Phase-field model of insulator-to-metal transition in ${\mathrm{VO}}_2$ under an electric field

The roles of an electric field and electronic doping in insulator-to-metal transitions are still not well understood. Here we formulated a phase-field model of insulator-to-metal transitions by taking into account both structural and electronic instabilities as well as free electrons and holes in ${\mathrm{VO}}_2$, a strongly correlated transition-metal oxide. Our phase-field simulations demonstrate that in a ${\mathrm{VO}}_2$ slab under a uniform electric field, an abrupt universal resistive transition occurs inside the supercooling region, in sharp contrast to the conventional Landau-Zener smooth electric breakdown. We also show that hole doping may decouple the structural and electronic phase transitions in ${\mathrm{VO}}_2$, leading to a metastable metallic monoclinic phase which could be stabilized through a geometrical confinement and the size effect. This work provides a general mesoscale thermodynamic framework for understanding the influences of electric field, electronic doping, and stress and strain on insulator-to-metal transitions and the corresponding mesoscale domain structure evolution in ${\mathrm{VO}}_2$ and related strongly correlated systems.

Figure 1

(a) Spatial profiles of the spin-correlation order parameter at an applied voltage 0.4 V, outside $(T=300$ K) and inside $(T=337$ K) the supercooling region, respectively. The inset is a schematic of the geometry. (b) The length of the metallic region as a function of the applied electric field at the two temperatures. The lines are guide to eyes. The inset shows the theory fit of the nonzero $l_{\text{M}}$ at $T=300$ K.

Figure 2

Total Landau potentials of the metal-insulator-metal sandwich (blue squares) and the metal (red diamonds) as a function of the electric field at 337 K. The lines are guide to eyes. Filled and empty markers indicate stable and metastable states, respectively. The threshold $E$ and $l_{\text{M}}$ of the sandwich at the crossing point of the Landau potentials are $4.5\times {10}^5$ V/m and 126 nm, respectively, as indicated in the figure.

Figure 3

Temperature-electric field phase diagram of ${\mathrm{VO}}_2$. The threshold voltage $\mathrm{\Delta }{V}_{\text{th}}=E_{\text{th}}L$ corresponding to each value of the threshold electric field is also shown on the right $y$ axis.

Figure 4

(a) Stable (solid lines) and metastable (empty lines) phases of the independent hole-doped $(N_a=5\times {10}^{-3}$ per unit cell) ${\mathrm{VO}}_2$ thin film at various temperatures. (b) Total Landau potential landscape of the same hole-doped ${\mathrm{VO}}_2$ thin film at 283 K, showing the stable M1 phase accompanied with the metastable MM phase.

Figure 5

Total Landau potentials (per unit cell) of the ${\mathrm{VO}}_2\text{−}{\mathrm{VO}}_{2-\delta }$ bilayer as a function of the ${\mathrm{VO}}_2$ layer thickness at 283 K for the M1-R (blue circles) and MM-R (green squares) configurations. The lines are guide to eyes. The insets are the stable profiles of the order parameters for a $t<t_c$ and a $t>t_c$. The bilayer is stacked along the $z$ direction. The arrows are the two-dimensional order parameter vector $(\eta ,\mu )$, and the color represents its norm $\sqrt{{\eta }^2+{\mu }^2}$.