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High-throughput design of magnetic materialsTo cite this article: Hongbin Zhang 2021 Electron. Struct. 3 033001

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Electron. Struct. 3 (2021) 033001 https://doi.org/10.1088/2516-1075/abbb25

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TOPICAL REVIEW

High-throughput design of magnetic materials

Hongbin ZhangInstitute of Materials Science, Technische Universität Darmstadt, 64287 Darmstadt, Germany

E-mail: [email protected]

Keywords: high-throughput, magnetic materials, magnetocaloric materials, magnetic topological materials, permanent magnets, two-dimensional magnets

AbstractMaterials design based on density functional theory (DFT) calculations is an emergent field ofgreat potential to accelerate the development and employment of novel materials. Magneticmaterials play an essential role in green energy applications as they provide efficient ways ofharvesting, converting, and utilizing energy. In this review, after a brief introduction to the majorfunctionalities of magnetic materials, we demonstrated how the fundamental properties can betackled via high-throughput DFT calculations, with a particular focus on the current challengesand feasible solutions. Successful case studies are summarized on several classes of magneticmaterials, followed by bird-view perspectives.

Glossary of acronyms2D Two-dimensionalAFM AntiferromagneticAHC Aomalous Hall conductivityANC Anomalous Nernst conductivityAMR Anisotropic magnetoresistanceARPES Angle-resolved photoemissionBZ Brillouin zoneDFT Density functional theoryDLM Disordered local momentDOS Density of statesDMFT Dynamical mean field theoryDMI Dzyaloshinskii–Moriya interactionDMS Dilute magnetic semiconductorsFiM FerrimagneticFL Fermi liquidFM FerromagneticFMR Ferromagnetic resonanceFOPT First-order phase transitionGMR Giant magnetoresistanceHM Half-metalHMFM Half-metallic ferromagnetHTP High-throughputICSD Inorganic crystal structure databaseiSGE inverse spin Galvanic effectIFM Itinerant ferromagnetIM Itinerant magnetMAE Magnetocrystalline anisotropy energyMBE Molecular beam epitaxyMCE Magnetocaloric effectMGI Materials genome initiativeML Machine learning

© 2021 The Author(s). Published by IOP Publishing Ltd

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Electron. Struct. 3 (2021) 033001 Topical Review

Mr Remanent magnetizationMs Saturation magnetizationMOKE Magneto-optical Kerr effectMRAM Magnetic random-access memoryMSMA Magnetic shape memory alloyMSME Magnetic shape memory effectNFL Non-Fermi liquidNM NonmagneticPM ParamagneticQAHE Quantum anomalous Hall effectQAHI Quantum anomalous Hall insulatorQCP Quantum critical pointQPI Quasipartical interferenceQPT Quantum phase transitionQSHE Quantum spin Hall effectRE Rare-earthRPA Random phase approximationSCR Self-consistent renormalizationSdH Shubnikov–de HaasSDW Spin density waveSGS Spin gapless semiconductorSHE Spin Hall effectSIC Self-interaction correctionSOC Spin–orbit couplingSOT Spin–orbit torqueSTS Scanning tunneling spectroscopySQS Special quasi-random structureSTT Spin-transfer torqueTC Curie temperatureTI Topological insulatorTRIM Time-reversal invariant momentaTM Transition metalTMR Tunneling magnetoresistanceTN Neel temperaturevdW van der WaalsXMCD X-ray magnetic circular dichroism

1. Introduction

Advanced materials play an essential role in the functioning and welfare of the society, particularly the mag-netic materials as one class of functional materials susceptible to external magnetic, electrical, and mechanicalstimuli. Such materials have a vast spectrum of applications thus are indispensable to resolve the current energyissue. For instance, permanent magnets can be applied for energy harvesting (e.g. wind turbine) and energyconversion (e.g. electric vehicles and robotics), thus are in high demand. According to the BCC research report[1], the global market for soft and permanent magnets reached $32.2 billion in 2016 and should reach $51.7billion by 2022. In addition, to go beyond the quantum limit of conventional electronic devices, spintronicsexploiting the spin degree of freedom of electrons provides a promising alternative for engineering energyefficient apparatus, which has attracted intensive attention in the last decades. Nonetheless there are a varietyof pending fundamental problems and emergent phenomena to be understood. Therefore, there is a strongimpetus to develop better understanding of magnetism, so that magnetic materials with optimal performancecan be designed.

The conventional way of discovering and employing materials is mostly based on the empirical struc-ture–property relationships, which is time costly. Early in 2011, the US government has launched the materialsgenome initiative (MGI), aiming at strategically exploring materials design [2]. The proposed synergisticparadigm integrating theory, modeling, and experiment has proven to be successful, while there are still plentyof open challenges [3]. From the theoretical point of view, as the material properties comprise the intrinsic(as given by the crystal structure) and extrinsic (as given by the microstructure) contributions, a multi-scalemodeling framework should be established and embraced, leading to the integrated computational materialsengineering approach [4] and the European materials modeling council program [5]. For both frameworks

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and the counterparts, density functional theory (DFT) is of vital importance, due to its capability to obtainthe accurate electronic structure and hence the intrinsic properties and essential parameters for multi-scalemodeling, ensuring the predictive power.

Till now, high-throughput (HTP) computations based on DFT have been applied to screening for vari-ous functional materials, such as electro-catalysts [6], thermoelectrics [7], and so on. Correspondingly, opendatabases such as Materials Project [8], AFLOWlib [9], NOMAD [10], and OQMD [11] have been established,with integrated platforms like AiiDA [12] and Atomate [13] available. This changes the way of performing DFTcalculations from monitoring jobs on a few compounds to defining and applying workflows on thousands ofcompounds, so that the desired properties get optimized, e.g. the thermoelectric figure of merit [7].

In contrast to the other physical properties such as band gaps, absorption energies for catalysts, and ther-moelectric properties, magnetic properties and their characterization based on DFT pose a series of challengesand there has been limited exploration of designing functional magnetic materials. For instance, there are threekey intrinsic magnetic properties, i.e. magnetization, magnetic anisotropy energy (MAE), and critical orderingtemperature, which are difficult to be evaluated in an HTP manner. Besides the intriguing origin of magnetiza-tion (e.g. localized or itinerant) where consistent treatment requires a universal theoretical framework beyondDFT, it is already a tricky problem to identify the magnetic ground states as the magnetic moments can getordered in ferromagnetic (FM), ferrimagnetic (FiM), antiferromagnetic (AFM), and even incommensuratenoncollinear magnetic configurations. Moreover, the dominant contribution to the MAE can be attributedto the relativistic effect, i.e. spin–orbit coupling (SOC), where the accurate evaluation demands good con-vergence with respect to the k-mesh and other numerical parameters, resulting in expensive computationalefforts. Lastly, the critical ordering temperature is driven by the magnetic excitations and the correspondingthermodynamic properties cannot in principle be addressed by the standard DFT without an extension tofinite temperature. Due to such challenges, to the best of our knowledge, there are only a few limited successfulstories about applying the HTP method on designing magnetic materials (cf section 4 for details).

In this review, we aim at illustrating the pending problems and discussing possible solutions to facilitateHTP design of magnetic materials, focusing particularly on the intrinsic properties for crystalline compoundswhich can be evaluated based on DFT calculations. Also, we prefer to draw mind maps with a priority on theconceptual aspects rather than the technical and numerical details, which will be referred to the relevant litera-ture. Correspondingly, the major applications of magnets and the criticality aspects will be briefly summarizedin section 2. In section 3, we will tackle several fundamental aspects which are essential for proper HTP com-putation on magnetic materials, with detailed discussions on a few representative classes of magnetic materials.The in-depth mathematical/physical justification will not be repeated but referred to necessary publicationsfor curious readers, e.g. the recently published ‘Handbook of Magnetism and Advanced Magnetic Materials’[14] is a good source for elaborated discussions on the physics of magnetic materials. In section 4 the successfulcases will be summarized, with the future perspectives given in section 5.

It is noted that there are a big variety of magnetic materials which are with intriguing physics and promisingfor applications, as listed in the magnetism roadmap [15]. We excuse ourselves for leaving out the discussionson quantum magnets [16–18], quantum spin liquid [19], (curvilinear) nanomagnets [20, 21], skyrmions [22],high entropy alloys [23], multiferroics [24], and ultrafast magnetism [25], which will be deferred to reviewsspecified. Also, we apologize for possible ignorance of specific publications due to the constraint on the manpower.

2. Main applications of magnetic materials

2.1. Main applicationsThe main applications of magnetic materials is compiled in table 1, together with the representative materials.We note that small magnetic nanoparticles (2–100 nm in diameter) can also be used in biology and medicinefor imaging, diagnostics, and therapy [26], which is beyond the scope of this review. Most remaining appli-cations are energy related. For instance, both permanent magnets (i.e. hard magnets) and soft magnets areused to convert the mechanical energy to electrical energy and vice versa. Whereas spintronics stands for noveldevices operating with the spin of electrons, which are more energy efficient than operating charges.

Permanent magnets are magnetic materials with significant magnetization (either FM or FiM) which aremostly applied in generating magnetic flux in a gap, with the corresponding figure of merit being the maxi-mum energy product (BH)max [27]. As marked in figure 1, it provides an estimation of the energy stored in themagnets and can be enhanced by maximizing the hysteresis, achieved by increasing the remanent magnetiza-tion Mr and coercivity Hc. Such extrinsic quantities like Mr and Hc are closely related to the microstructureof the materials, where the corresponding upper limits are given by the intrinsic quantities such as saturationmagnetization Ms and MAE. Nowadays, for commercially available Nd2Fe14B, 90% of the theoretical limit of

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Table 1. A summary of main applications of magnetic materials, with representative compounds.

Required properties Materials Applications

Permanent magnets High anisotropy, large Ms & Mr, AlNiCo, ferrite, Power generation, electrichigh coercivity Hc, Sm–Co, Nd–Fe–B motor, roboticslow permeability,

high Curie temperatureSoft magnets Low anisotropy, large Ms, Fe–Si, low-carbon steel, Transformer core, inductor,

high permeability, Fe–Co magnetic field shieldsmall coercivity, low hysteresis,

low eddy current lossesMSMA Magnetic phase transition, Ni–Mn–Ga Vibration damper, actuator,

structural reorientation sensor, energy harvesterMagnetocaloric Large entropy change ΔS, La–Fe–Si, Ni–Mn–X, Magnetic refrigerationmaterials large temperature change ΔT, Gd5(Si, Ge)4

Minimal hysteresism,strong magnetization,

magneto-structural transitionsSpintronics Strong spin polarization, Half metals, DMS Sensor, MRAM, spin transistor

efficient spin injection,long spin diffusion length,

controllable interfaces,high ordering temperature

Magnetic storage Medium coercivity, large Co–Cr, FePt Hard diskssignal-to-noise ratio,

short writing time 10−9 s,long stability time 10 years

Figure 1. Typical hysteresis curves of a FM material. The B–H and M–H loops are denoted by red dashed and blue solid lines,respectively. The black bullets marks the critical values of the key quantities such as the residual induction Br, remanentmagnetization Mr, coercivity Hc , intrinsic coercivity Hci and saturation magnetization Ms . The maximal energy product (BH)max

is indicated by the shaded region.

(BH)max can be attained [28], suggesting that there is a substantial space of further improvement for the othermaterial systems.

In contrast, soft magnets are magnetic materials with significant Ms as well, and they are easy to be magne-tized and demagnetized corresponding to high initial and maximal relative permeability μr = B/(μ0H) whereB = μ0(H + M) denotes the magnetic induction under a magnetic field H, and μ0 is the vacuum magneticpermeability. That is, the corresponding hysteresis loop as shown in figure 1 is ideally narrow for soft magnets.In this regard, most soft magnets are Fe-based alloys of cubic structures, because Fe has the largest averagemoment among the 3d series of elements in solids. The soft magnets are widely applied in power generation,transmission, and distribution (table 1). In addition to the Ms which is an intrinsic property, microstruc-tures play a significant role in developing soft magnets, e.g. the eddy current losses caused by the cyclicalrearrangements of magnetic domains in AC magnetic fields should be minimized [29].

Another interesting class of magnetic materials are those with phase transformations (either first-orderor second-order) driven by external magnetic fields, leading to the magnetocaloric effect (MCE) and mag-netic shape memory effect (MSME). Compared to 45% efficiency for the best gas-compressing refrigerators,the cooling efficiency of MCE devices based on Gd can reach 60% of the theoretical limit [30]. Moreover, theMCE devices are highly compact and noise-less, giving rise to environment-friendly solutions for ever-growingdemands of cooling on the global scale. Following the thermodynamic Maxwell relation ∂S

∂B

∣∣T= ∂M

∂T

∣∣B, opti-

mized MCE can be achieved upon phase transitions with significant changes in the magnetization, whichcan be easily realized in those compounds with first-order phase transitions (FOPTs). This causes a problem

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about how to reduce the concomitant hysteresis caused by the athermal nature of FOPTs [31]. On the otherhand, MSME is caused by the domain wall twinning induced by the magnetic fields during the FOPTs (mostlymartensitic transitions), and the corresponding magnetic shape memory alloys (MSMAs) such as Ni–Mn–Gaalloys can be applied as actuators and sensors [32].

Last but not least, magnetic materials play a pivotal role in the spintronic information technologies [33]. Asdetailed in section 3.7, the first generation of spintronic devices rely on the spin-dependent transport (eitherdiffusive or tunneling) phenomena which is best represented by the discovery and application of the giantmagnetoresistance (GMR) effect and the conjugate spin transfer torque (STT) [34]. Whereas the second gener-ation spintronics takes advantage of SOC (thus dubbed as spin–orbitronics) and functions via the generation,manipulation, and detection of pure spin current, engaging both FM and AFM materials [35]. Many materi-als have been investigated such as half metals [36], dilute magnetic semiconductors (DMSs) [37], and so on,leading to devices including magnetic random-access memory (MRAM), spin transistors, etc. Additionally,magnetic materials can also be applied for the storage of information in both analogue and digital forms, wherethe materials optimization is a trade-off between competing quantities like signal to noise ratio, write-abilitywith reasonable fields, and long-term stability against thermal fluctuations [38].

2.2. Criticality and sustainabilityMagnetic materials are a prime example where the supply risk of strategic metals, here most importantlyrare-earth (RE) elements, might inhibit the future development. In general, resource criticality and sustain-ability is understood as a concept to assess potentials and risks in using raw materials for certain technologies,particularly strategic metals and their functionalities in emerging technologies [39]. Such principles and eval-uation can be transferred to the other material classes subjected to HTP design and future applications. As theapplication and market for magnetic materials is expected to grow in future technologies of energy, factorssuch as geological availability, geopolitical situation, economic developments, recyclability, substitutability, ecolog-ical impacts, critical competing technologies [633] and the performance of the magnetic materials have to beconsidered for the start in the development of new materials and their constituent elements.

To be specific, the RE elements such as Sm, Dy, and Tb (the latter two are usually used to improve the coer-civity and thermal stability of the Nd–Fe–B systems [40]) are of a high supply risk with an expensive price,as specified in the ‘critical materials strategy’ report [41], leading to a source of concern due to the ‘RE crisis’[42]. Particularly, such RE metals in high demands have relatively low abundance, whereas the abundant lightRE elements such as La and Ce may also be utilized to design volume magnets with desired properties. Thescenario also applies to transition metal (TM) and main group elements such as Ga and Ge, which are suscep-tible to the sustainable availability and can probably be substituted with Mn, Fe, Ni, Al, etc. In addition to highprices and low abundance, toxic elements like As and P are another issue, which dictate complex processing tomake the resulting compounds useful. That is, not all elements in the periodic table are equally suitable choicesfor designing materials in practice.

3. Main challenges and possible solutions

In this section, the fundamental aspects of designing magnetic materials will be discussed, with the pendingproblems and possible solutions illustrated.

3.1. New compounds and phase diagramThere have been of the order of 105 inorganic compounds which are experimentally known, which amount toa few percent of all possible combinatorial compositions and crystal structures [43]. Therefore, a common taskfor materials design of any functionality is to screen for unreported compounds by evaluating the stabilities,which can be performed via HTP DFT calculations. Theoretically, the stabilities can be characterized in termsof the following criteria:

(a) Thermodynamical stability, which can be characterized by the formation energy and distance to the convexhull (figure 2), defined as

ΔG(T, P) = G(target) − G(comp. phases) � 0, (1)

where G(target) and G(comp.phases) denote the Gibbs free energy at finite temperature T and pressureP for the target and competing phases, respectively. In most cases, only the formation energy with respectto the constituent elements is evaluated, but not the convex hull with respect to the other competingphases. It is observed that evaluating the convex hull can reduce the number of stable compounds by oneorder of magnitude [44]. Thus, it is recommended to carry out the evaluation of convex hull routinelyfor reasonable predictions, using those relevant compounds collected in existing databases as competingphases. Certainly, there are always unknown phases which can jeopardize the predictions, but it is believed

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Figure 2. Sketch of the convex hull for a hypothetical binary metal(M) nitride. Each circled label (A-F) denotes a possiblecompound with different thermodynamic stability as noted. Reprinted with permission from [45]. Copyright (2017) AmericanChemical Society.

that the false positive predictions will be significantly reduced with the resulting candidates more accessiblefor further experimental validation.

A few comments are in order. First of all, DFT calculations are usually performed at 0 K and ambientpressure, thus cannot be directly applied to access the metastability at finite temperature and pressure.The pressure can be easily incorporated into calculations using most DFT codes, whereas the temperatureconstraint can also be remedied by combining the DFT and CALPHAD [46] methods, where the ther-modynamic stabilities can be evaluated in a more systematic manner particularly for multicomponentsystems. The resulting phase diagram will also provide valuable guidance for the experimental synthe-sis. Moreover, the stability of metastable phases can be further enhanced by modifying the experimentalprocesses. For instance, precursors can be used in order to reduce the thermodynamic barrier (figure 2),e.g. the more reactive nitride precursors like NH3 allows the synthesis of previously metastable phases[45]. Lastly, non-equilibrium synthesis techniques such as molecular beam epitaxy (MBE), melt spin-ning, mechanical alloying, and specific procedures to get nano-structured materials can also be applied toobtain metastable phases, as demonstrated for the ε-phase of MnAl [47].

Particularly for magnetic compounds, in most cases when evaluating the thermodynamic stability, theformation energies and distances to the convex hull are obtained assuming FM configurations, as donein Materials Project and OQMD. This is justified for the majority of the systems, but we observed thatthe magnetic states will change the energy landscape drastically for compounds with strong magneto-structural coupling, which will be discussed in detail in section 3.3. Another critical problem is how toobtain reliable evaluation of the formation energies for the RE-based intermetallic compounds, wheremixing DFT (for the intermetallics) and DFT + U (for the RE elements) calculations are required. Thisis similar to the case of TM oxides, where additional correction terms are needed to get a reasonableestimation of the formation energies [48]. To the best of our knowledge, the resolution is still missing forRE-based intermetallic compounds.

