Impact of disorder on the superconducting transition temperature near a Lifshitz transition

Multiband superconductivity is realized in a plethora of systems, from high-temperature superconductors to very diluted superconductors. While several properties of multiband superconductors can be understood as straightforward generalizations of their single-band counterparts, recent works have unveiled rather unusual behaviors unique to the former case. In this regard, a regime that has received significant attention is that near a Lifshitz transition, in which one of the bands crosses the Fermi level. In this paper, we investigate how impurity scattering ${\tau }^{-1}$ affects the superconducting transition temperature $T_c$ across a Lifshitz transition, in the regime where intraband pairing is dominant and interband pairing is subleading. This is accomplished by deriving analytic asymptotic expressions for $T_c$ and $\partial T_c/\partial {\tau }^{-1}$ in a two-dimensional two-band system. When the interband pairing interaction is repulsive, we find that, despite the incipient nature of the band crossing the Fermi level, interband impurity scattering is extremely effective in breaking Cooper pairs, making $\partial T_c/\partial {\tau }^{-1}$ quickly approach the limiting Abrikosov-Gor'kov value of the high-density regime. In contrast, when the interband pairing interaction is attractive, pair-breaking is much less efficient, affecting $T_c$ only mildly at the vicinity of the Lifshitz transition. The consequence of this general result is that the behavior of $T_c$ across a Lifshitz transition can be qualitatively changed in the presence of strong enough disorder: Instead of displaying a sharp increase across the Lifshitz transition, as in the clean case, $T_c$ can actually display a maximum and be suppressed at the Lifshitz transition. These results shed light on the nontrivial role of impurity scattering in multiband superconductors.

Figure 1

Illustration of the two-band model used in this paper. Two electronlike parabolic and concentric bands are displaced by an energy ${\varepsilon }_0>0$. Their occupations are controlled by the chemical potential $\mu >0$. When $\mu$ becomes larger than ${\varepsilon }_0$, the second band becomes populated, signaling a Lifshitz transition (LT).

Figure 2

Phase diagram of the clean two-band SC: the first two panels show $T_c$ as function of the occupation number $N$ for (a) 2D bands and (b) 3D bands, for several values of the parameter ${\lambda }_{12}$. In panel (c), we compare $T_c$ as a function of $N$ with $T_c$ as a function of the chemical potential $\mu \left(T_c\right)$ for the 2D case with $|{\lambda }_{12}|=0.013$. In all panels, ${\lambda }_{11}={\lambda }_{22}=0.13$ and ${\rho }_{2,0}={\rho }_{1,0}$. Note that $T_c$ is normalized by the energy displacement between the bands ${\varepsilon }_0$ and $N$ is normalized by the critical occupation number $N_c$ at which the LT takes place.

Figure 3

Regions of the $(\mu ,T)$ phase diagram for the calculation of the asymptotic behavior of $T_c\left(\mu \right)$ in the clean and dirty regimes. The size of the regions are exaggerated for schematic purposes. The precise definition of each region is given in the main text.

Figure 4

Comparison between the numerical (symbols) and asymptotic analytical results (solid curve) for $T_c$, as function of the chemical potential $\mu$, for the 2D clean system across the Lifshitz transition at $\mu ={\varepsilon }_0$. Panel (b) is a zoom of panel (a) that highlights the very narrow range of $\mu$ for which the asymptotic solutions start to fail (gray dashed area). The parameters used here are the same as in Fig. 2.

Figure 5

Panel (a) shows the diagrammatic expansion for the self-energy in the self-consistent Born approximation. Panel (b) shows the Dyson's equation for the total Green's function according to the self-consistent Born approximation. The solid single lines represent ${\widehat{\mathcal{G}}}_0(\mathbf{k},{\omega }_n)$, while the dashed lines refer to the impurity potential ${\widehat{W}}_{\mathbf{k},{\mathbf{k}}^{\prime }}$.

