Inflated nodes and surface states in superconducting half-Heusler compounds

Two topics of high current interest in the field of unconventional superconductivity are noncentrosymmetric superconductors and multiband superconductivity. Half-Heusler superconductors such as YPtBi exemplify both. In this paper, we study bulk and surface states in nodal superconducting phases of the half-Heusler compounds, belonging to the $A_1$ ($s+p$-like) and $T_2$ ($k_z{k}_x+i{k}_z{k}_y$-like) irreducible representations of the point group. These two phases preserve and break time-reversal symmetry, respectively. For the $A_1$ case, we find that flat surface bands persist in the multiband system. In addition, the system has dispersive surface bands with zero-energy crossings forming Fermi arcs, which are protected by mirror symmetries. For the $T_2$ case, there is an interesting coexistence of point and line nodes, known from the single-band case, with Bogoliubov Fermi surfaces (two-dimensional nodes). There are no flat-band surface states, as expected, but dispersive surface bands with Fermi arcs exist. If these arcs do not lie in high-symmetry planes, they are split by the antisymmetric spin-orbit coupling so that their number is doubled compared to the inversion-symmetric case.

Figure 1

(a) Two-dimensional Brillouin zone of a slab with (a) (111) and (b) (100) surfaces.

Figure 2

Band structure for the normal state of a slab with (100) surfaces along the $k_1$ axis. The thickness is $W=400$. The zero of energy is set to the Fermi energy (dashed horizontal line). The split-off surface bands are clearly visible.

Figure 3

Dispersion of surface states of YPtBi in the normal state for (a) the (111) surface and (b) the (100) surface, for a thickness of $W=2000$. The gray region in the center is the projection of the bulk Fermi sea, i.e., in this region the states at the Fermi energy are bulk like. Note that this does not preclude the presence of surface states away from the Fermi energy. In the white regions, there is no bulk Fermi sea but the states closest to the Fermi energy are still bulk like.

Figure 4

Superconducting gap for the $A_1$ pairing state with $p$-wave to $s$-wave ratio ${\mathrm{\Delta }}_p^0/{\mathrm{\Delta }}_{A_1}^0=7/3$ on (a) the smaller and (b) the larger normal-state Fermi surface. Infinitesimal pairing amplitudes have been used for reasons explained in the text.

Figure 5

Dispersion of surface states of YPtBi in the superconducting $A_1$ state for (a) the (111) surface and (b) the (100) surface, for a thickness of $W=4000$. At each momentum, the states occur in pairs with energies differing in their sign. The positive energy is plotted. Black regions correspond to flat surface bands. In the white regions, the states closest to the Fermi energy are bulk like.

Figure 6

Spin polarization in the $z$ direction, $\langle S_z\rangle =\langle J_z/3\rangle$ vs surface momentum ${\mathbf{k}}_{\parallel }$ of the lowest-energy state for a (111) slab with $W=4000$ of YPtBi in the $A_1$ state. At each momentum, the spectrum is symmetric. For states of nonzero energy, the negative-energy state is used, whereas for degenerate zero-energy states, the superposition localized at the $l=0$ surface is used.

Figure 7

Coefficient $g_2$ in the expansion of $det\mathcal{H}\left(\mathbf{k}\right)$ in the pairing amplitude, Eq. (50), for the $T_2$ pairing state with order parameter $\mathbf{l}=(1,i,0)$ on (a) the smaller and (b) the larger normal-state Fermi surface. The plots exhibit the nodal structure for infinitesimal pairing.

Figure 8

(a) Nodes for the $T_2$ pairing state with order parameter $\mathbf{l}=(1,i,0)$, which breaks time-reversal symmetry, and pairing amplitude ${\mathrm{\Delta }}_{T_2}^0=3\mathrm{meV}$. Red dots represent point nodes, blue lines line nodes, and orange surfaces Bogoliubov Fermi surfaces. Momentum axes are omitted for clarity. Note, however, that the point nodes coincide with the ones in Fig. 7. (b) Enlargement of the system of Fermi pockets and line nodes close to the $k_x$ and $k_y$ axes. (c) Enlargement of one of the sickle-shaped Fermi pockets close to the point nodes.

Figure 9

Dispersion of surface states of YPtBi in the superconducting $T_2$ state for (a) the (111) surface with a thickness of $W=16000$ and (b) the (100) surface with $W=4000$. A larger thickness was considered for the (111) surface because of large finite-size effects. The spectra at fixed momentum are not symmetric. The signful energy closest to the Fermi energy is plotted. The gray regions are the projections of the Bogoliubov Fermi pockets shown in Fig. 8 onto the surface Brillouin zone. In these regions, states exist at the Fermi energy and are bulk like. In the white regions, there are no Fermi pockets but the states closest to the Fermi energy are still bulk like.