Critical tunneling currents in quantum Hall superfluids: Pseudospin-transfer torque theory

At total filling factor $\nu =1$ quantum Hall bilayers can have an ordered ground state with spontaneous interlayer phase coherence. The ordered state is signaled experimentally by dramatically enhanced interlayer tunnel conductances at low-bias voltages; at larger bias voltages interlayer currents are similar to those of the disordered state. We associate this change in behavior with the existence of a critical current beyond which static interlayer phase differences cannot be maintained, and examine the dependence of this critical current on sample geometry, phase stiffness, and the coherent tunneling energy density. Our analysis is based in part on analogies between coherent bilayer behavior and spin-transfer torque physics in metallic ferromagnets. Comparison with recent experiments suggests that disorder can dramatically suppress critical currents.

Figure 1

(Color online) Separation of transport length scales in quantum Hall superfluid transport. This theory is intended to apply when the ordered state is well established and Hall angles are large because of the developing $\nu =1$ quantum Hall effect. At large Hall angles current enters and leaves the samples at the hot-spot corners, even when the source and drain contacts (gray) fully cover the ends of a Hall bar. When order is well established, the pseudospin orientation of the transport electrons achieves alignment with the condensate within a relatively small fraction of the sample area close to source and drain (solid yellow). In these areas pseudospin-transfer torques convert transport currents into condensate counterflow supercurrents. When source and drain are connected to opposite layers a net supercurrent is injected into the interior region of the sample. In the remaining sample area (shaded blue) collective interlayer tunneling can act as a sink for the counterflow supercurrent.

Figure 2

Magnetization change per time step $\delta m\times N$ vs applied voltage in a $500l\times 400l$ system for $2\pi l^{2}n{\Delta }_t={10}^{-5}{E}_0$. This curve was obtained by changing the bias voltage by $\delta V=2.5\times {10}^{-5}\mu \text{V}$ at each time step. $E_0=e^2/ϵl$, the energy unit used in all our calculations has a typical experimental value $\sim 7\text{meV}$. $\delta m$ develops large amplitude oscillation when the voltage exceeds the critical voltage $V_c$.

Figure 3

(Color online) Critical current vs $2\pi {\ell }^{2}n{\Delta }_t$ in a $400l\times 500l$ system. The dashed-dotted (red), long-dashed (blue), and short-dashed (black) curves plot the values of the bulk $\left(I_B^c\right)$, edge $\left(I_E^c\right)$, and corner $\left(I_C^c\right)$ limited critical currents discussed in the text for this sample geometry. The square dots plot LLS equation critical currents at a series of ${\Delta }_{t}n$ values. All the calculations in this paper were performed using pseudospin stiffness (exciton superfluid density) ${\rho }_s=0.005E_0$, where $E_0=e^2/ϵl$ is the energy unit. Currents are in units of $I_0=I_E^c=e{\rho }_s/\hslash$. For the value of ${\rho }_s$ used in these calculations $I_0\simeq 8\text{nA}$.

Figure 4

(Color online) Spatial distribution of the $\widehat{z}$ component of the pseudospin-transfer torque in a system with $500\times 400l^2$ area, $2\pi l^{2}n{\Delta }_t={10}^{-6}{E}_0$, and $I/I^c=0.29$. The pseudospin-transfer torque in the model studied here acts mainly in the hot-spot pixels. $E_0$ and $l$ are defined as in previous figures.

Figure 5

(Color online) Supercurrent distribution in system with area $500\times 400l^2$, $2\pi l^{2}n{\Delta }_t={10}^{-6}{E}_0$, and $I/I^c=0.29$. This plot is for a tunneling geometry in which the source is a top-layer contact and the drain is a bottom-layer contact. Supercurrents are generated near both hot spots and flow diagonally toward the sample center. $E_0$ and $l$ are defined as in previous figures.

Figure 6

(Color online) Supercurrent distribution in system with area $500\times 400l^2$, $2\pi l^{2}n{\Delta }_t={10}^{-6}{E}_0$. This plot is for the drag-geometry case in which the source and drain are both top-layer contacts, but other parameters of the calculation are identical to those used in the preceding tunnel-geometry figure. $E_0$ and $l$ are defined as in previous figures.

Figure 7

(Color online) Pseudospin phase distribution in a system with area $500\times 400l^2$, $2\pi l^{2}n{\Delta }_t={10}^{-8}{E}_0$, and $I/I^c=0.75$. The Josephson length at this value of ${\Delta }_t$ is $\sim 2500l$. Note that $\varphi$ is roughly constant through the system and that its value is close to $\pi /2$ because $I$ is close to $I_c$. The units used here are the same as in previous figures.

Figure 8

(Color online) Phase distribution of pseudospin in a system with area $500\times 400l^2$, $2\pi l^{2}n{\Delta }_t=1.2\times {10}^{-5}{E}_0$, and $I/I^c=0.97$. The Josephson length at this value of ${\Delta }_t$ is $\sim 70l$. Note that a steady state is reached even though $\varphi$ varies considerably across the sample and has values larger than $\pi /2$. The units used here are the same as in previous figures.

Figure 9

(Color online) Critical current vs system length $L$ for a narrow Hall bar with width $W=10l$. The single-particle tunneling amplitude $2\pi l^{2}n{\Delta }_t={10}^{-6}\left(E_0\right)$ corresponding to $\lambda \sim 250l$.

Figure 10

(Color online) Critical current vs system width $W$ for length $L=500l$. The single-particle tunneling amplitude $2\pi l^{2}n{\Delta }_t={10}^{-6}\left(E_0\right)$, corresponding to $\lambda \sim 250l$.