Spin-order dependent anomalous Hall effect and magneto-optical effect in noncollinear antiferromagnets Mn$_{3}X$N ($X$ = Ga, Zn, Ag, and Ni)
Xiaodong Zhou Jan-Philipp Hanke Wanxiang Feng Fei Li Guang-Yu Guo Yugui Yao Stefan Blügel Yuriy Mokrousov

	
	
	
	
	Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement, Ministry of Education, School of Physics, Beijing Institute of Technology, Beijing 100081, China
	
	
	Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
	
	
	wxfeng@bit.edu.cn
	Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement, Ministry of Education, School of Physics, Beijing Institute of Technology, Beijing 100081, China
	Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
	
	
	Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement, Ministry of Education, School of Physics, Beijing Institute of Technology, Beijing 100081, China
	
	
	Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan
	Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan
	
	
	Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement, Ministry of Education, School of Physics, Beijing Institute of Technology, Beijing 100081, China
	
	
	Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
	
	
	Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
	Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
	
	
	
	
	
		The anomalous Hall effect (AHE) and the magneto-optical effect (MOE) are two prominent manifestations of time-reversal symmetry breaking in magnetic materials. Noncollinear  antiferromagnets (AFMs) have recently attracted a lot of attention owing to the potential emergence of exotic spin orders on geometrically frustrated lattices, which can be characterized by corresponding spin chiralities.  By performing first-principles density functional calculations together with group-theory analysis and tight-binding modelling, here we systematically study the spin-order dependent AHE and MOE in representative noncollinear AFMs Mn_3XN (X = Ga, Zn, Ag, and Ni).  The symmetry-related tensor shape of the intrinsic anomalous Hall conductivity (IAHC) for different spin orders is determined by analyzing the relevant magnetic point groups.  We show that while only the xy component of the IAHC tensor is nonzero for right-handed spin chirality, all other elements - σ_xy, σ_yz, and σ_zx - are nonvanishing for a state with left-handed spin chirality owing to lowering of the symmetry.  Our tight-binding arguments reveal that the magnitude of IAHC relies on the details of the band structure and that σ_xy is periodically modulated as the spin rotates in-plane. The IAHC obtained from first principles is found to be rather large, e.g., it amounts to 359 S/cm in Mn_3AgN, which is comparable to other well-known noncollinear AFMs such as Mn_3Ir and Mn_3Ge.  We evaluate also the magnetic anisotropy energy and find that the evolution of spin order is related to the number of valence electrons in the X ion. Interestingly, the left-handed spin chirality could exist in Mn_3XN with some particular spin configurations.  By extending our analysis to finite frequencies, we calculate the optical isotropy [σ_xx(ω)≈σ_yy(ω)≈σ_zz(ω)] and the magneto-optical anisotropy [σ_xy(ω)≠σ_yz(ω)≠σ_zx(ω)] of Mn_3XN.  Similar to the IAHC, the magneto-optical Kerr and Faraday spectra depend strongly on the spin order.  The Kerr rotation angles in Mn_3XN are in the range of 0.3∼0.4 deg, which is large and comparable to other noncollinear AFMs like Mn_3Pt and Mn_3Sn. Our finding of large AHE and MOE in Mn_3XN suggests that these materials present an excellent antiferromagnetic platform for realizing novel spintronics and magneto-optical devices.  We argue that the spin-order dependent AHE and MOE are indispensable in detecting complex spin structures in noncollinear AFMs.
	
	
	Spin-order dependent anomalous Hall effect and magneto-optical effect in noncollinear antiferromagnets Mn_3XN (X = Ga, Zn, Ag, and Ni)
    Yuriy Mokrousov
    December 30, 2023
======================================================================================================================================



§ INTRODUCTION

	
	The anomalous Hall effect (AHE) <cit.> and the magneto-optical effect (MOE) <cit.> are fundamental phenomena in condensed matter physics and they have become appealing techniques to detect and measure magnetism by electric and optical means, respectively.  Usually occuring in ferromagnetic metals, the AHE is characterized by a transverse voltage drop resulting from a longitudinal charge current in the absence of applied magnetic fields. There are two distinct contributions to the AHE, that is, the extrinsic one <cit.> depending on scattering of electron off impurities or due to disorder, and the intrinsic one <cit.> solely determined by the Berry phase effect <cit.> in a pristine crystal. Both of these mechanisms originate from time-reversal (T) symmetry breaking in combination with spin-orbit coupling (SOC) <cit.>.  The intrinsic AHE can be accurately calculated from electronic-structure theory on the ab initio level, and examples include studies of Fe <cit.>, Co <cit.>, SrRuO_3 <cit.>, Mn_5Ge_3 <cit.>, and CuCr_2Se_4-xBr_x <cit.>.  Referring to the Kubo formula <cit.>, the intrinsic anomalous Hall conductivity (IAHC) can be straightforwardly extended to the ac case (as given by the optical Hall conductivity), which is intimately related to the magneto-optical Kerr and Faraday effects (MOKE and MOFE) [see Eqs. (<ref>) and (<ref>) below].  Phenomenally, the MOKE and MOFE refer to the rotation of the polarization plane when a linearly polarized light is reflected from, or transmitted through a magnetic material, respectively.  Owing to their similar physical nature, the intrinsic AHE is often studied together with MOKE and MOFE.
	
	
	
	As the AHE and the MOE are commonly considered to be proportional to the magnetization, most of the materials studied to date with respect to these phenomena are ferromagnets (FMs) and ferrimagnets (FiMs), while antiferromagnets (AFMs) are naively expected to have neither AHE nor MOE due to their vanishing net magnetization. Although T symmetry is broken in AFMs, its combination TS with other spatial symmetries S (e.g., fractional translations or inversion) can reinstate Kramers theorem such that AHE and MOE vanish. A simple example is the one-dimensional collinear bipartite antiferromagnet <cit.>, where S is the fractional translation by half of the vector connecting the two sublattices.  Another example is the two-dimensional honeycomb lattice with collinear Néel order (as realized, e.g., in the bilayer MnPSe_3) <cit.>, which has natively the combined symmetry TI although time-reversal symmetry T and spatial inversion symmetry I are both broken individually. The application of an electric field perpendicular to the film plane will manifest in broken TI symmetry and band exchange splitting that generates the MOKE <cit.>. Such electrically driven  MOKE has been realized, e.g., in multiferroic Cr-based metallorganic perovskites <cit.>.  Therefore, the AHE and the MOE, as the most fundamental fingerprints of T symmetry breaking in matter, can in principle exist in AFMs if certain crystal symmetries are absent, even though the net magnetization vanishes. Notably, the cluster multipole theory proposed by Suzuki et al. <cit.> has been recently applied to interpret the origin of AHE in AFMs.
	
	Leaving aside collinear AFMs, recent works <cit.> revealed that noncollinear AFMs can also host nonvanishing AHE and MOE.  Two types  of noncollinear AFMs can be considered: noncoplanar and coplanar, which are characterized by scalar and vector spin chiralities, respectively <cit.>.  On the one hand, the nonzero scalar spin chirality χ=S_i·(S_j×S_k)  (where S_i, S_j, and S_k denote three neighboring noncoplanar spins) will generate a fictitious magnetic field that makes the electrons feel a real-space Berry phase while hopping in the spin lattice <cit.>.  Consequently, the AHE can emerge in noncoplanar AFMs without SOC, which is referred to the topological Hall effect that has been theoretically predicted <cit.> and experimentally observed <cit.>, for instance, in disordered γ-Fe_xMn_1-x alloys.  Moreover, the quantized version of the topological Hall effect was reported in the layered noncoplanar noncollinear K_0.5RhO_2 AFM insulator <cit.>.  Extending these findings, Feng et al. <cit.> proposed that topological MOE and quantum topological MOE exist in γ-Fe_xMn_1-x and K_0.5RhO_2, respectively.
	
	Instead of the scalar spin chirality (which vanishes for coplanar spin configurations), the finite vector spin chirality <cit.>,
	
    κ=2/3√(3)∑_⟨ ij⟩[S_i×S_j]_z,

	where ⟨ ij⟩ runs over the nearest neighboring spins, is an important quantity in coplanar noncollinear AFMs such as cubic Mn_3X (X = Rh, Ir, Pt) and hexagonal Mn_3Y (Y = Ge, Sn, Ga).  The Mn atoms in the (111) plane of Mn_3X and in the (0001) plane of Mn_3Y are arranged into a kagome lattice, while Mn_3X and Mn_3Y have opposite vector spin chiralities <cit.> with κ=+1 (right-handed state) and κ=-1 (left-handed state) [see Figs. <ref>(a) and <ref>(b)], respectively.  The concept of right- and left-handed states adopted here follows the convention of Ref. Kawamura2001.  For both right- and left-handed spin chiralities, the spins can be simultaneously rotated within the plane, further resulting in different spin configurations [see Figs. <ref>(a) and <ref>(b)], e.g., the T1 and the T2 phases in Mn_3X <cit.> as well as the type-A and the type-B phases in Mn_3Y <cit.>.  The vector spin chirality and the spin rotation discussed here allow us to characterize coplanar AFMs that have a 120^∘ noncollinear magnetic ordering.  For the AHE, Chen et al. <cit.> discovered theoretically that Mn_3Ir has unexpectedly large IAHC and several other groups predicted the IAHC in Mn_3Y with comparable magnitudes <cit.>.  At the same time, the AHE in Mn_3Y has been experimentally confirmed <cit.>.  Because of the close relationship to AHE, Feng et al. <cit.> first predicted that large MOKE can emerge in Mn_3X even though the net magnetization is zero.  Eventually, Higo et al. <cit.> successfully measured large zero-field Kerr rotation angles in Mn_3Sn at room temperature.
	
