Generalization of Anderson's theorem for disordered superconductors

We show that at the level of BCS mean-field theory, the superconducting $T_c$ is always increased in the presence of disorder, regardless of order parameter symmetry, disorder strength, and spatial dimension. This result reflects the physics of rare events—formally analogous to the problem of Lifshitz tails in disordered semiconductors—and arises from considerations of spatially inhomogeneous solutions of the gap equation. So long as the clean-limit superconducting coherence length, ${\xi }_0$, is large compared to disorder correlation length, $a$, when fluctuations about mean-field theory are considered, the effects of such rare events are small (typically exponentially in ${[{\xi }_0/a]}^d$); however, when this ratio is $\sim 1$, these considerations are important. The linearized gap equation is solved numerically for various disorder ensembles to illustrate this general principle.

Figure 1

Effect of disorder on $s$-wave pairing: Log-linear plot of disorder-averaged $\rho \left(\lambda \right)$ with attractive on-site ($s$-wave) interactions ($V_1=0$) at temperature $T=0.05$. The blue dashed curve is for uniform $V_0=-U$ and Born scattering with $W=3\overline{t}$. The other curves are for the case of negative $U$ centers where $V_0=-U$ with probability $x=1/2$ and $V_0=0$ with probability $(1-x)$; the yellow dashed curve includes additional Born scattering with $W=3\overline{t}$ while the red solid curve has no additional single-particle disorder ($W=0$). The dotted vertical (black) line corresponds to the maximum eigenvalue of the disorder-free system, ${\lambda }_{\max }(W=0,V_0=-U,T=0.05)$, while the dashed vertical line represents ${\lambda }_{\max }(W=0,V_0=-Ux,T=0.05)$ with $x=1/2$.

Figure 2

Effect of disorder on $d$-wave pairing: Disorder-averaged $\rho \left(\lambda \right)$ with uniform interactions, $V_0=+U$, $V_1=-1.5U$ at $T=0.05$. The blue solid line is for Born scattering with $W=3$, and the red dashed line is for unitary scattering with $n_{\text{imp}}=10\%$. The dashed and dotted vertical lines represent maximum and minimum eigenvalues, respectively, for disorder-free system, $W=0$ and $n_{\text{imp}}=0$.

Figure 3

Effect of disorder on $s$-wave pairing: Log-linear plot of disorder-averaged $\rho \left(\lambda \right)$ with attractive on-site ($s$-wave) interactions ($V_1=0$) at temperature $T=1$. The blue dashed curve is for uniform $V_0=-U$ and Born scattering with $W=3\overline{t}$. The other curves are for the case of negative $U$ centers where $V_0=-U$ with probability $x=1/2$ and $V_0=0$ with probability $(1-x)$; the yellow dashed curve includes additional Born scattering with $W=3\overline{t}$ while the red solid curve has no additional single-particle disorder ($W=0$). The dotted vertical (black) line corresponds to the maximum eigenvalue of the disorder-free system, ${\lambda }_{\text{max}}(W=0,V_0=-U,T)$, while the dashed vertical line represents ${\lambda }_{\text{max}}(W=0,V_0=-Ux,T)$ with $x=1/2$.

Figure 4

Disorder-averaged $\rho \left(\lambda \right)$ (blue) for $x=1$, $V_0=-4$, and $V_1=0$ with $n_{\text{imp}}=10\%$ unitary scattering at $T=0.05$. The maximum clean eigenvalue ${\lambda }_{\text{max},0}$ is represented by the vertical dashed line (black), and the inverse participation ratio by circles (red). Solutions with largest $\lambda /V_0\sim 2.3$ have maximum inverse participation ratio $I_{\lambda }=1$ and correspond to the impurity band of states localized on a single site. A second impurity band appears around $\lambda /V_0\sim 1.25$ with $I_{\lambda }=1/2$ corresponding to a solution localized on two sites.

Figure 5

Chemical potential, LDOS, and on-site pairing solutions with $n_{\text{imp}}=10\%$ unitary scatterers for largest (${\lambda }_1=9.7$) and second largest (${\lambda }_2=3.4$) eigenvalues for a particular disorder realization at $T=0.05t$. The peak in the local density of states is associated with a rare impurity cluster.