Inhomogeneity and the metal-insulator transition for disordered systems

Abstract

The effect of both macroscopic stress inhomogeneity and doping inhomogeneity on the critical behavior of the conductivity in the vicinity of the metal-insulator transition are calculated. For the uniaxial stress case the inhomogeneity is calculated from the bending deflection $d(z)$ of a column under compression. It is the transverse variation in doping or stress S that can produce a substantial increase in the scaling exponent t of the $\stackrel{\ensuremath{\rightarrow}}{T}0$ conductivity and change the critical stress ${S}_{c}$ to an apparent critical stress ${S}_{c}^{*}.$ It is demonstrated the calculated results can explain the experimental results of uniaxial stress experiments for Si:P and Si:B. In these cases when $\ensuremath{\sigma}(S,\stackrel{\ensuremath{\rightarrow}}{T}0)$ is sufficiently large $\ensuremath{\sigma}(S,T=0)\ensuremath{\propto}|{S/S}_{c}\ensuremath{-}1{|}^{t}$ with $t\ensuremath{\sim}\frac{1}{2},$ but when $\ensuremath{\sigma}(S,T=0)$ is sufficiently small $\ensuremath{\sigma}(S,T=0)\ensuremath{\propto}|{S/S}_{c}^{*}\ensuremath{-}1{|}^{{t}_{\mathrm{eff}}}$ with ${t}_{\mathrm{eff}}$ between 1 and 1.6 dependent on geometrical factors. For the doping inhomogeneity case both uncorrelated dopant density variations and correlated linear dopant variations are considered. Correlated cases can either increase or decrease ${n}_{c}$ depending on the geometry of the doping gradients. An uncorrelated broad distribution can mask scaling behavior with an exponent $t=\frac{1}{2}$ and change the exponent to $t\ensuremath{\sim}1.$ This suggests that the microscopic physics may be the same for crystalline doped semiconductors Si:P, Ge:Ga, and the amorphous semiconductor-metal cases and that the difference in scaling exponents of the conductivity results from the breadth and shape of the distribution $P(n(\mathbf{r})\ensuremath{-}n).$ A large width of $P(n(\mathbf{r})\ensuremath{-}n)$ for $a\ensuremath{-}{\mathrm{S}}_{1\ensuremath{-}x}{M}_{x}$ alloys helps explain why the conductivity prefactors are comparable to Si:P; etc., even though the electron density is ${10}^{3}$ larger.