Study of the Ordered Magnetic State of Copper Formate Tetrahydrate by Antiferromagnetic Resonance

A brief account of antiferromagnetic-resonance (AFMR) measurements in single crystals of monoclinic Cu${\mathrm{}\left(\mathrm{HCOO}\right)\mathrm{}}_2$.4${\mathrm{H}}_2$O has been reported, and it was shown that a model of weak ferromagnetism of the easy-plane type is not consistent with the data. This paper presents in detail the AFMR data and includes calculations of the AFMR modes and the magnetization based on assuming that Cu${\mathrm{}\left(\mathrm{HCOO}\right)\mathrm{}}_2$4${\mathrm{H}}_2$O is very nearly an antiferromagnet in zero magnetic field. AFMR measurements have been made between 1.4 and 15°K. Resonance transitions have been observed at 9 GHz, from 32 to 36 GHz, from 50 to 80 GHz, and from 107 to 124 GHz in magnetic fields up to 26 kOe. From the angular dependence of the field positions of the resonance transitions, a set of "principal axes" $a^{\prime \prime }b{c}^{\prime \prime }$ of the magnetic crystal have been determined. $a^{\prime \prime }$, the antiferromagnetic axis, is shown to be 8.5° away from the $a$ axis toward the $c$ axis. The AFMR modes and the magnetization have been calculated on a two-sublattice model, based on a bilinear exchange interaction, assuming $a^{\prime \prime }$ as the antiferromagnetic axis, $b$ as the intermediate axis, and $c^{\prime \prime }$ as the hard axis. This calculation is an extension of one by Cinader by including the hard-plane anisotropy and an anisotropic antisymmetric $g$-tensor. It is found that a magnetic field (perpendicular to $a^{\prime \prime }$) larger than a critical field induces a weak ferromagnetic state. The calculation also shows that both antisymmetric exchange and the antisymmetric Zeeman interaction contribute to the effective canting fields. For the $c^{\prime \prime }$-axis case, the calculations are in good agreement (except for a constant gap in the low-frequency mode above 5.3 kOe) with the data and an effective canting field ${H_{\mathrm{DM}}}^{\prime }=84\mathrm{}$ kOe is obtained. For the $b$ axis, the experiment indicates an instability in the antiferromagnetic axis at 5.3 kOe. The calculations also suggest an instability in the antiferromagnetic axis if a second effective canting field ${h_{\mathrm{DM}}}^{\prime }$ is negative; however, a quantitative fit to the AFMR modes has not been possible. A simplified calculation for the $a^{\prime \prime }$ axis is in fair agreement with some of the $a^{\prime \prime }$-axis data. The temperature dependence of the low-frequency zero-field mode is found to deviate substantially from the prediction of the molecular field approximation. A temperature-dependent contribution to the AFMR line-width of $T^{3.3\pm 0.2}$ is found.