Insights into MnAl-C nano-twin defects by micromagnetic characterization

The removal or reduction of critical rare earth elements in permanent magnets is of utmost importance to ensure economically and ecologically sustainable applications. MnAl-C has promising magnetic characteristics filling the gap between strong rare earth based magnets and weak ferromagnets,¹ yet currently achieved coercivities are rather low.² Several intergranular defects induce pinning and nucleation sites.³ The various processes to fabricate MnAl-C compounds^4–6 influence the amount and density of such crystallographic defects and consequently the magnetic characteristics of the material.

During hot-extrusion of MnAl-C, a drastic reduction of the grain size can be observed due to a dynamic recrystallization process.⁷ Remaining larger grains contain a large amount of lamellar twin structures with a few nm in width (Fig. 2). The material composition at those twin boundaries is inhomogeneous as recently published in Ref. 8.

Kerr microscopy has been applied to bulk samples with fine recrystallized grains and large non-recrystallized grains with a high twin density.⁷ During demagnetization, the large grains with enclosed nano-twins reverse much faster than the fine recrystallized grains, which nucleate at much higher field values reaching switching fields as high as 0.74 T. Consequently, the nano-twin regions cause a drastic reduction of coercivity in the material.

Micromagnetic simulations are a fast and precise tool to obtain design guidelines for permanent magnets as well as to understand magnetic properties in the nanometer regime. The microstructure of the material can either be taken directly from microscopic data⁹ or synthetically generated from known geometric parameters.¹⁰ 

In this work, we analyze the influence of nano-twins on the coercivity and energy density product of MnAl-C in micromagnetic simulations. We calculate hysteresis loops of structures with crystallographic defects and compare them to results from ideal structures to quantify the role of the nano-twins.

In this section, we first describe our experimental setup and the production routine of MnAl-C. From these experiments, we obtain geometrical properties, such as mean grain diameter and lamella width for finite element models, which we analyze with micromagnetic simulations. We show the generation and the meshing of the underlying grain structure. A numerical optimization scheme will be fed by micromagnetic simulations to find promising designs of the magnet. The calculation of coercivity reducing effects and a reduced order model for bulk magnets are described as well, which are used to show the effect of suppressing crystallographic twin boundaries.

Alloys with the composition of $Mn53Al45C2$ (at. %) were prepared by induction melting of high purity elements (99.99% Mn, 99.99% Al, and 99.9% C) under argon atmosphere and casting into a cylindrical Cu mold with 10 mm diameter. The as-cast alloys were encapsulated in a quartz tube at a back pressure of 150 mbar Ar, homogenized for 2 days at 1100 $°$C, followed by air cooling to room temperature to obtain the ferromagnetic $τ$-phase. Samples with size 10 mm in diameter and 8 mm in length were then subjected to die upsetting at 700 $°$C with a logarithmic degree of deformation of 1.386 and a constant strain rate of $0.001s−1$⁠. Before the start of die upsetting, the temperature was held for 10 min to homogenize the temperature within the sample.

Thin slabs taken from the hot-deformed alloy were oriented so that the direction normal to the foil was perpendicular to the compression axis. The electron transparent foils for transmission electron microscopy (TEM) experiments were prepared by the mechanical grinding of the thin slabs to a thickness of 20–50 $μ$m, followed by Ar-ion milling at 4.5 kV under a low beam angle of 11–13$°$⁠. TEM studies were performed at 200 kV using a FEI Tecnai G2 microscope.

Finite element meshes for a specific material can be based on experimental images or synthetically generated with basic information of the grain structure. Experimental images can only be used to analyze thin layers because they usually lack depth information.⁹ If no 3D information is available from measurement, calculation of bulk magnets requires synthetically generated grains. We are using the software package Neper¹¹ to create the initial granular structure as well as the lamellar twin layers in the individual grains (Fig. 1).

For all simulations, in this work, we use the same granular structure but tune the mean grain diameter by the scaling of the cube edge length. The mean grain diameter ranges from 181 nm to 363 nm. It is given by a sphere diameter with the same volume as the simulation cube volume divided by the number of grains. Each grain contains certain lamellas separated by twin boundaries. The distance between the twin boundaries, the twin width, ranges from 25 nm to 65 nm. Each grain has an effective anisotropic easy axis parallel to its containing twin boundaries. The easy axes of the thin lamellar twin regions deviate from the main axis by $37.8°$⁠, thus resulting in a true twin misorientation of $75.6°$ between adjacent lamellas (white arrows in Fig. 1).

We use a graded mesh, where the mesh size gradually increases from 3 nm to 6 nm with increasing distance from interfaces. In Table I, we summarize the parameters for the optimization routine.

