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.

Signatures of Correlated Defects in an Ultraclean Wigner Crystal in the Extreme Quantum Limit.

Low-disorder two-dimensional electron systems in the presence of a strong, perpendicular magnetic field terminate at very small Landau level filling factors in a Wigner crystal (WC), where the electrons form an ordered array to minimize the Coulomb repulsion. The nature of this exotic, many-body, quantum phase is yet to be fully understood and experimentally revealed. Here we probe one of WC's most fundamental parameters, namely, the energy gap that determines its low-temperature conductivity, in record mobility, ultrahigh-purity, two-dimensional electrons confined to GaAs quantum wells. The WC domains in these samples contain ≃1000 electrons. The measured gaps are a factor of three larger than previously reported for lower quality samples, and agree remarkably well with values predicted for the lowest-energy, intrinsic, hypercorrelated bubble defects in a WC made of flux-electron composite fermions, rather than bare electrons. The agreement is particularly noteworthy, given that the calculations are done for disorder-free composite fermion WCs, and there are no adjustable parameters. The results reflect the exceptionally high quality of the samples, and suggest that composite fermion WCs are indeed more stable compared to their electron counterparts.

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.

Resistively loaded coplanar waveguide for microwave measurements of induced carriers.

We describe the use of a coplanar waveguide (CPW) whose slots are filled with a resistive film, a resistively loaded CPW (RLCPW), to measure two-dimensional electron systems (2DESs). The RLCPW applied to the sample hosting the 2DES provides a uniform metallic surface serving as a gate to control the areal charge density of the 2DES. As a demonstration of this technique, we present measurements on a Si metal-oxide-semiconductor field-effect transistor and a model that successfully converts microwave transmission coefficients into conductivity of a nearby 2DES capacitively coupled to the RLCPW. We also describe the process of fabricating the highly resistive metal film required for fabrication of the RLCPW.

Correlated States of 2D Electrons near the Landau Level Filling ν=1/7.

The ground state of two-dimensional electron systems (2DESs) at low Landau level filling factors (ν≲1/6) has long been a topic of interest and controversy in condensed matter. Following the recent breakthrough in the quality of ultrahigh-mobility GaAs 2DESs, we revisit this problem experimentally and investigate the impact of reduced disorder. In a GaAs 2DES sample with density n=6.1×10^{10}/cm^{2} and mobility µ=25×10^{6} cm^{2}/V s, we find a deep minimum in the longitudinal magnetoresistance (R_{xx}) at ν=1/7 when T≃104 mK. There is also a clear sign of a developing minimum in R_{xx} at ν=2/13. While insulating phases are still predominant when ν≲1/6, these minima strongly suggest the existence of fractional quantum Hall states at filling factors that comply with the Jain sequence ν=p/(2mp±1) even in the very low Landau level filling limit. The magnetic-field-dependent activation energies deduced from the relation R_{xx}∝e^{E_{A}/2kT} corroborate this view and imply the presence of pinned Wigner solid states when ν≠p/(2mp±1). Similar results are seen in another sample with a lower density, further generalizing our observations.

Probing the Melting of a Two-Dimensional Quantum Wigner Crystal via its Screening Efficiency.

One of the most fundamental and yet elusive collective phases of an interacting electron system is the quantum Wigner crystal (WC), an ordered array of electrons expected to form when the electrons' Coulomb repulsion energy eclipses their kinetic (Fermi) energy. In low-disorder, two-dimensional (2D) electron systems, the quantum WC is known to be favored at very low temperatures (T) and small Landau level filling factors (ν), near the termination of the fractional quantum Hall states. This WC phase exhibits an insulating behavior, reflecting its pinning by the small but finite disorder potential. An experimental determination of a T vs ν phase diagram for the melting of the WC, however, has proved to be challenging. Here we use capacitance measurements to probe the 2D WC through its effective screening as a function of T and ν. We find that, as expected, the screening efficiency of the pinned WC is very poor at very low T and improves at higher T once the WC melts. Surprisingly, however, rather than monotonically changing with increasing T, the screening efficiency shows a well-defined maximum at a T that is close to the previously reported melting temperature of the WC. Our experimental results suggest a new method to map out a T vs ν phase diagram of the magnetic-field-induced WC precisely.

