Spontaneous Interlayer Phase Coherence Research Articles

Layer pseudospin magnetism in a transition metal dichalcogenide double-moiré system

Spontaneous order of layer pseudospins in two-dimensional bilayers is common in quantum Hall systems, where it is responsible for hysteretic responses to gate fields in states with Ising order, and giant drag voltages in states with XY (spontaneous inter-layer phase coherence) order. In this article we predict that layer pseudospin order will also occur in double-moir\'e strongly correlated two-dimensional electron systems. We comment on similarities and differences in the competition between the two types of order in quantum Hall and double-moir\'e systems, and relate our findings to previous work on Falicov-Kimball models of electronic ferroelectrics.

Theory of two-dimensional spatially indirect equilibrium exciton condensates

We present a theory of bilayer two-dimensional electron systems that host a spatially indirect exciton condensate when in thermal equilibrium. Equilibrium bilayer exciton condensates (BXCs) are expected to form when two nearby semiconductor layers are electrically isolated, and when the conduction band of one layer is brought close to degeneracy with the valence band of a nearby layer by varying bias or gate voltages. BXCs are characterized by spontaneous inter-layer phase coherence and counterflow superfluidity. The bilayer system we consider is composed of two transition metal dichalcogenide monolayers separated and surrounded by hexagonal boron nitride. We use mean-field-theory and a bosonic weakly interacting exciton model to explore the BXC phase diagram, and time-dependent mean-field theory to address condensate collective mode spectra and quantum fluctuations. We find that a phase transition occurs between states containing one and two condensate components as the layer separation and the exciton density are varied, and derive simple approximate expressions for the exciton-exciton interaction strength which we show can be measured capacitively.

Exact-diagonalization study of exciton condensation in electron bilayers

We report on small-cluster exact-diagonalization calculations which prove the formation of electron-hole pairs (excitons) as a prerequisite for spontaneous interlayer phase coherence in double-layer systems described by the extended Falicov-Kimball model. Evaluating the anomalous Green's function and momentum distribution function of the pairs, and thereby analyzing the dependence of the exciton binding energy, condensation amplitude, and coherence length on the Coulomb interaction strength, we demonstrate a crossover between a BCS-like electron-hole pairing transition and a Bose-Einstein condensation of tightly bound preformed excitons. We furthermore show that a mass imbalance between electrons and holes tends to suppress the condensation of excitons.

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 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 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.

Superconductivity of electron–hole pairs in a bilayer graphene system in a quantizing magnetic field

A state with spontaneous interlayer phase coherence in a bilayer quantum Hall system based on graphene is studied. This state can be regarded as a gas of superfluid electron–hole pairs whose components belong to different layers. A superfluid flow of such pairs is equivalent to two electric supercurrents in the layers. It is shown that in a graphene system a state with interlayer phase coherence arises if a definite unbalance of the filling factors of the Landau levels in neighboring layers is created. The temperature of the transition into a superfluid state, the maximum interlayer distance for which phase coherence is possible, and the critical values of the supercurrent are found. The advantages of using graphene systems instead of GaAs heterostructures to realize bilayer electron–hole superconductivity are discussed.

Evolution of the spin-split quantum Hall states with magnetic field tilt in the InAs-based double quantum wells

Development of quantum Hall peculiarities due to mobility gap between spin-split magnetic levels with addition of the parallel magnetic field component B|| is analyzed in double quantum wells (DQW) created in InGaAs/GaAs and InAs/AlSb heterosystems chosen due to their relatively large bulk g-factors. In InGaAs/GaAs DQWs, the nonmonotonous behavior of these peculiarities is observed and explained within single-electron approach in terms of competition between enhanced spin splitting and localization of electrons in the layers of DQW with increased B||. In InAs/AlSb DQW, the tunneling connection between the layers is very weak due to high barrier, nevertheless the collective odd-numbered peculiarities are revealed that exist due to spontaneous interlayer phase coherence. B|| destroys these states that is manifested, in particular, in the suppression of the peculiarity for filling factor v = 3.

Influence of Disorder on Electron-Hole Pair Condensation in Graphene Bilayers

Graphene bilayers can condense into a state with spontaneous interlayer phase coherence that supports dissipationless counterflow supercurrents. Here, we address the influence of disorder on the graphene bilayer mean field and Kosterlitz-Thouless critical temperatures and report on a simple criteria for the survival of pair condensation.

