# Above-ordering-temperature large anomalous Hall effect in a triangular-lattice magnetic semiconductor

## Abstract

While anomalous Hall effect (AHE) has been extensively studied in the past, efforts for realizing large Hall response have been mainly limited within intrinsic mechanism. Lately, however, a theory of extrinsic mechanism has predicted that magnetic scattering by spin cluster can induce large AHE even above magnetic ordering temperature, particularly in magnetic semiconductors with low carrier density, strong exchange coupling, and finite spin chirality. Here, we find out a new magnetic semiconductor EuAs, where Eu

^{2+}ions with large magnetic moments form distorted triangular lattice. In addition to colossal magnetoresistance, EuAs exhibits large AHE with an anomalous Hall angle of 0.13 at temperatures far above antiferromagnetic ordering. As also demonstrated by model calculations, observed AHE can be explained by the spin cluster scattering in a hopping regime. Our findings shed light on magnetic semiconductors hosting topological spin textures, developing a field targeting diluted carriers strongly coupled to noncoplanar spin structures.## INTRODUCTION

The anomalous Hall effect (AHE) is one of the most fundamental phenomena in magnetic conductors (

*1*). In ferromagnets, the mechanisms of AHE are roughly classified into two groups: intrinsic mechanism related to the Berry curvature (*2*) and extrinsic mechanism by asymmetric scattering (*3*,*4*). The intrinsic mechanism is related to the Berry curvature of electronic bands, which appears as a consequence of spin-orbit interaction. A similar effect also occurs from spin Berry phase (*5*,*6*), in which case the Hall effect is related to the scalar spin chirality*S**· (*_{i}*S**×*_{j}*S**) of adjacent three spins; a related theory shows that the Hall conductivity is proportional to the scalar spin chirality in the weak coupling limit in dirty conductors (*_{k}*7*). Large Hall current is produced by the intrinsic mechanism especially in the metallic conduction regime, and thus, it has been extensively studied for many years (*8*–*11*). In recent years, large Hall angles exceeding 0.1 have been found in a variety of magnetic topological semimetals, which host topological singularities of the Berry curvature in momentum space (*12*–*18*).In contrast to the intrinsic mechanism, asymmetric scattering coupled with spin-orbit interaction is treated as small perturbation in the extrinsic mechanism, resulting in relatively small anomalous Hall angles of 10

^{−2}∼ 10^{−3}. Lately, however, it has been theoretically proposed that large AHE is induced by skew scattering by multiple spins with finite spin chirality (*19*,*20*). In this mechanism, anomalous Hall angles, roughly two orders of magnitude larger than in the case of single-spin scatterers, have been theoretically derived, especially in the limit of strong coupling between charge carriers and localized spins (*20*). Furthermore, this AHE can emerge even above the magnetic ordering temperature reflecting thermal average of the spin chirality, while the intrinsic AHE based on the Berry phase disappears above the ordering temperature. Actually, AHE above the magnetic ordering temperature has been observed in a metal hosting skyrmions (*21*,*22*), which has been ascribed to the skew scattering by fluctuating but locally correlated spins originally forming the skyrmion lattice below the ordering temperature. Moreover, large AHE ascribed to the spin cluster scattering has been observed in a Kagome metal even without magnetic ordering (*23*). In this theory (*20*), on the other hand, it turns out that anomalous Hall angles rapidly increase with decrease in charge carriers and also with increase in its exchange coupling to localized spins. In addition, quantization of AHE can also be expected, once the low-density charge carrier corresponds to a topological spin configuration one by one (*24*,*25*). Therefore, for realizing large Hall response by this mechanism, it is desirable to design magnetic conductors with more diluted carriers and stronger exchange coupling, while the study of AHE in the hopping conduction regime has been limited to conventional magnetic semiconductors so far (*26*–*28*).Here, we take note of a novel magnetic semiconductor EuAs. Regarding this compound, only its crystal structure has been reported in the 1970s (

