Data-driven design of a new class of rare-earth free permanent magnets
Alena Vishina, Daniel Hedlund, Vitalii Shtender, Erna K. Delczeg-Czirjak, Simon R. Larsen, Olga Yu. Vekilova, Shuo Huang, Levente Vitos, Peter Svedlindh, Martin Sahlberg, Olle Eriksson, Heike C. Herper
DData-driven design of a new class of rare-earth free permanent magnets
Alena Vishina, ∗ Daniel Hedlund, Vitalii Shtender, Erna K. Delczeg-Czirjak, Simon R. Larsen, Olga Yu. Vekilova,
4, 3
Shuo Huang, Levente Vitos,
5, 3
Peter Svedlindh, Martin Sahlberg, Olle Eriksson,
3, 6 and Heike C. Herper Department of Materials Science and Engineering,Uppsala University, Box 35, 751 03 Uppsala, Sweden Department of Chemistry - Ångström, Uppsala University, Box 538, 751 21, Uppsala, Sweden Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Uppsala, Sweden Department of Materials and Environmental Chemistry, Stockholm University, 10691 Stockholm, Sweden Applied Materials Physics, Department of Materials Science and Engineering,Royal Institute of Technology, Stockholm, SE-100 44, Sweden School of Science and Technology, Örebro University, SE-701 82 Örebro, Sweden (Dated: January 27, 2021)A new class of rare-earth-free permanent magnets is proposed. The parent compound of this classis Co Mn Ge, and its discovery is the result of first principles theory combined with experimen-tal synthesis and characterisation. The theory is based on a high-throughput/data-mining searchamong materials listed in the ICSD database. From ab-initio theory of the defect free material it ispredicted that the saturation magnetization is 1.71 T, the uniaxial magnetocrystalline anisotropyis 1.44 MJ/m , and the Curie temperature is 700 K. Co Mn Ge samples were then synthesized andcharacterised with respect to structure and magnetism. The crystal structure was found to be theMgZn -type, with partial disorder of Co and Ge on the crystallographic lattice sites. From mag-netization measurements a saturation polarization of 0.86 T at 10 K was detected, together with auniaxial magnetocrystalline anisotropy constant of 1.18 MJ/m , and the Curie temperature of T C = 359 K. These magnetic properties make Co Mn Ge a very promising material as a rare-earth freepermanent magnet, and since we can demonstrate that magnetism depends critically on the amountof disorder of the Co and Ge atoms, a further improvement of the magnetism is possible. Fromthe theoretical works, a substitution of Ge by neighboring elements suggest two other promisingmaterials - Co Mn Al and Co Mn Ga. We demonstrate here that the class of compounds based on T Mn X (T = Co or alloys between Fe and Ni; X=Ge, Al or Ga) in the MgZn structure type, forma new class of rare-earth free permanent magnets with very promising performance. I. INTRODUCTION
Rare earth (RE) permanent magnets dominate themarket where high-performance magnetic materials areneeded - areas such as green-energy generation, includ-ing wind and wave power, electric vehicle motors andgenerators, and many more. At the same time, the heav-ier rare-earth elements which are necessary for obtainingthe good magnetic characteristics of these materials (suchas Pr, Nd, Sm, Tb, or Dy) [1], can only be mined withmethods that leave an environmental footprint, are quiteexpensive, and are rapidly decreasing in availability. Asa consequence many rare-earth elements are labelled crit-ical. In order to pursue green technologies that rely onhigh-performance magnets, there is a growing interest infinding new magnetic materials containing cheaper andless critical elements, while maintaining a similar high-performance shown by their RE counterparts.In the last decades, many attempts have been made todesign both RE-lean [2, 3] and RE-free magnetic materi-als. The later include high magnetocrystalline anisotropyalloys (MnBi [4, 5] and MnAl [6, 7]), nanostructures [8– ∗ Department of Physics and Astronomy, Uppsala University, Box516, SE-75120, Uppsala, Sweden; [email protected] Mn Ge, which is a new material that previouslyhas not been considered as a permanent magnet. a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n II. HIGH-THROUGHPUT AND DATA-MININGSEARCH
An initial attempt to use high-throughput searches forRE-free permanent magnets was published in Ref.[18],employing a smaller subset of possible magnetic com-pounds. As described in this investigation, a high-performance permanent magnet must have ferromagneticordering, high saturation magnetization ( M S > T),high Curie temperature ( T C >
400 K), a large uniax-ial MAE ( > MJ/m ), and a magnetic hardness largerthan 1 (fo details see the Methods section). In the initialattempt of using high-throughput/data-mining searches,focus was put on materials with a stoichiometry neces-sarily containing one 3 d and one 5 d element [18]. Froma pure magnetic viewpoint, this study resulted in severalpromising materials. Unfortunately, none of the identi-fied compounds in Ref.[18] could serve as practical re-placements for the known rare-earth based compounds,since they all contained expensive elements like Pt.In the present study, we have made an extended searchto involve materials with a stoichiometry that containtwo types of 3d-elements, without any restriction on thetotal number of elements in the system. About a thou-sand materials from the ICSD database [30] contain twodifferent 3d-elements in the chemical formula. After thefirst high-throughput calculations of the electronic struc-ture and magnetic moments, about 45 systems were iden-tified to have a magnetic saturation field larger than avalue ∼ phase ofFeNi) or if the systems that were identified, are knownto not be ferromagnetic. These compounds were also re-moved from the list of potential new rare-earth free per-manent magnets. For the remaining materials, the MAEwas calculated. Compounds with planar or low magneticanisotropy (MAE < ∼ ) were not consideredfurther, leaving only five materials whose saturation mag-netization and MAE are high enough to be considered ashigh-performance permanent magnets.Note that the search criteria used here are somewhatless strict than the criteria of magnetic properties spec-ified from practical aspects of high performance perma-nent magnets. The reason for using less strict criteria,is to not miss new classes of compounds, that have thepotential to become technologically relevant, after e.g al-loying or structural refinements. Using these criteria wehave identified the following compounds as potential newpermanent magnets: ScFe P , Co Mn Ge, Mn V Si ,ScMnSi, and Cu Fe S and we report in Table I their cal-culated saturation magnetization, MAE, and magnetichardness. All data were obtained from calculations at T = 0 K and for a ferromagnetic configuration. For this final list of materials, we also calculated the Heisen-berg exchange interaction, and from Monte Carlo simu-lations we identified the ground state magnetic configu-ration and, for materials with ferromagnetic configura-tions, the Curie temperature. Out of the five materialslisted above, Co Mn Ge is the only one that is FM andhas a T C of 700 K. The rest of the materials were foundto have a non-collinear magnetic structure and were forthis reason not considered further. Some of the mate-rials, discarded during the search, can be found in the Appendix , Table III. We’d like to note, that Co Mn Gedoes have its magnetic hardness lower than 1. However,this parameter can be later on adjusted by alloying orother methods.TABLE I: Calculated MAE, saturation magnetization,Curie temperature, and magnetic hardness for thematerials that can be considered as candidates forRE-free permanent magnets. NC stands fornon-collinear spin structure obtained from Monte Carlosimulations.
