Role of midgap states in the inelastic tunneling between a d-wave superconductor and a normal metal

Abstract

Midgap states are a special type of quasiparticle states in a d-wave superconductor which are bound to any non-{n,0,m} surface (or interface) and have essentially zero energy with respect to the Fermi energy. We investigate how such midgap states play a role in the inelastic tunneling (IT) between a normal metal and a d-wave superconductor, assuming that there are molecular vibrational modes in the barrier that a tunneling electron can excite. To allow for the midgap states, a d-wave superconducting slab with two {110} surfaces is modeled by a tight-binding Hamiltonian. A discrete version of the Bogoliubov-de Gennes equations is solved numerically, and the spatially-varying order parameter is determined self-consistently by an iteration technique. After obtaining the eigenenergies and eigen-wavefuntions of all quasiparticles in this system, we then calculate the IT current between a normal metal and this system by using the tunneling Hamiltonian method. A model study is also made about IT mediated by the phonons in the barrier or of the electrodes at the barrier. Unlike in N-N tunneling, where the excitable mode frequencies are revealed in d² I/d²V, here dI/dV is already sufficient to reveal the same. Comparing with N-S (s-wave) tunneling, we find that IT involving the midgap states can give a sharp, essentially symmetric peak in dI/dV for each excitable mode, which permits a more direct observation of the effective density of states of these modes (including coupling constant variations), if in the right frequency range, whereas IT in N-S(s-wave) tunneling requires a non-trivial deconvolution, since the density-of-states peak due to the gap of an s-wave superconductor has an asymmetric shape and is also not narrow. Another advantage is that our method allows the inelastic peaks associated with low excitable-mode frequencies to appear much below the gap-induced peak in elastic tunneling where the elastic contribution is small, instead of appearing just above it where the elastic contribution is large and rapidly varying, as is the case in N-S(s-wave) tunneling.