Wigner solid pinning modes tuned by fractional quantum Hall states of a nearby layer.

We studied a bilayer system hosting two-dimensional electron systems (2DESs) in close proximity but isolated from one another by a thin barrier. One 2DES has low electron density and forms a Wigner solid (WS) at high magnetic fields. The other has much higher density and, in the same field, exhibits fractional quantum Hall states (FQHSs). The WS spectrum has resonances which are understood as pinning modes, oscillations of the WS within the residual disorder. We found the pinning mode frequencies of the WS are strongly affected by the FQHSs in the nearby layer. Analysis of the spectra indicates that the majority layer screens like a dielectric medium even when its Landau filling is ~1/2, at which the layer is essentially a composite fermion (CF) metal. Although the majority layer is only ~ one WS lattice constant away, a WS site only induces an image charge of ~0.1e in the CF metal.

Microwave spectroscopy of the low-filling-factor bilayer electron solid in a wide quantum well.

At the low Landau filling factor termination of the fractional quantum Hall effect series, two-dimensional electron systems exhibit an insulating phase that is understood as a form of pinned Wigner solid. Here we use microwave spectroscopy to probe the transition to the insulator for a wide quantum well sample that can support single-layer or bilayer states depending on its overall carrier density. We find that the insulator exhibits a resonance which is characteristic of a bilayer solid. The resonance also reveals a pair of transitions within the solid, which are not accessible to dc transport measurements. As density is biased deeper into the bilayer solid regime, the resonance grows in specific intensity, and the transitions within the insulator disappear. These behaviours are suggestive of a picture of the insulating phase as an emulsion of liquid and solid components.

Microwave spectroscopic observation of distinct electron solid phases in wide quantum wells.

In high magnetic fields, two-dimensional electron systems can form a number of phases in which interelectron repulsion plays the central role, since the kinetic energy is frozen out by Landau quantization. These phases include the well-known liquids of the fractional quantum Hall effect, as well as solid phases with broken spatial symmetry and crystalline order. Solids can occur at the low Landau-filling termination of the fractional quantum Hall effect series but also within integer quantum Hall effects. Here we present microwave spectroscopy studies of wide quantum wells that clearly reveal two distinct solid phases, hidden within what in d.c. transport would be the zero diagonal conductivity of an integer quantum-Hall-effect state. Explanation of these solids is not possible with the simple picture of a Wigner solid of ordinary (quasi) electrons or holes.

Colossal magnetoresistance in an ultraclean weakly interacting 2D Fermi liquid.

We report the observation of a new phenomenon of colossal magnetoresistance in a 40 nm wide GaAs quantum well in the presence of an external magnetic field applied parallel to the high-mobility 2D electron layer. In a strong magnetic field, the magnetoresistance is observed to increase by a factor of ∼300 from 0 to 45 T without the system undergoing any metal-insulator transition. We discuss how this colossal magnetoresistance effect cannot be attributed to the spin degree of freedom or localization physics, but most likely emanates from strong magneto-orbital coupling between the two-dimensional electron gas and the magnetic field. Our observation is consistent with a field-induced 2D-to-3D transition in the confined electronic system.

Pinning-mode resonance of a Skyrme crystal near Landau-level filling factor ν=1.

Microwave pinning-mode resonances found around integer quantum Hall effects, are a signature of crystallized quasiparticles or holes. Application of in-plane magnetic field to these crystals, increasing the Zeeman energy, has negligible effect on the resonances just below Landau-level filling ν=2, but increases the pinning frequencies near ν=1, particularly for smaller quasiparticle or hole densities. The charge dynamics near ν=1, characteristic of a crystal order, are affected by spin, in a manner consistent with a Skyrme crystal.

Observation of a pinning mode in a Wigner solid with ν=1/3 fractional quantum Hall excitations.

We report the observation of a resonance in the microwave spectra of the real diagonal conductivities of a two-dimensional electron system within a range of ∼ ± 0.015 from filling factor ν = 1/3. The resonance is remarkably similar to resonances previously observed near integer ν, and is interpreted as the collective pinning mode of a disorder-pinned Wigner solid phase of e/3-charged carriers.