Material ICSD Space MAE Sat. T C κ number group MJ/m magn., T KScFe P Mn Ge 52972 194 1.44 1.71 700 0.79Mn V Si Fe S III. EXPERIMENTAL RESULTSA. Synthesis and structural characterisation
Following the theoretical results presented above, fo-cus was then put on synthesis of Co Mn Ge. Initial tri-als revealed the magnetic Heusler phase Co MnGe [32]as being the main competing phase in that region of thephase diagram. Preliminary structural refinement of thehexagonal Co Mn Ge phase (MgZn -type) in a multi-phase sample indicated disordering on the 6 h and 2 a sitesimplying intermixing between Co and Ge. EDS analysison the multi-phase samples revealed the composition ofthe desired hexagonal phase to be in a small homogeneityregion spanning Co Mn Ge and Co . Mn . Ge . .Samples following these stoichiometries were then syn-thesized. The Co Mn Ge sample consisted of 95.4wt% of the hexagonal phase and 4.6 wt% of CoMn, aPauli paramagnet [33]. The diffractogram and accom-panying structural refinement is shown in Fig. 1. Thestructure of the compound (R Bragg = 4.20 %) is in goodagreement with previous studies [34] having similar unitcell dimensions (V = 154.60(1) ˚A compared to 154.61 ˚A [34]). The final refinement resulted in a compositionof Co . Mn Ge . with intermixing of Co and Ge onthe 6 h and 2 a sites.FIG. 1: (Color online) Refined powder diffraction dataof the synthesized Co Mn Ge alloy. Observed(Y obs ), calculated (Y calc ), difference (Y obs -Y calc )diffraction profiles and Bragg’s peaks positions forCo Mn Ge (95.4 wt.%) and CoMn (4.6 wt.%) phasesare shown.Results from XRPD refinements and microstructuralEDS results indicate that the structure is disordered witha tendency to contain more Co. Kuz’ma et al. [34] pro-posed both the ordered Mg Cu Si-type and disorderedMgZn -type structures from their XRPD results but aclear atomic distribution could not be established. Todetermine the atomic distribution for the theoretical cal-culations, precision single crystal studies were conducted.Three variants of the structure were considered during re-finement and are presented in Table II - the completelyordered structure, Co-Ge intermixing, and Mn-Ge inter-mixing. The structure solution obtained with SHELXT-2014 quickly provided an ordered model for Co Mn Ge.However, the wR value, the EDS results, and the dif-ference Fourier map all suggest that this model shouldbe rejected. The model of Mn-Ge intermixing showeda larger goodness-of-fit and the calculated compositionwas too far from the measured to be accepted. The fi-nal model of Co-Ge intermixing overall shows the bestparameters and is in agreement with other results. Allatoms were refined anisotropically. Detailed crystallo-graphic results of the SCXRD refinements can be foundin the Appendix , Table IV.
B. Experimental magnetism
The magnetization of the Co Mn Ge sample was mea-sured as a function of temperature and magnetic fieldto determine the magnetic ordering temperature T C ,the saturation magnetization, and the effective magneticanisotropy constant of the material. The temperaturedependent magnetization measurements were performed at several applied magnetic fields and in the temperatureregion between 10 K and 900 K. The results are shown inFig. 2. The magnetic ordering temperature, T C = 359 K,was determined from the Curie-Weiss law fit to the tem-perature dependence of the inverse magnetic susceptibil-ity (cf. inset in Fig. 2).FIG. 2: (Color online) Magnetization of Co Mn Ge as afunction of temperature in applied magnetic fields of µ H = 0.01 T (white open circles and white openrectangles) and µ H = 1 T (red filled rectangles). Theinset shows the Curie-Weiss fit for the inverse magneticsusceptibility with T C = 359 K. The cusp in µ H = 1 Tand drop in magnetization around 175 K is attributedto a spin–reorientation ( T srt ).The isothermal magnetization curves are shown in Fig-ure 3. Note that Figure 3(a) shows the result for bulkpowders whereas Figure 3(b–d) corresponds to singlecrystal measurements at different temperatures along ( (cid:107) )the crystallographic c–axis and perpendicular ( ⊥ ) to it.In Figure 3(a) the isothermal magnetization for the pow-der sample, measured at 10 K and 70 K, shows the sameapproach to saturation and reach the same saturation po-larization of 0.86 T. The isothermal magnetization mea-sured at 170 K reaches a saturation polarization close tothat recorded at lower temperature; whereas the satura-tion polarization measured at 300 K reaches a value of0.60 T.To analyze the behavior of the magnetic anisotropyand to compare to the value calculated theoretically, wehave used the law of approach to saturation to calcu-late the effective anisotropy constant at different temper-atures. The results are shown in Figure 4. At 10 K and70 K, the law of approach to saturation yields an effec-tive anisotropy constant of 1.18 MJ/m and the magnetichardness parameter, κ , becomes 1.42. The experimentalresults of magnetic configuration, saturation field, andmagnetic anisotropy hence seem consistent with the the-oretical predictions of this compound, which is gratifying.A closer inspection of the results presented in Fig. 2 re-veal that below ∼
175 K the magnetization drops consid-TABLE II: Results of single crystal refinements of the Co x Mn Ge − x compound. The Co-Ge disordered model(marked in bold) describes the measured data best. Parameters Ordered
Co-Ge disordered
Mn-Ge disorderedOccupation 6 h Co Co4 f Mn Mn Mn2 a Ge Mn Ge Co . Mn Ge . Co Mn . Ge . Calculated composition (at.%) Co Mn . Ge . Co Mn . Ge . Co Mn . Ge . Goodness-of-fit on F > = 0.0432 R = 0.0100 R = 0.0149wR = 0.1188 wR = 0.0264 wR = 0.0360R indices (all data) R = 0.0439 R = 0.0111 R = 0.0160wR = 0.1192 wR = 0.0268 wR = 0.0364Highest difference peak 2.277 -0.420 -1.0031- σ level 0.569 FIG. 3: (Color online) (a) Magnetization of a bulkpowder of Co Mn Ge, as a function of magnetic field at10 K, 70 K, 170 K and 300 K. (b,c,d) Isothermalmagnetization of single crystals of Co Mn Ge at 300 K,200 K and 100 K. Filled symbols show measurementswhere the magnetic field is perpendicular ( ⊥ ) to thecrystallographic c –axis whereas open symbols showmeasurements with the magnetic field parallell ( (cid:107) )withthe c –axis.erably when the applied field is small (0.01 T), whereasa cusp–like feature is seen in an applied magnetic fieldof 1 T, both for single crystal material and bulk sam-ples. Such features can be attributed to a change in mag-netocrystalline anisotropy with temperature and we in-terpret this to be a spin–reorientation temperature T srt ,from easy–axis to easy–cone anisotropy, a feature whichis not uncommon for permanent magnets (e.g. Fe SiB [35], MnBi [36], Cr . B . Te [37], Gd [38] and the cel-ebrated Nd Fe B [39]). Before presenting further re-sults supporting a claim of a temperature induced spin–reorientation, we draw attention to the fact that for a FIG. 4: (Color online) Approach to saturation plottedas the magnetization normalized with the valuemeasured at 5 T as a function of inverse applied fieldsquared for the bulk sample of Co Mn Ge.hexagonal uniaxial material the magnetic anisotropy en-ergy can be expressed as
