Correct Brillouin zone and electronic structure of BiPd

A promising route to the realization of Majorana fermions is in noncentrosymmetric superconductors, in which spin-orbit coupling lifts the spin degeneracy of both bulk and surface bands. A detailed assessment of the electronic structure is critical to evaluate their suitability for this through establishing the topological properties of the electronic structure. This requires correct identification of the time-reversal-invariant momenta. One such material is BiPd, a recently rediscovered noncentrosymmetric superconductor which can be grown in large, high-quality single crystals and has been studied by several groups using angular resolved photoemission to establish its surface electronic structure. Many of the published electronic structure studies on this material are based on a reciprocal unit cell which is not the actual Brillouin zone of the material. We show here the consequences of this for the electronic structures and show how the inferred topological nature of the material is affected.

Figure 1

The monoclinic $P{2}_1$ and the pseudo-orthorhombic $B{2}_1$ unit cells. Red lines show the (a), (b) $P{2}_1$ unit cell in real space and (c) its reciprocal unit cell, which is also the Brillouin zone of both the $P{2}_1$ and $B{2}_1$ cells. The blue lines indicate the doubled $B{2}_1$ unit cell and its smaller reciprocal unit cell as reported in Ref. [13]. This reciprocal cell is half the size of the Brillouin zone because of the doubled, nonprimitive unit cell in real space. The $z=0$ plane in $P{2}_1$ is shown in (a). (d) Comparison of the Brillouin zone and the $B{2}_1$ reciprocal cell with ARPES data [21].

Figure 2

Electronic structure in $P{2}_1$ and $B{2}_1$. (a) Band structure of BiPd calculated for $B{2}_1$ in its Brillouin zone (red dashed lines) and taking the reciprocal cell of the doubled unit cell as the Brillouin zone (blue solid line). For the latter, $X$ and $Z$ are at the zone boundary. (b) The band structure in the same energy range as in (a), obtained in the Brillouin zone for both crystal structures, i.e., $B{2}_1$ (red dashed lines) and $P{2}_1$ (black solid lines).

Figure 3

Effect of zone folding on the surface band structure. (a) Band structure obtained for a slab and shown in the surface Brillouin zone of the $P{2}_1$ crystal structure [19]. Circles mark the four surface states obtained in the slab calculation, two in the occupied states at S and two in the unoccupied states at $\mathrm{\Gamma }$. (b) Band structure shown in the reduced reciprocal unit cell for the $B{2}_1$ structure with the zone boundaries at X and Z, which is not a Brillouin zone. Shown in red are the bands from the unfolded band structure, the ones shown in blue appear in addition due to back-folding at X and Z. The Dirac-like surface states are identified with circles in (a) at their position in the Brillouin zone and in (b) where they appear after folding. The surface states above the Fermi energy near $\mathrm{\Gamma }$, which were originally in a directional band gap opened by spin-orbit coupling, are enveloped in folded bands, whereas the ones which were at $S$ in the Brillouin zone (a) now appear at $\mathrm{\Gamma }$.