	
	
	In addition to Mn_3X and Mn_3Y, the antiperovskite Mn_3XN (X = Ga, Zn, Ag, Ni, etc.) is another important class of coplanar noncollinear AFMs <cit.>, which was known since the 1970s <cit.>.  Compared to Mn_3X, the X atoms in Mn_3XN also occupy the corners of the cube [see Fig. <ref>(c)] and the face-centered Mn atoms are arranged into a kagome lattice in the (111) plane [see Fig. <ref>(d)], while there is an additional N atom located in the center of the cube [see Fig. <ref>(c)].  Despite the structural similarity, some unique physical properties have been found in Mn_3XN, such as magnetovolume effects <cit.> and magnetocaloric effects <cit.> that stem from a strong coupling between spin, lattice, and heat.  The most interesting discovery in Mn_3XN may be the giant negative thermal expansion that was observed in the first-order phase transition from a paramagnetic state to a noncollinear antiferromagnetic state with decreasing temperature  𝚃.  Below the Néel temperature (𝚃_𝙽), a second-order phase transition between two different noncollinear antiferromagnetic states, which are featured by a nearly constant volume but the change of spin configuration, possibly occurs.  
	
	Taking Mn_3NiN as an example <cit.>, all the spins point along the diagonals of the face if 𝚃<163 K (the so-called Γ^5g configuration), while in the temperature range of 163 K <𝚃< 266 K the spins can point to the center of the triangle formed by three nearest-neighboring Mn atoms (the so-called Γ^4g configuration).  The Γ^5g and the Γ^4g spin configurations are named as R1 (θ=0^∘) and R3 (θ=90^∘) phases in this work [see Figs. <ref>(e) and <ref>(g), where the azimuthal angle θ measures the rotation of the spins starting from the diagonals of the face], respectively.  An intermediate state (0^∘<θ<90^∘) between the R1 and R3 phases, referred to as the R2 phase (see Fig. <ref>(f) with θ=30^∘ as an example), was proposed to exist <cit.>.  Such nontrivial magnetic orders are also believed to occur in other Mn_3XN compounds <cit.>, as recently clarified by Mochizuki et al. <cit.> using a classical spin model together with the replica-exchange Monte Carlo simulation.  However, the details of the changes in spin configurations from R1 phase, passing through the R2 phase to R3 phase, and how they affect the relevant physical properties (e.g., AHE and MOE) are still unclear.  Moreover, although only the right-handed spin chirality was reported in the previous literature, the left-handed spin chirality as a counterpart [Fig. <ref>(h-j)] could also exist, e.g., in Mn_3NiN, because of the favorable total energy for a particular θ [see Fig. <ref>(a)].
	
	In this work, using  first-principles density functional theory together with group-theory analysis and tight-binding modelling, we systematically investigate the effect of spin order on the intrinsic AHE as well as the MOKE and the MOFE in coplanar noncollinear AFMs Mn_3XN (X = Ga, Zn, Ag, and Ni).  The spin order considered here has dual implications, i.e., spin chiralities (right- and left-handed states) and spin configurations [regarding the different spin orientations by simultaneously rotating the spins within the (111) plane].  In Sec. <ref>, we first identify the antisymmetric shape of the IAHC tensor (i.e., zero and nonzero elements) for different spin orders by a group theoretical analysis.  For the right-handed spin chirality, only σ_xy is nonzero (except for two particular spin configurations: θ=0^∘ and 180^∘); for the left-handed spin chirality, all three off-diagonal elements (σ_xy, σ_yz, and σ_zx) can be nonzero (except for some particular spin configurations, e.g., θ=0^∘ and 60^∘ for σ_xy, θ=30^∘ and 210^∘ for σ_yz, θ=120^∘ and 300^∘ for σ_zx).  The results of the group-theory analysis are further confirmed by both tight-binding modelling (Sec. <ref>) and first-principles calculations (Sec. <ref>).  In addition to the IAHC, the magnetic anisotropy energy (MAE) has also been accessed and the in-plane easy spin orientation is determined (Sec. <ref>).  
	
	Considering Mn_3NiN as a prototype, we extend the study of IAHC to the optical Hall conductivity [σ_xy(ω), σ_yz(ω) σ_zx(ω)] as well as the corresponding diagonal elements [σ_xx(ω), σ_yy(ω), and σ_zz(ω)] (Sec. <ref>).  The spin order hardly affects the diagonal elements, whereas a significant dependence on the spin order is observed in the off-diagonal elements akin to the IAHC.  Subsequently in Sec. <ref>, the MOKE and the MOFE are computed from the optical conductivity for all Mn_3XN (X = Ga, Zn, Ag, and Ni). Kerr and Faraday spectra exhibit a distinct dependence on the spin order, which they inherit from the optical Hall conductivity.  The computed Kerr and Faraday rotation angles in Mn_3XN are comparable to the ones in Mn_3X studied in our previous work <cit.>.  The magneto-optical anisotropy, originating from the nonequivalent off-diagonal elements of optical conductivity, is explored for both right- and left-handed spin chiralities.  Finally, the summary is drawn in Sec. <ref>.  Our work reveals that the AHE and the MOE depend strongly on the spin order in noncollinear AFMs Mn_3XN which suggests that complex noncollinear spin structures can be uniquely classified in experiments by measuring AHE and MOE.
	
	
	
	
	
	

§ GROUP THEORY ANALYSIS

	
	In this section, we determine the magnetic space and point groups of Mn_3XN for given spin orders, and then identify the nonzero elements of IAHC from group theory.  The magnetic groups computed with the isotropy code <cit.> are listed in Table <ref>, from which one can observe that the magnetic groups vary in the azimuthal angle θ with a period of π for right-handed spin chirality, but with a period of π/3 for left-handed spin chirality. This indicates that the magnetic groups that need to be analyzed are limited to a finite number.  Furthermore, it is sufficient to restrict the analysis to magnetic point groups since the IAHC <cit.>,
	
    σ_αβ = -e^2ħ∫_BZd^3k/(2π)^3Ω_αβ(k),

	is translationally invariant. In the above expression Ω_αβ(k)=∑_nf_n(k)Ω_n,αβ(k) is the momentum-space Berry curvature, with the Fermi-Dirac distribution function f_n(k) and the band-resolved Berry curvature
	
    Ω_n,αβ(k) = -2 Im∑_n^'≠ n⟨ψ_nk|v̂_α| ψ_n^'k⟩⟨ψ_n^'k|v̂_β|ψ_nk⟩/(ω_n^'k-ω_nk)^2.

	Here v̂_α is the velocity operator along the αth Cartesian direction, and ψ_nk (ħω_nk=ϵ_nk) is the eigenvector (eigenvalue) to the band index n and the momentum k.   Since the IAHC and the Berry curvature can be regarded as pseudovectors, just like spin, their vector-form notations σ=[σ^x,σ^y,σ^z]=[σ_yz,σ_zx,σ_xy] and Ω_n=[Ω_n^x,Ω_n^y,Ω_n^z]=[Ω_n,yz,Ω_n,zx,Ω_n,xy] are used here for convenience.
	
	Let us start with the right-handed spin chirality by considering the three non-repetitive magnetic point groups: 3̅1m [θ=nπ], 3̅1m^' [θ=(n +1/2)π], and 3̅ [θ≠ nπ and θ≠ (n +1/2)π] with n∈ℕ (see Tab. <ref>).  First, 3̅1m belongs to the type-I magnetic point group, i.e., it is identical to the crystallographic point group D_3d.  As seen from Fig. <ref>(e), it has one three-fold rotation axis (C_3), three two-fold rotation axes (C_2^(1), C_2^(2), and C_2^(3)) and three mirror planes (M^(1), M^(2), and M^(3)).  As mentioned before, Ω_n is a pseudovector, and the mirror operation M^(1) (parallel to the yz plane) changes the sign of Ω_n^y and Ω_n^z, but preserves Ω_n^x.  This indicates that Ω_n^y and Ω_n^z are odd functions along the k_x direction in momentum space, while Ω_n^x is an even function.  Correspondingly,  integrating the Berry curvature over the entire Brillouin zone should give  σ=[σ^x,0,0].  The role of C_2^(1) is the same as that of M^(1).   The other two mirror (two-fold rotation) symmetries are related to M^(1) (C_2^(1)) by the C_3 rotation, which transforms [σ^x,0,0] into [-1/2σ^x,-√(3)/2σ^x,0] and [-1/2σ^x,√(3)/2σ^x,0].  Therefore, all components of IAHC are zero, i.e., σ=[0,0,0], owing to the symmetries of the group 3̅1m.  Second, 3̅ is also a type-I magnetic point group, which is identical to the crystallographic point group C_3i.  Compared to D_3d, all C_2 and M operations are absent whereas only the C_3 operation is left [see Fig. <ref>(f)].  In this situation, the components of σ that are normal to the C_3 axis disappear due to the cancellations of Ω_n^x and Ω_n^y in the k_x–k_y plane.  This gives rise to σ=[0,0,σ^z].  Finally, 3̅1m^'=C_3i⊕ T(D_3d-C_3i) is a type-III magnetic point group as it contains operations combining time and space symmetries.  Here, T(D_3d-C_3i) is the set of three TM and three TC_2 operations depicted in Fig. <ref>(g).  With respect to the mirror symmetry M^(1), Ω_n^x is even but Ω_n^y and Ω_n^z are odd; with respect to the time-reversal symmetry T, all of Ω_n^x, Ω_n^y, and Ω_n^z are odd; hence, with respect to the TM^(1) symmetry, Ω_n^x is odd but Ω_n^y and Ω_n^z are even, resulting in σ=[0,σ^y,σ^z].  TC_2^(1) plays the same role, just like TM^(1) does.  The other two TM (TC_2) symmetries are related to TM^(1) (TC_2^(1)) by the C_3 rotation in the subgroup C_3i, which forces σ^y to be zero but allows finite σ^z. Thus, the IAHC tensor shape is σ=[0,0,σ^z] in the magnetic point group 3̅1m^'.  To summarize, for the right-handed spin chirality  only σ^z can be nonzero, except for θ=nπ where all components of the IAHC vanish.
	
