Abstract
A datadriven framework is presented for building magnetoelastic machinelearning interatomic potentials (MLIAPs) for largescale spinlattice dynamics simulations. The magnetoelastic MLIAPs are constructed by coupling a collective atomic spin model with an MLIAP. Together they represent a potential energy surface from which the mechanical forces on the atoms and the precession dynamics of the atomic spins are computed. Both the atomic spin model and the MLIAP are parametrized on data from firstprinciples calculations. We demonstrate the efficacy of our datadriven framework across magnetostructural phase transitions by generating a magnetoelastic MLIAP for αiron. The combined potential energy surface yields excellent agreement with firstprinciples magnetoelastic calculations and quantitative predictions of diverse materials properties including bulk modulus, magnetization, and specific heat across the ferromagnetic–paramagnetic phase transition.
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Introduction
Magnetism strongly influences thermomechanical properties in a large variety of materials, such as singleelement magnetic metals^{1,2}, steels^{3}, highentropy alloys^{4,5}, nuclear fuels such as uranium dioxide^{6}, magnetic oxides^{7,8}, and numerous other classes of functional materials^{9}. Despite the critical role of magnetism in the aforementioned materials classes, modeling efforts to study the interplay between structural and magnetic properties have been notably lacking. Furthermore, there are unanswered scientific questions regarding the significance of magnetism in matter that is shockcompressed^{10,11} or exposed to strong electromagnetic fields such as in coherent lights sources^{12,13}, pulsed power and high magnetic fields facilities^{14,15}. Properties of interest include phase transitions, thermal stability of magnetic defects, magnetomechanical couplings, but many of these subjects are challenging or prohibited by state of the art computational tools.
A prime but simple example of the computational advance made herein is the heatcapacity of αiron displayed in Fig. 1. The experimental measurement of the heat capacity C_{p} diverges at the magnetic Curie transition, characteristic of a secondorder phase transition^{16}. Without a scalable coupled spinlattice dynamics simulation environment, that properly accounts for thermal expansion and magnetic contribution to the pressure, reproducing the divergence of C_{p} (and of other thermomechanical properties) at the critical point is not possible.
Accurate numerical simulations are critical for enabling technological advances, as they shape our fundamental understanding of the underlying solid state physics that dictates material behavior. Developing high fidelity models, however, is challenging, because it necessitates capturing physical phenomena that occur across several length and time scales. This can only be achieved with sufficiently accurate multiscale simulation tools^{17,18}, which is the focus of this work.
Classical molecular dynamics (MD) simulations^{19} provide a useful framework for multiscale modeling by leveraging interatomic potentials (IAPs) to represent the dynamics of atoms on a BornOppenheimer potential energy surface (PES)^{20}. By utilizing massively parallel algorithms^{21} and long timescale methodologies^{22}, MD enables bridging firstprinciples with continuumscale simulations^{23}.
The absorption of machine learning (ML) techniques into the creation of interatomic potentials has lead to classical MD simulations that approach the accuracy of firstprinciples methods. A large number of these highly accurate MLIAPs^{24,25,26,27,28,29,30} have been developed. In general, they are parameterized on training data (configuration energy, atomic forces) from firstprinciples methods like density functional theory (DFT)^{31} and utilize different flavors of ML model forms to construct the PES. While they have proven to be useful for largescale simulations of materials properties^{32,33}, further progress in multiscale modeling is hampered by the limitation of MLIAPs to nonmagnetic materials phenomena. Even with highly accurate MLIAPs, stateoftheart MD simulations cannot reproduce the divergent behavior of C_{p} near the critical point (Fig. 1) because they fail to account for the magnetic degrees of freedom^{34}.
Coupling atomic spin dynamics with classical MD has been pioneered by Ma et al.^{35,36,37}. Herein, a classical magnetic spin is assigned to each atom in addition to its position leading to a 6Ndimensional PES (5N if the magnetic spin norms are fixed), instead of the common 3Ndimensional PES in classical MD:
where r_{ij} = r_{i} − r_{j} denotes the relative position between atoms i and j, s_{i} the classical spin assigned to atom i, and N the number of atoms in the system. In most classical spinlattice calculations, the 6Ndimensional PES is constructed by introducing an atomic spin model on top of a mechanical IAP^{35}. For example, a common approach is to combine a distancedependent Heisenberg Hamiltonian with an embeddedatommethod (EAM) potential^{37,38}.
While these prior approaches recover experimental properties on a qualitative level^{39,40}, their combined representation of phononic and magnetic degrees of freedom is not sufficiently consistent for providing quantitative predictions at the level of firstprinciples results. More recently, Ma et al. developed a magnetoelastic IAP for magnetic iron based on data from firstprinciples calculations^{41}. However, this remained an isolated attempt as there is no general methodology for generating a magnetoelastic PES in a classical context that enables largescale spinlattice dynamics simulations for any magnetic material.
In this work, we overcome this methodological obstacle by providing a datadriven framework for generating magnetoelastic MLIAPs that (1) provide a consistent representation of both mechanical and magnetic degrees of freedom and (2) achieve near firstprinciples accuracy. We refer to our new class of IAPs as magnetoelastic MLIAPs as they generate a consistent PES accurately representing the magnetic degrees of freedom and the interplay between magnetic and elastic phenomena. Our framework couples an atomic spin model (Heisenberg Hamiltonian) with an MLIAP and provides a unified magnetoelastic PES which yields the correct mechanical forces on the atoms in the MD framework. The Heisenberg Hamiltonian is parameterized with data from DFT spinspiral calculations at different degrees of lattice compression. In constructing the MLIAP, we leverage the flexible and datadriven spectral neighbor analysis potential (SNAP) methodology^{30} which is trained on a database of magnetic configurations generated using DFT calculations. We highlight the influence of magnetization dynamics on thermomechanical properties by assessing three different thermodynamic equilibration conditions. This allows us to conclude that a correction to the magnetization dynamics (achieved by adapting the temperaturerescaling method^{42}) is necessary for accurate elastic predictions.
