Competing Interactions and the Effects of Uniaxial Out-of-plane Perturbations in the Honeycomb Antiferromagnet Na${}_2$Co${}_2$TeO${}_6$

Despite exhibiting magnetic long-range order below $T_{\mathrm{N}}=26.7\mathrm{K}$, the honeycomb cobaltate Na${}_2$Co${}_2$TeO${}_6$ is predicted to enter a Kitaev spin liquid state when subjected to small external perturbations. While most of the reported literature investigates the effects of magnetic fields applied parallel to the honeycomb layers, we present high-resolution capacitance dilatometry studies for fields perpendicular to the Co-planes up to $15\mathrm{T}$. Grüneisen analysis reveals the effect of uniaxial out-of-plane strain and shows that antiferromagnetic order in Na${}_2$Co${}_2$TeO${}_6$ is stabilized at a rate of $\mathrm{\partial }{T}_{\mathrm{N}}\mathrm{/}\mathrm{\partial }{p}_{\mathrm{c}}=0.28(5)\mathrm{K}\mathrm{/}\mathrm{G}\mathrm{P}\mathrm{a}$. Further, failure of the Grüneisen scaling at low temperatures around $T_{\mathrm{c}\mathrm{r}}\simeq 7.5\mathrm{K}$ demonstrates the presence of competing energy scales. In contrast to an only weak field dependence of the anomaly at $T_{\mathrm{N}}$, a broad hump at $T_{\mathrm{c}\mathrm{r}}$ ($B=0\mathrm{T}$) evolves into a sharp peak at high fields applied $B\parallel c$. Our magnetostriction data show that a kink in the magnetisation at $B_{\mathrm{C}}\simeq 4.6\mathrm{T}$ is accompanied by an inflection point in the field-induced length changes, which is likely related to weak unequal spin canting. All observed phenomena leave their signatures in the magnetoelastic phase diagram as constructed by our experimental results.