(b) Mechanical stability, which describes the stability against distortions with respect to strain. It can beformulated as

ΔE = Edistorted − E0 =1

2

∑i,j

Ci,jεiεj > 0, (2)

where Cij denotes the elastic constant and ε the strain, and E0 denotes the ground state energy from DFTwith equilibrium lattices. Depending on the crystalline symmetry, equation (2) can be transformed intothe generic Born stability conditions [49], i.e. a set of relationships for the elastic constants which can bestraightforwardly evaluated based on DFT.

(c) Dynamical stability, which describes the stability against local atomic displacements. In the harmonicapproximation [50], it yields

E = E0 +1

2

∑R,σ

∑R′,σ′

DR,σΦσ,σ′

R,R′DR′,σ′

︸︷︷︸>0

, (3)

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where Φσ,σ′

R,R′ is the force constant matrix, DR,σ marks the displacement of atom at R in the Cartesian direc-

tion σ. The positive definiteness can be assured by diagonalizing the Fourier transformation of Φσ,σ′

R,R′ ,where no negative eigenvalue is allowed. That is, there should be no imaginary phonon mode in the wholeBrillouin zone (BZ). On the other hand, if there does exist imaginary phonon modes, particularly at afew specific q-points, it suggests that the compounds can probably be stabilized in the correspondinglydistorted structure.

Till now, the stabilities are evaluated assuming specific crystal structures. Such calculations can be eas-ily extended to include more structural prototypes based on the recently compiled libraries of prototypes [51,52]. This has been applied successfully to screen for stable ABO3 perovskite [53] and Heusler [54] compounds.One interesting question is whether there are phases with unknown crystal structures which might be stableor metastable, giving rise to a question of crystal structure prediction. It is noted that the number of pos-sible structures scales exponentially with respect to the number of atoms within the unit cell, e.g. there are1014 (1030) structures with 10 (20) atoms per unit cell for a binary compound [55]. There have been wellestablished methods as implemented in USPEX [56] and CALYPSO [57] to predict possible crystal structures.For instance, within the NOVAMAG project, the evolutionary algorithm has been applied to predict novelpermanent materials, leading to Fe3Ta and Fe5Ta as promising candidates [58].

To summarize, HTP calculations can be performed to validate the stability of known compounds and topredict possible new compounds. The pending challenges are (a) how to systematically address the stabilityand metastability, ideally with phase diagrams optimized incorporating existing experimental data, (b) how toextent to the multicomponent cases with chemical disorder and thus entropic free energy, so that the stabilityof high entropy alloys [59] is accessible, and (c) how to perform proper evaluation of the thermodynamicstability for correlated RE compounds and TM oxides.

3.2. Correlated nature of magnetismFor magnetic materials, based on (a) how the magnetic moments are formed and (b) the mechanism couplingthe magnetic moments, the corresponding theory has a bifurcation into localized and itinerant pictures [60].The former dates back to the work of Heisenberg [61], which applies particularly for RE-based materials withwell-localized f-electrons and strongly correlated TM insulators. The local atomic moments can be obtainedfollowing the Hund’s rule via

μeff = gJ

√J(J + 1)μB, (4)

where gJ is the Lande g-factor, J denotes the total angular momentum, and μB is the Bohr magneton. Themagnetic susceptibility above the critical ordering temperature (i.e. in the paramagnetic (PM) states) obeysthe Curie–Weiss law [62]:

χ(T) =C

T − θW, (5)

where C is a material-specific constant, and θW denotes the Curie–Weiss temperature where χ shows a singularbehavior. The Curie-Weiss temperature θW can have positive and negative values, depending on the resultingFM and AFM ordering, whereas its absolute value is comparable to the Curie/Neel temperature. The Curie-Weiss temperature corresponds to the molecular field as introduced by Weiss [63], which can be obtained viaa mean-field approximation of the general Heisenberg Hamiltonian:

H = −1

2

∑i,j

JijSi · Sj, (6)

where Jij denotes the interatomic exchange interaction. Specifically, Si · Sj is approximated to Szi 〈Sz

j 〉+ 〈Szi 〉Sz

j

by neglecting the fluctuations and hence the spin flip term S+i S−j , leading to an effective field proportional tothe magnetization [64].

On the other hand, for intermetallic compounds with mobile conduction electrons, the magnetic order-ing is caused by the competition between the kinetic energy and magnetization energy. Introducing theexchange energy I = J/N where J is the averaged exchange integral originated from the intra-atomic inter-orbital Coulomb interaction and N the number of atoms in the crystal, a FM state can be realized if [65]

Iν(EF) > 1, (7)

where ν(EF) is the density of states (DOS) at the Fermi energy EF in the nonmagnetic state. The so-obtaineditinerant picture has been successfully applied to understand the occurrence of ferromagnetism in Fe, Co, Ni,and many intermetallic compounds including such elements [66].

However, the real materials in general cannot be classified as either fully localized or fully itinerant and aremostly at crossovers between the two limits. As shown in figure 3, where the ratio of Curie–Weiss constant qc

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Figure 3. The Rhodes–Whlfarth ratio qc/qs for various magnetic materials. Reproduced from [62]. © IOP Publishing Ltd. Allrights reserved.

and the saturation magnetization qs is plotted with respect to the Curie temperature, the qc/qs ratio shouldbe equal to 1 for localized moments as indicated by EuO, whereas for the itinerant moments qc/qs � 1. It isinteresting that the qc/qs ratio for Ni is close to 1 while it is a well known itinerant magnet. More interestingly,there exists a class of materials such as TiAu and ZrZn2 where there is no partially filled d- or f-shells but theyare still displaying FM behavior [62].

It is noted that the localized moment picture based on the Heisenberg model has limited applicabilityon the intermetallic magnets, while the Stoner picture fails to quantitatively describe the finite temperaturemagnetism, e.g. it would significantly overestimate the Curie temperature and leads to vanishing moment andhence no Curie–Weiss behavior above TC. A universal picture can be developed based on the Hubbard model[67],

H =∑

ijσ

tija†iσajσ +

∑i

Uni↑ni↓, (8)

where tij is the hopping parameter between different sites, n is the number of electrons, and U denotes theon-site Coulomb interaction. As a matter of fact, the Stoner model is a mean field approximation of the Hub-bard model in the weakly correlated limit and the Heisenberg model can be derived in the strongly coupledlimit at half-filling [68]. Actually, another unified theory has been formulated by Moriya considering self-consistent renormalization of the spin fluctuations in the static and long-wave length limits [60], which worksremarkably for the weak itinerant magnets such as ZrZn2 [62]. In terms of the Hubbard model, which can befurther casted into the DFT + dynamical mean-field theory (DMFT) framework [69], where the hoppings areobtained at the DFT level and the onsite electron–electron correlations are evaluated accurately including allorders of Feynman diagrams. Recently, the DFT + DMFT methods have been applied on ZrZn2 [70], reveal-ing a Fermi liquid (FL) behavior of the Zr-4d electrons which are responsible for the formation of magneticmoments. In this regard, it provides a universal solution which can be applied for magnetic materials fromthe localized to itinerant limits, which is better than bare DFT. Importantly, for correlated TM oxides and REcompounds, the d and f moments are mostly localized, leading to narrow bands where the local quantum fluc-tuations are significant and hence the spin fluctuations [71], which are naturally included in DFT + DMFT.Detailed discussions will be presented in section 3.4. We noted that the classification of itinerant and localizedmagnetism is not absolute, e.g. the Ce-4f shell changes its nature depending on the crystalline environment[72].

In this sense, DFT as a mean field theory has been successfully applied to magnets of the itinerant andlocalized nature. In the latter case, the DFT + U [73] method is usually applied to account for the strong elec-tronic correlations. However, there are known cases where DFT fails, e.g. it predicts an FM state for FeAl whileexperimentally the system is PM [74]. Also, the local density approximation (LDA) gives wrong ground stateof Fe, i.e. LDA predicts the nonmagnetic fcc phase being more stable than the FM bcc phase [75]. Therefore,cautions are required when performing DFT calculations on magnetic materials and validation with experi-ments is always called for. On the other hand, the DFT + DMFT method is a valuable solution which coversthe whole range of electronic correlations, but the key problem is that the current implementations do notallow automated set-up and efficient computation on a number of compounds.

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3.3. Magnetic ordering and ground statesAssuming the local magnetic moments are well defined, i.e. the Heisenberg Hamiltonian equation (6) isvalid to describe the low temperature behavior, the interatomic exchange Jij is the key to understand theformation of various long-range ordered states. There are three types of mechanisms, based on the distancebetween the magnetic moments and how do they talk to each other.

(a) Direct exchange for atoms close enough so that the wave functions have a sufficient overlap. Assumingtwo atoms each with one unpaired electron, the Coulomb potential is reduced for two electrons localizedbetween the atoms when the atoms are very close to each other, leading to AFM coupling based on thePauli’s exclusion principle. On the other hand, when the distance between two atoms becomes larger,the electrons tend to stay separated from each other by gaining kinetic energy, resulting in FM coupling.Thus, the direct exchange is short ranged, which is dominant for the coupling between the first nearestneighbors. This leads to the Bethe–Slater curve, where the critical ratio between the atomic distance andthe spatial extent of the 3d-orbital is about 1.5 which separate the AFM and FM couplings [76].

(b) Indirect exchange mediated by the conduction electrons for atoms without a direct overlap of the wavefunctions. It can be understood based on the Friedel oscillation, where the conduction electrons try toscreen a local moment, thus forming long range oscillating FM/AFM exchange coupling, leading to theRKKY-interaction named after Ruderman, Kittel, Kasuya, and Yoshida [77]. It is the dominant exchangecoupling for atomic moments beyond the nearest neighbors in TM magnets and localized 4f-moments inRE magnets.

(c) Superexchange for insulating compounds with localized moments, which is driven by kinetic energygain via the virtual excitations of local spin between magnetic ions through the bridging nonmagneticelements, such as oxygen in TM oxides. The superexchange can favor both AFM and FM couplingsdepending on the orbital occupation and the local geometry, which can be understood based on theGoodenough–Kanamori–Anderson rules [78].

We note that in real materials, more than one type of exchange coupling should be considered, e.g. thedirect exchange and RKKY exchange for intermetallic compounds, and the competition of direct exchangeand superexchange in TM oxides.

Based on the discussions above, the exchange coupling between atomic moments can be either FM orAFM, depending on the active orbitals, distances, and geometries. Such AFM/FM exchange couplings inducevarious possible magnetic configurations, which give rise to a complication for HTP screening of magneticmaterials. That is, the total energy difference for the same compounds with different magnetic states can besignificant. This can be roughly estimated by the Curie temperature (TC), i.e. about 0.1 eV corresponding to theTC (1043 K) of bcc Fe. Thus, accurate evaluation of the thermodynamic stability should have the magneticground states considered, where further evaluation of the electronic structure can be based on. However, thenumber of possible AFM configurations which should be considered to define the magnetic ground state canbe big, depending on the crystal structures and the magnetic ions involved. Therefore, identifying the magneticground state is the most urgent problem to be solved for predictive HTP screening on magnetic materials.

There have been several attempts trying to develop a solution. The most straightforward way is to collectall possible magnetic configurations from the literature for a specific class of materials or consider a numberof most probable states, such as done for the half-Heuslers [80]. However, it is hard to be exhaustive and moreimportantly it is only applicable for one structural prototype. Horton et al developed a method to enumeratepossible magnetic configurations followed by HTP evaluation of the corresponding total energies [81]. It ismostly applicable to identify the FM ground state, as the success rate is about 60% for 64 selected oxides whichmight be due to the correlated nature of such materials. Another interesting method is the firefly algorithm, asdemonstrated for NiF2 and Mn3Pt [82], where the magnetic configurations are confined within the primitivecell. A systematic way to tackle the problem is to make use of maximal magnetic subgroups [79], where themagnetic configurations can be generated in a progressive way taking the propagation vectors as a controlparameter. We have implemented the algorithm and applied successfully on binary intermetallic compoundswith the Cu3Au-type structure. It is observed that the landscape of convex hull changes significantly afterconsidering the magnetic ground state, as shown in figure 4 for the Mn–Ir binary systems. That is, the magneticground state matters not only for the electronic structure but also for the thermodynamic stability of magneticmaterials, confirming our speculation in section 3.1. Recently, the genetic algorithm has been implemented togenerate magnetic configurations and combined with DFT calculations the energies for such configurationscan be obtained in order to search for the magnetic ground state [83]. It has been applied successfully on FeSe,CrI3 monolayers, and UO2, where not only the collinear but also noncollinear configurations can be generated,which is interesting for future exploration.

Furthermore, the interatomic exchange Jij in equation (6) can be evaluated based on DFT calculations,which can then be used to find out the magnetic ground states via Monte Carlo modeling. There are different

9


Figure 4. The binary convex hull for Mn-Ir. Dashed (solid) lines denotes the convex hull obtained assuming ferromagneticconfigurations (with real magnetic ground states obtained via systematic calculations by maximal magnetic subgroups). Themagnetic ground states for MnIr and Mn3Ir are display on top of the figure. ‘HTE’, ‘HTE-FM’, ‘MP’, and ‘OQMD’ denotes theconvex hull obtained using our high-throughput environment with proper magnetic ground states, high-throughputenvironment with only FM configurations, Materials Project, and Open Quantum Materials Database, respectively [79].Reproduced from [79]. CC BY 2.0.

methods on evaluating the Jij. The most straightforward one is the energy mapping, where the Jijs for eachpair of moments are obtained by the difference of total energies for the FM and AFM configurations [84].Such a method can be traced back to the broken symmetry method firstly proposed by Noodleman [85] andlater generalized by Yamaguchi [86], where one of the spin state used in energy mapping is not the eigenstateof the Heisenberg Hamiltonian (equation 6). In this regard, for magnetic materials with multiple magneticsublattices, Jij can be obtained by performing least square fitting of the DFT total energies of a finite numberof imposed magnetic configurations, e.g. using the four-state mapping method [84, 87].

Based on a two-site Hubbard model, it is demonstrated that the energy mapping method is only accuratein the strong coupling limit, i.e. insulating states with well defined local moments [88]. Additional possibleproblems for such an approach are (a) the supercells can be large in order to get the long-range interatomicexchange parameters and (b) the magnitude of local moments might change between the FM and AFM calcu-lations resulting in unwanted contribution to the total energies from the longitudinal excitations. Thus, thismethod is most applicable for systems with localized moments. In this case, Jij can also be evaluated based onthe perturbation theory, which can be done with the help of Wannier functions thus there is no need to gen-erate supercells [89]. The artificial longitudinal excitations can also be suppressed by performing constrainedDFT calculations [90].

There are two more systematic ways to evaluate Jij. One is based on the so-called frozen magnon method[91], where the total energies for systems imposed with spin-waves of various q vectors are calculated andafterward a back Fourier transformation is carried out in order to parameterize the real space Jij [92]. Thismethod demands the implementation of noncollinear magnetism in the DFT codes. Another one is basedon the magnetic force theorem [93], where the Jij is formulated as a linear response function which can beevaluated at both DFT and DFT + DMFT levels [94]. It is noted that the effective spin–spin interaction isactually a second order tensor [95], which yields

1

2

∑i,j

Si · Jij · Sj =1

2

∑i,j

[1

3Tr(Jij)Si · Sj + Si

(1

2(Jij + Jt

ij) − Jij

)Sj +

1

2(Jij − Jt

ij)Si × Sj

], (9)

where the Heisenberg term in equation (6) corresponds to the isotropic exchange 13 Tr(Jij), the symmetric

traceless part Jsymij = 1

2 (Jij + Jtij) − Jij is usually referred as anisotropic exchange, and the antisymmetric part

Jantisymij = 1

2 (Jij − Jtij) represents the Dzyaloshinsky–Moriya interaction (DMI). In order to address complex

magnetic orderings, e.g. in two-dimensional (2D) magnetic materials, the full tensor should be evaluatedconsistently.

In short, the essential challenges are (a) how to obtain the magnetic ground states in an HTP manner and(b) how to evaluate the exchange parameters. As we discussed, there are a few possible solutions for the formerquestion. Regarding the latter, as far as we are aware, there has been no reliable implementation where the

10


exchange parameters can be evaluated in an automated way, where either the DFT part of the codes needscareful adaption for materials with diverse crystal structures or only specific components of the exchangeparameters can be evaluated [96].

3.4. Magnetic fluctuationsThe discussions till now have been focusing on the magnetic properties at 0 K, however, the thermodynamicproperties and the magnetic excitations of various magnetic materials are also fundamental problems whichshould be addressed properly based on the microscopic theory. To this goal, the magnetic fluctuations drivenby temperature shall be accounted for, which will destroy the long-range magnetic ordering at TC/TN resultingin a PM state with fluctuating moments. The working horse DFT can in principle be generalized to finite tem-perature [97], but to the best of our knowledge there is no implementation into the DFT codes on the market.Specifically for the magnetic excitations, the ab initio spin dynamics formalism proposed by Antropov et al[98] is accurate but it is demanding to get implemented, thus it has only been applied to simple systems suchas Fe. In the following, we will focus on (a) the atomistic spin models, (b) the disordered local moment (DLM)approximation, and (c) the DFT + DMFT methods, with the thermodynamic properties at finite temperaturein mind.

From the physics point of view, there are two types of excitations, i.e. transversal and longitudinal exci-tations. For itinerant magnets, the former refers to the collective fluctuations of the magnetization directionswhich dominates at the low-temperature regime, whereas the latter is about the spin-flip excitations (i.e. Stonerexcitations) leading to the reduction of magnetic moments. Correspondingly for the localized moments, theoccupation of the atomic multiplets varies with respect to temperature, giving rise to the temperature depen-dent magnitude of local moments, while the interatomic exchange parameters will cause collective transversalspin waves as well.

One of the most essential intrinsic magnetic properties is the magnetic ordering temperature, i.e. TC

(TN) for FM (AFM) materials. The theoretical evaluation of TC/TN usually starts with the Heisenberg model(equation (6)) with the Jij parameters obtained from DFT calculations, as detailed in section 3.3. As the spinwave procession energy is smaller than the band width and the exchange splitting, it is justified to neglect theprocession of magnetization due to the spin waves when evaluating the electronic energies, i.e. in the adiabaticapproximation, the critical temperature can be obtained by statistical averaging starting from the Heisenbergmodel. Both mean-field approximation and random phase approximation (RPA) can be applied, where theformer fails to describe the low-temperature excitations [99]. In this way, the longitudinal excitations withcomparable energies [100] are not considered, which can be treated based on parameterized models [92, 101].Classical Monte Carlo is often used, leading to finite specific heat at 0 K [102], while quantum Monte Carlosuffers from the sign problem when long-range exchange parameters with alternative signs are considered.One solution to this problem is to introduce the quantum thermostat [103]. It is noted that more terms suchas external magnetic fields and magnetic anisotropy can be included, leading to atomistic spin dynamics whichis valuable to develop a multi-scale understanding of magnetic properties [104, 105].

