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Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism Hua Chen, Qian Niu, and A. H. MacDonald Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA (Received 3 October 2013; published 10 January 2014) As established in the very early work of Edwin Hall, ferromagnetic conductors have an anomalous Hall conductivity contribution that cannot be attributed to Lorentz forces and therefore survives in the absence of a magnetic field. These anomalous Hall conductivities are normally assumed to be proportional to magnetization. We use symmetry arguments and first-principles electronic structure calculations to counter this assumption and to predict that Mn3 Ir, a high-temperature antiferromagnet that is commonly employed in spin-valve devices, has a large anomalous Hall conductivity. DOI: 10.1103/PhysRevLett.112.017205
PACS numbers: 75.47.Np, 73.22.Gk, 75.50.Ee
In the presence of an external magnetic field, the Hall effect (current flow perpendicular to electric field) is present in all conductors. In ferromagnetic metals like Fe, Co, and Ni, however, the Hall effect is anomalous and controlled more by magnetization than by Lorentz forces [1,2]. Although the anomalous Hall effect is experimentally strong, it has stood alone among metallic transport effects for much of the last century because it lacked a usefully predictive, generally accepted theory. Progress over the past decade has explained why. It is now clear that the anomalous Hall effect in ferromagnets has contributions from both extrinsic scattering mechanisms similar to those that determine most transport coefficients, and from an intrinsic mechanism that is independent of scattering [2]. The intrinsic mechanism, first proposed by Karplus and Luttinger [3], has recently been reformulated in the language of Berry phases as a momentum-space geometrical effect [4] related to the quantum Hall effect and to topological insulators. In the quantized anomalous Hall effect [5] only the intrinsic component survives. The anomalous Hall effect is also observed in paramagnets, which have nonzero magnetization induced by an external magnetic field. Separately, studies of the anomalous Hall effect in frustrated and noncollinear magnetic systems have also been very active [2,6–13]. In these systems magnetic frustration may be removed by magnetocrystalline anisotropy or Dzyloshinskii-Moriya interactions, resulting in stable noncollinear magnetic order. For noncoplanar spin textures, the anomalous Hall effect in the strong local exchange coupling limit can then often be explained as a real-space Berry phase effect [2,6,9]. Although no explicit relationship has been established, the anomalous Hall effect in a particular material is usually assumed to be proportional to its magnetization. In this Letter, we point out that it is possible to have an anomalous Hall effect in a noncollinear antiferromagnet with zero net magnetization provided that certain common symmetries are absent, and predict that Mn3 Ir, a technologically important antiferromagnetic material with noncollinear order that 0031-9007=14=112(1)=017205(5)
survives to very high temperatures, has a surprisingly large anomalous Hall effect comparable in size to those of the elemental transition metal ferromagnets. Mn3 Ir can be viewed as an fcc crystal with Mn atoms on three of the four cubic sublattices [Fig. 1(a)], and is widely used as an exchange-bias material in spin-valve devices. Its Mn sublattice can be viewed as consisting of twodimensional kagome lattices [Fig. 1(b)] stacked along the (111) direction. Although isolated two-dimensional kagome lattices order weakly because of strong magnetic frustration [14–18], the Mn moments in Mn3 Ir have strong three-sublattice triangular (T1) magnetic order [Fig. 1(a)] [19,20] and a high magnetic transition temperature ∼950 K. The surprising stability is provided by interkagome coupling [21] combined with strong magnetic anisotropy [22,23]. Because the kagome structure, with appropriate exchange interactions, can support relatively simple noncollinear antiferromagnetic states similar to those of Mn3 Ir, we begin our discussion of the anomalous Hall effect in noncollinear antiferromagnets by focusing on a simple two-dimensional kagome model. The itinerant electron Hamiltonian for an s-d model on the kagome lattice is
FIG. 1 (color online). Structure of Mn3 Ir. (a) Unit cell of Mn3 Ir with triangular antiferromagnetic order. (b) An individual (111) plane of Mn3 Ir. The Mn atoms form a kagome lattice.
