Critical Supercurrents and Self-Organization in Quantum Hall Bilayers

We present a theory of interlayer tunneling in a disordered quantum Hall bilayer at total filling factor one, allowing for the effect of static vortices. In agreement with recent experiments [Phys. Rev. B 80, 165120 (2009); Phys. Rev. B 78, 075302 (2008)], we find that the critical current is proportional to the sample area and is comparable in magnitude to observed values. This reflects the formation of a Bean critical state as a result of current injection at the boundary. We predict a crossover to a critical current proportional to the square-root of the area in smaller samples. We also predict a peak in the critical current as the electron density varies at fixed layer separation.

Figure 1

Profile of counterflow current $I$ (lines) and interlayer voltage ($V\propto \dot{\varphi }$, vertical bars, nonzero in top curve only) from (4) on a one-dimensional lattice with current injection at both ends, at a time $\approx {10}^4/\lambda$ after the current is switched on. The injected counterflow currents at each boundary are 3, 6, 9, $12I_0$ for the four curves ($I_0=e{\rho }_s/\hslash \xi$). The interlayer voltages vanish below a critical current $I_c$. Here, $9<I_c/I_0<12$. $t{\xi }^2/{\rho }_s=0.4$, results averaged over 50 realizations.

Figure 2

Predicted critical current (solid, left axis) and domain size (dotted, right axis), for area $S=0.1{\mathrm{mm}}^2$, $d=28\mathrm{nm}$, $\xi (\approx d_d)=200\mathrm{nm}$, and bare tunneling ${\Delta }_0=150\mu \mathrm{K}$. The stiffness and tunneling renormalization are taken from theory [27], and scaled by the network geometrical factor (see text).