*29*–*31*). As illustrated in Fig. 1A, EuAs has a unique hexagonal structure (Na_{2}O_{2}type, space group: $\mathit{P}\stackrel{\u0304}{6}2\mathit{m}$) (*30*,*31*), where Eu forms a distorted triangular lattice with geometrical frustration, different from other europium monochalcogenides and rare earth monopnictides with cubic structure (*32*,*33*). Here, we report colossal magnetoresistance (CMR) and nonmonotonic AHE with large Hall angles observed even far above the Néel temperature. As also demonstrated by theoretical calculations, this large AHE can be explained by spin cluster skew scattering in the hopping regime.## RESULTS

### Structural characterization of EuAs films

Single-crystalline EuAs films were grown on lattice-matched Al

_{2}O_{3}substrates by molecular beam epitaxy. As shown in Fig. 1 (A and B), an Eu atom is located at the center of EuAs_{6}triangular prisms, which are alternately stacked along the*c*axis. A dimer expressed by [As-As]^{4−}is formed with short As─As bonds as in isostructural SrAs (*30*,*31*), and thus, Eu is expected to be divalent (Eu^{2+},*S*= 7/2). Corresponding to this dimerization, the Eu triangular lattice is distorted and composed of two large (blue) and small (red) regular triangles, as shown in Fig. 1A (insets) (see also fig. S4). X-ray diffraction θ − 2θ scan in Fig. 1C shows sharp peaks from the EuAs {001} planes without any impurity ones. As confirmed in atomically resolved scanning transmission electron microscopy image in Fig. 1D, the EuAs film is epitaxially grown with forming a sharp heterointerface to the Al_{2}O_{3}substrate. Further structural characterization was performed for examining the in-plane epitaxial relation and the Eu valence (Eu^{2+}) state (see note S1).### Colossal magnetoresistance

Figure 2 summarizes fundamental transport and magnetic properties of EuAs films. Longitudinal resistivity in Fig. 2A exhibits a semiconducting temperature dependence with a kink at

*T*_{N}= 23.0 K, which is interpreted as an antiferromagnetic (AFM) phase transition as detailed below. With applying the magnetic field, the resistivity markedly decreases by several orders of magnitude at the lowest temperature. This large negative magnetoresistance, which can be called CMR, appears even at far higher temperatures than*T*_{N}. CMR is confirmed also in the field sweep shown in Fig. 2B, where the drop of the resistivity at 1.4 K suddenly stops at*B*_{s}= 14.5 T. Magnetization in Fig. 2C saturates at the same field, exhibiting a transition to the forced ferromagnetic (FM) phase with a full magnetic moment of about 7μ_{B}. While temperature dependence of both the out-of-plane and in-plane magnetizations shows a sharp kink at*T*_{N}, the in-plane one is largely suppressed below*T*_{N}compared to the out-of-plane one, suggesting that Eu^{2+}Heisenberg spins are antiferromagnetically ordered on the*a-b*triangular lattice plane.Looking at the magnetic structures more closely, neutron diffraction experiments were also performed for a EuAs film. As shown in Fig. 2D, magnetic reflections indexed by wave vectors of

**= (1/2,1/2,0) and its equivalents were observed on the (***q**H*,*K*, 0) plane, which indicates AFM spin correlation on the triangular lattice plane. Temperature dependence of these reflections was measured using a polarized neutron beam as shown in Fig. 2E. The neutron spin polarization at the sample position was set parallel to the*c*axis, and so, non–spin-flip (NSF) and spin-flip (SF) scattering intensities are proportional to squares of Fourier-transformed spin components parallel to the*c*axis and those perpendicular to the*c*axis and*q*vector, respectively. We found that the magnetic scattering below*T*_{N}is dominated by the SF scattering, indicating that the magnetic moments are confined in the triangular lattice planes. A minimal model to explain the present results is a collinear AFM order, in which magnetic moments are pointing perpendicular to both the*c*axis and*q*vector. However, taking into account the symmetry of the crystal structure, it should be allowed to have spin components parallel to the*q*vector, which leads to a noncollinear spin arrangement on the triangular lattice plane. Consideration of Dzyaloshinskii-Moriya interaction, whose vector points along the*c*axis for adjacent Eu spins on the triangular lattice plane, also leads the same consequence. The obtained magnetic phase diagram of EuAs is presented in Fig. 2F. With increasing the out-of-plane magnetic field, the coplanar spin configuration suggested by the neutron scattering experiment is expected to be transformed into a noncoplanar one by canting and then eventually into the fully polarized one.It is naturally understood that CMR observed in the magnetic semiconductor EuAs is caused by diluted carriers coupled with the Eu