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PHYSICAL REVIEW LETTERS 10 JANUARY 2014 PRL 112, 017205 (2014) X X systems and is directly related to the anomalous Hall effect H¼t c†iαγ cjβγ − J c†iαγ ciατ σ γτ · Δα : (1) [4,25]. The momentum space Berry curvature of a Bloch iαγτ hiα;jβiγ band is defined as Ωn ðkÞ ¼ ih∇k unk j × j∇k unk i, where junk i is the Bloch eigenstate for band n at momentum k. Here α and β label kagome sublattices, the angle-brackets The intrinsic anomalous Hall effect is proportional to the restrict s-electron hopping to nearest neighbor sites, and Δα Brillouin-zone integration of the Berry curvature [2,25]. is an exchange field due to coupling to quenched d-electron If a system has both time reversal and spatial inversion local moments that have coplanar triangular magnetic order I symmetry, the Berry curvature satisfies Ωn ðkÞ ¼ [Fig. 1(b)]. The band structure of this model is shown in −Ωn ð−kÞ and Ωn ðkÞ ¼ Ωn ð−kÞ, implying that it vanishes Fig. 2(a). One interesting feature is that, in contrast to identically. In the present case, TM symmetry implies the case of simple collinear bipartite antiferromagnets that Ωzn ðkÞ ¼ −Ωzn ð−kÞ, and I symmetry that Ωzn ðkÞ ¼ [24], the bands are not doubly degenerate. In the latter case, Ωzn ð−kÞ, so the Berry curvature still vanishes identically. the time-reversal operation T in combination with a spatial An explicit evaluation of Ωzn ðkÞ for the model of Eq. (1) translation by the vector connecting the two sublattices T d confirms this conclusion, except at points in the is a good symmetry, supplying an analog of Kramers theoBrillouin zone where degeneracies occur and Ωzn ðkÞ is rem even though time-reversal symmetry on its own is bronot well defined [Fig. 2(a)]. The presence of the bandken. When combined with spatial inversion symmetry I, crossing (Dirac) points in [Fig. 2(a)], however, suggests this property implies double band degeneracy at every point that the system has the potential to become topologically in momentum space [24]. In contrast, in the three-sublattice nontrivial when TM symmetry is broken. kagome model there is no operation O that can make T O a We can conclude from the discussion above that the good symmetry while fulfilling the prerequisites for Kramers Berry curvature Ωzn ðkÞ will be an even nonzero function theorem, i.e., that is antiunitary and ðT OÞ2 ¼ −1. Mirror of k if we break TM symmetry, and that the intrinsic reflection M with respect to the kagome plane is a good anomalous Hall conductivity will therefore be nonzero. symmetry of the kagome lattice, but flips all in-plane spin A similar symmetry argument applies to the extrinsic components. M symmetry is therefore broken by the term anomalous Hall effect as well. One way to break TM symdescribing the interaction with local moments in Eq. (1). metry is to add an external magnetic field B perpendicular Since time reversal T reverses the in-plane spins a second to the plane, which breaks T symmetry but, since σ is a time, the Hamiltonian in Eq. (1) does have TM symmetry. pseudovector, not M symmetry. The influence of a magHowever, this symmetry does not imply double band degen2 netic field can therefore be captured by adding a Zeeman eracy because ðTMÞ ¼ 1 rather than −1. term to Eq. (1) (We ignore any change in the in-plane The fact that TM is a good symmetry of our model does components of Δα ), have other important consequences as we now show. X † Momentum space Berry curvature can be used to characterH ¼ −J ciαγ ciατ σ zγτ : (2) B z ize nontrivial topological properties of condensed matter iαγτ
FIG. 2 (color online). Noncollinear antiferromagnet on a kagome lattice. (a) Band structure of the model in Eq. (1), J ¼ 1.7t. (b) Vectors for the spin-orbit coupling term in Eq. (3). (c) Band structure with spin-orbit coupling, tSO ¼ 0.2t. (d) Berry curvature of the lowest band in (c).
