Melting of Wigner-like lattice of parallel polarized dipoles

Abstract

We show that a triangular lattice consisting of dipolar molecules pointing orthogonal to the plane undergoes a first-order defect melting transition. PACS numbers:About thirty years ago, Nelson and Halperin [1] extended the Kosterlitz-Thouless pair unbinding theory [2] of vortices in a thin layer of superfluid helium to the phase transitions of defects in two-dimensional crystals. They argued that melting would proceed by a sequence of two Kosterlitz-Thouless transition, the first when dislocations of opposite Burgers vector unbind, creating a hexatic phase, and a second in which disclinations of opposite Frank vector separate. However, they never specified the physical parameter of the crystal which would decide when this melting scenario happens, rather than a simple first-order melting transition which was previously expected on the basis of our three-dimensional experience. Such a parameter was found in Ref. [3], and developed further in [4], and in the textbook [5]. It was shown that a higher-gradient elastic constant called the angular stiffness determines which scenario takes place. Only for a high angular stiffness will the two-step melting process occur. Otherwise the melting transition would be a completely normal first-order process. Computer simulations of the simplest lattice defect model on a lattice confirmed the results [6].The theory was applied to a Lennard-Jones crystal and a Wigner crystal, and it was found that in both cases the angular stiffness was too small to separate the melting transition into two successive Kosterlitz-Thouless transitions.Here we investigate the angular stiffness for a crystal that is similar to the Wigner crystal, except that the repulsive forces are due to parallel magnetic dipoles. Thus the potential has the behavior 1/r 3 rather than 1/r. The angular stiffness parameter is defined as follows. Let µ and λ be the usual elastic constants of a crystal, then the usual elastic energy density depends on the displacement field u i (x) via the strain tensorThe angular stiffness is parametrized by the second of the higher-gradient energywhereis the local rotation field. The parameter ℓ 2 is the length scale of the angular stiffness. It was argued in [3] that for ℓ = 0, dislocations are indistinguishable from neighboring pairs of disclinations of opposite Frank vector, and disclinations can be built from strings of dislocations. There the transition is of first order. For high ℓ, on the other hand, beginning about with the lattice spacing a 0 , the disclinations could be suppressed with the consequence that the transition based on disclination unbinding would occur later than that of dislocation unbinding. The precise location of the critical ℓ was found by computer simulations [6], and is plotted in Fig. 1.In order to apply this criterion to the triangular lattice formed by dipoles we must calculate the elastic constants in E and ∆E. This can be done for any repulsive interatomic potential Φ(x) = 1/|x| p of power p, which has the value p = 1 for the Wigner crystal, and p =...

Electrostatic charge-charge and dipole-dipole interactions near the surface of a medium with screening non-locality (Review Article)

“…This is a first step in the consideration of the substrate influence on the adsorption processes and the kinetics of adsorbate arrangement on the surface. Anyway, the dipole-dipole interaction is a main input to be used in the problems of surface dipole-lattice structures, which serve as models of adsorbate patterns on various substrates [34,43,[64][65][66][67][68][69][70][71][72].…”

Non-free gas of dipoles of non-singular screw dislocations and the shear modulus near the melting

“…The equivalence in (35) occurs together with positivity and monotonic decreasing of f 1 (z), for instance, at z > 2.3. However, Figure 2 demonstrates us that f 1 (z) < βµd only for large enough z.…”

“…Proliferation of the dislocation dipoles is responsible for the renormalization of the elastic constants [14-17, 19, 34]. According to [35], the type of the defect mediated twodimensional melting transition is determined by the value of the higher-gradient elastic constant called the angular stiffness (see also [24]). Therefore it is attractive from the viewpoint of nano-physics to study the behavior of the elastic moduli using the modified dislocation solutions [26][27][28][29][30][31][32][33].…”