Superconductivity versus bound-state formation in a two-band superconductor with small Fermi energy: Applications to Fe pnictides/chalcogenides and doped ${\mathrm{SrTiO}}_3$

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 $E_F$ and weak attractive interaction, which extends to energies well above $E_F$. 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 $2E_0$ 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 $T_{\mathrm{ins}}$ and the quasiparticle dispersion at $T=0$. We then compute the superfluid stiffness ${\rho }_s(T=0)$ and obtain the actual $T_c$. For definiteness, we set $E_F$ in one band to be near zero and consider different ratios of $E_0$ and $E_F$ in the other band. We show that at $E_F\gg E_0$, the behavior of both two-band models is BCS-like in the sense that $T_c\approx T_{\mathrm{ins}}\ll E_F$ and $\mathrm{\Delta }\sim T_c$. At $E_F\ll E_0$, the two models behave differently: in the model with two hole/two electron bands, $T_{\mathrm{ins}}\sim E_0/ln\frac{E_0}{E_F}, \mathrm{\Delta }\sim {\left(E_0{E}_F\right)}^{1/2}$, and $T_c\sim E_F$, like in the one-band model. In between $T_{\mathrm{ins}}$ and $T_c$, the system displays a preformed pair behavior. In the model with one hole and one electron bands, $T_c$ remains of order $T_{\mathrm{ins}}$, and both remain finite at $E_F=0$ and of the order of $E_0$. The preformed pair behavior still does exist in this model because $T_c$ is numerically smaller than $T_{\mathrm{ins}}$. For both models, we reexpress $T_{\mathrm{ins}}$ 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 ${\mathrm{SrTiO}}_3$ and argue that our theory explains the rapid increase of $T_c$ when both bands start crossing the Fermi level.

Figure 1

Energy scales relevant to the interplay between the formation of bound pairs of fermions and true superconductivity in 2D fermionic systems with weak attractive pairing interaction. $\mathrm{\Lambda }$ is the upper energy cutoff, $E_F$ is the Fermi energy, and $E_0$ is the energy of a bound state of two fermions in a vacuum, i.e., at $\mu =0$. At weak coupling, $E_0\ll \mathrm{\Lambda }$. We assume that $E_F$ is also small and can be tuned by doping to be either larger or smaller than $E_0$. We show that in one-band model and in two-band model with two hole or two electron bands, the system displays BCS-like behavior at $E_F\gg E_0$ and BEC-like behavior at $E_F\ll E_0$. In the latter case, bound pais develop at $T_{\mathrm{ins}}\sim E_0/ln\frac{E_0}{E_F}$ but the true superconductivity with full phase coherence develops at $T_c\sim E_F$. In between $T_{\mathrm{ins}}$ and $T_c$, the system displays preformed pair behavior and the spectral function displays pseudogap behavior. In the two-band model with one hole and one electron pockets, $T_{\mathrm{ins}}$ and $T_c$ also split when $E_F$ gets smaller than $E_0$, but both remain of order $E_0$ even when $E_F$ vanishes. Still, the superconducting $T_c$ in this limit is several times smaller than $T_{\mathrm{ins}}$, so there is a wide temperature range of preformed pair behavior.

Figure 2

The onset temperature $T_{\mathrm{ins}}$ for the bound-state formation and the superconducting transition temperature $T_c$ in the three models that we consider—the model with one electron band (a), with one electron and one hole bands (b), and with two electron bands (c). For all three cases, we plot $T_{\mathrm{ins}}$ and $T_c$ in units of $E_F$, as functions of $E_0/E_F$, where $E_0$ is the bound-state energy for two free fermions in 2D. Dashed lines show $T_c$ and $T_{\mathrm{ins}}$ in the intermediate regime $E_F/E_0=O\left(1\right)$, where we did not obtain the explicit formulas. The highest superconducting $T_c$ at small $E_F$ is for the model with one hole and one electron pockets. In this model, both $T_{\mathrm{ins}}$ and $T_c$ are of order $E_0$ at large $E_0/E_F$.

Figure 3

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 nonzero $E_F$) and (b) a two-band model with one hole and 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.

Figure 4

The dispersion in the one-band model. Red dashed line—the bare dispersion (the one that the system would have at $T=0$ in the absence of the pairing). Black line: the dispersion right above $T_{\mathrm{ins}}$; blue line: the dispersion below $T_{\mathrm{ins}}$. The plot is for the case when the chemical potential $\mu$ is already negative at $T=T_{\mathrm{ins}}$. Observe that the minimal gap is $\sqrt{{\mathrm{\Delta }}^2+{\mu }^2}$ and the minimum of the dispersion is at $k=0$ rather than at $k_F$.

