Density functional theory study of energetics, local chemical environment and magnetic properties in a high-entropic MnNiSi0.2Ge0.2Sn0.2Al0.2Ga0.2 intermetallic magnet

Rare-earth-free magnetostructural MnNiSi-based solid solutions are considered as promising candidates for solid-state cooling applications. In this paper, we use density functional theory calculations to study the energetics, variations in atomic displacements and bond length, and magnetic properties of high-entropic, intermetallic MnNi-X (X = Si0.2Ge0.2Sn0.2Al0.2Ga0.2) magnet in both the low-symmetry Pnma and high-symmetry P63/mmc structures, where we confine the large configurational entropy to the non-magnetic X-site of the compound. Our calculations reveal that the high-entropic chemical substitution of Si0.2Ge0.2Sn0.2Al0.2Ga0.2 in the X-site carry fingerprints that favor a reduction in magnetostructural transition temperature with minimal impact of total magnetization. These results motivate a promising path of high-entropic X-site substitutions to tune the magnetostructural properties of MnNiSi-based solid solutions.


Introduction
Materials that undergo reversible magnetostructural phase transitions exhibit strong coupling between the crystal structure and magnetic ordering [1][2][3][4]. Such materials enable control of structural phase transformations via an external magnetic field. The promising capability of these materials to readily respond to multiple external stimuli provide additional knobs to simultaneously tailor the multitude of properties that must be met to be competitive with the current cooling and refrigeration technologies [5][6][7]. In these materials, the total isothermal entropy change (∆S) at the phase transition is due to the combined contributions from lattice and magnetic entropies [5]. According to the Landau theory of phase transitions, the ∆S at the magnetostructural transition is expected to decrease as the temperature difference between the magnetostructural transition and the magnetic transition increases [2]. As a result, there is interest in identifying optimal conditions (e.g. chemical substitutions, microstructure design via processing, hydrostatic pressure, and applied magnetic field) that will yield large total ∆S at the phase transition.
High-entropy materials have garnered significant interest in the last decade due to the vast chemical space and complex free energy landscape [8][9][10][11]. The combination of long-range structural order with short-range chemical order have shown to result in unique structural and functional properties not seen in traditional counterparts [12][13][14]. Large variation in local chemical environments have also been shown to lead to useful magnetic properties. For example, compositionally complex alloys, intermetallics, and oxides have been shown to have intriguing magnetic properties [15][16][17][18]. There is significant interest in leveraging these intriguing features to design novel magnets with low coercivity and high saturation magnetization (M s ) [19].
In addition, researchers have explored the potential of high configurational entropy in magnetocaloric materials [20][21][22]. In 2017, Yuan et al reported the synthesis of a series of rare-earth based crystalline high-entropy alloys composed of Gd, Dy, Er, Ho, and Tb that exhibited giant magnetocaloric behavior and low magnetic hysteresis [23]. The origin for enhanced magnetocaloric properties was attributed to the local chemical disorder (an intrinsic characteristic of high-entropy alloys) that have been shown to lead to complex magnetic ordering and discrete exchange coupling interactions between the magnetic atoms [16,22,[24][25][26][27].
Intermetallic compounds that form in the rare-earth-free MTX stoichiometry (where M = Mn, Fe, and Ni, T = Fe, Co, and Ni, and X = Si, Ge) are identified to undergo a martensitic phase transition from the low temperature TiNiSi-type Pnma orthorhombic phase to the high temperature Ni 2 In-type P6 3 /mmc hexagonal phase in the 1970's [28]. This original work also identified the susceptibility of the martensitic transition temperature (T t ) to changes in the chemistry and stoichiometry. In pure MnNiSi, T t is at 1206 K [28] whereas the ferromagnetic Curie temperature (T C ) is at 616 K [29]. In recent years, researchers have leveraged this sensitivity to chemistry and composition to design giant magneto-caloric alloys where (T t ) and (T C ) coincide, and the magnetostructural phase transition occurs near room temperature. Such coupled phase transitions generally have a large total ∆S at (T t ), which is desired for magnetocaloric applications [5,[30][31][32][33]. In addition to designing materials with a large ∆S, several challenges remain including reducing the thermal hysteresis (∆T hyst ) that negatively impacts the efficiency of cooling devices [34]. Further alloy design is required to identify promising compositions in this family to achieve the performance required for devices.
