Zero energy modes in a superconductor with ferromagnetic adatom chains and quantum phase transitions

Abstract

We study Majorana zero energy modes (MZEM) that occur in an s-wave superconducting surface, at the ends of a ferromagnetic (FM) chain of adatoms, in the presence of Rashba spin-orbit interaction (SOI) considering both non self-consistent and self-consistent superconducting order. We find that in the self-consistent solution, the average superconducting gap function over the adatom sites has a discontinuous drop with increasing exchange interaction at the same critical value where the topological phase transition occurs. We also study the MZEM for both treatments of superconducting order and find that the decay length is a linear function of the exchange coupling strength, chemical potential and superconducting order. For wider FM chains the MZEM occur at smaller exchange couplings and the slope of the decay length as a function of exchange coupling grows with chain width. Thus we suggest experimental detection of different delocalization of MZEM in chains of varying widths. We discuss similarities and differences between the MZEM for the two treatments of the superconducting order.

Supercurrent as a probe for topological superconductivity in magnetic adatom chains

“…The adatoms act as magnetic impurities for the host superconductor and give rise to Yu-Shiba-Rushinov (YSR) states which hybridize to form a band dispersing within the energy gap of the superconductor [17][18][19][20]. In suitable parameter regimes, SOC induces chiral TSC in the YSR impurity band [16,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]]. An adatom chain, therefore, mimics the physics of the onedimensional Kitaev model [36] which contains isolated MBS at the ends of the chain.…”

Majorana Zero Modes in Ferromagnetic Wires without Spin-Orbit Coupling

“…Superconductorferromagnet heterostructures do not require the application of an external magnetic field and, therefore, are advantageous with respect to other platforms. Following this line of argument, topological superconductivity and MZMs have been demonstrated to emerge in half-metal/superconductor [14] or ferromagnet/unconventional superconductor [15] heterostructures, in ferromagnetic wires proximised to conventional superconductors [16][17][18] and in ferromagnetically aligned chains of magnetic impurities embedded in conventional superconductors [19][20][21][22][23][24][25][26]. Spin-orbit coupling is an essential component in all of these proposals.…”

“…Based on Hamiltonian Equation ( 24) describing the FM wire, we derive, in this section, the criteria regarding the exchange field h FM and the chemical potential µ FM for which MZMs emerge localised at the edges of the FM. To this aim, we consider periodic boundary conditions and transform Hamiltonian Equation (24) in momenta k space. Introducing the spin-dependent Nambu spinor…”

Static and Dynamic Disorder in Topological Systems: Localized, Critical and Extended States

“…If the number of magnetic impurities is large enough there are bound states inside the gap that have zero energy and constitute self-conjugate Majorana fermions. It has been shown that the quantum phase transition that results from the closing of the gap as a function of the coupling between the local impurities and the conduction electron spin density coincides with the topological transition to a topological phase with the Majorana fermions [51].…”