Exchange narrowing of electron spin resonance in a two-dimensional system

Electron-spin-resonance (ESR) measurements are reported in the two-dimensional Heisenberg magnet ${\mathrm{K}}_2$Mn${\mathrm{F}}_4$ and compared with a theory developed here. Results are in excellent agreement with calculated values and, we feel, give strong confirmation of recent theories of spin dynamics. The theory treats the linewidth $\Delta H$ and line shape in a two-dimensional Heisenberg system by assuming diffusive motion for the long-time dependence of the time correlation functions. The short-time dependence is taken to be Gaussian, and the resulting short- and long-time parts are joined together in a manner similar to that used by Gulley, Hone, Scalapino, and Silbernagel. An angular dependence roughly of the form $\Delta H\propto {(3\mathrm{}{cos}^2\theta -1)}^2+(\mathrm{const}.\mathrm{})$ ($\theta$ is the angle of dc field with respect to the perpendicular to the plane) is observed at high temperature, as predicted by the theory. This angular dependence cannot be explained by either the secular or nonsecular parts of the second moment. Rather, it is due explicitly to the dominance of wave-vector $q\to 0$ modes in the long-time decay of correlations in a two-dimensional system. As temperature is lowered toward the antiferromagnetic ordering temperature $T_N=45\mathrm{}$ °K, the linewidth initially decreases, passes through a minimum, and then increases rapidly near $T_N$. The angular dependence is also temperature-dependent such that $\Delta H$ ($\theta =90\mathrm{}°$) becomes less than $\Delta H$ ($\theta =55\mathrm{}°$) below about 65 K. These features of the temperature dependence are consistent with the theory. Indeed, we find absolute agreement between theory and experiment to within 20% or better at all angles over a broad range of temperature. The theory contains no adjustable parameters since classical dipolar coupling is taken as the sole source of broadening and we use the same exchange constant $J$ as obtained from susceptibility measurements. The room-temperature line shape, which is Lorentzian at $\theta =55\mathrm{}°$ and non-Lorentzian at $\theta =90\mathrm{}°$, and the frequency dependence of $\Delta H$, measured at 9.8 and 23.4 GHz, are also in agreement with theory. Shift of the resonance field with angle has been measured as well. This effect can be explained quantitatively by the net dipolar field and, contrary to the other phenomena, does not, in the main, reflect two-dimensional spin dynamics.