Recently researchers have extended the concept of high-entropic chemical substitutions as a possible route to improve the magnetocaloric properties of MTX intermetallic compounds [22,27].  [32,37,38]. Developing a fundamental understanding of properties impacted by site-specific multi-principal element substitutions is crucial to further advance the materials design and discovery efforts in this high-entropy intermetallic magnets search space.
Inspired by these recent experimental findings [35,36], in this work we computationally explore the promise of a high-entropic MTX alloy where we fix the M-and T-sites to be fully occupied by Mn-and Ni-atoms, respectively. Unlike Law et al, we substitute the X-site with equiatomic proportions of five distinct non-magnetic, p-block elements from the periodic table, namely Si, Ge, Sn, Al and Ga. This formulation leads to a nominal chemical composition of MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 . Thus, the idea of high-entropy is confined solely to the X-site of the intermetallic compound. From our previous machine learning (ML) study of MTX compounds, we found that researchers have explored Si, Ge, Sn, Al and Ga as potential chemical substitutions in the MTX materials family [39]. However, there is no published work that investigated the collective influence of all five chemical elements in the same composition. In the same ML publication, an experimentally validated ML model predicted the T t for Mn 34 Ni 33 Si 7 Ge 7 Sn 7 Al 6 Ga 6 to occur at 111.2 ± 246.4 K [39]. The presence of large uncertainty is not entirely surprising because the training set lacked experimentally measured MTX alloy data that mimic the high-entropic MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 composition. Nonetheless, the observation of large uncertainty in this prediction, along with the suggested trend of massive decrease in T t indicated by the mean prediction (relative to (Mn,Fe,Ni) 66.7 (Si x Ge 1-x ) 33.3 ), makes this composition an excellent candidate for further exploration.
In this paper, we have performed electronic structure calculations in the Kohn-Sham DFT framework for the MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 compound in both the low-symmetry Pnma and high-symmetry P6 3 /mmc crystal structures. Previous work in the literature have used DFT to explore the magnetic properties of high-entropy alloys and compounds [40][41][42]. Our objective is to use DFT calculations to shed light on the energetics, local bond geometry, and magnetic properties of the chosen materials system. In order to make the high-entropic X-site substitution attractive for experimental study, it must reduce T t without significantly impacting the magnetic behavior so as to maximize the total ∆S at the phase transition. Our DFT calculations reveal that the Pnma structure is energetically favorable compared to the P6 3 /mmc structure, which is similar to what we have observed in the pure end-member compounds [37] and an encouraging sign for exploring the MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 composition further for magnetocaloric applications. However, the calculated total energy difference (∆E) between Pnma and P6 3 /mmc crystal structures for the MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 composition is much lower compared to that of the pure compounds. Intriguingly, this ∆E reduction did not impact the Mn-site atomic magnetic moments in any adverse way. Thus, the high-entropic X-site substitutions are predicted as a promising route for tailoring the T t of MnNiSi-based compounds.

