Pinned Wigner Crystal Research Articles

Coexistence of two hole phases in high-quality p−GaAs/AlGaAs in the vicinity of Landau-level filling factors ν=1 and ν=(1/3)

We focused on the transverse AC magnetoconductance of a high-mobility $p$-GaAs/AlGaAs quantum well $(p=1.2\ifmmode\times\else\texttimes\fi{}{10}^{11}{\mathrm{cm}}^{\ensuremath{-}2})$ in the vicinity of two values of the Landau-level filling factor $\ensuremath{\nu}$: $\ensuremath{\nu}=1$ (integer quantum Hall effect) and $\ensuremath{\nu}=1/3$ (fractional quantum Hall effect). The complex transverse AC conductance ${\ensuremath{\sigma}}_{xx}^{\mathrm{AC}}(\ensuremath{\omega})$ was found from simultaneous measurements of attenuation and velocity of surface acoustic waves propagating along the interface between a piezoelectric crystal and the two-dimensional hole system under investigation. We analyzed both the real and imaginary parts of the hole conductance and compared the similarities and differences between the results for filling factor 1 and filling factor 1/3. Both to the left and to the right of these values maxima of a specific shape, ``wings,'' arose in the $\ensuremath{\sigma}(\ensuremath{\nu})$ dependences at those two $\ensuremath{\nu}$. Analysis of the results of our acoustic measurements at different temperatures and surface acoustic wave frequencies allowed us to attribute these wings to the formation of collective localized states, namely, the domains of a pinned Wigner crystal, i.e., a Wigner solid. While the Wigner solid has been observed in 2D hole systems previously, we were able to detect it at the highest hole density and, therefore, the lowest hole-hole interaction reported.

Fermionic Many-Body Localization for Random and Quasiperiodic Systems in the Presence of Short- and Long-Range Interactions.

We study many-body localization (MBL) for interacting one-dimensional lattice fermions in random (Anderson) and quasiperiodic (Aubry-Andre) models, focusing on the role of interaction range. We obtain the MBL quantum phase diagrams by calculating the experimentally relevant inverse participation ratio (IPR) at half-filling using exact diagonalization methods and extrapolating to the infinite system size. For short-range interactions, our results produce in the phase diagram a qualitative symmetry between weak and strong interaction limits. For long-range interactions, no such symmetry exists as the strongly interacting system is always many-body localized, independent of the effective disorder strength, and the system is analogous to a pinned Wigner crystal. We obtain various scaling exponents for the IPR, suggesting conditions for different MBL regimes arising from interaction effects.

Crossover between localized states and pinned Wigner crystal in high-mobilityn-GaAs/AlGaAs heterostructures near filling factorν=1

We have measured magnetic field dependences of the attenuation and velocity of surface acoustic waves in a high-mobility $n$-GaAs/AlGaAs structure with a wide quantum well. The results allowed us to find the complex conductance, $\sigma(\omega)$, of the heterostructure for different frequencies, temperatures and magnetic fields near filling factors $\nu=1, 2$. Observed behavior of $\sigma(\omega)$ versus magnetic field outside close vicinities of integer fillings reveals an oscillation pattern indicative of the rich fractional quantum Hall effect. Our result is that in very close vicinities of integer filling factors the AC response of a high-mobility two-dimensional structures behaves as that of a two-dimensional system of localized electrons. Namely, both real and imaginary parts of the complex AC conductance at low temperatures agree with the predictions for the two-site model for a two-dimensional hopping system. Another result is the specific temperature dependences of $\sigma(\omega)$, which are extremely sensitive to the filling factor value. These dependences indicate a sharp crossover between the localized modes and a pinned Wigner crystal.

Connecting the Reentrant Insulating Phase and the Zero-Field Metal-Insulator Transition in a 2D Hole System

We present the transport and capacitance measurements of 10 nm wide GaAs quantum wells with hole densities around the critical point of the 2D metal-insulator transition (critical density p(c) down to 0.8 × 10(10)/cm2, r(s) ∼ 36). For metallic hole density p(c) < p < p(c) + 0.15 × 10(10)/cm2, a reentrant insulating phase (RIP) is observed between the ν = 1 quantum Hall state and the zero-field metallic state and it is attributed to the formation of pinned Wigner crystal. Through studying the evolution of the RIP versus 2D hole density, we show that the RIP is incompressible and continuously connected to the zero-field insulator, suggesting a similar origin for these two phases.

