Temperature dependence of the Dzyaloshinskii-Moriya interaction in ultrathin films

The Dzyaloshinskii-Moriya interaction gives rise to a chiral exchange between neighboring spins in the technologically relevant class of perpendicularly magnetized ultrathin film materials. In this paper, we study the temperature dependence of the Dzyaloshinskii-Moriya interaction based on extensive characterization of a thin film which hosts a skyrmion state using both bulk magnetometry and x-ray magnetic circular dichroism photoemission electron microscopy. A version of the Bloch law explicitly for thin film geometries is derived to extract the exchange stiffness. The strength of the Dzyaloshinskii-Moriya interaction, $D$, is found to have a dependence on the saturation magnetization, ${M}_{s}$ of $D\ensuremath{\propto}{M}_{s}^{1.86\ifmmode\pm\else\textpm\fi{}0.16}$. Further, by extracting the uniaxial anisotropy ${K}_{u}$ and the exchange stiffness $A$, we find that $D\ensuremath{\propto}{K}_{u}^{1.02\ifmmode\pm\else\textpm\fi{}0.11}$ and $D\ensuremath{\propto}{A}^{0.95\ifmmode\pm\else\textpm\fi{}0.07}$. Skyrmion radii are also used to extract the strength of the Dzyaloshinskii-Moriya interaction which is compared to that derived from measurements of stripe domains. The origins of the correlations between material parameters are discussed and consequences of these relationships for skyrmion devices are considered.