Opposite Thermal Expansion in Isostructural Noncollinear Antiferromagnetic Compounds of Mn₃A (A = Ge and Sn)

By virtue of the inherent anharmonic lattice dynamics, the volume of most solids expands upon heating. It is unconventional to find solids with negative thermal expansion (NTE), i.e., solids in which the distances between atoms decrease with increasing temperature.^1–3 NTE materials have crucial technological applications in the control of thermal expansion. In 1996, isotropic NTE was discovered in ZrW₂O₈.¹ In the past two decades, NTE has been found in various types of materials, such as oxides,^4–7 alloys,^8–13 nitrides,^14,15 fluorides,^16–18 and cyanides.^19–22 In essence, the thermal expansion of a solid is affected by a complex interplay between the electrons, phonons, and lattice.² The NTE phenomenon originates from diverse factors such as an open-framework,^17,19 the size effect,²³ charge transfer,^5,6 magnetism,^12–14,24 ferroelectricity,⁷ and superconductivity.^25,26 Obviously, most of the NTE solids are inorganics, which have relatively strong chemical bonds.

It is known that solids with weak bonds exhibit a strong positive thermal expansion (PTE).²⁷ For example, due to their weak metallic bonds, metals based compounds generally have a strong PTE, e.g., Al $\left({\alpha }_1=23.1\times {10}^{-6}/\text{K}\right)$, Cu $({\alpha }_1=17.7\times {10}^{-6}/\text{K})$, and Fe $({\alpha }_1=11.8\times {10}^{-6}/\text{K})$.²⁸ To meet the requirement for a low coefficient of thermal expansion (CTE), metal matrix composites (MMCs) are often fabricated by adding fillers with low or negative thermal expansion,²⁹ such as Cu/ZrW₂O₈ composites,³⁰ Al(Si)/Diamond,³¹ Al/stainless-invar,³² and Al/SiC.³³ It is well-known that metals based compounds are normally benefitted by advantages such as high mechanical strength and good electrical and thermal conductivities. Therefore, it is imperative that NTE metals based compounds be found. For example, the Invar alloy Fe_0.65Ni_0.35, which was found in 1897, has been a popular topic of fundamental studies and practical applications over the last century. Because of this discovery, Swiss scientist C. É. Guillaume won a Nobel Prize in Physics in 1920.³⁴ However, rare NTE metals based compounds have been found, such as Invar alloys,⁸ R₂Fe₁₇ (R = rare earth elements),⁹ (Hf,Nb)Fe₂,¹⁰ MnCoGe-based compounds,¹¹ La(Fe,Si,Co)₁₃,¹² and Tb-(Co,Fe)₂.¹³ Note that all of these NTE metals based compounds are ferromagnetic.

Here, we report an intriguing NTE in the noncollinear antiferromagnetic intermetallic compounds of Mn₃Ge. For a better understanding on the NTE mechanism of Mn₃Ge, the isostructural compound of Mn₃Sn showing PTE has also been studied for comparison. Their complicated triangular antiferromagnetic (AFM) structure and NTE mechanism are revealed via a combined analysis of the temperature dependence of neutron powder diffraction (NPD), synchrotron X-ray diffraction (SXRD) and macroscopic magnetic measurements. The direct link between the NTE behavior and the magnetic structure, which is correlated to the magnetovolume effect (MVE), is demonstrated. Particularly, the present NTE Mn₃Ge exhibits excellent mechanical properties and good thermal and electron conductivity behaviors.

