Conductance ofd-Wave Superconductor/Normal Metal/d-Wave Superconductor Junctions

Abstract

We develop a theory of the conductance of superconductor/normal metal/superconductor junctions in the case where the superconducting order parameter has d-wave symmetry. At low temperature the conductance is proportional to the square root of the inelastic electron relaxation time in the bulk of the superconductor. As a result it turns out to be much larger than the conductance of the normal part of the junction.PACS numbers: 74.45.+c, 74.50.+r, 74.20.Rp At small voltages V , the current I through a superconductor/normal metal/superconductor (SNS) junction can be written asHere φ is the phase difference between the superconductors, and J(φ) is a periodic function of φ with period 2π. The time dependence of φ is given by the Josephson relation:The first term in Eq.(1), describing the Josephson current, has been the subject of intensive experimental and theoretical studies over several decades, since the discovery of the Josephson effect. The second term, representing the dissipative current, has attracted relatively little attention, both on the theoretical and experimental sides. In the context of SNS junctions with s-wave symmetry of the order parameter in the leads, this problem was considered theoretically in [1][2][3][4][5][6][7], and experimentally in [8]. The common result of these works is that at low temperatures the conductance of the system is proportional to the inelastic relaxation time in the normal metal. A theory of conductance of a junction in the case when the superconducting order parameter has d-wave symmetry has not been developed. In this paper we show that at low temperatures T the conductance, G DN D , is proportional to the square root of the energy relaxation time in the bulk of superconductor, τ in , and that it is much larger than the conductance G N of the normal piece of the junction.The origin of the leading low-temperature contribution to G DN D is similar to the Debye relaxation mechanism in dielectrics, or the Mandelstam-Leontovich mechanism for sound absorption in liquids with internal degrees of freedom [9]. Due to the proximity effect the single particle density of states in the normal region, ν N (ε, φ), becomes ε and φ-dependent. Here ε is the energy of quasiparticle. According to Eq. (2), ν N (ε, φ(t)) changes in time. In the adiabatic approximation the electron population follows the motion of the levels. As a result, the quasiparticle distribution function becomes nonequilibrium. Its relaxation leads to the entropy production, and therefore contributes to the conductance:The equation for the entropy production readṡwhere f (ε, r, t) is the quasiparticle distribution function, f th (ε, r, t) = tanh ε/2T is the equilibrium distribution function, τ in is the inelastic relaxation time, D ij (ε) is the diffusion coefficient, and ν is the reduced density of states, measured in the units of the normal state density of states, ν 0 . We assume that the latter is the same in both metals.The kinetic equation describing the dynamics of quasiparticles has the following form: ν ∂f ∂...

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