Abstract
We apply an efficient numerical method to study inhomogeneous problems in superconductivity, such as vortices and Josephson junctions. The Chebyshev Bogoliubov-de Gennes method expands the Gorkov Green’s functions in Chebyshev polynomials. The time complexity is O(D) which is much lower than O(D3) of diagonalization, where D is the dimension of the tight-binding Hamiltonian, allowing studies of lattice with more than 109 atoms. We use this method to study vortex Majorana zero modes in topological materials and unconventional pairing in superconductor-ferromagnet heterostructures. We identify the signatures of hybridization for a new type of Majorana zero modes in topological crystalline insulator SnTe. In this system multiple Majorana zero modes coexist in a vortex protected by a magnetic mirror symmetry MT. Our simulations show that the spatial distribution of the Majorana zero modes differ drastically when a tilted magnetic field breaks or preserves MT. The hybridization of this new type of Majorana zero modes is easily controlled by altering the crystal symmetry, circumventing the difficulty of coupling isolated Majorana zero modes. We then study heterostructures of conventional superconductors and ferromagnets. The Chebyshev Bogoliubov-de Gennes method can efficiently calculate the Matsubara Green’s functions with fast Fourier transform. We show that even-frequency p-wave triplets and odd-frequency s-wave triplets can be generated in the ferromagnet. The unconventional pairings contribute to a spin-polarized supercurrent which can be applied for spintronics. These results demonstrate the power and versatility of the Chebyshev Bogoliubov-de Gennes method, it can be applied to study problems where traditional theories fail.