Gaussian Free Field in the background of correlated random clusters, formed by metallic nanoparticles

Abstract

The effect of metallic nano-particles (MNPs) on the electrostatic potential of a disordered 2D dielectric media is considered. The disorder in the media is assumed to be white-noise Coulomb impurities with normal distribution. To realize the correlations between the MNPs we have used the Ising model with an artificial temperature T that controls the number of MNPs as well as their correlations. In the T → 0 limit, one retrieves the Gaussian free field (GFF), and in the finite temperature the problem is equivalent to a GFF in iso-potential islands. The problem is argued to be equivalent to a scale-invariant random surface with some critical exponents which vary with T and correspondingly are correlation-dependent. Two type of observables have been considered: local and global quantities. We have observed that the MNPs soften the random potential and reduce its statistical fluctuations. This softening is observed in the local as well as the geometrical quantities. The correlation function of the electrostatic and its total variance are observed to be logarithmic just like the GFF, i.e. the roughness exponent remains zero for all temperatures, whereas the proportionality constants scale with T − Tc. The fractal dimension of iso-potential lines (D f ), the exponent of the distribution function of the gyration radius (τr), and the loop lengths (τ l ), and also the exponent of the loop Green function x l change in terms of T − Tc in a power-law fashion, with some critical exponents reported in the text. Importantly we have observed that, in which ξ(T ) is the spin correlation length in the Ising model.

Gaussian free field in the iso-height random islands tuned by percolation model

“…It is necessary to review some features of the scaleinvariant 2D random fields and rough surfaces. The methods employed in these systems have wide applications, ranging from the classical [4,35,36] to the quantum systems [38][39][40][41]. Let V (x, y) ≡ V (r) be the height profile (in this paper the electrostatic potential) of a scale invariant 2D random rough field.…”

“…If other kinds of disorder is present in the system (that is the case for any condensed matter system), the problem of the statistics of the electric potential becomes complicated, since the GFF is coupled to the other model which realizes the disorder. There are many experimental [14-21, 23, 25-27, 49] and theoretical [4] motivations to consider the metallic disorders in dielectric media. If the mentioned disorders are some metallic particles randomly distributed over the sample (which is the subject of the present paper), then the problem is simply finding the solution of the Poisson equation with some additional boundary conditions imposed by metallic regions.…”

“…The coupling of critical statistical models has recently been a subject of intense study [1][2][3][4][5][6][7][8][9][10]. One way of the coupling of models is to define a model on a host system with internal degrees of freedom whose arrangement is realized by another statistical model [2,4,6]. This type of coupling may help to understand the structure of fixed points of the combined model [11].…”

“…The coupling of GFF to the other models, is possible via (but not restricted to) the dilution of the host media whose pattern is tuned by the other statistical model, or by distributing some disorders in the regular lattice according to a model that yields the position pattern of the disorders. The example is the Poisson equation due to a white noise charged disorder (which is equivalent to GFF) in the presence of metallic regions whose formation pattern are modeled by the Ising model with an artificial temperature [4]. Such a study is expected to be relevant in understanding of many solid state systems which are doped with the metallic particles with a vast applications ranging from optical devices [14][15][16][17][18][19][20][21][22] and random lasers [23], to sensor technology [24] and solar cells [25][26][27].…”