We apply our framework to generate a magnetoelastic MLIAP for the α phase of iron. We demonstrate that our potential is transferable to an extended area of the phase diagram, corresponding to a temperature and pressure range of 0 to 1200 K and 0 to 13 GPa (up to the α → γ and α → ϵ transitions, respectively). The Curie temperature, which experimentally occurs at ~1045 K, lies within this parameter space. After presenting our training workflow, the “Results” section will probe the accuracy of our magnetoelastic MLIAP by performing magnetostatic comparisons to firstprinciples measurements. We then stress that our generated magnetoelastic MLIAP can also be directly used in the LAMMPS package^{21} to perform magnetodynamic simulations that take into account both the thermal expansion of the lattice and magnetic pressure due to spin disorder. This enables us to maintain a constant ambient pressure throughout all calculations of thermomechanical properties, consistent with conditions prevalent in experiments. As illustrated in Fig. 1, our framework allows us to perform pressurecontrolled quantitative prediction of the critical behavior across a secondorder phase transition within a classical spinlattice dynamics simulation.
Results
In this section we outline our advancements in magnetic materials modeling. We first present our training workflow and subsequently assess our results by comparing both static and dynamic properties in αiron against firstprinciples calculations and experiments.
Figure 2 displays our training workflow. Further details to each box in this diagram are presented as a subsection in the “Methods” section. All atomic configurations in the training set result from firstprinciples calculations performed with the same DFT setup (same pseudopotential and energy cutoff, similar kpoint densities) as detailed in the “Methods” section. In contrast to traditional forcematching approaches in the development of classical IAPs, we treat the magnetic and phononic degrees of freedom in the PES in a consistent and unified manner, as indicated by the exchange of information between spin Hamiltonian and SNAP potential parametrization steps. After parameterizing our atomic spin Hamiltonian by leveraging DFT spinspiral results, its energy, forces, and stress contributions are subtracted from each atomic configuration in the firstprinciples training set. The MLIAP is then trained to reproduce the nonmagnetic component of the firstprinciples data. Finally, both components of the magnetoelastic PES are recombined to construct a unified magnetoelastic MLIAP that is consistently trained on firstprinciples data. Optimization is handled by the DAKOTA software package^{43} in both fitting steps. For the SNAP component of the potential, DAKOTA optimizes the radial cutoff of the interaction along with the weights of each training data set (energy and force weights) to generate different candidate potentials. Those candidates potentials are then recombined with the spin Hamiltonian and tested against selected objective functions (meanabsolute errors (MAEs) in lattice constants, cohesive energies, elastic constants, forces and total energies). Table 1 summarizes the different groups of training data, the optimal weights obtained for each of those groups, and the corresponding energy and force MAEs. The target values for the objective functions are based on both experimental and DFT data, as outlined in Table 2. Objective function evaluations are done within LAMMPS^{21}.
Herein, the critical innovation that enables a leap forward in predictive simulations of magnetic materials is this datadriven workflow. Magnetic and phononic contributions to the PES are taken into account explicitly and any miscounting is avoided (for example, no double counting of the magnetic energy or contribution to the pressure). The obtained magnetoelastic MLIAP can directly be used to run spinlattice calculations in LAMMPS^{21,38,44}.
Magnetostatic accuracy
We first assess the quantitative agreement of our magnetoelastic MLIAP by comparing with DFT results where magnetic order and elastic deformations are coupled. This is done by leveraging a particular subset of spin configurations referred to as spinspirals, for which the energy and corresponding pressure can be evaluated from both DFT and classical magnetoelastic potential calculations. Details about definition and computation of spinspirals can be found in the “Methods” section. Equationofstate calculations (energy and pressure versus volume) are performed at the Γ point (corresponding to the purely ferromagnetic state) and for spinspirals corresponding to qvectors along the ΓH and ΓP highsymmetry lines. The calculations at the Γ point represent the magnetic ground state and, hence, serve as a point of reference for the spin spiral calculations. The geometric orientation of the various computed spin spirals is visualized in Fig. 3. The first set (q = 0.01 along ΓH and q = 0.07 along ΓP) represents long spirals, close to the Γ point, the second set (q = 0.1 along ΓH and q = 0.14 in ΓP) represents spirals with intermediate periodicity, and the last set (q = 0.2 along ΓH and q = 0.21 along ΓP) is chosen close to the borders of the magnetic training set (see red demarcation lines in Fig. 5 in the “Methods” section). The DFT results are obtained by leveraging the generalized Bloch theorem, whereas our classical spinlattice calculations were performed by generating the corresponding supercells (details given in the “Methods” section).
Excellent agreement between our classical spinlattice model and DFT is achieved at the Γ point and for the two first qvectors on each highsymmetry line (q = 0.01 and q = 0.1 along ΓH, q = 0.07 and q = 0.14 along ΓP) in the pressure range relevant for the αphase of iron (up to 13 GPa which corresponds to the α → ϵ transition). At higher qvector values, the energy and pressure predictions of our atomic spinlattice model still agree reasonably well with the DFT calculations. The observed deviation from the DFT results can be explained by the limitations of our atomic spinlattice model: as both the pressure and the relative angle between neighboring spins increase, fluctuations of the atomic spin norms become more important. As discussed in the “Methods” and “Discussion” sections, these are not included in the Hamiltonian of our atomic spinlattice model.
