Wigner Solid Research Articles

Origin of Pinning Disorder in Magnetic-Field-Induced Wigner Solids.

At low Landau level filling factors (ν), Wigner solid phases of two-dimensional electron systems in GaAs are pinned by disorder and exhibit a pinning mode, whose frequency is a measure of the disorder that pins the Wigner solid. Despite numerous studies spanning the past three decades, the origin of the disorder that causes the pinning and determines the pinning mode frequency remains unknown. Here, we present a study of the pinning mode resonance in the low-ν Wigner solid phases of a series of ultralow-disorder GaAs quantum wells which are similar except for their varying well widths d. The pinning mode frequencies f_{p} decrease strongly as d increases, with the widest well exhibiting f_{p} as low as ≃35 MHz. The amount of reduction of f_{p} with increasing d can be explained remarkably well by tails of the wave function impinging into the alloy-disordered Al_{x}Ga_{1-x}As barriers that contain the electrons. However, it is imperative that the model for the confinement and wave function includes the Coulomb repulsion in the growth direction between the electrons as they occupy the quantum well.

Moving Crystal Phases of a Quantum Wigner Solid in an Ultra-High-Quality 2D Electron System.

In low-disorder, two-dimensional electron systems (2DESs), the fractional quantum Hall states at very small Landau level fillings (ν) terminate in a Wigner solid (WS) phase, where electrons arrange themselves in a periodic array. The WS is typically pinned by the residual disorder sites and manifests an insulating behavior, with nonlinear current-voltage (I-V) and noise characteristics. We report here measurements on an ultralow-disorder, dilute 2DES, confined to a GaAs quantum well. In the ν<1/5 range, superimposed on a highly insulating longitudinal resistance, the 2DES exhibits a developing fractional quantum Hall state at ν=1/7, attesting to its exceptional high quality and dominance of electron-electron interaction in the low filling regime. In the nearby insulating phases, we observe remarkable nonlinear I-V and noise characteristics as a function of increasing current, with current thresholds delineating three distinct phases of the WS: a pinned phase (P1) with very small noise, a second phase (P2) in which dV/dI fluctuates between positive and negative values and is accompanied by very high noise, and a third phase (P3) where dV/dI is nearly constant and small, and noise is about an order of magnitude lower than in P2. In the depinned (P2 and P3) phases, the noise spectrum also reveals well-defined peaks at frequencies that vary linearly with the applied current, suggestive of washboard frequencies. We discuss the data in light of a recent theory that proposes different dynamic phases for a driven WS.

Thermal melting of a quantum electron solid in the presence of strong disorder: Anderson localization versus the Wigner crystal

We consider the temperature-induced melting of a Wigner solid in one-dimensional (1D) and two-dimensional (2D) lattices of electrons interacting via the long-range Coulomb interaction in the presence of strong disorder arising from charged impurities in the system. The system simulates semiconductor-based 2D electron layers where Wigner crystallization is often claimed to be observed experimentally. Using exact diagonalization and utilizing the inverse participation ratio as well as conductance to distinguish between the localized insulating solid phase and the extended metallic liquid phase, we find that the effective melting temperature may be strongly enhanced by disorder since the disordered crystal typically could be in a localized glassy state incorporating the combined nonperturbative physics of both Anderson localization and Wigner crystallization. This disorder-induced enhancement of the melting temperature may explain why experiments often manage to observe insulating disorder-pinned Wigner solids in spite of the experimental temperature being decisively far above the theoretical melting temperature of the pristine Wigner crystal phase in many cases.

Bragg–Cherenkov resonance and polaron-like decoupling of the Wigner solid on superfluid helium

Nonlinear polaron-like dynamics of the two-dimensional Wigner solid (WS) on superfluid 4He are theoretically analyzed in different models and transport regimes for their similarities and distinctions. The Bragg-Cherenkov (BC) resonant excitation of surface waves and WS decoupling from surface dimples were usually considered in terms of a dc transport model. At the same time, field-velocity characteristics of the WS are measured under ac conditions and presented for time-averaged quantities. Here the nonlinear equation of motion of the WS coupled to surface dimples is studied for ac conditions using two different approaches based on fixing the driving field or the output current. Both approaches are shown to give similar results for the first harmonics of major transport properties. In the ac theory, the BC resonances for dimple inertia and the momentum relaxation rate have asymmetrical shapes, which is in contrast with the results of dc models. Even a quite low driving frequency is shown to affect the amplitude of the BC resonance and decoupling of the WS. Above the BC threshold, the effective mass of surface dimples as a function of the velocity amplitude strongly oscillates indicating multiple recoupling processes.