M AE = K cos θ + K sin θ ,where V is the volume, K i are the anisotropy constants,and θ is the angle between the magnetization vector andthe hexagonal c–axis [40]. Depending on the values of K and K three cases can be distinguished; the ma-terial has an easy–axis, an easy–plane or an easy–conemagnetic anisotropy. The easy-cone state is the most fa-vorable one when − K ≤ K ≤ . The temperaturedependence of the anisotropy constants ( K i ( T ) ) may bestrong at low temperature and is usually described bya power–law behavior, K i ( T ) ∝ M ( T ) α [41, 42], where M ( T ) is the magnetization and α is a constant. The tem-perature dependence of the anisotropy constants can thenlead to a spin–reorientation, e.g. from an easy–axis to aneasy–plane or an easy–axis to an easy–cone anisotropy.The material under investigation here shows charac-teristics of an easy–axis to easy–cone spin–reorientationaround 175 K. We support this claim with the aid ofisothermal magnetization curves recorded at tempera-tures below and above T srt on the single crystals andbulk powders. The single crystal isotherms presented inFigure 3(b–d) indicate that the easy–cone state is presentat 100 K and below. Looking at the magnetization curvesat 300 K (Figure 3(b)) we see that it is easier to mag-netize the crystal along the c–axis than perpendicular toit. This shows that the material is magnetically uniax-ial at 300 K. The same applies at 200 K (Figure 3(c)),whereas at 100 K the measurements show that it is aseasy to magnetize the material parallel or perpendicu-lar to the crystallographic c –axis. However, it should benoted that at low field ( < . T) there is a deviationfrom a linear magnetic response in measurements per-formed parallel or perpendicular to the crystallographicc–axis. The non–linear behaviour at low fields, togetherwith the low field magnetization versus temperature mea-surements support the easy–cone state [43].Figure 4 supports the conclusion of a more complicatedbehaviour for the magnetization of Co Mn Ge. It shouldbe noted, that in order to extract an anisotropy constantusing the law of approach to saturation, averaging over arandom distribution of crystal directions yields the con-stant β = 4 / [44]. The effective anisotropy constantsand magnetic hardness parameter should thus be seenas the tentative descriptions of the magnetocrystallineanisotropy at low temperatures.Summarizing the experimental results so far we con-clude that the theoretical prediction of Co Mn Ge as apotential rare-earth free permanent magnet is likely. Thesaturation magnetization, uniaxial magnetic anisotropy,and magnetic hardness parameter obtained from experi-ments are consistent with the theoretical predictions. Itmust be noted that a more detailed picture is present inthe experiment, with a temperature stabilized easy-conestate. This state was not specified in the theoretical high-throughput screening search, and could therefore not beexpected from the theoretical prediction. However, theeasy axis, order of magnitude of the anisotropy, the sat-uration magnetization, and ordering temperature shouldagree, and here the theory has proven useful in finding anew material, with a potential to be used as a permanentmagnet. We return to the easy cone state, below, witha theory that takes disorder into account, and show thatcalculations then reproduce experimental observations.
IV. DISORDERED Co Mn Ge According to the experimental reports, there may be a50-50% intermixing of Co and Ge atoms on 6 h and 2 a [34]Wyckoff positions (see Fig. 5), in addition to the excessof Co in comparison to Ge (see Table II), in the exper-imental samples. This structure and composition differfrom the one reported in the ICSD [30] database, whereCo atoms are reported to occupy 6 h , Mn atoms take 4 f ,and Ge occupies 2 a atomic sites. It should be noted, that disorder of the type detected in our experiments isoften observed in the related ternary Laves phases withMgZn -type [45]. In this section, we, therefore, consider,from ab-initio alloy theory, the disorder of Co Mn Ge, inorder to analyze the effect it has on the magnetic state ofthe material and to compare the results with the theoret-ical data obtained for the ordered structure. The effect ofchemical and magnetic disorder on the stability and mag-netization of Co Mn Ge was investigated as described inthe
Methods section (Sec. VII).The results of these theoretical calculations is that theordered structure has a lower total energy, compared tothe disordered one. However, configurational entropycontributions to the free energy are reported in the ap-pendix to stabilize the disordered structure at tempera-tures 1700 K. The details can be found in
Appendix C .FIG. 5: (Color online) Ordered Co Mn Ge (as perICSD) with Co atoms occupying 6 h (light red balls),Mn atoms taking 4 f (violet balls), and Ge (grey balls)occupying 2 a atomic sites (top); and the disorderedCo Mn Ge structure obtained experimentally with50-50% intermixing between the Co and Ge sites(bottom).Following the experimental findings that point to aneasy-cone magnetic anisotropy at the lower temperatures,we calculated MAE for several magnetization directions( ◦ < θ < ◦ , φ = 0 ◦ ) for the ordered Co Mn Ge anddisordered Mn (Co . Ge . ) phases, see Fig. 6. In-deed, we can see that the disordered system shows apronounced deviation from the simple uniaxial behavior.Hence, the easy-cone anisotropy observed at lower tem-peratures is attributed to the Co-Ge disorder observedin the samples. At elevated temperature, the influenceof disorder evidently is less pronounced, and the uniax-ial anisotropy that was obtained from theory of orderedsamples, is recovered in the experiments.FIG. 6: (Color online) MAE for several magnetizationdirections ( φ = 0 ◦ ) for the ordered Co Mn Ge anddisordered Mn (Co . Ge . ) phases. V. CHEMICAL SUBSTITUTION OF Ge
In order to find related compounds with similar or im-proved magnetic properties compared to Co Mn Ge, weperformed additional theoretical work, where we replacedall Ge atoms in the unit cell by Al, Si, P, Ga, As, In, Sn,Sb, Tl, and Pb. All structures were relaxed and testedfor stability by calculating the formation enthalpy withrespect to their elemental components (for details see sec-tion
Methods ). The systems found to be stable with re-spect to the elemental components (Tl- and Pb-basedmaterials turned out to be unstable) were investigatedfurther for their magnetic characteristics. Their MAEand the details of the magnetic state are given in TableVII.FIG. 7: (Color online) Magnetic anisotropy energy ofCo Mn X, with X = Al, Si, P, Ga, Ge, As, In, Sn, Sb.Positive values indicate a uniaxial anisotrophy. Replacing Ge in Co Mn Ge by the neighboring ele-ments, does not change to the magnetic state much, butit has a large effect on the MAE, see Fig. 7. To de-termine the origin of the difference in MAE values weanalyzed the spin-orbit coupling energy (SOC) of theCo and Mn atoms with their spins oriented along the z and x directions as well as the partial density of states(DOS), similar to the analyses presented in Refs.[46–51].The details of that investigation can be found in Ap-pendix
D. Summarizing the results, we find that apartfrom the Co Mn Ge compound that is listed in the ICSDdatabase, Co Mn Al and Co Mn Ga are also expectedto have the properties of a good permanent magnet. Wehave shown that the main contribution to the large MAEof these compounds arises from Co atoms. A synthesisand charaterisation of these two compounds is outsidethe scope of this investigation, but represents clearly aninteresting avenue forward.
