Superconductivity in iron-based, magnesium diborides, and other novel superconducting materials has a strong multi-band and multi-gap character. Recent experiments support the possibillity for a BCS-BEC crossover induced by strong-coupling and proximity of the chemical potential to the band edge of one of the bands. Here we study the simplest theoretical model which accounts for the BCS-BEC crossover in a two-band superconductor, considering tunable interactions and tunable energy separations between the bands. Mean-field results for condensate fraction, correlation length, and superconducting gap are reported in typical crossover diagrams to locate the boundaries of the BCS, crossover, and BEC regimes. When the superconducting gap is of the order of the local chemical potential, superconductivity is in the crossover regime of the BCS-BEC crossover and the Fermi surface of the small band is smeared by the gap opening. In this situation, small and large Cooper pairs coexist in the total condensate, which is the optimal condition for high-Tc superconductivity. The ratio between the gap and the Fermi energy in a given band results to be the best detection parameter for experiments to locate the system in the BCS-BEC crossover. Using available experimental data, our analysis shows that iron-based superconductors have the partial condensate of the small Fermi surface in the crossover regime of the BCS-BEC crossover, supporting the recent ARPES findings.
Ultranarrow superconductors are in the strong quantum confinement regime with formation of multiple coherent condensates associated with the many subbands of the electronic structure. Here we analyze the multiband BCS-BEC crossover induced by the chemical potential tuned close to a subband bottom, in correspondence of a superconducting shape resonance. The evolution of the condensate fraction and of the pair correlation length in the ground state as functions of the chemical potential demonstrates the tunability of the BCS-BEC crossover for the condensate component of the selected subband. The extension of the crossover regime increases when the pairing strength and/or the characteristic energy of the interaction get larger. Our results indicate the coexistence of large and small Cooper pairs in the crossover regime, leading to the optimal parameter configuration for high transition temperature superconductivity.
We analyze the interplay between superconductivity and the formation of bound pairs of fermions (BCS-BEC crossover) in a 2D model of interacting fermions with small Fermi energy
EF and weak attractive interaction, which extends to energies well above
EF. The 2D case is special because a two-particle bound state forms at arbitrary weak interaction, and already at weak coupling, one has to distinguish between the bound-state formation and superconductivity. We briefly review the situation in the one-band model and then consider two different two-band models: one with one hole band and one electron band and another with two hole or two electron bands. In each case, we obtain the bound-state energy
2E0 for two fermions in a vacuum and solve the set of coupled equations for the pairing gaps and the chemical potentials to obtain the onset temperature of the pairing
Tins and the quasiparticle dispersion at
T=0. We then compute the superfluid stiffness ${\ensuremath{\rho}}_{s}(T=0)$ and obtain the actual
Tc. For definiteness, we set
EF in one band to be near zero and consider different ratios of
E0 and
EF in the other band. We show that at ${E}_{F}\ensuremath{\gg}{E}_{0}$, the behavior of both two-band models is BCS-like in the sense that ${T}_{c}\ensuremath{\approx}{T}_{\mathrm{ins}}\ensuremath{\ll}{E}_{F}$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{T}_{c}$. At ${E}_{F}\ensuremath{\ll}{E}_{0}$, the two models behave differently: in the model with two hole/two electron bands, ${T}_{\mathrm{ins}}\ensuremath{\sim}{E}_{0}/ln\frac{{E}_{0}}{{E}_{F}}, \mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{({E}_{0}{E}_{F})}^{1/2}$, and ${T}_{c}\ensuremath{\sim}{E}_{F}$, like in the one-band model. In between
Tins and
Tc, the system displays a preformed pair behavior. In the model with one hole and one electron bands,
Tc remains of order
Tins, and both remain finite at
EF=0 and of the order of
E0. The preformed pair behavior still does exist in this model because
Tc is numerically smaller than
Tins. For both models, we reexpress
Tins in terms of the fully renormalized two-particle scattering amplitude by extending to the two-band case (the method pioneered by Gorkov and Melik-Barkhudarov back in 1961). We apply our results for the model with a hole and an electron band to Fe pnictides and Fe chalcogenides in which a superconducting gap has been detected on the bands that do not cross the Fermi level, and to FeSe, in which the superconducting gap is comparable to the Fermi energy. We apply the results for the model with two electron bands to Nb-doped
SrTiO3 and argue that our theory explains the rapid increase of
Tc when both bands start crossing the Fermi level.
Understanding the material parameters that control the superconducting transition temperature T c is a problem of fundamental importance.In many novel superconductors phase fluctuations determine T c , rather than the collapse of the pairing amplitude.We derive rigorous upper bounds on the superfluid phase stiffness for multiband systems, valid in any dimension.This in turn leads to an upper bound on T c in two dimensions, which holds irrespective of pairing mechanism, interaction strength, or order-parameter symmetry.Our bound is particularly useful for the strongly correlated regime of low-density and narrowband systems, where mean-field theory fails.For a simple parabolic band in 2D with Fermi energy E F , we find that k B T c ≤ E F =8, an exact result that has direct implications for the 2D BCS-BEC crossover in ultracold Fermi gases.Applying our multiband bound to magic-angle twisted bilayer graphene, we find that band structure results constrain the maximum T c to be close to the experimentally observed value.Finally, we discuss the question of deriving rigorous upper bounds on T c in 3D.
Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering t-matrix, the low-energy form of which is expressible in terms of the s-wave scattering length
as. The BCS gap equation for s-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap \ensuremath{\Delta} as well as the chemical potential \ensuremath{\mu} as functions of the dimensionless coupling variable \ensuremath{\lambda}\ensuremath{\equiv}
kFas, where
kF is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of \ensuremath{\lambda} and compared with experimental values.
We have analyzed the doping and temperature dependencies of the physical properties of a square-lattice narrow-band superconductor with a relevant for systems with local Hubbard repulsion nematic pairing due to effective nearest-neighbor attraction. Our focus was a possible crossover of the system from the Bardeen–Cooper–Schrieffer (BCS) superconductivity to the Bose–Einstein condensation (BEC) regimes with increasing temperature. We have found that in the d-channel the crossover does not take place at realistic values of doping and temperature, and even at a large attraction, i.e., the system is always in the BCS regime, unlike the “usual” d-wave superconductor. Contrary to non-nematic d-wave superconductors, the chemical potential μ always grows with temperature increase that might indicate an unexpectedly large size of Cooper pairs. At the same time, in the s-channel, we have found a BCS–BEC crossover with temperature for a wide range of temperatures even at rather high dopings, nf ≈ 0.2. This result is different from the “usual” s-case, where a narrow-band (i.e., at strong coupling) system is always in the BEC state. We have found for a given large value of attraction that there is a value of doping when for a wide range of temperature the system is in a “pre-crossover” regime with μ ≈ 0, i.e., in a regime of coexistence of local and extended pairs. Thus, in narrow-band systems a “usual” s-superconductor is in a BEC state, but in the nematic case one can expect a BCS–to–BEC crossover with temperature growth at realistic system parameters, i.e., the crossover might be measured experimentally.