Magnetodynamic accuracy
Turning now to spinlattice dynamics calculations based on our magnetoelastic MLIAP (as detailed in the “Methods” section), we assess the quantitative accuracy with respect to experimental measurements of changes in magnetic and thermoelastic properties as the material is heated. In making this comparison, it is necessary to choose which thermodynamic state variables will be held fixed and which will be allowed to vary with temperature. Spinlattice dynamics algorithms have been developed for simulations in a canonical ensemble which preserves the number of particles, the volume, and the temperature in the system^{37}. Our first set of simulation conditions, referred to as fixedvolume conditions (FVC), hold the volume fixed while running dynamics in the canonical ensemble at specified values of the lattice and spin temperatures. A disadvantage of this choice is that the pressure steadily increases as heat is added to the material, in contradiction to the experimental observations, which are conducted at constant pressure. To this date, an isobaric spinlattice algorithm has not been developed (preserving the system’s pressure rather than its volume). However, our methodology as implemented in LAMMPS enables us to compute the magnetic contribution to the pressure. By alternating thermalization (coupled spinlattice dynamics in a canonical ensemble) and pressure equilibration (frozen spin configuration in an isobaric ensemble) steps, it is possible to control the pressure of our spinlattice system. Hence, we refer to calculations performed in this pressurecontrolled canonical ensemble as pressurecontrolled conditions (PCC). In both conditions, the temperature of the spin and lattice subsystems is set using two separate Langevin thermostats (one acting on the spins, the other on the lattice)^{37}. Finally, this enables us to define a third set of conditions: in addition to controlling the pressure, the spin thermostat can be set to match a given magnetization value (i.e., the experimental magnetization) rather than a temperature. We refer to this as pressurecontrolled and magnetizationcontrolled conditions (PCMCC). Figure 6 in the “Methods” section displays the different definitions of the spin temperature and the evolution of the pressure for those three different conditions.
In practice, FVC, PCC, and PCMCC only differ in their equilibration conditions (control of pressure and/or magnetization), as each of the corresponding simulations are performed in a canonical ensemble. We illustrate the predictive capability of our magnetoelastic MLIAP in αiron for these equilibration conditions in Fig. 4a–f (FVC: , PCC: , PCMCC: ). The agreement of the following magnetoelastic properties with experimental results is assessed: magnetization (Fig. 4a), heatcapacity C_{p} (Fig. 4b), thermal expansion (cell volume on Fig. 4c), bulk modulus (Fig. 4d), and two shear constants, (c_{11} − c_{12})/2 and c_{44} (Fig. 4e, f). The “Spinlattice dynamics” subsection of the Methods section details the computation of those temperaturedependent elastic constants.
We first work under the FVC (), keeping a constant volume and equal spin and lattice temperatures (Figs. 4c and 6). At constant volume, our model predicts a Curie temperature of ~716 K (Fig. 4a). Specific heat calculations shown in Fig. 4b were performed by computing the derivative of the internal energy, taking both the lattice and magnetic contributions into account. The SNAP contribution (lattice only) was first isolated and determined to be a constant value of 26.4 Jmol^{−1} K^{−1}, in good agreement with the DulongPetit value of 3R^{45}. The magnetic contribution offsets the total specific heat at low temperature, as the magnetization steadily decreases (thus steadily increasing the magnetic energy). Also at low temperature, deviation between simulations and experiment (highlighted by the semitransparent blue region in Fig. 4b) occurs due to quantum effects which reduce the experimental heat capacity below the 3R value. The FVC heatcapacity is determined at constant volume, although we use the symbol C_{p} on the axis label because the enhanced simulations described below are indeed conducted at constant pressure conditions. In those constant volume conditions, the pressure evolution with temperature increase is substantial (up to 12 GPa, almost corresponding to the α → ϵ transition, as can be seen on Fig. 6), which has a strong impact on the underlying elastic properties. Interestingly, at the Curie temperature (here 716 K), the increasing pressure exhibits an inflection point, confirming the importance of spin fluctuations on the thermoelastic properties. The temperature dependence of three elastic constants is shown in Fig. 4d–f. For the bulk modulus, FVC does not agree well with experimental data, especially at higher temperatures. The FVC results tend to overestimate the stiffness, which most likely arises from the buildup of thermal stresses in the material. Under these conditions a nearly temperatureinvariant c_{44} response is predicted, which is in strong contrast to trends in experiment. Despite these shortcomings, the FVC calculations actually match the experimental data for shear constant (c_{11} − c_{12})/2 relatively well throughout the entire temperature range. In general, the fixed volume assumption made under FVC fails to account for thermal expansion, leading to incorrect elastic predictions.
We correct this shortcoming of the model by working under PCC () which allows for thermal expansion. As can be seen on Fig. 4c, the cell volumes are relaxed at each finite temperature, until the pressure in the system drops to ~0 GPa. As shown in Fig. 4a, the thermal expansion incorrectly moves the onset of Curie transition to ~536 K. As the average interatomic distance increases, the strength of the exchange interaction is lowered, thus decreasing the transition temperature. The computed heatcapacity (Fig. 4b) now corresponds to the derivative of the free energy, and to an actual C_{p} measurement. However, as in the FVC, the low agreement between the experimental and computed magnetization evolution leads to an offset in the initial C_{p} and does not match the DulongPetit value at low temperature. The PCC fares better in reproducing the experimental bulk modulus up to the Curie transition (no hardening observed). PCC also does better in terms of the shear constant c_{44}, as it is able to reproduce the thermal softening seen in experiments. However, for shear constant (c_{11} − c_{12})/2, PCC underestimates the extent of the thermal softening. Overall, PCC does better than FVC in terms of elastic properties, but deviates more in terms of magnetic predictions compared to experiment. By shifting the Curie transition towards lower temperatures, it reduces the range of validity of our elastic calculations.
