Superconductivity on the surface of topological insulators and in two-dimensional noncentrosymmetric materials

We study the superconducting instabilities of a single species of two-dimensional Rashba-Dirac fermions, as it pertains to the surface of a three-dimensional time-reversal symmetric topological band insulator. We also discuss the similarities as well as the differences between this problem and that of superconductivity in two-dimensional time-reversal symmetric noncentrosymmetric materials with spin-orbit interactions. The superconducting order parameter has both $s$-wave and $p$-wave components, even when the superconducting pair potential only transfers either pure singlet or pure triplet pairs of electrons in and out of the condensate, a corollary to the nonconservation of spin due to the spin-orbit coupling. We identify one single superconducting regime in the case of superconductivity in the topological surface states (Rashba-Dirac limit), irrespective of the relative strength between singlet and triplet pair potentials. In contrast, in the Fermi limit relevant to the noncentrosymmetric materials we find two regimes depending on the value of the chemical potential and the relative strength between singlet and triplet potentials. We construct explicitly the Majorana bound states in these regimes. In the single regime for the case of the Rashba-Dirac limit, there exists one and only one Majorana fermion bound to the core of an isolated vortex. In the Fermi limit, there are always an even number (0 or 2 depending on the regime) of Majorana fermions bound to the core of an isolated vortex. In all cases, the vorticity required to bind Majorana fermions is quantized in units of the flux quantum, in contrast to the half flux in the case of two-dimensional $p_x\pm i{p}_y$ superconductors that break time-reversal symmetry.

Figure 1

(Color online) (a) Schematic picture of the surface states of the topological insulator ${\text{Bi}}_2{\text{Se}}_3$ (black lines). The chemical potential $\mu$ is far from the Rashba-Dirac (nodal) point and close to the conduction-band continuum (upper gray region) while the Rashba-Dirac (nodal) point is close to the valence-band continuum (lower gray region). (b) Left: one-dimensional cut of the dispersion of the Rashba-Dirac model defined by Eq. (2.1). In particular, we study the case where $\mu$ is close to the Rashba-Dirac (nodal) point rather than close to the energy cutoff $\pm \Lambda$ that defines the onset of the conduction band and the valence band. Right: the expectation values of the electron spins are perpendicular to their momenta and oriented in opposite directions for the upper and lower cone [see Eqs. (3.3b, 3.3c) at $B=0$].

Figure 2

(Color online) Temperature dependence of the self-consistent superconducting gap (red, normalized by the value at zero temperature) and the in-plane and out-of-plane Pauli magnetic susceptibility (red, normalized by the maximum of the out-of-plane susceptibility) for $\mu =5{\omega }_{\text{D}}$ and $\left|V\right|\nu \left({\omega }_{\text{D}}\right)=1$. Here, ${\omega }_{\text{D}}$ is the Debye cutoff used for the gap equations.

Figure 3

(Color online) Mean-field phase boundary in the Rashba-Dirac limit, Eq. (5.19).

Figure 4

(Color online) Mean-field phase boundary away from the Rashba-Dirac limit, i.e., when Eq. (5.20) holds.

Figure 5

(Color online) Qualitative comparison of the bound-state spectra of a deep and narrow vortex (left) and a wide and shallow vortex (right). While the former supports only few finite-energy bound states (blue), in the spectrum of the latter more states are allowed. However, the existence of a zero-energy mode (red) is independent of the details of the regime.