^{2+}localized spins over a wide range of temperatures. In the double exchange model for strong coupling between charge carriers and localized spins, the transfer integral*t*for hopping carriers is expressed by*t*∝ cos (θ/2) with the angle θ between adjacent localized spins (*34*,*35*), explaining metallic conduction in the spin polarized state. In simple FM systems such as La_{1–x}Sr*MnO*_{x}_{3}(*36*–*38*) and EuO (*39*), fluctuating spins near the FM transition are forcedly polarized by the applied magnetic field, and thus, CMR appears mainly above the Curie temperature. In the present case of AFM systems, on the other hand, CMR appears also below*T*_{N}as observed in NaCr_{2}O_{4}(*40*) and EuTe (*41*), because antiferromagnetically ordered spins are gradually polarized by the applied magnetic field. In EuAs, CMR begins to emerge at temperatures far above*T*_{N}, suggesting that the in-plane AFM spin fluctuation evolves from such high temperatures on the frustrated triangular lattice.### Anomalous hall signal above Néel temperature

In addition to CMR, EuAs exhibits unconventionally large AHE for a magnetic semiconductor. As shown in Fig. 3, Hall resistivity is almost proportional to

*B*at 300 K, but a hump starts to appear at about 15 T, and it significantly increases with lowering temperature. It shows a maximum approximately at 70 K, a temperature far higher than*T*_{N}. With further lowering temperature toward*T*_{N}, the hump shifts to lower fields, and then below*T*_{N}, the hump is partially seen in the canted AFM phase below*B*_{s}and the Hall resistivity above*B*_{s}becomes almost proportional to*B*again. Apparently, this nonmonotonic Hall signal is not proportional to the field induced magnetization (*M*) as represented by the Brillouin function. In addition, the*M*-proportional component, estimated by linear extrapolation of the forced FM phase (10 K) data to zero field, is negligibly small. As seen in the above temperature and field dependences, this anomalous Hall signal is enhanced in the region where the Eu^{2+}spins on the triangular lattice are forcedly canted but not yet fully polarized. This implies that the observed AHE is mainly caused by noncoplanar spin texture or local fluctuations with finite spin chirality under the out-of-plane magnetic field.The with ordinary Hall conductivity σandfrom the total Hall conductivity (for details, see the Supplementary Materials). Here,

*M*-nonproportional anomalous Hall component is extracted using the following equation$${\mathrm{\sigma}}_{\mathit{yx}}={\mathrm{\sigma}}_{\text{OHE}}+{\mathrm{\sigma}}_{\text{AHE},\mathit{M}}+{\mathrm{\sigma}}_{\text{AHE},\text{non}\mathit{M}}$$

(1)

_{OHE},*M*-proportional anomalous Hall conductivity σ_{AHE, M}, and*M*-nonproportional anomalous Hall conductivity σ_{AHE, nonM}. σ*is obtained by inverting the ρ*_{yx}*and ρ*_{xx}*matrix, and σ*_{yx}_{AHE, nonM}can be estimated by subtracting σ_{OHE}and σ_{AHE, M}defined by$${\mathrm{\sigma}}_{\text{OHE}}=\frac{-\frac{1}{\mathit{pe}}\mathit{B}}{{\mathrm{\rho}}_{\mathit{xx}}^{2}+{\left(\frac{1}{\mathit{pe}}\mathit{B}\right)}^{2}}$$

