Universal Scaling near Band-Tuned Metal-Insulator Phase Transitions

We present a theory for band-tuned metal-insulator transitions based on the Kubo formalism. Such a transition exhibits scaling of the resistivity curves in the regime where $T\tau >1$ or $\mu \tau >1$, where $\tau$ is the scattering time and $\mu$ the chemical potential. At the critical value of the chemical potential, the resistivity diverges as a power law, $R_c\sim 1/T$. Consequently, on the metallic side there is a regime with negative $dR/dT$, which is often misinterpreted as insulating. We show that scaling and this “fake insulator” regime are observed in a wide range of experimental systems. In particular, we show that Mooij correlations in high-temperature metals with negative $dR/dT$ can be quantitatively understood with our scaling theory in the presence of $T$-linear scattering.

Figure 1

In a band-tuned metal-insulator transition (MIT), the system changes from having overlapping valence (blue) and conduction (red) bands in the metallic side (right) to having a gap on the insulating side (left). The tuning parameter is the chemical potential $\mu$. When either $T\tau >1$ or $\mu \tau >1$, the resistivity (in shades on the background) can be described by a scaling form, as shown in Fig. 2. This scaling relation breaks down very close to the transition, where localization and interaction effects will change the picture.

Figure 2

Theoretical resistance curves close to a band-tuned metal-insulator transition. (a) Resistance calculated using Eq. (3), for a constant scattering time $\tau =25{\mathrm{eV}}^{-1}$, and chemical potential $\mu$ ranging from $-0.8$ to $+0.8\mathrm{eV}$. The resistance at the critical point $\mu =0$ diverges as $R_c(T)\sim 1/T$. On the metallic side, the resistance decreases as a function of temperature (a “fake” insulator), whereas on the insulating side the resistance is activated. (b) Resistance for a temperature-dependent scattering rate ${\tau }^{-1}={\tau }_0^{-1}+bT$ with ${\tau }_0=25{\mathrm{eV}}^{-1}$ and $b=0.1$. On the metallic side, a resistivity maximum arises at a temperature dependent on the distance from the transition. At high temperatures, this gives rise to Mooij correlations (see Fig. 4). (c) When $T\tau >1$ or $\mu \tau >1$, the resistance curves follow a simple scaling law, $R(T,\mu )=R_c(T)f(\mu /T)$. This can be verified by plotting $R/R_c$ versus $T/|\mu |$. All data points collapse onto one of the two curves, associated with either metallic or insulating behavior.

Figure 3

Scaling near the band-tuned MIT is observed in a range of materials. Here, we apply our scaling analysis to three material systems [19]: (a) the moiré heterobilayer ${\mathrm{MoTe}}_2/{\mathrm{WSe}}_2$ [5], (b) the heterostructure ${\mathrm{WSe}}_2/\text{bilayer graphene}/{\mathrm{WSe}}_2$ [22], and (c) GST amorphous phase change materials [23]. The measured resistivities are shown in insets. In panels (a) and (c) we see a genuine MIT, with data collapse on both an insulating and conducting branch. The theoretical scaling curve of Eq. (4) is shown as a dashed black line, and shows remarkable agreement with the experimental results.

Figure 4

The dimensionless temperature coefficient of the resistance versus the dimensionless resistance, for a variety of materials [5, 9, 23, 27, 28, 29, 30, 31, 32] (for details, see Ref. [19]), compared to our theoretical result of Eq. (6) (solid black line). For the experimental data the only fitting parameter is $R^{\infty }$, the limit of the critical resistance at high temperature. We find an excellent agreement of the experimental Mooij correlations and our theory.