Structural coupling and magnetic tuning in Mn_2−xCo_xP magnetocalorics for thermomagnetic power generation

By several estimates, over 60% of all energy produced is rejected as waste heat.^1–4 Low-grade waste heat (below 500 K) is usually difficult to recover because other inefficiencies besides the Carnot limit (e.g., friction) result in heat engines performing very poorly when the temperature difference is small. However, low-grade waste heat constitutes more than 50% of rejected heat, more than 10⁴ TWh in the United States alone. If even a small percentage of this energy could be harnessed using economically and environmentally friendly technologies, this would represent a tremendous new source of green energy. To this end, thermomagnetic power generation has emerged as a promising avenue for the conversion of waste heat into mechanical or electrical power, including for the conversion of very low-grade waste heat, where other technologies, such as thermoelectrics, are ineffective.⁵ 

The concept of thermomagnetic generation is closely related to the magnetocaloric effect, which provides a path to energy-efficient and environmentally friendly refrigeration. In magnetocaloric cooling, a magnetized ferromagnetic material near its Curie temperature is demagnetized adiabatically, leading to an increase in the magnetic entropy (spins are randomized) that is countered by a decrease in the lattice entropy and, therefore, the temperature.⁶ A thermodynamic cycle for cooling can be implemented by alternating adiabatic and isothermal magnetization and demagnetization, analogous to conventional gas compression cooling.

Conversely, using a temperature difference to traverse the ferromagnetic ordering temperature of a material results in a change in magnetization (ΔM) that can be used to produce useful work.⁷ This reverse magnetocaloric effect or thermomagnetic effect is the principle behind thermomagnetic power generators, proposed since the 19th century by Tesla^8,9 and Edison.^10,11 Interest in these has recently increased as a variety of promising prototype devices have been built.^12–17 These devices function by cycling the temperature of a magnetic material above and below its transition temperature, leading to changes in magnetization, which are converted into mechanical or electrical energy achieving realistic device efficiencies of around 1%–3%.⁴ Even with these low efficiencies, the ability to recapture abundant and currently un-utilized waste heat could result in an enormous new reserve of useful energy. Numerous contributions have described design strategies aimed at higher efficiencies.^{3,4,15,16,18–20}

It is advantageous to design materials whose transition temperatures can be controlled around and above room temperature through compositional tuning. While the common figure of merit used for the initial evaluation of a magnetocaloric is ΔS_M(T, H), the magnetic entropy change experienced by the material upon isothermal magnetization to a certain magnetic field H and at a certain temperature T, for thermomagnetic generators, the reversible change in magnetization ΔM(ΔT, H) may be a better metric. As ΔT and H are determined by the specific application (i.e., the temperatures of the hot and cold reservoirs available, and the magnetic field achievable), in this contribution, we characterize ΔM in a range of temperatures, temperature spans, and magnetic fields in order to provide a map of the conditions under which the reported materials may be useful. We also note that ΔS_M and ΔM are in fact closely related, as each of these parameters relies on having a sharp magnetic transition.

For magnetocaloric applications, attention has been paid to materials with first-order magnetostructural phase transitions due to the large (or “giant”) magnetocaloric effects possible at these phase transitions, as seen in Gd₅(Si,Ge)₄,²¹ (Mn,Fe)₂(P,Si,As,Ge),^22–26 La(Fe,Si)₁₃,^27,28 and a few other systems. This enhanced magnetocaloric effect arises from magnetostructural coupling, leading to an easily switchable magnetic state and a sharp magnetic transition. However, a first-order transition involves thermal hysteresis and a volume change during the transition, which are especially detrimental to the performance of a thermomagnetic generator. Interestingly, magnetostructural coupling can lead to an enhanced magnetocaloric effect even in materials without first-order magnetostructural transitions,^29–31 which is promising for high-performance materials for thermomagnetic power generation.

We initially identified MnCoP as a promising material to show strong magnetocaloric effects on the basis of a high-throughput computational search for ferromagnets with strong magnetostructural coupling.³⁰ This search introduced a simple density functional theory (DFT)-based proxy called the magnetic deformation Σ_M, which is a quantification of the difference of the DFT-optimized unit cells with and without the inclusion of spin polarization in the calculation. This parameter correlates well with experimentally determined values of peak ΔS_M. Σ_M for MnCoP was calculated to be large (3.03%), suggesting the potential for a large magnetocaloric effect near its magnetic Curie temperature of 578 K. Indeed, we measured an experimental peak ΔS_M to be −6.0 J kg⁻¹ K⁻¹ for an applied field of 5 T, which is large for a material without a first-order magnetostructural transition.