Another fundamental quantity is the Gibbs free energy which is crucial to elucidate the magneto-structuralphase transitions. In general, the Gibbs free energy can be formulated as [106]

G(T, P, em) = H(T, V , em) + PV = E(V , em) + Felectronic(V , T) + Flattice(V , T) + Fmagnetic(V , T) + PV ,(10)

where em denotes the magnetization direction, T, P, and V refer to the temperature, pressure, and volume,respectively. The electronic, lattice, and magnetic contributions to the free energy can be considered as inde-pendent of each other, because the typical time scale of electronic, lattice, and magnetic dynamics is about10−15 s, 10−12 s, and 10−13 s, respectively. This approximation is not justified anymore for the PM states wherethe spin decoherence time is about 10−14 s [107], which will be discussed in detail below.

To evaluate F magnetic, the DLM method [108] provides a valuable solution for both magnetically orderedand disordered (i.e. PM) states. The DLM method also adopts the adiabatic approximation, and it includestwo conceptual steps [109]: (1) the evolution of the local moment orientations and (2) the electronic structurecorresponding to each specific configuration. A huge number of orientational configurations are requiredin order to accurately evaluate the partition function and hence the thermodynamic potential, which canbe conveniently realized based on the multiple scattering theory formalism, namely the KKR method [110,111]. The recent generalization [112] into the relativistic limit enables accurate evaluation of the temperature-dependent MAE for both TM-based [112] and 4f–3d magnets [113], and also the magnetic free energies witheven frustrated magnetic structures [109]. Nevertheless, the codes with the KKR method implemented requiressophisticated tuning for complex crystalline geometries and the evaluation of forces is not straightforward.

The DLM method can also be performed using supercells by generating magnetic configurations like specialquasi-random structures (SQS), which is invented to model the chemical disorder [114, 115]. It has beenapplied to simulate the PM states with supercells of equal number of up and down moments, particularly

11


to evaluate the lattice dynamics as the forces can be obtained. Moreover, configurational averaging can beachieved by generating a number of supercells with randomly distributed moments which sum up to zero[115]. Spin-space averaging can be performed on top to interpolate between the magnetically ordered andthe PM states, as done for bcc Fe [116]. In this way, the spin–lattice dynamics can be performed to obtainthe thermodynamic properties for the PM states [117], which cannot be decoupled as discussed above. Itis noted that usually the spin–lattice dynamics is performed based on model parameters [118], where theinstantaneous electronic response is not considered. The DLM method has also been recently combined withthe atomistic spin dynamics with in general noncollinear spin configurations [119]. It is noted that the localmoments should be very robust in order to perform such calculations, otherwise significant contribution fromlongitudinal fluctuations might cause trouble.

Another promising method to address all the issues discussed above is the DFT + DMFT methods. TheDMFT method works by mapping the lattice Hubbard model onto the single-site Anderson impurity model,where the crystalline environment acts on the impurity via the hybridization function [69]. Continuous timequantum Monte Carlo has been considered to be the state-of-the-art impurity solver [120]. Very recently,both the total energies [121] and forces [122] can be evaluated accurately by performing charge self-consistentDFT + DMFT calculations, with a great potential addressing the PM states. Nevertheless, the local Coulombparameters are still approximated, though there is a possibility to evaluate such quantities based on constrainedRPA calculations [123]. One additional problem is that the mostly applied single-site DFT + DMFT doesnot consider the intersite magnetic exchange, which will overestimate the magnetic critical temperature. Wesuspect that maybe atomistic modeling of the long wavelength spin waves will cure the problem, instead ofusing expensive cluster DMFT.

Overall, the key challenge to obtain the thermodynamic properties of magnetic materials is the accurateevaluation of the total Gibbs free energy including both the longitudinal and transversal fluctuations, togetherwith coupling to the lattice degree of freedom. DFT + DMFT is again a decent solution but there is no properimplementation yet ready for HTP calculations.

3.5. Magnetic anisotropy and permanent magnets3.5.1. Origin of magnetocrystalline anisotropyAmong the intrinsic magnetic properties, MAE can be mostly attributed to SOC [124], which couples thespontaneous magnetization direction to the underlying crystal structure, i.e. breaks the continuous symme-try of magnetization by developing anisotropic energy surfaces [125]. Like spin, SOC is originated from therelativistic correction to the kinetic energy [126], which can be formulated as (in the Pauli two-componentformalism):

HSOC = − ∇V

2m2c2s · l = ξs · l, (11)

where ∇V marks the derivative of a scalar electrostatic potential, m the mass of electrons, c the speed of light, s(l) the spin (orbital) angular momentum operator. ξ indicates the magnitude of SOC, which is defined for eachl-shell of an atom, except that for the s-orbitals where l = 0 leads to zero SOC. The strength of atomic SOC canbe estimated by ξ ≈ 2

5 (εd5/2− εd3/2

) for the d-orbitals and ξ ≈ 27 (εd7/2

− εd5/2) for the f-orbitals, respectively

[127]. This leads to the average of ξ about 50 meV for the 3d-orbitals of 3d atoms while that of the 4f-orbitals foractinide elements can be as large as 0.3 eV. Additionally, the magnitude of ξ is proportional to ∇V. Therefore,the larger the atomic number is, the larger the magnitude of ξ, e.g. enhanced SOC in 4d- and 5d-elementsinduces many fascinating phenomena [128]. The electrostatic potential V can be modulated significantly atthe surfaces or interfaces, leading to the Rashba effect which is interesting for spintronics [129].

Conceptually, MAE can be understood in such a way that SOC tries to recover the orbital momentwhich is quenched in solids, i.e. it can be attributed to the interplay of exchange splitting, crystal fields, andSOC. Based on the perturbation theory, the MAE driven by SOC can be expressed as [130]:

MAE = −1

4ξ−→m · [δ〈l↓〉 − δ〈l↑〉]︸︷︷︸

orbital moment

+ξ2

ΔEex[10.5

−→m · δ〈T〉+ 2δ〈(l ζ sζ)2〉]︸︷︷︸

spin flip

, (12)

where−→m stands for the unit vector along the magnetization direction. T =

−→m − 3�r(�r · m) ≈ − 2

7 Q · −→mdenotes the anisotropic spin distribution, with�r indicating the position unit vector, Q = l2 − 1

3 l 2 the chargequadrupole moment. It is noted that the spin-flip contribution is of a higher order with respect to ξ, whichis significant for elements with strong SOC. On the other hand, for the orbital momentum term, it will bereduced to MAE = − 1

4ξ−→m · δ〈l↓〉 for strong magnets with the majority channel fully occupied [131], suggest-

ing that the magnetization prefers to be aligned along with the direction where the orbital moment is of largermagnitude.

12


Following the perturbation theory, MAE can be enhanced by engineering the local symmetry of the mag-netic ions. It is well understood that the orbital magnetic moments are originated from the degeneracy ofthe atomic orbitals which are coupled by SOC [130]. Therefore, when the crystal fields on the magnetic ionslead to the degeneracy within the {dxy, dx2+y2 } or {dyz, dxz} orbitals, MAE will be significantly enhanced. Thissuggests, local crystalline environments of the linear, trigonal, trigonal bipyramid, pentagonal bipyramid, andtricapped trigonal prism types are ideal to host a possible large MAE, because the resulting degeneracy in both{dxy, dx2+y2 } and {dyz, dxz} orbital pairs [132]. For instance, Fe atoms in Li2Li1−xFexN located on linear chainsbehave like RE elements with a giant anisotropy [133], and Fe monolayers on InN substrates exhibit a giantMAE as large as 54 meV/u.c. due to the underlying trigonal symmetry [134]. This leads to an effective wayto tailor MAE by adjusting the local crystalline geometries, as demonstrated recently for magnetic molecules[135].

Technically, MAE is defined and can be evaluated as the difference of total energies with SOC between twodifferent magnetization directions (

−→m 1 and

−→m 2):

MAE = E−→m 1

− E−→m 2

≈occ.∑

k,i=1

εi(k,−→m 1) −

occ.∑k,i=1

εi(k,−→m 2). (13)

In general, MAE is a small quantity with a typical magnitude between μeV and meV, as a difference of twonumbers which are orders of magnitude larger. In this regard, it is very sensitive to the numerical details suchas exchange–correlation functionals, k-point convergence, implementation of SOC, and so on. Thus, the fullrelativistic calculations are expensive, particularly for compounds with more than 10 atoms per unit cell. To geta good estimation, force theorem is widely used (equation (13)), where MAE is approximated as the differenceof the sum of DFT energies for the occupied states obtained by one-step SOC calculations [136]. We want toemphasize that the k-point integration over the full BZ is usually required, suggesting (a) symmetry shouldbe applied with caution, i.e. the actual symmetry depends on the magnetization direction and (b) there isno hot zone with dominant contributions [137]. In this regard, the recently developed Wannier interpolationtechnique may provide a promising solution to the numerical problem [138, 139].

3.5.2. Permanent magnets: RE or not?There are two classes of widely used permanent magnets, namely, the TM-based ferrite and AlNiCo, and thehigh performance RE-based materials like Nd2Fe14B and SmCo5. One crucial factor to distinguish such twoclasses of materials is whether the RE elements are included, which causes significant MAE due to the enhancedSOC in the 4f-shell [140]. Hereafter these two classes of permanent magnets will be referred as RE-free and4f–3d magnets. Given the fact that the research on permanent magnets based on these four systems is relativelymature, the current interest from the materials design point of view is to identify the so-called gap magnets[141], i.e. material systems with performances lying between these two classes.

It is noted that SOC is an inert atomic property, i.e. its strength cannot be easily tuned by applying externalforces which are negligible comparing to the gradient of the electrostatic potential around the nuclei. In addi-tion, the exchange splitting is mostly determined by the Coulomb interaction [66]. Thus, the knob to tailorMAE in magnetic materials is the hybridization of the atomic wave functions with those of the neighboringatoms, which can be wrapped up as crystal fields. On the one hand, for the 4f–3d magnets, MAE is mainlyoriginated from the non-spherical distribution of the 4f-electrons [142], which can be expressed based on thelocal crystal fields on the 4f-ions [143]. On the other hand, for TM-based magnetic materials, the orbital degreeof freedom can be engineered to enhance MAE by manipulating the crystal fields via fine tuning the crystallinesymmetries, as discussed above following the perturbation theory. For instance, the MAE of Co atoms can beenhanced by three orders of magnitude, from 0.06 meV/atom in bulk hcp Co [144] to about 60 meV for Coadatoms on MgO [145] or Co dimers on benzene [146]. Such a concept can also be applied to improve theMAE of bulk magnetic materials, such as in Li2Li1−xFexN [133] and FeCo alloys [147], where the d-orbitalsstrongly coupled by SOC (such as dxy and dx2−y2 orbitals) are adjusted to be degenerate and half-occupied bymodifying the local crystalline geometry and doping.

For the RE-free magnets, there is a theoretical upper limit [148] of MAE as high as 3.6 MJ m−3, i.e. thereis a strong hope that RE-free materials can be engineered for high performance permanent magnets. Therehave also been a number of compounds investigated following this line, focusing on Mn-, Fe-, and Co-basedcompounds [149]. The Mn-based systems are particularly interesting, as Mn has the largest atomic momentsof 5 μB which will be reduced in solid materials due to the hybridization. Another feature to be considered isthat the interatomic exchange between Mn moments has a strong dependence on the interatomic distance, e.g.the Mn moments tend to couple antiferromagnetically (ferromagnetically) if the distance is about 2.5–2.8 Å(>2.9 Å) [27]. Nevertheless, there are a few systems such as MnAl [150], MnBi [151], and Mn–Ga [152]showing promising magnetic properties to be optimized for PM applications. On the other hand, for the

13


Fe-based systems, FePt [153] and FeNi [154] have been extensively investigated. One key problem for thesetwo compounds is that there exists strong chemical disorder with depends on the processing. For instance, theorder–disorder transition temperature for FeNi is about 593 K, which demands extra long time of annealingto reach the ordered phase with a strong anisotropy [155].

Given that Ms and MAE can be obtained in a straightforward way for most TM-based magnetic materials,the main strategy for searching novel RE-free permanent magnets is to screen over a number of possible chem-ical compositions and crystal structures with the targeted properties. One particularly interesting direction isto characterize the metastable phases, such as Fe16N2 [156, 157] and MnAl [47]. Such phases can be obtainedvia non-equilibrium synthesis, such as MBE, melt spinning and mechanical alloying. It is also noted that pureDFT might not be enough to account for the magnetism in tricky cases due to the missing spin fluctuations asdemonstrated in (FeCo)2B [158].

For the 4f–3d compounds, it is fair to say that there is still no consistent quantitative theory at the DFTlevel to evaluate their intrinsic properties, though both SmCo5 and Nd2Fe14B were discovered decades ago. Themost salient feature common to the 4f–3d magnetic compounds is the energy hierarchy, driven by the couplingbetween the RE-4f and TM-3d sublattices [159]. First of all, TC is ensured by the strong TM–TM exchangecoupling. Furthermore, MAE is originated from the interplay of RE and TM sublattices. For instance, the Y-based compounds usually exhibit significant MAE, such as YCo5 [160], whereas for the RE atoms with nonzero4f moments, MAE is significantly enhanced due to non-spherical nature of the RE-4f charges [142] caused bythe crystal fields [143]. Lastly, the exchange coupling between RE-4f and TM-3d moments is activated viathe RE-5d orbitals, which enhances MAE by entangling the RE and TM sublattices. The intra-atomic 4f–5dcoupling on the RE sites is strongly FM, while the interatomic 3d–5d between TM and RE atoms is AFM. Thisleads to the AFM coupling between the spin moments of the TM-3d and RE-4f moments, as observed in most4f–3d magnets [161]. In this regard, light RE elements are more preferred [162] for engineering permanentmagnets, because the orbital moments of heavy RE elements will reduce the magnetization as they are parallelto the spin moments following the third Hund’s rule.

Such multiple sublattice couplings have been utilized to construct atomistic spin models to understand thefinite temperature magnetism of 4f –3d materials, such as RECo5 [163] and recently Nd2Fe14B [164]. Suchmodels are ideally predictive, if the parameters are obtained from accurate electronic structure calculationsbased on DFT. One typical problem of such phenomenological modeling is the ill-description of the temper-ature dependency of the physical properties, e.g. the magnitude of moments is assumed to be temperatureindependent [113]. This issue cannot be solved using the standard DFT which is valid at 0 K. For instance,both the longitudinal and transversal fluctuations are missing in conventional DFT calculations. Additionally,for the 4f–3d magnets, MAE and the inter-sublattice exchange energy are of the same order of magnitude, thusthe experimental measurement along the hard axis does not correspond to a simplified (anti-)parallel config-uration of 4f and 3d moments [159]. This renders the traditional way of evaluating the MAE by total energydifferences not applicable [165].

Another challenge is how to treat the strongly correlated 4f-electrons. Taking SmCo5 as an example, thereis clear experimental evidence that the ground state multiplet J = 5/2 is mixed with the excited multipletsJ = 5/2 and J = 7/2, leading to a strong temperature dependence of both spin and orbital moments of Sm[166]. The problem arises how to perform calculations with several multiplets, which requires more than oneSlater determinants. Furthermore, Ce is an abundant element which is preferred for developing novel RE-leanpermanent magnets. The single 4f electron in trivalent Ce can be localized or delocalized, leading to the mixedvalence state, as represented by the abnormal properties of CeCo5 [167]. Such features have been confirmedby the x-ray magnetic circular dichroism (XMCD) measurements in a series of Ce-based compounds [168].

Therefore, the strongly correlated nature of 4f-electrons calls for a theoretical framework which is beyondthe standard DFT, where a consistent theoretical framework being able to treat the correlated 4f -electronswith SOC and spin fluctuations considered consistently is required. Most DFT calculations [169, 170] treatthe 4f-electrons using the open-core approximation which does not allow the hybridization of 4f-shell withthe valence bands. Moreover, the spin moment of RE elements are usually considered as parallel to that of theTM atoms [171]. It is worthy mentioning that Patrick and Staunton proposed a relativistic DFT method basedon the DLM approximation to deal with finite temperature magnetism where the self-interaction correction(SIC) is used for the 4f-electrons [113]. It is observed that the resulting total moment (7.13 μB) of SmCo5

is significantly underestimated compared to the experimental value of about 12 μB [166]. In this regard, themethods such as SIC and DFT + U [73] based on a single Slater determinant may not be sufficient to captureall the features of the correlated 4f-shells.

It is believed that an approach based on multiconfigurational wave functions including SOC and accu-rate hybridization is needed, in order to treat many close-lying electronic states for correlated open shells (i.e.d- and f-shells) driven by the interplay of SOC, Coulomb interaction, and crystal fields with a comparativemagnitude. From the quantum chemistry point of view, the full configurational interaction method is exact

14


Figure 5. (color online) Homologous structures of Sm–Co intermetallic compounds. (a) SmCo5 (CaCu5-type), (b) SmCo12

(ThMn12-type), (c) Sm2Co17 (Th2Ni17-type, 2:17H), (d) Sm2Co17 (Th2Zn17-type, 2:17R), (e) Sm2Co7 (Ce2Ni7-type), (f) SmCo4B(CeCo4B-type), (g) SmCo3B2 (Co3GdB2-type), and (h) SmCo2 (MgZn2-type, C14 Laves). (a)–(d) are the members of the seriesREm−nTM5m+2n (n/m < 1/2), with (m, n) = (1, 0), (2, 1), hexagonal-(3, 1), and rhombohedra-(3, 1), respectively. (a) and (e) arethe members of the series REn+2TM5n+4 with n =∞ and 2, respectively. (a), (f), and (g) are the members of the seriesREn+1TM3n+5B2n with n = 0, 1, and ∞, respectively.

but can only be applied on cases with small basis sets. The complete active space self-consistent field methodbased on optimally chosen effective states is promising, where a multiconfigurational approach can be furtherdeveloped with SOC explicitly considered [172]. Such methods have been successfully applied on the evalua-tion of MAE for magnetic molecules [173], and recently extended to the four-component relativistic regime[174]. A similar method based on exact diagonalization with optimized bath functions has been developedas a potential impurity solver for DMFT calculations [175]. Thus, with the hybridization accounted for at theDFT level, such impurity solvers based on the multiconfigurational methods will make DFT+DMFT with fullcharge self-consistency a valuable solution to develop a thorough understanding of the magnetism in 4f–3dintermetallic compounds. Whereas previous DFT + DMFT calculations on GdCo5 [176] and YCo5 [177] havebeen performed using primitive analytical impurity solvers. It is noted that SOC is off-diagonal for the Ander-son impurities with real spherical harmonics as the basis, leading to possible severe sign problem when thequantum Monte Carlo methods are used as the impurity solver. For such cases, either the hybridization func-tion is diagonalized to obtain an effective basis such as demonstrated for iridates in the 5d5 J = 1/2 states [178,179], or the variational cluster approach can be applied [180, 181]. A recently developed bonding-antibondingbasis offers also a possible general solution to perform DFT + DMFT on such materials with strongSOC [182].

