Kosterlitz-Thouless vortex-scaling equations with nonzero current drives

The Kosterlitz-Thouless (KT) scaling procedure for the two-dimensional planar spin model is generalized to include an x-axis-applied current density I. Scaling equations for vortex coupling ${\mathit{K}}_{\mathit{l}}$ and vortex pair fugacity ${\mathit{y}}_{\mathit{l}}$ at a general minimum scale a≡${\mathit{e}}^{\mathit{l}}$ are derived, with current density acting as a y-axis ‘‘topological electric field’’ ${\mathit{E}}_{\mathit{l}}$ on the ±1 vortex ‘‘topological charges.’’ A vortex-unbinding onset scale l=${\mathit{l}}_{\mathit{c}}$ is defined by ${\mathit{E}}_{\mathit{l}\mathit{c}}$=${\mathit{K}}_{\mathit{l}\mathit{c}}$, where the current-driven repulsion of the ±1 vortex pairs begins to exceed their attraction. The nonlinear resistance R(T¯,I¯)=2π${\mathit{R}}_{\mathit{l}\mathit{c}}$ is related to the finite-scale phase-slip resistance ${\mathit{R}}_{\mathit{l}}$ that has a minimum at l=${\mathit{l}}_{\mathit{c}}$. Above transition, the zero-current resistance R(T¯,I¯=0) shows KT-like exponential inverse square-root temperature dependence, and is a universal function of dimensionless temperature T¯. The current-voltage exponent α(T¯,I¯) curves (where V∼${\mathit{I}}^{1+\mathrm{\alpha }}$) are universal in T¯ and I¯≡ħI/(2${\mathit{ek}}_{\mathit{B}}$T). Below transition, the α curves are weakly dependent on I¯, with α(T¯,I¯) close to π${\mathit{K}}_{\mathrm{\infty }}$(T¯). Above transition, non-Ohmic behavior α(T¯,I¯)≠0 is predicted for strong current.