VI. DISCUSSION AND CONCLUSIONS
Using the high-throughput and data-mining approachwe filtered through the RE-free materials of the ICSDdatabase that contain necessarily two 3 d -elements. Onlyone of those approximately thousand structures satis-fied our requirements for a strong permanent magnet,Co Mn Ge, with the saturation magnetization of 1.71T, MAE equal to 1.44 MJ/m , and T C of 700 K. Toreduce the cost of this material, we attempted to re-place Ge by other elements. Two materials, Co Mn Aland Co Mn Ga, were found to have sufficient magneti-zation, MAE, and Curie temperature to be used as high-performance permanent magnets. The former of the twocan be problematic to produce due to the competingHeusler compound being considerably more stable; thelatter doesn’t have a similar drawback.Co Mn Ge was successfully synthesized using induc-tion melting followed by annealing at 1073 K. The re-sulting crystal structure was in good agreement with thepreviously published results. However, when the order-ing of the system was investigated with precision singlecrystal analysis, it indicated that the disordered struc-ture type MgZn with intermixing Co and Ge on the 6 h and 2 a sites was preferred. The structure taken from theICSD database and used in the high-throughput calcula-tion was, unlike the experimental one, ordered.Magnetic characteristics of the synthesized compoundwere measured and produced the saturation magnetiza-tion of 0.86 T at the temperature of 10 K, uniaxial mag-netic anisotropy equal to 1.18 MJ/m above around 175K (with the easy-cone character of anisotropy below),and T C =359 K. These values, even though lower thanthe theoretical predictions, make Co Mn Ge a promis-ing candidate for a high-performance permanent magnet.Further analysis of its magnetic state as well as the waysto tune some of these numbers is highly desirable andrepresents ongoing work.Calculations were also performed for the disorderedcrystal structure of Co Mn Ge obtained in the experi-ment. We found that the crystallographic ordered struc-ture with FM ordering is more stable than both ferro-magnetically and antiferromagnetically ordered config-urations of the structurally disordered compound, in atemperature range of 0 – 1735 K. Calculations of themagnetocrystalline anisotropy were also performed forthe disordered Mn (Co . Ge . ) phase. These calcu-lations show a pronounced deviation from the uniaxialcharacter of magnetic anisotropy, in agreement with ourexperimental low temperature data. Combining the re-sults of the theoretical and experimental works, leadsto the conclusion that structural disorder influences themagnetic properties, including the MAE, of Co Mn Gein a detrimental way, and that the best properties areexpected for the most ordered samples.In the
Appendix
H we list the properties of the allthe previously known Co–Mn–Ge systems. Based onthat information we can conclude, that the disorder isquite common in all the Co–Mn–Ge compounds anddepends strongly on the sample preparation and ex-perimental procedures. With that in mind, we be-lieve that further investigation into the sample prepa-ration or some possible doping alternatives is necessaryfor an attempt to stabilize Co Mn Ge in its orderedform, which is expected to have higher magnetizationand magnetic anisotropy. Having said that, even the dis-ordered Mn (Co . Ge . ) phase, with saturation po-larization of 0.86 T and uniaxial magnetic anisotropyof 1.18 MJ/m , possesses the characteristics of a goodpermanent magnet, although a further investigation intoincreasing the T C = 359 is desirable.It is also worth mentioning, that orientation of magne-tocrystalline anisotropy and the magnetic moment of Co–Mn–Ge systems is strongly affected by Co:Mn ratio. Thesynthesized sample of Co Mn Ge has the actual compo-sition of Co Mn Ge which might result in the dis-crepancy between the magnetic results obtained experi-mentally and the ones predicted in the high-throughputsearch. It was proven that the easy-cone magnetocrys-talline anisotropy, observed at the temperatures belowaround 175 K, is the result of the Co–Ge disorder. Wewould like to point out, however, that one of the bestcurrent permanent magnets, Nd Fe B [39], possesses asimilar feature. Nevertheless, this fact is another incen-tive to look for the way of stabilizing the ordered phaseof Co Mn Ge.The current investigation is a promising example ofthe material found in the theoretical data-mining searchbeing synthesized and showing the desired characteris-tics of a good permanent magnet. Further experimentalinvestigation into the magnetic state of Co Mn Ge willbe performed as well as the computational search for im-proving its price-performance. With its Curie tempera-ture close to the room temperature, we can also considerfine-tuning Co Mn Ge for magnetocaloric applications.
VII. METHODSA. High-throughput DFT
The high-throughput screening step to calculate themagnetic moment of the materials was performed usingthe full-potential linear muffin-tin orbital method (FP-LMTO) including spin-orbit interaction as implementedin the RSPt code [52, 53]. For this step the initial mag-netic configuration for all the materials was ferromag-netic (FM).The magnetic state of materials (FM or antiferro-magnetic - AFM), unless previously known, was de-termined using Vienna Ab Initio Simulation Package(VASP) [54–57] within the Projector Augmented Wave(PAW) method [58], along with the Generalized Gradi-ent Approximation (GGA) in Perdew, Burke, and Ernz-erhof (PBE) form [59]. VASP was also used for structurerelaxation at the post-high-throughput stage as well asto calculate spin-orbit coupling energies, for the analysispresented in the Appendix.The magnetic anisotropy energy (MAE) was calculatedusing the RSPt code as ∆ E = E pl − E c ; here E c and E pl are the total energies with the magnetization directedalong and perpendicular to the c -axis. Calculations wereperformed with the tetrahedron method with Blöchl cor-rection for the Brillouin zone integration [60]. A positivesign of the MAE corresponds to the required uniaxialanisotropy. For the disordered Co Mn Ge, the MAE wasevaluated for several polar angles from the force theorem[61, 62] as the difference of the eigenvalue sums for thetwo magnetization directions e θ and e c ( θ is a polar anglewhile azimuthal angle is equal to zero), while keeping theeffective potential fixed. From the first principles calcu-lations, the magnetic hardness parameter was evaluatedfrom the expression κ = (cid:112) ∆ E/µ M S [63], where M S issaturation magnetization and µ is the vacuum perme-ability.The Curie temperature T C was calculated using MonteCarlo simulations implemented within the Uppsala atom-istic spin dynamics (UppASD) software [64]. Atom-istic spin dynamics calculations were performed on a × × supercell with periodic boundary conditions.The required exchange parameters were calculated withthe RSPt code within the first nine coordination shells[65].Formation enthalpies of the materials were calculatedwith respect to their elemental components as ∆ H = H Co Mn X − H Co − H Mn − H X , for X= Al, Si, P, Ga, As, In, Sn, Sb, Tl, and Pb.In this expression H Co Mn X , H Co , H Mn , and H X are the enthalpies of formation for Co Mn X, hexago-nal close-packed (hcp) cobalt, body-centered cubic (bcc)manganese, and element X (such as, for example, cu-bic close-packed (ccp) aluminum for Co Mn Al), respec-tively. Formation enthalpies with respect to Heusler al-loys Co MnX were obtained according to the followingformula ∆ H = H Co Mn X − H Co MnX − H Co − H Mn , where H Co , H Mn , and H Co MnX are the enthalpiesof hcp cobalt, bcc manganese, and the correspondingHeusler alloy.The effect of chemical and magnetic disorder on thestability and magnetization of Co Mn Ge was investi-gated by means of the coherent potential approximation[66, 67] as implemented in the Exact Muffin-Tin Or-bitals (EMTO) method [68, 69]. We used s , p , d and f orbitals in the basis set. The one-electron equationswere solved within the soft-core and scalar-relativisticapproximations. The Green’s function was calculated for16 complex energy points distributed exponentially on asemi-circular contour including states within 1.1 Ry be-low the Fermi level. For the one-center expansion of thefull charge density a l h max =8 cutoff was used. The elec-trostatic correction to the single-site coherent potentialapproximation was described using the screened impu-rity model [70] with a screening parameter of 0.6. Totalenergies were calculated using the PBE [59] exchange-correlation functional, while local density approximation[71, 72] was used to calculate the magnetic moments andexchange interactions. The latter was calculated withinthe magnetic force theorem [73] for the ferromagneticand disordered local moment (DLM)[74, 75] configura-tions as implemented in the EMTO code. DLM repre-sents the high temperature paramagnetic (PM) phase. Inthis model, the paramagnetic phase of Co Mn Ge readsas (Co ↑ . Co ↓ . ) (Mn ↑ . Mn ↓ . ) Ge. Similar formulation isapplied to the paramagnetic phase of the alloy as well.
