Electrons Wigner crystal

Of the situations where the above theory, based on the use of uniform electron-gas relations locally, is too crude, the electron Wigner crystal at zero temperature (Lein the completely degenerate limit) constitutes one example. This has been explored in the work of Senatore and Pastore [45]. 

In the low-density limit (r, — ), correlation and exchange are of comparable strength, and are together independent of exc is then nearly equal to the electrostatic energy per electron of the Wigner crystal [33-36] ... 

For the case where the bandwidth or the warping, i. e., the transfer integral tb, respectively fi in (2.2)), is small the Coulomb repulsion between the electrons becomes important. A limited screening of the electron charge in a narrow band due to restricted electron movement can lead to a localized electron lattice, a so-called Wigner crystal. This, in fact, has been observed in the strongly ID material TTF-TCNQ where in addition to the 2A p Peierls lattice distortion a 4fcp modulation was found [48, 49, 50]. The estimated value for the on-site Coulomb repulsion U in TTF-TCNQ is U/Atb — 0.9 extracted from the frequency dependence of the NMR relaxation time [51] and the susceptibility above the Peierls transition [52]. 

For V W> 1 electrons are localized and the metallic conduction is forbidden. The electrons will form a crystal in which they are arranged with equal distance. This is called the Wigner crystal. In one-dimension the distance between neighboring electrons is given as l/4kp because the distance is 1/n and kp = ttnll. 

In the diffuse X-ray measurements of TTF-TCNQ the superlattice reflection was found with the wave number 4kp = 0.59 b [56]. It is observed even at room temperature and suggests the absence of the interchain correlation above 49 K. A set of superlattice reflection was found below 49 K suggesting the formation of an ordered structure of three-dimension. This superstructure is ascribed to the molecular displacement caused by the Wigner crystal of electrons through the electron-lattice interaction [67]. The 4 p structure is considered to be formed predominantly on the TTF stacks. The 2kp superstructure is rather ascribed to TCNQ stacks. This is suggested [68] by detailed analyses of the results of X-ray, neutron, EPR and NMR measurements. 

When the long-range Coulomb interaction is small, one can take account of only the on-site Coulomb interaction energy U between two electrons on a molecule. This electronic system is described in terms of the Hubbard model. Theoretical studies have shown that for UtW > 1 the electrons undergo the Mott transition which does not necessarily involve any structural changes. The electrons are localized with equal distance. They are apparently the same as the Wigner crystal described above. It is shown that the Mott transition is easy to occur when the charge density is l/molecule or I/site. 

Figure 21 shows temperature dependence of electrical conductivity and magnetic susceptibility of MEM(Af-methyl-iV-ethyl-morpholinium)-(TCNQ)2 [70]. At about 335 K it undergoes a metal-insulator transition accompanied by the onset of a two-fold superstructure and a temperature dependent magnetic susceptibility characteristic of localized moments. It is considered as depicted in Fig. 22(a) that a dimerized TCNQ accepts an electron localized by, for example, the Mott transition or the Wigner crystallization. The solid curve shown in Fig. 21(b) denotes the theoretical prediction for the magnetic susceptibility of a one-... 

The spontaneous localization of geminals may also be important in extended systems, where long-range correlation effects may appear in the form of localization. For example, there is no way to describe the so called Wigner-crystal in a free electron gas (see e.g. [140]) at the HF level, while in principle it should be possible with a geminal wave function. 

Though the above argument can leave no doubt that in the jellium model there will be a localized assembly of electrons, i.e. a Wigner crystal, in the extremely low density limit, the actual analytic calculation of when the electron liquid, at absolute zero of temperature, freezes as the density is lowered has proved very delicate [20]. Eventually, this matter was settled using quantum Monte Carlo computer simulation by Ceperley and Alder [38], They found in this way that the crystallization first occurred at rs = 100. Herman and March [39] subsequently pointed out that, for the Wigner crystal phase, the theoretical expression [40,41]... 

Here it should be noted that the function E( ) in the electrostatic approximation (1) is not a function in the usual sense and its continuity is broken up at any rational point =Q/p (where p and q are mutually simple numbers). This discontinuity appears by the following reason. At zero temperature the system of electrons and holes is arranged in the Wigner crystal with a cell consisting of TCNQ (or TTP) molecules. Let us study Ejj( ) with a little deviation of from Q p i.e. = q/P + where 1. Assume that /> 0. ... 

In the extremely low density limit, a system of elecfions will form a regular lattice, with each electron occupying a unit cell this is known as the Wigner crystal. The energy of this crystal has been calculated to be... 

Anderson localization is only one possible way of inducing a transition from a localized to a delocalized state. Since the localized state is associated with insulating behavior and the delocalized state is associated with metallic behavior, this transition is also referred to as the metal-insulator transition. In the case of Anderson localization this is purely a consequence of disorder. The metal-insulator transition can be observed in other physical situations as well, but is due to more subtle many-body effects, such as correlations between electrons depending on the precise nature of the transition, these situations are referred to as the Mott transition or the Wigner crystallization. 

Mott transition. Wigner (1938) introduced the idea of electron-electron interactions and suggested that at low densities, a free-electron gas should crystallize ... 

The first attempt to calculate realistic wave functions for electrons in metals is that of Wigner and Seitz (1933). These authors pointed out that space in a body-or face-centred cubic crystal could be divided into polyhedra surrounding each atom, that these polyhedra could be replaced without large error by spheres of radius r0, so that for the lowest state one has to find spherically symmetrical solutions of the Schrodinger equation (6) subject to the boundary condition that... 

To obtain a rough estimate of the multiple scattering effects a model proposed by M. H. Cohen is useful (18). This model is based on the application of the Wigner-Seitz scheme to an electron in a helium crystal. Each helium atom is represented as a hard sphere characterized by a radius equal to the scattering length. The electron wave function will then be... 

The failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (19). The metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E. Wigner showed in 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an electron crystal wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. But one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity in terms of, say, effective mass at present. The effective mass may be introduced to account for errors in the density of states—not in the electron correlations. 

The problem of phase separation in cuprates superconductors has been longely debated [1-8], Recently, several experiments show the formation of electronic crystals at critical densities [9, 10], These results provide a strong experimental support for the scenario proposed some years ago (11-14) for the phase diagram of cuprate superconductors where generalized Wigner... 

The Voronoi deformation density approach, is based on the partitioning of space into the Voronoi cells of each atom A, that is, the region of space that is closer to that atom than to any other atom (cf. Wigner-Seitz cells in crystals see Chapter 1 of Ref. 202). The VDD charge of an atom A is then calculated as the difference between the (numerical) integral of the electron density p of the real molecule and the superposition of atomic densities SpB of the promolecule in its Voronoi cell (Eq. [42]) ... 

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