	Next, we turn to the left-handed spin chirality, which also has three non-repetitive magnetic point groups: 2/m [θ=nπ/3], 2^'/m^' [θ=(n +1/2)π/3], and 1̅ [θ≠ nπ/3 and θ≠ (n +1/2)π/3] with n∈ℕ (see Tab. <ref>).  First, 2/m belongs to the type-I magnetic point group, which is identical to the crystallographic point group C_2h that contains one two-fold rotation axis (C_2) and one mirror plane (M) [see Fig. <ref>(h)].  As mentioned before, the M symmetry allows only for those components of the IAHC that are perpendicular to the mirror plane (i.e., along the current C_2 axis), therefore, σ^z should be always zero but σ^x and σ^y are generally finite for θ= 0^∘ (for  current Cartesian coordinates).  If θ= 2π/3 or 5π/3, the mirror plane is parallel to the yz plane and renders only σ^x potentially nonzero.  Similarly, 1̅ is also a type-I magnetic point group that is identical to the crystallographic group C_i.  Since all components Ω_n^x, Ω_n^y, and Ω_n^z are even with respect to the spatial inversion symmetry I, the group C_i imposes no restrictions on the shape of σ, allowing all components to be finite.  Finally, 2^'/m^'=C_i⊕ T(C_2h-C_i) is a type-III magnetic point group containing one TM and one TC_2 operation [see Figs. <ref>(i) and <ref>(j)].  There are two scenarios: if θ= π/6 [Fig. <ref>(i)], TM (or TC_2) symmetry forces σ^x to vanish but facilitates nonzero σ^y and σ^z; if θ= π/2 [Fig. <ref>(j)], the principal axis of both symmetry operations changes (M is neither parallel to yz nor zx plane) such that all entries σ^x, σ^y and σ^z are finite.  The other cases of θ= 7π/6 and 5π/6 are identical to θ= π/6 and π/2, respectively.  In summary, all tensor components of σ are allowed (except for some particular θ) for the left-handed spin chirality owing to the reduced symmetry as compared to the systems with right-handed spin chirality.
	
	In the above discussion, all zero and potentially nonzero elements of the IAHC tensor are identified based on the underlying magnetic point groups. Alternatively, these results can also be obtained by following the Neumann principle, i.e., by applying all symmetry operations of the corresponding point group to the conductivity tensor <cit.>. This method has been implemented in a computer program <cit.>, which generates the shape of linear response tensors (IAHC or intrinsic spin Hall conductivity) in a given coordinate system. Another useful analysis tool is the so-called cluster multipole theory <cit.>, which is capable of uncovering the hidden AHE in AFMs by evaluating the cluster multipole moment that behaves as a macroscopic magnetic order.  For instance, although the cluster dipole moments (i.e., the net magnetization from the conventional understanding) vanish in noncollinear AFMs (e.g., Mn_3X and Mn_3Y), the emerging cluster octupole moments lead to a finite AHE.
		
	
	
	
	
	

§ TIGHT-BINDING MODEL

	
	Group theory is particularly powerful to identify the tensor shape of the IAHC, but it provides no insights into the magnitude of the allowed elements, which will depend strongly on details of the electronic structure. In this light, tight-binding models and first-principles calculations are valuable tools to arrive at quantitative predictions.  In this section, we consider a double-exchange s-d model that describes itinerant s electrons interacting with local d magnetic moments on the kagome lattice, which refers to the (111) plane of cubic Mn_3XN.  Following Ref. H-Chen2014, the Hamiltonian is written as
	
    H     =     t∑_<ij>αc_iα^†c_jα-J∑_iαβ(τ_αβ·S_i)c_iα^†c_iβ
           + it_SO∑_<ij>αβν_ij(τ_αβ·n_ij)c_iα^†c_iβ,

	where c_iα^† (c_iα) is the electron creation (annihilation) operator on site i with spin α, and τ is the vector of Pauli matrices, and ⟨ ij⟩ restricts the summation to nearest-neighbor sites.  The first term is the nearest-neighbor hopping with the transfer integral t.  The second term is the on-site exchange coupling between the conduction electron and the localized spin moment S_i, and J is the Hund's coupling strength.  The third term is the SOC effect with coupling strength t_SO, ν_ij is the antisymmetric 2D Levi-Civita symbol (with ν_12=ν_23=ν_31=1), and n_ij is an in-plane vector perpendicular to the line from site j to site i <cit.>.  In the following calculations, we set J=1.7t and t_SO=0.2t.
	
	We first discuss the band structure, the IAHC, and the Berry curvature of the system with right-handed spin chirality (κ=+1), plotted in Figs. <ref>(a), <ref>(c), and <ref>(d-g), respectively. The band structure significantly changes from θ=0^∘ (R1 phase, 3̅1m), to 30^∘ (R2 phase, 3̅), and to 90^∘ (R3 phase, 3̅1m^').  If θ=0^∘, two band crossings around 1.8 eV appear at the K point and along the M-K path, respectively.  This band structure is identical to the one without SOC <cit.>, because the SOC term in Eq. (<ref>) plays no effect in the spin configuration of θ=0^∘ in the sense that the left-handed and right-handed environments of an electron hopping between nearest neighbors are uniform.  Accordingly, the Berry curvature Ω_xy(k) vanishes everywhere in the Brillouin zone [Fig. <ref>(d)].  The band degeneracy is split when θ≠0^∘ and with increasing θ, the band gap at the K point enlarges significantly, while the one at the M point shrinks slightly.  In order to disentangle the dependence of the IAHC on the band structure, the IAHC is calculated at different Fermi energies (E_F) including 1.8 eV, 0 eV, and -1.8 eV, shown in Fig. <ref>(c).  In all three cases, the IAHC exhibits a period of 2π in θ, and the values for E_F=±1.8 eV are two order of magnitude larger than the ones at E_F=0 eV.  The large IAHC originates from the small band gap at the M point since the Berry curvature shows sharp peaks there [see Figs. <ref>(e-g)].  For E_F=-1.8 eV and 0 eV, the largest IAHC occurs for θ=90^∘ and 270^∘. The case of E_F=1.8 eV is special since the IAHC is quantized to ±2e^2/ħ in a broad range of θ, revealing the presence of quantum anomalous Hall state in coplanar noncollinear AFMs.
	
	For the left-handed spin chirality (κ=-1), the band structure, the IAHC, and the Berry curvature are plotted in Figs. <ref>(b), <ref>(c), and <ref>(h-k), respectively.  The band structure hardly changes from θ=0^∘ (2/m), to 15^∘ (1̅), and to 30^∘ (2^'/m^').  If θ=0^∘, the Berry curvature Ω_xy(k) is odd for the group 2/m [Fig. <ref>(h)] such that the IAHC σ_xy is zero when integrating the Ω_xy(k) over the entire Brillouin zone.  With increasing θ, the IAHC reaches its maximum at θ=30^∘ and exhibits a period of 2π/3 [Fig. <ref>(c)].  Similarly to the κ=+1 state, the IAHC at E_F=±1.8 eV is two orders of magnitude larger than at E_F=0 eV.  However, the IAHC of κ=-1 state is much smaller than that of κ=+1 state [Fig. <ref>(c)].  This is understood based on the Berry curvature shown in Figs. <ref>(i-k), which reveals that Ω_xy(k) at the three M points has different signs (two negative and one positive, or two positive and one negative) due to the reduced symmetry in the κ=-1 state, in contrast to the same sign in the κ=+1 state [Figs. <ref>(e-g)].
	
	The tight-binding model used here is constructed on a two-dimensional kagome lattice, for which the  σ_yz and σ_zx components vanish. Although the model is rather simple, the following qualitative results are useful: (1) the IAHC turns out to be large if the Fermi energy lies in a small band gap as outlined in previous theoretical work <cit.>; (2) σ_xy has a period of 2π (2π/3) in θ for right-handed (left-handed) spin chirality; (3) For structures with right-handed spin chirality, σ_xy is much larger than for the left-handed case.
	
	

§ FIRST-PRINCIPLES CALCULATIONS

	
	In this section, by computing explicitly the electronic structure of the Mn_3XN compounds with different spin orders, we first demonstrate that key properties of these systems follow the qualitative conclusions drawn from the discussed tight-binding model.  Then, we present the values of the computed magnetic anisotropy energy (MAE) and the IAHC of the Mn_3XN compounds.  The obtained in-plane easy spin orientations are comparable to the previous reports <cit.>, while the IAHC is found to depend strongly on the spin order, in agreement with the above tight-binding results.  Taking Mn_3NiN as an example, we further discuss the longitudinal and transverse optical conductivity, which are key to evaluating the MOE.  Finally, the spin-order dependent MOKE and MOFE as well as their anisotropy are explored. Computational details of the first-principles calculations are given in Appendix <ref>.
	
