Ab Initio calculations of phonons in metallic rutile and insulating monoclinic M1 and M2 VO$_2$*

Vanadium dioxide (VO$_2$) undergoes a first-order metal-insulator transition (MIT) at 340\,K from a metallic, high-temperature rutile phase (R) to an insulating, low-temperature monoclinic phase (M1). Under tensile strain, two other insulating phases, a second monoclinic phase (M2) and a low symmetry triclinic phase (T), are also known to exist. Recently, Park {\em et al.}\footnote{J. H. Park, Nature {\bf 500}, 431 (2013).} observed a solid-state triple point of these phases in strained VO$_2$ nanobeams. More recently, phonon frequencies for strain-stabilized M2 have been observed.\footnote{M. M. Qazilbash, private communication} Understanding the vibrational properties of these phases may help resolve questions surrounding the long-debated issue of the respective roles of electronic correlation and Peierls mechanisms in driving the MIT. We will present {\em ab initio} DFT and DFT+U calculations of phonon frequencies for the M2 phase and compare these to measured results and to previous calculations and measurements for the R and M1 phases.\footnote{T. J. Huffman et al., PRB {\bf 87},115121 (2013).}