In order to improve the magnetic predictions of αiron, we finally consider the PCMCC scheme (). The present calculations are aimed at probing the influence of the magnetization dynamics on thermomechanical properties of materials. For FVC and PCC, two main physical limitations associated to the classical spin dynamics model can be observed: an offset of the predicted Curie transition, and the magnetization versus temperature trend (as clearly displayed on Fig. 4a, b). The Curie transition offset is mainly attributed to our choice of parametrization of the magnetic interactions (see “Methods” section). Compared to experimental measurements, our low temperature magnetization trend is immediately decreasing. This, for example, strongly impacts the C_{p} measurements (Fig. 4b), as it generates a large magnetic energy decrease, leading to an offset from the DulongPetit value at low temperature. In order to improve this limitation of classical spin dynamics and better observe the influence of magnetization dynamics on the thermomechanical properties predicted by our model, we follow the approach developed by Evans et al.^{42}. However, our simulations account for thermal expansion, so that our magnetic temperature rescaling has to be slightly modified compared to their approach. A full magnetization versus temperature calculation in the PCC has to be performed before evaluating which spin temperature corresponds to a give lattice temperature (see “Methods” section for more details). Figure 4a shows that the obtained magnetization under PCMCC closely matches that of experiment. Most prominently, the resulting C_{p} agrees well with experiments (Fig. 4b). The DulongPetit value is recovered at low temperature, and the C_{p} discontinuity at the Curie transition is well captured. The thermal expansion trend is also in much better agreement with experiments, with very comparable slopes between ~200 and 750 K (Fig. 4c). Up to ~600 K, PCMCC agree very well with the experimental values for (c_{11} − c_{12})/2 (Fig. 4e) but at 800–1000 K a slight hardening is observed, which contradicts experimental data. For the bulk modulus, PCMCC correctly predicts the nearly linear trend up to the Curie temperature.
We note that in all three sets of conditions, a rapid increase of about 25–30 GPa in the bulk modulus is observed as we move across the critical point. This jump was found to be strongly impacted by the underlying mechanical potential. The prediction accuracy could possibly be improved by including additional, finitetemperature objective functions in the fitting procedure. The PCMCC prediction of the shear constant c_{44} closely matches the PCC data. This tends to indicate that this shear constant c_{44} is not impacted significantly by the spin dynamics. For both pressure controlled conditions (PCC and PCMCC) the maximum deviation from experiments occurs near 700 K and is ~14%.
Discussion
We presented a datadriven framework for automated generation of magnetoelastic MLIAPs which enable largescale spinlattice dynamics simulations for any magnetic material in LAMMPS. This framework was demonstrated by generating a robust magnetoelastic MLIAP for αiron. First we investigated the magnetostatic accuracy (energy and pressure) with respect to equivalent firstprinciples calculations. It was demonstrated that the generated magnetoelastic MLIAP (which represents the corresponding 5N dimensional PES) is in close agreement with firstprinciples magnetoelastic calculations. This was achieved by properly partitioning the PES into magnetic and mechanical degrees of freedom. Subsequently, we investigated the magnetodynamic accuracy by comparing predicted finite temperature magnetoelastic properties (magnetization, heatcapacity, thermalexpansion, bulk modulus, and shear constants) across the ferromagnetic–paramagnetic phase transition from spinlattice dynamics simulations against data from experiments. In the course of this, we analyzed the choice of simulation conditions (control of pressure and magnetization) and highlighted the importance of thermal and magnetic pressure contributions. This is an important advance over traditional classical magnetization dynamics methods, where contributions from thermal expansion or spin pressure due to disorder are negated. We demonstrated that spinlattice dynamics simulations of controlled pressure and constrained magnetization yields qualitative agreement with the measured magnetoelastic properties.
Our framework enables predictions of critical properties across the secondorder phase transition within classical spinlattice dynamics simulations, such as the divergent behavior of the heat capacity around the Curie temperature (Figs. 1 and 4b). We provide a more comprehensive perspective on our results by comparing them within the context of other firstprinciples and classical methods. At low temperature, firstprinciples methods can capture the electronic component of the heatcapacity, up to the DulongPetit value^{45,46} (the difference with our model is highlighted by the blue area on Fig. 4b). However, computing C_{p} across the Curie transition requires a dynamic treatment of large spinensembles whose calculation is computationally expensive in terms of firstprinciples methods. Classical IAPs do not explicitly treat magnetic degrees of freedom and, thus, cannot reproduce the effects of this magnetic phasetransition^{47}. An empirical model which is based on firstprinciples calculations and accounts for electronic, phononic and magnetic degrees of freedom gave excellent agreement with the experimental C_{p} curve of αiron up to the Curie temperature^{48}. However, this model does not extend above the Curie temperature, does not account for the pressure generated by the corresponding spin configurations, and cannot be easily extended to other thermomechanical properties. Thus, for a range of temperature from about 250 to 1200 K, our model provides with a set of very good predictions, obtained for the computational cost of classical MD calculations only.
We conclude the discussion of our results by pointing out limitations of the present method and future prospects. First, note that the agreement to the experimental Curie transition (T_{c} ≈ 716 K in a fixed volume calculation) could have been adjusted by parameterizing the spin potential on a smaller range of the highsymmetry lines (see Fig. 5), or by adding an objective function aimed at matching the experimental value in the spinpotential fitting procedure. However, this additional constraint would have worsened the agreement of our model with the DFT energy and pressure results (as displayed on Fig. 3) and would contradict the overall objective of this work.