Straightforward calculations show that the Berry curvature and the anomalous Hall effect indeed becomes nonzero due to this perturbation (Fig. S1 [26]). In contrast to the case of ferromagnetic metals, this anomalous Hall effect mechanism does not require spin-orbit coupling, and is in this sense reminiscent of the real-space spin chirality topological Hall effect scenario [2,6,9]. However, it is TM symmetry breaking by the Zeeman term Eq. (2) rather than the noncoplanar configuration of the background moments that is responsible for the anomalous Hall effect. We also note that a nonzero total magnetization is also induced by the Zeeman term. We now show that broken mirror symmetry combined with spin-orbit coupling leads to an anomalous Hall effect, but not necessarily to a net magnetization. Spin-orbit coupling is T invariant, but can break M symmetry and therefore the combined TM symmetry on a kagome lattice. For example, consider the following spin-orbit coupling term, X HSO ¼ itSO ναβ σ γτ · nαβ c†iαγ cjβτ ; (3)
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where tSO is related to the atomic spin-orbit coupling, pffiffiffiffiffiffi i ¼ −1, ναβ is antisymmetric with respect to its sublattice indices and ν12 ¼ ν23 ¼ ν31 ¼ 1, and nαβ are a set of coplanar vectors shown in Fig. 2(b). The spin-orbit coupling term identified in Eq. (3) is analogous to the corresponding term in the graphene model of [27], and preserves the inversion symmetry of the kagome lattice, but breaks the mirror symmetry. It accounts for the difference between the lefthand and right-hand environments of an electron hopping between nearest neighbors on a kagome lattice. When HSO is added to the model of Eq. (1), gaps open at the Dirac points [Fig. 2(c)], as in the external magnetic field case (Fig. S1). Figure 2(d) shows that the Berry curvature is again nonzero throughout momentum space, and that it is large near the gapped Dirac points. This calculation proves by explicit example that antiferromagnets can have nonzero anomalous Hall conductivities simply because they break time reversal symmetry. This property has largely [12,28–30] escaped notice because most antiferromagnets have other symmetries that force their Hall conductivities to vanish [7], with the collinear antiferromagnets on bipartite lattices being the simplest example. In general, by recognizing Berry curvature as a (pseudo)vector defined on the Brillouin zone, the existence of anomalous Hall effect in broken-time-reversal systems can be established by analyzing the transformation properties of such a (pseudo)vector under all the symmetry operations of the specific system. What are the implications of these considerations for Mn3 Ir, which has inversion symmetry, but lacks (111) plane mirror symmetry because of the fcc stacking of its kagome layers [Fig. 1(a)]? Ir is a heavy element whose large atomic spin-orbit coupling will be transferred to the Mn atoms with which it hybridizes. Our symmetry considerations therefore suggest that antiferromagnetic Mn3 Ir should have a sizeable anomalous Hall effect in the absence of a magnetic field. To explore this possibility we have used first-principles methods to evaluate its intrinsic anomalous Hall conductivity. (Technical details of this calculation are explained in the Supplemental Material [26]). First of all we found that fully relativistic electronic structure calculations predict that the three Mn moments in the Mn3 Ir unit cell form a very small angle ≈ 0.1° relative to the (111) plane, leading to a total magnetization of ∼0.02 μB per formula unit along the (111) direction. This moment rotation can be traced to magneto-crystalline anisotropy effects [22,23]. From a symmetry point of view, these small rotations, like the anomalous Hall effect discussed below, are a combined consequence of mirror symmetry breaking and spin-orbit coupling. By performing the Brillouin-zone integral of the (111) direction Berry curvature component, we find that the corresponding intrinsic anomalous Hall conductivity is σ ð111Þ ¼ 218 Ω−1 cm−1 , the same order of magnitude as for the ferromagnetic transition metals Fe, Co, Ni. As in the ferromagnetic metal case [31],
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the band sum of the Berry curvature is strongly peaked at points in momentum space where degeneracies [Fig. 3(a)] are lifted by spin-orbit coupling and states are split to opposite sides of the Fermi energy [Fig. 3(b)]. To show that the anomalous Hall conductivity is not due to the small net magnetization, we have constrained the Mn moments to be strictly coplanar, and found that σ ð111Þ changes by only 1 Ω−1 cm−1 . When spin-orbit coupling is not included, the anomalous Hall conductivity vanishes. These results are consistent with the symmetry arguments explained above using the kagome lattice example, and show that the symmetry allowed anomalous Hall effect in Mn3 Ir is surprisingly large. In the absence of spin-orbit coupling the broken symmetry ground state of Mn3 Ir is invariant under any global spin rotation. With spin-orbit coupling, the ground state has a discrete twofold degeneracy. The magnetic configuration sketched in Fig. 1(a), which we call A phase from now on, is degenerate with its time-reversal counterpart, which we call B phase, and with other phases associated with other (111)-type planes. The