Anisotropic Two-Dimensional Disordered Wigner Solid.

The interplay between the Fermi sea anisotropy, electron-electron interaction, and localization phenomena can give rise to exotic many-body phases. An exciting example is an anisotropic two-dimensional (2D) Wigner solid (WS), where electrons form an ordered array with an anisotropic lattice structure. Such a state has eluded experiments up to now as its realization is extremely demanding: First, a WS entails very low densities where the Coulomb interaction dominates over the kinetic (Fermi) energy. Attaining such low densities while keeping the disorder low is very challenging. Second, the low-density requirement has to be fulfilled in a material that hosts an anisotropic Fermi sea. Here, we report transport measurements in a clean (low-disorder) 2D electron system with anisotropic effective mass and Fermi sea. The data reveal that at extremely low electron densities, when the r_{s} parameter, the ratio of the Coulomb to the Fermi energy, exceeds ≃38, the current-voltage characteristics become strongly nonlinear at small dc biases. Several key features of the nonlinear characteristics, including their anisotropic voltage thresholds, are consistent with the formation of a disordered, anisotropic WS pinned by the ubiquitous disorder potential.

Observation of the Rydberg Resonance in Surface Electrons on Superfluid Helium Confined in a 4-$$\mu$$m Deep Channel

We report the first observation of the microwave-induced intersubband (Rydberg) resonance in the surface-bound electrons on superfluid helium confined in a single 4-$\mu$m deep channel. The resonance signal from a few thousand of surface electrons comprising the Wigner Solid (WS) is detected by observing WS melting due to the microwave absorption. The observed transition frequency for the two lowest Rydberg states, in the range of 0.4-0.5 THz, is determined by the image charges induced by the surface electrons in conducting electrodes of the microchannel and the potentials due to applied voltages, and is in a good agreement with our calculations. The observed large broadening of the resonance on the order of 10 GHz, which is due to inhomegeneous distribution of the pressing electric field in the microchannel, is also in a reasonable agreement with our finite-element modeling calculations. This study of the confined electrons is motivated by their potential use for charge and spin qubits.

Particle-hole symmetry and the reentrant integer quantum Hall Wigner solid

The interplay of strong Coulomb interactions and of topology is currently under intense scrutiny in various condensed matter and atomic systems. One example of this interplay is the phase competition of fractional quantum Hall states and the Wigner solid in the two-dimensional electron gas. Here we report a Wigner solid at ν = 1.79 and its melting due to fractional correlations occurring at ν = 9/5. This Wigner solid, that we call the reentrant integer quantum Hall Wigner solid, develops in a range of Landau level filling factors that is related by particle-hole symmetry to the so called reentrant Wigner solid. We thus find that the Wigner solid in the GaAs/AlGaAs system straddles the partial filling factor 1/5 not only at the lowest filling factors, but also near ν = 9/5. Our results highlight the particle-hole symmetry as a fundamental symmetry of the extended family of Wigner solids and paint a complex picture of the competition of the Wigner solid with fractional quantum Hall states.

Thermal and Quantum Melting Phase Diagrams for a Magnetic-Field-Induced Wigner Solid.

A sufficiently large perpendicular magnetic field quenches the kinetic (Fermi) energy of an interacting two-dimensional (2D) system of fermions, making them susceptible to the formation of a Wigner solid (WS) phase in which the charged carriers organize themselves in a periodic array in order to minimize their Coulomb repulsion energy. In low-disorder 2D electron systems confined to modulation-doped GaAs heterostructures, signatures of a magnetic-field-induced WS appear at low temperatures and very small Landau level filling factors (ν≃1/5). In dilute GaAs 2D hole systems, on the other hand, thanks to the larger hole effective mass and the ensuing Landau level mixing, the WS forms at relatively higher fillings (ν≃1/3). Here we report our measurements of the fundamental temperature vs filling phase diagram for the 2D holes' WS-liquid thermal melting. Moreover, via changing the 2D hole density, we also probe their Landau level mixing vs filling WS-liquid quantum melting phase diagram. We find our data to be in good agreement with the results of very recent calculations, although intriguing subtleties remain.