Here, we investigate the magnetostructural coupling in stoichiometric MnCoP by monitoring the structure through its magnetic transition temperature using variable-temperature high-resolution synchrotron powder x-ray diffraction. We find pronounced effects on the lattice at the magnetic ordering temperature, helping to explain the notable magnetocaloric effect. In order to establish the application of MnCoP as a material for thermomagnetic power generation, we characterize the solid solution Mn_2−xCo_xP (0.6 < x < 1.4) with respect to the structural and magnetic properties, finding good agreement with previous observations by Fruchart et al.^32,33 Mn_2−xCo_xP with x < 1.6 forms in the orthorhombic Co₂P structure, while x > 1.6 forms in the hexagonal Mn₂P (Fe₂P-type) structure. The maximum saturation magnetization and Curie temperature are found in the stoichiometric MnCoP compound, with both quantities falling off as the composition is varied. For 0.68 < x < 0.88, a metamagnetic transition exists at decreasing temperature for increasing x. The composition Mn_1.3Co_0.7P was found to have a helical magnetic order at low temperatures.³⁴ Sun et al. explored the magnetocaloric response of this composition finding ΔS_M = −2.3 J kg⁻¹ K⁻¹ for the second-order ferro-to-paramagnetic transition near room temperature.³⁵ 

We find that this system allows for the tuning of the Curie temperature between 334 K and 578 K while maintaining substantial ΔS_M and ΔM values, which makes this system a good candidate for these applications. Density functional theory calculations across the solid solution, along with the observed structural and magnetic properties across the series, explain how the magnetic and magnetostructural behavior of the system arises from the behavior of the two transition metal sites in the MnCoP structure.

Polycrystalline samples of Mn_2−xCo_xP (x = 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, and 1.4) were prepared from stoichiometric mixtures of Mn₃P₂, Co, and red phosphorus via assisted-microwave preparation,³⁶ followed by a furnace anneal. The samples were characterized using synchrotron x-ray diffraction (beamline 11-BM, Advanced Photon Source), x-ray fluorescence (XRF) compositional analysis, and superconducting quantum interference device (SQUID) magnetometry. Details on sample preparation and characterization may be found in the supplementary material.

In addition, the magnetic and magnetostructural coupling behavior of the solid solution Mn_2−xCo_xP was investigated using density functional theory (DFT) as implemented in the Vienna ab initio Simulation Package (VASP) code^38–40 using a recently described method,⁴¹ which involves calculating the magnetic deformation Σ_M on enumerated ordered supercells of the random alloy. Σ_M is a computational proxy that correlates with the strength of magnetostructural coupling, and has been shown to correlate as well with experimental peak ΔS_M.²⁹ Supercells up to size 24 atoms (two times the primitive TiNiSi-structure cell) were enumerated using the clusters approach to statistical mechanics (CASM) code,^42–44 by either substituting Mn atoms onto the Co site in MnCoP (x < 1) or substituting Co atoms onto the Mn site (x > 1). More details of the calculations may be found in the supplementary material.

Rietveld refinement of synchrotron x-ray diffraction data for stoichiometric MnCoP is shown in Fig. 1. At room temperature, MnCoP adopts the orthorhombic Pnma space group (TiNiSi-type) shown in the inset. This structure is isomorphous with Co₂P, while Mn₂P forms in the hexagonal Fe₂P-type structure.³³ In MnCoP, the cobalt–phosphorus network forms layers of corrugated edge-sharing six-membered rings in the bc plane with close bond distances (<2.24 Å). These rings are distorted into the half-chair shape described by Landrum, Hoffmann, and co-workers.⁴⁵ Each cobalt is tetrahedrally coordinated by phosphorus, while manganese is pyramidally coordinated with bond distances to phosphorus of >2.44 Å. The manganese forms zig-zag chains along the b direction with a Mn–Mn nearest neighbor distance of 2.89 Å. Each atom occupies a 4c Wyckoff position.