Static and dynamic properties of type-II composite fermion Wigner crystals

The Wigner crystal of composite fermions is a strongly correlated state of complex emergent particles in two dimensions, and therefore its unambiguous detection would be of significant importance. A recent observation of optical resonances in the vicinity of filling factor $\ensuremath{\nu}=1/3$ has been interpreted as evidence for a pinned Wigner crystal of composite fermions [Phys. Rev. Lett. 105, 126803 (2010)]. We evaluate in a microscopic theory the shear modulus and the magnetophonon and magnetoplasmon dispersions of the composite fermion Wigner crystal in the vicinity of filling factors 1/3, 2/5, and 3/7. We determine the region of stability of the crystal phase, and also relate the frequency of its pinning mode to that of the corresponding electron crystal near integer fillings. These results are in good semiquantitative agreement with experiment, and therefore support the identification of the optical resonance as the pinning mode of the composite fermion Wigner crystal. Our calculations also bring out certain puzzling features, such as a relatively small melting temperature for the composite fermion Wigner crystal, and also suggest a higher asymmetry between Wigner crystals of composite fermion particles and holes than that observed experimentally.

Breakdown of theN=0quantum Hall state in graphene: Two insulating regimes

We studied the unusual Quantum Hall Effect (QHE) near the charge neutrality point (CNP) in high-mobility graphene sample for magnetic fields up to 18 T. We observe breakdown of the delocalized QHE transport and strong increase in resistivities $\rho_{xx},|\rho_{xy}|$ with decreasing Landau level filling for $\nu < 2$, where we identify two insulating regimes. For $1 \gtrsim |\nu| \gtrsim 1/2$ we find an exponential increase of $\rho_{xx,xy} \sim e^{a(H-H_c)}$ within the range up to several resistance quanta $R_K$, while the Hall effect gradually disappears, consistent with the Hall insulator (HI) with local transport. Then, at $\nu \approx 1/2$ a cusp in $\rho_{xx}(H)$ followed by an onset of even faster growth indicates transition to a collective insulator (CI) state. The likely candidate for this state is a pinned Wigner crystal.

Wigner crystallization about ν=3

We measure a resonance in the frequency dependence of the real diagonal conductivity, Re[ σ xx ], near integer filling factor, ν=3. This resonance depends strongly on ν, with peak frequency f pk ≈1.7 GHz at ν=3.04 or 2.92 close to integer ν, but f pk ≈600 MHz at ν=3.19 or 2.82, the extremes of where the resonance is visible. The dependence of f pk upon n ∗ , the density of electrons in the partially filled level, is discussed and compared with similar measurements by Chen et al. (Phys. Rev. Lett., 91, 016801) about ν=1 and 2. We interpret the resonance as due to a pinned Wigner crystal phase with density n ∗ about the ν=3 state.

Microwave resonance of the 2D Wigner crystal around integer Landau fillings.

We have observed a resonance in the real part of the finite frequency diagonal conductivity using microwave absorption measurements in high quality 2D electron systems near integer fillings. The resonance exists in some neighborhood of filling factor around corresponding integers and is qualitatively similar to previously observed resonance of weakly pinned Wigner crystal in high B and very small filling factor regime. Data measured around both nu=1 and nu=2 are presented. We interpret the resonance as the signature of the Wigner crystal state around integer Landau levels.

Correlation lengths of the wigner-crystal order in a two-dimensional electron system at high magnetic fields.

The insulator terminating the fractional quantum Hall series at low Landau level filling nu is generally taken to be a pinned Wigner crystal (WC), and exhibits a microwave resonance that is interpreted as a WC pinning mode. For a high quality sample at several densities, n, we find maxima in resonance peak frequency, f(pk), vs magnetic field, B. L, the correlation length of WC order, is calculated from f(pk). For each n, L vs nu tends at low nu toward a line with positive intercept; the fit is accurate over as much as a factor of 5 range of nu. The linear behavior is interpreted as due to B compressing the electron wave functions, to alter the effective electron-impurity interaction.