small net magnetizations and the large anomalous Hall conductivities of the A and B phases are opposite. If one adds an external magnetic field along the (111) direction, the nearly coplanar Mn moments will rotate toward the field, which changes the anomalous Hall conductivity only slightly as discussed above. To develop a comprehensive picture of all these effects, we constrained the Mn moments of both A and B phases to have different angles relative to the (111) plane. Figure 3(c) shows the dependence of the total energy (relative to the ground states) of the two phases on the rotation angle. The large energy changes, by hundreds of meV per formula unit, reflect the strong exchange coupling between Mn moments [22], and imply that only small rotation angles can be achieved using magnetic fields. The energy differences between A and B phases at a given rotation angle [Fig. 3(d)] reflect the absence of mirror symmetry. Figure 3(e) plots intrinsic anomalous Hall conductivities calculated for various magnetic configurations. The detailed shape of these curves is sensitively dependent on band structure (Figs. S3 and S4 [26]); large changes in σ ð111Þ signal that the Fermi energy has entered or exited the energy interval between spin-orbit split states. Note that the anomalous Hall effect is due almost entirely to spinorbit coupling even when the moments are tilted out of (111) planes. To experimentally observe the anomalous Hall effect in Mn3 Ir, it will be necessary to prepare a sample in which one magnetic phase is dominant. Since all phases have small but nonzero magnetizations, one strategy would be to apply an external magnetic field oriented along a (111) direction at the highest convenient temperature and then cool. Another strategy would be to study instead the closely related material Mn3 Pt, which has collinear and noncollinear antiferromagnetic phases in different temperature
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FIG. 3 (color online). Anomalous Hall effect of Mn3 Ir. (a) Band structure of the A phase near the Fermi level. (b) Berry curvature along the same k-point path as (a). (c) Relative energies of the A and B phases with Mn moment angle relative to the (111) plane constrained. The ground states at 0.1° are shifted horizontally for illustration purpose. (d) Energy differences between A and B phases at different angles. (e) Anomalous Hall conductivities of A and B phases (black squares) with and without spin-orbit coupling (red triangles) as a function of the Mn moment tilt angle.
ranges [32]. Crossing a collinear to noncollinear phase boundary under the influence of a magnetic field could also selects a dominant noncollinear phase. If single-phase materials could be prepared, their magnetization configuration could be read electrically, or, since the anomalous Hall effect is always accompanied by optical gyrotropy [33], optically. Our study shows that the Karplus-Luttinger mechanism and the spin-chirality mechanism of the anomalous Hall effect are consistent with the same symmetry considerations, and indicates that it is in general unreasonable to assume the anomalous Hall effect to be small in the broad intermediate region between the two extreme regimes in which one of the two mechanisms dominates. The authors thank Lifa Zhang, Xiao Li, Karin Everschor, and Guang-Yu Guo for valuable discussions. H. C. and
A. H. M. were supported by the Welch Foundation under Grant No. TBF1473 and by the National Science Foundation under Grant No. DMR-1122603. Q. N. was supported by the US Department of Energy Division of Materials Sciences and Engineering under Grant No. DE-FG03-02ER45958. The calculations were mainly performed at the Texas Advanced Computing Center.
[1] E. M. Pugh and N. Rostoker, Rev. Mod. Phys. 25, 151 (1953). [2] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010). [3] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954). [4] G. Sundaram and Q. Niu, Phys. Rev. B 59, 14 915 (1999).
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[5] C.-Z. Chang et al., Science 340, 167 (2013). [6] K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B 62, R6065 (2000). [7] R. Shindou and N. Nagaosa, Phys. Rev. Lett. 87, 116801 (2001). [8] G. Tatara and H. Kawamura, J. Phys. Soc. Jpn. 71, 2613 (2002). [9] P. Bruno, V. K. Dugaev, and M. Taillefumier, Phys. Rev. Lett. 93, 096806 (2004). [10] M. Taillefumier, B. Canals, C. Lacroix, V. K. Dugaev, and P. Bruno, Phys. Rev. B 74, 085105 (2006). [11] A. Kalitsov, B. Canals, and C. Lacroix, J. Phys. Conf. Ser. 145, 012020 (2009). [12] H. Takatsu, S. Yonezawa, S. Fujimoto, and Y. Maeno, Phys. Rev. Lett. 105, 137201 (2010). [13] M. Udagawa and R. Moessner, Phys. Rev. Lett. 111, 036602 (2013). [14] A. Chubukov, Phys. Rev. Lett. 69, 832 (1992). [15] A. B. Harris, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 45, 2899 (1992). [16] D. A. Huse and A. D. Rutenberg, Phys. Rev. B 45, 7536 (1992). [17] S. Sachdev, Phys. Rev. B 45, 12377 (1992). [18] I. Ritchey, P. Chandra, and P. Coleman, Phys. Rev. B 47, 15 342 (1993). [19] I. Tomeno, H. N. Fuke, H. Iwasaki, M. Sahashi, and Y. Tsunoda, J. Appl. Phys. 86, 3853 (1999). [20] A. Kohn, A. Kovacs, R. Fan, G. J. McIntyre, R. C. C. Ward, and J. P. Goff, Sci. Rep. 3, 2412 (2013). [21] V. Hemmati, M. L. Plumer, J. P. Whitehead, and B. W. Southern, Phys. Rev. B 86, 104419 (2012).
