On the possibility of quasi small-world nanomaterials

Abstract

The possibility of materials that are governed by a fixed point related to small world networks is discussed. In particular, large-scale Monte Carlo simulations are performed on Ising ferromagnetic models on two different small-world networks generated from a one-dimensional spin chain. One has the small-world bond strengths independent of the length, and exhibits a finite-temperature phase transition. The other has small-world bonds built from atoms, and although there is no finite-temperature phase transition the system shows a slow power-law change of the effective critical temperature of a finite system as a function of the system size. An outline of a possible synthesis route for quasi small-world nanomaterials is presented. PACS numbers: * Electronic address: man40@ra.MsState.edu; URL: http://www.msstate.edu/dept/physics/profs/novotny.html 1 I. INTRODUCTION Many materials have novel physical properties that are attributed to the quasi-lowdimensional nature of their structure, as can be seen in recent research in a number of areas. Examples of quasi-one-dimensional materials include conductors and superconductors [1]. Two-dimensional thin films lead to novel effects, including the quantum Hall effect [2] and low-dimensional magnetism [3]. The change from one effective dimension to another may lead to interesting physical effects. For example, it may be responsible for the onset of high-temperature superconductivity [4]. This type of dimensional crossover can lead to effective non-integer dimensional fixed points [5]. However, quasi-low-dimensional materials have constraints due to their low-dimensional behavior. These include the absence of a phase transition both in one-dimensional systems and in some two-dimensional systems where two is below the lower critical dimension.Recently there has been a great deal of interest in networks that are not regular lattices,

Critical behavior of ising models with random long-range (small-world) interactions

“…This study is motivated by physical small-world networks [22][23][24] where the number of short-cut bonds (which are SW bonds) cannot grow as fast as N. In [22,23] it was shown that the properties of such systems for small N can nevertheless show some of the mean field properties of a full small-world system. Fig.…”

“…There are two ways to obtain physical SW networks [22][23][24] to study the effect of SW bonds that can be built from physical building blocks. These physical SW networks must satisfy certain constraints so that they can be built from physical building blocks.…”

“…The second way to obtain a physical SW network is to build the SW bonds from the underlying building blocks. This was done in [22] starting from a d=1 chain. Each SW bond is then composed of a string of spins with ferromagnetic coupling constant J 3 , with the number of spins along the chain equal to the Euclidean distance between the randomly chosen sites.…”

“…The physical SW networks either have a number of SW bonds that vanish as N→∞ (here z=4+ 2 √ N ), or are constructed from a linear chain of spins with the number of spins along the chain equal to the Euclidean distance between the randomly chosen spins on the square lattice. The study here can be compared to physical SW studies in [22][23][24], as well as the study with power-law SW bond strengths [18]. The summary is twofold.…”

“…The reason is that certain constraints due to the fixed size of atoms and atomic bonds, as well as the necessity of embedding the 'ball-and-stick' graph in three dimensional space, restricts the types of SW graphs that are of interest to materials researchers. SW graphs with such constraints are called physical SW graphs [22][23][24], and the critical behavior of Ising models on physical SW graphs has been studied. This study started with a linear graph, with added SW bonds.…”