Heusler 4.0: Tunable Materials · MR47CH12-Felser ARI 12 June 2017 12:3 Heusler 4.0: Tunable...

MR47CH12-Felser ARI 12 June 2017 12:3

Heusler 4.0: Tunable MaterialsLukas Wollmann,1 Ajaya K. Nayak,1,2

Stuart S.P. Parkin,2 and Claudia Felser11Max Planck Institute for Chemical Physics of Solids, Dresden, Germany, D-01187;email: [email protected] Planck Institute of Microstructure Physics, Halle (Saale), Germany, D-06120; email:[email protected]

Annu. Rev. Mater. Res. 2017. 47:247–70

The Annual Review of Materials Research is online atmatsci.annualreviews.org

https://doi.org/10.1146/annurev-matsci-070616-123928

Copyright c© 2017 by Annual Reviews.All rights reserved

Keywords

half-metallic ferromagnetism, half-Heusler compounds, tetragonalHeusler compounds, topological materials, Weyl semimetals, noncollinearmagnetic order

Abstract

Heusler compounds are a large family of binary, ternary, and quaternarycompounds that exhibit a wide range of properties of both fundamentaland potential technological interest. The extensive tunability of the Heuslercompounds through chemical substitutions and structural motifs makes thefamily especially interesting. In this article we highlight recent major devel-opments in the field of Heusler compounds and put these in the historicalcontext. The evolution of the Heusler compounds can be described by fourmajor periods of research. In the latest period, Heusler 4.0 has led to theobservation of a variety of properties derived from topology that includestopological metals with Weyl and Dirac points; a variety of noncollinear spintextures, including the very recent observation of skyrmions at room temper-ature; and giant anomalous Hall effects in antiferromagnetic Heuslers withtriangular magnetic structures. Here we give a comprehensive overview ofthese major achievements and set research into Heusler materials within thecontext of recent emerging trends in condensed matter physics.

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ANNUAL REVIEWS Further

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https://doi.org/10.1146/annurev-matsci-070616-123928https://doi.org/10.1146/annurev-matsci-070616-123928http://annualreviews.org/doi/full/10.1146/annurev-matsci-070616-123928


1. HEUSLER 1.0: TURNING NONMAGNETIC ELEMENTS INTOFERROMAGNETIC MATERIALS

The history of Heusler compounds dates back to 1903, the year of Friedrich Heusler’s seminalcontributions (1, 2) in the Verhandlungen der deutschen physikalischen Gesellschaft, in which heannounced the discovery of a ferromagnetic material at room temperature that is surprisinglyformed from the elements Cu, Mn, and Al, which show no magnetism at room temperature.Later, ferromagnetism was found in other compounds formed from Cu and Mn, but with severalother elements Z = Sb, Bi, Sn. Today, this perhaps does not seem so surprising, especially becauseof the concepts of antiferromagnetism and ferrimagnetism that were introduced by Louis Néel inthe 1930s–1940s (3, 4). However, these phenomena were unknown in 1903, which made Heusler’sdiscovery a major finding. The structure of the compound that Heusler prepared was also unknownin 1903, although Heusler realized that a chemical compound must have been formed. He thusanticipated what is today widely understood and accepted: Heusler compounds form a special classof materials that are located at the border between compounds and alloys and that combine featuresof both, namely the chemical stability of a covalent lattice from which the Heusler compound isconstructed, whereas single sites within the lattice can be substituted by different species andthereby behave as single-site alloys. In a nutshell, covalency and tunability best describe theuniqueness of this material’s class. It was not until 1934 that Otto Heusler, Heusler’s son (5),and Bradley & Rodgers (6) determined the crystal structure of Cu2MnAl. Otto Heusler notedthe possibility of another type of crystalline order that is now termed an inverse/inverted Heuslercompound with the space group T d2 (5) in the notation of Schoenflies or 216 in today’s space groupclassification. The Heusler structure can be described as intertwined cubic and rocksalt lattices oras four interpenetrating fcc sublattices, of which two are formed from the same element.

Shortly after Heusler’s discovery, Nowotny and Juza published results on a different group ofmaterials, all main-group element compounds, namely LiMgAs (Nowotny) and CuMgAs ( Juza),that are now referred to as Nowotny-Juza phases (7, 8). The connection between the Nowotny-Juzaphases and the Heusler compounds was established by Castelliz, who first synthesized NiMnSb (9,10), as part of the compositional series Ni2−xMnSb (0 ≤ x ≤ 1). NiMnSb and the Nowotny-Juzaphases are now described as half-Heusler compounds in which one of the four fcc sublattices ofthe full Heusler is empty. By filling this fourth sublattice, a series of compounds can be formedbetween half- and full Heuslers, which we can describe as XYZ and X2YZ, respectively, whereX , Y are transition metal elements and Z is a main-group element. The full-Heusler compoundshave several variants, including the inverse structure, in which one of the X elements is swappedwith Y, and quaternary Heuslers, in which one of the X elements is replaced by a fourth distinctelement.

Thus, the Heusler name now encompasses a broad and extensive family of compounds. Fur-thermore, all these variants can be subjected to various structural distortions, including tetragonalelongation or compression along one of the cubic crystal axes, as well as distortion along the [111]direction that leads to a hexagonal structure (11). Finally, superstructures are sometimes found,and these can arise from chemical ordering in nonstoichiometric compounds or from modulationof the structure and structural phase transitions, for example, in the case of shape memory alloys.The key differences between these various Heusler compounds are highlighted in Table 1.

The three main prototypical Heusler stoichiometric, chemically ordered structural types,namely half, regular, and inverse, are shown in Figure 1.

Early reports on Heusler phases (12) appear in the context of X2MnZ alloys, where X =Ni, Pd, Au, and Z = Al, Si, Ga, Ge, In, Sn, Sb, with a particular focus on their chemical and mag-netic order, and on an ordering transition that can be induced by composition, temperature, or

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Table 1 Site occupations in cubic Heuslers and Heusler-related structures

Type Space group 4d 4c 4b 4a Example

Fluorite C1 A E E CaHCF2 (133)

Juza C1b E A A ′ LiMgNHC (7)Nowotny C1b E T A MgAgAsHC (134)

Nowotny C1b T A E MgCuHCSb (134)

Half-Heusler C1b T T ′ E MnCuHCSb (9, 10)Full Heusler L21 T T T ′ E Cu2HCMnAl (5)Inverse Heusler Xa A T A E Li2AgSb (135)

A, element of main group 1; E, element of main group 2; T, T ′, transition metal element. Sites 4c and 4d are located in a heterocubic environment (HC)spanned by the nearest neighbors. The chemical order is given in terms of the Strukturbericht designation.

pressure (9, 13, 14). Remarkably, a ferromagnetic-to-antiferromagnetic transition was observed inPd2MnIn1−xSnx (15). Webster and coworkers were the first to explore the magnetic properties ofCo2MnZ compounds, which they stated were quite different from previously studied Heuslers, asthe former compounds incorporate elements other than Mn that carry a substantial local magneticmoment (12). These researchers did not anticipate the great interest of the scientific communityin these materials that has occurred over the past decade.

