Critical Tunneling Currents in Quantum Hall Superfluids: Pseudospin-Transfer Torque Theory arXiv:1001.2923v1 [cond-mat.mes-hall] 18 Jan 2010 Jung-Jung Su 1 , 2 and Allan H. MacDonald 1 1 Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA and 2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545,USA (Dated: October 31, 2018) At total filling factor ν = 1 quantum Hall bilayers can have an ordered ground state with spon- taneous interlayer phase coherence. The ordered state is signaled experimentally by dramatically enhanced interlayer tunnel conductances at low bias voltages; at larger bias voltages inter-layer 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 inter-layer phase differences cannot be main- tained, 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. PACS numbers: At Landau level filling factor ν = 1 bilayer two- dimensional electron systems in the quantum Hall regime can have broken symmetry ground states 1–4 with sponta- neous inter-layer phase coherence. These ordered states can be viewed as excitonic superfluids, 5,6 or as XY pseudospin ferromagnets 7 in which the pseudospin is formed from the two-valued which layer quantum de- gree of freedom. The most robust experimental signa- ture of these states, a vastly enhanced inter-layer tunnel conductance 8–11 at small bias voltages, is still poorly un- derstood from a quantitative point of view. Two types of ideas, which differ most essentially in how the bias voltage is introduced in the theory, have been ex- plored in an effort to understand the height and width of the tunnel conductance peak. In one approach 12–15 the bias voltage is introduced as an effective magnetic field, uniform across the bilayer, which induces pseudospin pre- cession around the ˆ z axis, driving the interlayer phase difference at a steady rate and inducing a purely oscil- lating inter-layer current. When the microscopic inter- layer tunneling amplitude is treated as a perturbation, thermal and disorder fluctuations of the condensate are then responsible for a finite dc conductance peak. We refer to this type of theory below as the weak-coupling theory of the tunneling anomaly. Weak-coupling theo- ries predict 12–15 splitting of the zero voltage tunnel con- ductance peak into separate finite voltage peaks in the presence of a magnetic field component parallel to the two-dimensional layers, an effect that is not 16 seen ex- perimentally. The second type of transport theory 17–21 is formulated in terms of local chemical potentials of fermionic quasiparticles which may be altered by the or- dered state condensate but are still responsible for charge conduction. In this type of theory the tunnel conduc- tance is finite even in the absence of disorder and ther- mal fluctuations because charge has to be driven between normal-metal source and drain contacts. The resistance generally depends 18,21 on how the fermionic degrees of freedom which carry charge between leads are influenced by order parameter and disorder configurations and can- not be described in terms of condensate dynamics alone. In the second approach the width in voltage of the con- ductance peak is simply equal to the product of this re- sistance and the maximum current between source and drain at which the order parameter can maintain a time- independent steady-state value. In this article we expand on the second type of the- ory of the conductance peak, using pseudospin-transfer torque ideas 22,23 borrowed from recent ferromagnetic- metal spintronics literature to model the influence of the transport current on the pseudospin magnetization. We find that the critical current depends in general on de- tails of the sample geometry and on how disorder and lo- calization physics influence transport inside the system. Generally speaking however, the critical current is pro- portional to system area and to the inter-layer tunnel- ing amplitude when the condensate’s Josephson length is larger than the system perimeter, and proportional to the system perimeter and to the square root of the inter-layer tunneling amplitude when it is shorter. Our paper is organized as follows. In Section II we in- troduce the pseudospin-transfer torque theory of order- parameter dynamics in a bilayer quantum Hall ferromag- net. We use this theory in Section III to discuss critical current values from a qualitative point of view. In Section IV we report on a series of numerical studies which take into account the two-dimensional nature of the systems of interest and the edge dominated current-paths typical of strong magnetic fields. The pseudospin transfer torque theory enables us to assess the influence of sample geom- etry on critical currents. Finally in Section V we discuss the significance of our findings in relation to recent exper- iments. We find that experimental critical currents are several orders of magnitude smaller than theoretical ones and argue that vortex-like disorder-induced pseudospin textures must be largely responsible for this discrepancy. I. INTRODUCTION