B. Synthesis
Samples of Co Mn Ge were synthesized by melting Co(Alfa Aesar, 99.9%), Mn (Höganäs AB, 99.9%) and Ge(Kurt J. Lesker, 99.999%) together in an induction fur-nace under Ar (purity 99.999%) atmosphere. The re-sulting ingots were placed in Al O crucibles, sealed inevacuated quartz glass tubes and annealed at 1073 K for14 days after which they were quenched in water. Afterthe final composition had been established by energy dis-persive X-ray spectroscopy analyses (see below), startingmaterials in stoichiometry of Co Mn Ge were pre-pared using the established protocol to yield the finalsamples. Samples were manually ground and powderstaken for analysis. C. Crystal structure analysis
The crystal structure was investigated using X-raypowder diffraction (XRPD), single crystal X-ray diffrac-tion (SCXRD) and scanning electron microscopy (SEM)coupled with energy dispersive X-ray spectrocopy (EDS). The powders were mounted on single-crystal Si sampleholders and X-ray diffraction patterns were collected us-ing a Bruker D8 Advance with monochromatized Cu-K α ( λ = 1.540598 Å) radiation at room temperature. Full-Prof was used with the Rietveld refinement method toanalyse the data [76]. A Bruker D8 single-crystal X-raydiffractometer with Mo K α radiation ( λ = 0.71073 Å) up-graded with an Incoatec Microfocus Source (I µ S, beamsize ≈ µ m at the sample position) and an APEX IICCD area detector (6cm × and H O was used.
D. Magnetic measurements
Magnetization versus field and temperature measure-ments were performed using a Quantum Design MPMSXL system. Isothermal magnetization curves wererecorded at several temperatures in applied magneticfields up to 5 T. Magnetization measurements were per-formed on bulk samples as well as on single crystals. Thesingle crystals used for these measurement were rathersmall hexagons, with a height of µ m and a side lengthof µ m, resulting in a magnetic moment of − – − Am . Care was done to avoid common artifacts intro-duced when using this system [78]. The small hexagonswere too small to accurately measure the weight of thesample, and thus the hysteresis curves from single crys-tals were scaled to match the magnetization at the sametemperature for bulk samples. The temperature depen-dent magnetization was measured between 10 K and390 K in the applied magnetic fields of 0.01 T and 1T. The temperature dependent magnetization was alsomeasured between 300 and 900 K and back to 300 K inan LakeShore VSM equipped with a furnace. The hightemperature measurements were performed in an appliedmagnetic field of 0.01 T using a heating/cooling rate of3 K/min. The magnetization in SI units was calculatedfrom the measured magnetic moment by using the sampleweight and density obtained from XRD measurements at298 K. The law of approach to saturation [44, 79] wasused to calculate the effective anisotropy constant of thematerial, | K eff | assuming the material to be uniaxial. VIII. ACKNOWLEDGEMENT
The authors would like to acknowledge the supportof the Swedish Foundation for Strategic Research, theSwedish Energy Agency (SweGRIDS), the Swedish Re-search Council, The Knut and Alice Wallenberg Foun-dation, STandUPP and the CSC IT Centre for Science,and the Swedish National Infrastructure for Computing(SNIC) for the computation resources. E. K. D.-Cz. ac-knowledges A. V. Ruban for valuable discussions. O.Yu. V. acknowledges the support of Sweden’s InnovationAgency (Vinnova).
IX. CONTRIBUTIONS
O.E., H.C.H, P.S. and M.S. initiated the research.A.V. performed the high-throughput search and data analysis. V.S. and S.R.L synthesized the samples. V.S.carried out the crystal structure analysis and SEM/EDS.D.H. performed magnetic measurements. DFT disor-der CPA calculations were performed and analysed byE.K.D.-C. and S.H. A.V., V.S., and D.H. drafted themanuscript, which was reviewed and edited by all theauthors. All authors contributed to discussions and anal-ysed the data.
X. COMPETING INTERESTS
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Some of the materials with the magnetic moment above the set-up threshold of 1.0 µ B /f.u. were discarded laterdue to the low or planar MAE or saturation magnetization being much lower than 1 T. These materials are listed inTable III.TABLE III: Systems with magnetic moment calculated to be higher than 1.0 µ B /f.u. during the high-throughputstep, which did not fulfill the requirements of a good permanent magnet on the further steps of data-mining. Material ICSD Space Mag. MAE (th) Sat. magn.number group state MJ/m TCrNiAs 43913 189 FM 0.37 0.86CrNiP 43913 189 FM 0.21 0.83Fe Ti 103663 194 FM 0.10 0.84CoCrGe 409451 194 FM -0.18 0.75CoGeMn 623495 194 FM -0.16 1.10Cu Co(SnSe ) 99296 121 FM 0.17Cu CoGeS CoSiS CoSnS Co C 44353 123 FM -2.21 0.43MnCoAs 610084 62 FM -0.88 0.90MnCoP 16483 62 FM -0.41 1.00Ni Sc 646468 191 FM 0.20GeMnSc 600156 189 FM 0.24 0.51Sc FeRh B Appendix B: Crystallographic data
Here we list the crystallographic and structural data for the hexagonal Co . Mn Ge . compound along with someof the measurement details.TABLE IV: Crystallographic data and experimental details of the single crystal structure refinement for thehexagonal Co . Mn Ge . compound. Measurements were carried out at 296 K with Mo K α radiation. Empirical formula Co Mn GeCalculated formula Co . Mn Ge . Structure type MgZn Formula weight, Mr (g/mol) 177.92Space group (No.) P / mmc (194)Pearson symbol, Z hP
12, 4Unit cell dimensions: a , Å 4.8032(2) b , Å 4.8032(2) c , Å 7.7378(4) V , Å 154.60(1)Calculated density, ρ (g · cm − ) 7.64Absorption coefficient, µ (mm − ) 31.835Theta range for data collection ( ◦ ) 4.901 ÷ F (000) h k l -8 ≤ h ≤ ≤ k ≤ ≤ l ≤ R eq = 0.0264)No. of reflections with I > σ ( I ) 174 ( R σ = 0.0096)Data/parameters 180/13Weighting details w=1/[ σ F o +0.5284P]where P=( F o + 2 F C )/3Goodness-of-fit on F R indices [ I > σ I ] R = 0.0100;w R = 0.0264 R indices (all data) R = 0.0111;w R = 0.0268Extinction coefficient 0.00135(13) TABLE V: Atomic coordinates, anisotropic displacement parameters and selected interatomic distances for thehexagonal Co . Mn Ge . compound. Atom Site x y z U eq (Å )0.837Co1+0.163Ge1 6 h f a U U U U d d | Ge1–Co2 | Ge2 2.4039(2) 3Co1 | Ge1–Mn1 2.7978(3)2Co1 | Ge1–Mn1 2.7815(5) 5Co2 | Ge2–Mn1 2.8178(2) U eq is defined as one third of the trace of the orthogonalized U ij tensor. U , U = 0.4 Appendix C: Ordering temperature, magnetic configuration, and crystal structure of the disorderedCo Mn Ge In this section, we present the additional data for the ordered Co Mn Ge and disordered Mn (Co . Ge . ) structures relaxed with EMTO code in the FM (Fig. 8) and PM (Fig. 9) phases. Table VI lists the crystal structure andmagnetic parameters obtained for these structures for both FM and PM magnetic configurations. Fig. 10 summarizesthe temperature effects for the FM and PM phases for the c/a-ratio fixed to the experimental value.To calculate the temperature which makes the ordered and