For temperatures below ~250 K, our classical framework cannot access the quantized free energy, and is thus unable to accurately reproduce the trends of all the quantities being its derivatives (C_{p}, elastic constants, ...). This is reducing the agreement versus experiments of the magnetodynamic accuracy measurements displayed on Fig. 4 at low temperature, and can be seen as a limitation of our classical approach^{49}.
Another limitation of our work lies in the simplicity of the spin Hamiltonian model used. The Heinsenberg Hamiltonian (as well as its extended forms) is a robust model that can be transposed to a large number of magnetic systems and applications^{7,39,50,51}. However, finding a unique lattice dependence (herein the BetheSlater function) and its’ associated accurate parametrization is very challenging. Comparing Fig. 4a and Fig. 5 reveals some of this difficulty: although the predicted spinspiral energies are in good agreement with the DFT calculations, the corresponding pressure dependence and the Curie temperature predictions (in the FVC and PCC) are departing from their expected values. Other parametrizations of the spin Hamiltonian could have been performed, either improving the Curie temperature predictions, or the generated magnetic pressure, but worsening the comparison versus DFT of the spinspiral energies. In this study, we decided to focus our fit on the spinspiral energies for two main reasons:

(i)
Our approach aims at developing magnetoelastic MLIAPs trained on firstprinciples data without double counting of the magnetic component of the PES. Thus, obtaining an energetic dependence of spin configurations close to the DFT results should remain the priority.

(ii)
The approximations of the model could be better controlled this way. Indeed, the spin model we used was developed to reproduce magnon energetics, and does not account for fluctuations of the spin norms (although this is not a limitation of the presented framework). Thus, a stronger emphasis was set on reproducing the spinspiral energies for which the spin norm fluctuations remain under 5% fluctuations from the Γ point value (as displayed on Fig. 5 in the “Methods” section). Different forms of spin Hamiltonians, such as spincluster expansions, might be a promising route to improving the accuracy of the magnetic component of the PES by both accounting for the fluctuation of the magnetic moment magnitudes and manybody spin interactions^{52,53}. A straightforward extension of this work could combine recently developed extended spin Hamiltonians with firstprinciples studies, and apply our formalism to extend our αiron magnetoelastic MLIAP to account for defect configurations^{54,55}, Cr clustering^{56,57}, and magnetostructural phasetransitions^{11,58}.
Our study emphasizes the important influence of magnetization dynamics on thermomechanical properties, even for a simple ferromagnet such as iron where the magnetoelasticity is rather weak. We showed that the wellknown departures from the experimental magnetization curve observed in classical spin dynamics have a strong influence on the predictions of the model. In this work, we chose to follow the approach proposed by Evans et al.^{42} to correct this shortcoming of the approach. This allowed us to accurately represent the temperature influence on thermomechanical properties of iron through its αphase and across the Curie transition. Enhanced magnetic thermostats have been proposed in order to better match the experimental magnetic transition versus temperature^{59,60}. Such thermostats could be implemented in LAMMPS and used to replace the magnetizationcontrolled conditions defined in the “Results” section. This could extend the range validity of our framework to areas of phasediagrams where the magnetization distribution is not well measured (for example in the ϵ phase of iron).
A recent study added a magnetic contribution to the set of descriptors used in a momenttensor MLIAP^{61}. Although this approach does not explicitly simulate the magnetization dynamics (and its effects on thermomechanical properties), the authors demonstrated remarkable improvement in terms of error convergence. At this stage of our work, we believe improving the modeling of the magnetic component of the PES remains our first priority (and thus implementing and fitting improved spin Hamiltonians, as discussed above). However, depending on the success of this first effort, this complementary approach could be leveraged to improve the accuracy of our magnetoelastic MLIAPs.
In summary, we have presented a new computational framework for simulations of magnetoelastic materials properties near firstprinciples accuracy. By leveraging the flexibility of MLIAPs, our datadriven workflow enables to model the interplay between magnetic and phononic dynamics for a large class of magnetic materials. Furthermore, our straightforward connection to the LAMMPS package makes it possible to perform largescale quantitative magnetoelastic predictions over controlled pressure and temperature spaces, hitherto study unexplored magnetodynamics properties of materials.
Methods
Density functional theory calculations
Parameterizing both the MLIAP and the magnetic Heisenberg Hamiltonian relies on data computed using spindependent DFT calculations. They were performed using VASP^{62,63}. In all calculations the PBE^{64} exchangecorrelation functional was employed. We used PAW pseudopotentials^{65} with 8 valence electrons and a core radius of r_{c} = 2.3a_{B}. The plane wave cutoff was set to 320 eV and the convergence in each selfconsistency cycle was set to 10^{−8}. The FermiDirac smearing scheme with a width of 0.026 eV was used. The Brillouin zone was sampled on a 10 × 10 × 10 grid of kpoints. The number of bands used was 224 per atom.
Spinspiral calculations
Spinspirals define a subset of noncollinear magnetic states. In this work, we leverage spinspirals as a convenient tool to perform onetoone comparisons between firstprinciples and classical magnetoelastic calculations. They can be defined as follows:
where q is the spinspiral vector, R_{0j} is the position of atom j relative to a central atom 0, s_{j} is the spin on atom j, and θ is a constant angle between the spins and the spinspiral vector (often referred to as cone angle)^{51}. \({{{\hat{\boldsymbol{x}}}}}\), \({{{\hat{\boldsymbol{y}}}}}\), and \({{{\hat{\boldsymbol{z}}}}}\) are the unit vectors along [100], [010], and [001], respectively. Our calculations are restricted to θ = π/2, corresponding to flat spinspirals in the (001) plane.