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[22] L. Szunyogh, B. Lazarovits, L. Udvardi, J. Jackson, and U. Nowak, Phys. Rev. B 79, 020403(R) (2009). [23] M. D. LeBlanc, M. L. Plumer, J. P. Whitehead, and B. W. Southern, Phys. Rev. B 88, 094406 (2013). [24] C. Herring, in Magnetism: A Treatise on Modern Theory and Materials, ed. by G. T. Rado and H. Suhl (Academic Press, New York, 1966), Vol. IV. [25] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). [26] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.017205 for a detailed description of the calculation methods and Supplemental Figures S1–S4. [27] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). [28] Tomizawa and Kontani have previously studied the anomalous Hall effect in Nd2 Mo2 O7 and Pr2 Ir2 O7 , which cannot be satisfyingly accounted for by either the k-space Berry phase or the spin chirality mechanism. They interpreted the anomalous Hall effect in these systems in terms of a real-space Berry phase associated with site-dependent t2g orbital states [29,30]. [29] T. Tomizawa and H. Kontani, Phys. Rev. B 80, 100401(R) (2009). [30] T. Tomizawa and H. Kontani, Phys. Rev. B 82, 104412 (2010). [31] Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-S. Wang, E.-G. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204 (2004). [32] E. Krén, G. Kádár, L. Pál, J. Sólyom, P. Szabó, and T. Tarnóczi, Phys. Rev. 171, 574 (1968). [33] J. Orenstein and J. E. Moore, Phys. Rev. B 87, 165110 (2013).
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Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism Hua Chen, Qian Niu, and A. H. MacDonald Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA (Received 3 October 2013; published 10 January 2014) As established in the very early work of Edwin Hall, ferromagnetic conductors have an anomalous Hall conductivity contribution that cannot be attributed to Lorentz forces and therefore survives in the absence of a magnetic field. These anomalous Hall conductivities are normally assumed to be proportional to magnetization. We use symmetry arguments and first-principles electronic structure calculations to counter this assumption and to predict that Mn3 Ir, a high-temperature antiferromagnet that is commonly employed in spin-valve devices, has a large anomalous Hall conductivity. DOI: 10.1103/PhysRevLett.112.017205
PACS numbers: 75.47.Np, 73.22.Gk, 75.50.Ee
In the presence of an external magnetic field, the Hall effect (current flow perpendicular to electric field) is present in all conductors. In ferromagnetic metals like Fe, Co, and Ni, however, the Hall effect is anomalous and controlled more by magnetization than by Lorentz forces [1,2]. Although the anomalous Hall effect is experimentally strong, it has stood alone among metallic transport effects for much of the last century because it lacked a usefully predictive, generally accepted theory. Progress over the past decade has explained why. It is now clear that the anomalous Hall effect in ferromagnets has contributions from both extrinsic scattering mechanisms similar to those that determine most transport coefficients, and from an intrinsic mechanism that is independent of scattering [2]. The intrinsic mechanism, first proposed by Karplus and Luttinger [3], has recently been reformulated in the language of Berry phases as a momentum-space geometrical effect [4] related to the quantum Hall effect and to topological insulators. In the quantized anomalous Hall effect [5] only the intrinsic component survives. The anomalous Hall effect is also observed in paramagnets, which have nonzero magnetization induced by an external magnetic field. Separately, studies of the anomalous Hall effect in frustrated and noncollinear magnetic systems have also been very active [2,6–13]. In these systems magnetic frustration may be removed by magnetocrystalline anisotropy or Dzyloshinskii-Moriya interactions, resulting in stable noncollinear magnetic order. For noncoplanar spin textures, the anomalous Hall effect in the strong local exchange coupling limit can then often be explained as a real-space Berry phase effect [2,6,9]. Although no explicit relationship has been established, the anomalous Hall effect in a particular material is usually assumed to be proportional to its magnetization. In this Letter, we point out that it is possible to have an anomalous Hall effect in a noncollinear antiferromagnet with zero net magnetization provided that certain common symmetries are absent, and predict that Mn3 Ir, a technologically important antiferromagnetic material with noncollinear order that 0031-9007=14=112(1)=017205(5)
