Majorana fermions in a topological-insulator nanowire proximity-coupled to an $s$-wave superconductor

A finite-length topological-insulator nanowire, proximity-coupled to an ordinary bulk $s$-wave superconductor and subject to a longitudinal applied magnetic field, is shown to realize a one-dimensional topological superconductor with unpaired Majorana fermions localized at both ends. This situation occurs under a wide range of conditions and constitutes an easily accessible physical realization of the elusive Majorana particle in a solid-state system.

Figure 1

Schematic of the proposed device. Magnetic field $\mathbf{B}$ is applied along the axis of the wire taken to coincide with the $z$ direction.

Figure 2

(a) Surface-state excitation spectra $E_{kl}$ for various values of magnetic flux $\Phi =\eta {\Phi }_0$. The solid and dashed lines indicate doubly degenerate and nondegenerate bands, respectively. (b) Kitaev's Majorana number: $\mathcal{M}=-1(+1)$ in shaded (white) regions. The numerals inside the squares indicate the number of Fermi points for $k>0$. (c) A possible shape of the SC and magnetic domain wall located near $z=0$. The dashed line shows the exact zero-mode solution $u\left(z\right)$ for this domain wall.

Figure 3

(a) Energy dispersion for an infinitely long TI wire with a $20\times 20$ base described by lattice Hamiltonian (9) in the normal state with $\eta =0.52$. For clarity, only the low-energy portion of the spectrum is displayed in a part of the Brillouin zone. The inset shows the spin expectation values for the gapless state at small positive $k$. The length of the arrow is proportional to the wave-function amplitude. (b) Lines separating regions with different Majorana numbers $\mathcal{M}=-1(+1)$ extracted from the spectrum. All energies are in units of $\lambda =150$ meV and we use parameters ${\lambda }_z=t=1$, $\epsilon =4$, and $g=32$, corresponding to the strong TI phase with a $Z_2$ index (1;000) and a bulk band gap $2\lambda =300$ meV.

Figure 4

(a) Energy bands for an infinitely long TI wire with a $6\times 6$ base in the SC state with $\eta =0.49$, $\mu =0.09$, ${\Delta }_0=0.08$, and $g=0$ (solid lines), and the energy levels for a $L=36$ finite-length wire with open boundary conditions (red circles) obtained by exact numerical diagonalization. (b) Three lowest positive energy eigenvalues obtained by the Lanczos method as a function of $L$. The Dashed line represents the envelope function $0.089e^{-L/\xi }$ with $\xi =17.5$.