Superconducting Phase with a Chiral $f$-Wave Pairing Symmetry and Majorana Fermions Induced in a Hole-Doped Semiconductor

We show that a chiral ($f+if$)-wave superconducting pairing may be induced in the lowest heavy hole band of a hole-doped semiconductor thin film through proximity contact with an $s$-wave superconductor. The chirality of the pairing originates from the $3\pi$ Berry phase accumulated for a heavy hole moving along a close path on the Fermi surface. There exist three chiral gapless Majorana edge states, in consistence with the chiral ($f+if$)-wave pairing. We show the existence of zero-energy Majorana fermions in vortices in the semiconductor-superconductor heterostructure by solving the Bogoliubov–de Gennes equations numerically as well as analytically in the strong confinement limit.

Figure 1

(a) A schematic illustration of the heterostructure composed of a hole-doped semiconductor thin film, an $s$-wave superconductor (SC), and a magnetic insulator. (b) The band structure of the hole-doped semiconductor. The lower (upper) two bands are the HH (LH). $\alpha =2\times {10}^5\mathrm{m}/\mathrm{s}$, $\hslash \sqrt{⟨k_z^2⟩}=3\times {10}^{-26}\mathrm{kg}·\mathrm{m}/\mathrm{s}$, ${\gamma }_1=6.92$, ${\gamma }_2=2.1$, $h_0=1.75\mathrm{meV}$. The parameters are chosen from GaAs [other materials (e.g., InAs, InSb) yield the same physics]. The hole density is $\sim 4\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 2

(a) Plot of $A(k)$ with respect to $k$. (b) Plot of the bulk quasiparticle energy $E_n$ of Eq. (7). Solid, dashed, dotted, and dash-dotted lines correspond to the lowest to the highest bands in Fig. 1b. $\mu =-32.5\mathrm{eV}$. The other parameters are the same as in Fig. 1.

Figure 3

Plot of the wave function of the zero-energy state. $\mu =-32.5\mathrm{meV}$. The other parameters are the same as in Fig. 1. Inset: Plot of the $s$-wave pairing gap with a vortex.

Figure 4

Plot of the quasiparticle energies in a vortex core with respect to the chemical potential $\mu$. Solid curve, zero-energy state; dashed curve, the minigap; dotted curve, the bulk excitation gap. The parameters are the same as in Fig. 1.