Wigner Crystallization Of Electrons Research Articles

Theory of electron states in two-dimensional Wigner clusters

The results of modeling a finite system of interacting two-dimensional electrons placed at zero temperature into an external parabolic potential well are presented. The structure of the 2D Wigner cluster is found. The distribution functions of interelectron distances and the coordination numbers of internal and external electrons are obtained. The ”string-zigzag” transition is analyzed. The low-temperature specific heat of the system is determined. An electronic Wigner crystal on a sphere is also considered.

Wigner Crystallization of Electrons in a One-Dimensional Lattice: A Condensation in the Space of States.

We study the ground state of a system of spinless electrons interacting through a screened Coulomb potential in a lattice ring. By using analytical arguments, we show that, when the effective interaction compares with the kinetic energy, the system forms a Wigner crystal undergoing a first-order quantum phase transition. This transition is a condensation in the space of the states and belongs to the class of quantum phase transitions discussed in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021).JPAMB51751-811310.1088/1751-8121/aba144]. The transition takes place at a critical value r_{s}_{c} of the usual dimensionless parameter r_{s} (radius of the volume available to each electron divided by effective Bohr radius) for which we are able to provide rigorous lower and upper bounds. For large screening length these bounds can be expressed in a closed analytical form. Demanding MonteCarlo simulations allow to estimate r_{s}_{c}≃2.3±0.2 at lattice filling 3/10 and screening length 10 lattice constants. This value is well within the rigorous bounds 0.7≤r_{s}_{c}≤4.3. Finally, we show that if screening is removed after the thermodynamic limit has been taken, r_{s}_{c} tends to zero. In contrast, in a bare unscreened Coulomb potential, Wigner crystallization always takes place as a smooth crossover, not as a quantum phase transition.

Signatures of Wigner crystal of electrons in a monolayer semiconductor.

When the Coulomb repulsion between electrons dominates over their kinetic energy, electrons in two-dimensional systems are predicted to spontaneously break continuous-translation symmetry and form a quantum crystal1. Efforts to observe2-12 this elusive state of matter, termed a Wigner crystal, in two-dimensional extended systems have primarily focused on conductivity measurements on electrons confined to a single Landau level at high magnetic fields. Here we use optical spectroscopy to demonstrate that electrons in a monolayer semiconductor with density lower than 3×1011per centimetre squared form a Wigner crystal. The combination of a high electron effective mass and reduced dielectric screening enables us to observe electronic charge order even in the absence of a moiré potential or an external magnetic field. The interactions between a resonantly injected exciton and electrons arranged in a periodic lattice modify the exciton bandstructure so that an umklapp resonance arises in the optical reflection spectrum, heralding the presence of charge order13. Our findings demonstrate that charge-tunable transition metal dichalcogenide monolayers14 enable the investigation of previously uncharted territory for many-body physics where interaction energy dominates over kinetic energy.

Imaging the electronic Wigner crystal in one dimension

The quantum crystal of electrons, predicted more than 80 years ago by Eugene Wigner, remains one of the most elusive states of matter. In this study, we observed the one-dimensional Wigner crystal directly by imaging its charge density in real space. To image, with minimal invasiveness, the many-body electronic density of a carbon nanotube, we used another nanotube as a scanning-charge perturbation. The images we obtained of a few electrons confined in one dimension match the theoretical predictions for strongly interacting crystals. The quantum nature of the crystal emerges in the observed collective tunneling through a potential barrier. These experiments provide the direct evidence for the formation of small Wigner crystals and open the way for studying other fragile interacting states by imaging their many-body density in real space.

Resonance Tunneling Phenomena in Two-Dimensional Multilayer van der Waals Crystalline Systems

Works, mostly experimental, concerning the most interesting features of application of the resonant tunneling spectroscopy to a new type of heterosystems, van der Waals heterostructures, have been briefly reviewed. These heterostructures appeared after the recent discovery of two-dimensional crystals, which are a new class of materials beginning with graphene. The role of the angular matching of crystal lattices of conducting graphene electrodes of van der Waals systems in carrier tunneling between them has been analyzed together with the closely related problems of satisfaction of conservation laws in tunneling transitions. Manifestations of multiparticle correlation interactions between carriers in van der Waals systems such as Wigner crystallization of electrons in a two-dimensional electron gas in a magnetic field and Bose condensation of excitons in parallel two-dimensional electron gases have been briefly discussed.