survives to very high temperatures, has a surprisingly large anomalous Hall effect comparable in size to those of the elemental transition metal ferromagnets. Mn3 Ir can be viewed as an fcc crystal with Mn atoms on three of the four cubic sublattices [Fig. 1(a)], and is widely used as an exchange-bias material in spin-valve devices. Its Mn sublattice can be viewed as consisting of twodimensional kagome lattices [Fig. 1(b)] stacked along the (111) direction. Although isolated two-dimensional kagome lattices order weakly because of strong magnetic frustration [14–18], the Mn moments in Mn3 Ir have strong three-sublattice triangular (T1) magnetic order [Fig. 1(a)] [19,20] and a high magnetic transition temperature ∼950 K. The surprising stability is provided by interkagome coupling [21] combined with strong magnetic anisotropy [22,23]. Because the kagome structure, with appropriate exchange interactions, can support relatively simple noncollinear antiferromagnetic states similar to those of Mn3 Ir, we begin our discussion of the anomalous Hall effect in noncollinear antiferromagnets by focusing on a simple two-dimensional kagome model. The itinerant electron Hamiltonian for an s-d model on the kagome lattice is
FIG. 1 (color online). Structure of Mn3 Ir. (a) Unit cell of Mn3 Ir with triangular antiferromagnetic order. (b) An individual (111) plane of Mn3 Ir. The Mn atoms form a kagome lattice.
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PHYSICAL REVIEW LETTERS 10 JANUARY 2014 PRL 112, 017205 (2014) X X systems and is directly related to the anomalous Hall effect H¼t c†iαγ cjβγ − J c†iαγ ciατ σ γτ · Δα : (1) [4,25]. The momentum space Berry curvature of a Bloch iαγτ hiα;jβiγ band is defined as Ωn ðkÞ ¼ ih∇k unk j × j∇k unk i, where junk i is the Bloch eigenstate for band n at momentum k. Here α and β label kagome sublattices, the angle-brackets The intrinsic anomalous Hall effect is proportional to the restrict s-electron hopping to nearest neighbor sites, and Δα Brillouin-zone integration of the Berry curvature [2,25]. is an exchange field due to coupling to quenched d-electron If a system has both time reversal and spatial inversion local moments that have coplanar triangular magnetic order I symmetry, the Berry curvature satisfies Ωn ðkÞ ¼ [Fig. 1(b)]. The band structure of this model is shown in −Ωn ð−kÞ and Ωn ðkÞ ¼ Ωn ð−kÞ, implying that it vanishes Fig. 2(a). One interesting feature is that, in contrast to identically. In the present case, TM symmetry implies the case of simple collinear bipartite antiferromagnets that Ωzn ðkÞ ¼ −Ωzn ð−kÞ, and I symmetry that Ωzn ðkÞ ¼ [24], the bands are not doubly degenerate. In the latter case, Ωzn ð−kÞ, so the Berry curvature still vanishes identically. the time-reversal operation T in combination with a spatial An explicit evaluation of Ωzn ðkÞ for the model of Eq. (1) translation by the vector connecting the two sublattices T d confirms this conclusion, except at points in the is a good symmetry, supplying an analog of Kramers theoBrillouin zone where degeneracies occur and Ωzn ðkÞ is rem even though time-reversal symmetry on its own is bronot well defined [Fig. 2(a)]. The presence of the bandken. When combined with spatial inversion symmetry I, crossing (Dirac) points in [Fig. 2(a)], however, suggests this property implies double band degeneracy at every point that the system has the potential to become topologically in momentum space [24]. In contrast, in the three-sublattice nontrivial when TM symmetry is broken. kagome model there is no operation O that can make T O a We can conclude from the discussion above that the good symmetry while fulfilling the prerequisites for Kramers Berry curvature Ωzn ðkÞ will be an even nonzero function theorem, i.e., that is antiunitary and ðT OÞ2 ¼ −1. Mirror of k if we break TM symmetry, and that the intrinsic reflection M with respect to the kagome plane is a good anomalous Hall conductivity will therefore be nonzero. symmetry of the kagome lattice, but flips all in-plane spin A similar symmetry argument applies to the extrinsic components. M symmetry is therefore broken by the term anomalous Hall effect as well. One way to break TM symdescribing the interaction with local moments in Eq. (1). metry is to add an external magnetic field B perpendicular Since time reversal T reverses the in-plane spins a second to the plane, which breaks T symmetry but, since σ is a time, the Hamiltonian in Eq. (1) does have TM symmetry. pseudovector, not M symmetry. The influence of a magHowever, this symmetry does not imply double band degen2 netic field can therefore be captured by adding a Zeeman eracy because ðTMÞ ¼ 1 rather than −1. term to Eq. (1) (We ignore any change in the in-plane The fact that TM is a good symmetry of our model does components of Δα ), have other important consequences as we now show. X † Momentum space Berry curvature can be used to characterH ¼ −J ciαγ ciατ σ zγτ : (2) B z ize nontrivial topological properties of condensed matter iαγτ
FIG. 2 (color online). Noncollinear antiferromagnet on a kagome lattice. (a) Band structure of the model in Eq. (1), J ¼ 1.7t. (b) Vectors for the spin-orbit coupling term in Eq. (3). (c) Band structure with spin-orbit coupling, tSO ¼ 0.2t. (d) Berry curvature of the lowest band in (c).
