Signatures of nematic superconductivity in doped $\mathrm{Bi}{}_2\mathrm{Se}{}_3$ under applied stress

The $M_x{\text{Bi}}_2{\text{Se}}_3$ family are candidates for topological superconductors, where $M$ could be Cu, Sr, or Nb. Twofold anisotropy has been observed in various experiments, prompting the interpretation that the superconducting state is nematic. However, it has since been recognized in the literature that a twofold anisotropy in the upper critical field $H_{c2}$ is incompatible with the naive nematic hypothesis. In this paper we study the Ginzburg-Landau theory of a nematic order parameter coupled with an applied stress, and classify possible phase diagrams. Assuming that the $H_{c2}$ puzzle is explained by a preexisting “pinning field,” we indicate how a stress can be applied to probe an extended region of the phase diagram, and verify if the superconducting order parameter is indeed nematic. We also explore the Josephson tunneling between the proposed nematic superconducting state and an $s$-wave superconductor. The externally applied stress is predicted to serve as an on/off switch to the tunneling current, and in a certain regime the temperature dependence of the critical current can be markedly different from that between two conventional $s$-wave superconductors.

Figure 1

The positions of the selenium atoms within a quintuple layer of ${\text{Bi}}_2{\text{Se}}_3$. The superconducting order parameter $\vec{\eta }$ is shown here pointing in an arbitrary direction.

Figure 2

The three allowed phases when the stress is oriented at $\mathrm{\Phi }=0$ as shown. (a) Phase A: immediately below the upper transition. (b) Phase B: between the middle and the lower transition if applicable. The two orientations depicted in phase B are degenerate and coexisting, and the dotted lines mark $\theta =\pm \pi /6$. (c) Phase C: below the possible lower transition.

Figure 3

The three possible phase diagrams when $\mathrm{\Phi }=0$. The dashed lines denote second order transitions, and the dash-dotted lines denote first order transitions. N denotes the normal phase. Note that the critical temperature $T_2$ of the middle transition is asymptotically $(T_2-T_0)\propto \sqrt{\varepsilon }$ for small $\varepsilon$, as required by thermodynamics. (a) The lower transition is absent. (b) The lower transition is present. (c) The lower transition is present, and the middle transition merges with it at $\varepsilon ={\varepsilon }_{*}$.

Figure 4

The two allowed phases when the stress is oriented at $\mathrm{\Phi }=\pi /6$. (a) Phase D: immediately below the upper transition. (b) Phase E: the phase below the lower transition if it is present. The dotted line marks the direction perpendicular to the stress.

Figure 5

The two possible phase diagrams when $\mathrm{\Phi }=\pi /6$. The dashed lines represent second order transitions, and the dash-dotted line represents first order transition. N denotes the normal phase. (a) The phase diagram in the absence of a lower transition; and (b) the phase diagram in the presence of a lower transition. The lower transition change from first to second order at $\varepsilon ={\varepsilon }_c$.

Figure 6

The possible phase diagrams, with the $\mathrm{\Phi }$ axis added. The upper shaded surface is the second order upper transition that separate normal and superconducting phases. The other shaded surfaces are first order coexistence surfaces, while the thick dashed line represents second order critical end lines. (a) Lower transition is absent. (b) Lower transition is present. (c) Lower transition is present, and the middle transition line ends on it.