Incorporation of a frequency-dependent dielectric response for the barrier material in the josephson junction circuit model

We extend the resistively shunted Josephson (RSJ) junction circuit model originally proposed by Stewart and McCumber to incorporate a frequency-dependent dielectric response so that the influence of free carriers in the barriers can be taken into account. The methodology that we have developed uses an iterative numerical technique to calculate the current-voltage (I-V) characteristics of a Josephson junction with a barrier exhibiting both dissipation and dispersion. We give detailed results for two barrier materials with conductivities near the metal-insulator transition: a conventional semiconductor with a relatively high mobility and a strongly scattered defect solid. We show that the incorporation of the dynamic response of free carriers in the barriers of superconductor- normal-superconductor (SNS) junctions significantly influences the dc I-V characteristics for the case of material near the metal-insulator transition with high mobility. Hysteretic anomalies occur at nonzero voltages in the I-V characteristics associated with the barrier layer's plasma frequency. The resulting features, which we call critical regions, occur when the dc junction voltage 〈V〉 is equal to ℏ/2en√ω̃ _p² - Γ², where ω̃_p is the barrier's plasma frequency, Γ is the quasi-particle scattering rate, n is an integer, and ℏ is the reduced Planck's constant. We also show that our results for SNS junctions with a low-mobility barrier material are essentially identical to the predictions of the simpler RSJ model. Since the method we develope can solve the nonlinear junction equations for a barrier with an arbitrary complex conductivity, it is also capable of including other relevant processes within the barrier, including the influence of excitation from shallow defects or very soft phonon modes, as well as boundary resistances.