For the analysis of the energy density product with respect to the twin density in MnAl-C in Sec. III B, we are varying the twin width and the mean grain diameter of the initial granular structure. The numerical optimization tool Dakota¹² chooses from the free parameters given in Table I and maximizes the energy density product of the given microstructure with the efficient global optimization algorithm.¹³ We keep the crystallographic orientation for all the simulations in the optimization routine. The exact limit of the free parameters in the optimization routine can be seen in Fig. 6 by the dashed line. This limit originated by geometrical parameters of the grain structure seen in experiments as well as from needed computational resources. For instance, a simulation model with a large box length and a small twin width could have too many finite elements to be calculated within a reasonable time.¹⁴ 

For the optimization routine in Sec. III B, we are computing the magnetic states by minimizing the Gibbs free energy using a nonlinear conjugate gradient method.¹⁵ The Gibbs free energy is given by the sum of the energies of exchange, anisotropy, demagnetization field, and external field. We discretize the energy of the magnetic structure using linear tetrahedral finite elements. The resulting algebraic optimization problem is solved using a conjugate gradient method. Using a standard finite element method,¹⁶ we compute the demagnetizing field, which is obtained by a magnetic scalar potential. In Table II(a), the parameters for the micromagnetic simulations are given. The intrinsic magnetic properties are taken from first principle and atomistic spin dynamics simulations at 300 K.¹⁷ 

We initially saturate the magnet with an external field $μ0Hext$ ascending from 0 T to 2 T. Afterward, we compute the demagnetization curve $M(Hext)$ by gradually reducing the external field $μ0Hext$ from 2 T to $−2$ T. We correct the computed demagnetization curve with the demagnetizing factor of a cube, $N=1/3$⁠, and compute the energy density product.

In a magnetic system, the coercive field $Hc$ is usually lower than the ideal nucleation field $Hn$⁠, i.e., $Hn$ is the theoretical limit for $Hc$⁠. Using micromagnetic simulations, the various origins of this discrepancy can be separated and analyzed individually.¹⁹ The discrepancy can be attributed to the misalignment of grains, to defects, and to demagnetization effects.

In classical micromagnetic theory, thermal activation in the system is considered only by the temperature-dependent intrinsic magnetic properties. By including thermal activation, the energy barrier between the metastable state before the switch and the reversed state can be passed at a lower external field. During magnetization reversal, an increasing opposing field reduces the energy barrier, the system follows the local minima until the energy barrier vanishes and the magnetization switches irreversibly.²⁰ 

We implemented the climbing string method²¹ for searching saddle points in the micromagnetic energy landscape.¹⁹ With an increasing external field, we look for the first saddle point, which gives the peak of the energy barrier and the exact location in the microstructure where the magnetization reversal is initiated. For a typical measurement time of 1 s, the magnetic system may overcome an energy barrier of 25 $kBT$⁠,²² where $kB$ is the Boltzmann constant and $T$ is the temperature. When the energy overcomes this 25 $kBT$⁠, we denote the field as the computed coercivity at elevated temperature.

In Sec. III, we analyze the coercive field and its reduction due to the described effects. We examine three different samples of MnAl-C listed in Table II. The procedure to separate the different effects is done as follows:

• Calculate ideal nucleation field $Hn=2K/(μ0Ms)$⁠.

• Calculate the Stoner–Wohlfarth switching field of a single grain, which takes into account misalignment, with an external field applied $1°$ off the easy axis.

• Compute classical micromagnetic simulations, which incorporates the magnetostatic field to determine the effect of the demagnetization field.

• Finally, consider the reduction of coercivity by thermal activation.

Using micromagnetic simulations, bulk magnets cannot be computed within a reasonable time and computational costs. A few hundred nanometers are typically the limit for the extension of the model magnet. In order to compute larger microstructures, e.g., 1000 grains with an extension of 20 $μ$m in Fig. 9, we are using a reduced order model by computing the switching fields with the embedded Stoner–Wohlfarth model.²³ At specific locations in the microstructure, Brown’s micromagnetic equation²⁴ is solved incorporating the applied external field, the anisotropy field, the demagnetization field, and an approximated exchange field. Each grain consists of such virtual particles that are located close to the grain surface. If the magnetization of one virtual particle inside a grain is switched, the whole corresponding grain is assumed to be switched. We consider the grains to be defect free and separated by a thin nonmagnetic grain boundary phase.