Nucleation of merons in double quantum dots with spontaneous interlayer phase coherence

We have investigated how merons nucleate in the spontaneous interlayer phase coherent state of double quantum dots when a distortion with a reflection symmetry is present in the lateral confinement potential. When the lateral confinement strength is large enough merons with a negative topological charge nucleate in the ground state. For weaker values of the lateral confinement strength both density depletions and positively charged merons are present. These density depletions form an ordered structure. When the strength of the distortion increases some of these density depletions can transform into merons. In the presence of merons the resulting electron density in one of the layers is the mirror image of the electron density of the other layer. Even when a random disorder potential is present both density depletions and merons can coexist.

Spin-flip excitations in bilayer quantum Hall ferromagnets

Bilayer quantum Hall systems at ν=1 are believed to have a ground state with spontaneous spin-polarization and spontaneous interlayer phase coherence. In this paper, we address the spectrum of spin-flip collective excitations using a generalized lattice spin model that appeals to the quantum Hall gap to justify neglect of charge fluctuations. We find that the long wavelength spin-flip excitation spectrum consists of the global Larmor spin-wave mode and a higher energy mode in which spin-waves in the two-layers are out of phase. We address the dependence of the energy splitting of the two modes on quantum fluctuations in the bilayer ground state.

Spontaneous interlayer coherence in bilayer Kondo systems

Bilayer Kondo systems present interesting models to illustrate the competition between the Kondo effect and intermoment exchange. Such bilayers can exhibit two sharply distinct Fermi liquid phases which are distinguished by whether or not the local moments participate in the Fermi sea. We study these phases and the evolution from one to the other upon changing Kondo coupling. We argue that an ordered state with spontaneous interlayer phase coherence generically intervenes between the two Fermi liquids. Such a condensate phase breaks a U(1) symmetry and is bounded by a finite-temperature Kosterlitz-Thouless transition. Based on general arguments and mean-field calculations we investigate the phase diagram and associated quantum phase transitions.

Interlayer phase coherence and superfluidity of electron–hole pairs in multilayer quantum Hall systems

The states with spontaneous interlayer phase coherence in multilayer quantum Hall systems are studied. It is shown that in systems with the average filling factor per layer equal to one half the phase coherence establishes only in bilayer complexes of adjacent layers, and there is no coherence between layers belonging to different complexes. Conditions of stability for such a state against transitions to the uniform charge-ordered state and the charge density wave state are analyzed. It is shown that in systems with finite number of layers N the state with interlayer phase coherence may exist at interlayer distances d < d c , and the critical distance d c decreases under increasing N.

Onset of interlayer phase coherence in a bilayer two-dimensional electron system: Effect of layer density imbalance

Tunneling and Coulomb drag are sensitive probes of spontaneous interlayer phase coherence in bilayer two-dimensional electron systems at total Landau level filling factor ${\ensuremath{\nu}}_{T}=1$. We find that the phase boundary between the interlayer phase coherent state and the weakly coupled compressible phase moves to larger layer separations as the electron density distribution in the bilayer is imbalanced. The critical layer separation increases quadratically with layer density difference.

Vanishing Hall resistance at high magnetic field in a double-layer two-dimensional electron system.

At total Landau level filling factor nu(tot)=1 a double-layer two-dimensional electron system with small interlayer separation supports a collective state possessing spontaneous interlayer phase coherence. This state exhibits the quantized Hall effect when equal electrical currents flow in parallel through the two layers. In contrast, if the currents in the two layers are equal, but oppositely directed, both the longitudinal and Hall resistances of each layer vanish in the low-temperature limit. This finding supports the prediction that the ground state at nu(tot)=1 is an excitonic superfluid.

Tunneling, dissipation, and superfluid transition in quantum Hall bilayers.

We study bilayer quantum Hall systems at total Landau level filling factor nu=1 in the presence of interlayer tunneling and coupling to a dissipative normal fluid. Describing the dynamics of the interlayer phase by an effective quantum dissipative XY model, we show that there exists a critical dissipation sigma(c) set by the conductance of the normal fluid. For sigma>sigma(c), interlayer tunnel splitting drives the system to a nu=1 quantum Hall state. For sigma<sigma(c), interlayer tunneling is irrelevant at low temperatures; the system exhibits an excitonic superfluid transition to a collective quantum Hall state supported by spontaneous interlayer phase coherence. The resulting phase structure and the behavior of the in-plane and tunneling currents are studied in connection to experiments.

