Microscopic Theory of the Current-Voltage Characteristics of Josephson Tunnel Junctions

Deep theoretical understanding of the electrical response of Josephson junctions is indispensable regarding both recent discoveries of new kinds of superconductivity and technological advances such as superconducting quantum computers. Here, we study the microscopic theory of the dc current-biased $I\text{−}V$ characteristics of Josephson tunnel junctions. We derive an analytical formula of the $I\text{−}V$ characteristics of generic junctions. We identify subharmonics of the $I\text{−}V$ characteristics and their underlying mechanism as the feedback effect of intrinsic ac currents generated by voltage pulses in the past. We apply our theory to analytically solve the Werthamer equation and describe various dc current-biased $I\text{−}V$ characteristics as a function of softening of the superconducting gap. Strikingly, we identify voltage staircases of the $I\text{−}V$ characteristics in a genuine Josephson junction without ac current bias or qubit dynamics. Our general analytical formalism opens new avenues for a microscopic understanding of $I\text{−}V$ characteristics of Josephson junctions that have been limited to phenomenological models so far.

Figure 1

Analytical estimation and numerical calculation of the modulating height $J(t)$ of the dynamical washboard potential $\mathcal{U}(\varphi ,t)$. We depict a numerically calculated color map of $J(t)$ with respect to the memory time $t_m$ in (a). The time ranges where the voltage pulse modulates $\mathcal{U}(\varphi ,t)$ are indicated as a guide for the eye (dashed lines). In the right panels, we present plots of $J(t)$ for several $t_m$ for a better comparison of analytics (dashed lines) to numerics (solid lines). The height $J(t)$ of $\mathcal{U}(\varphi ,t)$ oscillates around $I_c$. We use ${\varphi }_0(t)$ exhibiting $\delta \tau /\tau =0.05$ and the memory kernels from Ref. [32]. Note that $\mathcal{U}(\varphi ,t+\tau )=\mathcal{U}(\varphi ,t)$ and $\varphi (t+\tau )=\varphi (t)+2\pi$ [30].

Figure 2

dc current-biased $I\text{−}V$ characteristics based on the Werthamer equation with smeared DOS $N(E)$. In (a), our analytical result (solid line) is provided together with a numerical calculation (red dots) using Eq. (1) without smearing. Analytical calculations for $\mathrm{\Gamma }\ll \mathrm{\Delta }$ and $\mathrm{\Gamma }\sim \mathrm{\Delta }$ are shown in (b) and (c), respectively. The voltage staircases become visible as peaks in the color map of $dI/dV$ in (d). $V/V_g=1,1/3,1/5,1/7$ are indicated with arrows from top to bottom. $V=\mathrm{\Gamma }/e$ is drawn as a guide to the eye (dashed line) and $I_{c0}=\pi \mathrm{\Delta }/(2e{R}_n)$. Insets: Quasiparticle DOS $N(E)$ corresponding to the chosen $\mathrm{\Gamma }$ in each panel [33].