Firstprinciples calculations of the peratom energy and the pressure corresponding to spinspiral states are performed using DFT by leveraging the frozenmagnon approach^{66,67} and the generalized Bloch theorem^{68} as implemented in VASP^{69}. We consider a primitive cell of one atom. A 10 × 10 × 10 kpoint grid, an energy cutoff of 320 eV, and 224 bands proved sufficient to reach the level of accuracy expected in our model (as can be seen in Fig. 5).
Classical calculations are performed by using Eq. (2) to generate supercells accommodating the spinspirals corresponding to the qvectors used in the DFT calculations. Based on a given supercell and a spin Hamiltonian, the peratom energy and pressure are computed using the SPIN package of LAMMPS^{21,38}.
Spin Hamiltonian
A spin Hamiltonian is used to model the energy, mechanical forces, and pressure contributions of magnetic configurations. Rosengaard and Johansson^{50} and Szilva et al.^{70} showed that adding a biquadratic term to the classical Heisenberg Hamiltonian improves the accuracy of magnetic excitations in 3d transition ferromagnets. We adopted their Hamiltonian form:
where s_{i} and s_{j} are classical atomic spins of unit length located on atoms i and j, \(J\left({r}_{ij}\right)\) and \(K\left({r}_{ij}\right)\) (in eV) are magnetic exchange functions, and r_{ij} is the interatomic distance between magnetic atoms i and j. The two terms in Eq. (3) are offset by subtracting the spin ground state (corresponding to a purely ferromagnetic situation), as detailed in Ma et al.^{35}. Although this offset of the exchange energy does not affect the precession dynamics of the spins, it allows to offset the corresponding mechanical forces. Without this additional term, the magnetic contribution to the forces and the pressure are not zero at the energy ground state. For the exchange interaction terms \(J\left({r}_{ij}\right)\,\) and \(K\left({r}_{ij}\right)\), the interatomic dependence is taken into account through the following function based on an approximation of the BetheSlater curve^{71,72}:
where α denotes the interaction energy, δ the interaction decay length, γ a dimensionless curvature parameter, r = r_{ij} is the radial distance between atoms i and j, and \({{\Theta }}\left({R}_{c}r\right)\) a Heaviside step function for the radial cutoff R_{c}. This assumes that the interaction decays rapidly with the interatomic distance, consistent with former calculations^{70,73}. We set R_{c} = 5 Å to include five neighbor shells, as Pajda et al.^{73} showed that the exchange interaction decays slower along the [111] direction in αiron.
Using Eq. (3) and leveraging the generalized spinlattice Poisson bracket as defined by Yang et al.^{74}, the magnetic precession vectors (ω_{i}), mechanical forces (F_{i}), and their corresponding virial components (\(W\left({{{{\boldsymbol{r}}}}}^{N}\right)\)) are derived:
where r^{N} denotes a 3N size vector of all atomic positions and r_{i} the position vector of atom i. The primed sum in the above expression for the virial indicates that force contributions on atoms that are periodic images must be summed separately^{75}. The total pressure is obtained by combining this virial with thermal and mechanical contributions. The precession vectors (ω_{i}) are defined following the definitions of Evans^{76}. In Eq. (5), γ is the gyromagnetic ratio (γ ≈ 0.176 rad ⋅ THz ⋅ T^{−1}), and μ_{i} is the atomic spin norm, in Bohr magneton. This yields a precession frequency in rad ⋅ THz (corresponding to the metal units of LAMMPS).
The spin Hamiltonian is used to reproduce spinspiral energy and pressure reference results obtained from DFT. They are sampled along two highsymmetry lines, ΓH and ΓP, and for two different lattice constant values (corresponding to the equilibrium bulk value and to a lattice compression of 2%). This allows us to encapsulate in the model the influence of lattice compression on the spin stiffness and the Curie temperature, which was experimentally and theoretically predicted to be small^{77,78,79}. Figure 5 displays the excellent agreement obtained between our firstprinciples spinspiral energies and experimental measurements.
Our current spin Hamiltonian does not account for fluctuations of the magnetic moment magnitudes, i.e., the norm of atomic spins remains constant in our calculations. As can be seen in Fig. 5, this is not the case for our DFT results, as those fluctuations can become important when departing from the Γ point. We thus decided to parameterize our model only on spinspirals corresponding to qvectors for which the spin norm deviates from the ferromagnetic value (≈2.2μ_{B}/atom at the Γ point) by <5%. The red dashed lines in Fig. 5 delimit this qvector range.
Finally, we used the single objective genetic algorithm within the DAKOTA software package^{43} to optimize the six coefficients of \(J\left({r}_{ij}\right)\) and \(K\left({r}_{ij}\right)\) in order to obtain the best possible agreement between our reference DFT spinspiral energy and pressure results and our spin model. Figure 5 displays the obtained result. As can be seen in Fig. 4, for a fixedvolume calculation, our spin Hamiltonian predicts a Curie temperature of 716 K. Note that a better match of the DFT spinspiral energies would yield a larger spinstiffness, and thus a better agreement for the Curie temperature. However, this would worsen the pressure agreement.
Spin–orbit coupling effects were included by accounting for an irontype cubic anisotropy^{80}:
with K_{1} = 0.001 eV and \({K}_{2}^{(c)}=0.0005\) eV the intensity coefficients corresponding to αiron. The cubic anisotropy was only included to run calculations, but ignored in the fitting procedure as its intensity is below the range of accuracy of our MLIAP.
In all our classical spinlattice dynamics calculations, our system size remained small compared to the typical magnetic domainwall width in iron^{80}. Thus, longrange dipole–dipole interactions could safely be neglected.
The parameters if this optimized spin Hamiltonian are contained in the Supplementary Table 1, along with LAMMPS input scripts used in the following section. The Supplementary Figure 3 also reports a comparison between our spin Hamiltonian and DFT data on the ΓN highsymmetry line, on which it was not parametrized. This additional calculation aims at probing the ability of our spin Hamiltonian to account for spin configurations that are outside of its training set.