Collective transport in bilayer quantum Hall systems

Filling factor ν=1 incompressible states in ideal bilayer quantum Hall systems have spontaneous interlayer phase coherence and can be regarded either as easy-plane pseudospin ferromagnets or as condensates of excitons formed from electrons in one layer and holes in the other layer. In this paper we discuss efforts to achieve an understanding of the two different types of transport measurements (which we refer to as drag and tunneling experiments, respectively) that have been carried out in bilayer quantum Hall systems by the group of Jim Eisenstein at the California Institute of Technology. In a drag experiment, current is sent through one of the two-layers and the voltage drop is measured in the other layer. We will argue that the finding of these experiments that the voltage drop in the drag layer is different from that in the drive layer, is an experimental proof that these bilayers do not have quasi-long-range excitonic order. The property that at ν=1 the longitudinal drag voltage increases from near zero when spontaneous coherence is initially established, then falls back toward zero as it becomes well established, can be explained as a competition between the broken symmetry and the gap to which it gives rise. In the tunneling experiment, current is injected in one layer and removed from the other layer. The absence of quasi-long-range order likely explains the relatively small tunneling conductance per area found in the these measurements.

Charge ordering and interlayer phase coherence in quantum Hall superlattices

The possibility of the existence of states with a spontaneous interlayer phase coherence in multilayer electron systems in a high perpendicular magnetic field is investigated. It is shown that phase coherence can be established in such systems only within individual pairs of adjacent layers, while such coherence does not exist between layers of different pairs. The conditions for stability of a state with interlayer phase coherence against transition to a charge-ordered state are determined. It is shown that in a system with N⩽10 layers there is stability at any value of the interlayer distance d. For N&amp;gt;10 there are two intervals of stability: at sufficiently large and at sufficiently small d. For N→∞ the stability interval in the region of small d vanishes.

Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron–hole pairing

In n–p bilayer systems an exotic phase-coherent state emerges due to Coulomb pairing of n-layer electrons with p-layer holes. Unlike Josephson junctions, the order parameter phase may be locked by matrix elements of interlayer tunneling in n–p bilayers. Here we show how the phase locking phenomenon specifies the response of the electron–hole condensate to interlayer voltages. In the absence of an applied magnetic field, the phase is steady-state (locked) at low interlayer voltages, V<Vc, but the phase increases monotonically with time (is unlocked) at V>Vc. The change in the system dynamics at V=Vc gives rise to a peak in the differential tunneling conductance. The peak width Vc is proportional to the absolute value of the tunneling matrix element |T12|, but its height does not depend on |T12|; thus the peak is sharp for small |T12|. An in-plane magnetic field reduces the peak height considerably. The present results are in qualitative agreement with the zero-bias peak behavior that has recently been observed in bilayer quantum Hall pseudoferromagnets with spontaneous interlayer phase coherence.

Bilayer 2D electron systems at νtot=1: phase boundary between weak and strong coupling

The collapse of the excitonic quantized Hall state at ν tot=1 in double layer two-dimensional electron systems is studied via interlayer tunneling and Coulomb drag. We find that spontaneous interlayer phase coherence, perhaps the most important hallmark of the excitonic phase and directly detected via a strong resonant enhancement of the zero-bias tunneling conductance, collapses very rapidly above a critical layer separation. In contrast, a related anomaly in the longitudinal component of Coulomb drag at ν tot=1 subsides much more slowly as the layer separation is increased beyond the critical point.

Even-odd effect in spontaneously coherent bilayer quantum Hall droplets.

Using exact diagonalization in the disk geometry we predict a novel even-odd effect in the Coulomb-blockade spectra of vertically coupled double quantum dots under an external magnetic field. The even-odd effect in the tunneling conductance is a direct manifestation of spontaneous interlayer phase coherence, and is similar to the even-odd resonance in the Cooper pair box problem in mesoscopic superconducting grains. Coherent fluctuations in the number of Cooper pairs in superconductors are analogous to the fluctuations in the relative number difference between the two layers in quantum Hall droplets.

Evidence for spontaneous interlayer phase coherence in a bilayer quantum Hall exciton condensate

Recent experimental work on the quantized Hall state at total filling factor ν T=1 in bilayer 2D electron systems has revealed a number of striking phenomena, including a giant and sharply resonant enhancement of the interlayer tunneling conductance at zero bias. The tunneling enhancement is a compelling indicator of spontaneous interlayer phase coherence among the electrons in the system. Such phase coherence is perhaps the single most important attribute of the excitonic Bose condensate which describes this remarkable quantum Hall state.