A series of single-phase ${\text{Mn}}_3\left({\text{Ge}}_{1-x}{\text{Sn}}_x\right)$ $(x=0.0,0.2,0.4,0.6,0.8,\text{ and }1.0)$ samples were prepared via arc melting. The crystal structure was identified via high-intensity SXRD (Figure S1a). To determine the precise magnetic and crystal structure of both Mn₃Ge and Mn₃Sn, the temperature dependence of NPD has been carried. Mn₃Ge remains in a single phase with a hexagonal structure (space group: P6₃/mmc) over the whole temperature range. Mn and Ge atoms occupy Wyckoff sites 6h $\left(x,\hspace{0.17em}2x,\hspace{0.17em}1/4\right)$ and $2\text{c}\left(1/3,2/3,1/4\right)$, respectively (inset of Figure 1). By comparing the NPD patterns at typical low and high temperatures (150 and 500 K, respectively), the magnetic structure makes only an additional intensity contribution to the nuclear peaks, and no additional peaks can be observed. Therefore, the propagation vector of the Mn₃Ge magnetic structure is $k=\left(0,0,0\right)$ (Figure S2). Its crystal structure consists of alternating layers of manganese triangles, stacked parallel to the c axis. Because of the geometrical frustration, three neighboring spins cannot be pairwise antialigned on a triangular lattice.³⁵ Therefore, only the triangular antiferromagnetic structures are compatible. Some of the possible magnetic structural models allowed by the symmetry of the P6₃/mmc structure and its subgroups are listed in Figure S3. Among these AFM spin configurations, the structural model of Figure S 3f gives the best refinement. Figure 1 presents the magnetic structure refinements of the NPD pattern for Mn₃Ge. The Mn moments of inverse triangular antiferromagnetic order are along the [110] direction, and the spins rotate 120° counterclockwise in the $ab$ plane (the inset of Figure 1). It needs to note that the magnetic structure of Mn₃Ge is similar to the previous studies.^36,37

Figure 1.

Magnetic structure of NTE Mn₃Ge, determined via neutron powder diffraction. The space group is P6₃/mmc. The observed (red circles), calculated (black line), and differential (gray line at bottom of figure) patterns are shown for the full-profile refinement of Mn₃Ge at $T=150\text{ K}$. The vertical ticks mark the calculated positions of nuclear and magnetic reflections. The inset shows the magnetic and crystal structures of Mn₃Ge.

Intriguingly, Mn₃Ge exhibit a strong NTE property. No structure phase transition occurs in the NTE temperature range. The average linear CTE of Mn₃Ge is ${\alpha }_1=-7.58\times {10}^{-6}{\text{K}}^{-1}$ $\left(297-374\text{ K}\right)$, which was determined via a macroscopic thermo-dilatometer measurement (Figure 2). However, the isostructural compound of Mn₃Sn shows an opposite PTE at the whole temperature range. Two inflection points are associated with the magnetic transitions. The temperature dependence of the unit cell volume, which was determined via NPD, can also confirm the reliability of NTE (Figure 4b). As a comparison, the present Mn₃Ge exhibits an NTE magnitude similar to that of other typical NTE inorganics, such as ScF₃ (${\alpha }_1=-7.47\times {10}^{-6}{\text{K}}^{-1}$, $150-425\hspace{0.17em}\text{K}$)^16,38 and ZrW₂O₈ (${\alpha }_1=-9.1\times {10}^{-6}{\text{K}}^{-1}$, $10-300\hspace{0.33em}\text{K}$).¹ Note that most available NTE alloys are ferromagnetic (FM), whereas the present Mn₃Ge exhibits an antiferromagnetic structure. Furthermore, even a high Neel temperature $\left(T_{\text{N}}\right)$ is found in the present Mn₃Ge $(T_{\text{N}}=380\hspace{0.33em}\text{K})$ when compared with other antiferromagnetic intermetallic compounds, such as YMn_{2 $(T_{\text{N}}=100\hspace{0.33em}\text{K})$},³⁹ Gd₅Ge_{4 $(T_{\text{N}}=125\hspace{0.33em}\text{K})$},⁴⁰ and CeFe_{2 $\left(T_{\text{N}}\sim 90\hspace{0.33em}\text{K}\right)$}.⁴¹ Because of the magnetostriction effect, the dimensions of FM alloys can be influenced by the external magnetic field. However, AFM compounds can avoid the negative effect of the external magnetic field on the precise dimension.

Figure 2.

Opposite thermal expansion in the isostructural antiferromagnetic intermetallics of Mn₃Ge and Mn₃Sn determined by a dilatometer $\left(\text{Δ}l/l_0\right)$. NTE is observed in Mn₃Ge, whereas PTE in the isostructural Mn₃Sn. $T_{\text{S}}$ and $T_{\text{N}}$ are the tempratures of magnetic transition.

Figure 4.

NTE mechanism of Mn₃Ge. Temperature dependence of (a) the Mn moments of Mn₃Ge and Mn₃Sn, (b) the relative change in unit cell volume of Mn₃Ge and Mn₃Sn, (c) the Mn moment and lattice parameter $a$ of Mn₃Ge, and (d) $dM/dT$ and $\text{d}\left(\text{Δ}l/l\right)/\text{d}T$ of Mn₃Ge and Mn₃Sn.

