Transport Properties in Unconventional Superconductors

This dissertation investigates the transport properties of unconventional superconductors which differ from the conventional superconductors on two aspects, one is the pairing symmetry of the order parameter, the other is the net momentum of the Cooper pair. The former ones are discovered in high-$T_c$ cuprates, heavy-fermion, Sr$_2$RuO$_4$ and so on. The latter ones can be realized by splitting the Fermi surfaces of spin-up and -down electrons under Zeeman field and are known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states. This work is consisted of two parts. In the first part, we present results of numerical studies of spin quantum Hall transitions in disordered superconductors, in which the pairing order parameter breaks time-reversal symmetry. We focus mainly on $p$-wave superconductors in which one of the spin components is conserved. The transport properties of the system are studied by numerically diagonalizing pairing Hamiltonians on a lattice, and by calculating the Chern and Thouless numbers of the quasiparticle states. We find that in the presence of disorder, (spin-)current carrying states exist only at discrete critical energies in the thermodynamic limit, and the spin-quantum Hall transition driven by an external Zeeman field has the same critical behavior as the usual integer quantum Hall transition of non-interacting electrons. These critical energies merge and disappear as disorder strength increases, in a manner similar to those in lattice models for integer quantum Hall transition. The second part is a proposal of identifying the FFLO state based on its transport properties in the normal metal/superconductor junction (NSJ). The 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 examine the possibility of detecting the phase structure of the order parameter directly using conductance spectroscopy through NSJ, 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 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 $z$.