Theory of magnetism-driven negative thermal expansion in inverse perovskite antiferromagnets

Magnetism-induced negative thermal expansion (NTE) observed in inverse perovskite antiferromagnets ${\mathrm{Mn}}_{3}A\mathrm{N}(A=\mathrm{Zn},\mathrm{Ga}$, etc.) is theoretically studied by a classical spin model with competing bond-length-dependent exchange interactions. We numerically reproduce the crystal-volume expansion upon cooling triggered by a noncoplanar antiferromagnetic order and show that the expansion occurs so as to maximize an energy gain of the nearest-neighbor antiferromagnetic interactions. This mechanism is not specific to inverse perovskite magnets and might also be expected in magnets with other crystal structures. We propose other candidate crystal structures that might exhibit NTE through this mechanism.

Figure 1

(a) Schematic figure of the negative thermal expansion where the crystal volume shows a pronounced expansion upon cooling at the magnetic transition. (b) Exchange interactions considered for the classical Heisenberg model in Eq. (4). (c) ${\mathrm{\Gamma }}^{5g}$-type antiferromagnetic order in ${\mathrm{Mn}}_{3}A\mathrm{N}$. (d) Two opposite contributions to the nearest-neighbor exchange interaction $J_1$, i.e., the antiferromagnetic contribution $J^{\mathrm{AF}}$ and the ferromagnetic contribution $J^{\mathrm{FM}}$. (e) Easy magnetization plane for each Mn sublattice.

Figure 2

(a) [(b)] Calculated specific heat $C_{\mathrm{s}}$, averaged difference of normalized bond length $\overline{\delta }=(1/N){\sum }_k{\delta }_k$, and spin correlation $\widehat{S}=(-1/N_{\mathrm{pair}}){\sum }_{\langle i,\mu ;j,\nu \rangle }\langle {\mathit{S}}_{i,\mu }·{\mathit{S}}_{j,\nu }\rangle$ as functions of temperature in the presence [absence] of the anharmonic component with $K_2=4800[K_2=0]$ in the lattice elastic-energy term. The quantity $\overline{\delta }$ is related to the linearized crystal volume as $\overline{\delta }\propto {\left[V\left(T\right)/V_0\right]}^{1/3}-1$ where $V_0$ is the original volume. Insets show $\delta$ dependence of the lattice elastic energy per octahedron, which shows an anharmonic [a harmonic] behavior for (a) [(b)] with $K_2=0[K_2=4800]$.

Figure 3

(a) [(b)] Calculated thermal averages of the normalized bond-length difference $\overline{\delta }$ as functions of temperature for several values of $|J^{\mathrm{FM}}|/J^{\mathrm{AF}}$ (≤$0.9)$ [(≥$1.1\left)\right]$. A positive (negative) value of $\overline{\delta }$ indicates expansion (contraction) of the crystal from its original volume. Upturns after the magnetic transition upon cooling indicates that the NTE is triggered by the magnetic transitions as seen in the cases of $|J^{\mathrm{FM}}|/J^{\mathrm{AF}}=0.8$ and 0.9. The magnetic transition temperatures are indicated by inverted triangles. (c) Calculated $\overline{\delta }$ at ${\mathrm{k}}_{\mathrm{B}}T=0.01J^{\mathrm{AF}}$ as a function of $|J^{\mathrm{FM}}|/J^{\mathrm{AF}}$, which shows that the ${\mathrm{\Gamma }}^{5g}$-type antiferromagnetic order does not necessarily host the magnetism-driven NTE, but it occurs only in the limited area of the ${\mathrm{\Gamma }}^{5g}$-type antiferromagnetic phase. The NTE occurs in the range $0.7\lesssim |J^{\mathrm{FM}}|/J^{\mathrm{AF}}\lesssim 1$, whereas the ${\mathrm{\Gamma }}^{5g}$-type order takes place in $|J^{\mathrm{FM}}|/J^{\mathrm{AF}}<1$.

Figure 4

Phase diagram of the spin model in Eq. (4) and color map of the averaged difference of normalized bond length $\overline{\delta }$ in plane of temperature $k_{\mathrm{B}}T/J^{\mathrm{AF}}$ and the ratio $|J^{\mathrm{FM}}|/J^{\mathrm{AF}}$, which indicates the occurrence of NTE in the limited area of the ${\mathrm{\Gamma }}^{5g}$-type antiferromagnetic phase with $0.7\lesssim |J^{\mathrm{FM}}|/J^{\mathrm{AF}}\lesssim 1$.

Figure 5

Crystal structures possibly hosting the NTE phenomenon triggered by the antiferromagnetic order, in which keen competition between antiferromagnetic and ferromagnetic contributions to the nearest-neighbor exchange interactions is realized.