The macroscopic magnetic behavior of Mn₃Ge was studied by measuring the temperature dependence of magnetization (zero-field-cooling (ZFC) and field-cooling (FC) modes) (Figure S5a). An obvious peak of the ZFC curve (100 Oe) can be seen, which reveals the feature of the AFM transition. The Neel temperature $T_{\text{N}}$ is determined to be 370 K. With increasing external magnetic field, the magnetization of the residual FM component is increased, which overwhelms the signal of the AFM structure. Note that the very small residual FM component is caused by a small distortion of the hexagonal structure.⁴² Furthermore, the magnetic transition temperature is hardly changed by the external magnetic field, which verifies the stability of the triangular antiferromagnetic structure. Isothermal magnetization curves of Mn₃Ge measured at various temperatures are shown in Figure S6. The dominating AFM nature is clearly indicated by the linear and unsaturated magnetization behavior at higher fields. In Figure S5b, there are two obvious magnetic transitions at 255 K and above 400 K for Mn₃Sn, which are consistent with the dilatometer result. In addition, the magnetic transition temperature of Mn₃Ge is close to the disappearing temperatures of NTE (Figure 2 and Figure S5), indicating that the anomalous thermal expansion phenomenon of Mn₃A (A = Ge, Sn) is entangled with the magnetic behavior.

In the purpose of explaining opposite thermal expansion in the isostructural noncollinear antiferromagnetic intermetallics of Mn₃A (A = Ge and Sn), the microscopic thermal expansion information was extracted from the results of magnetic structure refinement of high-intensity NPD (Figure S4). The temperature dependence of the lattice parameters extracted from the NPD data is shown in Figure S7. Obviously, the NTE of Mn₃Ge is dominated by the shrinking of the $a\left(b\right)$ axis; nevertheless, the $c$ axis linearly expands. However, in Mn₃Sn both $a\left(b\right)$ and $c$ axes linearly expand. It means that such opposite thermal expansion is correlated to the $ab$ plane.

To study the detailed behavior for the shrinkage of the $ab$ plane, the bond distances between different magnetic atoms (Mn) were extracted from variable temperature NPD data. Figure 3a shows a $4\times 4\times 1$ supercell of the $ab$ plane for Mn₃A (A = Ge and Sn), ${\text{d}}_1$ and ${\text{d}}_2$ sign the bond distances of Mn–Mn (I) and Mn–Mn (II), respectively. As shown in Figure 3b,c, ${\text{d}}_1$ always increases in the whole temperature range for both Mn₃Ge and Mn₃Sn. However, ${\text{d}}_2$ takes place to shrink at different level below magnetic transition temperature. The total change of ${\text{d}}_1$ and ${\text{d}}_2\left(\text{Δ}{\text{d}}_1+\text{Δ}{\text{d}}_2\right)$ shows the same behavior with that of the $a$ axis, which exhibits NTE for Mn₃Ge but PTE for Mn₃Sn below magnetic transition temperatures (Figure 3d). Therefore, the NTE of Mn₃Ge is due to the decrease in the bond distance ${\text{d}}_2$, which is caused by MVE.

Figure 3.

Temperature dependence of bond distances between different sites of magnetic Mn atoms for NTE Mn₃Ge and PTE Mn₃Sn. (a) A layer of atoms in ab plane of $4\times 4\times 1$ supercell for Mn₃A (A = Ge and Sn), ${\text{d}}_1$ and ${\text{d}}_2$ mean the bond distances of Mn–Mn (I) and Mn–Mn (II) in the ab plane, respectively. Temperature dependence of ${\text{d}}_1$ and ${\text{d}}_2$ for (b) Mn₃Ge, and (c) Mn₃Sn. (d) Temperature dependence of relative change of $\left(\text{Δ}{\text{d}}_1+\text{Δ}{\text{d}}_2\right)$ Δd₁+Δd₂ for Mn₃A (A = Ge and Sn).