disordered state degenerate in the FM and PM phase,the crystal structures of Co Mn Ge and Mn (Co . Ge . ) were relaxed in volume and c/a ratio using the EMTOcode, as shown in Fig. 8 and Fig. 9. Table VI lists the crystal structure and magnetic parameters obtained for thesestates for both FM and PM magnetic configurations. As can be seen, the chemical disorder does not have significanteffect on the magnetic properties in the FM state. However, magnetic configuration (FM or DLM) does stronglyaffect the atomic magnetic moments, especially in the case of Co (Table VI).The order-disorder transition can be estimated by the cross point of the free energies of the ordered and disorderedstructures, i.e. ∆ F = F dis - F ord . The temperature dependent free energy of the FM phase F ( T ) FM is estimated as F ( T ) FM ≈ E FM0 + F conf , where E FM0 is the internal energy for FM state at 0 K and F conf is configurational free energyevaluated at different temperatures. The energy difference of Co Mn Ge between the FM ordered and FM disordered(Mn (Co . Ge . ) ) states is about 4.124 mRy; the difference in the configurational entropy of the alloy betweenthe ordered and disordered phases equals to 0.375 k B . Since F conf = - T S conf we find the transition temperature to bearound 1735 K in the FM state.Temperature effects are summarized for the FM and PM phases in Figure 10 for c/a -ratio fixed to the experimentalvalue. As we can see, there is no transition up to either the measured (359 K) or the theoretically predicted (700 K)magnetic transition temperature (see left panel of Fig. 10). The ordered and disordered structures become degeneratein energy at ≈ T C for fixed c/a -ratio. None of the antiferromagnetic states considered in the calculations are stabilized by the configurationalentropy up to the T C .The Curie temperature for the ordered system and for Mn (Co . Ge . ) was estimated from J ij ’s calculated forthe ferromagnetic reference state and performing subsequent Monte-Carlo (MC) simulations. We obtained 720 K forCo Mn Ge and 760 K for the disordered case using the theoretical lattice parameters given in Table VI. It is satisfyingto note that the T C for the ordered system, obtained with EMTO method, is in good agreement with the RSPt results.The Curie temperature increase with disorder is due to the decrease of the antiferromagnetic Mn-Mn nearest neighborinteractions as chemical disorder is applied. T C estimated for the experimental structure and composition given inTable II is 820 K. We can conclude that the discrepancy between the predicted T C and the measured one does not comefrom the order-disorder effect alone. To address this issue, DLM calculations were also performed. These result in theconsiderable drop of Co local magnetic moment in the DLM state compared to the FM moments, while Mn momentdoes not change (see Table VI). Magnetic moment of any atomic species, that is reduced in the DLM configurationcompared to a FM configuration, will lead to the reduced strength of the inter-atomic exchange interactions, andreduce of the ordering temperature. Appendix D: Origin of magnetic anisotropy in Co Mn Ge The magnetocrystalline anisotropy originates from the SOC, since it is the only term in the Hamiltonian thatcouples spin- and real-space, something that was first suggested by Van Vleck [80]. In the case of transition metalswhere SOC is much smaller than the bandwidth or crystal field, it can be treated as a perturbation. This lead tothe possibility to connect the MAE with anisotropy in orbital moment [81]. The original expression for this coupling,made by Bruno, was subsequently extended, approximated, and used for various applications in [48, 50, 51, 82–86].It has been shown previously [50, 51, 82, 85] that the coupling between the occupied and unoccupied states closeto the Fermi energy (cid:15) F dominates the spin-orbit induced change of the total energy. Matrix elements (cid:104) µσ | L · S | µ (cid:48) σ (cid:48) (cid:105) [87, 88] determine the spin quantization axis direction which modifies the eigenvalues of the Kohn-Sham Hamiltonian.In this expression, µ represents a d-orbital (with symmetry { xy, yz, zx, x − y , z r } ), σ denotes spin, while L and S areorbital and spin angular momentum operators. For states within the same spin channel, the couplings d xy −→ d x − y , d yz −→ d xz promote the uniaxial anisotropy, while d xy −→ d xz , d xy −→ d yz , d xz −→ d z , d yz −→ d z , d xz −→ d x − y , and d yz −→ d x − y favour the easy-plane magnetocrystalline anisotropy [48, 50, 51, 82, 83, 85]. The situation is reversedfor the couplings between opposite spin channels. Table VIII (in Appendix E ) lists the transitions that contributeeither to easy-plane or uniaxial anisotropy along with their relative weights.5FIG. 8: (Color online) Volume and c/a relaxation of the ordered Co Mn Ge (left) and disordered (right)Mn (Co . Ge . ) structures in the FM phase. r WS denotes the Wigner-Seitz radius. Yellow lines follow theminimum of E( r WS ) curve for a specific value of c/a and the minimum of E ( c/a ) curve for each r WS .FIG. 9: (Color online) Volume and c/a relaxation of the ordered Co Mn Ge (left) and disordered (right)Mn (Co . Ge . ) structures in PM phase. r WS denotes the Wigner-Seitz radius. Yellow lines follow the minimumof E( r WS ) curve for a specific value of c/a and the minimum of E ( c/a ) curve for each r WS .To understand the origin of the difference in MAE for Co Mn X (X = Al, Si, P, Ga, As, In, Sn, Sb, Tl, and Pb), itis electron states close to the Fermi level ( (cid:15) F ) one should focus on, since spin-orbit interaction that couples states justbelow (cid:15) F to states just above (cid:15) F , are particularly important in deciding the magnetic anisotropy [22]. To undertakethis analysis we inspected partial densities of states (pDOS) around (cid:15) F . Figure 11 shows the pDOS for Co Mn Ge,which has a large uniaxial anisotropy of 1.44 MJ/m , and Fig. 12 shows the pDOS for Co Mn As, that has a largeeasy-plane anisotropy of -1.2 MJ/m . The pDOS curves for the other Co Mn X materials can be found in