Methods
DFT calculation of configurational entropy is a non-trivial task. In this approach, one typically simulates a finite number of fixed atoms at 0 K, where the computational complexity increases as O(N 3 ), where N is the number of valence electrons in the system [43]. In order to account for configurational entropy, the total energies of supercells consisting of large number of atoms would be needed in every possible configuration to approximate the chemical composition. Since this is not a computationally tractable solution, other methods have been devised. For our problem we use the alloy theoretic automated toolkit to generate special quasi-random structures (SQS) of MnNi-X, with equal compositions of all p-block elements in both the Pnma and P6 3 /mmc crystal structures [40,[44][45][46][47]. Our previous DFT work has used MnNiSi as the parent lattice in this materials family [37], so the lattice parameters of MnNiSi were used in the construction of the SQS supercells. This supercell consists of 120 atoms for both the Pnma and All electronic structure calculations were performed using the planewave pseudopotential code as implemented in the QUANTUM ESPRESSO package [48][49][50]. The atom positions of the SQS generated structures were relaxed using the Broyden-Fletcher-Goldfarb-Shanno quasi-Newtonian algorithm until the sum of force between the atoms was less than 0.04 eV Å −1 . At each stage of the relaxation a self-consistent field (SCF) calculation was performed with an energy cutoff of 1×10 −9 eV using Perdew-Burke-Ernzerhof optimized for solids, generalized gradient approximation exchange-correlation functionals [51] and ultrasoft pseudopotentials [52]. During the relaxation stage the DFT calculations were performed with a plane-wave kinetic energy cutoff of 60 Ry considering only the Γ-point for k-point sampling. After the cells were full relaxed the k-mesh density was increased to a uniform grid of 2×2×2 k-points [53] for a final SCF calculation. Magnetism was considered through the collinear local spin density approximation. All Mn-Mn, Mn-Ni and Ni-Ni interactions are assumed to be of ferromagnetic nature in both the Pnma and P6 3 /mmc SQS structures. Figure 1 shows the relaxed atomic arrangements in both the low-symmetry Pnma and high-symmetry P6 3 /mmc crystal structures. The relaxed atomic coordinates are given in the supplemental information. Atomic displacements and complex local or nearest-neighbor bonding environments have long been attributed to the interesting behavior of high-entropy alloys. Recent work found that the average displacement of constituent atoms from regular lattice positions can be used to rationalize the high solid solution strengthening observed in CrMnFeCoNi solid solution [54]. To gain insights into the local site disorder in high-entropy materials, it is common to use characterization techniques such as neutron diffraction and pair distribution function analysis [55]. To emulate this in our DFT study and provide atomic structure data that may be validated experimentally, the distribution of atomic displacements was calculated as the euclidean distance from the final relaxed structure to the initial SQS structure, which is based on the ideal location for atoms in the MnNiSi structures. The distributions of different X-site elements are shown in figure 2, where the form of the probability density function was approximated through the use of kernel density estimation (details are given in supplemental information section S1) [56]. In the low-symmetry Pnma SQS structure, the Al-, Si-, and Ga-atoms appear to have the largest displacements while Ge-and Sn-atoms have relatively smaller displacements ( figure 2(a)). The Si-and Al-atoms, in particular, appear to have some multimodal characteristics. The atomic displacements trend shown in figure 2(a) correlates well with the atomic weight trend, i.e. lighter elements displace relatively more compared to the heavier elements. Similar atomic displacements trend are observed in high-entropy alloys [54]. Future work can include several chemical compositions and/or stoichiometries to rigorously test this correlation and improve our understanding of the complex local structure and chemical bonding in high-entropy MTX alloys. Comparatively, the X-site atoms in the high-symmetry P6 3 /mmc SQS structure have distributions that appear centered around ∼1.2 Å, with the exception of Al that exhibits a relatively broader distribution (shown in figure 2(b)). In the P6 3 /mmc structure, the displacement distribution of Sn-atoms is much narrow compared to the other X-site elements.