Pinning mode resonances of 2D electron stripe phases: effect of an in-plane magnetic field.

We study the anisotropic pinning-mode resonances in the rf conductivity spectra of the stripe phase of 2D electron systems around a Landau level filling of 9/2, in the presence of an in-plane magnetic field B(ip). The polarization along which the resonance is observed switches as B(ip) is applied, consistent with the reorientation of the stripes. The resonance frequency, a measure of the pinning interaction between the 2D electron systems and disorder, increases with B(ip). The magnitude of this increase indicates that disorder interaction is playing an important role in determining the stripe orientation.

Observation of pinning mode of stripe phases of 2D systems in high Landau levels.

We study the radio-frequency diagonal conductivities of the anisotropic stripe phases of higher Landau levels near half-integer fillings. In the hard direction, in which larger dc resistivity occurs, the spectrum exhibits a striking resonance, while in the orthogonal, easy direction, no resonance is discernible. The resonance is interpreted as a pinning mode of the stripe phase.

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.

Experimental evidence for a collective insulating state in two-dimensional superconductors.

We present the results of an experimental study of the current-voltage characteristics in a strong magnetic field (B) of disordered, superconducting, thin films of amorphous indium oxide. As the B strength is increased superconductivity degrades, until a critical field (B(c)) where the system is forced into an insulating state. We show that the differential conductance measured in the insulating phase vanishes abruptly below a well-defined temperature, resulting in a clear threshold for conduction. Our results indicate that a new collective state emerges in two-dimensional superconductors at high B.

Evidence for two different solid phases of two-dimensional electrons in high magnetic fields.

We have observed two different rf resonances in the frequency dependent real diagonal conductivity of very high quality two-dimensional electron systems in the high magnetic field insulating phase and interpret them as coming from two different pinned electron solid phases (labeled as "A" and "B"). The "A" resonance is observable for Landau level filling nu<2/9 [reentrant around the nu=1/5 fractional quantum Hall effect (FQHE)] and then crosses over to the different "B" resonance which dominates at sufficiently low nu. Moreover, the "A" resonance is found to show dispersion with respect to the size of the transmission line, indicating that the "A" phase has a large correlation length. We suggest that quantum correlations such as those responsible for FQHE may play an important role in giving rise to such different solids.

Evidence of a first-order phase transition between Wigner-crystal and bubble phases of 2D electrons in higher Landau Levels.

For filling factors nu in the range between 4.16 and 4.28, we simultaneously detect two resonances in the real diagonal microwave conductivity of a two-dimensional electron system (2DES) at low temperature T approximately 35 mK. We attribute the resonance to Wigner-crystal and Bubble phases of the 2DES in higher Landau Levels. For nu below and above this range, only single resonances are observed. The coexistence of both phases is taken as evidence of a first-order phase transition. We estimate the transition point as nu=4.22.

Superconductivity-related insulating behavior.

We present the results of an experimental study of superconducting, disordered, thin films of amorphous indium oxide. These films can be driven from the superconducting phase to a reentrant insulating state by the application of a perpendicular magnetic field (B). We find that the high-B insulator exhibits activated transport with a characteristic temperature, TI. TI has a maximum value (TpI) that is close to the superconducting transition temperature (Tc) at B=0, suggesting a possible relation between the conduction mechanisms in the superconducting and insulating phases. Tp(I) and Tc display opposite dependences on the disorder strength.

Competition between a fractional quantum hall liquid and bubble and Wigner crystal phases in the third Landau level.

Magnetotransport measurements were performed in an ultrahigh mobility GaAs/AlGaAs quantum well of density approximately 3.0 x 10(11) cm(-2). The temperature dependence of the magnetoresistance Rxx was studied in detail in the vicinity of nu=9/2. In particular, we discovered new minima in Rxx at a filling factor nu approximately 41/5 and 44/5, but only at intermediate temperatures 80 approximately less than T approximately less than 120 mK. We interpret these as evidence for a fractional quantum Hall liquid forming in the N=2 Landau level and competing with bubble and Wigner crystal phases favored at lower temperatures. Our data suggest that a magnetically driven insulator-insulator quantum phase transition occurs between the bubble and Wigner crystal phases at T=0.