It is worth mentioning that cousins of the original Heusler type were found in the 1960s–1970sat IBM Research Laboratory (San Jose, CA). Jim Suits explored Rh2-based Heusler compoundsand found a structural transformation at temperatures of approximately 700–800 K, with a verylarge deformation amplitude ε = c tet(c cub −1) of approximately 17%, which is the same size as theMn3+ Jahn-Teller ion–containing spinels (16, 17). Some of these compounds were nonmagneticand some antiferromagnetic, and some showed tetragonally distorted derivatives that have becomeof considerable interest today.

Not until much later did the Co2-based Heusler compounds receive renewed attention inthe context of half-metallic ferromagnetism (HMF). Half-metallic ferromagnets exhibit metallicbehavior in one spin channel and an insulating behavior in the other and are thus of great interestbecause they should intrinsically have fully spin polarized electronic states at the Fermi energy (18).Figure 2 shows the band structure of an archetypal half-metallic ferromagnic Heusler.

a cb

Figure 1The three prototypical Heusler structures, wherein X atoms are represented by red spheres, Y atoms byblue/light blue spheres, and Z atoms by green spheres: (a) half-Heusler, (b) regular Heusler, and(c) inverse-Heusler structures.

www.annualreviews.org • Heusler 4.0 249

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Momentum kDensity of states (E) (eV–1)

Ener

gy E

(k) (

eV)

t2geg

eg

a1 a1

eg

eg

eg

eg

t2g

t2g

t2g

t1ut1u

Momentum kΓ ΓK W L X

–2

0

2

–4

–6

–8

–10

10 1055 0ΓΓ KWLX

a Minority b Co2MnAl c Majority

Figure 2Band structure of Co2MnAl depicting the E(k) dispersion relation for the minority (a) and majority (c) spin channels, together with thedensity of states (b).

The first example of the Heusler HMF was first realized by de Groot et al. (19) for the caseof NiMnSb. Such materials are also of great interest for spintronic applications, for example, asmagnetic electrodes in magnetic tunnel junctions. Magnetic tunnel junctions are composed of twomagnetic electrodes separated by a thin tunnel barrier. The current crossing the tunnel barrierfrom one electrode to the other depends on the magnetic configuration of the electrodes. Whenthe moments of the two electrodes are aligned parallel to one another, the current can flow easily,but when the moments of the two electrodes are aligned exactly antiparallel to one another, thetunneling current will be reduced and, for the case of HMF, should go to zero.

Although semiconducting Heusler compounds are usually not magnetic, magnetism can beintroduced by the introduction of rare-earth (RE) elements, REPtBi or Mn on the 4b posi-tion. Whereas the Mn-containing compounds become metallic, the REYZ compounds remainsemiconducting due to the localization of the 4 f n electrons. The semiconducting behavior ofMnYZ (commonly written as Y MnZ) is conserved only in one spin channel, and therefore thesecompounds have been coined half-metallic ferromagnets, in which one spin channel exhibitssemiconducting or insulating characteristics caused by a minority gap, whereas the majority spinchannel is metallic. NiMnSb is of this type and has 22 valence electrons. This compound hasfour excess electrons, in contrast to the semiconducting Heuslers, which have a valence electroncount of N V = 18. Interestingly, these excess electrons fill one spin channel and thereby form aspin magnetic moment of M spin = 4 μB. In this way these compounds follow almost exactly theSlater-Pauling (SP) rule, whereby M spin = N V − 18. We return to this point below (Section 2.2).The early work on NiMnSb emphasized its magneto-optical properties stimulated by the largemagneto-optical Kerr angle of 1.27◦ found in PtMnSb, one of the other early HMFs (19).

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2. HEUSLER 2.0: HALF-METALLICITY

2.1. Co-Based Half-Metallic Ferromagnets

Ishida et al. (20) first began exploring HMF in Co2-based Heusler compounds, such as Co2MnSn,Co2TiAl, and Co2TiSn, using LDA methods to calculate these compounds’ electronic structures.However, Ishida et al. did not obtain HMF in these compounds (20) because the LDA (21)formalism did not allow a minority gap to open. This was later confirmed in 1983 by Kübleret al. (22), who noted, however, that peculiar transport properties are to be expected for thesenearly HMF-type compounds. In the same year, 1983, Kübler studied the formation and couplingof the magnetic moments in Co2MnSn and similar Co-based alloys and remarked how the alloys’spin magnetic moments follow linear behavior with the number of valence electrons. Kübleret al. quickly recognized the peculiar role of Mn in X2MnZ-type compounds. They reproducedthe ferromagnetic-to-antiferromagnetic transition from Pd2MnIn to Pd2MnSn and elaborated onthe formation of localized magnetic moments in purely itinerant magnets, triggered by the strongexchange splitting of the d states attributed to the Mn site (22). These findings were soon putinto context, and chemical trends were established, of which the backbone is the SP rule. The SPrule was first expressed as a function of the magnetic valence but was soon reformulated in a moregeneral manner. The SP rule originally covered elements and binary alloys (23), but the extensionto ternary alloys led to the wide success of the SP rule as a very useful tool for assessing the expectedmagnetic moment of a ternary HMF. The SP rule describes the interplay of electron filling andthe resulting magnetic moment (see Figure 3). In Co-based full-Heusler alloys, a valence electroncount of 24 leads to a zero net moment, as each spin channel is occupied by 12 valence electrons.Filling or depleting electrons leads to a net integer spin moment M = N V − 24.

Further contributions to the theory of Heusler alloys showed how the SP rule could be describedusing molecular orbital coupling schemes and symmetry analysis. Fecher and colleagues (24)emphasized that an upper bound of approximately 6 μB is unlikely to be exceeded, as the s statesthat were to be populated in the majority spin channel are strongly dispersed, which would lead

fccitinerant

bcclocalized

4

8

12

0

–4

Mag

neti

c m

omen

t m (μ

B/f.u

.)

ScTi

VCrCrCr

Fe NiCu

Co

Mn3Ga

16 20 24 28 32 36 40 44 48

Co

Ni

Fe

Number of valence electrons NV

Mn2Y(3d)Ga

Co2FeSiCo2MnSi

Co2MnAlCo2CrGa

Co2Y(3d)Z

Figure 3Schematic representation of Slater-Pauling behavior for a set of selected Mn2Y Ga and Co2Y Z compounds.f.u. denotes formula unit. Based on References (40) and (41).