Transport Properties of a Quasi-1D Wigner Solid on Liquid Helium Confined in a Microchannel with Periodic Potential

We present transport measurements in a quasi-1D system of surface electrons on liquid helium confined in a 101-upmu m-long and 5-upmu m-wide microchannel where an electrostatic potential with periodicity of 1-upmu m along the channel is introduced. In particular, we investigate the influence of such a potential on the nonlinear transport properties of a quasi-1D Wigner solid (WS) by varying the amplitude of the periodic potential in a wide range. At zero and small values of amplitude, the quasi-1D WS in the microchannel shows expected features such as the Bragg–Cherenkov scattering of ripplons and re-entrant melting. As the amplitude of the potential increases, the above features are strongly suppressed. This behavior suggests the loss of long-range positional order in the electron system, which is reminiscent of re-entrant melting behavior due to lateral confinement of the WS in the channel.

Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium

Nonlinear dynamics and transport properties of a 2D Wigner solid (WS) on the free surface of superfluid helium are theoretically studied. The analysis is nonperturbative in the amplitude of the WS velocity. An anomalous nonlinear response of the liquid helium surface to the oscillating motion of the WS is shown to appear when the driving frequency is close to subharmonics of the frequency of a capillary wave (ripplon) whose wave vector coincides with a reciprocal-lattice vector. As a result, the effective mass of surface dimples formed under electrons and the kinetic friction acquire sharp anomalies in the low-frequency range, which affects the mobility and magnetoconductivity of the WS. The results obtained here explain a variety of experimental observations reported previously.

Bistable transport properties of a quasi-one-dimensional Wigner solid on liquid helium under continuous driving

We investigate low-frequency fluctuations in the transport characteristics of a quasi-1D Wigner solid (WS) moving along a liquid helium substrate in response to a sinusoidal driving voltage. The fluctuations occur between distinct transport modes in which the decoupling of the WS from ripplonic polarons (or 'dimple lattice', DL) formed on the helium surface does, or does not, occur during each ac cycle. We demonstrate that a Gaussian-like distribution in the decoupling threshold force gives rise to this bistability, as the low-frequency switching occurs when the probability of decoupling during each ac cycle is small but finite. We attribute the distribution in the decoupling threshold force to the range of structural configurations allowed for the quasi-1D electron lattice, which influences the strength of the WS-DL coupling. Hence, the switching rate between the ac transport modes is extremely sensitive to the microscopic properties of the electron solid.

Microwave spectroscopic observation of a Wigner solid within the ν=1/2 fractional quantum Hall effect

The fractional quantum Hall effect (FQHE) states at half integer Landau fillings ($\nu$) have long been of great interest, since they have correlations that differ from those of the fundamental Laughlin states found at odd denominators. At $\nu=1/2$ the FQHE has been observed in wide or double quantum wells, and is ascribed to the two-component Halperin-Laughlin $\Psi_{331}$ state. $\Psi_{331}$ excitations carry charge $\pm e/4$, like the carriers of $\nu=5/2$ states which are of interest in quantum computation. Further, such an excitation (quasiparticle or -hole) of $\Psi_{331}$ has unequal, opposite charge in the top and bottom layers, and hence an up or down dipole moment. Here we report evidence for a Wigner solid (WS) of such dipolar quasiholes from a quantitative study of the microwave spectra of a wide quantum well (WQW) at $\nu$ close to 1/2.

Nonlinear transport of the inhomogeneous Wigner solid in a channel geometry

The transport properties of an electronic Wigner solid (WS) on the surface of superfluid helium-4 are investigated in a geometry that allows WS systems of significantly different sizes to be defined. We probe and compare transport phenomena attributed to the electronic order, such as Bragg-Cherenkov scattering and the WS sliding transition, in the cases of a long homogeneous WS, a small WS island, and a long inhomogeneous WS. We find no significant WS size effects on the transport properties, in contrast to predictions of theoretical works, which indicates an absence of long-range order in the WS systems under consideration. For the inhomogeneous WS, consisting of two distinct WSs connected in series, a complex interplay of individual WS transport properties is observed, as revealed in the observation of two separate Bragg-Cherenkov plateaus and WS sliding transitions. A simple model is proposed that provides a qualitative explanation of this behavior.

Structural order and melting of a quasi-one-dimensional electron system

We investigate the influence of confinement on the positional order of a quasi-1D electron system trapped on the surface of liquid helium. We find evidence that the melting of the Wigner solid (WS) depends on the confinement strength, as well as electron density and temperature. A reentrant solid-liquid-solid transition is observed for increasing electron density under constant electrostatic confinement. As the electron row number $N_y$ changes, varying commensurability results in a modulation of the WS order, even when $N_y$ is large (several tens). This is confirmed by Monte Carlo simulations.