SNAP potential
For this work, an interatomic potential for iron was developed that is specifically parameterized for use in coupled spin and molecular dynamics simulations. Training data for a Spectral Neighborhood Analysis Potential (SNAP)^{44,81,82} was collected to constrain the fit to the pressure and temperature phase space of <20 GPa and <2000 K. The set of noncolinear, spinpolarized VASP calculations includes α (BCC), ϵ (HCP) and liquidiron, Table 1 displays the quantity of each training type and target properties that are captured therein. Optimization of a SNAP potential necessitates that the generated training database be broken into these groups (rows in Table 1) such that the weighted linear regression can (de)emphasize different parts in search of a global minima in objective function errors. Each training group is assigned a unique weight for its’ associated energies and atomic forces for each candidate potential, optimization of these weights is controlled by DAKOTA. Regression is carried out using singular value decomposition with a squared loss function (L2 norm). In order to avoid double counting, and properly simulate the magnetic properties of iron in classical MD, we have adapted the SNAP fitting protocol^{44} to isolate the nonmagnetic energy and forces from the generated training data. To do so, the fitted biquadratic spin Hamiltonian is used to evaluate the magnetic energy and forces for every atom in the training set, as well as the generated stress tensor pressure on the corresponding cell. Those quantities are then subtracted from the corresponding total DFT quantities. This is similar to previous uses of an ion core repulsion^{83} or electrostatic interaction term^{84} as a reference potential while fitting SNAP models. A key distinction, however, is that the inclusion of spin dynamics increase the dimensionality of the PES, which is not the case for Coulombic interactions. As a result, determination of magnetic energies and forces necessitates running a concurrent simulation for the spins within a standard MD run. The intricacies in evolving this 5Ndimensional system forward in time far exceed those encountered in the evaluation of static charge Coulombic interactions^{38}.
The energy for a SNAP interatomic potential, \({E}_{{\mathrm{SNAP}}}^{i}\) for each atom i from its’ neighboring atom positions, r^{N}, is expressed as a sum of the bispectrum components B^{i}
where the vector β are constant linear coefficients whose values are trained to reproduce energies and forces obtained from DFT training structures. Bispectrum components map the density correlations of neighboring atoms in a rotational and translation invariant manner, making them well posed descriptors of atomic energies and forces. By construction, the bispectrum components of an isolated atom are nonzero so the descriptors are shifted by the term B_{0} in order to force the potential energy to zero for an atom with no neighbors within the radial cutoff distance. Similarly, the forces on each atom k are expressed in terms of the derivative of atomic energies with respect to the position of atom k, where N is the total number of atoms in the structure
Optimization of the β terms in the SNAP potential was achieved using a single objective genetic algorithm within the DAKOTA software package^{43}. Radial cutoff distance, training group weights and number of bispectrum descriptors were varied to minimize a set of objective functions, as percent error to available DFT or experimental^{85,86} data, that encapsulate the desired mechanical properties of Fe. These objective functions specific to αiron are listed in Table 2, and the RMSE energy and force regression errors are included in optimization as well. In all objectives, our linear SNAP model with 30 bispectrum descriptors achieves accuracy in all mechanical properties within a few percent of experiment/DFT. Additionally, lattice constants and cohesive energies of γ (FCC) and ϵiron (HCP) phases were fit, but given far less priority with respect to the αiron mechanical properties resulting in ~6–7% errors with respect to DFT. Importantly, each of the objective functions were evaluated including the magnetic spin contributions to avoid unforeseen changes in property predictions. A full breakdown of the optimal training group weights and mean absolute energy/force errors are given in Table 1. Group weights listed have been adjusted by the number of configurations or forces they are applied to, therefore allowing for larger group weights to be (cautiously) interpreted more valuable at meeting the set of targeted objective functions. This optimized FeSNAP interatomic potential is detailed in the Supplementary Tables 1, 2, and 3 along with LAMMPS input scripts used in the following section.
Spinlattice dynamics
Calculations are performed following the spinlattice dynamics approach as implemented in the SPIN package of LAMMPS^{21,38}, and set by the spinlattice Hamiltonian below:
where \({{{{\mathcal{H}}}}}_{{\mathrm{mag}}}\) is the spin Hamiltonian defined by the combination of Eqs. (3) and (8). The term V_{SNAP}(r_{ij}) is our SNAP MLIAP. The second term on the right in Eq. (11), represents the kinetic energy, where the particle momentum is given as p and the mass of particle i is m_{i}. Based on this spinlattice Hamiltonian and leveraging the generalized spinlattice Poisson bracket as defined by Yang et al.^{74}, the equations of motion can be defined as:
Particle positions are advanced according to Eq. (12). The derivative of the momentum, given in Eq. (13), is dependent not only on the mechanical potential but the magnetic exchange functions as well. Here γ_{L} is the Langevin damping constant for the lattice and f is a fluctuating force following Gaussian statistics given below^{38}.
The fluctuating force f is coupled to γ_{L} via the fluctuation dissipation theorem as shown in Eq. (16). Here k_{B} is the Boltzmann constant, T_{l} is the lattice temperature, and α and β are coordinates. Shown in Eq. (14) is the stochastic LandauLifshitzGilbert equation which describes the precessional motion of spins under the influence of thermal noise. In Eq. (14), λ is the transverse damping constant and ω_{i} is a spin force analog as shown in Eq. (5). Note that the gyromagnetic ratio is included in the calculation of the precession vectors (see Eq. (5)). The variable η(t) is a random vector whose components are drawn from a Gaussian probability distribution given below:
where the amplitude of the noise D_{S} can be related to the temperature of the external spin bath T_{s} according to D_{S} = 2πλk_{B}T_{s}/ℏ^{37}.
