Strong magnetoelastic coupling in ${\mathrm{Mn}}_{3}X$ ($X=\mathrm{Ge}$, Sn)

We measure the full elastic tensors of ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$ as a function of temperature through their respective antiferromagnetic phase transitions. Large discontinuities in the bulk moduli at the Néel transitions indicate strong magnetoelastic coupling in both compounds. Strikingly, the discontinuities are nearly a factor of 10 larger in ${\mathrm{Mn}}_3\mathrm{Ge}$ than in ${\mathrm{Mn}}_3\mathrm{Sn}$. We use the magnitudes of the discontinuities to calculate the pressure derivatives of the Néel temperature, which are 39 K/GPa 14.3 K/GPa for ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, respectively. We measured the in-plane shear modulus $c_{66}$, which couples strongly to the magnetic order, in magnetic fields up to 18 T and found quantitatively similar behavior in both compounds. Recent measurements have demonstrated strong piezomagnetism in ${\mathrm{Mn}}_3\mathrm{Sn}$ : Our results suggest that ${\mathrm{Mn}}_3\mathrm{Ge}$ may be an even better candidate for this effect.

Figure 1

Crystal structure and irreducible strains of ${\mathrm{Mn}}_{3}X$. (a) ${\mathrm{Mn}}_{3}X$ crystal structure. A hexagonal unit cell consists of AB-stacked Kagome planes of Mn atoms. Different shades of green indicate A and B planes, respectively. (b) View of one Mn Kagome layer, with purple arrows that illustrate one possible ordered-state spin configuration. (c) Visualization of the irreducible representations of strain. The definition of irreducible strains are given in terms of ${\varepsilon }_{ij}$, alongside the symmetry representations and the corresponding elastic moduli.

Figure 2

Change in elastic moduli as a function of temperature for ${\mathrm{Mn}}_3\mathrm{Ge}$ (left) and ${\mathrm{Mn}}_3\mathrm{Sn}$ (right). Upper and lower panels show changes compressional and shear elastic moduli, respectively. The change is defined as $\mathrm{\Delta }\mathrm{c}\left(T\right)=\mathrm{c}\left(T\right)-\mathrm{c}\left(387K\right)$ for ${\mathrm{Mn}}_3\mathrm{Ge}$ and $\mathrm{\Delta }\mathrm{c}\left(T\right)=\mathrm{c}\left(T\right)-\mathrm{c}\left(438K\right)$ for ${\mathrm{Mn}}_3\mathrm{Sn}$. The Néel temperatures are indicated by vertical-dashed lines.

Figure 3

Ehrenfest scaling for the bulk moduli and specific heat of ${\mathrm{Mn}}_{3}X$. Blue points are the specific heat of ${\mathrm{Mn}}_3\mathrm{Ge}$ taken from [19] divided by the Néel temperature $T_N$ and scaled by a factor with units of ${(\mathrm{kelvin}/\mathrm{GPa})}^2$. The bulk modulus of ${\mathrm{Mn}}_3\mathrm{Ge}$, divided by $B_{T_N}^2$—the square of the value of the bulk modulus at $T_N$—is shown as black points in the main panel. Both data sets are given in units of 1/GPa. The scaling factor used here corresponds to a value of $d{T}_N/dP=39\mathrm{K}/\mathrm{GPa}$, and the shaded region corresponds to deviations of $\pm 3\mathrm{K}/\mathrm{GPa}$. In the inset, this analysis is repeated for the bulk modulus of ${\mathrm{Mn}}_3\mathrm{Sn}$. It reflects a value of $d{T}_N/dP\approx (14.3\pm 2)\mathrm{K}/\mathrm{GPa}$.

Figure 4

The field dependence of $c_{66}$ in ${\mathrm{Mn}}_{3}X$. The changes in $c_{66}$ for ${\mathrm{Mn}}_3\mathrm{Ge}$ (solid lines) and ${\mathrm{Mn}}_3\mathrm{Sn}$ (dashed lines) at different fields with respect to the zero-field elastic moduli are shown as a function of the reduced temperature. The data were taken at 1, 2, 5, 9, and 15 T for ${\mathrm{Mn}}_3\mathrm{Ge}$ and at 1, 2.7, 5, 10, 14, and 18 T for ${\mathrm{Mn}}_3\mathrm{Sn}$. The inset shows the zero-field data for both compounds.

Figure 5

Custom-built high temperature RUS setup. Panel (a) shows the entire setup covered by insulating firebricks. In panel (b), one of the firebricks has been removed to give a better view of the copper mount. Panel (c) shows a sample being mounted on its corners between two piezoelectric transducers.

Figure 6

Raw RUS signal. Amplitude of the response of the receiving transducer in the RUS measurement of ${\mathrm{Mn}}_3\mathrm{Sn}$ at room temperature.

Figure 7

Poisson ratios. Poisson ratios ${\nu }_{xy}$ (left panel) and ${\nu }_{zx}$ for ${\mathrm{Mn}}_3\mathrm{Ge}$ (purple) and ${\mathrm{Mn}}_3\mathrm{Sn}$ (red) as a function of reduced temperature $(T-T_N)/T_N$. In a hexagonal crystal ${\nu }_{zx}={\nu }_{zy}$.

Figure 8

Bulk moduli. Bulk moduli for ${\mathrm{Mn}}_3\mathrm{Ge}$ (purple) and ${\mathrm{Mn}}_3\mathrm{Sn}$ (red) as a function of reduced temperature $(T-T_N)/T_N$.