From the materials point of view, compounds including Fe or Co with crystal structures derived from theCaCu5-type are particularly important. It is noted that Nd2Fe14B has also a structure related to the CaCu5-type,but its maximal permanent magnets performance has reached the theoretical limit, leaving not much space toimprove further [183]. Interestingly, as shown in figure 5, various crystal structures can be derived from theCaCu5-type structure. Taking Sm–Co as an example, a homologous series REm−nTM5m+2n can be obtained,where n out of m Sm atoms are substituted by the dumbbells of Co–Co pairs. For instance, the ThMn12-typestructure (figure 5(b)) corresponds to (m = 2, n = 1), i.e. one Sm out of two is replaced by a Co–Co pair. ForSm2Co17 (m = 3, n = 1), both the hexagonal Th2Ni17-type (2:17H) (figure 5(c)) and rhombohedra Th2Zn17-type (2:17R) (figure 5(d)) structures can be obtained depending on the stacking of Co dumbbells. Anotherhomologous series REn+2TM5n+4 can be obtained by intermixing the SmCo2 (MgZn2-type) (figure 5(h)) andthe SmCo5 phases, leading to Sm2Co7 (figure 5(e)) with n = 2. Lastly, by substituting B for preferentially Coon the 2c-sites, yet another homologous series REn+1TM3n+5B2n can be obtained, leading to SmCo4B (n = 1,figure 5(f)), and SmCo3B2 (n = ∞, figure 5(g)). For all the derived structures, the Kagome layers formed bythe Co atoms on the 3g sites of SmCo5 are slightly distorted, while the changes occur mostly in the SmCo2

layers due to a large chemical pressure [184].Such a plenty of phases offer an arena to understand the structure–property relationship and to help us

to design new materials, particularly the mechanical understanding based on the local atomic structures. Forinstance, Sm2Co17 has uniaxial anisotropy while all the other early RE–Co 2:17 compounds show a planar

15


behavior [185]. This is caused by the competition between the positive contribution from the 2c-derived Coand the negative contribution from the dumbbell Co pairs [185]. As a matter of fact, the 2c-Co atoms inthe original CaCu5-type structure have both enhanced spin and orbital moments [186]. Thus, insight on thestructure–property relationship at the atomic level with defined driving forces from the electronic structurewill be valuable to engineer new permanent materials.

One particularly interesting class of compounds which has drawn intensive attention recently is the Fe-richmetastable materials with the ThMn12-type structure [183]. There have been many compounds REFe11X orREFe10X2 stabilized by substituting a third element such as Ti and V [187]. Moreover, motivated by the success-ful preparation of NdFe12 [188] and SmFe12 [189] thin films with outstanding permanent magnet properties,there have been many follow-up experimental and theoretical investigations [183]. The key problem thoughis to obtain bulk samples with large coercive fields, instead of thin films fabricated under non-equilibriumconditions. In order to guide experimental exploration, detailed thermodynamic optimization of the multi-component phase diagram is needed, where the relative formation energies in most previous DFT calculationsare of minor help, e.g. the prediction of NdFe11Co [190] is disproved by detailed experiments [191]. Anotherchallenge for the metastable Fe-rich compounds is that the TC is moderate. This can be cured by partially sub-stituting Co for Fe. Additionally, interstitial doping with H, C, B, and N can also be helpful, e.g. Sm2Fe17N2.1

has a TC as high as 743 K, increased by 91% compared to pristine Sm2Fe17 [192]. Interestingly, the MAEfor Co-rich ThMn12-type compounds shows an interesting behavior. For instance, for both RECo11Ti andRECo10Mo2, almost all compounds have uniaxial anisotropy, except that Sm-based compounds show a basal-plane MAE [193]. This echoes that only Sm2Co17 has a uniaxial anisotropy [185]. Such observations can beeasily understood based on the crystal structure as shown in figures 5(b) and (d), where the uniaxial c-axis ofthe ThMn12-type is parallel to the Kagome plane of 3g-Co atoms, but perpendicular in the case of 2:17 types.That is, from the MAE point of view, the 1:12 structure conjugates with the 2:17 structures. Thus, we suspectthat substitutional and interstitial doping can be utilized extensively to engineer MAE of the 2:17 and 1:12compounds.

Overall, although MAE is numerically expensive to evaluate, it is a straightforward task to perform calcula-tions with SOC considered which has been reliably implemented in many DFT codes. The key issue to performHTP screening of permanent magnets is to properly account for the magnetism such as the spin fluctuationsin RE-free and strongly correlated 4f electrons in RE-based materials. In addition, a systematic investigation ofthe intermetallic structure maps and the tunability of crystal structures via interstitial and substitutional dop-ing will be valuable to design potential compounds. Cautions are needed particularly when trying to evaluateMAE for doped cases where the supercells might impose unwanted symmetry and also that a FM ground stateshould be confirmed before evaluating MAE.

3.6. Magneto-structural transitionsIn general, the functionalities of ferroic materials are mostly driven by the interplay of different ferroic orderssuch as ferroelasticity, ferromagnetism, and ferroelectricity, which are coupled to the external mechanical,magnetic, and electrical stimuli [195]. Correspondingly, the caloric effect can be induced by the thermalresponse as a function of generalized displacements such as magnetization, electric polarization, strain (vol-ume), driven by the conjugate generalized forces such as magnetic field, electric fields, and stress (pressure),leading to magnetocaloric, electrocaloric, and elastocaloric (barocaloric) effects [196]. In order to obtain opti-mized response, such materials are typically tuned to approach/cross a FOPT. Therefore, it is critical to under-stand the dynamics of FOPT in functional ferroic materials. Most FOPTs occur without long-range atomicdiffusion, i.e. of the martensitic type, leading to intriguing and complex kinetics. For instance, such solid-solidstructural transitions are typically athermal, i.e. occurring not at thermal equilibria, due to the high energy bar-riers and the existence of associated metastable states. This leads to the broadening of the otherwise sharpnessof FOPTs with respect to the driving parameters and hence hysteresis [31]. Therefore, mechanistic understand-ing of such a thermal hysteresis entails accurate evaluation of the thermodynamic properties, where successfulcontrol can reduce the energy loss during the cycling [197].

Two kinds of fascinating properties upon such FOPTs in ferroic functional materials are the shape memory[198] and caloric effects [196], e.g. MSME and MCE in the context of magnetic materials. Note that MCEcan be induced in three different ways, namely, the ordering of magnetic moments (second order phase tran-sitions) [199], the magneto-structural phase transitions [200], or the rotation of magnetization directions[201], all driven by external magnetic fields. Taking Gd [202] as an example compound with a second-orderphase transition, under adiabatic conditions (i.e. no thermal exchange between the samples and the envi-ronment) where the total entropy is conserved, applying (removing) the external magnetic fields leads to areduction (increase) in the magnetic entropy, as the magnetic moments get more ordered (disordered), lead-ing to increased (decreased) temperature. That is, the temperature change is mostly due to the entropy changeassociated with the ordering/disorder of the magnetic moments. In contrast, for materials where MCE is driven

16


by the magneto-structural transitions of the first-order nature, additional enthalpy difference between twophases before and after the transition should be considered, although the total Gibbs free energy is continuousat the transition point. In this case, the resulting contribution to the entropy change comprise the lattice andmagnetic parts, which can be cooperative or competitive. A prototype compound is the Ni–Mn–Sn Heuslerswith the inverse MCE [203].

Therefore, to get enhanced magnetocaloric performance which is best achieved in the vicinity of the phasetransitions, the ideal candidates should exhibit (1) a large change of the magnetization ΔM, trigged by lowmagnetic field strengths. This makes such materials also promising to harvest waste heat based on the thermo-magnetic effect [204]; (2) tunable transition temperature (e.g. via composition) for the first-order magneto-structural transitions, because the transition is sharp so that a wide working temperature range can be achievedby successive compositions; (3) a good thermal conductivity to enable efficient heat exchange.

To search for potential candidates with significant MCEs, it is critical to develop mechanistic understandingof the existing cases. For instance, Ni2MnGa is the only Heusler compounds with the stoichiometric composi-tion that undergoes a martensitic transition at 200 K between the L21 austenite and DO22 martensitic phases,which can be optimized as a magnetocaloric material. DFT calculations show that the martensitic transitioncan be attributed to the band Jahn–Teller effect [205], which is confirmed by neutron [206] and photoemissionmeasurements [206, 207]. Detailed investigation to understand the concomitant magnetic field induced strainas large as 10% [206] reveals that there exists a premartensitic phase between 200 K and 260 K [206] and thatboth the martensitic and premartensitic phases have modulated structures. Nevertheless, it is still under inten-sive debate about the ground state structure and its origin. Kaufmann et al introduced the adaptive martensiticmodel, arguing that the ground state is the L10 type tetragonal phase which develops into modulated structuresvia nanotwinning [208, 209]. On the other hand, Singh et al argued based on the synchrotron x-ray mea-surements that the 7M-like incommensurate phase is the ground state, which is caused by phonon softening[210].

To shed light on the nature of the magneto-structural coupling, the Gibbs free energy G(T, V; P) (equation(10)) as a function of temperature and volume (T, V) for different phases P should be evaluated, whereF ele., F lat., and Fmag. denote the electronic, lattice, and magnetic contributions, respectively, Fele. can be easilyobtained using the methods as detailed in reference [211]. To evaluate Flat., quasi harmonic approximationcan be applied [212], which has been implemented as a standard routine in the phonopy code [213]. It isnoted that the harmonic approximation is only valid around the local minima on the potential energy sur-face, whereas a proper consideration on the transition paths and kinetics requires an accurate evaluation ofthe anharmonic effects [214]. It is noted that there exists strong spin–phonon interaction, e.g. the phononspectra should be evaluated for the PM states, which is a challenge task. The spin space average techniquehas been developed and applied for bcc Fe [116] and we suspect the recent implementation of forces in theDMFT regime is another promising solution [122]. For the magnetic free energy Fmag., the classical MonteCarlo simulations, as usually done based on the Heisenberg model (equation (6)) with DFT derived exchangecoupling Jij, are not enough, as the specific heat at 0 K stays finite instead of the expected zero, due to thecontinuous symmetry of spin rotations. Additionally, quantum Monte Carlo suffers from the sign problemas Jij is long range and changes its sign with respect to the distance. One solution is to rescale the classicalMonte Carlo results, as done for bcc Fe [102]. An alternative is to map the free energy based on the fittedtemperature dependence of the magnetization [215]. This has been applied successfully to get the marten-sitic transition temperature and the stability of premartensitic phases in Ni–Mn–Ga [216]. The recentlydeveloped spin dynamics with quantum thermostat [103], the Wang–Landau technique to sample the ther-modynamic DOS in the phase space directly [217], and the atomistic spin–lattice dynamics are interestingto explore in the future [218]. Last but not least, magnetic materials with noncollinear magnetic orderingand thus vanishing net magnetization can also be used for caloric effects such as barocaloric [219] and elas-tocaloric [220] effects, it is essential to evaluate the free energy properly which has been investigated in a recentwork [109].

It is noted that the magnetocaloric performance of stoichiometric Ni2MnGa is not significant, as themartensitic transition occurs between the FM L21 and modulated martensitic phase. As shown in figure 6(a),excess Ni will bring the martensitic phase transition temperature and TC closer, and hence enhance the magne-tocaloric performance [221]. Interestingly, such a phase diagram can be applied to elucidate the transitions inseveral Ni–Mn–X (X = Ga, In, Sn, Sb) alloys, where the dependence with respect to the valence electron con-centration can be formulated (figure 6). The valence electron concentration accounts for the weighted numberof the s/p/d electrons, which can be tuned by excess Mn atoms. For instance, the martensitic transition tem-perature for the Ni–Mn–X alloys interpolates between the stoichiometric Ni2MnX and the AFM NiMn alloys.In such cases, the martensitic transition occurs usually between the FM L21 and the PM L10 phases, leading tothe metamagnetic transitions and hence giant magnetocaloric performances [200, 203]. The magnetocaloricperformance can be further enhanced by Co doping [222–224], which enhances the magnetization of the

17


Figure 6. Phase diagram of Ni-Mn-X. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [SpringerNature] [JOM Journal of the Minerals, Metals and Materials Society] [194], (2013)

Figure 7. (color online) Library of MCE materials. Reproduced from [226]. CC BY 2.0.

austenite phase. However, Co-doping modifies the martensitic transition temperature significantly, leading toa parasitic dilemma [225].

From the materials point of view, there are many other classes of materials displaying significant magne-tocaloric performance beyond the well-known Heusler compounds, as summarized in figure 7 [226]. Threeof them, e.g. Gd alloys [227], La–Fe–Si [228, 229], and MnFe(Si, P) [230] have been integrated into devices,where La(FeSi)13 is considered as the most promising compounds for applications, driven by the first-ordermetamagnetic phase transition at TC [231]. For such materials, research has been focusing on the fine tuningof the magnetocaloric performance by chemical doping. As discussed above, the key problem to overcomewhen designing MCE materials is to reduce the hysteresis in order to minimize the energy losses [232], thisis a challenging task for HTP design where the Gibbs free energies for not only the end phases but alsothe intermediate phases are needed in order to have a complete thermodynamic description of the phase

18


transitions. In this regard, a particularly promising idea is to make use of the hysteresis rather than to avoidit, which can be achieved in the multicaloric regime engaging multiple stimuli. This has been demonstratedrecently in Ni–Mn–In [233] combining magnetic fields and uniaxial strain.

Therefore, designing novel MCE materials entails joint theoretical and experimental endeavor. HTP calcu-lations can be applied to firstly search for compounds with polymorphs connected by possible first-order struc-tural transitions, and then to evaluate the Gibbs free energies for the most promising candidates. Nevertheless,for such materials with first-order magneto-structural transitions, hysteresis typically arises as a consequenceof nucleation, in caloric materials it occurs primarily due to the domain-wall pinning, which is the net result oflong-range elastic strain associated with phase transitions. Thus, experimental optimization on the microstruc-tures [234] (section 5.1) is unavoidable in order to obtain a practical material ready for technicalapplications.

3.7. SpintronicsTill now we have been focusing on the physical properties of the equilibrium states, whereas the non-equilibrium transport properties of magnetic materials leads to spintronics. In contrast to the conventionalelectronic devices, the spin degree of freedom of electrons is explored to engineer more energy efficient devicesin spintronics [33]. The first generation of spintronic applications are mostly based on the FM materials, ini-tiated by the discovery of GMR in magnetic multilayers in 1988 [235, 236], e.g. with the MR ratio as largeas 85% between parallel and antiparallel configurations of Fe layers separated by the Cr layers in-between.The underlying mechanism of GMR can be understood based on the two-current model [237], where spin-polarized currents play a deterministic role. Two important follow-up developments of GMR are the tunnelingmagnetoresistance (TMR) [238] and STT [239, 240]. The TMR effect is achieved in multilayers where theFM metals are separated by large gap insulators such as MgO, and the MR ratio can be as large as 600% inCoFeB/MgO/CoFeB by improving the interfacial atomic morphology [241]. In the case of STT, by drivingelectric currents through the reference layer, finite torque can be exerted on the freestanding layers and henceswitch their magnetization directions. This enables also engineering spin transfer oscillator [242], racetrackmemory [243], and nonvolatile random-access memory [244].

The second generation of spintronics comprise phenomena driven by SOC, dubbed spin–orbitronics,which came around 2000 [34]. For FM materials, SOC gives rise to an anisotropic magnetoresistance (AMR)[245], where the resistivity depends on the magnetization direction, in analogy to GMR but without the neces-sity to form multilayers. Nevertheless, the most essential advantage of spin–orbitronics is to work with spincurrent (i.e. a flow of spin angular momentum ideally without a charge current), instead of the spin polarizedcurrents [246]. Correspondingly, the central subject of spin–orbitronics is the generation, manipulation, anddetection of spin currents (table 2). Note that after considering SOC, spin is not any more a good quantumnumber, thus it is still a pending question how to define the spin current properly [247].

Two most important phenomena for spin–orbitronics are the spin Hall effect (SHE) [286] and spin–orbittorque (SOT) [287]. SHE deals with the spin-charge conversion, where a transversal spin current can be gen-erated by a longitudinal charge current. It has been observed in PM metals such as Ta [288] and Pt [289], FMmetals like FePt [290], AFM metals including MnPt [291], Mn80Ir20 [292], and Mn3Sn [293], semiconductorssuch as GaAs [294], and topological materials such as Bi2Se3 [295] and TaAs [296]. It is noted that the recip-rocal effect, i.e. the inverse SHE (iSHE), can be applied to detect the spin current by measuring the resultingcharge current.

SOT deals with the interaction of magnetization dynamics and the spin current, leading to magnetizationswitching and spin–orbit pumping. Following the Landau–Lifshitz–Gilbert (LLG) equation [287],

dm

dt= −γm × B︸︷︷︸

precession

+ αm × dm

dt︸︷︷︸relaxation

+γ

MsT︸︷︷︸

torque

, (14)

where γ, α, and Ms mark the gyromagnetic ratio, the Gilbert damping parameter, and the saturation magneti-zation, respectively. m, B, and T indicate the magnetization unit vector, the effective field, and the total torque,respectively. The torque T is perpendicular to m, which can be generally expressed as [287]

T = τFLm × ε︸︷︷︸field-like

+ τDLm × (m × ε)︸︷︷︸damping-like

, (15)

where ε is the unit vector of the torque. Note that the field-like and damping-like terms act on the magnetizationlike the precession and relaxation terms in the LLG equation (equation (14)), respectively.

To induce finite SOT, effective non-equilibrium spin polarization is required, which can be obtained via (a)SHE and (b) Edelstein effect [297] (i.e. inverse spin Galvanic effect (iSGE) [298]). From the symmetry point

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Table 2. Selected systems implementing spin-orbitronics and AFM spintronics. Reprinted by permission from Springer NatureCustomer Service Centre GmbH: [Springer Nature] [Nature] [248], (2010). Reprinted by permission from Springer Nature CustomerService Centre GmbH: [Springer Nature] [Springer eBook] [249] (2017). Reproduced from [250]. CC BY 2.0.

spin transport spin–orbit torque spin pumping

spin current [248] SOT in NM|FM bilayers [249] spin pumping in FM|NM bilayers [250]

FM metal review [251] Pt|Co [252, 253] NiFe|Pt [250, 254, 255]Ta|CoFeB [253, 256] NiFe|GaAs [257] (metal|semiconductor)SrIrO3|NiFe [258] (oxides) NiFe|YIG [259] (metal|insulator)WTe2|NiFe [260] (out-of-plane)NiMnSb [261] (bulk)

FM insulator YIG [262] GaMnAs|Fe [263] GaMnAs|GaAs [264]GaMnAs [265] (bulk) YIG|Pt [248, 266]

AFM metal IrMn|CoFeB [267] NiFe|IrMn [268]IrMn|NiFe [269]MnPt|Co/Ni [270, 271]Mn3Ir|NiFe [272]CuMnAs [273] (bulk)Mn2Au [274, 275] (bulk)

AFM insulator Cr2O3 [276] Pt|NiO [277–279] YIG|NiO [280, 281]Fe2O3 [282] Pt|TmIG [283] YIG|CoO [284]

Bi2Se3|NiO|NiFe [285]

of view, the occurrence of SOT requires noncentrosymmetric symmetry, thus the heterostructures of NM/FMmaterials provide a rich playground to investigate SOT, where materials with significant SHE have been thedominant source for inducing spin injection and SOT in FM materials. For instance, SOT has been investigatedin Ta/CoFeB [256], Pt/Co [252, 253], SrIrO3/NiFe [258], and Bi2Se3/BaFe12O19 [299]. On the other hand, theEdelstein effect requires Rashba splitting, which can also be applied for spin-charge conversion and hence SOT.For instance, the spin-charge conversion has been demonstrated for 2DEG at the interfaces of LaAlO3/SrTiO3

[300], Ag/Bi [301], and Cu/Bi2O3/NiFe [302]. Correspondingly, SOT is recently observed in NiFe/CuOx drivenby the Rashba splitting at the interfaces [303]. We note that the SHE–SOT and iSGE–SOT (Edelstein) areentangled, e.g. can be competitive or cooperative with each other, as demonstrated in (GaMn)As/Fe [263].