SDMD calculations are carried out using a 20 × 20 × 20 BCC cell. The BCC lattice is oriented along each of the coordinate directions. The MD timestep in all cases is set to 0.1 fs. The damping constants are set to 0.1 (Gilbert damping, no units) for the spin thermostat, and to 0.1 ps for the lattice thermostat. Initially all spins start out aligned in the zdirection. To measure the magnetic properties for the canonical ensemble we initially thermalize the system under NVT dynamics at the target spin/lattice temperatures for 40 ps and then sample the target properties for 10 ps using a sample interval of 0.001 ps. For pressurecontrolled simulations (see PCC and MCPCC in the “Results” section), after the initial 40 ps of temperature equilibration we freeze the spin configuration and run isobaricisothermal NPT dynamics in order to allow the system to thermally expand (still accounting for the effect of the magnetic pressure, generated by the spin Hamiltonian). The pressure damping parameter is set to 10 ps. The pressure equilibration run is terminated once the system pressure drops below 0.05 GPa. After this, the spin configuration is unfrozen and another equilibration run is carried out under NVT dynamics for 20 ps. Unfreezing the spin configurations causes a small jump in the pressure, typically within the range of +/−2 GPa. To reduce this pressure fluctuation, a series of uniform isotropic box deformations are performed under the NVE ensemble. During this procedure the box is deformed in 0.02% increments every 2 ps until the magnitude of the pressure is reduced to negligible values (<10 MPa). Figure 6 displays the pressure profiles obtained within the FVC and PCMCC (similar to the PCC).
For the magnetizationcontrolled conditions (PCMCC in the “Results” section), the spin temperature is adjusted to match the experimental magnetization values. Spin temperature adjustments are made based on the magnetization curve obtained in the pressurecontrolled conditions (PCC in the “Results” section). The corresponding spinlattice temperature relationship is shown in Eqs. ((19)–(22)). Here the fitting coefficients are given as a_{1} = 471.6, a_{2} = 0.1, and α = 2.73, respectively. The functions T_{s,pre} and T_{s,post} prescribe how the spin temperature varies before and after the critical point. The form of T_{s,pre} is adopted from the spin temperature rescaling done by Evans et al.^{42}. In our case we account for the shift in the Curie temperature when T_{s} = T_{l} by including the T_{c,PCC} = 576 K term in Eq. (20). The value of α found here agrees reasonably well with the work of Evans et al. where for iron α was found to be 2.876. At the critical point we use a switching function f_{sw} to smoothly transition from T_{s,pre} to T_{s,post}. Above the experimental Curie temperature T_{c,exp} we do not assume T_{l} = T_{s} like Evans et al. The reason for this is that T_{c,PCC} ≠ T_{c,exp} hence if we assume T_{l} = T_{s} there will be a discontinuity in the effective temperature near the critical point which would also lead to unphysical discontinuities in the lattice constant and mechanical properties.
Figure 6 displays the spin temperature profiles for the FVC (and, similarly, the PCC), and the PCMCC. After the magnetic measurements we compute elastic constants by performing both uniaxial and shear deformations along each of the coordinate directions and planes. The magnitude of these deformations in all cases is 2% of the box length. Following each deformation the box is relaxed for 3 ps. After this relaxation the stresses are sampled for 2 ps using a sampling interval of 0.001 ps. For statistical averaging every spinlattice dynamics simulation is ran six times using different random seeds. The final elastic and magnetic properties are averaged across these six simulations. The maximum uncertainty in all cases (FVC, PCC, and PCMCC) occurs near the critical points. For the elastic measurements and magnetization the maximum standard deviation is ~2% of the reported mean value. For the specific heat measurements the maximum standard of deviation occurs at the critical point and is ~25% of the reported mean value. Away from the critical point the standard of deviation for C_{p} is in the range of 5–10%.
In the Supplemental Note, we probe the ability of our magnetoelastic MLIAP to be transferred to other material phases of iron. Supplementary Figures 1 and 2 display results obtained for liquid phase calculations as well as free energy measurements in the epsilon hcp phase of iron. Our result are in good agreement with DFT data and experimental observations. Those calculations constitute extrapolation of our potential (performed out of his training range), and are therefore not reported as main results of this manuscript.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code which was used to train the SNAP potential is available from https://github.com/FitSNAP/FitSNAP.
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Acknowledgements
All authors thank Mark Wilson for his detailed review and edits. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DENA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. A.C. acknowledges funding from the Center for Advanced Systems Understanding (CASUS) which is financed by the German Federal Ministry of Education and Research (BMBF) and by the Saxon State Ministry for Science, Art, and Tourism (SMWK) with tax funds on the basis of the budget approved by the Saxon State Parliament.
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A.C., M.A.W., M.P.D., and J.T. performed the DFT calculations. J.B.M., M.C.M., J.T., and M.A.W. generated the Database of configurations. J.T. implemented the extended spin Hamiltonian and the magnetic pressure computation in LAMMPS, and parametrized it on firstprinciples calculations. M.A.W. and S.N. trained the SNAP potential. J.T. and S.N. performed the magnetostatic calculations. S.N., A.P.T., M.A.W., and J.T. performed the magnetodynamics calculations. All authors participated in conceiving the research and writing the manuscript.
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Nikolov, S., Wood, M.A., Cangi, A. et al. Datadriven magnetoelastic predictions with scalable classical spinlattice dynamics. npj Comput Mater 7, 153 (2021). https://doi.org/10.1038/s41524021006172
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DOI: https://doi.org/10.1038/s41524021006172
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