The more detailed magnetic and crystal structure information was extracted from NPD data to reveal the detailed MVE of Mn₃Ge and Mn₃Sn. As shown in Figure 4a, it is interesting to note that the Mn moments of Mn₃Ge and Mn₃Sn are nearly identical at 275 K, but nonlinearly decrease and vanish at $T_{\text{N}}=380\text{ K}$ for Mn₃Ge and $T_{\text{N}}=420\text{ K}$ for Mn₃Sn. It seems that the AFM order can be more easily disturbed in Mn₃Ge than Mn₃Sn. Figure 4c shows the strong coupling role between magnetism and lattice in the NTE Mn₃Ge. As the Mn magnetic moment drops slowly at low temperatures (below 275 K), the lattice parameter $a$ expands, i.e., PTE occurs; however, when the magnetic moment drops quickly (275–375 K), $a$ begins to shrinks, i.e., NTE happens. To further study the relation between magnetism and thermal expansion, the trends of $\text{d}M/\text{d}T$ and $\text{d}\left(\text{Δ}l/l\right)/\text{d}T$ are calculated for both Mn₃Ge and Mn₃Sn (Figure 4d). Clearly, a similar tendency can be observed in both magnetic moment and macroscopical thermal expansion of Mn₃A (A = Ge and Sn) as a function of temperature. With increasing temperature, the magnetic moment decreases rapidly or slowly, the property of thermal expansion changes accordingly. The rate of magnetic transition dominates the thermal expansion of Mn₃A (A = Ge and Sn), which is directly associated with the Mn moments lying the $ab$ plane. The rate of magnetic transition $\left(\text{d}M/\text{d}T\right)$ in NTE Mn₃Ge is stronger than that of PTE Mn₃Sn, which means a stronger contribution of spontaneous magnetostriction to thermal contraction. Therefore, the bonds of Mn–Mn $\left({\text{d}}_2\right)$ in Mn₃Ge shrink more obviously than Mn₃Sn, which exceeds the expansion of the bond distance ${\text{d}}_1$. The NTE occurs in Mn₃Ge (Figure 3).

In other words, the Wigner–Seitz volume of Mn₃Sn is expanded due to the existence of bigger Sn atoms, resulting in the increase of Mn–Mn bond length (Figure S1). The longer of Mn–Mn distance makes the antiferromagnetic exchange interaction stronger in Mn₃Sn, which is also evidenced by the higher $T_{\text{N}}=420\text{ K}$ in Mn₃Sn compared with $T_{\text{N}}=380\text{ K}$ in Mn₃Ge. Theoretically, when the exchange interaction is enhanced, the compound needs more energy to break magnetic ordering. Therefore, the spins of Mn₃Sn depart from parallel alignment more difficultly than Mn₃Ge with increasing temperature, i.e., the rate of magnetic transition $\left(\text{d}M/\text{d}T\right)$ in Mn₃Sn is slower than that of Mn₃Ge (Figure 4d). It is known that magnetic configuration with aligned spins has a larger volume than that with disordered spins,²⁴ which means the magnetic transition brings a negative contribution to thermal expansion. Because of the faster magnetic transition, the contribution to lattice change from magnetic order overwhelms that from lattice thermal vibration. As a result, the overall thermal expansion of Mn₃Ge becomes negative (Figure 4b). However, the spontaneous magnetostriction of Mn₃Sn is relatively weak, which cannot overcome positive contribution to thermal expansion from lattice vibration. PTE happens at the whole temperature range. The change of magnetic moment control the NTE property should be the common feature for most magnetic NTE compounds, such as Mn₃AN.^43–45

To quantify the relationship between magnetic contribution and thermal expansion, spontaneous magnetostriction $\left({\omega }_{\text{s}}\right)$ is used to quantitatively describe the contribution of MVE to anomalous thermal expansion.^46,47 The value of ${\omega }_{\text{s}}$ is calculated by ${\omega }_{\text{s}}={\omega }_{\text{exp}}-{\omega }_{\text{nm}}$, in which ${\omega }_{\text{exp}}$ is the experimental linear thermal expansion measured by thermos-dilatometer, and ${\omega }_{\text{nm}}$ is the linear thermal expansion of a nonmagnetic reference.⁴⁸ As shown in Figure S8, the NTE property of Mn₃Ge can be regarded as a combined result of ${\omega }_{\text{s}}$ and lattice vibration $\left({\omega }_{\text{nm}}\right)$. Here, temperature dependence of both the square of Mn moment $\left(M^2\right)$ and ${\omega }_{\text{s}}$ is performed for the detailed analysis of NTE mechanism of Mn₃Ge. As depicted in Figure 5a,b, it is obvious to reveal a strong linear correlation between $M^2$ and ${\omega }_{\text{s}}$, which can be quantitatively ascribed by the equation of

$${\omega }_{\text{s}}\left(T\right)=kCM{\left(T\right)}^2$$ (1)

Figure 5.