We show the formation of nanoscale defects created during hot-deformation of MnAl-C. Using numerical optimization routines, we tune the density of twin boundaries for a high energy density product. Further on, we discuss the reduction of coercivity with respect to a misalignment of grains, demagnetization effects, and thermal activation. In addition, we compare the results for different intrinsic magnetic properties. Finally, the switching field distribution of an ideal bulk structure with 1000 grains is analyzed using a reduced order model.

Several production routes exist to obtain the magnetically attractive $τ$-phase of MnAl-C. During hot-extrusion, the materials crystal structure transforms to regions with fine recrystallized grains and to some extent to large non-recrystallized grains. These non-recrystallized grains contain a high density of true twins in a lamellar configuration (Fig. 2), which are one of the reasons for the poor magnetic properties of current hot-extruded MnAl-C magnets.⁷ During demagnetization, the regions with high twin density reverse much faster than the fine recrystallized grains and cause a drastic reduction of coercivity in the material.

Twin boundaries are crystallographic defects that can trigger nucleation of reversed domains and pinning. Due to their strong misalignment, MnAl-C based magnets are hard to saturate. With true twin misorientation, the angle between the easy axes of adjacent crystals changes by 75.6$°$⁠. Figure 3 shows the initial and demagnetization curves of a MnAl-C magnet consisting of five grains using the model depicted in Fig. 1. A 2D cross section and a 3D transparent model including domain walls with (w) and without (w/o) twin defects demonstrate typical magnetic states along the curves. The orange letters a–i for the model without twins and the red letters a–n for the twin model correspond to the respective images in Figs. 4 and 5. In the beginning, an artificial domain wall is set in the center of the cube because we are especially interested in the domain wall movement. The simulation parameters are given in Table II(a).

The resulting hysteresis loops demonstrate that the model without the twin defects performs better compared to the model containing the twins. Both models jump to the negative magnetization direction right in the beginning at the zero field (a) due to the initial magnetic configuration and their preferred energy minimum. The twin model is slightly harder to saturate due to pinning of the domain wall on twin and grain boundaries (w, a–h). The magnet without the twins is slightly easier to saturate (w/o, a–f) and shows a higher remanence and coercive field (w/o, f–h), which agrees with the experimental results of Jia et al.⁶ The coercivity of the twin model is reduced by the magnetic configuration of the defects. The bright lamellas’ easy axes (w, h–i) are close to $90°$ with respect to the external field, i.e., the magnetization easily rotates in these lamellas. Through exchange interactions, the other half of the twins switch at a lower external field. Early nucleation can already be observed at a positive external field value (3D w, i). For a similar case of exchange coupled neighboring grains, the mechanism can be explained with analytic micromagnetic simulations.²⁵ For a misalignment angle of $90°$ between neighboring grains, the switching field is 0.45 $Hn$⁠, which is lower than the minimum Stoner–Wohlfarth switching field. Similarly, the saddle point for the nucleation of reversed domains is formed at the twin boundaries in Sec. III C. Note that close to saturation, the magnetization is on a reversible path (w/o, e–g) and (w, f–j).

The presence of twins reduces the switching field, especially easy reversible rotation may trigger irreversible switching in neighboring lamellas. Therefore, the coercive field in the model with twins is smaller than in the model without twins as shown in Fig. 3. After a significant portion of the twin sample is reversed, some lamellas remain unswitched. Pinning at twin boundaries hinders the expansion of a reversed domain through the entire magnet. An example can be seen in (w, k). When the reversed field is further increased, a reversed domain is nucleated at the grain boundary in the lamella and the domain wall moves through the lamella (2D, w, k–m).

The amount of twins within a grain is influencing the energy density product of a MnAl-C magnet. A bayesian search algorithm evaluates the MnAl-C model by varying the twin width and the mean grain diameter. In Fig. 6, all sampling points of the optimization routine are shown while the objective was to maximize the energy density product $(BH)max$⁠. The best energy density product of about $70kJ/m3$ is reached with thin twins in small grains (dark green). Further reducing grain size and twin width beyond the given optimization limits might improve this result. Coercivity values are in the range of 0.16–0.59 T and show no clear trend. Due to the misalignment of the twin crystals, and a consequently low remanence, these values are rather low in comparison to its ideal nucleation field $Hn$⁠. When twins are present, the remanence increases with decreasing grain size and decreasing width of the twins. In all cases, the coercive field is larger than half of the remanence. Therefore, remanence enhancement causes an increase in the energy density product as shown in Fig. 6. This situation is similar to remanence enhancement in isotropic nanocrystalline permanent magnets.²⁶ 

During irreversible switching, the magnetic system overcomes an energy barrier. Thermally activated switching over a finite barrier gives insight into the different effects that reduce the coercive field in a magnetic material as compared to the ideal nucleation field. If such a simulation is done for an ideal, small, and defect free grain, we obtain the maximum possible coercive field. We compute the maximum possible coercive field for a cube of MnAl-C with an edge length of 40 nm, and the external field applied at $1°$ with respect to the easy axis. In this section, we perform this analysis for three different intrinsic magnetic properties listed in Table II.