Stick-Slip Motion of the Wigner Solid on Liquid Helium.

We present time-resolved transport measurements of a Wigner solid (WS) on the surface of liquid helium confined in a micron-scale channel. At rest, the WS is "dressed" by a cloud of quantized capillary waves (ripplons). Under a driving force, we find that repeated WS-ripplon decoupling leads to stick-slip current oscillations, the frequency of which can be tuned by adjusting the temperature, pressing electric field, or electron density. The WS on liquid He is a promising system for the study of polaronlike decoupling dynamics.

Pinning modes of high-magnetic-field Wigner solids with controlled alloy disorder

For a series of samples with 2D electron systems in dilute Al_x Ga_{1-x}As, with varying x from 0 to 0.8 %, we survey the pinning mode resonances of Wigner solids at the low Landau filling termination of the fractional quantum Hall effect (FQHE) series. For all x studied, the pinning modes are present with frequencies, f_pk, that are consistent with collective weak pinning. For x>=0.22% we find f_pk vs B exhibits a rapid increase that is not present for x=0. We find the observed f_pk is much smaller than values calculated with a simple Wigner solid model which neglects the effects of the disorder on the charge distribution of a carrier.

Transport Properties of the Two-Dimensional Wigner Solid on Liquid Helium in the Presence of a High-Frequency Damaging Electric Field

Experimental studies of transport properties of the two-dimensional (2D) Wigner solid (WS) over a liquid helium surface are performed in the presence of an additional (damaging) high frequency electric field. Surface electrons are in the linear transport regime with regard to the driving electric field created by a potential of a small amplitude (about 1 mV) whose frequency varies in the range 3–8 MHz. The damaging potential applied simultaneously has substantially higher amplitudes (30–300 mV) and frequency 36 MHz. The evolution of resonance spectra of the coupled phonon-ripplon system, electron resistivity, and the effective mass of surface dimples caused by an increase in the damaging electric field amplitude is observed. At a certain damaging potential of about 300 mV the major electron-ripplon resonance is shown to disappear, which is accompanied by a sharp decrease in the dimple effective mass, and electron resistivity. Experimental data are explained by a theoretical model of high-frequency WS decoupling from surface dimples caused by the breakdown of the balance of forces applied to the WS.

Electrons on the Surface of Superfluid 3He

A review of recent investigations of transport properties of surface state electrons on superfluid 3He is given. The surface state electrons in this temperature region form the Wigner solid (WS), a triangular lattice of electrons with a typical lattice constant of 1 μm. The WS is accompanied with a shallow corrugation of He surface commensurate with the WS. A model is introduced to interpret the observed WS resistivity. The model takes into account specular quasiparticle (QP) reflection from the moderately corrugated free surface, and treats the QP as if it is a quasiclassical particle. After adopting anisotropic properties of superfluid 3He order parameter and QP energy, the model provides satisfactory account of the observed properties. Preliminary results of mobility measurements of ions trapped below superfluid 3He-B are also given.

Melting of Two-Dimensional Electron Crystal on Liquid 3He Induced by Resonance Microwave Absorption

AbstractMicrowave-resonance induced increase of magneto-resistivity σ xx −1 of the surface-state electrons (SSE) on liquid 3He is observed at low temperatures at which SSE crystallize into the Wigner solid (WS). At the resonance, the relative change of σ xx −1 of SSE in the solid phase is found to be several times of the corresponding change in the liquid phase measured under similar experimental conditions. We attribute this effect to melting of WS induced by electron heating which accompanies the microwave absorption. σ xx −1 is found to increase gradually with microwave power, which is rather different from the sharp resistivity increase induced by direct heating of the substrate in the absence of microwaves. The effect of microwave resonance-induced heating on the sliding of WS from the periodic surface deformation of liquid is also investigated.

Pinning Modes and Interlayer Correlation in High-Magnetic-Field Bilayer Wigner Solids

We report studies of pinning mode resonances in the low total Landau filling (nu) Wigner solid of a series of bilayer hole samples with negligible interlayer tunneling and with varying interlayer separation d. Comparison of states with equal layer densities (p,p) to single layer states (p,0) produced in situ by biasing, indicates that there is interlayer quantum correlation in the solid at small d. Also, the resonance frequency at small d is decreased just near nu = 1/2 and 2/3, indicating the importance in the solid of correlations related to those in the fractional quantum Hall effects.