Local density of states in a vortex at the surface of a topological insulator in a magnetic field

Fermion bound states at the center of the core of a vortex in a two-dimensional superconductor are investigated by analytically solving the Bogoliubov–de Gennes equations in a magnetic field. The metallic surface states of a strong topological insulator become superconducting via proximity effect with an s-wave superconductor. Due to the magnetic field, the states undergo Landau quantization. A zero-energy Majorana state arises for the Landau level $$n=0$$ n = 0 together with an equally spaced ( $$\Delta ^2_{\infty }/E_F$$ Δ ∞ 2 / E F ) sequence of fermion excitations. The spin-momentum locking due to the Rashba spin–orbit coupling is key to the formation of the Majorana state. Extending previous results in zero magnetic field, we present analytical expressions for the energy spectrum and the wave functions of the bound states in a finite magnetic field. The solutions consist of harmonic oscillator wave functions (associated Laguerre polynomials) times a function that falls off exponentially with distance $$\rho $$ ρ from the core of the vortex as $$\exp [-\int _0^{\rho } d\rho ' \Delta (\rho ')/v_F]$$ exp [ - ∫ 0 ρ d ρ ′ Δ ( ρ ′ ) / v F ] . An analytic expression for the local density of states (LDOS) for the bound states is obtained. It depends on two length scales, $$1/k_F$$ 1 / k F and the magnetic length, $$l_H$$ l H , and the angular momentum index $$\mu $$ μ . The particle-hole symmetry is broken in the LDOS as a consequence of the spin–orbit coupling and the chirality of the vortex. We also discuss the spin polarization of the bound states.