	
	
	
	
	
	
	
	

 §.§ Electronic structure

	Figure <ref> illustrates the first-principles band structures of the Mn_3XN systems, taking Mn_3ZnN and Mn_3NiN as two prototypical examples. While the electronic structure of the left-handed state with κ=-1 hardly changes as the spin-rotation angle θ is tuned, the right-handed state of opposite vector spin chirality is rather sensitive to details of the noncollinear spin configuration. Specifically, the calculated electronic structure for the κ=+1 state reveals that the band degeneracy (e.g., at the Γ point) is lifted for θ≠0^∘, and the magnitude of the band splitting increases with the spin-rotation angle. These features are in a very good qualitative agreement with the tight-binding results [see Figs. <ref>(a) and <ref>(b)], which roots in the fact that the (111) planes of Mn_3XN compounds and the 2D Kagome lattice considered in the previous sections have common symmetries.
	
	
	

 §.§ Intrinsic anomalous Hall conductivity and magnetic anisotropy energy

	
	The MAE is one of the most important parameters that characterizes a magnetic material.  In FMs, the MAE refers to the total energy difference between easy- and hard-axis magnetization directions. In the noncollinear AFMs that we consider here, we define the MAE as the total energy difference between different spin orders, given by
	
    MAE(θ)=E_κ=±1,θ≠0^∘-E_κ=+1,θ=0^∘,

	where the spin order with κ=+1 and θ=0^∘ is set as the reference state.  The calculated MAE of Mn_3XN is plotted in Fig. <ref>(a). For the κ=+1 state, the MAE can be fitted well to the uniaxial anisotropy K_effsin^2(θ), where K_eff is the magnetic anisotropy constant listed in Tab. <ref>. Compared to traditional Mn-based alloys, the value of K_eff in Mn_3XN is comparable in magnitude MnPt (0.51 meV/cell) <cit.>, MnPd (-0.57 meV/cell) <cit.>, MnNi (-0.29 meV/cell) <cit.>, and MnRh (-0.63 meV/cell) <cit.>, but are one order of magnitude smaller than in MnIr (-7.05 meV/cell) <cit.>, Mn_3Pt (2.8 meV/cell) <cit.>, and Mn_3Ir (10.42 meV/cell) <cit.>.  For the κ=-1 state, the MAE is approximately constant with a value of K_eff/2, indicating the vanishing in-plane anisotropy energy that leads to a relatively easy rotation of the spins within the (111) plane.  This feature has also been found in other noncollinear AFMs such as Mn_3Ir <cit.>, Mn_3Ge <cit.>, and Mn_3Sn <cit.>.
	
	
	
	Fig. <ref>(a) reveals that MAE(θ)=MAE(θ+π), implying that the ground state of 120^∘ triangular spin order has a discrete two-fold degeneracy <cit.>.  For the κ=+1 state, Mn_3GaN and Mn_3ZnN obviously prefer the R1 phase (θ=0^∘ or 180^∘), which is in full accordance with the Γ^5g spin configuration identified in Ref. Mochizuki2018 using a classical spin model with frustrated exchange interactions and magnetic anisotropy.  As the spin configuration is closely related to the number of valence electrons n_ν in the X ion, Mochizuki et al. <cit.> propose a mixture of the Γ^5g and the Γ^4g spin patterns in Mn_3AgN and Mn_3NiN due to the smaller n_ν (weaker X-ion crystal field) as compared to that of Mn_3GaN and Mn_3ZnN.  In the present calculations, Mn_3AgN still hosts the Γ^5g spin configuration but has a much smaller MAE compared to Mn_3GaN and Mn_3ZnN, while Mn_3NiN favors the Γ^4g spin configuration (R3 phase, θ=90^∘ or 270^∘).  Our calculated MAE is a monotonic function of n_ν, i.e., Ni (n_ν=0)< Ag (n_ν=1)< Zn (n_ν=2)< Ga (n_ν=3), which provides a clear interpretation for the systematic evolution of the magnetic orders in Mn_3XN.  On the other hand, the κ=-1 state of Mn_3XN has not been considered in previous works, while we find that it could exist for particular values of θ.  For example, the κ=-1 state in Mn_3NiN has the favorable energy in three segments of θ: [0^∘,45^∘), (135^∘,225^∘), and (315^∘,360^∘].  In the light of recent experiments on Mn_3Sn <cit.> and Mn_3Ge <cit.>, an external magnetic field may be used to tune the spin orientation by coupling to the weak in-plane magnetic moment. This finding  enriches the spectrum of possible magnetic orders in Mn_3XN compounds.
	
	The IAHC of Mn_3ZnN and Mn_3NiN with different spin orders is illustrated in Figs. <ref>(b) and <ref>(c), respectively.  The component σ_xy displays a period of 2π (2π/3) in θ for the κ=+1 (κ=-1) state, and its magnitude in the κ=+1 state is much larger than that of the κ=-1 state, in excellent agreement with the tight-binding results. From the group theoretical analysis  we showed that σ_yz and σ_zx are allowed in κ=-1 state, which is confirmed by our first-principles results. Moreover, we observe that both σ_yz and σ_zx display a period of 2π in θ and their magnitudes are much larger than that of σ_xy.  The maximum of IAHC for κ=±1 states is summarized in Tab. <ref>. Overall, the obtained magnitude of the IAHC in the studied family of compounds is comparable or even larger than that in other noncollinear AFMs like Mn_3X <cit.> and Mn_3Y <cit.>.  In contrast to the MAE, the IAHC follows the relation  σ(θ)=-σ(θ+π), which manifests that the spin state at θ+π is the time-reversed counterpart of the order at θ and the IAHC is odd under time-reversal symmetry.
	
	
	
	
	
	

 §.§ Optical conductivity

	
	Before proceeding to the MOE, we evaluate the optical conductivity as a key quantity that comprises the MOE. Expanding on the expressions for the IAHC [Eqs. (<ref>) and (<ref>)], the optical conductivity can be written down as
	
    σ_αβ(ω)     =    σ^'_αβ(ω) + i σ^''_αβ(ω) 
        =    ħ e^2∫d^3k/(2π)^3∑_n≠ n^'[f_n(k)-f_n^'(k)] 
          ×Im[⟨ψ_nk|v_α|ψ_n^'k⟩⟨ψ_n^'k|v_β|ψ_nk⟩] /(ħω_nk-ħω_n^'k)^2-(ħω+iη)^2,

	where the superscript ^' (^'') of σ_αβ denotes its the real (imaginary) part, η is an adjustable smearing parameter in units of energy, and ħω is the photon energy.  Due to the found similarity in the results among all studied systems, we take  Mn_3NiN as a representative example for discussing the optical conductivity (Fig. <ref>).  The real part of the diagonal element, σ^'_xx [see Figs. <ref>(a) and. <ref>(e)], measures the average in the absorption of left- and right-circularly polarized light. The spectrum exhibits one absorptive peak at 1.8 eV with a shoulder at 1.1 eV and another absorptive peak at 3.9 eV.   The imaginary part of the diagonal element, σ^''_xx [see Figs. <ref>(b) and. <ref>(f)], is the dispersive part of the optical conductivity, revealing two distinct valleys at 0.6 eV and 3.4 eV.  Obviously, σ_xx is not affected by the spin order (neither spin chirality κ nor azimuthal angle θ).  A similar behavior has been found in Mn_3X <cit.>, where σ_xx is identical for T1 and T2 spin structures.  From the symmetry analysis <cit.>, it should hold that σ_xx=σ_yy≠σ_zz for the magnetic point groups  3̅1m, 3̅, and 3̅1m^' in the κ=+1 state, whereas σ_xx≠σ_yy≠σ_zz for the magnetic point groups of 2/m, 1̅, and 2^'/m^' in the κ=-1 state.  However, all diagonal elements are approximately equal in our calculations, i.e., we observe that σ_xx≈σ_yy≈σ_zz. This promotes the optical isotropy in the Mn_3XN family.
	
	In contrast to the diagonal entries, the off-diagonal elements displayed in Figs. <ref>(c,d) and <ref>(g–l) depend significantly on the spin order.  For the κ=+1 state [Figs. <ref>(c,d)], σ_xy(ω) vanishes if θ=0^∘, but it increases with the increasing θ and reaches its maximum at θ=90^∘.  For the κ=-1 state [Figs. <ref>(g–l)], all three off-diagonal elements - σ_xy(ω), σ_yz(ω), and σ_zx(ω) - can be nonzero and they peak at θ=30^∘, 120^∘, and 30^∘, respectively. Furthermore, σ_xy(ω) is at least two orders of magnitude smaller than σ_yz(ω) and σ_zx(ω).  The overall trend of σ_xy(ω) depending on the spin order is very similar to that of the IAHC in Fig. <ref>(c). 
	
