Low-temperature thermal transport at the interface of a topological insulator and a $d$-wave superconductor

We consider the low-temperature thermal transport properties of the two-dimensional (2D) proximity-induced superconducting state formed at the interface between a three-dimensional strong topological insulator (TI) and a $d$-wave superconductor (dSC). This system is a playground for studying massless Dirac fermions, because they enter both as quasiparticles of the dSC and as surface states of the TI. For TI surface states with a single Dirac point, the four nodes in the interface-state quasiparticle excitation spectrum coalesce into a single node as the chemical potential $\mu$ is tuned from above the impurity scattering rate $\left(\left|\mu \right|\gg {\Gamma }_0\right)$ to below $\left(\left|\mu \right|\ll {\Gamma }_0\right)$. We calculate, via Kubo formula, the universal-limit $\left(T\to 0\right)$ thermal conductivity ${\kappa }_0$ as a function of $\mu$ as it is tuned through this transition. In the large- and small-$\left|\mu \right|$ limits, we obtain disorder-independent, closed-form expressions for ${\kappa }_0/T$. The large-$\left|\mu \right|$ expression is exactly half the value expected for a $d$-wave superconductor, a demonstration of the sense in which the TI surface topological metal is half of an ordinary 2D electron gas. Our numerical results for intermediate $\left|\mu \right|$ illustrate the nature of the transition between these limits, which is shown to depend on disorder in a well-defined manner.

Figure 1

Feynman diagram representing the bare bubble thermal current-current correlation function ${{\displaystyle \overleftrightarrow{\Pi }}}_{\kappa }\left(i\Omega \right)$. On each vertex sits a thermal current density operator ${\mathbf{j}}^{\kappa }$. Each propagator line denotes a Green's function dressed with disorder self-energy, $G(\mathbf{k},i\omega )$ and $G(\mathbf{k},i\omega +i\Omega )$.

Figure 2

Calculated universal-limit thermal conductivity, ${\kappa }_0/T$, as a function of chemical potential $\mu$. Solid curve denotes numerical solution of Eqs. (81, 82, 83) for parameter values ${\Delta }_0=0.1v/a$ and ${\Gamma }_0=0.01v/a$. The solution matches our large-$\left|\mu \right|$ expression (dashed) for $\left|\mu \right|\gg {\Gamma }_0$, then reaches a maximum at an intermediate value of $\left|\mu \right|$ before decreasing toward the value of our small-$\left|\mu \right|$ expression (dotted) for $\left|\mu \right|\ll {\Gamma }_0$.

Figure 3

Evolution of the $k$-space structure of the ${\kappa }_0/T$ integrand. (a) $[\mu =\frac{\pi }{\sqrt{2}}\frac{v}{a}]$: Large-$\left|\mu \right|$ limit. Four, well-separated, elliptical peaks within ${\Gamma }_0$ of the nodal points. (b) $[\mu =\frac{1}{2}\frac{\pi }{\sqrt{2}}\frac{v}{a}]$: Nodal peaks closer to the origin, more anisotropic, and curving around the Fermi circle. (c) $[\mu =\frac{1}{20}\frac{\pi }{\sqrt{2}}\frac{v}{a}]$: Nodal peaks have merged into an annular peak of width ${\Gamma }_0$.

Figure 4

Further evolution of the $k$-space structure of the ${\kappa }_0/T$ integrand, zoomed in by a factor of 20. (a) $[\mu =\frac{1}{20}\frac{\pi }{\sqrt{2}}\frac{v}{a}]$: Closeup view of the same annular peak shown in Fig. 3. (b) $[\mu =\frac{1}{200}\frac{\pi }{\sqrt{2}}\frac{v}{a}]$: Width and radius of the annular peak are now nearly equal. (c) $[\mu =0]$ Small-$\left|\mu \right|$ limit. Annular peak is blurred into single, isotropic peak within ${\Gamma }_0$ of the origin.

Figure 5

Disorder dependence of calculated universal-limit thermal conductivity, ${\kappa }_0/T$, as a function of chemical potential $\mu$. We plot numerical solutions of Eqs. (81, 82, 83) for ${\Delta }_0=0.1v/a$ and five values of the impurity scattering rate ${\Gamma }_0$. Results are disorder independent in both the large-$\left|\mu \right|$ and small-$\left|\mu \right|$ limits. The transition between limits depends on disorder, with the peaks more prominent for smaller ${\Gamma }_0$, smoothing out with increasing disorder.