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to an energetically unfavorable situation. Furthermore, it could be shown that some compoundsinvolving late transition metals Y = Cu, Zn, e.g., Mn2ZnSi, do not obey the SP rule as M =N V − 24, but another situation occurs in which doubly degenerate d states are shifted below a setof triply degenerate states caused by the symmetry of the crystal structure (Xa, no inversion) suchthat the equation M = N V − 28 guides the emerging magnetic moments (25).

The SP rule is a simple yet powerful tool for the prediction of half-metallic ferromagnets. Inprinciple, one would not expect a simple relationship between a compound’s magnetic moment andthe critical magnetic ordering temperature, which depends sensitively on the electronic structure.In the Heisenberg picture, the ordering temperature TC depends on an atom’s nearest neighbors.In Co-based Heusler compounds, a simple relationship wherein TC ∝ M has been found (26).This is another beautiful example of the fundamental tunability of Heusler compounds that islimited only by their chemical stability. The evolution of the Curie temperature is traced backto two competing factors in terms of the spherical approximation, namely the decreasing averageexchange energy counterbalanced by an increasing total moment with an increasing number ofvalence electrons. It was concluded that Zener’s double/kinetic exchange mechanism provides thekey concept for understanding the observed behavior (26, 27).

The 1980s saw the beginning of a phase of intense research on the Heusler class of com-pounds that lay the foundation for the current perspective of Heusler compounds as multifunc-tional materials, possibly providing answers for many materials science challenges. The use ofhalf-metallic ferromagnets in spin-valve structures should lead to out-of-scale magnetoresistance(MR), from which the field of spintronics received an activity boost. Spin-valve structures basedon NiMnSb did not appear to be promising in the beginning, so mechanisms leading to a possiblesuppression of the expected high TMR values were explored. Relativistic first-principles calcu-lations were used to show that spin-orbit coupling induces a finite but negligibly low density ofstates for 3d-based Heusler compounds, whereas this effect is more pronounced for PdMnSband PtMnSb. After these less successful attempts, the incorporation of Co2-based Heusler alloys,such as Co2Cr0.6Fe0.4Al (28) and Co2MnSi, into spin-valve structures succeeded in delivering theexpected MR values in the range of MR ≈ 2,000%, after many years of materials and inter-face engineering (29). In 2014, spin-polarized photoemission experiments conducted by Jourdanet al. (30), supported by theoretical calculations, provided proof of HMF in Co2-based Heusleralloys in the exemplary compound Co2MnSi (within experimental accuracy).

2.2. Mn-Based Half-Metallic Ferrimagnets

In addition to NiMnSb and Co2MnSi, Mn2VAl has attracted a lot of attention, as it was identifiedearly on as a HMF by numerical methods (31). Nevertheless, experimental and theoretical studieshave focused largely on the Co-based compounds. Renewed interest in the Mn-based compoundswas triggered by the discovery of structurally distorted cousins of the cubic systems: namely thetetragonally distorted Heusler compounds. The most renowned member, Mn3Ga (32), was alreadybeing studied in the 1970s, yet the potential for spintronic applications was not recognized untilthe late 2000s (33, 34).

The first example of a member of the Mn2 family is without doubt Mn2VAl. It was synthesizedby Kopp & Wachtel (35) and Nakamichi & Itoh (36) but was misinterpreted as a ferromagnet[this misinterpretation was later corrected by neutron diffraction measurements (37)] before itwas studied within the scope of half-metallicity (31). Mn-based materials received renewed atten-tion in the late 2000s, when ferrimagnetic Heusler compounds were proposed as free-magneticlayers for spin-valve structures such as, for example, magnetic tunnel junctions, in which a spin-polarized current is used to trigger the switching of the free layer. For a long time, Mn2VAl was the

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only known member of that class until low moment magnets for spintronics/electrodes/memoryreceived renewed attention, which led to the discovery of ferrimagnetic Heuslers with low mag-netization. Further half-metallic ferrimagnetic Mn2VZ-Heusler compounds were predicted byDFT-based computations (38–40). With these compounds it was realized that SP behavior is pos-sible even when the valence electron count is less than 24. Then negative moments are formallycomputed, which reflects the relative magnitudes of the individual sublattice moments, when theorientation of these sublattices is fixed with respect to the crystal lattice. This can lead to a com-pensation point at which the two sublattice moments are identical. Systematically varying thecomposition within a given compositional series is particularly interesting, as a change in sign inthe net magnetic moment may also result in a change in sign of other properties.

The formation of magnetic moments on Mn atoms at site 4b was explained by Kübler et al. (22),and this finding is correct even within the Mn2-Heusler family. Mn atoms at the X (4c /4d ) site havesmall moments comparable to those of Co atoms at the same positions within the Co2-Heuslerclass. A high moment of approximately 3–4 μB at site Y (4b) (40) was observed from the veryearliest of studies on these compounds and was identified as a characteristic feature of the MnZlayer. The Mn2-Heuslers can also form another type of order that is today termed inverted. Thisterm emphasizes the fact that an atomic swap/exchange is observed, in contrast to the prototypicalCu2MnAl type, within which the X atoms evenly occupy site 8c in the inversion symmetric spacegroup 225. Independently of these types of order, the SP rule remains valid, even for the invertedtype.

Additional ferrimagnetic and possibly half-metallic systems have been studied for their mag-netic ground states, although these studies often refer to simple numerical calculations, neglectingother types of magnetic and structural order. Mn3Ga is a material that exhibits features of reg-ular X2YZ and inverse XYXZ ordered Heusler compounds at the same time. At first, the evendistribution of Mn atoms at the 8c position resembles the regular order in X2YZ, whereas theoccupation of Mn atoms of the 4b position is indeed a feature of inverse order in the Mn2-Heuslerclass as, for instance, in Mn2CoAl, where Mn and Co are located at 4d and 4c and Mn and Al at4b and 4a , respectively. In theory, Mn3Ga crystallizes in the centrosymmetric space group 225(prototype BiF3), showing compensated ferrimagnetism (42). That said, various studies have foundthat systems incorporating early transition metals as Y elements prefer the regular order, whereaswhen late transition metals are chosen, the inverse order is preferred (40, 43, 44), where the earlyand late transition metals are separated by Mn. This observation is known as Burch’s rule (43). Inexperimental studies, bulk Mn3Ga stabilizes in a tetragonal lattice (see Section 3.3).