(2)

$${\mathrm{\sigma}}_{\text{AHE},\mathit{M}}=-\mathit{A}{\mathrm{\sigma}}_{\mathit{xx}}^{\mathrm{\alpha}}\mathit{M}$$

(3)

*p*denotes the hole carrier density,*A*denotes a constant independent of magnetic field and temperature, and α denotes a scaling factor relating σ*and σ*_{xx}_{AHE, M}(*28*). σ_{AHE, M}is calculated using α = 1.6 and*A*determined in the forced FM phase at 10 K, and σ_{OHE}is determined on the assumption that it is the only contribution to the remaining component at 56 T (see also note S2). As confirmed in the Hall resistivity components converted from σ_{OHE}, σ_{AHE, M}, and σ_{AHE, nonM}, the*M*-nonproportional component is more dominant than the*M*-proportional one. Although the spin fluctuation may not be fully suppressed and σ_{AHE, nonM}may still slightly contribute at a high field of 56 T, this assumption gives the most conservative estimate of σ_{AHE, nonM}. The carrier density obtained from σ_{OHE}is nearly constant with temperature at*p*∼ 3 × 10^{17}cm^{−3}(see fig. S6). It is also confirmed that the nonmonotonic Hall signal cannot be explained by a multicarrier model (see fig. S7).### AHE in the hopping regime

Figure 4A maps ρ

_{AHE, nonM}on the magnetic field–temperature phase diagram. ρ_{AHE, nonM}remains over a wide high-temperature region spread from the AFM critical point, where the Eu^{2+}spins on the triangular lattice are expected to form local noncoplanar spin structures or fluctuations with finite spin chirality. Despite the hopping conduction regime, anomalous Hall angle defined by ρ_{AHE, nonM}/ρ*rises up to 0.13 at 70 K. This is a very large value for the extrinsic mechanism and also comparable to those recently achieved by the intrinsic mechanism (*_{xx}*12*–*17*). At lower temperatures, ρ_{AHE, nonM}is largely suppressed with approaching the forced FM phase. At higher temperatures, on the other hand, ρ_{AHE, nonM}also gradually decreases with elevating temperature, reflecting more randomly oriented spin states.To discuss possible origins of the observed large AHE, we consider how the local spin correlation with finite spin chirality affects electron conduction in the hopping regime. Electron conduction in this regime is described by phonon-assisted hopping of electrons localized on a site, as schematically shown in Fig. 4B. The electron trajectory skews when they are coupled strongly to the underlying localized moments (see also the Materials and Methods). Figure 4 (C and D) shows typical temperature dependence of σ

*and σ*_{xx}_{AHE, cl. skew}calculated assuming constant spin chirality, with comparing to the measured ones in Fig. 4 (E and F). A weak temperature dependence of σ*characteristic in the hopping conduction is well reproduced in this model (Fig. 4, C and E). On the other hand, the calculation shows that σ*_{xx}_{AHE, cl. skew}rapidly increases with elevating temperature (Fig. 4D) and then decreases above*T*/ω_{ph}∼ 5, while σ_{AHE, nonM}in EuAs reaches a maximum at about 70 K (Fig. 4F). This indicates that the local spin chirality in EuAs is gradually reduced by thermal fluctuation above*T*_{N}and also suggests future possibility that anomalous Hall angles even exceeding the present value could be achieved in other semiconductor systems that host finite spin chirality at much higher temperatures.## DISCUSSION

In the hopping conduction regime, to be precise, the extrinsic mechanism cannot be distinguished from the intrinsic one because the band picture of magnetotransport is not preserved anymore. On the other hand, its temperature dependence greatly differs from the one in the intrinsic mechanism. In the spin cluster skew scattering, thermal average of the spin chirality 〈

