Effects of disorder on coexistence and competition between superconducting and insulating states

We study effects of nonmagnetic impurities on the competition between the superconducting and electron-hole pairing. We show that disorder can result in coexistence of these two types of ordering in a uniform state, even when in clean materials they are mutually exclusive.

Figure 1

Schematic picture of the holelike (a) and electronlike (b) bands.

Figure 2

The dependence of the superconducting (solid line) and excitonic (dashed line) order parameters on the Matsubara frequency $\omega$ (see Appendix for details).

Figure 3

The temperature vs disorder strength phase diagram of the model Eq. (1) for ${\lambda }_1^{-1}-{\lambda }_2^{-1}=0.5$. The dimensionless disorder strength is $\Gamma ∕{\Delta }_2(T=0,\Gamma =0)$ and temperature is measured in units of $T_{c2}(\Gamma =0)$. The EI and SC state are separated by a thin domain, in which the two order parameters coexist.

Figure 4

The zero-temperature phase diagram. Close to the quantum critical point $\eta =\Gamma =0$, the coexistence region is extremely thin. For $\eta ⪡1$ it practically coincides with the disorder interval, in which the EI is gapless.

Figure 5

The plots show the dependence of the superconducting order parameter ${\Delta }_1$ (dashed line) and the excitonic order parameter ${\Delta }_2$ (thin solid line) on the disorder strength $\Gamma$ in the region of coexistence of the two phases for $\eta =0.1$ and $T=0$. For ${\Gamma }_1<\Gamma <{\Gamma }_2,{\Delta }_1$ and ${\Delta }_2$ vary very fast, while $\Delta =\sqrt{{\Delta }_1^2+{\Delta }_2^2}$ (thick line) stays approximately constant.

Figure 6

The plot shows the energy minimum versus the angle $\varphi$ at $\eta =0.5$ and for different values of $\Gamma :\Gamma =0.234<{\Gamma }_1$ (solid line), $\Gamma \sim {\Gamma }_{fo}=0.237$ (dashed line), and $\Gamma =0.240>{\Gamma }_2$ (dot-dashed line).