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n Δ$_{t}$ and large n Δ$_{t}$ limits. In the former case our expecta- tion that sin φ should be nearly constant except near the hot spots is confirmed in Fig. (7). Collective tunneling acts sinks the injected counterflow supercurrent at a rate that is nearly constant across the sample area. The large n Δ$_{t}$ case is more complex. φ varies over a large range and sin( φ ) changes sign in different parts of the sample. Near the sample corners φ varies rapidly with position because of the large counterflow supercurrents. In the interior of the sample the phase changes less rapidly because the 

flow pattern spreads and because coherent tunneling is providing the required current sink. The critical current in the large n Δ$_{t}$ limit, depends in a complex way on the geometry of the sample and on the spatial distribution of the pseudospin transfer torques. Nevertheless, the crit- ical current saturation we find in our numerical studies suggests that once the Josephson length is smaller com- pared to both the width and the length of a Hall bar sample with a large Hall angle, it is no longer relevant to the critical current value. 

Finally, we briefly discuss critical current dependence on Hall bar dimensions at fixed n Δ$_{t}$. In Fig.( 9) we plot I c vs. Hall bar length in a series of model samples with a fixed single pixel width W = 10 l and single-particle tun- neling amplitude 2 πl $^{2}$n Δ$_{t}$ = 10 − $^{6}$E$_{0}$ . For these param- eters the W is much smaller than the Josephson length. The critical current increases linearly with sample length and is therefore proportional to sample area until it sat- urates at L ∼ 1000 l . The length at which the current saturates is a bit longer than the Josephson length and the value of the critical current is accordingly somewhat larger than the 1D estimate 21 I c $_{E}$. The large L behavior is however consistent with the expectation that the crit- ical current should not increase with system length once L is substantially larger than the Josephson length λ . In Fig. (10) we plot I$_{c}$ vs. system width W with the length fixed at 500 l and the same Δ$_{t}$ as above. The range of W covered is limited somewhat by numerical practicalities and goes from a width much smaller than the Josephson length to a width which is somewhat larger. Over this range deviations from the 1D model in which the critical current is proportional to Hall bar width are small. 

## V. DISCUSSION AND CONCLUSIONS 

We start our discussion by commenting briefly on some essential differences between the critical current of a Josephson junction and critical currents for coherent bi- layer tunneling. In a Josephson junction, current can flow without dissipation across a thin insulating layer that separates two superconductors. The difference in condensate phase across the junction φ$_{J}$ is normally zero in equilibrium but can be driven to a non-zero steady state value when biased by current flow I$_{J}$ in the cir- cuit in which the junction is placed. For thick insulating layers the current is related to the phase difference by 

where I c J is the junction’s critical current. Eq.( 20) should be compared with Eq.( 7). The most obvious difference is the appearance of the lateral 2D coordinate in the co- herent bilayer case. In the Josephson junction case, the 

## C. I c vs. System Geometry 

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