*S*· (_{i}*S*×_{j}*S*)〉, not the spin chirality of averaged spins 〈_{k}*S*〉 · (〈_{i}*S*〉 × 〈_{j}*S*〉) as in the intrinsic mechanism, contributes to the anomalous Hall conductivity, and this remains finite even above the magnetic ordering temperature (_{k}*19*). This is consistent with our observation that in EuAs, the*M*-nonproportional AHE emerges up to high temperatures much above*T*_{N}, while the intrinsic AHE disappears above the magnetic ordering temperature.It is also instructive to mention that

*M*-nonproportional AHE above the magnetic ordering temperature has been reported for La_{1–x}Sr*MnO*_{x}_{3}(*36*) and EuTe (*41*) in the past, which also exhibit CMR in the hopping regime. While this AHE has been also ascribed to thermally fluctuating spins forming local noncoplanar structures near the magnetic ordering temperature (*36*), their Hall angles were very small due to simple FM or AFM ordering without frustration at the ground state. In EuAs, noncoplanar fluctuation may survive much above the magnetic ordering temperature, largely enhancing the spin cluster skew scattering. In this context, triangular-lattice magnetic semiconductors forming magnetic sublattices with finite spin chirality (*42*) may be promising as further targets.In summary, we have studied magnetotransport of novel magnetic semiconductor EuAs with large Heisenberg Eu

^{2+}spin moments on the distorted triangular lattice. EuAs exhibits CMR even at far higher temperatures than*T*_{N}, indicating strong coupling between the localized spins and diluted carriers and also AFM spin fluctuations evolving on the frustrated lattice. In the Hall transport, nonmonotonic AHE with large Hall angles is observed up to high temperatures much above*T*_{N}. Considering that the signal appears centered at the paramagnetic phase, skew scattering by multiple spins with finite spin chirality is a possible origin of the large AHE observed in the hopping regime, as also supported by the model calculations of polaron motion on the noncoplanar spin texture. Our findings pave the way for studying magnetic semiconductors hosting topological spin textures. Emergent magnetotransport that originates in low-density charge carriers strongly coupled to short-period noncoplanar spin structures includes large nonmonotonic AHE and its quantization (*24*,*25*), which are expected to be markedly controlled by semiconductor techniques such as electrostatic gating and chemical doping.## MATERIALS AND METHODS

### Epitaxial film growth

EuAs films were grown in an EpiQuest RC1100 MBE chamber on single-crystalline (0001) Al

_{2}O_{3}substrates. Al_{2}O_{3}has a trigonal structure, and the lattice mismatch between EuAs and Al_{2}O_{3}is only 1.1%, as illustrated in Fig. 1B. The molecular beams were provided from a conventional Knudsen cell containing Eu (3N, Nippon Yttrium Co. Ltd.) and an MBE-Komponenten valved cracker source containing As (7N5, Furukawa Denshi Co. Ltd.), respectively. The reservoir temperature of the cracker source was set to 950°C to sublime arsenic as As_{2}. The growth temperature was set to 1000°C, and the beam equivalent pressures, measured by an ionization gauge, were set to 3 × 10^{−5}Pa for Eu and 1.2 × 10^{−4}Pa for As_{2}during the growth. The film thicknesses were typically designed at 100 nm for structural characterization, 400 nm for magnetotransport measurements, and 1 μm for magnetization measurements and neutron diffraction experiments. The growth rate was about 0.4 Å̊/s.### Magnetotransport measurements

Longitudinal resistivity ρ

*and Hall resistivity ρ*_{xx}*were measured with a standard four probe method. Nickel and gold electrode was deposited on terminals of the sample for better ohmic contact, and then, SiO*_{yx}_{2}capping layer was entirely deposited for preventing oxidization of EuAs. Aluminum wire was connected to the terminals by using an ultrasonic bonding machine, and these connections were reinforced by applying silver paste. Transport measurements up to 56 T were performed using a nondestructive pulsed magnet with a pulse duration of 37 ms at the International MegaGauss Science Laboratory at the Institute for Solid State Physics, The University of Tokyo. Longitudinal resistivity ρ*below 14 T was also measured using a Quantum Design Physical Properties Measurement System.*_{xx}### Magnetization measurements

Magnetization curves up to 27 T were taken for 1-μm-thick EuAs films using a nondestructive pulsed magnet with a pulse duration of 4 ms at the International MegaGauss Science Laboratory in the Institute for Solid State Physics, The University of Tokyo. Temperature dependence at 0.1 T was measured using a superconducting quantum interference device magnetometer in a Quantum Design Magnetic Property Measurement System.