Crystallization in the Fractional Quantum Hall Regime Induced by Landau-Level Mixing.

The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the nonperturbative fixed-phase diffusion MonteCarlo method, we determine the phase diagram of the Wigner crystal in the ν-κ plane, where ν is the filling factor and κ is the strength of Landau-level (LL) mixing. The phase boundary is seen to exhibit a striking ν dependence, with the states away from the magic filling factors ν=n/(2pn+1) being much more susceptible to crystallization due to Landau-level mixing than those at ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n- and p-doped gallium arsenide quantum wells and, in particular, the existence of an insulating state for ν<1/3 and also for 1/3<ν<2/5 in low-density p-doped systems. We predict that, in the vicinity of ν=1/5 and ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal, but rather into a strongly correlated crystal of composite fermions carrying two vortices.

Anomalous Quasiparticle Reflection from the Surface of a ^{3}He-^{4}He Dilute Solution.

A free surface of a dilute ^{3}He-^{4}He liquid mixture is a unique system where two Fermi liquids with distinct dimensions coexist: a three-dimensional (3D) ^{3}He Fermi liquid in the bulk and a two-dimensional (2D) ^{3}He Fermi liquid at the surface. To investigate a novel effect generated by the interaction between the two Fermi liquids, the mobility of a Wigner crystal of electrons formed on the free surface of the mixture is studied. An anomalous enhancement of the mobility, compared with the case where the 3D and 2D systems do not interact with each other, is observed. The enhancement is explained by the nontrivial reflection of 3D quasiparticles from the surface covered with the 2D ^{3}He system.

Effects of interedge scattering on the Wigner crystallization in graphene nanoribbons

We investigate the effects of coupling between the two zigzag edges of graphene nanoribbons on the Wigner crystallization of electrons and holes using a combination of tight-binding, mean field Hubbard and many-body configuration interaction methods. We show that the thickness of the nanoribbon plays a crucial role in the formation of Wigner crystal. For ribbon widths smaller than 16 \mbox{\AA}, increased kinetic energy overcomes the long-range Coulomb repulsion and suppresses the Wigner crystallization. For wider ribbons up to 38 \mbox{\AA} wide, strong Wigner localization is observed for even number of electrons, revealing an even-odd effect also found in Coulomb blockade addition spectrum. Interedge correlations are found to be strong enough to allow simultaneous crystallization on both edges, although an applied electric field can decouple the two edges. Finally, we show that Wigner crystallization can also occurs for holes, albeit weaker than for electrons.

Sharp tunnelling resonance from the vibrations of an electronic Wigner crystal

Photoemission and tunneling spectroscopies measure the energies at which single electrons can be added to or removed from an electronic system. Features observed in such spectra have revealed electrons coupling to vibrational modes of ions both in solids and in individual molecules. Here we report the discovery of a sharp resonance in the tunneling spectrum of a 2D electron system. Its behavior suggests that it originates from vibrational modes, not involving ionic motion, but instead arising from vibrations of spatial ordering of the electrons themselves. In a two-dimensional electronic system at very low temperatures and high magnetic fields, electrons can either condense into a variety of quantum Hall phases or arrange themselves into a highly ordered Wigner crystal lattice. Such spatially ordered phases of electrons are often electrically insulating and delicate and have proven very challenging to probe with conventional methods. Using a unique pulsed tunneling method capable of probing electron tunneling into insulating phases, we observe a sharp peak with dependencies on energy and other parameters that fit to models for vibrations of a Wigner crystal. The remarkable sharpness of the structure presents strong evidence of the existence of a Wigner crystal with long correlation length.