	
	
	
	
	

 §.§ Magneto-optical Kerr and Faraday effects

	
	We now turn to the magneto-optical Kerr and Faraday effects (MOKE and MOFE).  The characteristic Kerr rotation angle θ_K and the ellipticity ε_K are typically combined into the complex Kerr angle given by <cit.>
	
    ϕ^γ_K    =    θ^γ_K+iε^γ_K
        =    -ν_αβγσ_αβ/σ_0√(1+i(4π/ω)σ_0),

	where ν_αβγ is the 3D Levi-Civita symbol with the Cartesian coordinates α,β,γ∈{x,y,z} and σ_0=1/2(σ_αα+σ_ββ)≈σ_αα.  The complex Kerr angle expressed here, similarly to the IAHC, holds a pseudovector form, i.e., ϕ_K=[ϕ^x_K,ϕ^y_K,ϕ^z_K], which differentiates the Kerr angles when the incident light propagates along different crystallographic axes.  One can read from Eq. (<ref>) that the longitudinal optical conductivity σ_αα modulates the magnitude of the Kerr spectrum, while the transverse optical conductivity σ_αβ determines key features of the Kerr spectrum.  For example, only the component ϕ^z_K is finite for the κ=+1 state, whereas all components ϕ^x_K, ϕ^y_K, and ϕ^z_K are nonzero in the κ=-1 configuration.  More importantly, ϕ^x_K≠ϕ^y_K≠ϕ^z_K implies the presence of magneto-optical anisotropy if the incident light propagates along x (011̅), y (2̅11), and z (111) axes (see Fig. <ref>), respectively.  Similarly, the complex Faraday angle can be expressed as <cit.>
	
    ϕ^γ_F    =    θ^γ_F+iε^γ_F
        =    ν_αβγ(n_+-n_-)ω l/2c,

	where n_±=[1+4π i/ω(σ_αα± iσ_αβ)]^1/2 are the complex refractive indices and l is the thickness of the thin film.  Since σ_αα is generally much larger than σ_αβ (see Fig. <ref>),  n_±≈[1+4π i/ωσ_αα]^1/2∓2π/ωσ_αβ[1+4π i/ωσ_αα]^-1/2 (Ref. YM-Fang2018) and consequently, the complex Faraday angle can be approximated as θ^γ_F+iε^γ_F = -ν_αβγ2π l/cσ_αβ[1+4π i/ωσ_αα]^-1/2.  Therefore, the Faraday spectrum is also determined by σ_αβ.
	
	The magneto-optical Kerr and Faraday spectra for the spin order with κ=+1 in Mn_3XN are plotted in Fig. <ref>, where only ϕ^z_K,F are shown since all other components vanish.  Taking Mn_3NiN as an example [Fig. <ref>(d)], one can observe that the Kerr and Faraday spectra indeed inherit the behavior of the optical conductivity σ_xy(ω) [Figs. <ref>(c) and <ref>(d)].  For example, the Kerr and Faraday angles are zero when θ=0^∘, increase with increasing θ, and reach their maximum at θ=90^∘.  This indicates that the symmetry requirements for MOKE and MOFE are the same as that for the optical Hall conductivity.  In addition, all Mn_3XN compounds considered here have similar Kerr and Faraday spectra, primarily due to their isostructural nature.  The Kerr rotation angles in Mn_3XN are comparable to the theoretical values in Mn_3X (0.2∼0.6 deg) <cit.> and are larger than the experimental value in Mn_3Sn (0.02 deg) <cit.>.  The largest Kerr and Faraday rotation angles of respectively 0.42 deg and 4×10^5 deg/cm emerge in Mn_3AgN. This roots potentially in the stronger SOC of the Ag atom as compared to other lighter X atoms.
	
	Figure <ref> shows the magneto-optical Kerr and Faraday spectra for the κ=-1 state of Mn_3NiN.  Since all off-diagonal elements σ_yz(ω), σ_zx(ω), and σ_xy(ω) of the optical conductivity are nonzero for the κ=-1 state, the Kerr and Faraday effects will appear if the incident light propagates along any Cartesian axes.  This is in contrast to the case of the κ=+1 configuration, for which only the incident light along the z axis generates finite ϕ^z_K,F.  In Fig. <ref>(a), ϕ^x_K,F are zero at θ=30^∘ but have the largest values at θ=120^∘, owing to the features in σ_yz [Figs. <ref>(i) and <ref>(j)].  Moreover, the Kerr and Faraday rotation angles (θ^x_K and θ^x_F) and the ellipticity (ε^x_K and ε^x_F) resemble, respectively, the real part (σ^'_yz) and imaginary part (σ^''_yz) of the corresponding off-diagonal conductivity element.  Compared to ϕ^x_K,F, the angle ϕ^y_K,F in Fig. <ref>(b) displays an opposite behavior in the sense that it has the largest values at θ=30^∘ but vanishes at θ=120^∘.  This is not surprising as the periods of σ_yz and σ_zx as a function of θ differ by π/2, which can be read from Fig. <ref>(c) and Figs. <ref>(i-l).  The angles ϕ^z_K,F shown in Fig. <ref>(c) are two orders of magnitude smaller than ϕ^x_K,F and ϕ^y_K,F, implying that very weak Kerr and Faraday effects are expected for the incident light along the z axis.  From Figs. <ref> and <ref>, we conclude that the MOKE and MOFE depend strongly on the spin order, as in the case of the IAHC.
	
	
	
	

§ SUMMARY

	
	In summary, using a group theoretical analysis, tight-binding modelling, and first-principles calculations, we have systematically investigated the spin-order dependent intrinsic anomalous Hall effect and magneto-optical Kerr and Faraday effects in Mn_3XN (X = Ga, Zn, Ag, and Ni)  compounds, which are considered to be an important class of noncollinear antiferromagnets.  The symmetry-imposed shape of the anomalous Hall conductivity tensor is determined via the analysis of magnetic point groups, that is, only σ_xy can be nonzero for the right-handed spin chirality (κ=+1) while finite σ_xy, σ_yz, and σ_zx exist for the left-handed spin chirality (κ=-1).  Our tight-binding modelling confirms these results and further reveals that σ_xy is a sine-like function of the azimuthal angle θ with a period of 2π (2π/3) for the κ=+1 (κ=-1) state.  By examining the k-resolved Berry curvature, we uncovered that the intrinsic anomalous Hall conductivity is generally large if the Fermi energy enters into the region with small band gaps formed at anticrossings.  The first-principles calculations reproduce all features of σ_xy and further verify that σ_yz and σ_zx have a period of 2π for the κ=-1 state.  The intrinsic anomalous Hall conductivity shows a distinct relation of σ(θ)=-σ(θ+π) due to its odd nature under time-reversal symmetry.  In addition, we have calculated the magnetic anisotropy energy which manifests as K_eff sin^2(θ) for the κ=+1 state but remains nearly constant at K_eff/2 for the κ=-1 state.  A discrete two-fold energy degeneracy, i.e., MAE(θ)=MAE(θ+π), is found in the noncollinear antiferromagnetic Mn_3XN.  Strikingly, our first-principles calculations reveal that the κ=-1 state could exist in Mn_3XN for certain values of θ.
	
	The optical conductivities for κ=±1 states were explored, considering Mn_3NiN as a prototypical example.  We find that the spin order hardly affects the diagonal elements whereas it influences strongly the off-diagonal entries.  The optical isotropy is established since  σ_xx(ω)≈σ_yy(ω)≈σ_zz(ω), while magneto-optical anisotropy occurs inevitably as σ_xy(ω)≠σ_yz(ω)≠σ_zx(ω).  Finally, magneto-optical Kerr and Faraday effects are evaluated based on the optical conductivity.  The largest Kerr rotation angles in Mn_3XN amount to 0.4 deg, which is comparable to other noncollinear antiferromagnets, e.g., Mn_3X <cit.> and Mn_3Sn <cit.>.  Since the optical Hall conductivity plays a major role for magneto-optical effects, the Kerr and Faraday spectra also display a spin-order dependent behavior.  Our work illustrates that complex noncollinear spin structures could be probed via anomalous Hall and magneto-optical effects measurements.
	
	
	
		W. F. and Y. Y. acknowledge the support from the National Natural Science Foundation of China (Nos. 11874085 and 11734003) and the National Key R&D Program of China (No. 2016YFA0300600).  W. F. also acknowledges the funding through an Alexander von Humboldt Fellowship.  Y.M., J.-P. H. and S.B. acknowledge  funding under SPP 2137 “Skyrmionics" (project MO 1731/7-1), collaborative Research Center SFB 1238, and Y.M. acknowledges  funding  from project  MO  1731/5-1 of Deutsche Forschungsgemeinschaft (DFG).  G.-Y. G. is supported by the Ministry of Science and Technology and the Academia Sinica as well as NCTS and Kenda Foundation in Taiwan.  We acknowledge computing time on the supercomputers JUQUEEN and JURECA at Jülich Supercomputing Centre and JARA-HPC of RWTH Aachen University.
	
	
	
	

§ THE DETAILS OF FIRST-PRINCIPLES CALCULATIONS

	
	First-principles density functional theory calculations are performed using the Vienna ab initio simulation package (vasp) <cit.>, in which the accurate frozen-core full-potential projector augmented wave (PAW) method is used <cit.>.  The exchange-correlation potential is treated by the generalized-gradient approximation with the parameters of Perdew-Burke-Ernzerhof <cit.>. The energy cut-off of 500 eV, the energy criterion of 10^-6 eV, and the k-mesh of 16×16×16 are used in the self-consistent calculations.  Spin-orbit interaction is explicitly included in fully relativistic projector augmented potentials.  A penalty functional is added to total energy in order to constrain the local spin moments along a desired direction.  The experimental lattice constants of 3.898 Å, 3.890 Å, 4.013 Å, and 3.886 Å are adopted for Mn_3XN (X = Ga, Zn, Ag, and Ni), respectively <cit.>.
	