Appendix F ,Figs. 13-14. The majority spin channel of the d-states is essentially fully occupied for all the materials considered inthis work which shows its inertness for the magnetic anisotropy. Instead significant contributions are expected for theminority spin channel, that has states on either side of (cid:15) F . In this spin channel there are large peaks correspondingto the d yz , d xz , and d xy states of Co, close to the Fermi energy. These peaks are mostly empty for X = Al, Ga, Ge,In, and Sn, while they lie directly on (cid:15) F for X = Si and Sb, and are mostly occupied for X = P, As. All the materialswith uniaxial magnetic anisotropy (X = Al, Ga, Ge, In) have these large peaks of the DOS occupied, and we concludethat from a microscopic point of view, these electrons are decisive for the magnetic anisotropy.To analyze the MAE further we consider the difference in SOC energies with spin orientation along the z and x axes, ∆ E so = E z so − E x so , separately for all Co and Mn atoms, see Table VII (negative sign marks a contribution touniaxial magnetic anisotropy). As expected, for most of X , ∆ E so (Mn) is considerably smaller than that of Co atoms.The latter can be divided into two groups (noted by the subscripts in Table VII); the contribution to ∆ E so from the6TABLE VI: Optimized crystal structure parameters ( a , c/a , and volume) and magnetic moments per unit cellcalculated for the ordered Co Mn Ge and disordered Mn (Co . Ge . ) phases using the EMTO code, along withthe low temperature experimental lattice parameters and magnetic saturation field. Third column containsexperimental data. Fourth column (labeled ”Co-excess”) contains experimental data for the low temperaturestructure, and calculated magnetic moments for this structure and for a composition given in Table II (54 % Co). Ordered Disordered Experiment Co-excessFM a , ˚A c/a V , ˚A m h Co , µ B m Mn , µ B m a Co , µ B M tot /cell, T 1.66 1.65 0.86 1.75DLM a , ˚A c/a V , ˚A m h Co , µ B m Mn , µ B m a Co , µ B FIG. 10: (Color online) Left panel: Normalized total energies ( E ) (full lines) and free energies ( F ( T ) ) (dashed anddotted lines) calculated for the ordered (black squares) and disordered (red circles) FM state of Co Mn Ge at 0 K,359 K (experimental T C ) and 700 K (theoretical T C ). Right panel: Normalized total energies ( E ) (full lines) andfree energies ( F ( T ) ) calculated for the ordered (black squares) and disordered (red circles) Co Mn Ge at 0 K and2300 K in the PM state. Dashed line stands for F ( T ) = E + F conf , dotted line denotes F ( T ) = E + F conf + F mag .On the x-axis, r WS denotes the Wigner-Seitz radius calculated as r WS = (cid:112) V atom / /π , where V atom is the averagevolume of an atom in the unit cell. F conf = - k B T (cid:80) i c i ln(c i ) and F mag =- k B T (cid:80) i c i ln(1+ m i ).two Co atoms positioned at (0.667, 0.833, 0.75) and (0.333, 0.166, 0.25) 6 h (we denote them type 2 ) is different tothat of the remaining four Co atoms ( type 1 ). All Co atoms contribute to the uniaxial magnetic anisotropy for X =Al, Ga, and In. The values of ∆ E so are extremely low for both Co and Mn atoms in Co Mn P, which results in thenegligibly small value of MAE. In the case of Co Mn Si and Co Mn As, all Co atoms give rise to the large valuescorresponding to easy-plane magnetocrystalline anisotropy. For the remaining materials, the two groups of Co atomshave opposite signs of ∆ E so .We can also determine the d-orbitals that give the largest change to the SOC matrix element (cid:104) µσ | (cid:98) H so | µ (cid:48) σ (cid:48) (cid:105) asthe magnetization direction changes from z to x . Co Mn X with X = Al, Ga, Ge, and In, that exhibit uniaxialanisotropy, all present a similar picture, see Fig. 19 (
Appendix G ) for Co Mn Ge (the rest of the figures can be7FIG. 11: (Color online) Contribution to the density of states of Co Mn Ge from the d x − y and d z ( e g set); d xy , d yz , and d xz ( t g set) orbitals without SOC interaction. Fermi energy is set at zero.FIG. 12: (Color online) Contribution to the density of states of Co Mn As from the d x − y and d z ( e g set); d xy , d yz , and d xz ( t g set) orbitals without SOC interaction.found in Appendix G , Fig. 15-21). The main contribution to the uniaxial anisotropy comes from (cid:104) d x + y | (cid:98) H so | d xy (cid:105) for all the Co atoms, even though there are additional contributions which are different for the Co atoms of type 1 (Fig. 19, top) and type 2 (Fig. 19, bottom). As expected (Table VIII), for these transitions to contribute to uniaxialanisotropy the states must be within the same spin channel. It is more difficult to distinguish any specific transitionsfor materials with the easy-plane anisotropy, see Fig. 20 ( Appendix G ) for Co Mn As (the other materials can befound in
Appendix G , Fig. 15-21) - there are several matrix elements promoting the negative sign in MAE. The mostsignificant are (cid:104) d x + y | (cid:98) H so | d yz (cid:105) , (cid:104) d xz | (cid:98) H so | d xy (cid:105) , and (cid:104) d z | (cid:98) H so | d yz (cid:105) and their size varies depending on X. Appendix E: d-orbitals contribution to MAE
Matrix elements (cid:104) µσ | L · S | µ (cid:48) σ (cid:48) (cid:105) [87, 88] determine the preferable direction of spin quantization axis. Table VIIIlists the transitions between the states below and above (cid:15) F which favour either z-axis (uniaxial) or xy-plane (planar)magnetic anisotropy along with the relative weight of each transition.8TABLE VII: Calculated MAE (RSPt), saturation magnetization (RSPt), the average magnetic moment per Co andMn atoms, Curie temperature (UppASD), formation enthalpies with respect to the elemental components for therelaxed Co Mn X structures ( ∆ H el ), and formation enthalpies with respect to the Co MnX Heusler compounds (X= Al, Si, P, Ga, Ge, As, In, Sn, Sb, Tl, and Pb) (VASP), marked ∆ H Heus . The last three columns contain thedifference in SOC energy with spin orientation along z and x axis (negative sign corresponds to uniaxial magneticanisotropy) for Mn and Co atoms (VASP). Subscript notes two cobalt atoms positioned at (0.667, 0.833, 0.75) and(0.333, 0.166, 0.25) 6 h sites, subscript points at the remaining four Co atoms. Material MAE Sat. M (Co) M (Mn) T C ∆ H el ∆ H Heus ∆ E so (Mn) ∆ E so (Co ) ∆ E so (Co )MJ/m magn., T µ B µ B K eV/f.u. eV/f.u. meV meV meVCo Mn Al 1.38 1.77 1.45 3.37 820 -1.78 0.274 0.016 -0.42 -0.46Co Mn Si -0.64 1.63 1.14 3.32 -2.34 0.090 -0.057 0.11 1.20Co Mn P 0.047 1.50 0.76 3.35 -2.62 0.018 -0.04 0.10Co Mn Ga 0.67 1.75 1.50 3.47 800 -1.49 0.061 0.100 -0.25 -0.30Co Mn Ge 1.44 1.71 1.44 3.52 700 -1.57 0.070 0.005 -0.26 0.13Co Mn As -1.2 1.58 1.08 3.50 -1.60 -0.011 0.01 1.13Co Mn In 0.36 1.65 1.60 3.62 -0.14 0.160 -0.11 -0.25Co Mn Sn -0.42 1.63 1.53 3.59 -0.57 0.064 -0.21 0.03Co Mn Sb -0.81 1.56 1.35 3.56 -0.67 0.047 -0.19 0.68Co Mn Tl -0.21 1.63 1.63 3.67 0.87Co Mn Pb -2.7 1.58 1.58 3.66 0.93
TABLE VIII: Transitions between the d-orbitals below and above (cid:15) F which contribute to either z-axis (uniaxial) orxy-plane (planar) magnetic anisotropy along with the relative weight of each transition, ω Contributes Same spin ω Opposite spin ω transition transitionUniaxial d xy −→ d x − y d xy −→ d xz , d xy −→ d yz d yz −→ d xz d xz −→ d z , d yz −→ d z d xz −→ d x − y , d yz −→ d x − y d xy −→ d xz , d xy −→ d yz d xy −→ d x − y d xz −→ d z , d yz −→ d z d yz −→ d xz d xz −→ d x − y , d yz −→ d x − y Appendix F: DOS for Co Mn X (X = Al, Si, P, Ga, In, Sn, Sb)
