Effect of gap suppression by superfluid current on the nonlinear microwave response of $d$-wave superconductors

Effect of gap suppression by superfluid current on the nonlinear microwave response of $d$ -wave superconductors

E. J. Nicol and J. P. Carbotte

Phys. Rev. B 73, 174510 – Published 11 May 2006

Abstract

Recently several works have focused on the intrinsic nonlinear current in passive microwave filters as a tool for identifying $d$ -wave order parameter symmetry in the high- $T_{c}$ cuprates. Evidence has been found for $d$ -wave pairing in YBCO and further work has ensued. Most of the theoretical work has been limited to low temperatures because it has not included the effect of the superfluid current on the energy gap. We find that this effect leads to important corrections above $T \sim 0.2 T_{c}$ , while leaving the $1 ∕ T$ low-temperature behavior intact. A twofold increase in the nonlinear coefficient at temperatures of the order of $\sim 0.75 T_{c}$ is found, and as $T \to T_{c}$ the nonlinearity comes entirely from the effects of the superfluid current on the gap. Impurity scattering has been included and, in addition, signatures for the case of $d + s$ wave are presented.

Received 3 January 2006

DOI:https://doi.org/10.1103/PhysRevB.73.174510

Authors & Affiliations

E. J. Nicol^*

Department of Physics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1

J. P. Carbotte^†

Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada, L8S 4M1

^*Electronic address: nicol@physics.uoguelph.ca
^†Electronic address: carbotte@mcmaster.ca

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Vol. 73, Iss. 17 — 1 May 2006

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Images

Figure 1
The normalized current $j_{s} ∕ j_{0}$ as a function of $q_{s} v_{F} ∕ Δ_{0}$ , where $j_{0} = n e Δ_{0} ∕ (m v_{F})$ and $Δ_{0}$ is the energy gap at $T = 0$ . Shown are the low-temperature BCS curves for two directions: the current in the antinodal direction with $α = 0$ (dashed line) and the nodal direction with $α = π ∕ 4$ (solid line), given for a reduced temperature $t = T ∕ T_{c} = 0.1$ . The inset shows that the curves overlap for $T$ near $T_{c}$ (in this case, $t = 0.95$ ).Reuse & Permissions
Figure 2
The nonlinear coefficient $β (T)$ as a function of $T ∕ T_{c}$ , shown for two directions: $α = π ∕ 4$ (solid line) and $α = 0$ (short-dashed line). The dotted curves are for the same two directions but with the approximation of neglecting the $q_{s}$ dependence in the gap—i.e., $Δ (q_{s} = 0)$ . The inset illustrates via a log-log plot that the low-temperature behavior varies as $Δ_{0} ∕ 24 T$ for $α = 0$ and $Δ_{0} ∕ 12 T$ for $α = π ∕ 4$ (these expressions are shown as the long-dashed lines in both cases).Reuse & Permissions
Figure 3
The nonlinear coefficient $β (T)$ versus $T ∕ T_{c 0}$ for varying impurity scattering in the unitary limit $(c = 0)$ and with $α = π ∕ 4$ . The inset shows the corresponding curves for the superfluid density $n_{s} (T) ∕ n$ . Curves are given for the pure limit (solid line), $Γ^{+} ∕ T_{c 0} = 0.0082$ (short-dashed line), 0.0404 (long-dashed line), 0.0697 (dot-short-dashed line), and 0.1432 (dot-long-dashed line).Reuse & Permissions
Figure 4
Plot of $\log_{10} [A b_{a v}^{2} (T)]$ versus $\log_{10} (T ∕ T_{c})$ for $s$ -wave (dashed curve), BCS $d$ -wave with $2 Δ_{0} ∕ k T_{c} = 4.28$ (solid line), and strong-coupling (SC) $d$ -wave with $2 Δ_{0} ∕ k T_{c} = 5$ (dotted line). The curves are compared with the experimental data (solid dots) taken from Ref. 9. The inset shows the same curves for $b (T)$ versus $T ∕ T_{c}$ .Reuse & Permissions
Figure 5
Plot of $\log_{10} (j_{2} (T) ∕ j_{c})$ versus $λ (T) ∕ λ_{0}$ for $T = 0.75 T_{c 0}$ . Andersen et al. (Ref. 13) have defined a quantity ${(j_{2} ∕ j_{c})}^{2}$ which is equal to our $1 ∕ b (T)$ . The data (open circles) is taken from Ref. 13. The solid curve is for BCS $d$ -wave with $Δ (q_{s})$ and the dashed line is for the case where the approximation $Δ (q_{s} = 0)$ has been used. The dotted curve shows the $Δ (q_{s} = 0)$ curve shifted down to overlap with the data. The inset shows the reduction in $T_{c}$ from the pure value of $T_{c 0}$ that is implied by the values of $λ (T)$ at $T = 0.75 T_{c 0}$ normalized to the pure value for $T = 0$ , $λ_{0}$ .Reuse & Permissions
Figure 6
The effect of a $(d + s)$ -wave gap on the anisotropy near $T_{c}$ as measured via $\bar{b} (T_{c})$ (upper frame) and at low temperature as measured by $β (T)$ normalized to $Δ_{0} ∕ 24 T$ (lower frame). The anisotropy is plotted as a function of the current direction relative to the antinode, $α$ . Curves are shown for $s$ -wave (dotted line), $d$ -wave (short-dashed line), and varying percentage of $s$ -wave: 10% (dot-dashed line), 15% (solid line), and 20% (long-dashed line).Reuse & Permissions

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