It is noteworthy pointing out that from the symmetry perspective [253], SOT generated at the interfacesof FM metals and heavy metals lies in-plane for both the anti-damping and field-like contributions, thus canonly switch the in-plane magnetization direction efficiently. Nevertheless, it is demonstrated that in WTe2/NiFe[260], out-of-plane SOT can be induced by a current along a low-symmetry axis, which is promising for futureSOT switching of FM materials with perpendicular magnetic anisotropy. Although SOT discussed above takesplace at the interfaces, it does not mean that SOT is prohibited in bulk materials. For instance, SOT has beenconfirmed in half Heusler NiMnSb [261]. Thus, the key is to break the inversion symmetry in order to obtainfinite SOT.

As SOT can be interpreted as torque induced by spin current, the corresponding Onsager reciprocal effectis spin pumping, i.e. the generation of spin current via induced magnetization dynamics. Usually the spinpumping is induced by generating non-equilibrium magnetization dynamics using the ferromagnetic reso-nance (FMR) and hence creates a pure spin current which can be injected into the adjacent NM layers (i.e.spin sink) without a charge flow under zero bias voltage. For instance, it is demonstrated that spin pumpingcan be achieved in NiFe|Pt bilayers [254], with Pt being the spin sink, where the resulting spin current con-veniently detected via iSHE. In this regard, many materials have been considered as spin sink, such as heavymetals like Au and Mo [255] and semiconductors as GaAs [257]. An intriguing aspect of spin pumping isthat the polarization of the spin current is time-dependent, leading to both dc- and ac-components [304]. Itis demonstrated that the ac-component is at least one order of magnitude larger than the dc-component forNiFe|Pt [250], which can lead to future ac spintronic devices.

An emergent field of great interest is AFM spintronics, which possesses several advantages over the FMcounterpart [35, 305, 306]. For instance, there is no stray field for AFM materials thus the materials are insen-sitive to the neighboring unit, which allows denser integration of memory bits. Moreover, the typical resonance

20


frequency of FM materials is in the GHz range which is mostly driven by MAE [307], while that for AFM mate-rials can be in the THz range due to the exchange interaction between the moments [308]. This leads to ultrafastmagnetization dynamics in AFM materials. All the phenomena of FM materials have been observed in AFMmaterials, such as AMR in FeRh [309] and MnTe [310], SHE in MnPt [291], and tunneling AMR [311, 312].Nevertheless, one of the challenges for AFM spintronics is how to control and to detect the AFM ordering.It turns out that SOT can be applied to switch the magnetic ordering of AFM compounds. For instance, theiSGE–SOT can lead to staggered Neel SOTs, creating an effective field of opposite sign on each magnetic sub-lattice. The SOT induced switching has been observed in CuMnAs [273] and Mn2Au [274, 275]. It is notedthat the staggered SOT has a special requirement on the symmetry of the materials, i.e. the inversion symmetryconnecting the AFM magnetic sublattices in the crystal structure is broken by the magnetic configuration [35].Thus, it is an interesting question whether there are more materials with the desired magnetic configurationsfor bulk AFM spintronics. To detect the AFM magnetic ordering, spin Hall magnetoresistance effect can beused [313].

One particularly interesting subject is SOT and spin pumping for the magnetic junctions combining FMand AFM materials. For instance, as AFM materials display also significant SHE, SOT has been observedin, e.g. IrMn|CoFeB [267], IrMn|NiFe [269], MnPt|Co [270], and Mn3Ir|NiFe [272]. It is demonstrated inMnPt|Co/Ni bilayers, the resulting SOT switching behaves like an artificial synapses, and a programmable net-work of 36 SOT devices can be trained to identify 3 × 3 patterns [271]. This leads to an intriguing applicationof spintronic devices for neuromorphic computation [314]. Furthermore, as spin pumping depends on themagnetic susceptibility at the interfaces and thus the magnetic fluctuations, it is observed that the spin pump-ing in NiFe|IrMn bilayers is significantly enhanced around the AFM phase transition temperature of IrMn[268], consistent with the theoretical model based on enhanced interfacial spin mixing conductance [315].In addition to enhancing the spin pumping efficiency, such types of experiments can be applied to detect theAFM ordering without being engaged with the neutron scattering facilities.

Another emerging field of spintronics is magnonic spintronics [316] realized based on both FM and AFMinsulators, as each electron carriers an angular momentum of �/2 whereas each magnon as the quasiparticle ofspin wave excitations carries an angular momentum of �. Theoretically, it is predicted that at the interfaces ofNM|FM-insulator bilayers, spin gets accumulated which is accompanied by the conversion of spin current tomagnon current [317]. In the case of FM (FiM) insulators, YIG|Pt is a prototype bilayer system to demonstratethe transmission of spin current at the interfaces [248] and spin pumping [266, 318]. It is also demonstratedthe SOT can be applied to switch the magnetization directions of FM insulators, as in Pt|TmIG [283]. TheSOT and spin pumping have also been observed in AFM insulators. For instance, SOT leads to the switchingof the AFM ordering of NiO in Pt|NiO bilayers [277–279]. The origin can be attributed to the non-staggeredSHE–SOT which acts as an anti-damping-like torque exerted by the spin accumulation at the interface, butthis is a question under intensive debate [279]. Furthermore, spin pumping can be induced in bilayers ofFM|AFM-insulators such as YIG|NiO [280]. One intriguing aspect is the non-locality of the spin current whichcan propagate 100 nm in NiO [281]. In general, the number of magnons is not conserved, leading to a finitetransport length for magnon mediated spin current determined by the Gilbert damping coefficient which is anintrinsic character of compounds. In contrast to the FM materials [251], AFM materials allow long range spintransport, as observed in Cr2O3 [276] and Fe2O3 [282] which might be attributed to the spin superfluidity[319]. Particularly, the damping of magnons can be compensated by SOT, which leads to zero damping asobserved recently in YIG [262]. Lastly, it is a fascinating idea to combine SOT and long diffusion length of spincurrent in AFM materials, e.g. switching magnetism using the SOT carried by magnons in Bi2Se3|NiO|NiFe[285].

In short, spintronics is a vast field but there is an obvious trend that more phenomena driven by SOCare being investigated. Moreover, materials to generate, manipulate/conduct, and detect spin current shouldbe integrated for devices, thus the interfacial engineering is a critical issue. Following table 2, both FM andAFM metals/insulators can be incorporated into the spintronic devices, resulting in flexibility but also timeconsuming combinatorial optimizations. For instance, a few FM metals like Fe/Co/Ni, permalloy (NiFe), andCoFeB have been widely applied in the spintronic heterostructures, as the experimental techniques to fabricatesuch systems are well established. In this regard, there is a significant barrier between theoretical predictionsand experimental implementations, though HTP calculations can be carried out in a straightforward way (cfsection 4).

3.8. Magnetic topological materialsMaterials with nontrivial topological nature have been firstly predicted in graphene [320] and HgTe quantumwells [321], and shortly afterwards confirmed experimentally in many 2D and 3D systems [322, 323]. Thekey concept lies in the Berry phase, which measures the global geometric phase of the electronic wave func-tion accumulated through adiabatic evolutions [324]. Such materials host many fascinating properties such as

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Table 3. An incomplete list of magnetic topological materials.

Compounds Phase Space group Measurements

Cr0.15(Bi0.1Sb0.9)1.85Te3 QAHI R3m Transport at 30 mK [337]Ba2Cr7O14 QAHI R3m DFT [338]MnBi2Te4 AFM TI R3m DFT + ARPES [339]EuSn2P2 AFM TI R3m ARPES [340]EuIn2As2 AFM axion P63/mmc DFT [341], ARPES [342, 343]EuSn2As2 AFM TI R3m DFT + ARPES [344]FeSe monolayers 2D AFM TI P4/nmm (bulk) DFT + ARPES + STS [345]HgCr2Se4 FM Weyl Fd3m DFT [346, 347]Co2MnGa FM Weyl/nodal-line Fm3m DFT [348], ARPES [349]EuCd2As2 Ideal Weyl P3m1 DFT + ARPES [350] + SdH [351]GdPtBi AFM Weyl F43m DFT + transport [352]YbPtBi AFM Weyl F43m DFT + ARPES + transport [353]Co3Sn2S2 FM Weyl R3m DFT + transport [354]Mn3Sn AFM Weyl P63/mmc DFT + APRES [355]Fe3Sn2 FM Weyl R3m ARPES [356], STS + QPI [357](Y/RE)2Ir2O7 AFM Weyl/axion TI Fd3m DFT [358]CeAlGe Weyl I41md Transport [359]CuMnAs AFM massive Dirac Pnma DFT [360, 361], transport [362]GdSbTe AFM Dirac P4/nmm DFT + ARPES [363]BaFe2As2 AFM massless Dirac I4/mmm DFT + infrared [364], ARPES [365]CaIrO3 AFM Dirac Pnma Transport + SdH [366]FeSn AFM massless Dirac P6/mmm DFT + ARPES [367, 368]CaMnBi2 AFM massive Dirac P4/nmm ARPES [369](Sr/Ba)MnBi2 AFM massive Dirac I4/mmm DFT + ARPES [369, 370]EuMnBi2 AFM massive Dirac I4/mmm Transport + SdH [371]YbMnBi2 AFM massive Dirac P4/nmm DFT + ARPES [372]GdAg2 2D nodal-line — DFT + ARPES [373]GdPtTe AFM nodal-line P4/nmm DFT + ARPES [363]MnPd2 AFM nodal-line Pnma DFT [374]

quantum spin Hall effect (QSHE), which is promising for future applications such as topological spintronics[322, 323]. Particularly, introducing the magnetic degree of freedom not only enriches their functionalities, e.g.quantum anomalous Hall effect (QAHE) [325], axion electromagnetic dynamics [326, 327], and chiral Majo-rana fermions [328], but also enables more flexible tunability. The relevant materials as compiled in table 3can be categorized based on the dimensionality of the resulting band touchings, namely, topological insulator,nodal point semimetals (including Weyl, Dirac, and manyfold nodal points [329]), and node line semimetals[330].

As symmetry plays an essential role in the topological nature of such materials [331], we put forward abrief discussion before getting into particular material systems. It is clear that the time-reversal symmetry Θ isbroken for all magnetically ordered compounds, and hence they are good candidates to search for the QAHEand Weyl semimetal phases, which are topologically protected rather than symmetry protected. In addition,the Weyl node is a general concept which is closely related to the accidental degeneracy of electronic bands[332], e.g. there are many Weyl points in bcc Fe [333]. Thus, the ultimate goal there is to search for magneticWeyl semimetals where the low energy electronic structure around the Fermi energy are dominated solely (ormostly) by the linear dispersions around the Weyl points.

The occurrence of other topological phases such as TI and Dirac fermions in magnetic materials requiresextra symmetry. For instance, the Kramers degeneracy is necessary to define the Z2 index for nonmagneticTI [320]. However, the electronic bands in magnetic materials are generally singly degenerate except at thetime-reversal invariant momenta (TRIMs), therefore additional symmetry is required in order to restore theKramers degeneracy. This leads to the prediction of AFM topological insulators [334], which can be protectedby an anti-unitary product symmetry S = PΘ, where P can be a point group symmetry [335] or a non-symmorphic symmetry operator which reverses the magnetization direction of all moments. Furthermore,it is noted that the inversion symmetry I respects the magnetization direction, thus parity as the resultingeigenvalues of I can still be applied to characterize the topological phase if the compound is centrosymmetric.Nevertheless, the generalized Z4 character should be used [336], e.g. 4n + 2 of odd parities from the occu-pied states at TRIMs corresponds to a nontrivial axion insulator. In the same way, for topological nodal-linesemimetallic phases [329], additional crystalline line symmetry should be present, as will be discussed forparticular materials below.

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Turning now to the first class of materials exhibiting QAHE, which was predicted by Haldane [375]. Sucha phenomena can be realized in magnetic topological insulators [376], with magnetism introduced by dop-ing as observed in Cr0.15(Bi0.1Sb0.9)1.85Te3 at 30 mK [337]. There have been many other proposals based onDFT calculations to predict finite temperature QAHE [325], awaiting further experimental validations. Partic-ularly, compounds with honeycomb lattices are a promising playground for QAHE, including functionalizedgraphene [377, 378], Mn2C18H12 [379], etc. We want to point out that QAHE is defined for 2D systems [375],an interesting question is whether there exists 3D QAHI. By making an analogue to the 3D integer quantumHall effect [380], a 3D QAHI can be obtained by stacking 2D QAHI on top of each other, or by stacking Weylsemimetals [381]. In the former case, if the Chern number for each kz-plane (i.e. in the stacking direction)changes, one will end up with Weyl semimetals [382]. It is proposed recently that for Ba2Cr7O14 and h-Fe3O4

[338], 3D QAHI can be obtained due to the presence of inversion symmetry.The 2D nature of QAHE observed in Cr-doped TIs arises two questions: (1) whether the gap opening in

the Dirac surface states required for QAHE can be confirmed experimentally and (2) whether it is possibleto engineer axion insulators by forming heterostructures of magnetic-TI/normal-TI/magnetic-TI. For the for-mer, a band gap of 21 meV was observed in Mn-doped Bi2Se3 [383] but proved not originated from magnetism[384]. Only recently, a comparative study of Mn-doped Bi2Te3 and Bi2Se3 reveals that the perpendicular MAEfor the Mn moments in Bi2Te3 is critical to induce a finite band gap of 90 meV opening in the Dirac surfacestates [385]. In the latter case, if the magnetization directions for two magnetic-TIs are antiparrallel and thesandwiching normal-TI is thick enough, the axion insulator phase can be achieved with quantized ac responsesuch as magneto-optical effect and topological magnetoelectric effect [386]; whereas QAHE with quantizeddc-response corresponds to the parallel magnetization of two magnetic-TIs [387]. Several experiments havebeen done in this direction with convincing results [388–390].

Focusing from now on only the intrinsic (bulk, non-doped, non-heterostructure) magnetic topologicalmaterials, an inspiring system is MnBi2Te4 which is confirmed to be an AFM TI recently [339]. It is an orderedphase with one Mn layer every septuple layer in the Bi2Te3 geometry and the Mn atoms coupled ferromagneti-cally (antiferromagnetically) within (between) the Mn layers, leading to a Neel temperature of 24 K. Thus, thenontrivial topological phase is protected by the combined symmetry S = ΘT with T being the translationaloperator connecting two AFM Mn-layers [339]. It is worthy mentioning that the MnBi2Te4 phase is actuallythe reason why the observed band gap is as large as 90 meV in Mn-doped Bi2Te3 [385]. In this regard, it is stillan open issue about whether the Dirac surface states of MnBi2Te4 are gapped [391–393] or gapless [344, 394,395] and the corresponding magnetic origin as the gap survives above the Neel temperature [393]. Neverthe-less, MnBi2Te4 stands for a family of compounds with a general chemical formula MnBi2n(Se/Te)3n+1, whichare predicted to host many interesting topological phases [396].

It seems that the topological materials can be designed based on the chemistry in systems with comparablecrystal structures [397, 398]. Taking three Eu-based compounds listed in table 3 as examples, the electronicstates around the Fermi energy are mainly derived from the In-5s/Sn-5p and P-3p/As-4p states, whereas theEu-4f electrons are located more than 1 eV below the Fermi energy [340, 341, 344]. Certainly the hybridizationbetween 4f and the valence electronic states around the Fermi energy is mandatory in order to drive the systemsinto the topological phases by the corresponding magnetic ordering. Nevertheless, it is suspected that givenappropriate crystal structures, there is still free space to choose a the chemical composition, i.e. via substitutingchemically similar magnetic RE elements to fine tune the electronic structure and thus design novel materials.This philosophy is best manifested in the class of Dirac semimetals AMnBi2 (A = Ca, Sr, Bi, Eu, and Yb) aslisted in table 3. For such compounds, the Dirac cone can be attributed to the square lattice of Bi atoms, whichis common to all the compounds following the DFT prediction [399]. It is noted that the Dirac cones in all theAMnBi2 compounds are massive due to SOC, but still leads to high mobility as the resulting effective mass ofelectrons is small [400]. Substituting Sb for Bi leads to another class of Dirac semimetals AMnSb2 as predictedby DFT [401] and confirmed experimentally [400, 402–405].

As discussed above, the occurrence of Dirac points (with four-fold degeneracy) in magnetic materialsdemands symmetry protection. Taking CuMnAs as an example, it is demonstrated that the Dirac points areprotected via the combined symmetry of S = IΘ as a product of inversion I and Θ [360]. Considering addi-tionally SOC leads to lifted degeneracies depending on the magnetization directions because of the screwrotation operator S2z. The same arguments applies to FeSn [367] and MnPd2 [374]. Three comments are inthe way. Firstly, it is exactly due to the combined symmetry S = IΘ why there exists staggered SOT for CuM-nAs, leading to a descriptor to design AFM spintronic materials [361]. Secondly, the magnetization directiondependent electronic structure is a general feature for all magnetic topological materials, such as surface statesof AFM TIs like MnBi2Te4 [339] and EuSn2As2 [344], and also the nodal line and Weyl semimetals [406].This offers another handle to tailor the application of the magnetic topological materials for spintronics [407].Thirdly, the Dirac (Weyl) features prevail in AFM (FM) materials as listed in table 3, thus we suspect that thereare good chances to design AFM materials with high mobility.

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The relevant orbitals responsible for the topological properties of most materials discussed are the p-orbitals of the covalent elements, where the TM-d and RE-f states play an auxiliary role to introduce magneticordering. A particularly interesting subject is to investigate the topological features in bands derived from morecorrelated d/f-orbitals, which has recently been explored extensively in FM Kagome metallic compounds inthe context of Weyl semimetals. We note that the Kagome lattice with AFM nearest neighbor coupling hasbeen an intensively studied field for the quantum spin liquid [408]. On the one hand, Weyl semimetals hostanomalous Hall conductivity [409] and negative magnetoresistance driven by the chiral anomaly [410, 411].On the other hand, due to the destructive interference of the Bloch wave functions in the Kagome lattice, flatbands and Dirac bands can be formed with nontrivial Chern numbers [412, 413]. Therefore, intriguing physicsis expected no matter whether the Fermi energy is located in the vicinity of the flat band or the Dirac gap, asin the TmXn compounds (T = Mn, Fe, and Co; X = Sn and Ge) [368]. Particularly, for Fe3Sn2, the interplayof gauge flux driven by spin chirality and orbital flux due to SOC leads to giant nematic energy shift depend-ing on the magnetization direction, leading to the spin–orbit entangled correlated electronic structure [357].This offers a fascinating arena to investigate the emergent phenomena of topology and correlations in suchquantum materials, which can be generalized to superconducting materials such as Fe-based superconductors[345, 364, 365].

