Electronic transport in alkali-tungsten bronzes

In this paper we present a coherent physical picture of the electronic structure and the transport properties of alkali-tungsten bronzes, $M_X\mathrm{W}{\mathrm{O}}_3$, over the entire concentration range $X=1\mathrm{}-0$ of the alkali atoms. We propose a model based on nonrandom clustering of the alkali atoms into metallic regions characterized by the local atomic fraction $X_{\mathrm{local}}=1\mathrm{}$. Provided that the $M-M$ correlation length $b$ considerably exceeds the lattice spacing, we can define a percolation problem in which a volume fraction $C=X$ of the material is occupied by the metallic regions, the remainder consisting of semiconducting regions. The threshold $C^{*}\simeq 0.17$ for continuous percolation marks the onset of the continuous metal-nonmetal transition in this microscopically inhomogeneous material. This physical picture is borne out by the available magnetic data. A semiquantitative calculation of the cluster stabilization energy with respect to the dispersed phase suggests that a negative surface energy exists between $M\mathrm{W}{\mathrm{O}}_3$ and W${\mathrm{O}}_3$ within a coherent W${\mathrm{O}}_3$ lattice. $b$ is determined by the cluster size at which the metallic clusters begin to lose that stabilization energy. The Madelung energy provides a barrier against penetration of the electrons into the nonmetallic regions. Tunneling corrections are therefore negligible and we can define local electronic structure and transport properties. An analysis of the electrical conductivity data was carried out by utilizing the results of numerical simulations of the conductivity in simple-cubic lattices which incorporated correlation between metallic bonds. The numerical results were modified to account for scattering from the boundaries of the metallic regions. For low values of the conductivity ratio (∼ ${10}^{-4\mathrm{}}$) between the nonmetallic and the metallic regions, the effective-medium theory, modified to account for boundary scattering effects, faithfully reproduces the results of the numerical simulation for $C>\mathrm{}0.4\mathrm{}$. An excellent fit of the available experimental conductivity data in the range $C=0.9\mathrm{}-0.22$ and over the temperature range 4-770°K has been obtained. In the process we found that the $M-M$ correlation length is temperature independent and has the value of $b\simeq 45$ Å. The Hall effect and the Hall mobility were successfully analyzed utilizing a modified effective-medium theory.