Here we provide the DOS for the crystal structures Co Mn X (X = Al, Si, P, Ga, In, Sn, Sb, Tl, and Pb). These werecalculated by replacing Ge in Co Mn Ge structure by the neighboring elements and relaxing their crystal structures.DOS of Co Mn Ge and Co Mn As are given above in the text.9 (a) Density of states of Co Mn Al (MAE = 1.38MJ/m ). (b) Density of states of Co Mn Si (MAE = -0.64MJ/m ).(c) Density of states of Co Mn P (MAE = 0.047MJ/m ). (d) Density of states of Co Mn Ga (MAE = 0.67MJ/m ).(e) Density of states of Co Mn In (MAE = 0.36MJ/m ). (f) Density of states of Co Mn Sn (MAE = -0.42MJ/m ). FIG. 13: (Color online) Density of states of Co Mn X, with X = Al, Si, P, Ga, Ge, As, In, Sn, Sb
Appendix G: Change of the SOC matrix elements of Co when magnetization changes direction from z to x forCo Mn X (X = Al, Si, P, Ga, In)
To determine the main orbital contribution to MAE we calculate the change in SOC matrix element (cid:104) µσ | (cid:98) H so | µ (cid:48) σ (cid:48) (cid:105) for transitions between different d -orbitals as magnetization direction changes from z to x . Fig. 15-21 show contribu-tions from type 1 Co atoms (top) and type 2
Co atoms (bottom).0FIG. 14: (Color online) Density of states of Co Mn Sb (MAE = -0.81 MJ/m ).FIG. 15: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1 (left), and Co, type 2 (right) inCo Mn Al when magnetization changes direction from z to x .FIG. 16: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1 (left), and Co, type 2 (right) inCo Mn Si when magnetization changes direction from z to x . Appendix H: The properties of the previously reported phases for Co-Mn-Ge system
The previously reported crystallographic phases for the Co–Mn–Ge system are presented in Table IX together withtheir crystal structure and some of the magnetic properties. We will briefly describe some of the key features of thestructural and magnetic properties of Co–Mn–Ge systems reported in the literature, which are expected to be relevant1FIG. 17: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1 (left), and Co, type 2 (right) inCo Mn P when magnetization changes direction from z to x .FIG. 18: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1 (left), and Co, type 2 (right) inCo Mn Ga when magnetization changes direction from z to x .FIG. 19: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1 (top), and Co, type 2 (bottom)in Co Mn Ge when magnetization changes direction from z to x .to the Co Mn Ge. It should be noted that this outline is not intended as a review, but rather as a summary of theknown properties of Co–Mn–Ge phases, which we find relevant to understanding the properties of Co Mn Ge.CoMnGe exists in two stable phases; the low-temperature orthorhombic CoMnGe (TiNiSi type) is stable below themartensitic temperature ( T M ) of 650 K [89] while the high-temperature hexagonal CoMnGe (BeZrSi type) is stableabove it [89]. The reported T C -values of the high- and low-temperature phases vary slightly, for instance, Kaprzyk andNiziol [90] report T C = 337 K for CoMnGe (TiNiSi) and T C = 287 K for CoMnGe (BeZrSi). As it is possible to tune2FIG. 20: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1 (top), and Co, type 2 (bottom)in Co Mn As when magnetization changes direction from z to x .FIG. 21: (Color online) Change of spin-orbit coupling matrix elements of Co, type 1 (left), and Co, type 2 (right) inCo Mn In when magnetization changes direction from z to x . T M to coincide with T C [89, 102, 103], a large/giant magnetocaloric effect can be achieved at the room temperature[92–94].CoGeMn is reported to exhibit a collinear [91] magnetic state. However, depending on the Co:Mn ratio,Co x Mn − x Ge samples can possess not only an easy-axis or easy-plane magnetic anisotropy but also an interme-diate state [95], i.e. spin reorients with Co:Mn ratio. The substitution of Ni (Co x Ni − x MnGe) produces the evenmore diverse results; the samples can be AFM, collinear FM, and non-collinear FM, depending on the amount of Niand the annealing temperature [100].Co MnGe is a full Heusler compound which exhibits some common features with the prototypical full HeuslerCu MnAl. These include a high T C and certain peculiarities related to its structural and magnetic properties derivedfrom the range of chemical environments, disorder, vacancies, and stacking faults. Using anomalous X-ray diffractiontechniques Collins et al. [97] were able to demonstrate that thin films of Co MnGe grown on Ge (111) show thepreference of Mn–Ge site swapping along with the small ( < . %) site-interchange of Co–Mn. The authors were alsoable to show that over–stoichiometry of Ge leads to the decrease in chemical disorder, vacancies, and stacking faults.It is important to note, that they were not able to grow the Co Mn Ge hexagonal phases due to a too large latticemismatch with the Ge (111) surface.Kogachi et al. [96] showed that the increase in sample quenching temperature gives rise to Mn–Ge disorder andresults in a lower magnetization at 4.2 K. Okubo et al. [98] were able to demonstrate that the aforementioned disorderbrings on a considerable decrease in T C . Webster [99], on the other hand, had earlier reported that there was nosign of chemical disorder in Co MnGe and that magnetic moment originated primarily from Mn (3.58 µ B ); Co onlycontributes 0.75 µ B only. Some of these ”discrepancies” reported previously are thus likely due to the difference in thepreparation of the samples, but they also show that Co MnGe exists in a number of states ranging from a collinearferromagnet with a high T C to a virtually non-magnetic material, depending on the chemical disorder.3TABLE IX: Co–Mn–Ge phases reported previously, with their crystal structures and transition temperatures. Compound Entry prototype, RemarksSGR Symbol and number and commentsCoMnGe TiNiSi,
Pnma (62) Low-temperature phase, stable below 650 K [89]; T C = 337 [90].CoMnGe BeZrSi, P / mmc (194) High-temperature phase, stable above 650 K [89];collinear [91] with T C = 283 K [90], 334 K [89].Transformation with T M = T C leads to large/giant magnetocaloriceffect [92–94].Co x Mn − x Ge Ni In, P / mmc (194) Spin reorientation depending on the amount of Co; can be easy axis,easy plane, as well as hard/easy equally in all directions [95].Co MnGe Cu MnAl, Fm ¯3 m (225) The increase in quenching temperature leads to increase in Mn-Gedisorder [96]. Disorder causes the decrease in the magnetization [96];higher quenching temperature rates produce lower magnetization [96].High Mn-Ge disorder, thin films [97], enrichment of 5 at % of Ge inCo . Mn . Ge . gives the highest degree of chemical ordering [97].Chemical disorder in Co MnGe decreases T C [98].Strong ordering in Co MnGe [99], almost all magnetic moment comesfrom Mn (3.58 µ B vs. 0.75 µ B for Co).Co x Ni − x MnGe TiNiSi,
Pnma (62) Samples can be either AFM or FM or non-collinear FM depending on x and the annealing temperature [100].Co Mn Ge Mg Cu Si, P / mmc (194) Ordered crystallographic model [34]Co Mn Ge MgZn , P / mmc (194) Disordered crystallographic model [34]Co Ge Mn Mn Co Ge , R h , (155) No magnetic properties reported. [101] Lastly, Co Ge Mn9