Effects of strain on the stability of the metallic rutile and insulating M1 phases of vanadium dioxide

TL;DR

This work presents a systematic first-principles analysis of how strain biases the relative stability of VO$_2$'s rutile R phase and monoclinic M1 phase. Using a $1\times2\times2$ orthorhombic supercell within DFT+$V$ (with $V=2$ eV) and Wannier-based extraction of hopping parameters, it maps how epitaxial strain along the $c$-axis and in the basal plane modifies V–V dimerization, orbital occupations, and the Peierls-like energy gain. The key finding is that $c$-axis strain dominantly controls phase stability, with a softening lattice stiffness along the dimerization direction offsetting changes in hopping, while basal-plane strain primarily influences stability through Poisson-driven changes in $c$. These results provide a quantitative framework for strain-engineering VO$_2$ MIT and reconcile diverse experimental observations across substrate orientations.

Abstract

We present a systematic density-functional theory study of the effects of strain on the structural and electronic properties in vanadium dioxide (VO$_2$), with particular emphasis on its effect on the relative stability of the metallic rutile and the insulating monoclinic M1 phases. We consider various strain conditions that can be related to epitaxial strain present in VO$_2$ films grown on different lattice planes. Our calculations confirm the dominant role of $c$ axis strain, i.e., along the direction of the V-V dimerization in the M1 phase. Our analysis suggests that this effect stems primarily from the weakening of the lattice stiffness, with the hopping along the $c$ axis playing a minor role. We also confirm that, in strain scenarios that deform the basal plane, the $c$ axis strain still has a dominant effect on the phase stability.

Figures (5)

• Figure 1: (a, b) Supercell used in the calculations to describe both the (a) R and (b) M1 structures. The dashed lines in (a) indicate the underlying primitive rutile cell. Arrows indicate the definition and orientation of the corresponding orthorhombic axes. V (O) atoms shown in light (dark) gray. (c-e) The three V $t_{2g}$ orbitals, $d_{x^2-y^2}$, $d_{xz}$, and $d_{yz}$, respectively, highlighting their orientation with respect to the local environment. Yellow (cyan) colors indicate positive (negative) phase of the orbitals. (f, g) The two epitaxial strain orientations considered in this work, corresponding to growth along the (f) [001] and (g) [010] directions. Axes shown in red are free to relax, while the axes shown in black are fixed by the substrate.
• Figure 2: (a) Energy difference between the R and M1 structures as a function of strain, $\epsilon_{xx}=\epsilon_{yy}$, calculated with (without) the $+V$ correction in black (gray). (b) The $c/a$ ratio as a function of strain for the R and M1 phases marked by triangles and squares, respectively. Strain is defined with respect to the relaxed R phase lattice parameter, $a_0=4.630$ Å.
• Figure 3: (a) Energy difference between the R and M1 phases. Triangle (square) indicates the relaxed R (M1) lattice parameters. Numbered lines indicate cuts along different directions. (b) Nearest-neighbor V--V distance, $d$, as a function of $c$ (Cut 1, left) or $a$ (Cut 2, right) lattice parameter, and (c) ratio of the nearest-neighbor V--V distance to $c$, $r=d/c$. Triangular markers indicate R and filled (empty) square markers indicate M1 SB (LB). (d) Energy levels, $\varepsilon$, relative to the average energy level, $\langle \varepsilon \rangle$, and (e) nearest neighbor hopping amplitudes along $c$, $t_c$, both corresponding to $d_{x^2-y^2}$ (cyan), $d_{xz}$ (green), $d_{yz}$ (yellow) orbitals, as function of $a$ or $c$ in the R phase.
• Figure 4: Evolution of the PDOS of the different $t_{2g}$ orbitals within the R phase (a) along Cut 1 and (b) along Cut 2. Color gradient indicates increasing lattice parameter. The vertical dashed line indicates the Fermi level.
• Figure 5: (a) Energies of the R and M1 phases as a function of the $b/a$ ratio, indicated by triangular and square markers, respectively, with $a$ and $c$ strained to values corresponding to a (010)-oriented TiO$_2$ substrate. (b) Energy difference between the two phases, with the dashed line indicating the corresponding energy difference at $c=c(\text{TiO}_2)$ and the corresponding equilibrium $a$.