	After the ground-state charge density is obtained, a total of 80 maximally localized Wannier functions, including the s, p, and d orbitals of Mn and X atoms as well as the s and p orbitals of N atom, are disentangled from 144 Bloch bands on a uniform k-mesh of 10×10×10, using the wannier90 package <cit.>.  Then, using the Kubo formula, intrinsic anomalous Hall conductivity [Eqs. (<ref>) and (<ref>)] and optical conductivity [Eq. (<ref>)] are evaluated on a denser k-mesh of 100×100×100.  Finally, the Kerr [Eq. (<ref>)] and Faraday [Eq. (<ref>)] spectra are derived from the optical conductivity.
	
	
	
	
	
	
	
	
	
	
	
	
	76
		fxundefined [1]
			ifx#1
		
		fnum [1]
			#1firstoftwo
			secondoftwo
			
		fx [1]
			#1firstoftwo
			secondoftwo
			
		
		
		
		
		
		noop [0]secondoftwo
		
		ref[1]@startlink#1@href
		href[1]#1@endlink
		anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
		startlink[1]
		endlink[0]
		
		rl [1]href #1
		
		
		
		
		
		
		
		
		
		
		
		
		@bib@innerbibempty
		
		[Nagaosa et al.(2010)Nagaosa, Sinova, Onoda, MacDonald, and Ong]Nagaosa2010
		author author N. Nagaosa, author J. Sinova,
			author S. Onoda, author A. H. MacDonald,  and author N. P. Ong, http://doi.org/10.1103/RevModPhys.82.1539 journal journal Rev. Mod. Phys. volume 82, pages 1539 (year 2010)NoStop
		[Ebert(1996)]Ebert1996
		author author H. Ebert, https://doi.org/10.1088/0034-4885/59/12/003 journal journal Rep. Prog. Phys. volume 59, pages 1665 (year
			1996)NoStop
		[Oppeneer(2001)]Oppeneer2001
		author author P. M. Oppeneer, @noop  title Handbook of Magnetic
					Materials, Vol. volume 13 (publisher Elsevier,
			Amsterdam, year 2001) Chap. chapter
		3NoStop
		[Antonov et al.(2004)Antonov, Harmon, and Yaresko]Antonov2004
		author author V. Antonov, author B. Harmon, 			and author A. Yaresko,		@noop  title Electronic Structure and
					Magneto-Optical Properties of Solids (publisher Kluwer
			Academic Publishers, Dordrecht, year 2004) Chap. chapter 1.4NoStop
		[Smit(1955)]Smit1955
		author author J. Smit, https://doi.org/10.1016/S0031-8914(55)92596-9 journal journal Physica volume 21,			pages 877 (year 1955)NoStop
		[Smit(1958)]Smit1958
		author author J. Smit, https://doi.org/10.1016/S0031-8914(58)93541-9 journal journal Physica volume 24,			pages 39 (year 1958)NoStop
		[Berger(1970)]Berger1970
		author author L. Berger, https://doi.org/10.1103/PhysRevB.2.4559 journal journal Phys. Rev. B volume
				2, pages 4559 (year 1970)NoStop
		[Sundaram and Niu(1999)]Sundaram1999
		author author G. Sundaram and author Q. Niu,		https://link.aps.org/doi/10.1103/PhysRevB.59.14915 journal journal Phys. Rev. B volume
				59, pages 14915 (year 1999)NoStop
		[Yao et al.(2004)Yao,
			Kleinman, MacDonald, Sinova,
			Jungwirth, Wang, Wang, and			Niu]YG-Yao2004
		author author Y. Yao, author L. Kleinman,
			author A. H. MacDonald, author J. Sinova, author
			T. Jungwirth, author
			D.-S. Wang, author
			E. Wang,  and author
			Q. Niu, https://doi.org/10.1103/PhysRevLett.92.037204 journal
			journal Phys. Rev. Lett. volume 92,			pages 037204 (year 2004)NoStop
		[Xiao et al.(2010)Xiao,
			Chang, and Niu]D-Xiao2010
		author author D. Xiao, author M.-C. Chang, 			and author Q. Niu, https://doi.org/10.1103/RevModPhys.82.1959 journal journal Rev. Mod. Phys. volume 82, pages 1959 (year 2010)NoStop
		[Wang et al.(2006)Wang,
			Yates, Souza, and Vanderbilt]XJ-Wang2006
		author author X. Wang, author J. R. Yates,
			author I. Souza,  and author D. Vanderbilt, https://link.aps.org/doi/10.1103/PhysRevB.74.195118 journal
			journal Phys. Rev. B volume 74,			pages 195118 (year 2006)NoStop
		[Wang et al.(2007)Wang,
			Vanderbilt, Yates, and Souza]XJ-Wang2007
		author author X. Wang, author D. Vanderbilt,
			author J. R. Yates,  and			author I. Souza, https://link.aps.org/doi/10.1103/PhysRevB.76.195109 journal
			journal Phys. Rev. B volume 76,			pages 195109 (year 2007)NoStop
		[Roman et al.(2009)Roman,
			Mokrousov, and Souza]Roman2009
		author author E. Roman, author Y. Mokrousov, 			and author I. Souza, 10.1103/PhysRevLett.103.097203 journal journal Phys. Rev. Lett. volume 103, pages 097203 (year 2009)NoStop
		[Fang et al.(2003)Fang,
			Nagaosa, Takahashi, Asamitsu,
			Mathieu, Ogasawara, Yamada,
			Kawasaki, Tokura, and Terakura]Fang2003
		author author Z. Fang, author N. Nagaosa,
			author K. S. Takahashi, author A. Asamitsu, author
			R. Mathieu, author T. Ogasawara, author H. Yamada, author M. Kawasaki, author Y. Tokura,  and author K. Terakura, https://doi.org/10.1126/science.1089408 journal journal Science volume 302, pages
			92 (year 2003)NoStop
		[Mathieu et al.(2004)Mathieu, Asamitsu, Yamada, Takahashi, Kawasaki, Fang, Nagaosa, and Tokura]Mathieu2004
		author author R. Mathieu, author A. Asamitsu,
			author H. Yamada, author K. S. Takahashi, author
			M. Kawasaki, author
			Z. Fang, author N. Nagaosa,  and author Y. Tokura, https://link.aps.org/doi/10.1103/PhysRevLett.93.016602 journal journal Phys. Rev. Lett. volume 93, pages 016602 (year
			2004)NoStop
		[Zeng et al.(2006)Zeng,
			Yao, Niu, and Weitering]CG-Zeng2006
		author author C. Zeng, author Y. Yao, author Q. Niu,  and author
			H. H. Weitering, https://link.aps.org/doi/10.1103/PhysRevLett.96.037204 journal journal Phys. Rev. Lett. volume 96, pages 037204 (year
			2006)NoStop
		[Yao et al.(2007)Yao,
			Liang, Xiao, Niu,
			Shen, Dai, and Fang]YG-Yao2007
		author author Y. Yao, author Y. Liang, author D. Xiao, author
			Q. Niu, author S.-Q. Shen, author X. Dai,  and author Z. Fang, https://link.aps.org/doi/10.1103/PhysRevB.75.020401 journal
			journal Phys. Rev. B volume 75,			pages 020401 (year 2007)NoStop
		[Kubo(1957)]Kubo1957
		author author R. Kubo, https://doi.org/10.1143/JPSJ.12.570 journal
			journal J. Phys. Soc. Jpn. volume
				12, pages 570 (year 1957)NoStop
		[Wang and Callaway(1974)]CS-Wang1974
		author author C. S. Wang and author J. Callaway, 10.1103/physrevb.9.4897 journal
			journal Phys. Rev. B volume 9,			pages 4897 (year 1974)NoStop
		[Herring(1966)]Herring1966
		author author C. Herring, @noop  title Magnetism: A Treatise
					on Modern Theory and Materials, edited by editor
		G. T. Rado and editor
		H. Suhl, Vol. volume IV		(publisher Academic Press, New York, year
		1966)NoStop
		[Sivadas et al.(2016)Sivadas, Okamoto, and Xiao]Sivadas2016
		author author N. Sivadas, author S. Okamoto, 			and author D. Xiao, https://doi.org/10.1103/PhysRevLett.117.267203 journal
			journal Phys. Rev. Lett. volume 117,			eid 267203 (year 2016)NoStop
		[Fan et al.(2017)Fan,
			Wu, Nabok, Hu, Ren, Draxl, and Stroppa]FR-Fan2017
		author author F.-R. Fan, author H. Wu, author D. Nabok, author
			S. Hu, author W. Ren, author C. Draxl,  and author A. Stroppa, https://doi.org/10.1021/jacs.7b04911 journal journal J. Am. Chem. Soc. volume 139, pages 12883 (year
			2017)NoStop
		[Suzuki et al.(2017)Suzuki,
			Koretsune, Ochi, and Arita]Suzuki2017
		author author M.-T. Suzuki, author T. Koretsune,
			author M. Ochi,  and author R. Arita, https://doi.org/10.1103/PhysRevB.95.094406 journal journal Phys. Rev. B volume 95, eid
			094406 (year 2017)NoStop
		[Suzuki et al.(2018)Suzuki,
			Ikeda, and Oppeneer]Suzuki2018
		author author M.-T. Suzuki, author H. Ikeda, 			and author P. M. Oppeneer,		http://dx.doi.org/10.7566/JPSJ.87.041008 journal
			journal J. Phys. Soc. Jpn. volume
				87, pages 041008 (year 2018)NoStop