From the materials point of view, there are many promising candidates to explore. For instance, the recentlydiscovered FM Co3Sn2S2 shows giant anomalous Hall conductivity [354] and significant anomalous Nernstconductivity [414], driven by the Weyl nodes around the Fermi energy. In addition, various Heusler com-pounds display also Weyl features in the electronic structure, including Co2MnGa [348, 349], Co2XSn [415,416], Fe2MnX [416], and Cr2CoAl [417]. Nevertheless, the most interesting class of compounds are those withnoncollinear magnetic configurations. It is confirmed both theoretically and experimentally that Mn3Sn is aWeyl semimetal [355, 418], leading to significant AHC [419] and SHC [293]. It is expected that the Weyl pointscan also be found in the other noncollinear systems such as Mn3Pt [420] and antiperovskite [44], awaitingfurther investigation.

Iridates are another class of materials which are predicted to host different kind of topological phases [421,422]. Most insulating iridates contain Ir4+ ions with five 5d electrons, forming IrO6 octahedra with variousconnectivity. Due to strong atomic SOC, the t2g shell is split into the low-lying J = 3/2 and high-lying half-filled J = 1/2 states. Therefore, moderate correlations can lead to an insulating state, and such an uniqueSOC-assisted insulating phase have been observed in Ruddlesden–Popper iridates [423]. Pyrochlore iridatesRE2Ir2O7 can also host such insulating states, which were predicted to be topologically nontrivial [358]. Manytheoretical studies based on models have been carried out, supporting the nontrivial topological nature [424,425], but no smoking-gun evidence, e.g. surface states with spin-momentum locking using ARPES, has beenobserved experimentally. Our DFT + DMFT calculations with proper evaluation of the topological charactersuggest that the insulating RE2Ir2O7 are likely topologically trivial [179], consistent with the ARPES measure-ments [426]. Nevertheless, proximity to the Weyl semimetal phase is suggested with the Weyl semimetallicphase existing in a small phase space upon a second-order phase transition [427], where magnetic fields canbe applied to manipulate the magnetic ordering of the RE-sublattices and thus the topological character of theelectronic states [428–430]. Last but not least, it is observed that there exist robust Dirac points in CaIrO3 witha post-perovskite structure [366], leading to high mobility which is rare for correlated oxides.

Obviously, almost all known topological phases can be achieved in magnetic materials, which offer moredegrees of freedom compared to the nonmagnetic cases to tailor the topological properties. Symmetry playsa critical role, e.g. semimetals with nodal line [363, 373, 374] and many-fold nodal points [329] can also beengineered beyond the Dirac nodes and AFM TIs discussed above. Unlike nonmagnetic materials where thepossible topological phases depend only on the crystal structure and there are well compiled databases such asICSD, there is unfortunately no extensive database with the magnetic structures collected. Also, it is fair to saythat no ideal Weyl semimetals have been discovered till now, e.g. there are other bands around the Fermi energyfor all the Weyl semimetal materials listed in table 3. Therefore, we foresee a strong impetus to carry out moresystematic research on screening and designing magnetic materials with nontrivial topological properties.

3.9. Two-dimensional magnetic materialsPioneered by the discovery of graphene [431], there has been recently a surge of interest on 2D materials,due to a vast spectrum of functionalities such as mechanical [432], electrical [433], optoelectronic [434], andsuperconducting [435] properties and thus immense potential in engineering miniaturized devices. However,the 2D materials with intrinsic long-range magnetic ordering have been missing until the recently confirmedsystems like CrI3 [436], Cr2Ge2Te6 [437], and Fe3GeTe2 [438] in the monolayer limit. This enables the pos-sibility to fabricate van der Waals (vdW) heterostructures based on 2D magnets and hence paves the way to

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Figure 8. Dependence of the magnetic ordering temperature with respect to the number of layers for typical 2D magnets. Thevalues are normalized using the corresponding bulk Curie/Neel temperature, e.g. CrCl3 (TC = 17 K) [448], CrBr3 (TC = 37 K)[448], CrI3 (TC = 61 K) [436], Cr2Ge2Te6 (TC = 68 K) [437], MnPS3 (TN = 78 K)[454], FePS3 (TN = 118 K) [455], NiPS3

(TN = 155 K) [444], and Fe3GeTe2 (TC = 207 K) [438]. Lines are guide for the eyes. Adapted by permission from Springer NatureCustomer Service Centre GmbH: [Springer Nature] [Nature Nanotechnology] [477], (2019).

engineer novel spintronic devices. Nevertheless, the field of 2D magnetism is still in its infancy, with manypending problems such as the dimensional crossover of magnetic ordering, tunability, and so on.

According to the Mermin–Wagner theorem [439], the long-range ordering is strongly suppressed at finitetemperature for systems with short-range interactions of continuous symmetry in reduced dimensions, due tothe divergent thermal fluctuations. Nevertheless, as proven analytically by Onsager [440], the 2D Ising modelguarantees an ordered phase, protected by a gap in the spin-wave spectra originated from magnetic anisotropy[441]. In this sense, the rotational invariance of spins can be broken by dipolar interaction, SOC effects (such assingle-ion anisotropy, anisotropic exchange interactions, and band structure anisotropy), or external magneticfields, which will lead to magnetic ordering at finite temperature. The complex magnetic phase diagram is bestrepresented by that of TM thiophosphates MPS3 (M = Mn, Fe, and Ni) [442, 443], which are of the 2D-Heisenberg nature for MnPS3, 2D-Ising for FePS3, and 2D-XXZ for NiPS3, respectively. As shown in figure 8for few-layer systems of both FePS3 and MnPS3, the transition temperature remains almost the same as thatof the corresponding bulk phase. However, for NiPS3, the magnetic ordering is suppressed for the monolayerswhile few-layer slabs have slightly reduced Neel temperature compared to that of the bulk [444]. This can beattributed to the 2D-XXZ nature of the effective Hamiltonian, where in NiPS3 monolayers it may be possibleto realize the Berezinskii-Kosterlitz-Thouless topological phase transition [445]. This arises also an interestingquestion whether there exist novel quantum phases when magnetic ordering is suppressed and how to tune themagnetic phase diagram. Also, it is suspected that the interlayer exchange coupling does not play a significantrole for such 2D systems with intralayer AFM ordering.

The interplay of interlayer exchange and magnetic anisotropy leads to more interesting magnetic proper-ties for CrX3 (X = Cl, Br, and I). The Cr3+ ions with octahedral environment in such compounds have the3d3 configuration occupying the t2g orbitals in the majority spin channel. SOC is supposed to be quenchedleading to negligible single-ion MAE. Nevertheless, large MAE can be induced by the strong atomic SOC onthe I atoms, i.e. intersite SOC (i.e. band structure anisotropy), as elaborated in references [446, 447]. As theatomic SOC strength scales as Z4 with Z being the atomic number following the hydrogenic arguments, theMAE favors in-plane (out-of-plane) magnetization for CrCl3 (CrBr3 and CrI3), respectively. Note that for thebulk phases, the intralayer exchange is FM for all three compounds, while the interlayer exchange is FM forCrBr3 and CrI3, and AFM for CrCl3 [448]. In this regard, the most surprising observation for the few-layersystems is that CrI3 bilayers have AFM interlayer coupling [436] with a reduced Neel temperature of 46 K incomparison to the bulk value of 61 K (figure 8). Additionally, the FM/AFM interlayer coupling can be furthertuned by pressure, as showed experimentally recently [449, 450]. Such a transition from FM to AFM interlayercoupling was suggested to be induced by the stacking fault based on DFT calculations, namely the rhombohe-dral (monoclinic) stacking favors FM (AFM) interlayer exchange [451]. It is noted that CrX3 has the structuralphase transition in the bulk phases from monoclinic to rhombohedral around 200 K, e.g. 220 K for CrI3 [452].Thus, a very interesting question is why there is a structural phase transition from the bulk rhombohedralphase to the monoclinic phase in bilayers. A recent experiment suggests that such a stacking fault is inducedduring exfoliation, where it is observed that the interlayer exchange is enhanced by one order of magnitudefor CrCl3 bilayers [453]. It is noted that it is still not clear how the easy-plane XY magnetic ordering in CrCl3

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would behave in the monolayer regime. Therefore, for such 2D magnets, an intriguing question to explore isthe dimensional crossover, i.e. how the magnetic ordering changes for the few-layer and monolayer cases incomparison to the bulk phases, as we summarized in figure 8 for the most representative 2D magnets.

The key to understand the magnetic behavior of such 2D magnets is to construct a reliable spin Hamil-tonian based on DFT calculations. Different ways to evaluate the interatomic exchange parameters have beenscrutinized in reference [454] for CrCl3 and CrI3. It is pointed out that the energy mapping method, whichhas been widely used, leads to only semi-quantitative estimation. Whereas the linear response theory includ-ing the ligand states give more reasonable results, ideally carried out in a self-consistently way. Particularly,exchange parameters beyond the nearest neighbors should be considered, as indicated by fitting the inelasticneutron scattering on FePS3 [455]. Moreover, for the interlayer exchange coupling, DFT calculations predictAFM coupling while the ground state is FM for Cr2Si2Te6 [456], which may be due to the dipole–dipole inter-action which is missing in most DFT calculations which can be considered by including the Breit correction[457]. Note that the interlayer exchange coupling is very sensitive to the distance, stacking, gating, and probablytwisting [458], which can be used to engineer the spin Hamiltonian which should be evaluated quantitatively.Furthermore, as most of the 2D magnets have the honeycomb lattice with edge-sharing octahedra like thelayered Na2IrO3 and α-RuCl3 which are both good candidates to realize the quantum spin liquid states [459],the Kitaev physics becomes relevant, e.g. a comparative investigation on CrI3 and Cr2Ge2Te6 demonstratedthat there do exist significant Kitaev interactions in both compounds [460]. Thus, for 2D magnets, the full 3× 3 exchange matrix as specified in equation (9) has to be considered beyond the nearest neighbors, in orderto understand the magnetic ordering and emergent properties. This will enable us to engineer the parameterspace to achieve the desired phases, e.g. the antisymmetric DMI can be tuned to stabilize skyrmions in 2Dmagnets, as observed in recent experiments on Fe3GeTe2 nanolayers [461, 462].

Due to the spatial confinement and reduced dielectric screening, the Coulomb interaction is less screenedin 2D materials, which bring forth the electronic correlation physics. Taking Cr2Ge2Te6 as an example, recentARPES measurements demonstrate that the band gap is about 0.38 eV at 50 K [463], in comparison to 0.2 eVat 150 K [464], while the band gap is only about 0.15 eV in the DFT + U calculations with U = 2.0 eV whichreproduce the MAE [465]. As both the magnetic ordering and electronic correlations can open up finite bandgaps, recent DFT + DMFT calculations reveal that the band gap in Cr2Si2Te6 is mostly originated from theelectronic correlations [466]. Additionally, both the d–d transitions and molecular ligand states are crucial forthe helical photoluminescence of CrI3 monolayers [467]. Moreover, it is also demonstrated that the FM phasecoexists with the Kondo lattice behavior in Fe3GeTe2 [468]. The correlated nature of the electronic states in 2Dmagnets calls for further experimental investigation and detailed theoretical modeling. 2D magnetic materialsprovide also an interesting playground to study the interplay between electronic correlations and magnetism,which is in competition with other emergent phases.

To tailor the 2D magnetism, voltage control of magnetism has many advantages over the mainstream meth-ods for magnetic thin films such as externally magnetic fields and electric currents, because they suffer frompoor energy efficiency, slow operating speed, and appalling size compactness. For biased CrI3 bilayers, externalelectric fields can be applied to switch between the FM and AFM interlayer coupling [469, 470]. Such a gatingeffect can be further enhanced by sandwiching dielectric BN layers between the electrode and the CrI3 lay-ers [471]. Moreover, ionic gating on Fe3GeTe2 monolayers can boost their Curie temperature up to the roomtemperature, probably due to the enhanced MAE but with quite irregular dependence with respect to the gat-ing voltage [472]. An interesting question is to understand the variation in the spin Hamiltonian and thusthe Curie temperature in such materials under finite electric fields. Additionally, strain has also been appliedto tailor 2D magnets. For instance, it is demonstrated theoretically that 2% biaxial compressive strain leadsto a magnetic state transition from AFM to FM for FePS3 monolayers [473], and it is found experimentallythat 1.8% tensile strain changes the FM ground state into AFM for CrI3 monolayers [474]. Moreover, straincan also induce significant modification of MAE, e.g. 4% tensile strain results in a 73% increase of MAE forFe3GeTe2 with a monotonous dependence [475]. Thus, it is suspected that biaxial strain can be applied to tunethe magnetic properties of vdW magnets effectively.

2D magnets exhibit a wide spectrum of functionalities. For instance, TMR in heterostructures of CrI3 hasbeen observed in several set-ups incorporating CrI3 multilayers [477–479], with the magnetoresistance ratio aslarge as 1 000 000% [480]. Such a high ratio can be attributed to perfect interfaces for the vdW heterostructures,which is challenging to achieve for conventional heterostructures of magnetic metals and insulators such asFe/MgO. The tunneling can be further tuned via gating [481], leading to the implementation of spin tunnelfield-effect transistors [482]. Moreover, metallic 2D magnets displays also fascinating spintronic properties,e.g. spin valve has been implemented based on the sandwich Fe3GeTe2/BN/Fe3GeTe2 geometry, where a TMRof 160% with 66% polarization has been achieved [483].

Such transport properties are based on the charge and spin degrees of freedom of electrons, whereas emer-gent degree of freedom such as valley has attracted also intensive attention, leading to valleytronics [484]. As

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most of the compounds discussed above have the honeycomb lattice, the valley-degeneracy in the momentumspace will be lifted when the inversion symmetry is broken, resulting in valley-dependent (opto-)electronicproperties. Moreover, considering SOC causes the spin–valley coupling [485], i.e. valley-dependent opticalselection rules become also spin-dependent, which can be activated by circularly polarized lights. Such phe-nomena can be realized in the vdW heterostructures such as the CrI3/WSe2 heterostructures [486] via theproximity effect or in AFM 2D magnets such as MnPS3 [487].

From the materials perspective, 2D materials have been extensively investigated both experimentally andtheoretically. There exist several 2D material databases [488–491] based on HTP DFT calculations, leadingto predictions of many interesting 2D magnets [492]. Nevertheless, it is still a fast developing field, e.g. theCurie temperature of 2D FM compounds has been pushed experimentally up to the room temperature inVTe2 [493] and CrTe2 [494]. It is noted that the DFT predictions should go hand-in-hand with detailed exper-imental investigation. For instance, VSe2 monolayers are predicted to be a room temperature magnet [495]supported by a follow-up experiment [496]. However, recent experiments reveal that the occurrence of thecharge density wave phase will prohibit the magnetic ordering [497, 498]. To go beyond the known prototypesto design new 2D materials, one can either predict more exfoliable bulk compounds with layered structure orlaminated compounds where etching can be applied. For instance, MXene is a new class of 2D materials whichcan be obtained from the MAX compounds [499], where the HTP calculations can be helpful. It is particularlyinteresting to explore such systems with 4d/5d TM or RE elements, where the interplay of electronic correla-tions, exchange coupling, and SOC will leads to more intriguing properties, such as recently reported MoCl5[500] and EuGe2 [501]. Last but not least, vdW materials can also be obtained using the bottom-up approachlike MBE down to the monolayer limit, such as MnSe2 on GaSe substrates [502]. This broadens significantlythe materials phase space to those non-cleavable or metastable vdW materials and provides a straightforwardway to in situ engineer the vdW heterostructures.

Like the 3D magnetic materials, 2D magnets display comparable properties but with a salient advantagethat they can form vdW heterostructures without enforcing lattice matching [503]. This leads to a wide rangeof combinations to further tailor their properties, e.g. the emergent twistronics [504] exampled by the super-conductivity in magic angle bilayer graphene [505]. In comparison to the interfaces in conventional magneticheterostructures, which are impeded by dangling bonds, chemical diffusion, and all possible intrinsic/extrinsicdefects, vdW heterostructures with well-defined interfacial atomic structures are optimal for further engineer-ing of spintronic devices [506]. Nevertheless, to the best of our knowledge, there is no freestanding 2D magnetshowing magnetic ordering at room temperature, which is a challenge among many as discussed above forfuture HTP design.

4. Case studies

4.1. High-throughput workflowsFigure 9 displays a typical HTP workflow to perform screening on stable magnetic materials, highlighting howthe most fundamental target magnetic properties can be evaluated. As DFT calculations need only the chem-ical compositions and crystal structures as the input, HTP calculations mostly start with such informationwhich can be obtained from the known databases such as ICSD https://icsd.fiz-karlsruhe.de/index.xhtml andCOD http://crystallography.net/cod/, or generated based on evolutionary algorithms using USPEX [507] andCALYPSO [57]. The thermodynamic stability (both the formation energy and the convex hull) can be obtainedafter obtaining the DFT total energies by comparing with those for relevant compounds in existing databasessuch as Materials Project [8], AFLOW [9], and OQMD [11]. Note that the data from such databases shouldbe used with caution, e.g. validated based on the input parameters. The mechanical and vibrational stabilitiesand the corresponding properties can be obtained by evaluating the elastic constants and the phonon spec-tra, followed by the calculation of the other magnetic properties such as magnetization, electronic structure,Curie/Neel temperature, and MAE (please refer to the previous sections for detailed discussions). It is wor-thy pointing out that a systematic evaluation of the thermodynamic properties dictates accurate Gibbs freeenergies, as illustrated for the Fe–N systems [508].

It is straightforward to set up such HTP workflows, either based on integrated platforms or starting fromscratch, as many ab initio codes have Python interfaces. In this regard, the atomic simulation environment[509] is a convenient tool, which has been interfaced to more than 30 software packages performing DFT andmolecular simulations. Nevertheless, the following aspects deserve meticulosity, which essentially distinguishHTP from conventional DFT calculations:

(a) The workflow should be automated, though it may not have to include all the steps specified in figure 9.For instance, a typical practice is to generate input files for thousands of cases combining crystal struc-tures and compositions and to perform DFT calculations to optimize the crystal structures and then assess

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Figure 9. A typical HTP workflow for screening magnetic materials. Blue, yellow, green, and purple blocks denote the processesof crystal structure identification, evaluation of stabilities, characterization of magnetic properties, and database curation,respectively.

the thermodynamic stability. The workflows to achieve such goals should be robust, which can be imple-mented using stand along python packages like fireworks [510], or using the integrated platforms such asAiiDa [12] and Atomate [13].