TRIONS IN COUPLED QUANTUM WELLS AND WIGNER CRYSTALLIZATION

We consider a restricted three-body problem, where two interacted particles are located in a two-dimensional (2D) plane and interact with the third one located in the parallel spatially separated plane. The system of such type can be formed in the semiconductor coupled quantum wells, where the electrons (holes) and direct excitons spatially separated in different parallel neighboring quantum wells are sufficiently close to interact and form negative X- or positive X+ indirect trions. It is shown that at large interwell separations, when the interwell separation is much greater than the exciton Bohr radius, this problem can be solved analytically using the cluster approach. Analytical results for the energy spectrum and the wavefunctions of the spatially indirect trion are obtained, their dependencies on the interwell separations is analyzed and a conditional probability distribution is calculated. The formation of 2D Wigner crystal of trions at the low densities is predicted. It is shown that the critical density of the formation of the trion Wigner crystal is sufficiently greater than the critical density of the electron Wigner crystal in the same material.

Investigation of Electron Crystallization in a HCP Structure

The ground state energies of non-magnetic, ferromagnetic and antiferromagnetic phases of 3D electron crystals corresponding to hcp structure are computed. In each case, the possibility of the Wigner electrons having cubic or spherical constant energy surface (the region of integration in momentum space) is investigated. The role of correlation energy is suitably taken into account. The range of the low density region favourable for Wigner electron crystallization is found. The structure dependent Wannier functions, which give proper localized representation for the Wigner electrons in the crystal, are employed in the calculation.

Wigner crystallization of electrons in deep traps in a two-dimensional dielectric

A two-dimensional model is used to examine the spatial distribution of electrons in deep traps in a two-dimensional dielectric. When the trap concentration is much higher than the trapped electron concentration, Coulomb repulsion leads to the formation of a two-dimensional quasi-periodic hexagonal lattice of localized electrons (Wigner glass).

Direct observation of ferroelectric domains created by Wigner crystallization of electrons in α-[bis(ethylenedithio)tetrathiafulvalene]2I3

The organic conductor α-[bis(ethylenedithio)tetrathiafulvalene]2I3 undergoes a ferroelectric transition due to the Wigner-crystal-like charge ordering. The present paper reports the direct observation of 180° polar domains growing in the ferroelectric phase by means of optical second-harmonic generation interferometry. Mapping observation of the nonlinear optical interference revealed the development of large polar domains, illustrating a strong tendency of single-domain growth in the ferroelectric compound. The domain structures were varied when the crystal was annealed above the transition temperature and recooled into the ferroelectric phase. The variation indicates the mobile nature of the domains in the ferroelectric organic conductor.

Ground-plane screening of Coulomb interactions by a nearby two-dimensional system

The use of a nearby metallic ground-plane to limit the range of the Coulomb interactions between carriers is a useful approach in studying the physics of two-dimensional (2D) systems. This approach has been used to study Wigner crystallization of electrons on the surface of liquid helium, and insulating behaviour of semiconductor two-dimensional systems. We recently reported a variation of this technique where an adjacent 2D system was used as the ground plane instead of a metal gate. In this paper, we perform calculations of the ground plane screening effect of a 2D system and show that it is as effective as a metal in screening Coulomb interactions.

Ground-plane screening of Coulomb interactions in two-dimensional systems: How effectively can one two-dimensional system screen interactions in another

The use of a nearby metallic ground-plane to limit the range of the Coulomb interactions between carriers is a useful approach in studying the physics of two-dimensional (2D) systems. This approach has been used to study Wigner crystallization of electrons on the surface of liquid helium, and most recently, the insulating and metallic states of semiconductor-based two-dimensional systems. In this paper, we perform calculations of the screening effect of one 2D system on another and show that a 2D system is at least as effective as a metal in screening Coulomb interactions. We also show that the recent observation of the reduced effect of the ground-plane when the 2D system is in the metallic regime is due to intralayer screening.