		[Ohgushi et al.(2000)Ohgushi, Murakami, and Nagaosa]Ohgushi2000
		author author K. Ohgushi, author S. Murakami, 			and author N. Nagaosa, https://doi.org/10.1103/PhysRevB.62.R6065 journal journal Phys. Rev. B volume 62, pages R6065 (year 2000)NoStop
		[Shindou and Nagaosa(2001)]Shindou2001
		author author R. Shindou and author N. Nagaosa, https://doi.org/10.1103/PhysRevLett.87.116801
		journal journal Phys. Rev. Lett. volume 87, eid 116801 (year
			2001)NoStop
		[Hanke et al.(2017)Hanke,
			Freimuth, Blügel, and Mokrousov]Hanke2017
		author author J.-P. Hanke, author F. Freimuth,
			author S. Blügel,  and			author Y. Mokrousov, http://dx.doi.org/10.1038/srep41078 journal journal Sci. Rep. volume 7, pages
			41078 (year 2017)NoStop
		[Shiomi(2018)]Shiomi2018
		author author Y. Shiomi, https://doi.org/10.1063/1.5044594 journal
			journal AIP Adv. volume 8, pages 085018 (year 2018)NoStop
		[Zhou et al.(2016)Zhou,
			Liang, Weng, Chen,
			Yao, Chen, Dong, and			Guo]J-Zhou2016
		author author J. Zhou, author Q.-F. Liang,
			author H. Weng, author
			Y. B. Chen, author
			S.-H. Yao, author Y.-F. Chen, author J. Dong,  and author G.-Y.				Guo, https://link.aps.org/doi/10.1103/PhysRevLett.116.256601 journal journal Phys. Rev. Lett. volume 116, pages 256601 (year
			2016)NoStop
		[Feng et al.()Feng,
			Zhou, Hanke, Guo,
			Blügel, Mokrousov, and Yao]WX-Feng2018
		author author W. Feng, author X. Zhou, author J.-P. Hanke, author
			G.-Y. Guo, author S. Blügel, author Y. Mokrousov,  and author Y. Yao, https://arxiv.org/abs/1811.05803
		journal arXiv: 1811.05803 NoStop
		[Chen et al.(2014)Chen,
			Niu, and MacDonald]H-Chen2014
		journal   author author H. Chen, author Q. Niu,  and author A. H. MacDonald, https://doi.org/10.1103/PhysRevLett.112.017205
		journal journal Phys. Rev. Lett. volume 112, pages 017205 (year
			2014)NoStop
		[Kübler and Felser(2014)]Kubler2014
		author author J. Kübler and author C. Felser, https://doi.org/10.1209/0295-5075/108/67001 journal journal EPL volume 108,			eid 67001 (year 2014)NoStop
		[Guo and Wang(2017)]GY-Guo2017
		author author G.-Y. Guo and author T.-C. Wang,		https://doi.org/10.1103/PhysRevB.96.224415 journal
			journal Phys. Rev. B volume 96,			eid 224415 (year 2017)NoStop
		[Zhang et al.(2017)Zhang,
			Sun, Yang, Železný,
			Parkin, Felser, and Yan]Y-Zhang2017
		author author Y. Zhang, author Y. Sun, author H. Yang, author
			J. Železný, author
			S. P. P. Parkin, author
			C. Felser,  and author
			B. Yan, https://doi.org/10.1103/PhysRevB.95.075128 journal journal Phys. Rev. B volume 95, eid
			075128 (year 2017)NoStop
		[Nakatsuji et al.(2015)Nakatsuji, Kiyohara, and Higo]Nakatsuji2015
		author author S. Nakatsuji, author N. Kiyohara,
			 and author T. Higo, https://doi.org/10.1038/nature15723 journal journal Nature volume 527, pages
			212 (year 2015)NoStop
		[Nayak et al.(2016)Nayak,
			Fischer, Sun, Yan,
			Karel, Komarek, Shekhar,
			Kumar, Schnelle, Kübler,
			Felser, and Parkin]Nayak2016
		author author A. K. Nayak, author J. E. Fischer,
			author Y. Sun, author
			B. Yan, author J. Karel, author A. C.				Komarek, author C. Shekhar, author N. Kumar,
			author W. Schnelle, author J. Kübler, author
			C. Felser,  and author
			S. S. P. Parkin, https://doi.org/10.1126/sciadv.1501870 journal journal Sci. Adv. volume 2, pages
			e1501870 (year 2016)NoStop
		[Kiyohara et al.(2016)Kiyohara, Tomita, and Nakatsuji]Kiyohara2016
		author author N. Kiyohara, author T. Tomita, 			and author S. Nakatsuji, https://doi.org/10.1103/PhysRevApplied.5.064009 journal
			journal Phys. Rev. Appl volume 5,			eid 064009 (year 2016)NoStop
		[Ikhlas et al.(2017)Ikhlas,
			Tomita, Koretsune, Suzuki,
			Nishio-Hamane, Arita, Otani,			and Nakatsuji]Ikhlas2017
		author author M. Ikhlas, author T. Tomita,
			author T. Koretsune, author M.-T. Suzuki, author
			D. Nishio-Hamane, author
			R. Arita, author Y. Otani,  and author S. Nakatsuji, https://doi.org/10.1038/nphys4181
		journal journal Nat. Phys. volume 13, pages 1085 (year
			2017)NoStop
		[Feng et al.(2015)Feng,
			Guo, Zhou, Yao, and			Niu]WX-Feng2015
		author author W. Feng, author G.-Y. Guo,
			author J. Zhou, author
			Y. Yao,  and author
			Q. Niu, https://doi.org/10.1103/PhysRevB.92.144426 journal journal Phys. Rev. B volume 92, pages 144426 (year 2015)NoStop
		[Higo et al.(2018)Higo,
			Man, Gopman, Wu,
			Koretsune, van't Erve, Kabanov, Rees, Li, Suzuki,
			Patankar, Ikhlas, Chien,
			Arita, Shull, Orenstein,			and Nakatsuji]Higo2018
		author author T. Higo, author H. Man, author D. B. Gopman, author
			L. Wu, author T. Koretsune, author O. M. J.				van't Erve, author Y. P.				Kabanov, author D. Rees, author Y. Li, author M.-T. Suzuki, author
			S. Patankar, author
			M. Ikhlas, author C. L. Chien, author R. Arita, author R. D.				Shull, author J. Orenstein,  and author S. Nakatsuji, https://doi.org/10.1038/s41566-017-0086-z journal journal Nat. Photonics volume
				12, pages 73 (year 2018)NoStop
		[Kawamura(2001)]Kawamura2001
		author author H. Kawamura, https://doi.org/10.1139/p01-111 journal
			journal Can. J. Phys volume 79,			pages 1447 (year 2001)NoStop
		[Singh et al.(2018)Singh,
			Zhang, Opahle, Ohmer,
			Yao, and Zhang]Singh2018
		author author H. K. Singh, author Z. Zhang,
			author I. Opahle, author D. Ohmer, author
			Y. Yao,  and author
			H. Zhang, http://dx.doi.org/10.1021/acs.chemmater.8b01618 journal
			journal Chem. Mater. volume 30,			pages 6983 (year 2018)NoStop
		[Bertaut et al.(1968)Bertaut, Fruchart, Bouchaud, and			Fruchart]Bertaut1968
		author author E. F. Bertaut, author D. Fruchart,
			author J. P. Bouchaud,  and			author R. Fruchart, https://doi.org/10.1016/0038-1098(68)90098-7 journal
			journal Solid State Commun. volume
				6, pages 251 (year 1968)NoStop
		[Fruchart and F. Bertaut(1978)]Fruchart1978
		author author D. Fruchart and author E. F. Bertaut, https://doi.org/10.1143/JPSJ.44.781 journal journal J. Phys. Soc. Jpn. volume 44, pages 781 (year 1978)NoStop
		[Gomonaj(1989)]Gomonaj1989
		author author E. V. Gomonaj, https://doi.org/10.1080/01411598908206858 journal journal Phase Trans. volume
				18, pages 93 (year 1989)NoStop
		[Gomonaj and L'vov(1992)]Gomonaj1992
		author author E. V. Gomonaj and author V. A. L'vov, https://doi.org/10.1080/01411599208203457 journal journal Phase Trans. volume
				38, pages 15 (year 1992)NoStop
		[Kim et al.(2003)Kim,
			Chi, Kim, Hur, Lee, and Choi]WS-Kim2003
		author author W. S. Kim, author E. O. Chi,
			author J. C. Kim, author N. H. Hur, author
			K. W. Lee,  and author
			Y. N. Choi, https://doi.org/10.1103/PhysRevB.68.172402 journal journal Phys. Rev. B volume 68, pages 172402 (year 2003)NoStop
		[Lukashev et al.(2008)Lukashev, Sabirianov, and Belashchenko]Lukashev2008
		author author P. Lukashev, author R. F. Sabirianov,  and author K. Belashchenko, https://doi.org/10.1103/PhysRevB.78.184414
		journal journal Phys. Rev. B volume 78, pages 184414 (year
			2008)NoStop
		[Lukashev and Sabirianov(2010)]Lukashev2010
		author author P. Lukashev and author R. F. Sabirianov, https://doi.org/10.1103/PhysRevB.82.094417
		journal journal Phys. Rev. B volume 82, pages 094417 (year
			2010)NoStop
		[Takenaka et al.(2014)Takenaka, Ichigo, Hamada, Ozawa, Shibayama, Inagaki, and			Asano]Takenaka2014
		author author K. Takenaka, author M. Ichigo,
			author T. Hamada, author A. Ozawa, author