(b) Another critical point is job management and error handling. Ideally, a stand-by thread shall monitor thesubmitted jobs, check the finished ones, and (re-)submit further jobs with proper error (due to eitherhardware or software) recovery. In this regard, custodian (https://materialsproject.github.io/custodian/)is a good option, with integrated error handling for the VASP, NwChem, and QChem codes. On the otherhand, the set-up and maintenance of such job management depends on the computational environment,and also requires expertise with the scientific softwares.

(c) A data infrastructure is needed in order to avoid repetitive calculations, and more importantly to sharedata. Non-SQL databases (e.g. MongoDB) are mostly used because of their flexibility with the data struc-ture, which can be adjusted for compounds with various complexity and add-on properties evaluatedin an asynchronized way. Moreover, the data should be findable, accessible, interoperable, and reusable[511], so that they can be shared within the community with approved provenance. Last but not least, suchdatabases are indispensable for data mining using machine learning techniques, as discussed in section 5.2.

4.2. Heusler compoundsHeusler compounds are an intriguing class of intermetallic systems possessing a vast variety of physical prop-erties including half metals, MSME, TI, superconductivity, and Seebeck effect (please refer to reference [512]for a comprehensive review). Such versatility is originated from the tunable metallic and insulating nature oftheir electronic structure due to the flexibility in the chemical compositions. As shown in figure 10, there aretwo main groups of Heusler compounds, namely, the half Heusler with the chemical composition XYZ and thefull-Heusler X2YZ, where X and Y are TM or RE elements, and Z being a main group element. For both halfHeusler and full Heusler compounds, the crystal structures can be considered as a skeleton of the ZnS-type(zinc blende) formed by covalent bonding between the X and Z elements, where the Y cations will occupy theoctahedral sites and the vacancies. This gives rise to the flexibility to tune the electronic structure by playingwith the X, Y, and Z elements [512].

The full Heuslers X2YZ can end up with either the regular (L21-type) or the inverse (Xα-type) Heuslerstructure, as shown in figure 10. It is noted that both the regular and inverse full Heusler crystal structuresare derived from the closely packed fcc structure, which is one of the most preferred structures for ternaryintermetallic systems [513]. Depending on the relative electronegativity of the X and Y elements, the empirical

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Figure 10. Illustration of the Heusler crystal structures for (a) (regular) full Heusler X2YZ, (b) inverse (full) Heusler X2YZ, (c)half Heusler XYZ, (d) quaternary Heusler XX′YZ, and (e) tetragonal (full) Heusler X2YZ. The conventional unit cell of 16 atomsare sketched, using the Wyckoff positions of the F43m to give a general description for the cubic (a–d) cases.

Burch’s rule states that the inverse Heusler structure is preferred if the valence of Y is larger than that of X,e.g. Y is to the right of X if they are in the same row of the periodic table [514]. Several HTP calculationshave been done to assess the Burch’s rule, such as in Sc- [515] and Pd-based [516, 517] Heuslers. In terms ofHTP calculations, the nonmagnetic L21 Heusler compounds as high-strength alloys [518] and thermoelectric[519] materials have been studied. For the magnetic properties, Sanvito et al [54] performed HTP calculationson 236 115 X2YZ compounds and found that 35 602 compounds are stable based on the formation energy,where 6778 compounds are magnetic. Unfortunately, the relative stability with respect to competing binaryand ternary phases is expensive thus demanding to accomplish. In a recent work [520], Ma et al carried outDFT calculations on a small set of 405 inverse Heusler compounds, with the relative stability to the regularHeusler and the convex hull considered systematically, including also a few possible magnetic configurations[520]. Ten HMs or nearly HMs have been identified, which can be explored for spintronic applications.

Focusing particularly on the regular Heusler systems, Balluff et al [521] performed systematic evaluationof the convex hull based on the available data in AFLOW and it is found that the number of magnetic Heuslerswill be reduced from 5000 to 291. That is, the convex hull construction will reduced the number of stablecompounds by one order of magnitude, consistent with our observation on magnetic antiperovskites [44].Therefore, although the database of competing phases might not be complete, it is highly recommended toevaluate the thermodynamic stability including both the formation energy and convex hull, if not the mechan-ical and dynamic stabilities. Furthermore, to identify the magnetic ground states, Balluff et al carried outcalculations on two AFM configurations and obtained 70 compounds with AFM ground states. The Neeltemperature of such compounds are then computed using the Monte Carlo method with exchange param-eters calculated explicitly using the SPRKKR (https://ebert.cup.uni-muenchen.de) code, resulting in 21 AFMHeusler compounds with Neel temperature higher than 290 K. Such compounds are ready to be explored forapplications on AFM spintronics.

Interestingly, for both regular and inverse Heusler compounds, tetragonal distortions can take place drivenby the band Jahn–Teller effect associated with high DOS at the Fermi energy [522]. HTP calculations revealthat 62% of the 286 Heusler compounds investigated prefer the tetragonal phase, due to the van Hove singu-larities associated with DOS at the Fermi energy [523]. It is further suggested that tetragonal Fe2YZ and Co2YZcompounds show large MAE, e.g. 5.19 MJ m−3 for Fe2PtGe and 1.09 MJ m−3 for Fe2NiSn, making them inter-esting candidates as permanent magnets and STT–MRAM materials [524]. To search for potential materialsfor STT applications, Al-, Ga-, and Sn-based inverse Heusler compounds in both cubic and tetragonal struc-tures have been investigated, aiming at optimizing the spin polarization and Gilbert damping for materialswith perpendicular magnetic anisotropy [525]. Additionally, such tetragonal Heusler compounds can also beused to engineer magnetic heterostructures with an enhanced TMR [526].

Significant MAE can also be obtained by inducing tetragonal distortions on the cubic Heusler compounds.For instance, the MAE of Ni2YZ compounds can be as large as 1 MJ m−3 by imposing c/a �= 1 [527]. Tostabilize such tetragonal distortions, we examined the effect of light elements (H, B, C, and N) as intersti-tial dopants and found that tetragonal distortions can be universally stabilized due to the anisotropic crys-talline environments for the interstitials preferentially occupying the octahedral center [528]. This leads to an

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effective way to design RE-free permanent magnets. Two pending problems though are the solubility andpossible disordered distribution of the light interstitials in the Heusler structures. Nevertheless, it is demon-strated that for the Fe–C alloys, due to interplay of anharmonicity and segregation, collective ordering ispreferred [529], entailing further studies on the behavior of interstitials in the Heusler compounds. We notethat the structural phase transition between the cubic and tetragonal phases in Ni2MnGa induces MSME andin Ni2MnX (X = In, Al, and Sn) MCE. In this regard, there are many more compounds with such martensitictransitions which can host MSME and MCE, awaiting further theoretical and experimental investigations.

To go beyond the p–d covalent bonding between the X and Z atoms which stabilizes the Heusler structures,Wei et al first explored the d–d hybridization in the Ni–Mn–Ti systems and observed that there exist stablephases with significant MSME, which offers also the possibility to improve the mechanical properties overthe main-group-element based conventional Heusler compounds [530]. Considering only the cubic phases,follow-up HTP calculations revealed that 248 compounds are thermodynamically stable out of 36 540 proto-types and 22 of them have magnetic ground states compatible with the primitive unit cell [54]. The predictionsare validated by successful synthesis of Co2MnTi and Mn2PtPd, and interestingly Mn2PtPd adopts a tetrago-nal structure with the space group I4/mmm experimentally [54]. As the mechanical properties of all-d-metalHeusler compounds are expected to be better due to the d–d bonding, an interesting question is whetherthere exist promising candidates with enhanced caloric performance beyond the known Ni–Mn–Ti case [531].There have been several studies focusing on Zn- [532], V- [533], and Cd-based [534] Heuslers, where possi-ble martensitic transformation is assessed based on the evaluation of Bain paths. We note that in order to getenhanced MCE, the martensitic phase transition temperature is better aligned within the Curie temperaturewindow [535]. Therefore, to predict novel all-d-metal Heuslers with magneto-structural functionalities, moresystematic HTP characterization on the thermodynamic and magnetic properties for both cubic and tetragonalphases is required.

The flexibility in the chemical composition for the intermetallic Heusler compounds can be furtherexplored by substituting a fourth element or vacancy in X2YZ, leading to quaternary XX′YZ and half XYZHeusler (figure 10). Following the empirical rule, we performed HTP calculations on magnetic quaternaryHeusler compounds with 21, 26, and 28 valence electrons to search for spin gapless semiconductors (SGSs)[536]. Considering both structural polymorphs by shuffling atomic positions and magnetic ordering compati-ble with the primitive cell, we identified 70 unreported candidates covering all four types of SGSs out of 12 000chemical compositions, with all the 22 known cases validated. Such SGSs are promising for spintronic applica-tions, as they display significant AMR, and tunable AHC and TMR [536, 537]. We note that the semimetallicphase can also be found in quaternary Heuslers with the other numbers of electrons [538], which is interestingfor future exploration.

Regarding half Heusler compounds, there have been extensive HTP calculations to screen for nonmagneticsystems as thermoelectric materials [539, 540], where the formation of defects has also been addressed in anHTP manner [541]. Ma et al carried out HTP calculations on 378 half Heuslers and identified 26 semicon-ductors, 45 half metals, and 34 nearly half metals, which are thermodynamically stable [80]. Another recentwork investigated the alkaline element based half Heuslers and found 28 FM materials out of 90 compositions[542]. Importantly, for half Heusler materials, it is not enough to consider only the binary competing phases,but also possible ternary compounds as there are many crystal structures which can be stabilized based on the18-electron rule for the 1:1:1 composition [543]. For instance, the so-called hexagonal Heusler compounds area class of materials which can host topological insulator [544] and (anti-)ferroelectric phases [545, 546]. Par-ticularly, the magnetic counterparts hexagonal Heuslers host martensitic transitions, making them promisingfor MCE applications [547]. This leads to two recent work where the composition of the hexagonal Heuslercompounds is optimized to further improve MCE [548, 549], whereas more systematic HTP calculations arestill missing. Additionally, it is demonstrated that the half Heuslers with the F43m structure can be distortedinto either P63mc or Pnma structures, giving rise to an interesting question whether magnetic materials withsignificant MCE can be identified based on HTP screening. Last but not least, like the quaternary Heuslers, theflexibility in composition can also be realized in the half Heusler structure. For instance, 131 quaternary dou-ble half Heusler compounds are predicted to be stable where Ti2FeNiSb2 has been experimentally synthesizedshowing low thermal conductivity as predicted.

Furthermore, motivated by the recently discovered Weyl semimetal Co2MnGa [349], an enlightening workevaluated the topological transport properties (e.g. AHC, ANC, and MOKE which obey the same symmetryrules) of 255 cubic (both regular and inverse) Heusler compounds in an HTP way [550]. By comparing theresults for full and inverse Heusler systems, it is observed that the mirror symmetry present in the full but notthe inverse Heusler compounds plays an essential role in inducing significant linear response properties. It isnoted that only one FM (AFM) configuration is considered for the full (inverse) Heusler compounds, whichmight not be enough as the electronic structure and the derived physical properties are subjected to changesdepending on the magnetic configurations.

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Overall, it is obvious that Heusler compounds are a class of multifunctional magnetic materials, but thecurrent HTP calculations are still far away from being complete. One additional issue is how to properly treatthe chemical and magnetic disorders in such compounds in order to predict the electronic and thermody-namic properties. For instance, MCE takes place only for Mn-rich Ni–Mn–Ti [530], which has not yet beenexplained based on DFT calculations. We believe substitutions and chemically disordered systems deserve moresystematic treatments in the future HTP studies, particularly for Heusler alloys.

4.3. Permanent magnetsAs discussed above, the main task for HTP screening of permanent magnets is to evaluate the intrinsic magneticproperties the known and predicted phases, including both RE-free and RE-based systems. A recent work[551] focuses on the RE-free cases, where starting from 10 000 compounds in the ICSD database containingCr, Mn, Fe, Co, and Ni, step-by-step screening is carried out based on the chemical composition, Ms, crystalstructure, magnetic ground state, and MAE. Three compounds, namely, Pt2FeNi, Pt2FeCu, and W2FeB2 arefinally recommended. It is noted that the magnetic ground state for the compounds during the screening isdetermined by the literature survey, which might be labor intensive and definitely cannot work for ‘unreported’cases.

In addition to the HTP characterization of the known phases and prediction of novel compounds, tailoringthe existing phases on the boundary of being permanent magnets is another way, which can be done via sub-stitutional and interstitial doping. For instance, theoretical predictions suggest that Ir/Re doping can enhancethe MAE of (Fe1−xCox)2B, which is confirmed by experiments [552]. Nevertheless, cautions are deserved whenevaluating the MAE for the doped compounds in terms of the supercell method, where the symmetry of thesolution phases may be broken leading to erroneous predictions. Special techniques such as VCA and CPA areneeded to obtain MAE as done in reference [552], which are not generally available in the main stream codes.Furthermore, the interstitial atoms such as H, B, C, and N can also be incorporated into the existing inter-metallic phases, giving rise to structural distortions and thus significant MAE as demonstrated for the FeCoalloys [553, 554]. Following this line, we have performed HTP calculations to investigate the effects of inter-stitials on the intermetallic compounds of both the Cu3Au [79] and Heusler [528] types of crystal structures,and identified quite a few candidates interesting for further exploration.

Turning now to the RE-based permanent magnets, due to the challenging task to treat the 4f-electrons,approximations are usually made. For instance, in a series of work, the tight-binding linear Muffin–Tin orbitalmethod in the atom sphere approximation has been applied to screen the 1-5/2-17/2-14-1 type [555], 1-12 type[556, 557], 1-11-X type [169], and 1-13-X type [558] classes of materials, where the 4f-electrons are consideredin the spin-polarized core approximation. One apparent problem is that the magnetization and MAE cannotbe obtained consistently based on such a method. Additionally, it is difficult to access the thermodynamicstability of the RE-based compounds studied, which is a general problem for all the DFT codes on the market.Nevertheless, the trend with respect to chemical compositions can be obtained, which can be validated withfurther experiments. It is note that the Staunton group has developed a consistent theoretical framework toevaluate both the magnetization and MAE [559, 560], which is interesting if further HTP calculations can beperformed.

Therefore, due to the challenges such as the interplay of SOC and electronic correlations, HTP design ofpermanent magnets is still a subject requiring further improvement on the computing methodology. As suchmaterials are applied in big volume, it might be strategic to consider firstly the criticality of the constituteelements, e.g. the high cost and supply risk on Co, Nd, Dy, and Tb. In this regard, compounds based on Fe andMn combined with cheap elements in the periodic table should be considered with a high priority. In addition,all three intrinsic properties (Ms, MAE, and TC) should be optimized which is numerically expensive and thusnot usually done. In this regard, a consistent figure of merit should be adopted, e.g. the suggestion by Coey

to define and optimize the dimensionless magnetic hardness parameter κ =(

K1μ0M2

s

)1/2> 1, which should be

implemented for future HTP screening [27].

4.4. Magnetocaloric materialsAs discussed in section 3.6, MCE is driven by the interplay of various interactions leading to challenges inperforming HTP design of novel MCE materials. For instance, rigorous evaluation of the Gibbs free energiesincluding the lattice, spin, and electronic degrees of freedom is a formidable and numerically expensive task,not to mention the complex nature of magnetism and phase transitions. Thus, it is of great interest to estab-lish a computational ‘proxy’ which correlates the MCE performance with quantities easily accessible via DFTcalculations.

Following the concept that MCE is significant when the magneto-structural phase transitions occur,Bocarsly et al proposed a proxy in terms of the magnetic deformation ΣM = 1

3 (η21 + η2

2 + η33)1/2 × 100 and

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η = 12 (PTP − I) where P = A−1

nonmag · Amag with Anonmag and Amag being the lattice constants of the nonmag-netic and magnetic unit cells [561]. Assuming the high temperature PM phase can be described with thenonmagnetic phase at 0 K, the magnetic deformationΣM measures ‘the degree to which structural and magneticdegrees of freedom are coupled in a material’. In fact, there is a universal correlation between the ΔS and ΣM

no matter whether MCE is driven by FOPT or not by examining the known compounds. Nevertheless, thereis no direct scaling between ΔS and ΣM and it is suggested that ΣM > 1.5% is a reasonable cutoff to selectthe promising compounds. Further screening on the known FM materials reveals 30 compounds out of 134systems are good candidates, where one of them MnCoP is validated by experimental measurements showingΔS = −3.1 J kg−1 K−1 under an applied field of 2 T. Recently, this proxy has been successfully applied to eval-uate the MCE behavior of two solid solutions Mn(Co1−xFex)Ge and (Mn1−yNiy)CoGe, where the predictedoptimal compositions x = 0.2 and y = 0.1 are in good agreement with the experiments [549]. It is interestingthat such a consistency is obtained by evaluating the corresponding quantities via configurational average oversupercells with Boltzmann weights, instead of using the SQS method. One point to be verified is whether thenonmagnetic geometries are good enough to be compared with the PM states, and our results on a serious ofAPVs with noncollinear magnetic ground states indicate that the negative thermal expansion associated withthe magneto-structural transition and hence MCE is overestimated using the magnetic deformation betweenthe FM and nonmagnetic states [632].

Another more systematic and computationally involved workflow is the CaloriCool approach [232], basedon a two-step screening. It is suggested that the metallic alloys and intermetallic compounds are more promis-ing candidates than the oxides, which ‘suffer from intrinsically low isentropic (a.k.a. adiabatic) temperaturechanges due to large molar lattice specific heat’. The first step fast screening is based on the phase diagramsand crystal structures from the literature and known databases, which results in compounds with the samechemical composition but different crystal structures. In the second step, the physical properties includingmechanical, electronic, thermodynamic, kinetic properties will be evaluated, combining DFT with multi-scaleand thermodynamic methods. For instance, the thermodynamic properties can be evaluated following themethods proposed in reference [562], which is foreseeably expensive and better done one after another. Also,the success of the approach depends significantly on the database used for the first screening. Nevertheless, it isproposed based on such design principles that there should be a solution phase Zr1−xYbxMn6Sn6 with signifi-cant MCE due to the fact that the pristine ZrMn6Sn6 and YbMn6Sn6 are with AFM and FM ground states withcritical temperatures being 580 K and 300 K, respectively. This idea is very similar to the morphotropic phaseboundary concept in the community of ferroelectric materials, where the piezoelectric response associatedwith first order ferroelectric phase transition can be greatly enhanced at the critical compositions [563].

4.5. Topological materialsAs discussed in section 3.8, insulators and semimetals of nontrivial nature are an emergent class of materialsfrom both fundamental physics and practical applications points of view. To design such materials, symme-try plays an essential role as explained in detail in section 3.8 for a few specific compounds. Particularly formagnetic materials, the occurrence of QAHI and Weyl semimetals is not constrained to compounds of specificsymmetries, whereas the AFM TIs entail a product symmetry S = PΘ which can give rise to the demandedKramers degeneracy. It is noted that the symmetry argument applies to the nonmagnetic topological materialsas well, where based on topological quantum chemistry [564] and filling constraints [565, 566] HTP char-acterization of nonmagnetic materials have been systematically performed [567–570]. Such screening canalso be performed in a more brute-force way by calculating the surface states [571] and spin–orbit spillage[572, 573].

