Phonon anisotropy in ZnCr$_2$O$_4$ from first principles

The geometrically frustrated spinel ZnCr$_2$O$_4$ undergoes a cubic-to-tetragonal structural transition simultaneously as it enters a Neel state at T$_c$=12.5K. A recent experiment\footnote{A.B. Sushkov et al., Phys. Rev. Letters 94 (2005) 137202.} using infrared spectroscopy measured a large splitting of an optical phonon frequency at T$_c$; it was argued that this splitting was due to a direct spin-phonon coupling. Here, we describe a first-principles study of the zone-center optical phonons in metrically cubic ZnCr$_2$O$_4$, with the LSDA+U as implemented in VASP. For selected collinear magnetically ordered structures, the influence of magnetic order on the phonon frequencies is identified, confirming the recent experimental results. The origin of this large phonon anisotropy will be discussed.