First-principles calculation of the full orbital magnetoelectric response

The possibility of a quantized magnetoelectric (ME) effect in $Z_2$ topological insulators suggests that the orbital part of the ME response can, at least in principle, be comparable in magnitude to the total response of known ME materials.\footnote{S. Coh {\it et al.}, arXiv:1010.6071.} A band theory of the orbital ME response of generic insulators was recently developed,\footnote{A. Malashevich {\it et al.}, New J. Phys. {\bf 12}, 053032 (2010)} paving the way for first-principles calculations. Two types of terms contribute to the response. The Chern-Simons term, which only depends on the unperturbed valence Bloch states, was the subject of a recent Wannier-based calculation.$^1$ The Kubo terms require a knowledge of the Bloch states at first order in the electric field and can be calculated by finite differences from the change in orbital magnetization induced by small electric fields.$^2$ We present preliminary results of such a calculation for Cr$_2$O$_3$, using the Berry-phase approach to calculate the electronic structure under a finite electrical bias. By monitoring the field-induced changes in orbital and spin magnetization and comparing the results obtained with and without structural relaxation, all contributions to the ME coupling can be computed.