Quantum-mechanical wigner electron crystallization with and without magnetic fields

More than 50 years ago, Wigner argued that Coulombically interacting electrons moving in a uniform, nonresponsive, neutralizing background would exhibit a phase transition in the ground state, at sufficiently low density, to a localized electron crystal. For jellium, Ceperley and Alder demonstrated, in the ′80s, by quantum computer simulation, that the Wigner electron crystal formed when the mean interelectronic spacing rs was ˜80–100 Bohr radii. The electron crystal is (a) an insulator, (b) magnetic, probably with long-range Neel-type antiferromagnetism, (c) phononlike in its low-lying non-current-carrying excited states, and (d) defective at elevated temperatures, with hopping conduction likely to occur. Durkan, Elliott, and March pointed out, again some 30 years after Wigner, that, under suitable conditions, Wigner electron crystallization could be aided by appropriate application of magnetic fields. These workers considered n-type InSb in a magnetic field, and stressed the importance of observing Bragg reflections in X-ray or neutron experiments. While their considerations were immediately relevant to 3-dimensional Wigner electron crystals, advances in semiconductor technology have now led Andrei et al. to use a high-quality GaAs/GaAlAs heterojunction to study 2-dimensional Wigner crystallization, as induced by application of a magnetic field. Here, after a brief review of these experiments, some discussion is given of the melting curve of a 2-dimensional Wigner crystal in a magnetic field.

INVESTIGATION OF NON-MAGNETIC AND FERROMAGNETIC PHASES OF 3D ELECTRON CRYSTAL WITH NaCl AND CsCl STRUCTURES

The non-magnetic and ferromagnetic phases of 3D Wigner electron crystal are investigated using a localized representation of the electrons with NaCl and CsCl structures. The ground state energies of ferromagnetic and non-magnetic phases of Wigner electron crystal are computed in the range 10 ≤ rs ≥ 130. The role of correlation energy is suitably taken into account. The low density region favorable for the ferromagnetic phase is found to be 4.8 × 1020 electrons/cm3 and for the non-magnetic phase, it is 2.03 × 1020 electrons/cm3. It is found that the ground state energy of ferromagnetic phase is less than that of the non-magnetic phase of the Wigner electron crystal. The structure-dependent Wannier functions, which give proper localized representation for Wigner electrons, are employed in the calculation.

Strong Optical Nonlinearity and its Ultrafast Response Associated with Electron Ferroelectricity in an Organic Conductor

We report experimental evidence for the generation of ferroelectric polarization in an organic conductor α-[bis(ethylenedithio)tetrathiafulvalene] 2 I 3 obtained by optical second-harmonic generation. The spontaneous polarization emerges along with a metal-to-insulator transition that is driven by the Wigner crystallization of electrons. The strong optical nonlinearity and its ultrafast photoresponse demonstrated by this study exemplify the nature of the ferroelectric polarization that originates from the electron ordering.

Wigner crystallization in low dimensional materials

We study a model that describes Wigner crystallization of electrons in low-dimensional solids. Restricting the electron motion into layers or chains, while keeping the full three-dimensional nature of the Coulomb interactions, can result in a sizeable stabilization of the crystallized phase as compared to the usual three-dimensional case. A sequence of structural transitions of the electron crystal is also predicted upon varying the density.

Fingerprints of quantal Wigner solid-like correlations in D-dimensional assemblies

After a brief summary of the mechanism of quantal Wigner electron crystallization, we emphasize experimental situations in three, two, and one dimensions in which fingerprints of Wigner solid-like correlations are in evidence. Special stress is placed on the way applied magnetic fields can aid Wigner localization, as first pointed out in relation to n-type InSb by Durkan et al. (1968). Semiconductor heterojunctions with a two-dimensional electron fluid in the presence of a transverse magnetic field are given some prominence in relation to melting of the Wigner solid and its connection with magnetic properties. It is also proposed that a Wigner “hole” solid may be induced by a magnetic field of 60 T in a specific high T c cuprate and that the fulleride phase diagram may also contain Wigner solids.