Results and discussion
When compared to the parent MnNiSi composition, in MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 the P6 3 /mmc structure is much closer in total energy to that of the Pnma structure. We have previously studied the impact of individual chemical substitutions on total energy difference (∆E) between the two structures for the pure end-member compounds [37]. While typically only indicative of relative phase stability, ∆E has been correlated to the T t of the system, both by other groups studying MTX systems and in our own ML modeling of T t [32,39]. The correlation between ∆E and T t can be expressed as ∆E ∝ k B T t , where k B is the Boltzmann constant [57][58][59][60][61]. Although this simple expression neglects vibrational and magnon  contributions to the free energy, it corresponds to a zeroth-order approximation for T t . The calculated total energy difference between the Pnma and P6 3 /mmc structures for the MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 system is 109.081 meV atom −1 . As stated above, this value is strikingly lower than for any individual X-site substituted pure compound in MnNi-X systems as shown in figure 3(a), indicating a relatively less stable Pnma phase at 0 K compared with pure compounds. The total and atom-projected density of states show that Mn-3 d and Ni-3 d electronic states dominate the Fermi level in both the low-and high-symmetry structures (see supplemental information figures S1 and S2).
Our recently published ML study that considered several MTX-based solid solutions identified the importance of T-X bond length in the Pnma structure (where T = Fe, Co, or Ni) as a key feature for the prediction of T t [39]. We found that in most cases, increasing the T-X bond length resulted in lower T t . Previous work by Kanematsu showed that the chemical bonding between the metal and metalloid atoms in a similar B8 2 crystal systems are responsible for the stability of the P6 3 /mmc phase [62]. Clearly, the local bonding environments are crucial to manipulate the relative phase stability, and thus the T t . Figure 3 figure 3(b). If we assume a normal distribution, the mean value for Ni-X bond distance is found to be larger than that of  any of the individual pure compounds. We surmise that this larger average bond distance provides a viable mechanism for lowering the ∆E in MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 , which is in turn key to reducing the T t . Future work can include combinations of three or four X-site elements to systematically investigate the relationship between Ni-X bond distances and ∆E.
While tuning bond distances may offer a pathway for impacting the relative phase stability and T t , we anticipate that any changes to the bond distances will also have a concurrent impact on the magnetic behavior of the system. Ideally, to make efficient cooling devices we would like to tune T t while maintaining large magnetization in the system. Typically, saturation magnetization (M s ) is used to understand the total magnetization for the system but since this is a volume-dependent quantity and we have not fully relaxed the simulation cell volume in our calculations, we instead focus on the site-specific atomic magnetic moments to extract insights about M s . Figure 4 shows the distribution of atomic magnetic moments (in Bohr magnetons) for the high-entropic intermetallic compound in the low-symmetry Pnma structure. Vertical dashed red lines show the relative Mn-atomic magnetic moments in the pure compounds. The Mn-atoms have large magnetic moments in both the high-entropic and the pure end-member compounds [37]. In addition, we infer from figure 4 that the substitution of multiple X-site elements in the Pnma structure did not have a large impact on the average magnetic moment of the Mn-atoms. However, the distribution of Mn-moments is bimodal. One of the origins of the peak near ≈2.7 µ B is attributed to the fewer Ni-atoms in the local coordination environment surrounding those specific Mn-atoms. The average atomic magnetic moment of Ni-atoms in the Pnma SQS structure is less than 0.35 µ B .

Conclusions
First principles-based DFT calculations reveal signatures indicating that high-entropic X-site chemical substitution of Si-, Ge-, Sn-, Al-and Ga-atoms in the MnNi-X intermetallic compound can potentially lead to reduced T t . This is supported by comparing two key calculated quantities between the high-entropic MnNiSi 0.2 Ge 0.2 Sn 0.2 Al 0.2 Ga 0.2 and pure end-member compounds: (i) relatively smaller total energy difference (∆E) between Pnma and P6 3 /mmc crystal structures, and (ii) relatively large Ni-X average bond lengths in the high-entropic Pnma structure. In addition, we show that substitution of five different non-magnetic X-site atoms does not significantly reduce the Mn-site atomic magnetic moments in the Pnma structure. Therefore, we conjecture that high-entropic X-substitution provide a potential path to selectively tune T t without significantly impacting the atomic magnetic moment of the Mn-and Ni-atoms. In addition, one can further extend the composition tuning by deviating from equiatomic concentrations. This will provide yet another knob to decrease the temperature difference between the T t and T C , which will enable a large total ∆S at the phase transition.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).