The earliest models used in the study of lattice structures are mean field theories, which do not contain structural dependence. The Lattice Compatibility Theory (LCT) proposes here a novel framework where the measure of the disorder is based on Urbach tailing features and lattice matching features between the host matrix and doping agent intrinsic structures. This study has been implemented on a particular compound (BTO:Co) and refers to the Simha-Somcynsky (SS) theory, a mean field theory where the measure of the disorder is stated as holes.
The knowledge of doping agents behaviors within host lattice matrix is of considerable importance for the optimal design for applications such as semiconductor windows functional glasses, transparent electrodes in flat panel displays, buffer layers, and solar cells [
The first theories based on mean field theory and independent from the design of lattice structures failed in the statistical thermodynamics of branched macromolecules. Studies on branched structures served as attempts to mathematically correct the mean field theories. One of the most important of these studies is the Lattice-Cluster Theory (LCT), developed by Freed and Bawendi [
For complex lattice systems, other theories have been developed. Dee and Walsh [
In the hole theory, a major change in volume is explained by the number of holes, and the change in cell size plays a minor role while in the cell theory [
Zhong et al. [
In this paper, it is outlined that SS theory was independent from the macromolecular architecture, and it exhibited differences in behavior concerning some doped lattices. Hence new elements for explanation of these differences are presented and discussed in the framework of the Lattice Compatibility Theory (LCT). The paper is organized in the following way. In Section
According to the generalized Simha-Somcynsky theory [
Simha-Somcynsky configuration (BTO undoped lattice).
In this theory’s framework, the most important aggregate which traduces entities repartition within the lattice is the volume fraction
The total occupied volume fraction is coupled with temperature through the minimization condition
The configurational partition function
The combinatorial factor
The Lattice Compatibility Theory, as mentioned in some recent studies [
An original formulation of the Lattice Compatibility Theory [
In the actual discussed case (cobalt-doped BTO lattice), the nature of the highest occupied band and the location of holes in BTO lattice structures have been demonstrated to be determinant. In this context, fundamental geometrical observations concerning the structure of BTO and the doping lattice were interpreted in terms of conventional orbitals patterns-linked geometry.
The Lattice Compatibility Theory tried to give a plausible understanding of the disparity concerning doping element incorporating dynamics starting from intrinsic doping element lattice properties in comparison to those of the host. In precedent study [
Urbach energy
The width of the localized states (band tail energy or Urbach energy
Plots of doped BTO optical coefficient versus energy (as guides to Urbach tailing quantification).
Co3+ and Co2+ ions outer shell configuration inside oxygen-dominant structures.
Using a complete set of measurements, main lattice constants of cobalt intrinsic lattice have been compared to those of BTO and revealed an obvious compatibility with the cubic main metric and angular parameters of Co2+ intrinsic lattice. It has been recorded that Co2+ incorporation in BTO matrix was followed by a loss of the hopping motion of electrons which decreased the piling up of electrons at host matrix vacancies hence impeding the buildup of space charge polarization. The variation in lattice parameters with Co3+ incorporation had been attributed to the increase of the unit cell volume of the host lattice with increasing Co3+ content. In fact, since Co3+ ions are trivalent and in order to maintain the charge balance during doping, incorporation had to be accompanied by an increase of unit cell volume either by a reduction of host cations valence or an oxygen content increase.
In the same context, and according to Muncaster et al. [
The most fundamental structure alteration which occurred in the host BTO lattice has been interpreted in terms of changes either in the configurational partition function
In the Simha-Somcynsky analysis, the structure-dependent coefficients (
Co2+ ions incorporation scheme inside BTO matrix.
Co3+ ions incorporation scheme inside BTO matrix.
In this context, for a molecule
The presented work showcases some fundaments of the Lattice Compatibility Theory (LCT) framework in relation with the precedent Simha-Somcynsky theory-linked analyses. The most probable explanations which can be provided to some intriguing nanoscale unexpected alterations, through the LCT theory, have been discussed in terms of the well-known Simha-Somcynsky (SS) lattice hole theory. What this theory and more sophisticated frameworks have in common is that they incorporate a description for holes on the lattice, acting as free volumes. This approach, which became necessary since the lattice-based arrangement of molecules became the most encountered in the recent literature, was confronted to the results which have been recorded in an unupdated compound (BTO:Co). The results were as accurate as relevant. The LCT analyses used in lattice linear structures can also be used together with the SS model for other similar structures.
Partial financial support by project of Shumen University (2013) is gratefully acknowledged.