  In order to explore the consequences of spin-orbit coupling on spin-phonon
interactions in a set of chemically-similar mixed metal oxides, we measured the
infrared vibrational properties of Co$_4B_2$O$_9$ ($B$ = Nb, Ta) as a function
of temperature and compared our findings with lattice dynamics calculations and
several different models of spin-phonon coupling. Frequency vs. temperature
trends for the Co$^{2+}$ shearing mode near 150 cm$^{-1}$ reveal significant
shifts across the magnetic ordering temperature that are especially large in
relative terms. Bringing these results together and accounting for
noncollinearity, we obtain spin-phonon coupling constants of -3.4 and -4.3
cm$^{-1}$ for Co$_4$Nb$_2$O$_9$ and the Ta analog, respectively. Analysis
reveals that these coupling constants derive from interlayer (rather than
intralayer) exchange interactions and that the interlayer interactions contain
competing antiferromagnetic and ferromagnetic contributions. At the same time,
beyond-Heisenberg terms are minimized due to fortuitous symmetry
considerations, different than most other 4$d$- and 5$d$-containing oxides.
Comparison with other contemporary oxides shows that spin-phonon coupling in
this family of materials is among the strongest ever reported, suggesting an
origin for magnetoelectric coupling.