Straightforward calculations show that the Berry curvature and the anomalous Hall effect indeed becomes nonzero due to this perturbation (Fig. S1 [26]). In contrast to the case of ferromagnetic metals, this anomalous Hall effect mechanism does not require spin-orbit coupling, and is in this sense reminiscent of the real-space spin chirality topological Hall effect scenario [2,6,9]. However, it is TM symmetry breaking by the Zeeman term Eq. (2) rather than the noncoplanar configuration of the background moments that is responsible for the anomalous Hall effect. We also note that a nonzero total magnetization is also induced by the Zeeman term. We now show that broken mirror symmetry combined with spin-orbit coupling leads to an anomalous Hall effect, but not necessarily to a net magnetization. Spin-orbit coupling is T invariant, but can break M symmetry and therefore the combined TM symmetry on a kagome lattice. For example, consider the following spin-orbit coupling term, X HSO ¼ itSO ναβ σ γτ · nαβ c†iαγ cjβτ ; (3)
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where tSO is related to the atomic spin-orbit coupling, pffiffiffiffiffiffi i ¼ −1, ναβ is antisymmetric with respect to its sublattice indices and ν12 ¼ ν23 ¼ ν31 ¼ 1, and nαβ are a set of coplanar vectors shown in Fig. 2(b). The spin-orbit coupling term identified in Eq. (3) is analogous to the corresponding term in the graphene model of [27], and preserves the inversion symmetry of the kagome lattice, but breaks the mirror symmetry. It accounts for the difference between the lefthand and right-hand environments of an electron hopping between nearest neighbors on a kagome lattice. When HSO is added to the model of Eq. (1), gaps open at the Dirac points [Fig. 2(c)], as in the external magnetic field case (Fig. S1). Figure 2(d) shows that the Berry curvature is again nonzero throughout momentum space, and that it is large near the gapped Dirac points. This calculation proves by explicit example that antiferromagnets can have nonzero anomalous Hall conductivities simply because they break time reversal symmetry. This property has largely [12,28–30] escaped notice because most antiferromagnets have other symmetries that force their Hall conductivities to vanish [7], with the collinear antiferromagnets on bipartite lattices being the simplest example. In general, by recognizing Berry curvature as a (pseudo)vector defined on the Brillouin zone, the existence of anomalous Hall effect in broken-time-reversal systems can be established by analyzing the transformation properties of such a (pseudo)vector under all the symmetry operations of the specific system. What are the implications of these considerations for Mn3 Ir, which has inversion symmetry, but lacks (111) plane mirror symmetry because of the fcc stacking of its kagome layers [Fig. 1(a)]? Ir is a heavy element whose large atomic spin-orbit coupling will be transferred to the Mn atoms with which it hybridizes. Our symmetry considerations therefore suggest that antiferromagnetic Mn3 Ir should have a sizeable anomalous Hall effect in the absence of a magnetic field. To explore this possibility we have used first-principles methods to evaluate its intrinsic anomalous Hall conductivity. (Technical details of this calculation are explained in the Supplemental Material [26]). First of all we found that fully relativistic electronic structure calculations predict that the three Mn moments in the Mn3 Ir unit cell form a very small angle ≈ 0.1° relative to the (111) plane, leading to a total magnetization of ∼0.02 μB per formula unit along the (111) direction. This moment rotation can be traced to magneto-crystalline anisotropy effects [22,23]. From a symmetry point of view, these small rotations, like the anomalous Hall effect discussed below, are a combined consequence of mirror symmetry breaking and spin-orbit coupling. By performing the Brillouin-zone integral of the (111) direction Berry curvature component, we find that the corresponding intrinsic anomalous Hall conductivity is σ ð111Þ ¼ 218 Ω−1 cm−1 , the same order of magnitude as for the ferromagnetic transition metals Fe, Co, Ni. As in the ferromagnetic metal case [31],