A very fascinating work done recently is to perform HTP screening of AFM topological materials basedon the magnetic topological quantum chemistry [574], as an extension to the topological quantum chem-istry [564]. For this approach, the irreducible co-representations of the occupied states are evaluated at thehigh symmetry points in the BZ, and the so-called compatibility relations are then taken to judge whetherthe compounds are topologically trivial or not [575]. In this way, six categories of band structures can bedefined, e.g. band representations, enforced semimetal with Fermi degeneracy, enforced semimetal, Smith-index semimetal, strong TI, and fragile TI, where only band representations are topologically trivial. Out of 403well converged cases based on DFT calculations of 707 compounds with experimentally available AFM mag-netic structures collected in the MAGNDATA database [576], it is observed that about 130 (≈32%) compoundsshowing nontrivial topological features, where NpBi, CaFe2As2, NpSe, CeCo2P2, MnGeO3, and Mn3ZnC arethe most promising cases. We note that one uncertainty is the U values used which are essential to get thecorrect band structure, and more sophisticated methods such as DFT + DMFT might be needed to properlyaccount for the correlations effect [179].

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4.6. 2D magnetsThe HTP screening of 2D magnetic materials has been initiated by examining firstly the bulk materials that areheld together by the vdW interaction, which can then be possibly obtained by chemical/mechanical exfoliationor deposition. Starting from the inorganic compounds in the ICSD database [577], two recent publicationstried to identify 2D materials by evaluating the packing ratio, which is defined as the ratio of the covalentvolume and the total volume of the unit cell [578, 579]. This leads to about 90 compounds which can beobtained in the 2D form, where about 10 compounds are found to be magnetic [579]. The screening criteriahave then been generalized using the so-called topological-scaling algorithm, which classifies the atoms intobonded clusters and further the dimensionality [488]. Combined with the DFT evaluation of the exfoliationenergy, 2D materials can be identified from the layered solids. This algorithm has been applied to compoundsin the Materials Project database, where 826 distinct 2D materials are obtained. Interestingly, 128 of them arewith finite magnetic moments >1.0 μB/u.c., and 30 of them showing half metal behavior [488]. Extendedcalculations starting from the compounds in the combined ICSD and COD databases identified 1825 possible2D materials, and 58 magnetic monolayers have been found out of 258 most promising cases [489]. Similarly,by evaluating the number of ‘covalently connected atoms particularly the ratio in supercell and primitive cells’,45 compounds are identified to be of layered structures out of 3688 systems containing one of (V, Cr, Mn, Fe,Co, Ni) in the ICSD database, leading to 15 magnetic 2D materials [580].

As discussed in section 3.1, the occurrence of 2D magnetism is a tricky problem, particularly the magneticordering temperature which is driven by the interplay of magnetic anisotropy and exchange parameters. Basedon the computational 2D materials database (C2DB) [490], HTP calculations are performed to obtain theexchange parameters, MAE, and critical temperature for 550 2D materials, and it is found that there are about150 (50) FM (AFM) compounds being stable [492]. Importantly, the critical temperatures for such compoundshave a strong dependence on the values of U , indicating that further experimental validation is indispensable.Nevertheless, the characterization of the other properties for 2D materials in the HTP manner is still limited.One exception is the HTP screening for QAHC with in-plane magnetization [581], where the prototype LaClmight not be a stable 2D compound. It is noted that there have been a big number of predictions on 2Dmagnetic and nonmagnetic materials hosting exotic properties but the feasibility to obtain such compoundshas not been systematically addressed. Therefore, we suspect that consistent evaluations of both stability andphysical properties are still missing, particularly in the HTP manner which can guide future experiments.

Despite the existing problems with stability for 2D (magnetic) materials, a systematic workflow to charac-terize their properties has been demonstrated in a recent work [582]. Focusing on the FM cases and particularlythe prediction of associated TC, it is found that only 53% of 786 compounds predicted to be FM [490], are actu-ally stable against AFM configurations generated automatically based on the method developed in reference[81]. In this regard, the parameterization of the Heisenberg exchange parameters should be scrutinized, whereexplicit comparison of the total energies for various magnetic configurations can be valuable. Follow-up MonteCarlo simulations based on fitted exchange parameters reveal 26 materials out of 157 would exhibit TC higherthan 400 K, and the results are modeled using ML with an accuracy of 73%. Unfortunately, the exact AFMground states are not addressed, which is interesting for future investigation.

It is well known that 2D materials offer an intriguing playground for various topological phases particularlyfor magnetic materials including the QAHE [581], AFM TI [583], and semimetals [407]. HTP calculationsbased on the spin–orbit spillage [572] has been carried out to screen for magnetic and nonmagnetic topologicalmaterials in 2D materials [584], resulting in four insulators with QAHE and seven magnetic semimetals. Suchcalculations are done on about 1000 compounds in the JARVIS database (https://ctcms.nist.gov/~knc6/JVASP.html), and we suspect that more extensive calculations together with screening on the magnetic ground statecan be interesting.

Beyond the calculations across various types of crystal structures, HTP calculations on materials of thesame structural type substituted by different elements provide also promising candidates and insights on the2D magnetism [585]. For instance, Chittari et al performed calculations on 54 compounds of the MAX3 type,where M=V, Cr, Mn, Fe, Co, Ni, A= Si, Ge, Sn, and X= S, Se, Te. It is observed that most of them are magnetic,hosting half metals, narrow gap semiconductors, etc. Importantly, strain is found to be effective to tailor thecompetition between the AFM and FM ordering in such compounds [586]. Similar observations exist in in-plane ordered MXene (i-MXene) [587], which can be derived from the in-plane ordered MAX compoundswith nano-laminated structures [588]. Moreover, our calculations on the 2D materials of the AB2 (A being TMand B = Cl, Br, I) indicate that FeI2 is a special case adopting the 2H-type structure while all the neighboringcases have the 1T-type [589]. This might be related to an exotic electronic state as linear combination of thet2g orbitals in FeI2 driven by the strong SOC of I atoms [590]. We found that the band structure of such AB2

compounds is very sensitive to the value of the effective local Coulomb interaction U applied on the d-bandsof TM atoms. This is by the way a common problem for calculations on 2D (magnetic) materials, where thescreening of Coulomb interaction differ from that in 3D bulk materials [591].

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https://ctcms.nist.gov/~knc6/JVASP.html

https://ctcms.nist.gov/~knc6/JVASP.html


5. Future perspectives

5.1. Multi-scale modelingAs discussed in previous sections, HTP calculations based on DFT are capable of evaluating the intrinsicphysical properties, which usually set the upper limits for the practical performance. To make more realisticpredictions and to directly validate with the experiments, multi-scale modeling is required.

The major challenge therein is how to make quantitative predictions for materials with structural featuresof large length scales beyond the unit cells, including both topological defects(e.g. domain walls) and structuraldefects (e.g. grain boundaries). Taking permanent magnets as an example, as indicated in figure 1, both Mr andHa cannot reach the corresponding theoretical limit of Ms and MAE. MAE (represented by the lowest-orderuniaxial anisotropy K) sets an upper limit for the intrinsic coercivity 2K/(μ0Ms) as defined for single-domainparticles via coherent switching [593], whereas the real coercivity is given by Ha = α2K/(μ0Ms) − βMs, whereα and β are the phenomenological parameters, driven by complex magnetization switching processes mostlydetermined by the microstructures. It is noted that α is smaller than one (about 0.1–0.3) due to the extrin-sic mechanisms such as inhomogeneity, misaligned grains, etc, and β accounts for the local demagnetizationeffects [594]. This leads to the so-called Brown’s paradox [595]. That is, the magnetization switching cannot beconsidered as ideally a uniform rotation of the magnetic moments in a single domain, but is significantly influ-enced by the extrinsic processes such as nucleation and domain wall pinning [596], which are associated withstructural imperfections at nano-, micro-, and macroscopic scales [594]. Therefore, in addition to screeningfor novel candidates for permanent magnets, current research is focusing on how to overcome the limitationsimposed by the microstructures across several length scales [597, 598].

Furthermore, the functionalities of ferroic materials are mostly enhanced when approaching/crossing thephase transition boundaries, particularly for the magneto-structural transitions of the first-order nature. Uponsuch structural phase transitions, microstructures will be developed to accommodate the crystal structurechange, e.g. formation of twin domains due to the loss of point-group symmetry. Moreover, most FOPT occurwithout long-range atomic diffusion, i.e. of the martensitic type, leading to intriguing and complex kinetics. Asindicated above, what is characteristic to the athermal FOPTs is the hysteresis, resulting in significant energyloss converted to waste heat during the cooling cycle for magnetic materials [31]. While hysteresis typicallyarises as a consequence of nucleation, in caloric materials it occurs primarily due to domain-wall pinning,which is the net result of long-range elastic strain associated with phase transitions of interest. When the hys-teresis is too large, the reversibility of MCE can be hindered. Therefore, there is a great impetus to decipher themicrostructures developed during the magneto-structural transitions, particularly the nucleation and growthof coexisting martensitic and austenite phases. For instance, it is observed recently that materials satisfyingthe geometric compatibility condition tend to exhibit lower hysteresis and thus high reversibility, leading tofurther stronger cofactor conditions [599].

Therefore, in order to engineer magnetic materials for practical applications, multi-scale modeling is indis-pensable. To this goal, accurate DFT calculations can be performed to obtain essential parameters such as theinterlayer exchange, DMI, and SOT, which will be fed into the atomistic [105] and micromagnetic modeling[20]. It is noted that such scale-bridging modeling is required to develop fundamental understanding of spin-tronic devices as well. For instance, for the SOT memristors based on the heterostructures composing AFMand FM materials [270], the key problems are to tackle the origin of the memristor-like switching behaviordriven possibly by the interplay of FM domain wall propagation/pinning with the randomly distributed AFMcrystalline grains, and to further engineer the interfacial coupling via combinatorial material combinations foroptimal device performance.

Given the fact that the computational facilities have been significantly improved nowadays, HTP predic-tions can be straightforwardly made, sometimes too fast in comparison to the more time-consuming experi-mental validation. In this regard, HTP experiments are valuable. Goll et al proposed to use reaction sinteringas an accelerating approach to develop new magnetic materials [600], which provides a promising solutionto verify the theoretical predictions and to further fabricate the materials. Such an HTP approach has alsobeen implemented to screen over the compositional space, in order to achieve optimal MSME performance[601]. Nevertheless, what is really important is a mutual responsive framework combining accurate theoreticalcalculations and efficient experimental validations.

5.2. Machine learningAnother emergent field is materials informatics [602] based on advanced analytical machine learn-ing techniques, which can be implemented as the fourth paradigm [603] to map out the pro-cess–(micro)structure–property relationship and thus to accelerate the development of materials including themagnetic ones. As the underlying machine learning is data hungry, it is critical to curate and manage databases.Although there is increasing availability of material databases such as Materials Project [8], OQMD [604], and

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Figure 11. Theoretical framework for future materials design.

NOMAD [10], the sheer lack of data is currently a limiting factor. For instance, the existing databases men-tioned above compile mostly the chemical compositions and crystal structures, whereas the physical propertiesparticularly the experimentally measured results are missing. In this regard, a recent work trying to collect theexperimental TC of FM materials is very interesting, which is achieved based on natural language processingon the collected literature [605].

Following this line, machine learning has been successfully applied to model the TC of FM materials, whichis still a challenge for explicit DFT calculations as discussed above. Dam et al collected the TC for 108 binaryRE-3d magnets and the database is regressed using the random forest algorithm [606]. 27 empirical features aretaken as descriptors with detailed analysis on the feature relevance. Two recent work [607, 608] started withthe AtomWork database [609] and used more universal chemical and structural descriptors. It is observedthat the Curie temperature is mostly driven by the chemical position, where compounds with polymorphs areto be studied in detail with structural features. The other properties such as MAE [610] and magnetocaloricperformance [611] can also be fitted, which makes it very interesting for the future.

HTP calculations generate a lot of data which can be further explored using machine learning. This hasbeen performed on various kinds of materials and physical properties, as summarized in a recent review [612].Such machine learning modeling has also been applied on magnetic materials. For instance, based on theformation energies calculated via HTP DFT calculations, machine learning has been used to model not onlythe thermodynamic stability of the full Heusler [513, 613], half Heusler [614], and quaternary Heusler [615]compounds, but also the spin polarization of such systems [616].

In this sense, machine learning offers a straightforward solution to model many intrinsic properties of mag-netic materials, but is still constrained by the lack of data. For instance, the TC and TN are collected for about10 000 compounds in the AtomWork data [609], with significant uncertainty for compounds with multipleexperimental values thus the database should be used with caution. In addition, both machine learning mod-elings of TC [607, 608] are done based on the random forest algorithm, whereas our test using the Gaussiankernel regression method leads to less satisfactory accuracy. This implies that the data are quite heterogeneous.Although our modeling can distinguish the FM and AFM ground state, it is still unclear about how to predictthe exact AFM configurations, which requires a database of magnetic structures. To the best of our knowledge,there is only one collection of AFM structures available in the MAGNDATA database, where there are 864compounds listed with the corresponding AFM ground states [576]. Therefore, there is a strong impetus tosort out the information in the literature and to compile a database of magnetic materials.

Importantly, it is suspected that the interplay of machine learning and multi-scale modeling will create evenmore significant impacts on identifying the process– structure–property relationships. For instance, phase fieldmodeling aided with computational thermodynamics can be applied to simulate the microstructure evolutionat the mesoscopic scale for both bulk and interfaces [617], leading to reliable process–structure mapping. Asdemonstrated in a recent work [618], the structure–property connection for permanent magnets can also beestablished based on massive micromagnetic simulations. Thus, HTP calculations performed in a quantitativeway can be applied to generate a large database where the process–structure–property relationship can be fur-ther addressed via machine learning, as there is no unified formalism based on math or physics to define suchlinkages explicitly. Additionally, machine learning interatomic potentials with chemistry accuracy have beenconstructed and tested on many material systems [619], which are helpful to bridge the DFT and moleculardynamics simulations. We believe an extension of such a scheme to parameterize the Heisenberg Hamiltonian(equation (6)) with on-top spin–lattice dynamics simulations [118, 218] will be valuable to obtain accurateevaluation of the thermodynamic properties for magnetic materials.

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From the experimental perspective, adaptive design (i.e. active learning) based on surrogate models hasbeen successfully applied to guide the optimization of NiTi shape memory alloys [620] and BaTiO3-basedpiezoelectric materials [621]. Such methods based on Bayesian optimization or Gaussian process work on smallsample sizes but with a large space of features, resulting in guidance on experimental prioritization with greatlyreduced number of experiments. This method can hopefully be applied to optimizing the performance of the2-14-1 type magnets, as the phase diagram and underlying magnetic switching mechanism based on nucle-ation is well understood. Furthermore, the machine learning techniques are valuable to automatize advancedcharacterization. Small angle neutron scattering is a powerful tool to determine the microstructure of mag-netic materials, where machine learning can be applied to accelerate the experiments [622]. Measurements onthe spectral properties such as XMCD can also be analyzed on-the-fly in an automated way based on Gaussianprocess modeling [623]. In this regard, the hyperspectral images obtained in scanning transmission electronicmicroscopy can be exploited by mapping the local atomic positions and phase decomposition driven by ther-modynamics. That is, not only the local structures but also the possible grain boundary phases assisted withdiffraction and EELS spectra can be obtained, as recently achieved in x-ray spectro-microscopy [624, 625].

6. Summary

In conclusion, despite the magnetism and magnetic materials have been investigated based on quantummechanics for almost a hundred years (if properly) marked by the discovery of electron spin in 1928, it isfair to say that they are not under full control of us due to the lack of thorough understanding, as we arefacing fundamental developments marked by progresses on AFM spintronics, magnetic topological materi-als, and 2D magnets, and we are exposed to complex multi-scale problems in magnetization reversal andmagneto-structural phase transitions. Thus, we believe systematic calculations based on the state-of-the-artfirst-principles methods on an extensive list of compounds will at least get the pending issues better defined.

Among them, we would emphasize a few urgent and important aspects as follows,

• Integrating the magnetic ground state searching into the available HTP platforms so that automatedworkflow can be defined. There are several solutions developed recently as discussed in section 3.3;

• Implementation of a feasible DFT + DMFT framework for medium- (if not high-) throughput calcu-lations. Two essential reasons are the interplay of Coulomb interaction, SOC, and hybridization for the4f–3d magnets and the treatment of magnetic fluctuations and PM states. This is also important for 2Dmagnets;

• Consistent framework to evaluate the transport properties which can be achieved based on accuratetight-binding-like models using Wannier functions [626];

• Quantitative multi-scale modeling of the magnetization reversal processes and the thermodynamic prop-erties upon magneto-structural transitions in order to master hysteresis for permanent magnets andmagnetocaloric materials;

• Curation and management of a flexible database of magnetic materials, ready for machine learning.

Additionally, a common problem not only applicable for magnetic materials but also generally true for theother classes of materials subjected to property-optimization is a consistent way to consider substitution inboth the dilute and concentrated limits. There are widely used methods such as virtual crystal approximation,coherent potential approximation [627], special quasirandom structures [114], cluster expansion [628], etc,but we have not yet seen HTP calculations done systematically using such methods to screen for substitutionaloptimization of magnetic materials.

Beyond the bare HTP design, we envision a strong interplay between HTP, machine learning, and multi-scale modeling, as sketched in figure 11. Machine learning is known to be data hungry, where HTP calculationsbased on DFT can provide enough data. In addition to the current forward predictions combining HTP and ML[612], we foresee that inverse design can be realized [629], as highlighted by a recent work on prediction newcrystal structures [630]. As mentioned above, the marriage between HTP and multi-scale modeling helps toconstruct accurate atomistic models and thus to make the multi-scale modeling more quantitative and predic-tive. Last but not least, it is suspected that the interaction between machine learning and multi-scale modelingwill be valuable to understand the microstructures, dubbed as microstructure informatics [631]. Such a the-oretical framework is generic, i.e. applicable to not only magnetic materials but also the other functional andstructural materials.

All in all, HTP calculations based on DFT are valuable to provide effective screening on interesting achiev-able compounds with promising properties, as demonstrated for magnetic cases in this short review. As theHTP methodology has been mostly developed in the last decade, we have still quite a few problems to besolved in order to get calculations done properly for magnetic materials, in order to be predictive. Concomitant

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with such an evolution of applying DFT to tackle materials properties, i.e. with a transformation from under-standing one compound to defining a workflow, more vibrant studies on (magnetic) materials are expected,particularly combined with the emergent machine learning and quantitative multi-scale modeling.

Acknowledgments

We appreciate careful proofreading and constructive suggestions from Manuel Richter, Ingo Opahle and NunoFortunato. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun-dation) Project-ID 405553726 TRR 270. Calculations for this research were conducted on the Lichtenberg highperformance computer of TU Darmstadt.

ORCID iDs

Hongbin Zhang https://orcid.org/0000-0002-1035-8861

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