Topological Metal of NaBi with Ultralow Lattice Thermal Conductivity and Electron-phonon Superconductivity

By means of first-principles and ab initio tight-binding calculations, we found that the compound of NaBi is a three-dimensional non-trivial topological metal. Its topological feature can be confirmed by the presence of band inversion, the derived effective Z2 invariant and the non-trivial surface states with the presence of Dirac cones. Interestingly, our calculations further demonstrated that NaBi exhibits the uniquely combined properties between the electron-phonon coupling superconductivity in nice agreement with recent experimental measurements and the obviously anisotropic but extremely low thermal conductivity. The spin-orbit coupling effects greatly affect those properties. NaBi may provide a rich platform to study the relationship among metal, topology, superconductivity and thermal conductivity.


Band gap of NaBi between No.2 and No.3 bands
To elucidate whether or not the continuous band gap exists between No.2 and No.3 bands in the whole BZ of NaBi, we have performed a detailed analysis via four steps. According to the symmetry of NaBi, we only need to analyze the corresponding band gaps in the one eighth part [k x (0,1/2), k y (0,1/2), k z (0,1/2)] of the whole BZ as illustrated in Fig. s1(c) (marked by thick lines).
In the first, we calculated electronic energy bands using a dense k-mesh (46×46×46 = 94336 k points) in the one eighth part [k x (0,1/2), k y (0,1/2), k z (0,1/2)] of the whole BZ. Figure s1 lists all band gaps between No.2 and No.3 bands as a function of the distance between any k point among those 94336 k-points and the centered  point. In order to clearly visualize the specified position of the smallest band gap, we further plot the three-dimensional (3D) band gap evolutions ( Fig. s2(a)) and its two-dimensional (2D) projection ( Fig. s2(b)) as a function of the k-space position of the k z =1/2 plane.
In the second, we calculated electronic energy bands in a smaller region [k x (1/3,0.4), k y (1/3,0.4), k z (0.4,1/2)] using the very dense k-mesh (40×40×40 = 64000 k points), as illustrated in Fig. 1s. In the third, we will reduce the scale of the k-space in the second step into a smaller one [k x (1/3,0.38), k y (1/3,0.38), k z (0.4792,1/2)]] using the very dense k-mesh (30×30×30 = 27000 k points), as illustrated in Fig. 1s. Finally, we perform a final dense calculations with a tiny k-space [k x (0.34,0.35), k y (0.34,0.35), k z =1/2] using the highly dense k-mesh (50×50×1 = 2500 k points), as shown in Fig According to the symmetry of NaBi, it can be further referred that in the whole BZ there should are four equivalent points (±0.3461, ±0.3484, 0.5) at which the band gap between No.2 and No.3 bands has the smallest value of 0.08 eV. Furthermore, we also plot the electronic band structure along the Z-(0.3461,0.3494,0.5)-A line (here Z and A are high-symmetric point of (0, 0, 0.5) and (0.5, 0.5, 0.5), respectively (see Fig. 2s(e)). From this figure, it can be clearly seen that the smallest band gap is about 0.08 eV at the (0.3461, 0.3484, 0.5) position, which has a distance of 0.1759 Å -1 away from the  point.

Analyzing Berry phase of No.band
We calculated the band gap using the very dense k-point in the whole of BZ region and we found that the minimum band gap between No.2 and No.3 should be 0.08 eV at four k points in the k z = plane. Although we used the much denser k-point number to doubly check the band gap between No.2 and No.3 in the k z = plane, from the viewpoint of the numerical calculations it is still a bit difficult to 100% guarantee whether or not we have found the k-point that exhibits the minimum band gap. Furthermore, we want to use the zero Berry phase to clarify further the fact that No.1 and No.3 bands ever touch with No2 band.
Firstly, we have defined a mirror (001) plane and its corresponding plane in the k space satisfies Note that here we only consider if there is any band crossing between No.2 and No.3 (without spin degeneration). For sake of the convenience, we calculated the Berry phase for n=2 and m=i, namely,