Conductance characteristics between a normal metal and a two-dimensional Fulde-Ferrell-Larkin-Ovchinnikov superconductor: the Fulde-Ferrell state

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state has received renewed interest recently due to the experimental indication of its presence in CeCoIn$_5$, a quasi 2-dimensional (2D) d-wave superconductor. However direct evidence of the spatial variation of the superconducting order parameter, which is the hallmark of the FFLO state, does not yet exist. In this work we explore the possibility of detecting the phase structure of the order parameter directly using conductance spectroscopy through micro-constrictions, which probes the phase sensitive surface Andreev bound states of d-wave superconductors. We employ the Blonder-Tinkham-Klapwijk formalism to calculate the conductance characteristics between a normal metal (N) and a 2D $s$- or $d_{x^2-y^2}$-wave superconductor in the Fulde-Ferrell state, for all barrier parameter $z$ from the point contact limit ($z=0$) to the tunneling limit ($z \gg 1$). We find that the zero-bias conductance peak due to these surface Andreev bound states observed in the uniform d-wave superconductor is split and shifted in the Fulde-Ferrell state. We also clarify what weighted bulk density of states is measured by the conductance in the limit of large $z$.