The temperature evolution of the lattice parameters of MnCoP (Fig. 2) determined by Rietveld refinement of synchrotron diffraction patterns (Fig. S1) at temperatures between 628 K and 538 K reveals highly anisotropic thermal expansion as well as clear anomalies at the magnetic transition (T_C = 578 K). Above T_C, the coefficient of thermal expansion in each direction is linear across the dataset (r is >0.995), and is an order of magnitude larger in the b direction (α_b = 151 ppm K⁻¹) than in the a or c direction (α_a = 76 ppm K⁻¹ and α_c = 84 ppm K⁻¹).

Upon cooling through the magnetic transition, the evolution of the a and b lattice parameters deviates from this linear trend at T_C, indicating the presence of magnetostructural coupling modifying the nonmagnetic (phonon-driven) thermal expansion of the material, a phenomenon known as spontaneous magnetostriction. Interestingly, the influence of magnetic order is to expand the a lattice parameter (positive magnetostriction) while contracting the b lattice parameter (negative magnetostriction) and leaving the c lattice parameter virtually unchanged. Since the negative effect in the b direction is much larger than the positive effect in the a direction, the material shows overall a sizable negative spontaneous volume magnetostriction. This is shown in Fig. 2(c), quantified as ω_V(T) = (V(T) −V_P(T))/V_P(T), where V(T) represents the experimentally observed volume and V_P(T) is the volume extrapolated from the paramagnetic regime (i.e., the dotted lines in Fig. 2).

Anisotropic coupling between magnetism and structure is also observed in several other promising magnetocalorics, including the Mn,Fe(P,Si) family^46,47 and MnB.²⁹ In these systems, the anisotropic coupling arises from competition between magnetic and structural degrees of freedom in the structure. After discussing the structural evolution of this system with compositional tuning, we will return to this point in order to provide an explanation for the anisotropic coupling in MnCoP.

Wavelength dispersive x-ray fluorescence spectroscopy (XRF) confirms that the manganese to cobalt ratio in all samples in the series Mn_2−xCo_xP (0.6 < x < 1.4) is the same as the nominal composition, with the exception of x = 1.4, which has a composition closer to Mn_0.5Co_1.5P. Rietveld refinement of x-ray diffraction determines that all compositions within the sampled range form in the same orthorhombic Pnma structure as stoichiometric MnCoP, in agreement with previous studies.^32,33 Some samples have minor amounts of MnO, which is antiferromagnetic below T_N = 118 K, as a secondary phase. Properties determined from Rietveld refinement may be found in Table I.

Lattice parameters refined from the diffraction data are shown as a function of composition in Fig. 3. The unit cell volume is found to linearly decrease as the Co content is increased, as expected by the smaller atomic radius of Co compared to Mn. However, as with the magnetostriction properties, the development of the individual lattice parameters is highly anisotropic. On the Mn-rich side, where Mn atoms are substituted onto the Co site, a large change in the b lattice parameter with composition is seen, while there is almost no change in a. On the Co-rich side, the a parameter is highly dependent on the composition, while b barely changes or even slightly increases despite the addition of a smaller atom. Meanwhile, the c lattice parameter uniformly decreases across the full series.

For x < 1, manganese atoms are being introduced into the CoP bonding network of corrugated hexagonal rings in the bc plane. As a result, Mn atoms cause the lattice to expand in the b and c directions [expansions of 3.75%(mol Mn)⁻¹ and 1.75%(mol Mn)⁻¹, respectively]. The a parameter, by comparison, experiences a much smaller expansion of 0.75%(mol Mn)⁻¹. In contrast, in the cobalt-rich compounds (x > 1), the a parameter decreases much faster than the b or c parameter as cobalt is substituted onto the Mn site. Therefore, we can conclude that the intra-layer spacing is controlled primarily by the Mn site, while the in-layer spacing is controlled primarily by the Co site.

This finding allows us to interpret the observed anisotropic magnetostriction in stoichiometric MnCoP. Below T_C, we expect the manganese to occupy a larger volume due to the magnetovolume effect, which is based on the theory of itinerant magnetism and therefore occurs in metallic magnets with (at least partly) itinerant character of the magnetism.^48–50 The magnetovolume effect may be thought of as arising from the localization of dispersive electrons into magnetic moments as the temperature is cooled. The expanded volume of the Mn site causes the observed positive magnetostriction in the a direction, but has little effect on the in-layer b and c directions. At the same time, there is negative magnetostriction in the b direction, which indicates the presence of an exchange-volume coupling⁵¹ within the Mn–Mn chains that run in the b direction (i.e., the Mn–Mn atoms move toward each other in order to optimize the magnetic exchange energy, rather than packing considerations based on the ion size).