2.3. Compensated Ferrimagnets

Despite the potential for HMFs to attain 100% spin polarization, they typically produce largemagnetic dipole fields that can hinder the performance of spintronic devices that contain them.Therefore, materials that display 100% spin polarization but with very low or, even more inter-esting, zero net magnetic moment are of special interest, both technologically and scientifically.According to the SP (45, 46) rule, which, as discussed above, also describes the HMF Heuslers, thetotal magnetic moment (M) of the L21 cubic Heusler compounds is given by M spin = N V − 24.Thus, Heusler compounds with 24 valence electrons should exhibit a zero net magnetic moment.The search for compensated ferrimagnetic Heuslers with 24 valence electrons has focused on theMn-based Heusler compounds because the Mn atoms sit in an octahedral environment that re-sults in a strongly localized magnetic moment. An example of a 24 valence electron–based Heuslercompound is Mn3Ga, which is predicted to display half-metallicity in the cubic L21 structure. Un-fortunately, cubic Mn3Ga does not exist in the bulk form due to a tetragonal distortion that destroys


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Energy E – εF (eV)

5

a

b

c

0

5

0 100 200 300 400

0.0

–0.1

0.1

0.00

0.02

0.04

0.06

0.08

μ0H (T)

T (K)

μ0H = 0.1 T

M (μ

B/f.u

.)M

(μB/

f.u.)

Den

sity

of s

tate

s n(

E) (e

V–1

)Fm3m

_

d

–4 –2 0 2 4

2 K263 K

Figure 4(a) Crystal structure of the L21-type cubic Heusler compound Mn1.5FeV0.5Al (space group Fm3̄m) and theorientation of the magnetic moments on different sublattices. Different atoms are represented by balls withdifferent colors: Al atoms ( green spheres), V atoms ( gray spheres), MnI atoms (red spheres), MnII atoms (bluespheres), and Fe atoms (beige spheres). (b) The spin-resolved density of states for Mn1.5FeV0.5Al.(c) Temperature dependence of magnetization [M (T )] measured in a field of 0.01 T forMn1.55V0.3Fe1.08Al0.7. (d ) Field dependence of magnetization [M (H )] measured at 2 K and 263 K forMn1.55V0.3Fe1.08Al0.7. Based on Reference 49.

both the compensated magnetic state and the half-metallic behavior. Investigators recently showedthat a compensated magnetic state can be achieved in cubic thin films of Mn2RuxGa (47, 48). Inthis case, the compensated magnetic state was obtained in off-stoichiometric thin films having 21valence electrons.

Stinshoff et al. (49) recently found that the cubic L21 compound Mn1.5FeV0.5Al with 24 valenceelectrons exhibits a fully compensated ferrimagnetic state. As shown in Figure 4a, Mn1.5FeV0.5Alcrystallizes in the regular L21 cubic Heusler structure with space group Fm3̄m. Al atoms occupythe 4a position, and V and MnI atoms equally occupy the 4b position, whereas the 8c positionis occupied equally by MnII and Fe atoms. A calculation of the site-specific magnetic momentyields a larger localized moment of approximately 2.57 μB for the Mn atoms that are sitting inthe octahedrally coordinated 4b position compared with a smaller moment of 1.25 μB for thetetrahedrally coordinated Mn atoms that sit in the 8c position. The V and Fe atoms exhibit muchsmaller moments of only 0.43 μB and 0.23 μB, respectively. In this configuration a total mo-ment of 3 μB for Mn at the 4b site and 2.96 μB for Mn at the 8c site is calculated. A nearly zeronet magnetic moment is expected due to the antiparallel alignment of the moments on the twodifferent sublattices. The calculation of the total density of states indicates the presence of a pseu-dogap in one of the spin directions, demonstrating the half-metallic character of Mn1.5FeV0.5Al(Figure 4). The experimental verification of the compensated magnetic structure was obtainedfrom the M (T ) measurements shown in Figure 4c. The M (T ) measurement performed in a

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Table 2 Some examplary spin-gapless semiconductors exclusively crystallizing in space group 216

NV Materials

21 FeVTiSi CoVScSi FeCrScSi FeMnScAl

26 Mn2CoAl CoFeCrAl CoMnCrSi CoFeVSi FeMnCrSb

28 CoFeMnSi

field of 0.01 T exhibits a Curie temperature (TC) of approximately 340 K. As the temperature isreduced, the magnetization drops to a nearly zero value at 2 K, indicating the presence of a fullycompensated magnetic state near zero temperature. The M (H ) loop measured at 2 K exhibitslinear behavior with nearly zero spontaneous magnetization. This confirms the completely com-pensated ferrimagnetic nature of Mn1.5FeV0.5Al. The M (H ) hysteresis curve measured at 263 Kdisplays a small (approximately 0.1 μB) residual moment that originates from a small compensationarising from the different temperature dependences of the sublattice magnetic moments.

2.4. Spin-Gapless Semiconductors

Within the subset of Mn-based Heusler compounds, a particularly interesting material isMn2CoAl, which is the first example of a spin-gapless semiconductor (SGS) in the Heusler familyof compounds (50). SGSs were originally proposed by Wang (51) in 2008 and are formed when agapless semiconductor is doped with magnetic ions. Examples of several SGSs are listed in Table 2.The fcc-type band structure, which is an inherent feature of Heusler compounds, allows for thispeculiar electronic feature. When there are between 18 and 30 valence electrons, several energywindows in the band structure have only weakly dispersed bands. A simplified molecular orbitaldiagram shows how a sequence of doubly and triply degenerate states are successively filled for18, 21, 24, and 26 valence electrons. Mn2CoAl with N V = 26 naturally fulfills this requirement.Özdoğan et al. (52) have studied a set of quaternary LiMgPdSn-type materials, predicting fourSGSs—CoFeCrAl, CoMnCrSi, CoFeVSi, and FeMnCrSb—while a fifth, FeTiVSi, is almost aSGS, as the Fermi energy touches the edges of the conduction and valence bands. Synthesized andcharacterized by Ouardi et al. (50), Mn2CoAl turned out to show a peculiar electronic structurethat was termed a spin-gapless semiconducting state (50, 51), which depicts the band gap in theminority spin channel, accompanied by an indirect zero band gap in the majority spin channel.The Curie temperature was found to be TC = 720 K, with a magnetic moment of approximately2 μB. The carrier concentration below 300 K is nearly independent of temperature, whereas theSeebeck coefficient is vanishingly small. SGSs are expected to find applications in spintronic de-vices, especially semiconductor spintronics, as the electronic excitations in the gapless state do notrequire a threshold energy but the carriers, whether holes or electrons, remain completely spinpolarized. These materials may thus serve, for example, as spin injectors.

