Ab Initio phonon calculations in metallic and insulating phases of VO$_2$

Vanadium dioxide (VO$_2$) undergoes a first-order metal-insulator transition (MIT) from the high-temperature rutile phase (R) to an insulating, low-temperature monoclinic phase (M1). Several competing insulating phases exist, with phase boundaries in a narrow temperature and strain range close to the MIT\footnote{J. H. Park, Nature {\bf 500}, 431 (2013).\label{3point}}\footnote{J. M. Atkin et al, PRB {\bf 85}, 020101 (2012).}. Recently, novel IR and Raman measurements of micro- and nano-structured VO$_2$ samples have become increasingly available; this allows the phases to be studied while avoiding many difficulties with bulk samples, such as twinning and cracking on cycling through the MIT. Theoretical calculations of vibrational properties can assist in the interpretation of such experiments\footnote{\label{huffman}T. J. Huffman et al., PRB {\bf 87}, 115121 (2013).}. We will present {\em ab initio} DFT+U calculations of phonon frequencies for the various phases and compare these to reported measurements for the R, M1\footnote{P. Schilbe et al., Mat. Sci. and Eng. A {\bf 370}, 449-452 (2004).}, M2 and T phases\footnote{C. Marini et al., PRB {\bf 77}, 235111 (2008).}.