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the band sum of the Berry curvature is strongly peaked at points in momentum space where degeneracies [Fig. 3(a)] are lifted by spin-orbit coupling and states are split to opposite sides of the Fermi energy [Fig. 3(b)]. To show that the anomalous Hall conductivity is not due to the small net magnetization, we have constrained the Mn moments to be strictly coplanar, and found that σ ð111Þ changes by only 1 Ω−1 cm−1 . When spin-orbit coupling is not included, the anomalous Hall conductivity vanishes. These results are consistent with the symmetry arguments explained above using the kagome lattice example, and show that the symmetry allowed anomalous Hall effect in Mn3 Ir is surprisingly large. In the absence of spin-orbit coupling the broken symmetry ground state of Mn3 Ir is invariant under any global spin rotation. With spin-orbit coupling, the ground state has a discrete twofold degeneracy. The magnetic configuration sketched in Fig. 1(a), which we call A phase from now on, is degenerate with its time-reversal counterpart, which we call B phase, and with other phases associated with other (111)-type planes. The small net magnetizations and the large anomalous Hall conductivities of the A and B phases are opposite. If one adds an external magnetic field along the (111) direction, the nearly coplanar Mn moments will rotate toward the field, which changes the anomalous Hall conductivity only slightly as discussed above. To develop a comprehensive picture of all these effects, we constrained the Mn moments of both A and B phases to have different angles relative to the (111) plane. Figure 3(c) shows the dependence of the total energy (relative to the ground states) of the two phases on the rotation angle. The large energy changes, by hundreds of meV per formula unit, reflect the strong exchange coupling between Mn moments [22], and imply that only small rotation angles can be achieved using magnetic fields. The energy differences between A and B phases at a given rotation angle [Fig. 3(d)] reflect the absence of mirror symmetry. Figure 3(e) plots intrinsic anomalous Hall conductivities calculated for various magnetic configurations. The detailed shape of these curves is sensitively dependent on band structure (Figs. S3 and S4 [26]); large changes in σ ð111Þ signal that the Fermi energy has entered or exited the energy interval between spin-orbit split states. Note that the anomalous Hall effect is due almost entirely to spinorbit coupling even when the moments are tilted out of (111) planes. To experimentally observe the anomalous Hall effect in Mn3 Ir, it will be necessary to prepare a sample in which one magnetic phase is dominant. Since all phases have small but nonzero magnetizations, one strategy would be to apply an external magnetic field oriented along a (111) direction at the highest convenient temperature and then cool. Another strategy would be to study instead the closely related material Mn3 Pt, which has collinear and noncollinear antiferromagnetic phases in different temperature
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FIG. 3 (color online). Anomalous Hall effect of Mn3 Ir. (a) Band structure of the A phase near the Fermi level. (b) Berry curvature along the same k-point path as (a). (c) Relative energies of the A and B phases with Mn moment angle relative to the (111) plane constrained. The ground states at 0.1° are shifted horizontally for illustration purpose. (d) Energy differences between A and B phases at different angles. (e) Anomalous Hall conductivities of A and B phases (black squares) with and without spin-orbit coupling (red triangles) as a function of the Mn moment tilt angle.
ranges [32]. Crossing a collinear to noncollinear phase boundary under the influence of a magnetic field could also selects a dominant noncollinear phase. If single-phase materials could be prepared, their magnetization configuration could be read electrically, or, since the anomalous Hall effect is always accompanied by optical gyrotropy [33], optically. Our study shows that the Karplus-Luttinger mechanism and the spin-chirality mechanism of the anomalous Hall effect are consistent with the same symmetry considerations, and indicates that it is in general unreasonable to assume the anomalous Hall effect to be small in the broad intermediate region between the two extreme regimes in which one of the two mechanisms dominates. The authors thank Lifa Zhang, Xiao Li, Karin Everschor, and Guang-Yu Guo for valuable discussions. H. C. and
A. H. M. were supported by the Welch Foundation under Grant No. TBF1473 and by the National Science Foundation under Grant No. DMR-1122603. Q. N. was supported by the US Department of Energy Division of Materials Sciences and Engineering under Grant No. DE-FG03-02ER45958. The calculations were mainly performed at the Texas Advanced Computing Center.
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