Pinned Wigner crystals

We study the effects of weak disorder on a Wigner crystal in a magnetic field. We show that an elastic description of the pinned Wigner crystal provides an excellent framework to obtain most of the physically relevant observables. Using such a description, we compute the static and dynamical properties. We find that, akin to the Bragg glass phase, a good degree of translational order survives (up to a large lengthscale in $d=2$, infinite in $d=3$). Using a gaussian variational method, we obtain the full frequency dependence of the conductivity tensor. The zero temperature Hall resistivity is independent of frequency and remains unaffected by disorder at its classical value. We show that the characteristic features of the conductivity in the pinned Wigner crystal are dramatically different from those arising from the naive extrapolations of Fukuyama-Lee type theories for charge density waves. We determine the relevant scales and find that the physical properties depend crucially on whether the disorder correlation length is larger than the cyclotron length or not. We analyse, in particular, the magnetic field and density dependence of the optical conductivity. Within our approach the pinning frequency can increase with increasing magnetic field and varies as $n^{-3/2}$ with the density $n$. We compare our predictions with recent experiments on transport in two dimensional electron gases under strong magnetic fields. Our theory allows for a consistent interpretation of these experiments in terms of a pinned WC.

Microwave resonance and weak pinning in two-dimensional hole systems at high magnetic fields

Microwave frequency conductivity $\mathrm{Re}({\ensuremath{\sigma}}_{\mathrm{xx}})$ of high quality two-dimensional hole systems (2DHS) in a large perpendicular magnetic field (B) is measured with the carrier density ${(n}_{s})$ of the 2DHS controlled by a backgate bias. The high-$B$ insulating phase of the 2DHS exhibits a microwave resonance that remains well defined, but shifts to higher peak frequency ${(f}_{\mathrm{pk}})$ as ${n}_{s}$ is reduced. In different regimes, ${f}_{\mathrm{pk}}$ vs ${n}_{s}$ can be fit to ${f}_{\mathrm{pk}}\ensuremath{\propto}{n}_{s}^{\ensuremath{-}1/2}$ or to ${f}_{\mathrm{pk}}\ensuremath{\propto}{n}_{s}^{\ensuremath{-}3/2}.$ The data clearly indicate that both carrier-carrier interactions and disorder are indispensable in determining the dynamics of the insulator. The ${n}_{s}$ dependence of ${f}_{\mathrm{pk}}$ is consistent with a weakly pinned Wigner crystal in which domain size increases with ${n}_{s},$ due to larger carrier-carrier interaction.

Electromagnetic absorption of a pinned Wigner crystal at finite temperatures

We investigate the microwave absorption of a pinned, two-dimensional Wigner crystal in a strong magnetic field at finite temperatures. Using a model of a uniform commensurate pinning potential, we analyze thermal broadening of the electromagnetic absorption resonance. Surprisingly, we find that the pinning resonance peak should remain sharp even when the temperature is comparable or greater than the peak frequency. This result agrees qualitatively with recent experimental observations of the ac conductivity in two-dimensional hole systems in a magnetically induced insulating state. It is shown, in analogy with Kohn's theorem, that the electron-electron interaction does not affect the response of a harmonically pinned Wigner crystal to a spatially uniform external field at any temperature. We thus focus on anharmonicity in the pinning potential as a source of broadening. Using a $1/N$ expansion technique, we show that the broadening is introduced through the self-energy corrections to the magnetophonon Green's functions.

Pinned Wigner crystals

We study the effects of weak Gaussian disorder on various physical and dynamical properties of the pinned Wigner crystal. We use the replica based Gaussian variational method to study the problem. The pinned crystal is the quantum analog of the Bragg glass phase seen in vortex lattices. We obtain the full frequency dependence of the dynamical magneto-conductivities. This allows us to extract various features of the pinning peak which can be compared directly with experimentally measured quantities. We also calculate the magnetic field and density dependences of these quantities. We find that the physical properties of the crystal depend crucially on the relative magnitudes of the disorder correlation length and the cyclotron length. This has dramatic experimental consequences. Our results are of extreme relevance to the recent experiments on two dimensional Ga-As quantum Hall samples and Si MOSFETS, where the insulating phase seen at low densities and/or high magnetic fields is expected to be a pinned Wigner crystal.

Electromagnetic response of a pinned Wigner crystal

A microscopic model for analyzing the microwave absorption properties of a pinned, two-dimensional Wigner crystal in a strong perpendicular magnetic field is developed. The method focuses on excitations within the lowest Landau level, and corresponds to a quantum version of the harmonic approximation. For pure systems (no disorder), the method reproduces known results for the collective mode density of states of this system, and clearly identifies the origin of previously unexplained structure in this quantity. The application of the method to a simple diagonal disorder model uncovers a surprising result: a sharp (\ensuremath{\delta}-function) response at zero temperature that is consistent with recent experiments. A simple spin lattice model is developed that reproduces the results of the quantum harmonic approximation, and shows that the sharp response is possible because the size scale ${L}_{c}$ of patches moving together in the lowest-frequency collective mode is extremely large compared to the sample size for physically relevant parameters. This result is found to be a direct repercussion of the long-range nature of the Coulomb interaction. Finally, the model is used to analyze different disorder potentials that may pin the Wigner crystal, and it is argued that interface disorder is likely to represent the dominant pinning source for the system. A simple model of the interface is shown to reproduce some of the experimental trends for the magnetic field dependence of the pinning resonance.