3. HEUSLER 3.0: UNIAXIAL HEUSLER COMPOUNDSAND NONCOLLINEAR SPIN STRUCTURES

3.1. Tetragonal Heusler Compounds for Spin-Transfer Torque

Besides cubic Heusler compounds, which show lots of interesting physics, their structural cousins,the tetragonal Heuslers, are widely studied today. The tetragonal structural modification lowersthe symmetry, and unforeseen effects emerge. Tetragonal Heusler compounds were describedby Suits (16, 17) in the 1970s, although major research began only in the 1990s, when research


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a b c

Figure 5A set of generalized tetragonal Heusler structures (space group 119) are shown depicting the four Wyckoffpositions, which are to be occupied according to Mn (red, dark blue), Y (light blue), and Ga ( green, Z) toachieve inverse-type order. (a) In-plane (KU < 0) and (b) out-of-plane (KU > 0) orientation of themagnetization. (c) Exemplary noncollinear order for competing in-plane and out-of-plane anisotropycontributions.

focused on reversible structural phase transitions from cubic to tetragonal lattices, i.e., from austen-ite to martensite. Shape memory materials rely on this transition, of which Ni2MnGa (53) andMn2NiGa (54) are the flagship materials. The latest boost in tetragonal Heusler compounds re-search began with the realization that the large magnetocrystalline anisotropy of Mn3Ga (32, 33)was of use for spintronic applications. The tetragonal distortion induces a preferred orientationof the magnetization toward the in-plane or out-of-plane directions (Figure 5). Perpendicularmagnetic anisotropy, with the magnetization pointing perpendicular to the film surface, is desiredfor high-density memory and storage devices to reduce the switching current needed to switch themagnetic state of the memory or storage layer and to guarantee thermal stability. Magnetic mem-ory bits can be switched via either magnetic field or spin-polarized currents through the concept ofspin-transfer torque (STT). STT refers to a torque that is exerted perpendicular to the magnetiza-tion, leading to a precessing magnetic moment that finally switches to the opposite direction. TheSlonczewski-Berger equation (55, 56) describes the dependence of the switching current densityon materials properties such as the magnetic moment M, anisotropy constant KU, and Gilbertdamping parameter α. As the STT technology requires materials with small switching currentswhile guaranteeing data retention/thermal stability, the Mn-based Heusler compounds have beenexplored in searches for new tetragonal phases. That said, Alijani et al. (57) have studied theMn-Co-Ga system (58), testing the stability of the tetragonal Mn3Ga phase with the substitutionof Mn with Co: Mn3−xCoxGa. A tetragonal phase is maintained until a critical Co concentrationof xCo = 0.5. The subsequent members with higher Co concentration exhibited the inverted cu-bic structure with mixed Mn-Co sites. In agreement with a subsequent theoretical treatment, themagnetic moments of the tetragonal phases decrease with increasing Co content. Surprisingly,an approximately linear relationship between the number of valence electrons and the magneticmoment was found for these and other tetragonal Heusler phases (41), such as Mn2CrGa, Mn3Ga,Mn2FeGa, and Mn2CoGa. These phases were modeled by density functional theory, and a localenergy minimum in the energy landscape could be observed. Nevertheless, the tetragonal phasedoes not show half-metallic behavior, which renders this observation particularly surprising. Butthis theoretical study provided evidence for the correlation of the tetragonal distortion with thepeak and valley structure of the density of states. For the case in which the Fermi edge residesat a peak in the density of states, a distorted structure is likely to occur (Mn-Ga, Mn-Ni-Ga,

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Mn-Fe-Ga), as a slight distortion either breaks the band’s degeneracy or changes the band disper-sion such that the electronic susceptibility is diminished. Mn2CoGa is found to be cubic, as theFermi edge is located at a local minimum in the density of states of both spin channels. Studying thetransport properties of the Mn-Co-Ga series revealed another effect that is termed spin-selectiveelectron localization (59, 60). Spin-selective electron localization describes the change in conduc-tivity polarization for transition metal–substituted Mn3Ga systems. The exchange of Mn with atransition metal (Y = Sc, Ti, V, Cr, Fe, Co, Ni) leads to an increase in spin polarization becauseonly one spin channel is perturbed by the substitution.

3.2. Tetragonal Compensated Ferrimagnetic Heusler Compounds

Tetragonal magnetic materials with out-of-plane magnetic anisotropy are of great interest forSTT and permanent magnet–related applications. It is well known that the Mn-based tetragonalHeuslers are ferrimagnetic in nature, with at least two magnetic sublattices, where the magneticmoments align antiparallel to each other (33, 34, 40, 41, 61–66). In addition, tetragonal magneticHeuslers can exhibit high magnetic ordering temperatures well above room temperature, whichis a necessary condition for any practical application. The best example of a tetragonal magneticHeusler is Mn3Ga, which shows a ferrimagnetic ordering with TC of approximately 750 K (33).Despite these ideal characteristics, ferro/ferrimagnets exhibit large unwanted stray fields that affectthe magnetic state of neighboring layers in multilayer spintronic devices or neighboring devicesin arrays of devices. We have proposed that a fully compensated ferrimagnetic state with a zeronet magnetic moment can be achieved by systematically tuning the sublattice magnetic momentsin Mn3Ga (67). From theoretical calculations, we have shown that the compensated magneticstate can be achieved for a wide range of Heusler materials by substituting a late transition metalelement, such as Ni, Cu, Rh, Pd, Ag, Ir, Pt, and Au (68), for Mn in Mn3−xYxGa.

Figure 6a shows a specific example of magnetic compensation when Y = Pt. Mn3Gaconsists of two inequivalent types of Mn atoms, one within the Mn-Ga planes and one

a b

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0.8

0.4

–0.4

–0.8

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0.00.0 0.2 0.4 0.6 0.8 1.0

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2

4

M (μ

B/f.u

.)

M (μ

B/f.u

.)μ0 HEB (T)

μ0H (T)x–4

Mn3–xPtxGa

Mn3–xPtxGa

Mn3Ga Mn2PtGa

x = 0.20x = 0.50x = 0.60

x = 0.70x = 0.65

T = 300 K

–2 0 2 4

Figure 6(a) Theoretical calculation of magnetic moment (solid squares, left axis) as a function of Pt content inMn3−xPtxGa. The line connecting the solid squares is a guide to the eye. Exchange bias field (H EB) as afunction of Pt concentration xPt (right axis). Solid circles show the experimental data, whereas the solid curvecorresponds to a model calculation. (b) Curves for Mn3−xPtxGa thin films measured at 300 K.


