid:90) Λ dx x larger E ≤ Λ, when λlogE /E = O(1), T is given
F F 0 ins
∆ = − ∆ tanh
h 2 e x 2T by
−µe ins
µe = T log11++ee−µµeh//TT Tins ∼Λ(cid:18)EΛF(cid:19)1/4e−√λ2 (32)
µ + µ =E . (25)
e h F
The ratio of the gaps on electron and hole bands at
The first two equations reduce to T =T −0 is
ins
4 =(cid:90) Λ dxtanh x ×(cid:90) Λ dxtanh x (26) ∆e ≈−(cid:18)1+ λlogEF(cid:19) (33)
λ2 −µe x 2Tins −µh x 2Tins ∆h 6 E0
We see that the gap on the electron band, which touches
BelowT ,∆ and∆ becomenon-zeroandonehasto
ins e h the Fermi level, is larger than the gap on the hole band,
consider non-linear gap equations and modify the equa-
which crosses Fermi level. This result indeed holds even
tion for the chemical potential. At T = 0 the set of
when E becomes negative, i.e., the electron band is
equations becomes [µ =µ (T =0),µ =µ (T =0)]: F
h h e e above the Fermi level. The ratio of ∆ /∆ increases
√e h
λ 2Λ when EF gets larger and approaches 2 when EF be-
∆h =−2∆elog(cid:112)µ2+∆2−µ comes of order Λ.
e e e At T = 0, solution of the set (27) at E (cid:29) E but
F 0
λ 2Λ
λlogE /E (cid:28) 1 shows that the ratio of ∆ and ∆
∆ =− ∆ log F 0 e h
e 2 h (cid:112)µ2 +∆2 −µ remains the same as in Eq. (33), i.e., up to subleading
h h h
(cid:113) (cid:112) terms ∆e(T = 0) = −∆h(T = 0) = ∆. Solving for the
µ2 +∆2 +µ −2E =µ + µ2+∆2
h h h F e e e chemical potentials we then find
µ +µ =E (27)
h e F ∆
µ (T =0)=− √ , µ (T =0)=E −µ (T =0) (34)
e h F e
2 2
A. The case E (cid:29)E
F 0 Substituting this into the first two equations in Eq. (27)
and solving for ∆ we obtain
We assume and then verify that in this situation
T (cid:28) E and for all T ≤ T , µ ≈ E , while ∆=25/6E1/3E2/3 =1.78E1/3E2/3 (35)
ins F ins h F F 0 F 0
µ ∼ T Under these assumptions, the equations on
e ins
the chemical potentials in (25) yield The minimum energy on the hole band is Eh = ∆, at
k ≈k . The minimum energy on the electron band is at
√ F (cid:112) √
5−1 k =0, and Ee = µ2e(T =0)+∆2 =3∆/2 2=1.06∆.
µ (T ) = T log =−0.48T ,
e ins ins 2 ins FortheratiooftheminimalenergytoTins wethenhave,
up to corrections of order λlogE /E ,
µ (T ) = E −µ (T ) (28) F 0
h ins F e ins
E E
Substituting these values of the chemical potentials into e =1.71, h =1.81 (36)
T T
the first two equations in (25) we obtain after simple ins ins
algebra
Note that both ratios are rather close to BCS values,
although our consideration includes the renormalization
(cid:20) (cid:21)
λ 1.13Λ 2 E
∆ = − ∆ 2log − +log F of the chemical potential, neglected in BCS theory.
e 2 h T λ E
ins 0
λ 1.13DΛ
∆ = − ∆ log (29)
h 2 e Tins B. The case EF =E0
where D = 0.79 (logD = −(cid:82)0|µe|/2T dxtanxhx). Combin- To establish the bridge to the case of small EF/E0,
ing the two equations and introducing Z = log1.13DΛ, consider the intermediate case when EF is compara-
we obtain at small λ, Tins ble to E0. To be specific, we just set EF = E0, al-
though the analysis can be easily extended to arbitrary
2 1 E E /E ∼ O(1). Because E is now the only relevant
Z = − log F +O(λ). (30) F 0 F
λ 3 D2E0 low-energy scale, we express Tins = aEF,µe(Tins) =