Figure 5

Diagrammatic representation of the kinetic and potential energy of a one-band superconductor. The sum $E_{\text{kin}}+E_{\text{pot}}(q=0)$ gives the condensation energy, and the prefactor for the $q^2$ term in $E_{\text{pot}}\left(q\right)$ determines the superfluid stiffness.

Figure 6

The onset temperature $T_{\mathrm{ins}}$ for the bound-state formation and the superconducting transition temperature $T_c$ in the one-band model as functions of $E_0/E_F$. The temperatures are normalized to $E_F$ (a) and $E_0$ (b). Dashed lines show $T_c$ and $T_{\mathrm{ins}}$ in the intermediate regime $E_F/E_0=O\left(1\right)$, where we did not obtain the explicit formulas. Observe that $T_{\mathrm{ins}}$ scales as $E_0/ln{E}_0/E_F$ at large $E_0/E_F$. This $T_{\mathrm{ins}}$ increases when plotted in units of $E_F$ and decreases when plotted in units of $E_0$.

Figure 7

The DOS in the single-band model at $T=0$ for $E_F\ll E_0$. We set $E_F=0.1E_0$, in which case $\mu =-9E_F$ and $\mathrm{\Delta }=2\sqrt{E_0{E}_F}\approx 6.3E_F$. We added the fermionic damping $\gamma =0.001E_F$. In the clean limit, the density of states vanishes at $\left|\omega \right|<\sqrt{{\mu }^2+{\mathrm{\Delta }}^2}$ and jumps to a finite value at $\left|\omega \right|=\sqrt{{\mu }^2+{\mathrm{\Delta }}^2}+0$ [see Eq. (24)]. Due to the difference between coherence factors, the DOS is strongly particle-hole anisotropic and has much larger value at positive frequencies, unobservable in photoemission experiments. At large $\left|\omega \right|$, the DOS tends to a finite value at positive $\omega$ and vanishes as $1/{\omega }^2$ at negative $\omega$. To make the power-law suppression of the DOS at negative $\omega$ more visible, we plot the negative frequency region separately in the inset.

Figure 8

Fermionic dispersions for the two-band model with one hole and one electron band in the limit $E_F\ll E_0$. Red dashed line: the bare dispersions (the one which the system would have at $T=0$ in the absence of the pairing). Black lines: the dispersions right above $T_{\mathrm{ins}}$, blue lines: the dispersion below $T_{\mathrm{ins}}$. The chemical potential for the electron band is negative, and the minimal gap is $\sqrt{{\mathrm{\Delta }}_e^2+{\mu }^2}$. For the hole band, the chemical potential is positive and the dispersion $\pm \sqrt{{\mathrm{\Delta }}_h^2+{(k^2/\left(2m\right)-{\mu }_h)}^2}$ is nonmonotonic, with the minimal energy ${\mathrm{\Delta }}_h$ at $k_F=\sqrt{2m{\mu }_h}$. Note that this $k_F$ is larger than the bare $k_{F,0}=\sqrt{2m{E}_F}$.

Figure 9

Diagrammatic representation of the two-loop diagram for the potential energy $E_{\mathrm{pot}}\left(q\right)$ in two-band models with a constant interband interaction $U$ (the dashed line). Dark and light lines represent fermions from two different bands. The prefactor for $q^2$ term in $E_{\mathrm{pot}}\left(q\right)$ determines the superfluid stiffness. For a constant (i.e., angle-independent) $U$, the $q^2$ term appear by expanding $GG$ or $FF$ terms either on the right side of on the left side of each diagram, but there is no cross-term from taking linear in $q$ terms on the right and on the left.

Figure 10

The onset temperature $T_{\mathrm{ins}}$ for the bound-state formation and the superconducting transition temperature $T_c$ in the two-band model with one hole and one electron Fermi surface, as functions of $E_0/E_F$. The temperatures are in units of $E_F$ (a) and $E_0$ (b). Observe that both $T_{\mathrm{ins}}$ and $T_c$ scale as $E_0$ at large $E_0/E_F$.