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within the Mn-Mn planes of the tetragonal structure with space group I4/mmm. TheMn sitting in the Mn-Ga planes possesses a larger localized moment of approximately 3.1 μB;in contrast, the Mn atoms sitting in the Mn-Mn planes have moments of 2.1 μB. Thus, a largertotal moment of 4.2 μB/f .u. (formula unit) is expected from the Mn atoms sitting in the Mn-Mnplane, as there are two Mn/f.u. in this plane. One way of reducing the magnetic moment of theMn atoms sitting in the Mn-Mn planes to match the moment of the single Mn in the Mn-Gaplanes is by partially substituting Mn in the Mn-Mn planes with a nonmagnetic element. We haveshown that, for a Pt concentration of approximately x = 0.6 in Mn3−xPtxGa, the Mn atoms in theMn-Mn(Pt) planes and the Mn atom in the Mn-Ga planes contribute nearly the same momentwith opposite alignment. As demonstrated in Figure 6a, with increasing Pt concentration, thetotal magnetic moment decreases and becomes zero at x = 0.6 before further increasing for higherPt concentrations. The theoretical concept is nicely verified by growing thin films with differentPt concentrations. As shown in Figure 6b, M (H ) loops measured at 300 K for different thin filmsexhibit the typical out-of-plane hysteretic behavior found in a tetragonal system. The sample witha Pt concentration x = 0.2 exhibits a magnetic moment of 0.7 μB/f .u. at 300 K. The magnetizationstarts decreasing with increasing Pt content and becomes nearly zero for x = 0.65, demonstratinga fully compensated ferrimagnetic state for Mn2.35Pt0.65Ga. As expected, the magnetization startsincreasing again with further increases in the Pt concentration. Most importantly, these thin filmsshow magnetic ordering temperatures well above room temperature. The feasibility of the practi-cal application of these tetragonally compensated ferrimagnets has been demonstrated by showingthe existence of a large exchange bias in both bulk and bilayer thin-film materials (67–69).

3.3. Noncollinear Magnetic Structure

There has been much recent interest in magnetic materials exhibiting noncollinear spin struc-tures. One of the most exciting uses of noncollinear spin structure is the motion of chiral domainwalls using spin-polarized currents that generate large chiral spin-orbit torques to drive the do-main walls (70, 71). This current-driven back-and-forth motion of the domain walls inside amagnetic nanowire forms a novel high-density, high-performance, solid-state storage memorydevice—Racetrack Memory—that was first proposed by Parkin et al. (72) in 2002 [and patented in2004 (73)] and that has the potential to even replace conventional magnetic data storage. The do-main wall can be replaced by other noncollinear spin textures such as skyrmions (74). In this regard,Heusler materials are perfect candidates for modification of the magnetic state via competing ex-change interactions between different sublattices (67). In addition, most of the Mn-based Heuslermaterials exhibit a noncentrosymmetric crystal structure, which is necessary for the realization ofthe Dzyaloshinskii-Moriya interaction that leads to the formation of skyrmions (75).

Direct evidence of a noncollinear magnetic state was found in the noncentrosymmetric tetrag-onal Heusler compound Mn2RhSn (75). As shown in Figure 7a, Mn2RhSn exhibits a TC ofapproximately 275 K, with a transition to a state with a comparatively higher magnetic momentat approximately 80 K. The magnetic ordering at low temperature has been assigned to a spin-reorientation transition. The M (H ) loop measured at 2 K shows a saturation magnetization ofapproximately 2 μB (inset of Figure 7), which cannot be accounted for by considering a collinearspin arrangement. In Mn2RhSn, Mn atoms occupy two distinct sublattices. The Mn sitting in theMn-Sn planes (MnI) shows a moment of nearly 3.6 μB, whereas the Mn in the Mn-Rh planes(MnII) exhibits a moment of approximately 3 μB. A simple antiparallel alignment between themoments of MnI and MnII will give a net moment of 0.6 μB. However, this simplified model doesnot match the experimentally determined magnetic moment of 2 μB. Similarly, a ferromagneticordering will result in a total moment of 6.6 μB, which also does not fit to the observed moment.

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0 0.0 0.2 0.4 0.650 100 150 200 250 300 350

0 2–2–4 4–2

–1

0

1

2

0

–2

–4

–6

b

T (K)

a

T = 2 K

20 K

50 K

10 K

120 K

30 K

80 K

100 K

2 K

1.2

1.0

0.8

0.6

0.4

0.2

0.0μ0H (T)

μ0H (T)

M (μ

B/f.u

.)

M (μ

B/f.u

.)

ρ xy

(nΩ

cm

)TH

E

μ0H = 0.1 T

Figure 7(a) M (T ) curve measured in a field of 0.1 T for the Heusler tetragonal compound Mn2RhSn. The insetshows an M (H ) loop measured at 2 K. (b) Topological Hall effect (ρTHExy ) at different temperatures for aMn2RhSn thin film. Based on Reference 78.

However, the experimental magnetic moment can be nicely explained by considering a non-collinear magnetic structure in Mn2RhSn. The theoretical calculations show that the competitionbetween the antiferromagnetic interaction between the Mn moments in nearest and next-nearestplanes can give rise to a canting of the MnII moment of approximately 55◦, thereby providing anadditional z component of the moment to the simple ferrimagnetic configuration. The theoreticalresults are supported by evidence of a noncollinear spin configuration from neutron diffractionmeasurements (75).

Theoretical calculations also show that the present class of materials with acentric crystalstructures should give rise to the formation of skyrmions under appropriate conditions. Recently,it was shown that thin films of Mn2RhSn exhibit a considerable topological Hall effect (THE),as shown in Figure 7b. The THE was obtained in temperatures of up to 100 K, which is thespin-reorientation transition in thin-film samples. Materials that have skyrmions also exhibit alarge THE due to their chiral noncollinear spin structures (76, 77). Hence, one can assume thatsome type of noncollinear spin structure such as skyrmions is present in Mn2RhSn (78).

4. HEUSLER 4.0: TOPOLOGICAL HEUSLER COMPOUNDS

The prediction of the quantum spin Hall (QSH) state (79) triggered tremendous interest in thecondensed matter community. Not only did this prediction of a novel state of matter lay the foun-dation for a completely new field of research, but the use and application of topological conceptsin a wide range of condensed matter research gained much attention. The QSH state was subse-quently realized in a HgTe/CdTe quantum well structure (80): Materials that crystallize in thesame space group (F 4̄3m) as the XZ binary semiconductors (HgTe, CdTe, ZnS) include diamondand half-Heusler compounds. Half-Heusler semiconductors with 8 or 18 valence electrons (VEs)exist with a wide range of band gaps (81, 82), and the possibility of finding half-Heuslers witha topologically nontrivial state was quickly realized (83, 84). A set of half-Heuslers that couldbe classified into topologically trivial materials and potentially nontrivial materials was proposed.Nontrivial HgTe exhibits a band inversion of the �6 and �8 states at the � point. Chadov et al. (83)


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k (2π/a) k (2π/a)

Γ7 Γ7

Γ8

Γ8Γ6

Γ6

εF

a b

Figure 8Schematic of a trivial and inverted band structure as found in (a) CdTe and (b) HgTe. Based on References79 and 80.