Metal–insulator transition at B=0 in an ultra-low density two-dimensional hole gas

We have observed a metal–insulator transition in an ultra-low density two-dimensional hole gas formed in a high-quality GaAs–AlGaAs heterostructure at B=0. At the highest carrier density studied ( p s =2.2×10 10 cm −2 , r s=16) the hole gas is strongly metallic, with an exceptional mobility of 425 000 cm 2 V −1 s −1 . The low disorder and strength of the many-body interactions in this sample are highlighted by the observation of re-entrant metal–insulator transitions in both the fractional ( ν< 1 3 ) and integer (2> ν>1) quantum Hall regimes. On reducing the carrier density the temperature and electric field dependence of the resistivity show that the sample is still metallic at p s =1.3×10 10 cm −2 ( r s=21), becoming insulating at p s ≃1×10 10 cm −2 . Our results indicate that electron–electron interactions are dominant at these low densities, pointing to the many-body origins of this metal–insulator transition. We note that the value of r s at the transition ( r s=23±2) is large enough to allow the formation of a weakly pinned Wigner crystal, and approaches the value calculated for the condensation of a pure Wigner crystal.

Dynamical Properties of the Pinned Wigner Crystal

We study various dynamical properties of the weakly pinned Wigner crystal in a high magnetic field. Using a Gaussian variational method we can compute the full frequency and field dependence of the real and imaginary parts of the diagonal and Hall conductivities. The zero temperature Hall resistivity is independent of frequency and remains unaffected by disorder at its classical value. We show that, depending on the inherent length scales of the system, the pinning peak and the threshold electric field exhibit strikingly different magnetic field dependences.

Microwave resonances in low-filling insulating phase of two-dimensional electron system

We present data on finite frequency (0.2 ≤ f ≤ 8.6 GHz) conductivity, Re( σ xx ), of the insulator terminating a fractional quantum Hall effect series in the high magnetic field ( B) limit. As B is increased within the insulator, at temperatures of 50 and 200 mK Re( σ xx ) vs f evolves from a decreasing curve into a peak. The height and f of the peak increase with B. Integrals of Re( σ xx ) vs f become B-independent at large enough B. The results are discussed within the framework of the Fukuyama-Lee model of the dynamics of a pinned Wigner crystal.

Density of states of localized phonons in a pinned Wigner crystal.

Using the method of optimal fluctuation, we investigate the asymptotic behavior of the phonon density of states in the low-frequency limit \ensuremath{\omega}\ensuremath{\rightarrow}0. If the pinning potential is not too strong, both the density of states g(\ensuremath{\omega}) and the ac conductivity at zero temperature \ensuremath{\sigma}(\ensuremath{\omega}) are found to be proportional to ${\mathrm{\ensuremath{\omega}}}^{4}$. This power law is valid in both one- and two-dimensional crystal and does not change in the presence of magnetic field.

Termination of the integer QHE by an electron solid in a dilute 2D electron system at mK-temperatures

Upon decreasing the density, the 2D electron system exhibits a transition from the metallic state, characterized by the Quantum Hall effect, to the collective insulating state (electron solid), interrupted by the Quantum Hall effect at integer filling factors ν=1, 2, and 6. The electron solid shows many features attributed to the pinned Wigner crystal. On the basis of the longitudinal and Hall resistance measurements through these reentrant transitions, we discuss the pathway for the disappearance of delocalized states.

Zero-temperature Hall coefficient of a localized-electron insulator: Wigner crystal versus Anderson localization.

The [ital zero][minus][ital temperature] Hall coefficient [ital R][sub [ital H]] of a localized-electron insulator is shown to reduce to the free-electron value when the external potential varies slowly on the scale of the localization length. For a general circularly symmetric potential, an exact quantum-mechanical expression for [ital R][sub [ital H]] is derived, and computed for electrons bound to hydrogenlike impurities. It is thus shown that measurements of [ital R][sub [ital H]] in two-dimensional magnetic-field-localized electron systems can distinguish between a pinned Wigner crystal and a strong Anderson insulator, both of which are, at [ital T]=0, Hall insulators.