Mechanism of Hopping Transport in Disordered Mott Insulators

By using a combination of detailed experimental studies and simple theoretical arguments, we identify a novel mechanism characterizing the hopping transport in the Mott insulating phase of ${\mathrm{C}\mathrm{a}}_{2-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{R}\mathrm{u}\mathrm{O}}_4$ near the metal-insulator transition. The hopping exponent $\alpha$ shows a systematic evolution from a value of $\alpha =1/2$ deeper in the insulator to the conventional Mott value $\alpha =1/3$ closer to the transition. This behavior, which we argue to be a universal feature of disordered Mott systems close to the metal-insulator transition, is shown to reflect the gradual emergence of disorder-induced localized electronic states populating the Mott-Hubbard gap.

Figure 1

(a) $T$-dependence of the in-plane resistivity ${\rho }_{\mathrm{a}\mathrm{b}}(T)$ measured on cooling for ${\mathrm{C}\mathrm{a}}_{2-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{R}\mathrm{u}\mathrm{O}}_4$. Vertical broken lines are guides to the eye. (b) Phase diagram for the region $0\le x\le 0.2$ with three different phases: paramagnetic metal (top), paramagnetic insulator (middle), and antiferromagnetic insulator (bottom). $T_{\mathrm{M}\mathrm{-}\mathrm{I}}$ and $T_{\mathrm{A}\mathrm{F}}$ are the MI and AF transition temperatures determined on cooling. Thick and thin lines indicate the first and second order transitions.

Figure 2

$T$-dependence of the in-plane susceptibility for ${\mathrm{C}\mathrm{a}}_{2-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{R}\mathrm{u}\mathrm{O}}_4$ with $x=0.06$, 0.09, and 0.15. All curves were measured in a field-cooled sequence. The solid and open symbols indicate the results measured on cooling and heating, respectively. Inset I: thermal hystereses observed in the paramagnetic phase for $x=0.06$ and 0.09. Inset II: $C_P/T$ vs $T^2$ for $x=0$ and 0.15. The solid lines represent linear fit.

Figure 3

(a) $\ln \rho$ vs $1/T^{1/2}$ and $1/T^{1/3}$ for the insulating phase of ${\mathrm{C}\mathrm{a}}_{2-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{R}\mathrm{u}\mathrm{O}}_4$. The results for $x\le 0.09$ are plotted against the lower horizontal axis $1/T^{1/2}$, while these for $x=0.15$ are plotted against both $1/T^{1/2}$ and the upper axis $1/T^{1/3}$. Straight lines indicate a fitting result to Eq. (2). (b) Corresponding plot for theoretical predictions. The results for $W=0.02$ and 0.08 eV are plotted against the lower axis $1/T^{1/2}$, while these for $W=0.16\mathrm{e}\mathrm{V}$ are plotted against both $1/T^{1/2}$ and the upper axis $1/T^{1/3}$. Inset I: $T_{\mathrm{0}}$ obtained by fitting of $\rho (T)$ to Eq. (2) with $\alpha =1/2$ for ${\mathrm{C}\mathrm{a}}_{2-x}{\mathrm{S}\mathrm{r}}_x{\mathrm{R}\mathrm{u}\mathrm{O}}_4$ (solid circle) as a function of $x$ (lower axis), and for theory (open circle) as a function of $W$ (upper axis). Inset II: Model band structure used in our calculation. As the disorder increases ($W\mathrm{\text{∶}}0.02\mathrm{e}\mathrm{V}\to 0.16\mathrm{e}\mathrm{V}$), the band-tails gradually fill in the Mott gap, producing a soft gap similar to that predicted by Efros and Shklovskii, but on a much larger energy scale.