Magnetization vs temperature measurements on this series, shown in Fig. 4(a), reveal that all of the samples are ferromagnetic at room temperature except for Mn_1.4Co_0.6P, which is antiferromagnetic with a Neél temperature of 213 K. The highest T_C is found for MnCoP (578 K), and compositional tuning in either direction allows for the Curie temperature to be brought down to nearly room temperature. In addition, the composition Mn_1.3Co_0.7P has a metamagnetic transition at 205 K, where it becomes antiferromagnetic below this temperature, as seen by the sudden decrease in magnetization with decreasing temperature. This behavior has been seen previously,³² and this transition has been investigated for its inverse magnetocaloric effect.⁵² 

Magnetization vs applied field measurements [Fig. 4(b)] reveal that stoichiometric MnCoP possesses the largest total moment at room temperature, and saturation magnetization decreases as the composition is changed in either direction. No visible hysteresis is observed for these samples, indicating soft ferromagnetic behavior. Values for T_C and M_sat may be found in Table II.

ΔS_M is calculated for the ferro-to-paramagnetic transition. ΔS_M for a magnetic field change of 0 T–5 T is given as a function of temperature for each composition. The ΔS_M peak is narrow for MnCoP and broadens as the composition is varied. Transition temperatures are widely tunable while maintaining significant ΔS_M by varying composition.

ΔM may be considered a more important metric for thermomagnetic generators than ΔS_M. Figures 5(a) and 5(c) show the maximum ΔM as a function of the temperature difference at different fixed fields, and 5(b) and 5(d) show maps of ΔM for ΔT = 50 K as a function of temperature and field for two compositions. Data for the other compositions may be found in the supplementary material, Fig. S4. MnCoP compares favorably to other non-rare-earth containing second-order intermetallics, which range from ΔM = 12–40 A m² kg⁻¹ for ΔT = 30 K.⁵³ 

As discussed in the Introduction, the temperature interval over which we calculate ΔM is determined by the application. The operation temperature of a device, as well as the temperature difference between hot and cold reservoirs, must be matched to the material. ΔM increases as the temperature difference is increased, but with diminishing returns for materials with a sharp magnetic transition, such as MnCoP. The transition in Mn_0.6Co_1.4P is much broader [as seen in the ΔS_M curve in Fig. 4(c)], and so ΔT must be larger to encompass the minimum and maximum moment states at a given magnetic field.

While ΔS_M increases with increasing field, this is not the case for ΔM in a material with a continuous magnetic transition. When the ΔT is low (i.e., in applications for thermomagnetic generators), a larger ΔM may be accessed at a smaller fixed field, where the magnetic transition is sharper. In practical applications, it is advantageous for the necessary applied field to be below 1 T because this is readily accessible with permanent magnets.

The magnetic deformation Σ_M was previously calculated for stoichiometric MnCoP,³⁰ suggesting the presence of strong magnetostructural coupling. Here, we extend these calculations to study the magnetostructural coupling across the solid solution Mn_2−xCo_xP using calculations on enumerated supercells (Fig. 6), as has been recently performed for MnCoGe-based solid solutions.⁴¹ For this study, all enumerated supercells were based on the orthorhombic TiNiSi-type structure of MnCoP. Therefore, the calculation for the very Mn-rich side of the phase diagram, where experimentally a Fe₂P-type structure is observed, is not expected to be physical.

Several cells are found to have negative formation energies with respect to Mn₂P, MnCoP, and Co₂P [Fig. 6(c)], with most other cells within about 20 meV atom⁻¹ of the convex hull, supporting the experimentally observed formation of a homogeneous solid solution. The maximum Σ_M is found for MnCoP, with the predicted magnetostructural coupling strength smoothly decreasing as the composition is changed in either direction. This calculation corresponds well with the observed peak ΔS_M values [Fig. 6(a)], while the maximum ΔM value for a field of 0.5 T and ΔT of 50 K actually falls slightly on the Mn-rich side of the solid solution, at Mn_1.1Co_0.9P. However, the difference in ΔM between Mn_1.1Co_0.9P and MnCoP is very small, and therefore, Σ_M performs as a reasonable predictor for ΔM.

