Uniaxial pressure effects in the two-dimensional van-der-Waals ferromagnet CrI$_3$

Magnetoelastic coupling and uniaxial pressure dependencies of the ferromagnetic ordering temperature in the quasi-two-dimensional layered van-der-Waals material CrI$_3$ are experimentally studied and quantified by high-resolution dilatometry. Clear anomalies in the thermal expansion coefficients at $T_{\rm C}$ imply positive (negative) pressure dependencies $\partial T_{\rm C}/\partial p_{\rm i}$ for pressure applied along (perpendicular to) the $c$ axis. The experimental results are backed up by numerical studies showing that the dominant, intra-layer magnetic coupling increases upon compression along the $c$ direction and decreases with negative in-plane strain. In contrast, inter-layer exchange is shown to initially increase and subsequently decrease upon the application of both out-of-plane and in-plane compression.