			T. Shibayama, author
			T. Inagaki,  and author
			K. Asano, https://doi.org/10.1088/1468-6996/15/1/015009 journal
			journal Sci. Technol. Adv. Mater. volume
				15, pages 015009 (year 2014)NoStop
		[Deng et al.(2015)Deng,
			Sun, Wu, Huang, Yan, Shi, Malik, Lu,
			Wang, Huang, Li, and			Wang]SH-Deng2015
		author author S. Deng, author Y. Sun, author H. Wu, author
			Q. Huang, author J. Yan, author K. Shi, author M. I. Malik,
			author H. Lu, author
			L. Wang, author R. Huang, author L. Li,  and author C. Wang, https://doi.org/10.1021/cm504702m
		journal journal Chem. Mater. volume 27, pages 2495 (year
			2015)NoStop
		[Zemen et al.(2017a)Zemen, Gercsi,			and Sandeman]Zemen2017a
		author author J. Zemen, author Z. Gercsi, 			and author K. G. Sandeman,		https://doi.org/10.1103/PhysRevB.96.024451 journal
			journal Phys. Rev. B volume 96,			pages 024451 (year 2017a)NoStop
		[Sun et al.(2012)Sun,
			Wang, Huang, Guo,
			Chu, Arai, and Yamaura]Y-Sun2012
		author author Y. Sun, author C. Wang, author Q. Huang, author
			Y. Guo, author L. Chu, author M. Arai,  and author K. Yamaura, https://doi.org/10.1021/ic300978x journal
			journal Inorg. Chem. volume 51,			pages 7232 (year 2012)NoStop
		[Matsunami et al.(2014)Matsunami, Fujita, Takenaka, and			Kano]Matsunami2014
		author author D. Matsunami, author A. Fujita,
			author K. Takenaka,  and			author M. Kano, http://dx.doi.org/10.1038/nmat4117 journal journal Nat. Mater. volume 14, pages
			73 (year 2014)NoStop
		[Shi et al.(2016)Shi,
			Sun, Yan, Deng, Wang, Wu, Hu, Lu,
			Malik, Huang, and Wang]KW-Shi2016
		author author K. Shi, author Y. Sun, author J. Yan, author
			S. Deng, author L. Wang, author H. Wu, author P. Hu, author H. Lu, author
			M. I. Malik, author
			Q. Huang,  and author
			C. Wang, https://doi.org/10.1002/adma.201600310 journal journal Adv. Mater. volume 28, pages
			3761 (year 2016)NoStop
		[Zemen et al.(2017b)Zemen, Mendive-Tapia, Gercsi, Banerjee,
			Staunton, and Sandeman]Zemen2017
		author author J. Zemen, author E. Mendive-Tapia, author Z. Gercsi, author R. Banerjee,
			author J. B. Staunton,  and			author K. G. Sandeman, https://doi.org/10.1103/PhysRevB.95.184438 journal journal Phys. Rev. B volume 95, pages 184438 (year 2017b)NoStop
		[Mochizuki et al.(2018)Mochizuki, Kobayashi, Okabe, and			Yamamoto]Mochizuki2018
		author author M. Mochizuki, author M. Kobayashi, author R. Okabe, 			and author D. Yamamoto, https://doi.org/10.1103/PhysRevB.97.060401 journal journal Phys. Rev. B volume 97, pages 060401 (year 2018)NoStop
		[Stokes et al.()Stokes,
			Hatch, and Campbell]Stokes
		author author H. T. Stokes, author D. M. Hatch, 			and author B. J. Campbell,		http://stokes.byu.edu/iso/isotropy.php journal
			ISOTROPY Software Suite NoStop
		[Stokes and Hatch(2005)]Stokes2005
		journal   author author H. T. Stokes and author D. M. Hatch, https://doi.org/10.1107/S0021889804031528 journal journal J. Appl. Cryst volume 38, pages 237 (year 2005)NoStop
		[Seemann et al.(2015)Seemann, Ködderitzsch, Wimmer, and			Ebert]Seemann2015
		author author M. Seemann, author D. Ködderitzsch, author S. Wimmer,  and author H. Ebert, https://doi.org/10.1103/PhysRevB.92.155138 journal journal Phys. Rev. B volume
				92, eid 155138 (year 2015)NoStop
		[Železný et al.(2017)Železný, Gao, Manchon,
			Freimuth, Mokrousov, Zemen,
			Mašek, Sinova, and Jungwirth]Zelezny2017a
		author author J. Železný, author H. Gao,
			author A. Manchon, author F. Freimuth, author
			Y. Mokrousov, author
			J. Zemen, author J. Mašek, author J. Sinova,  and author T. Jungwirth, https://doi.org/10.1103/PhysRevB.95.014403 journal journal Phys. Rev. B volume 95, eid
			014403 (year 2017)NoStop
		[Železný()]Zelezny2018a
		author author J. Železný, https://bitbucket.org/zeleznyj/linear-response-symmetry journal https://bitbucket.org/zeleznyj/linear-response-symmetry		NoStop
		[Umetsu et al.(2006)Umetsu,
			Sakuma, and Fukamichi]Umetsu2006
		journal   author author R. Y. Umetsu, author A. Sakuma,  and author K. Fukamichi, http://dx.doi.org/10.1063/1.2236103
		journal journal Appl. Phys. Lett. volume 89, pages 052504 (year
			2006)NoStop
		[Kota et al.(2008)Kota,
			Tsuchiura, and Sakuma]Kota2008
		author author Y. Kota, author H. Tsuchiura, 			and author A. Sakuma, http://dx.doi.org/10.1109/TMAG.2008.2001678 journal journal IEEE Trans. Magn. volume 44, pages 3131 (year 2008)NoStop
		[Szunyogh et al.(2009)Szunyogh, Lazarovits, Udvardi,
			Jackson, and Nowak]Szunyogh2009
		author author L. Szunyogh, author B. Lazarovits, author L. Udvardi,
			author J. Jackson,  and			author U. Nowak, https://doi.org/10.1103/PhysRevB.79.020403 journal journal Phys. Rev. B volume 79, pages 020403 (year 2009)NoStop
		[Nagamiya et al.(1982)Nagamiya, Tomiyoshi, and Yamaguchi]Nagamiya1982
		author author T. Nagamiya, author S. Tomiyoshi,
			 and author Y. Yamaguchi,		http://doi.org/10.1016/0038-1098(82)90159-4 journal
			journal Solid State Commun. volume
				42, pages 385 (year 1982)NoStop
		[Tomiyoshi and Yamaguchi(1982)]Tomiyoshi1982
		author author S. Tomiyoshi and author Y. Yamaguchi, http://dx.doi.org/10.1143/JPSJ.51.2478 journal journal J. Phys. Soc. Jpn. volume 51, pages 2478 (year
			1982)NoStop
		[Kahn et al.(1969)Kahn,
			Pershan, and Remeika]Kahn1969
		author author F. J. Kahn, author P. S. Pershan, 			and author J. P. Remeika,		https://doi.org/10.1103/PhysRev.186.891 journal
			journal Phys. Rev. volume 186,			pages 891 (year 1969)NoStop
		[Guo and Ebert(1994)]GY-Guo1994
		author author G. Y. Guo and author H. Ebert,		10.1103/PhysRevB.50.10377 journal journal Phys. Rev. B volume 50, pages 10377 (year 1994)NoStop
		[Guo and Ebert(1995)]GY-Guo1995
		author author G. Y. Guo and author H. Ebert,		http://dx.doi.org/10.1103/physrevb.51.12633 journal
			journal Phys. Rev. B volume 51,			pages 12633 (year 1995)NoStop
		[Reim and Schoenes()]Reim1990
		author author W. Reim and author J. Schoenes, @noop  title Handbook of Magnetic
					Materials, edited by editor E. P.			Wohlfarth and editor K. H. J.			Buschow, Vol. volume 5 (publisher
		Elsevier, New York, year 1990) Chap. chapter
		2NoStop
		[Fang et al.(2018)Fang,
			Wu, Zhu, and Guo]YM-Fang2018
		author author Y. Fang, author S. Wu, author Z.-Z. Zhu,  and author G.-Y. Guo, https://doi.org/10.1103/PhysRevB.98.125416 journal journal Phys. Rev. B volume 98, pages 125416 (year 2018)NoStop
		[Kresse and Furthmüller(1996)]Kresse1996a
		author author G. Kresse and author J. Furthmüller, 10.1016/0927-0256(96)00008-0
		journal journal Comput. Mater. Sci. volume 6, pages 15 (year
			1996)NoStop
		[Blöchl(1994)]Bloechl1994
		author author P. E. Blöchl, http://dx.doi.org/10.1103/physrevb.50.17953
		journal journal Phys. Rev. B volume 50, pages 17953 (year
			1994)NoStop
		[Perdew et al.(1996)Perdew,
			Burke, and Ernzerhof]Perdew1996
		author author J. P. Perdew, author K. Burke, 			and author M. Ernzerhof, http://dx.doi.org/10.1103/physrevlett.77.3865 journal
			journal Phys. Rev. Lett. volume 77,			pages 3865 (year 1996)NoStop
		[Mostofi et al.(2008)Mostofi, Yates, Lee, Souza,
			Vanderbilt, and Marzari]Mostofi2008
		author author A. A. Mostofi, author J. R. Yates,
			author Y.-S. Lee, author I. Souza, author
			D. Vanderbilt,  and author
			N. Marzari, http://dx.doi.org/10.1016/j.cpc.2007.11.016 journal journal Comput. Phys. Commun. volume 178,			pages 685 (year 2008)NoStop
	
	