Figure 7 shows the ideal nucleation field $Hn=2K/(μ0Ms)$⁠, and the expected switching fields of the cube according to (1) the Stoner–Wohlfarth model that takes into account misalignment, (2) classical micromagnetics, and (3) thermally activated switching. The difference in the switching fields between (1) and (2) gives the reduction owing to demagnetizing fields. The difference between (2) and (3) gives the reduction owing to thermal activation. Depending on the set of intrinsic material, parameters used for the simulation the maximum possible coercivity for MnAl-C in an ideal structure range from 1.6 T to 4.4 T.

Huang et al. measured thin films of MnAl-C and achieved coercivity values up to 1.2 T.²⁷ In their work, the microstructure has not been characterized, but if grown from an amorphous phase, there could be fewer twins and a very small grain size. MnAl-C has a low saturation magnetization compared to other magnetic phases. Therefore, demagnetizing effects are less pronounced.

In a simulation model with five grains including the twinning defects (Fig. 8, mean grain diameter = 181 nm, twin width = 39 nm) the first nucleation is formed at an external field of 0.4 T. Several nucleation sites are appearing in one grain at every second lamella. This effect is also seen in Fig. 5, where the easy axes stand almost normal to the external field and magnetization is easily rotated. Only a small fraction of the theoretically reachable nucleation field $Hn$ is reached. The value is only slightly higher than achieved by good quality extruded samples (⁠$Hc≈0.3$ T), which are composed mostly of fine, recrystallized grains with few defects.^2,5 Probably, most of these grains act as single-domain particles.

We computed the switching field of grains in a granular structure using a reduced order micromagnetic model. We assume defect free grains which are separated by a nonmagnetic grain boundary phase. A decrease in the grain size increases the minimum switching field of each grain. This can be seen in Fig. 9, where switching field distributions of four different grain size models are presented. For grain sizes up to 2 $μ$m, a mean switching field greater than 1 T is expected using the anisotropy $K=0.7MJ/m3$ [Table II(a)]. In Fig. 9, on the right, the switching field is color-coded onto the grains with the 2 $μ$m grain size. Bright areas show weak spots of the cube with low switching fields, which indicates that nucleation typically starts at edges and corners of the magnet.²⁸ 

In Fig. 10, we demonstrate the drastic increase of the switching field distributions with higher anisotropies in the simulations.

Accurate micromagnetic simulations rely on realistic input parameters, the intrinsic magnetic properties, as well as the microstructure of the material. For the latter, proper information is obtained from transmission electron microscopy. For the first, there are discrepancies between atomistic theory and bulk measurements. For various sets of intrinsic magnetic properties, an ideal MnAl-C magnet with small, defect free grains could show a promising coercive field. While intrinsic properties gained by experiments suggest a high energy density product, the twin defects in the microstructure mitigate their performance. Hence, measurements of well prepared samples show only a fraction of the ideal nucleation fields.

A simple comparison of a MnAl-C cube with 400 nm edge length containing five grains with and without lamella twin structures confirms the decrease of coercivity in realistic structures. A cube containing twin defects is harder to saturate and shows a reduction in coercivity. While the twin defects introduce pinning sites for domain walls, twin lamellas with easy axes normal to the external field cause the magnet to nucleate at lower external fields. Hence, the pinning effect is dominated by the latter.

It is often difficult to compare absolute values between simulations and experimental measurements. Right now, the calculated values for twinned systems come close to the best extruded materials, which contain mostly fine recrystallized grains and only a small fraction of non-recrystallized grains. When twins cannot be avoided, numerical optimization suggests that small grains and thin lamellas improve the energy density product.

To conclude the results in this work, MnAl-C is a suitable material for making bulk granular magnets. Energy density products greater than $100kJ/m3$ may be achieved if crystallographic twinning defects can be suppressed sufficiently during production. If they cannot be completely removed, it is advisable to try reducing the grain size and the twin width of the lamellar structures. The smaller the twinning defects and the smaller the grains, the higher the energy density product can be expected.

This work was supported by the EU H2020 project NOVAMAG (Project No. 686056), the Austrian Science Fund (FWF) (Project No. I 3288-N36), and the German Research Foundation (DFG) (Project No. 326646134).

The data that support the findings of this study are available from the corresponding author upon reasonable request.