Wigner crystals

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] ... 

Wigner crystallization is a subject on which there is little theoretical work, and it is not certain whether the phenomenon has been observed. 

The collective behavior of condensed modulated structures like charge or spin density waves (CDWs/SDWs) [23, 22, 4], flux line lattices [2, 36] and Wigner crystals [4] in random environments has been the subject of detailed investigations since the early 1970s. These were motivated by the drastic influence... 

For p values with rational fractions (p 1/2,2/3), a Wigner crystal [56] can occur The charges alternate regularly in the crystal, yielding a "frozen CDW", so that one site has p 0, the next p 1, and so on. A Wigner crystal is thus the antithesis of a mixed-valent [38] state. 

Two competing models are the so-called bistripe model and the Wigner crystal, both of which involve ordered regions... 

Figure 24 Possible models for charge ordering near x 2/3 in Lni j Aj Mn03, the Wigner crystal versus the bistripe model. (Reprinted with permission from P.G. Radaelli, D.E. Cox, L. Capogna, S.-W. Cheong, M. Marezio, Phys. Rev., 1999, B59, 14440. 1999 by the American Physical Society)...

To verify the syimnetry and identify the sensitivity of CBED to CO syimnetry, dynamic simulations using the Bloch wave method were examined to see the difference between the two models. The atomic positions within the unit cells for the two models from RadaeUi etal. are very close to each other. To avoid the possible pseudo-symmetry generated in the Bi-stripe model, dynamic simulations from the CO stmctures described by the Wigner-crystal model and Bistripe model are calculated and compared for the thickness of 300 nm in Figure 15(d) and (e). The difference between the two simulations is that the G-M lines exist in four (303) reflections and 2n +, 0, 0) reflections simulated by the Wigner-crystal model, as they are in the experimental CBED patterns, but do not show up in four (303) reflections simulated by the Bi-stripe model. 

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]... 

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]. 

For the generic models this would be compatible with the hexagonal symmetry along one direction imposed by the rods, with which the Wigner crystal has to arrange. 

At first sight this finding might appear inevitable. Nevertheless, since r0A < ve0 for the investigated cases, the inequality from the beginning of this section does in fact hold, albeit not in the strong version. For this specific situation veJA is 1.9 for the divalent systems and 2.9 for the trivalent ones. Observe that the inequality does not specify the density of counterions or rods, at which the three-dimensional Wigner crystal is to appear. Rather, Ref. 47 assumes only that a bundle of rods has formed and... 

Shklovskii BI. Wigner crystal model of counterion induced bundle formation of rodlike polyelectrolytes. Phys. Rev. Lett. 1999 82 3268-3271. 

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