showed that zero-gap semiconductors within the half-Heusler family exhibit similar features (seeFigure 8), and an odd number of band inversions was observed in some systems. The flexibil-ity of the Heusler compounds with respect to site occupation allows for tuning these materialsfrom trivial to nontrivial states by means of lattice parameter adjustment through substitutionof isoelectronic elements or adequate hybridization strength. Because the band inversion resultsfrom spin-orbit coupling, another way is by strengthening of the average spin-orbit interactionvia, for example, substitution by heavy transition metals. Some of the zero-gap semiconductorsthat have been experimentally realized show a surprising range of multifunctional properties inaddition to the nontrivial electronic structure when RE elements are introduced, as, for example,GdPtBi–antiferromagnetism (85), LaPtBi–superconductivity (T crit. < 0.9 K) (86), YPtBi–heavyfermion behavior and superconductivity (T crit. < 0.9 K) (87, 88), and ErPtBi–antiferromagnetism(TN < 1.2 K) (85). The zero-gap state, which is a prerequisite for topological insulators, is alsoa useful feature for thermoelectric materials. Consequently, a connection between topologicalinsulators, the zero-gap state, and thermoelectric performance is profound (89).

4.1. GdPtBi: A Weyl Semimetal in a Magnetic Field

Weyl semimetals are a class of topological semimetals, beyond topological insulators, in whichthe conduction and valence bands cross in the vicinity of the Fermi energy (90–102). The crossingpoints, termed Weyl points, are separated in momentum space. These Weyl points, which act asmagnetic monopoles in momentum space, always appear in pairs and are connected by an unusualsurface state termed the Fermi arc (99–101, 103–105). In general, semimetals exhibit ultrahighcarrier mobilities and large transverse MR values, which are also promising effects for spintronicapplications (106–108). However, the nontrivial electronic structure of a Weyl semimetal can giverise to additional unusual phenomena such as negative MR, which is connected to chiral magneticeffects (92, 98). The chiral anomaly comes from the charge pumping between two Weyl pointsconnected through the Fermi arc. In most three-dimensional Weyl semimetals, in which thebreaking of either time reversal symmetry or inversion symmetry occurs, the Weyl points resultfrom accidental touching/crossing of the conduction and valence bands. In recent work, however,experimental evidence of an unusual topological surface state (109) and a chiral anomaly (110) wasfound in lanthanide half-Heusler semimetals.

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40 0.25

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0 10 20 30

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a bM

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Hal

l ang

le

0 15

15 K40 K 100 K 60 K

50 K

30 K

2 K

20 K

30 0 2 4 6 8

2.5 K

I⊥BI⊥B

I,B

B

[111]

[111]

[110]

[110]

I

[111][110]

MR

T = 2.5 K

B

μ0H (T) μ0H (T)

μ0H (T)

Figure 9(a) Field dependence of longitudinal magnetoresistance (MR) measured at different temperatures for fieldsof up to 33 T in GdPtBi for the crystal directions shown. The inset shows the transverse MR measured at2.5 K. (b) Anomalous Hall angle at different temperatures for GdPtBi. Based on Reference 114.

The existence of the negative MR associated with a chiral anomaly in these lanthanide half-Heuslers was attributed to the formation of Weyl points due to external field–induced Zeemansplitting. However, it is presumed that the Zeeman splitting is negligible in comparison to themuch larger exchange field coming from the 4f electrons in REPtBi compounds (RE = Gd, Nd).Both GdPtBi and NdPtBi are antiferromagnetic below their Néel temperatures of 9.0 K and 2.1 K,respectively (111–113). In the absence of an external field, the exchange fields originating fromthe magnetic moments at different sublattices cancel, and hence no Weyl points are observed.With the application of a modest external magnetic field, the Gd moments in different sublatticesalign parallel to each other, resulting in ferromagnetic ordering. Thus, exchange splitting of theconduction bands can give rise to the formation of Weyl points.

Figure 9a shows the existence of large, unsaturated, longitudinal negative MR in GdPtBi. TheMR measurements were performed in fields of up to 33 T with I parallel to B with the configurationshown in the figure. A large negative MR of 68% is observed at 2.5 K. The negative MR persistsup to temperatures greater than 100 K, which is much greater than the TN of GdPtBi (114).However, the field required to give a negative MR increases with increasing temperature due tothermal effects. As shown in the inset of Figure 9a, a large transverse positive MR is observedwhen the magnetic field is applied perpendicular to the current. These experiments confirm thatthe negative MR in GdPtBi originates from the formation of Weyl points due to the field-drivensplitting of the conduction band. Another proof of the chiral anomaly is the finding of a largeanomalous Hall effect (AHE) in GdPtBi. A large anomalous Hall angle (AHA) of approximately0.23 is calculated in a field of 2.2 T at 2 K. The AHA decreases with increasing temperature, andthe field where a maximum AHE is observed shifts to higher magnetic fields. This indicates thatthe AHE is closely related to the presence of negative MR originating from the chiral anomaly.

4.2. Co2TiSn: A Magnetic, Centrosymmetric Weyl Semimetal

Crossing points in the electronic band structure, such as Weyl and Dirac points, are not rare, butvery often these points are not at the Fermi energy, or many other bands also cross the Fermi


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energy. Weyl and Dirac points have a strong influence on the transport properties and other prop-erties if they are very close to the Fermi energy in a semimetal. The review by Nagaosa et al. (115)established that these crossing points in the electronic structure lead to an enhanced Berry phaseand, therefore, to a large AHE in magnetic systems and to a spin Hall effect in paramagneticmetals or diamagnetic topological semimetals. It is reported that these points act as a magneticmonopole for the Berry curvature in momentum space (116). A large AHE in connection with alarge Berry phase in Heusler compounds was predicted by Kübler & Felser (117) for Co2MnAl.The predicted value of the giant AHE was in excellent agreement with the experimentally deter-mined value (118). Husmann et al. (119) investigated Co2CrAl and found that the intrinsic AHEdominates. An intrinsic contribution comes from the Berry phase, whereas extrinsic contributionsoriginate from scattering from impurities through skew scattering and side jump contributions.Significantly, Kübler & Felser had already noted a Dirac point below the Fermi energy in Co2VSn,but a direct connection between the giant anomalous Hall conductivity and the Dirac point was notmade (117). In the context of recent searches for Weyl and Dirac points in the electronic structureof semimetals, Wang et al. (120) and Chang et al. (121) proposed that electron-doped Co2TiSnshould be a magnetic Weyl semimetal. On the basis of the work on AHE, Kübler & Felser recog-nized that Co2MnAl is a Weyl metal even when undoped. There seems to be a relation betweenthe Weyl semimetals and spin-gapless semimetals. SGSs appear only in the noncentrosymmetricspace group 216 (F 4̄3m), whereas for the same number of valence electrons, the correspondingHeusler compounds in space group 225 (Fm3̄m) are Weyl semimetals. The minority band struc-tures for the respective materials are identical in both space groups, but the bands are not allowedto cross in the space group 225 and result in a forbidden crossing, and therefore a semiconductingmajority spin channel is obtained instead of a Weyl semimetal (see Figure 10). This conceptcan help to identify Weyl semiconductors and noncentrosymmetric semiconductors via simpleelectron-counting rules.