(a) Temperature dependence of the square of Mn moment $\left(M^2\right)$ and spontaneous magnetostriction $\left({\omega }_{\text{s}}\right)$ in Mn₃Ge. (b) Quantitative relationship between $M^2$ and ${\omega }_{\text{s}}$.

where $k$ and $\text{C}$ are the compressibility and the magnetovolume coupling constant, and $M\left(T\right)$ is the amplitudes of Mn magnetic moment.^49,50 It means that the magnetic contribution and the thermal expansion have a quantitative relationship, and thus thermal expansion can be tuned by changing the magnetic ordering.

From the practical application viewpoint, better performances in terms of high mechanical strength and good thermal and electron conductivity properties are demanded for NTE materials. The engineering stress–strain curves of Mn₃Ge ingots are shown in Figure S9. The maximum compressive strength can be as large as 204 MPa, together with a total elongation of 5.4%. As a comparison, the compressive strength of the Nd–Fe–B-based permanent magnets and that of the La(Fe,Si)₁₃-based magnetic refrigeration materials are 112 MPa, and 120 MPa, respectively.^51,52 It is well-known that intermetallic compounds are generally very fragile, and are thus unsuitable for compressive or tensile tests. In general, to improve the mechanical property of NTE intermetallic compounds, binder is often introduced to bond powders of alloys. For example, the as-prepared MnCoGe-based alloys are brittle and spontaneously fragment into powders. Their compressive strength increases to 70.4 MPa after using epoxy-binder as a bonder.¹¹ The temperature dependence of the metallic property of Mn₃Ge is also depicted in Figure S10. At room temperature, the thermal conductivity is $6.8\hspace{0.33em}\text{W}·{(\text{m}·\text{K})}^{-1}$, which is much larger than that of most NTE inorganic compounds, such as the prototype NTE oxide of ZrW₂O₈ $\left(0.5\hspace{0.33em}\text{W}·{(\text{m}·\text{K})}^{-1}\right)$ Furthermore, the electrical conductivity of the present NTE Mn₃Ge is $1-2\times {10}^3{(\text{Ω}·\text{cm})}^{-1}$. As a comparison, most NTE inorganic solids are insulating. Their electrical conductivity is linearly reduced upon heating, which reveals the characteristic of metallic conductivity.

The present Mn₃Ge exhibit a new and simple structure type for the NTE property, which yields a new class of antiferromagnetic NTE compounds. The control of thermal expansion can be achieved in Mn₃Ge-based compounds via chemical modification similar to that often utilized for NTE materials, such as ScF₃^16,17,38 and antiperovskites.^3,14,15 For example, the solid solutions of ${\text{Mn}}_3\left({\text{Ge}}_{1-x}{\text{Sn}}_x\right)$ have been investigated (Figure S1). Thermal expansion of ${\text{Mn}}_3\left({\text{Ge}}_{1-x}{\text{Sn}}_x\right)$ gradually transforms from NTE to PTE with increasing Sn concentration.

In summary, a new and simple NTE structure type has been found in antiferromagnetic compounds of Mn₃Ge. The unusual inverse triangular antiferromagnetic structure of Mn₃Ge was confirmed via variable temperature NPD and macroscopic magnetic measurements. The direct experimental evidence reveals the contribution of the magnetovolume effect to the anomalous thermal expansion of Mn₃Ge, compared with the isostructural PTE Mn₃Sn. The NTE property of Mn₃Ge originates from the quick decrease of Mn moments which causes the shrink of the bond distances of Mn–Mn $\left({\text{d}}_2\right)$. The present NTE Mn₃Ge exhibits the antiferromagnetic property which can thus avoid the influence of the external magnetic field on the dimensions. Furthermore, the high mechanical properties and good metallic property of the present NTE Mn₃Ge provide a greater possibility for better industrial applications in the future.