Figure 6

Superconducting transition temperature $T_c$ as a function of the occupation number $N$ of a dirty two-band SC with 2D bands. The dashed line corresponds to finite interband impurity scattering (dirty system), whereas the solid line corresponds to the clean system. The parameters are the same as in Fig. 2; here, the intraband impurity scattering is set to zero.

Figure 7

The rate of suppression of $T_c$ by interband nonmagnetic impurity scattering ${\tau }_{\mathrm{inter}}^{-1}, {\left.\frac{\partial T_c}{\partial {\tau }_{\mathrm{inter}}^{-1}}\right|}_{{\tau }_{\mathrm{inter}}^{-1}=0}$ for repulsive (${\lambda }_{12}<0$, red curves) and attractive (${\lambda }_{12}>0$, blue curves) interband pairing interactions, in the high-density regime. In panel (a), the density of states of the two bands are set to be the same, but the intraband pairing interactions of the two bands, ${\lambda }_{11}$ and ${\lambda }_{22}$, are allowed to be different. In panel (b), ${\lambda }_{11}$ is set to be the same as ${\lambda }_{22}$, but the two density of states are allowed to be different, with $r={\rho }_{2,F}/{\rho }_{1,F}$. In both panels, the suppression rates are normalized by the magnitude of the Abrikosov-Gor'kov value of $-\pi /4$, corresponding to the suppression rate of $T_c$ of a single-band superconductor by magnetic impurity scattering.

Figure 8

Comparison between the numerical (symbols) and asymptotic analytical results (solid curves) for $T_c$, as function of the chemical potential $\mu$, for the 2D dirty system across the Lifshitz transition at $\mu ={\varepsilon }_0$. The parameters are the same as in Fig. 4 but with ${\lambda }_{12}<0$ (repulsive interband pairing interaction) in panel (a) and ${\lambda }_{12}>0$ (attractive interband pairing interaction) in panel (b). The interband impurity scattering ${\tau }_{\mathrm{inter}}^{-1}$ is set to ${\tau }_{\mathrm{inter}}^{-1}/{\varepsilon }_0={10}^{-3}$.

Figure 9

The rate of suppression of $T_c$ by interband impurity scattering, ${\left.\frac{\partial T_c}{\partial {\tau }_{\mathrm{inter}}^{-1}}\right|}_{{\tau }_{\mathrm{inter}}^{-1}=0}$ for attractive $[{\lambda }_{12}>0$, panel (a)] and repulsive $[{\lambda }_{12}<0$, panel (b)] interband pairing interactions, in the dilute regime. The insets highlight the asymptotic behaviors across the boundaries of regions II, III, and IV of Fig. 3. In both panels, the suppression rates are normalized by the absolute value of the Abrikosov-Gor'kov suppression rate of $-\pi /4$, corresponding to the case of a single-band superconductor by magnetic impurity scattering. The parameters used here are ${\rho }_{1,0}={\rho }_{2,0}, {\lambda }_{11}={\lambda }_{22}$, and ${\lambda }_{12}={\lambda }_{21}$.

Figure 10

Rate of enhancement of $T_c$ by changes in the chemical potential, $\frac{\partial T_c}{\partial \mu }$, for the clean 2D system. The parameters are the same as those used in Fig. 9. To make the comparison with that figure more transparent, we also normalize the rate of change of $T_c$ by $\pi /4$.

Figure 11

Numerical and asymptotic solutions for the Matsubara sum Eq. (A1). The dot-dashed blue and red lines are the asymptotic solutions for $\left|x\right|\ll 1$ and $\left|x\right|\gg 1$, while the solid line is the numerical result. The dashed vertical lines delimit the region where the asymptotic approximation begins to fail.

Figure 12

Different regions of the two-dimensional parameter space $|x_1|\times |x_2|$ in which the analytic expansions are performed. In region 3, the silver area around the line $|x_1|=|x_2|$ indicates the region where the approximations lose precision, since the neglected terms of order $\mathcal{O}\left(\frac{1}{\left|x_j\right|}{\left(\frac{\left|x_{<}\right|}{\left|x_{>}\right|}\right)}^2\right), j=1,2$, become more important.