Superconductivity in three-dimensional spin-orbit coupled semimetals

Motivated by the experimental detection of superconductivity in the low-carrier density half-Heusler compound YPtBi, we study the pairing instabilities of three-dimensional strongly spin-orbit coupled semimetals with a quadratic band touching point. In these semimetals the electronic structure at the Fermi energy is described by spin $j=\frac{3}{2}$ quasiparticles, which are fundamentally different from those in ordinary metals with spin $j=\frac{1}{2}$. For both local and nonlocal pairing channels in $j=\frac{3}{2}$ materials we develop a general approach to analyzing pairing instabilities, thereby providing the computational tools needed to investigate the physics of these systems beyond phenomenological considerations. Furthermore, applying our method to a generic density-density interaction, we establish that: (i) The pairing strengths in the different symmetry channels uniquely encode the $j=\frac{3}{2}$ nature of the Fermi surface band structure—a manifestation of the fundamental difference with ordinary metals. (ii) The leading odd-parity pairing instabilities are different for electron doping and hole doping. Finally, we argue that polar phonons, i.e., Coulomb interactions mediated by the long-ranged electric polarization of the optical phonon modes, provide a coupling strength large enough to account for a Kelvin-range transition temperature in the $s$-wave channel, and are likely to play an important role in the overall attraction in non-$s$-wave channels. Moreover, the explicit calculation of the coupling strengths allows us to conclude that the two largest non-$s$-wave contributions occur in nonlocal channels, in contrast with what has been commonly assumed.

Figure 1

(a) Schematic electronic band structure of quadratic band crossing semimetals such as YPtBi. The touching of the ${\mathrm{\Gamma }}_8$ bands at the Gamma point is protected by symmetry. In the presence of strong spin-orbit coupling, i.e., coupling of the quasiparticle spin and crystal momentum, see Eqs. (3) and (4), and with inversion symmetry the ${\mathrm{\Gamma }}_8$ bands are split into two twofold degenerate bands away from $\mathrm{\Gamma }$. Motivated by YPtBi, we assume that one of these bands curves upward, forming the electron band, and one curves downward, forming the hole band. In YPtBi, when the Fermi energy is in the hole band, corresponding to hole doping, the quasiparticle states on the Fermi surface are spin $\pm \frac{3}{2}$ states, in the spherical approximation. (b) In the case of electron doping, which we also consider, the quasiparticle states on the Fermi surface are spin $\pm \frac{1}{2}$ states.

Figure 2

Diagrammatic representation of the Eliashberg equation. The solid lines are fermion propagators ${\langle C_{k,a}{C}_{k,b}^{†}\rangle }_{{\mathcal{S}}_0^{\prime }}$, where $C_k$ is the Nambu spinor of Eq. (31). The dashed line represents the interaction $V$.

Figure 3

The value $\frac{1}{2}{f}_{0,1}A$ which controls the coupling strength of the Eliashberg equations as a function of the parameter $\eta$ defined in Eq. (44) for the screened Coulomb interaction $V(\mathbf{q},\omega )=4\pi e^2/\left\{\left[{\varepsilon }_{\infty }+{\varepsilon }_c\left(\omega \right)\right]q^2+8\pi e^{2}N\left(0\right)\right\}$. We assume spherical symmetry and particle-hole symmetric bands. For the usual non-spin-orbit coupled parabolic bands, the large-$\eta$ limit of $\frac{1}{2}{f}_0{A}_0$ is 0.5, as compared to 0.25 in the present case.