Using the microscopic model Hamiltonian, the equations for order parameter and chemical potential have been obtained for a system of Fermi-particles interacting with phonons which permit an analysis of its superconducting properties to be made for an arbitrary fermion concentration. The solution of the equations is given that makes it possible to trace the crossover from a local pair regime to Cooper pairing. The type of superconducting behavior is shown to depend on both the position of the Fermi level with respect to a maximum phonon energy and strength of indirect interactions between fermions.
An attempt is made to analyze the state-of-the-art in the description of the superconducting phase in 2D metals with an arbitrary carrier density. It is shown that a decrease in the density leads to a nontrivial equation for chemical potential μ, in particular, to the inequality μ ≠ εF including μ < 0 (εF is the Fermi energy). In this case, the region of μ ≃ 0 corresponds to a transition (crossover) from Cooper pairing (with strongly overlapping pairs) and from the description of superconductivity in terms of the BCS–Bogoliubov theory to local (nonoverlapping) pairs and their Bose–Einstein condensation. Some examples of such a crossover are discussed.
The superconducting properties of the 2D fermion system with local and different types of the indirect boson-exchange attractions in the cases of s-wave and d-wave pairing are reviewed and analysed at T=0. In particular, the possibility of the crossover from the Bose-Einstein condensation regime to the BCS-like superconductivity with the carrier density and coupling changing in the case of different pairing channels is discussed. Gaussian fluctuations of the order parameter are taken into account and the carrier density dependence of the gap is studied in this case. The role of the interaction form in the physical behavior of the systems is also discussed.
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.
The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε 0 /ε F , where ε 0 is the two-fermion binding energy and ε F is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε 0 /ε F > 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε 0 /ε F «1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum q c ≃ l/ξ 0 (where ξ 0 is the Cooper pair size), and (ii) pair excitations with q ≃ (8 m μ) ½ , where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε 0 /ε F ≃¼. Introduction Since the discovery of high- T c superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ 0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- T c superconductors, most of which have a layered structure, have a short coherence length, so that k F ξ 0 ≃1, where k F is the Fermi momentum.
One- and two-dimensional attractive Hubbard models in the presence of magnetic field h are studied in a wide range of on-site interaction strength U, temperature T and electron concentration n by means of the generalized self-consistent field (GSCF) approach with renormalized chemical potential. Pairing, the smooth crossover from the BCS regime into the Bose condensation state and the inhomogeneous superconductivity with the non-uniform order parameter are investigated. The excitation spectrum, the momentum distribution function and thermodynamic properties, including the concentration of double occupied sites, the kinetic energy, and the chemical potential, are calculated. The numerical calculations are found in a good agreement with the Bethe-ansatz results and quantum Monte-Carlo calculations in a wide range of system parameters U, n and h. The GSCF theory correctly displays the separation of the energy gap from the order parameter (crossover) as n or |U| decrease. The concentration of double occupied sites and the bound electron pairs n pair are calculated and we found a simple relationship between the order parameter and n pair , valid for the arbitrary U and all n in any dimension. The developed GSCF approach in magnetic field suggests a smooth transition into the inhomogeneous superconducting state.