At the Co-rich side of the phase diagram, Σ_M falls to 0% for Co₂P as the DFT magnetic moment falls to nearly zero, consistent with the experimental observation that Co₂P is paramagnetic and does not show ordered ferromagnetism.⁵⁴ On the other hand, at x = 0, a hypothetical ferromagnetic Mn₂P compound with the TiNiSi structure shows stable magnetic moments and a finite Σ_M of 1.5%.

Figure 7 shows the behavior of the local magnetic moments extracted from the enumerated DFT calculations. As expected, the behavior of the local Mn and Co moments depends greatly on the atom identity as well as the site on which it sits. Mn on the Mn site has a nearly constant moment of 2.7 μ_B across the full solid solution, while Co on the Co site shows only a small moment of between 0.1 μ_B and 0.3 μ_B. This large difference in local moment is expected based on the larger size of the Mn site and the presence of strong bonding between the Co site and P atoms, which competes with the formation of local moments. Indeed, when Co is substituted onto the Mn site, it holds an increased moment of up to 1.1 μ_B. However, the local moment of each Co atom on the Mn site decreases as more and more Co is added, eventually yielding the almost nonmagnetic Co₂P composition.

On the other side of the series, as Mn is added onto the Co site, it is found to hold a moment of about −0.4 μ_B, with a strong preference for the antiparallel orientation to the moments of the Mn site. Analogous behavior has been observed using neutron diffraction in the compound Mn_1+xSb, which has the same structure as MnCoP except with vacancies at the Co positions. At finite values of x, Mn interstitials sit on the vacant Co site, and hold moments antiparallel to the larger ferromagnetic Mn atoms on the main site.⁵⁵ These calculations explain why the experimentally observed magnetic moment decreases as the composition is varied from MnCoP in either direction. On the Co-rich side, the decrease is caused by the replacement of high-moment Mn atoms with lower moment Co atoms. On the Mn-rich side, conversely, low-moment Co atoms are replaced with higher-moment Mn atoms, which orient themselves counter to the ferromagnetic moment and therefore decrease the net moment.

MnCoP has previously been predicted to have strong magnetostructural coupling (and therefore large |ΔS_M|) and is, indeed, found to display characteristic discontinuities in the thermal expansion coinciding with the magnetic transition temperature T_C suggestive of such coupling. This is despite the absence of any first-order phase transition. Mn_2−xCo_xP forms a solid solution within the sampled range (0.6 ≤ x ≤ 1.4), allowing magnetic properties to be tuned by varying x. Substitution on the Co site controls the b and c lattice parameters, while substitution on the Mn site controls the a lattice parameter. This is consistent with the positive spontaneous magnetostriction in a observed when MnCoP is cooled through its magnetic transition. T_C, saturation magnetization, and ΔS_M are all largest for stoichiometric MnCoP, but the transition temperature is widely tunable down to nearly room temperature by varying composition. Changes in |ΔS_M| trace computed changes in the ensemble-averaged values of Σ_M for the solid solution, underpinning the utility of this proxy. We find ΔM to be large across a range of compositions for ΔT = 50 K, and increasing the temperature differential has a larger effect on the off-stoichiometric materials with broader transitions. The results point to the potential use of this material system for thermomagnetic power generation.

See the supplementary material for details of experimental methods and characterization techniques, crystal structure refinement information, x-ray fluorescence analysis of composition, thermal evolution of synchrotron x-ray diffraction, Rietveld fits across compositions, detailed magnetic measurement data, and examples of sample microstructure from scanning electron microscopy.

The research reported here was supported by the National Science Foundation (NSF) through Grant No. DMR-1710638 and partially by the Materials Research Science and Engineering Center (MRSEC) program under Grant No. DMR-1720256 (IRG-1). We gratefully acknowledge the use of the computing facilities of the Center for Scientific Computing at UC Santa Barbara, supported by Grant No. NSF CNS-1725797 and by the NSFMRSEC Program (Grant No. DMR-1720256). The use of the Advanced Photon Source at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. We thank Dr. Saul Lapidus for help with synchrotron data collection at beamline 11-BM.