The existence of a noncollinear magnetic structure in tetragonal Heusler materials is discussedin Section 3.3 (75, 78). Many of these Mn-based Heusler materials also exhibit a stable hexag-onal crystal structure (122–125). By varying the preparation conditions, cubic, tetragonal, andhexagonal phase can be stabilized in one system (126). Most hexagonal materials display anti-ferromagnetic ordering (122). Neutron diffraction studies of the hexagonal Mn3Sn and Mn3Gecompounds reveal the presence of noncollinear antiferromagnetic ordering (124). In particular,hexagonal Mn3Ge, which consists of two layers of Mn triangles stacked along the c axis, shows a120◦-triangular antiferromagnetic structure (124). As shown in Figure 11a, the Mn atoms form aKagomé lattice with Ge sitting at the center of a hexagon. The noncollinear spin structure arisesdue from geometrical frustration of the Mn spins arranged in the triangular spin structure. Recenttheoretical works have demonstrated that materials with noncollinear antiferromagnetic structuresand with some special symmetries should exhibit a large AHE (127, 128). The AHE is an intrinsicproperty of all ferromagnets and roughly scales with magnetization (115). Therefore, antiferro-magnets with a zero net magnetic moment should not, in general, display an AHE. Because theintrinsic AHE is a manifestation of the Berry curvature, we have calculated the Berry curvature inMn3Ge by considering the noncollinear antiferromagnetic spin structure shown in Figure 11a.The distribution of Berry curvature in the momentum space is depicted in Figure 11b. It can beclearly seen that a nonvanishing Berry curvature is obtained only in the kx-kz plane, and almostzero amplitudes of the Berry curvature are observed in the other two planes (129). Our theoreticalpredictions were supported by measuring the experimental AHE in Mn3Ge, as shown in panels cand d of Figure 11. A large anomalous Hall resistivity (ρH) of 5.1 μ� cm was found in the x-z plane(ρ yxz), whereas a small ρH of approximately 5.1 μ� was measured in the x-y plane (ρ

zxy ). Similarly,

an extremely large AHC of approximately 500 �cm−1 and almost zero AHC are calculated in the

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Co2T

iSn

Minority

a

b

Majority

eg

eg

egt2g

t2g

t1u

eg

εF

εF

eg

egt2g

t2g

t1u

e

e

e

e

et2

Mn 2

CoA

l

Density of states E (eV–1)Momentum k Momentum kΓ ΓK W L X

Energy E(k) (eV)

Energy E(k) (eV)

–2

0

2

–4

–2

0

2

55 0ΓΓ KWLX

t2

t2

t2

t2

t2

– 4

Figure 10Spin-resolved band structure and density of states of (a) the magnetic Weyl semimetal Co2TiSn and (b) thespin-gapless semiconductor Mn2CoAl, both with 26 valence electrons.

x-z (σ yxz) and x-y (σzxy ) planes, respectively (129). A similarly large AHE has also been found in

hexagonal noncollinear antiferromagnetic Mn3Sn (130). The ferromagnetic Heusler compoundsCo2MnSi and Co2MnGe with a net magnetic moment of approximately 5 μB display a maximumρH of 4 μ� (131). Thus, the finding of such a large AHE in antiferromagnetic Mn3Ge and Mn3Sncan be further explored for their possible use in antiferromagnetic spintronics (132).

5. SUMMARY

The Heusler compounds have a rich history dating back more than a century, and yet fascinat-ing new properties continue to emerge even today. In this review we elaborate on the historyof Heuslers in four distinct development phases. In Heusler 1.0 the discovery of combinationsof essentially nonmagnetic elements that form ferromagnetic compounds well above room tem-perature was a remarkable finding. In Heusler 2.0 the theoretical prediction and experimentalfinding of half-metallicity in certain classes of Heusler materials was another major discovery thatwould lead several decades later to the finding of very large values of tunneling MR. In Heusler3.0 an entirely new world of Heusler compounds was revealed by the application of newly devel-oped notions of topology that led to the prediction and later experimental proof of topological


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b

a c

d

b

y

x

a

kz

kx

ky

Ge

ρxzy

σxzy

σxzy σxyz

σxyz

ρxyzρ H (μ

Ω c

m)

σ H (Ω

–1 c

m–1

)

μ0H (T)

Mn

6

4

2

0

–2

–4

–6

600

400

200

0

0

High

–200

–400

–4 –2 0 2 4

μ0H (T)–4 –2 0 2 4

–600

AHC (σij )k(a.u.)

Figure 11(a) Triangular noncollinear antiferromagnetic configuration for hexagonal Mn3Ge. (b) Berry curvatureintegrated along the kx , ky , and kz axes, plotted in the ky kz, kxkz, and kxky planes, respectively. For the spinconfiguration shown in panel a, only a nonzero anomalous Hall conductivity (σ yxz) is obtained in the kx-kzplane. (c) Hall resistivity (ρH) as a function of magnetic field (H ) measured at 2 K. (d ) Field dependence ofHall conductivity (σH) obtained from the corresponding Hall resistivities shown in panel c. Based onReference 129.

insulators and, more recently, Weyl semimetallic Heuslers. In Heusler 4.0 more complex mag-netic structures in which the magnetic moments are aligned noncollinearly have been discoveredin a range of Heusler compounds. What is perhaps even more remarkable is that many of theseproperties evolve from simple concepts of electron counting. By changing the number of valenceelectrons, the magnetization and Curie temperature of magnetic Heuslers can be systematicallyvaried, or the Weyl points in their band structure can be tuned though the Fermi energy. In thisreview there was no space to describe other properties of Heuslers that include, for example, giantthermoelectricity and magnetocaloric applications. Heusler compounds are promising for a widerange of applications that have been focused to date on spintronics. The future for Heuslers seemsvery bright.

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings thatmight be perceived as affecting the objectivity of this review.

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ACKNOWLEDGMENTS

Financial support from a European Research Council Advanced Grant (ERC-AG; number291472, IDEA Heusler!) is gratefully acknowledged. The authors thank Gerhard Fecher for pro-viding us with Figures 2 and 10.

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