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We present a theory for the regime of coherent interlayer tunneling in a disordered quantum Hall bilayer at total filling factor one, allowing for the effect of static vortices. We find that the system consists of domains of polarized superfluid phase. Injected currents introduce phase slips between the polarized domains which are pinned by disorder. We present a model of saturated tunneling domains that predicts a critical current for the breakdown of coherent tunneling that is extensive in the system size. This theory is supported by numerical results from a disordered phase model in two dimensions. We also discuss how our picture might be used to interpret experiments in the counterflow geometry and in two-terminal measurements.

In a quantum Hall bilayer at total Landau level filling

This counterflow superfluidity can be probed in tunneling experiments. In the tunneling geometry (Figure

Schematic diagram of tunneling experiment.

The Josephson-like regime persists for interlayer currents up to a critical value

In this paper, we focus on the Josephson regime below the critical current and present a physical picture of its breakdown. We have previously presented, in a short paper [

A similar puzzle is found in the original observation of dissipationless counterflow [

The resolution of this puzzle lies in the presence of disorder. We shall see (Figure

Spatial distribution of tunneling currents (left column) and interlayer voltages (right column), in a lattice model of 200 × 20 sites, with current injection at the two lower corners. The injected counterflow currents are

Schematic diagram of a counterflow experiment with a short circuit to complete current loop for counterflow.

This paper is organized as follows. We will discuss the origin of disorder in the bilayer in Section

Weak disorder, such as a spatially varying tunneling splitting, does not affect the tunneling properties of the system dramatically [

We consider here a bilayer with charge disorder in the bulk. One common source for this disorder is the electrostatic potential due to disordered dopant layers. We expect the incompressible quantum Hall phase to occupy only a fraction of the sample, with the remainder occupied by puddles of compressible electron liquid. Thus, the incompressible phase forms a network of channels separating puddles of size

In a quantum Hall superfluid, excess charge nucleates vortices in the exciton superfluid [

The above scenario provides a specific physical model for quenched vortices with short-ranged correlations in the exciton superfluid. The theory we present below depends on the existence of trapped fractional

In the previous section, we have outlined a model of disorder which induced quenched vortices in a quantum Hall state. To describe this exciton superfluid with quenched vortices, we start with an effective Hamiltonian for the phase

In the Josephson regime, there is no quasiparticle flow at zero temperature. All currents are accounted for by superflow and coherent tunneling. The counterflow supercurrent density above the ground state,

A time-varying superfluid phase

We expect that the counterflow current injected at the boundary will decay into the sample because interlayer tunneling will recombine electrons and holes across the two layers, as depicted in Figure

Since the phase angle is compact, this implies a maximum injected current density of

Note that this picture of current penetration into the clean system gives a penetration depth as a microscopic length scale independent of the injected current. We will see below that the disordered system behaves qualitatively differently—the current can penetrate into an indefinitely large area of the system. The reason is that injected phase slips are pinned by disorder, and therefore, a static solution to (

We will now review the heuristic theory of pinning presented in our previous work [

We will borrow from the Fukuyama-Lee theory [

In two dimensions,

Consider now the effect of an injected counterflow which imposes a phase twist at the boundary. The phase will therefore twist away from its equilibrium configuration. We assume that the domain at the boundary remains polarized at short distances and so will rotate uniformly on the scale of

This picture allows us to average over each domain. The total tunneling current in a domain consists of a similar random sum to that for the tunneling energy,

We will now present numerical results to support the theory in the previous section. Our numerical results are obtained using the dissipative model

The boundary conditions for (

The ground state of the system is found by evolving from a random state using the dissipative dynamics (

We expect that the counterflow current injected at the boundary will decay into the sample, because interlayer tunneling will recombine electrons and holes across the two layers. We find that the manner in which this occurs is qualitatively different in clean and disordered bilayers. As mentioned in Section

At a high enough injected current (

We emphasize that this interpretation of the threshold for the breakdown of the stationary solutions is qualitatively different from the clean case. In the clean model, the breakdown can be understood in terms of the injection of phase solitons at the boundary [

We have so far focused our discussion on the bilayer in the tunneling geometry. Finally, we will discuss how two other experimental situations can be interpreted in our theory. The first setup is the transport in the bilayer in a counterflow geometry, where the source and drain contacts are on the same side of the bilayer, while the other end is short-circuited to allow the current to flow from the top layer to bottom without the need for tunneling. This is depicted schematically in Figure

In our theory, this situation can be simulated by solving (

The second situation we wish to discuss is the two-terminal measurements of zero-bias interlayer conductance,

In summary, we have presented a theory of the Josephson regime of coherent tunneling in a disordered quantum Hall bilayer with static pinned vortices. We find that in the tunneling geometry, there are two current-carrying regions emanating separately from the source and drain contacts. In these regions, coherent tunneling is saturated. All injected counterflow current is lost by tunneling by the edge of these regions. The area of the saturated region

This picture tells us that the system reaches the critical current when the whole sample is saturated with coherent tunneling. This results in a critical current that is extensive for sufficiently large samples that contain many domains of polarized phase. In contrast, the clean limit [

Theoretically, our results are qualitatively different from clean theories [

The authors thank P. B. Littlewood for helpful discussions. This work was supported by EPSRC-GB (EP/C546814/01) and Science Foundation Ireland (SFI/09/SIRG/I1592).