Spontaneous surface current in multicomponent cubic superconductors with time-reversal symmetry breaking

In this paper we present a comprehensive study of the spontaneous currents in time-reversal symmetry breaking (TRSB) multicomponent superconductors with cubic crystalline symmetry. We argue, not limiting to cubic lattices, that spontaneous current on certain high-symmetry surfaces can exist only if the TRSB pairing simultaneously breaks a certain pair of mirror symmetries. This is shown to have exact correspondence with the Gingzburg-Landau (GL) theory and is verified by numerical Bogoliubov de-Gennes (BdG) calculations. In the course we extend the BdG to include effects of gap anisotropy and surface disorder, both of which could lead to much suppressed current. The GL theory has been known to describe well the spontaneous current. However, we highlight a special case where it becomes less adequate and show that a refined effective theory for low temperatures is needed. These results could shed light on the phenomenology of cubic superconductors such as ${\mathrm{U}}_{1-x}{\mathrm{Th}}_x{\mathrm{Be}}_{13}$, the filled skutterudites ${\mathrm{PrOs}}_4{\mathrm{Sb}}_{12}$, ${\mathrm{PrPt}}_4{\mathrm{Ge}}_{12}$, and related compounds.

Figure 1

Left: sketch of a mirror-invariant surface (MIS) defined in the main text, the associated pair of mirror planes (yellow) used to judge whether a particular component of the spontaneous surface current (red arrow) can exist. One of the two mirror planes is parallel to the surface (${\mathcal{M}}_x$), while the other is perpendicular to the direction of the surface current component in question (${\mathcal{M}}_y$). The blue block represents the superconductor under consideration, which is periodic in both $y$ and $z$ directions. Right: top view of the left panel. Note that the current on opposite surfaces flows in opposite directions, as the pairing possesses inversion symmetry in the present study. Spontaneous current along $y$ (on the [100] surface) is prohibited if the TRSB pairing preserves the reflection symmetry about either of the two mirror planes, i.e., ${\mathcal{M}}_x$ and ${\mathcal{M}}_y$ in the left panel.

Figure 2

Total surface current of various TRSB superconducting states as a function of chemical potential in a tight-binding BdG calculation. The $x$ axis is the chemical potential measured w.r.t. the band bottom ${\mu }_0=-6t$. For the two-dimensional representations, we have chosen $\mathbf{\varphi }={\mathrm{\Delta }}_0(1,i)$, and for the three-dimensional representations, we select only those states with $\mathbf{\varphi }={\mathrm{\Delta }}_0(1,w,w^2)$, where $w=\pm i2\pi /3$ and ${\mathrm{\Delta }}_0=0.2t$. All calculations were performed at $T=0$. The calculations with effective [110] surface are indicated.

Figure 3

Effect of gap anisotropy and surface disorder on the magnitude of the spontaneous current of the $E_u(1,i)$ states at the [110] surface. The simple and anisotropic $E_u$ pairing gap functions are both described in the text. Surface disorder is simulated by setting the pairing to be zero in the surface regime between the sites $i=1$ and 10. The calculations were carried out using the same parameters as in Fig. 2, except that the temperature here is $T=0.1{\mathrm{\Delta }}_0$. Note with the anisotropic pairing the total current changes sign at around $\mu -{\mu }_0=3.4t$.

Figure 4

BdG results: $y$ component of the total [100]-surface current of $(1,w,w^2)$ and $(1,i,0)$ states in the $T_{1u}$ and $T_{2u}$ representations. The gap functions and parameters used here are the same as in Fig. 2.

Figure 5

Contours of the gap structure of the $(1,w,w^2)$ states in the (a),(b) $T_{1u}$ and (c),(d) $T_{2u}$ representations. The calculations assume gap functions given by the simple bases as in Table 1 and a spherical Fermi surface. Two gaps appear for each of the nonunitary states.

Figure 6

Coefficients $b_i$ and $b_{ij}$ of the free energy (23) at low fillings of the lattice models (near the continuum limit) as in the BdG calculations. The $x$ axis is the chemical potential measured w.r.t. the band bottom. Calculation is performed at a somewhat elevated temperature $T={\mathrm{\Delta }}_0/5$ with ${\mathrm{\Delta }}_0=0.4t$ for better convergence.