### Neutron diffraction experiments

Unpolarized and polarized neutron scattering experiments were performed at the small- and wide-angle neutron scattering instrument TAIKAN (BL15) in Materials and Life Science Experimental Facility in Japan Proton Accelerator Research Complex. An EuAs film with a thickness of 1 μm was loaded in an aluminum cell with He gas and then installed in a closed-cycle 4He refrigerator and a horizontal field cryomagnet for unpolarized and polarized experiments, respectively. Time-of-flight neutron diffraction patterns were taken using a pulsed incident neutron beam with a wavelength range from 0.7 to 7.7 Å̊. For the polarized neutron scattering experiment, the incident neutron beam was spin polarized by a supermirror polarizer. To maintain spin polarization of the incident neutrons, we applied a magnetic field of 0.05 T parallel to the

*c*axis at the sample position. We also used a supermirror to analyze the spin state of the scattered neutrons, in which wavelength was 3.25 Å̊ when measuring magnetic Bragg reflections in the first Brillouin zone. The polarization direction of the incident neutrons was set to be parallel or antiparallel to the*c*axis using a spin flipper. We then measured intensities of the magnetic Bragg reflection for SF and NSF scattering processes. The beam polarization for neutrons with the wavelength of 3.25 Å̊ was 0.92. The SF and NSF intensities shown in Fig. 2E have been corrected taking into account the imperfection of the beam polarization.### Theoretical calculation of hall current

The Hall conductivity was calculated according to the method developed for polaron motion (

*26*,*43*). Later, this method was used to evaluate the spin chirality–related Hall effect in the CMR manganites (*36*). Following these studies, we consider a double-exchange model coupled to phonons$$\mathit{H}={\displaystyle \sum _{\mathit{i},\mathit{j}}}{\mathit{t}}_{\mathit{ij}}{\mathit{c}}_{\mathit{i}}^{\u2020}{\mathit{c}}_{\mathit{j}}+{\displaystyle \sum _{\mathit{q}}}{\mathrm{\omega}}_{\mathit{q}}{\mathit{a}}_{\mathit{q}}^{\u2020}{\mathit{a}}_{\mathit{q}}+{\displaystyle \sum _{\mathit{i},\mathit{q}}}{\mathit{M}}_{\mathit{q}}{\mathit{e}}^{\mathit{i}\mathit{q}\xb7{\mathit{r}}_{\mathit{i}}}{\mathit{c}}_{\mathit{i}}^{\u2020}{\mathit{c}}_{\mathit{i}}({\mathit{a}}_{\mathit{q}}+{\mathit{a}}_{-\mathit{q}}^{\u2020})-{\displaystyle \sum _{\mathit{i}}}\mathit{e}\mathit{E}\xb7{\mathit{r}}_{\mathit{i}}{\mathit{c}}_{\mathit{i}}^{\u2020}{\mathit{c}}_{\mathit{i}}$$

(4)

Here,

*c*$\left({\mathit{c}}_{\mathit{i}}^{\u2020}\right)$ is the annihilation (creation) operator of an electron at site_{i}*i*,*a*_{q}$\left({\mathit{a}}_{\mathit{q}}^{\u2020}\right)$ is the annihilation (creation) operator for phonons,*M*_{q}is the electron-phononcoupling, and*e***·***E**r**is the electric potential due to the external electric field*_{i}**= (***E**E*,0). Here, the electrons are effectively spinless, as the electron spins are polarized along the underlying magnetic texture. Instead, the effect of magnetic correlation appears in the hopping integral*t*(_{ij}*S**,*_{i}*S**) = cos (θ*_{j}*/2) cos (θ*_{i}*/2) + sin (θ*_{j}*/2) sin (θ*_{i}*/2)*_{j}*e*^{i(ϕj − ϕi)}(*34*,*35*). In what follows, we assume only nearest-neighbor hopping, and the phonons are Einstein phonons, i.e.,*M*_{q}=*M*and ω_{q}= ω_{ph}.In the polaron theory, the leading order in electron conduction appears from the second order in hopping (whereThis term gives the longitudinal currentwhen the electron field is along the