Figure 11

The DOS at $T=0$ for the model with one hole and one electron pockets. (a) $E_0=E_F$ and (b) $E_0=10E_F$. For $E_0=E_F, {\mu }_e=-0.34E_F,{\mu }_h=1.34E_F$, and ${\mathrm{\Delta }}_h\approx -{\mathrm{\Delta }}_e=2.43E_F$. For $E_F=0.1E_0, {\mu }_h\approx \frac{3E_F}{2}, {\mu }_e\approx -\frac{E_F}{2}$, and ${\mathrm{\Delta }}_e\approx -{\mathrm{\Delta }}_h\approx 20.3E_F$. To cut the singularities and make other features of DOS visible, we added fermionic damping $\gamma =0.001E_F$. Main figures: the DOS separately for the hole band (dashed blue) and the electron band (dashed red). For the hole band, the DOS has a square-root singularity at $\left|\omega \right|=|{\mathrm{\Delta }}_h|+0$, symmetric between negative and positive frequencies, and van Hove singularity at $\left|\omega \right|=\sqrt{{\mathrm{\Delta }}_h^2+{\mu }_h^2}+0$ [see Eq. (22); for hole dispersion positive and negative frequencies in (22) have to be interchanged]. The latter is stronger for positive frequencies, due to anisotropy of coherence factors. For $E_F=0.1E_0$, the singularity and the discontinuity are almost undistinguishable. For the electron band, the DOS jumps to finite value at $\left|\omega \right|=\sqrt{{\mathrm{\Delta }}_e^2+{\mu }_e^2}+0$ and is highly anisotropic between negative and positive frequencies, again due to anisotropy of the coherence factors [see Eq. (24)]. For $E_F=0.1E_0, \mathrm{\Delta }>>{\mu }_e$, and the DOS right after the jumps are large, of order $\mathrm{\Delta }/{\mu }_e$. (Insets) The total DOS. Observe that at large frequencies the total DOS tends to a finite value for both positive and negative $\omega$.

Figure 12

Fermionic dispersions for the two-band model with two electron bands in the limit $E_F\ll E_0$. Red dashed line – the bare dispersions (the one which the system would have at $T=0$ in the absence of the pairing). Black lines – the dispersions right above $T_{ins}$, blue lines – the dispersion below $T_{ins}$. The chemical potentials for both bands are now negative at $T\ge T_{ins}$ and in the BEC state below $T_{ins}$, and the minimal gaps on each band is $\sqrt{{\mathrm{\Delta }}^2+{\mu }^2}$. The minimum of dispersion on both bands is at $k=0$. This is very similar to the behavior of the one-band model (see Fig. 4).

Figure 13

The onset temperature $T_{\mathrm{ins}}$ for the bound-state formation and the superconducting transition temperature $T_c$ in the two-band model with two electron bands, as functions of $E_0/E_F$. The temperatures are normalized to $E_F$ (a) and to $E_0$ (b). Like in the case of one-band model, $T_{\mathrm{ins}}$ scales as $E_0/ln{E}_0/E_F$ at large $E_0/E_F$, while $T_c$ scales as $E_F$.

Figure 14

The DOS at $T=0$ for the model with two electron bands. (a) $E_0=E_F$ and (b) $E_0=10E_B$. For $E_0=E_F, {\mu }_1=-0.05E_F,{\mu }_2=-1.05E_F$, and ${\mathrm{\Delta }}_1\approx -{\mathrm{\Delta }}_e=1.38E_F$. For $E_F=0.1E_0, {\mu }_1\approx -24.6E_F, {\mu }_2\approx -25.6E_F$, and ${\mathrm{\Delta }}_1\approx {\mathrm{\Delta }}_2=4.47E_F$. We introduced the damping $\gamma =0.001E_F$ to make all features of the DOS visible. Main figures: the DOS for each band (dashed red and blue lines). (Insets) The total DOS. In the clean limit, the DOS has a discontinuity at $\left|\omega \right|=\sqrt{{\mathrm{\Delta }}_i^2+{\mu }_i^2}+0$ ($i=1,2$), like in the one-band model. At $E_0=E_F$, DOS on band 1 is large immediately after the jump because ${\mu }_1$ is very small. The DOS for each band is anisotropic between negative and positive frequencies due to the anisotropy of the coherence factors. At large frequencies, the DOS tends to a finite value for positive $\omega$ and scales as $1/{\omega }^2$ for negative $\omega$. The arrow in (b) indicates the position of the would be discontinuity at a negative frequency.

Figure 15

Hartree-Fock diagrams for the self-energy.

Figure 16

Second-order diagrams for the renormalization of the irreducible pairing interaction due to contributions from the particle-hole channel.