Intercalation of the layer type transition metal dichalcogenides by a variety of organic molecules, alkali metals, or ‘3d’ transition metals, provides a powerful way to finely tune the electron occupation of the relatively narrow ‘d’ bands met in these solids These transition metal dichalcogenides are highly anisotropic solids, sometimes referred to as ‘two-dimensional’ solids, and the intercalant molecules which are electron donors enter between the layers This can result in profound changes in the electronic properties of the host lattice, and these changes can be understood in terms of charge transfer and increased interlayer separation The phenomena discussed include optical properties, transport properties, super-conductivity, order-disorder phenomena and phase changes, staging, magnetic properties, metal-insulator transitions, Anderson localization, and fast-ion conduction Some possible practical applications are also considered

Electronic properties of intercalation complexes of the transition metal dichalcogenides

Publisher Summary This chapter presents the formalism and applications of cluster expansions to the problem of ab initio calculations of thermodynamic properties of crystalline alloys. The success of the method owes much to the parallel developments of statistical and quantum mechanical methods and, of course, to the availability of powerful computational hardware and software. The technique of orthogonal expansions in cluster functions has revolutionized the treatment of configurationally disordered systems. In addition to offering the most rational and general state-of-order description of alloys, cluster methods provide the rigorous and essential link between quantum and statistical mechanical aspects of first-principles thermodynamic calculations. Cluster functions constitute a complete orthonormal set. This basis set produces a rigorous cluster algebra that is used for systems containing an arbitrary degree of configurational order (or disorder). Cluster methods are, in principle, ideally suited for calculating alloy phase equilibria.

Cluster Approach to Order-Disorder Transformations in Alloys

Physical approach to complex systems

The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics. This important physical theory has been generalized [1] in 1988 by using the entropy $S_{q} = \frac{1-\sum_{i} p^{q}_{i}}{q-1}$$({\rm with}\,q\,\in {{{\mathcal{R}}}})$ instead of its particular BG case $S_{1} = S_{BG} = - \sum_{i} p_{i}\,{\rm ln}\,p_{i}$. The theory which emerges is usually referred to as nonextensive statistical mechanics and recovers the standard theory for q = 1. During the last two decades, this q-generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. A conjecture[2] and numerical indications available in the literature have been, for a few years, suggesting the possibility of q-versions of the standard central limit theorem by allowing the random variables that are being summed to be strongly correlated in some special manner, the case q= 1 corresponding to standard probabilistic independence. This is what we prove in the present paper for $1{\leqslant}\,q 0$, and normalizing constant Cq. These distributions, sometimes referred to as q-Gaussians, are known to make, under appropriate constraints, extremal the functional Sq (in its continuous version). Their q = 1 and q = 2 particular cases recover respectively Gaussian and Cauchy distributions.

On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

Chalcopyrite-type CuInS2-based alloyed fluorescent nanocrystals (NCs), which contain no regulated heavy metal ions, were synthesized by heating an organometallic solution to demonstrate optical property tunability. Introduction of Zn into the CuInS2 system enhanced their photoluminescence (PL) intensity. The resultant particles were 3−6 nm; they varied with experimental conditions and were discrete and colloidally stable. The band-gap energy and PL wavelength of Zn-Cu-In-S (ZCIS) NCs varied with Zn content and particle size. Their PL was controllable within 570−800 nm by altering the band-gap energy. Furthermore, indium substitution with gallium was shown to control band-gap energy toward ∼3.1 eV, 500 nm of PL wavelength. In addition, ZnS coating of this nanocrystal can approximately double the PL strength. Finally, surface treatment with mercaptoundecanoic acid dispersed hydrophilic ZCIS NCs into water.

Tunable Photoluminescence Wavelength of Chalcopyrite CuInS2-Based Semiconductor Nanocrystals Synthesized in a Colloidal System

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann–Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accommodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy S BG =−k∑ i p i ln p i , and typically provides exponential laws for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form S q =k(1−∑ i p i q )/(q−1) (with S1=SBG), and typically provides, for the same type of description, (asymptotic) power laws.

Nonextensive statistical mechanics and economics

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann-Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accomodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy $S_{BG}=-k \sum_i p_i \ln p_i$, and typically provides {\it exponential laws} for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$), and typically provides, for the same type of description, (asymptotic) {\it power laws}.

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Distributions of High-Frequency Stock-Market Observables

Although bulk III-V alloys exhibit phase separation, vapor-phase epitaxial growth of Ga 0.5 In 0.5 P/GaAs (001) at ≃900-1000 K shows spontaneous ordering into the (111)-oriented mono-layer (GaP) 1 (InP) 1 superlattice (the «CuPt» structure). Only two superlattice directions ([111] and [111], which define the CuPt B variant) out of four possible are seen. Both [111] and [111] subvariants are observed on flat surfaces or when surface steps are perpendicular to the cation dimers

Ordering thermodynamics of surface and subsurface layers in the Ga1-xInxP alloy.

Nonisovalent ({ital A}{sup III}{ital BV}){sub 1{minus}{ital x}C2{ital x}}{sup IV} semiconductor alloys exhibit a transition as a function of composition between a phase with the zinc-blende symmetry (where the two fcc sublattices of the diamond lattice are unequally occupied by {ital A}{sup III} and {ital B}{sup V} atoms) and a phase with the diamond symmetry (where the two sublattices have equal occupations). Previous thermodynamic models of this transition have considered only nearest-neighbor interactions between {ital neutral} atoms. This approach ignores the important electrostatic interactions associated with electron transfer between the electron-rich {ital C}{sup IV}-{ital B}{sup V} ( donor'') bonds and the electron-deficient {ital A}{sup III}-{ital C}{sup IV} ( acceptor'') bonds. We have reexamined the validity of a three-dimensional bulk thermodynamic model for (GaAs){sub 1{minus}{ital x}}Ge{sub 2{ital x}} and (GaP){sub 1{minus}{ital x}}Si{sub 2{ital x}} using an energy model that includes such electrostatic interactions and pairwise energies extracted from first-principles local-density total-energy calculations.

Roberto Osório

Papers

Nonextensive statistical mechanics and economics

Nonextensive statistical mechanics and economics

Distributions of High-Frequency Stock-Market Observables

Ordering thermodynamics of surface and subsurface layers in the Ga1-xInxP alloy.

Structural phase transition in (GaAs)1-xGe2x and (GaP)1-xSi2x alloys: Test of the bulk thermodynamic description.