*43*)$${\mathit{J}}_{\mathit{i}\to \mathit{j}}=-\mathit{e}\frac{{\mid {\mathit{t}}_{\mathit{ij}}\mid}^{2}{\mathit{E}}_{\mathit{x}}({\mathit{r}}_{\mathit{j}}^{\mathit{x}}-{\mathit{r}}_{\mathit{i}}^{\mathit{x}})}{\mathrm{\u0127}{\mathrm{\omega}}_{\text{ph}}^{2}}2{\mathit{F}}_{0}(\frac{\mathit{M}}{{\mathrm{\omega}}_{\text{ph}}},\frac{{\mathrm{V}}_{0}}{{\mathrm{\omega}}_{\text{ph}}})$$

(5)

$${\mathit{F}}_{0}(\mathit{x},\mathit{y})={\displaystyle {\int}_{-\infty}^{\infty}}\mathit{d}{\mathrm{\tau}}^{\prime}\mathit{i}{\mathrm{\tau}}^{\prime}{\mathit{e}}^{-{\mathit{y}}^{2}\tilde{\mathrm{\varphi}}({\mathrm{\tau}}^{\prime})-{\mathit{x}}^{2}{{\mathrm{\tau}}^{\prime}}^{2}}$$

(6)

$$\stackrel{~}{\mathrm{\varphi}}(\mathrm{\tau})=2[1-\text{cos}(\mathrm{\tau})][2{\mathit{n}}_{\text{ph}}(\frac{{\mathit{k}}_{\mathrm{B}}\mathit{T}}{{\mathrm{\omega}}_{\text{ph}}})+1]+\mathit{i}\text{sin}(\mathrm{\tau})$$

(7)

$${\mathit{n}}_{\text{ph}}(\mathit{x})=\frac{1}{\text{exp}(1/\mathit{x})-1}$$

(8)

$${J}_{x}=\frac{1+\mathrm{\langle}{S}_{i}\xb7{S}_{j}\mathrm{\rangle}}{2}\frac{\sqrt{3}{e}^{2}{\mathit{nt}}^{2}E}{\mathrm{\u0127}{\mathrm{\omega}}_{\text{ph}}^{2}}{F}_{0}(\frac{M}{{\mathrm{\omega}}_{\text{ph}}},\frac{{V}_{0}}{{\mathrm{\omega}}_{\text{ph}}})$$

(9)

*x*axis. Here, 〈*S*·_{i}*S*〉 is the thermal average of the dot product of spins on_{j}*i*and*j*sites.The Hall current appears from the cubic order in hopping. The electric current flowing from site with and

*i*to*j*reads$$\begin{array}{c}{J}_{i\to j}=\frac{4\mathit{eE}({r}_{j}^{x}-{r}_{i}^{x})}{\u0127}\sum _{m}\frac{\text{Re}({t}_{\mathit{ij}}{t}_{\mathit{jm}}{t}_{\mathit{mi}})}{{\mathrm{\omega}}_{\text{ph}}^{3}}\\ \text{Re}\left[{F}_{1}\right(\frac{M}{{\mathrm{\omega}}_{\text{ph}}},\frac{{V}_{0}}{{\mathrm{\omega}}_{\text{ph}}})-{F}_{2}(\frac{M}{{\mathrm{\omega}}_{\text{ph}}},\frac{{V}_{0}}{{\mathrm{\omega}}_{\text{ph}}}\left)\right]\\ +\sum _{m}\frac{2\mathit{eE}(2{r}_{m}^{x}-{r}_{i}^{x}-{r}_{j}^{x})}{\u0127}\frac{\text{Im}({t}_{\mathit{ij}}{t}_{\mathit{jm}}{t}_{\mathit{mi}})}{{\mathrm{\omega}}_{\text{ph}}^{3}}\text{Im}\left[{F}_{2}\right(\frac{M}{{\mathrm{\omega}}_{\text{ph}}},\frac{{V}_{0}}{{\mathrm{\omega}}_{\text{ph}}}\left)\right]\end{array}$$

