Aspects of bulk and surface magnetism of magnetoelectric Fe$_2$TeO$_6$*

Magnetoelectric antiferromagnets can be used to implement voltage-controlled magnetism, but materials design for above room-temperature operation is a challenge. Here we focus on the trirutile Fe$_2$TeO$_6$ magnetoelectric and use first-principles calculations to develop several strategies for increasing its N\'eel temperature $T_N$ above the bulk 210 K value. We find that substitution of larger ions like Zr or Hf for Te increases $T_N$ by increasing the superexchange angles. The compensating O vacancies tend to form bound complexes with such dopants, preserving the electronic band gap. Substitution of N for O is favorable due to the decreased charge-transfer gap. $T_N$ is also increased by compressive [001] epitaxial strain. To help interpret the XMCD signal observed from the (110) surface of Fe$_2$TeO$_6$,\footnote{Wang \emph{et al.}, J. Phys: Condens. Matter 26, 055012 (2014)} we compare the energies of several terminations of this surface and find the known TiO$_2$-like termination is the most stable. The perpendicular magnetic moment at this surface, which appears through spin canting due to spin-orbit coupling, is found to be only 0.015 $\mu_B$ per surface Fe. The XMCD signal likely originates from the lowered symmetry of the combined surface and X-ray beam configuration.