(10)

*V*_{0}, where$${\mathit{F}}_{1}(\mathit{x},\mathit{y})={\displaystyle {\int}_{-\infty}^{\infty}}\mathit{d}\mathrm{\tau}{\displaystyle {\int}_{-\infty}^{\infty}}\mathit{d}{\mathrm{\tau}}^{\prime}\mathrm{\tau}{\mathit{e}}^{-{\mathit{x}}^{2}{\tilde{\mathrm{\varphi}}}^{(3)}(\mathrm{\tau},0,{\mathrm{\tau}}^{\prime})-{\mathit{y}}^{2}({\mathrm{\tau}}^{2}+{{\mathrm{\tau}}^{\prime}}^{2}-\mathrm{\tau}{\mathrm{\tau}}^{\prime})}$$

(11)

$${\mathit{F}}_{2}(\mathit{x},\mathit{y})={\displaystyle {\int}_{-\infty}^{\infty}}\mathit{d}\mathrm{\tau}{\displaystyle {\int}_{-\infty}^{0}}\mathit{d}{\mathrm{\tau}}^{\prime}{\mathrm{\tau}}^{\prime}{\mathit{e}}^{-{\mathit{x}}^{2}{\tilde{\mathrm{\varphi}}}^{(3)}(\mathrm{\tau},0,{\mathrm{\tau}}^{\prime})-{\mathit{y}}^{2}({\mathrm{\tau}}^{2}+{{\mathrm{\tau}}^{\prime}}^{2}-\mathrm{\tau}{\mathrm{\tau}}^{\prime})}$$

(12)

$$\begin{array}{c}{\tilde{\mathrm{\varphi}}}^{(3)}({\mathrm{\tau}}_{1},{\mathrm{\tau}}_{2},{\mathrm{\tau}}_{3})=(3-{\mathit{e}}^{\mathit{i}({\mathrm{\tau}}_{2}-{\mathrm{\tau}}_{1})}-{\mathit{e}}^{\mathit{i}({\mathrm{\tau}}_{3}-{\mathrm{\tau}}_{1})}-{\mathit{e}}^{\mathit{i}({\mathrm{\tau}}_{3}-{\mathrm{\tau}}_{2})})\left[{\mathit{n}}_{\text{ph}}\right(\frac{{\mathit{k}}_{\mathrm{B}}\mathit{T}}{{\mathrm{\omega}}_{\text{ph}}})+1]\\ +(3-{\mathit{e}}^{-\mathit{i}({\mathrm{\tau}}_{2}-{\mathrm{\tau}}_{1})}-{\mathit{e}}^{-\mathit{i}({\mathrm{\tau}}_{3}-{\mathrm{\tau}}_{1})}-{\mathit{e}}^{-\mathit{i}({\mathrm{\tau}}_{3}-{\mathrm{\tau}}_{2})}){\mathit{n}}_{\text{ph}}\left(\frac{{\mathit{k}}_{\mathrm{B}}\mathit{T}}{{\mathrm{\omega}}_{\text{ph}}}\right)\end{array}$$

(13)

Using the above formula for the current, the transverse current readswhere χ

$${\mathit{J}}_{\mathit{y}}=-\frac{3\sqrt{3}}{2}\frac{{\mathit{e}}^{2}\mathit{n}{\mathit{t}}^{3}\mathit{E}}{\mathrm{\u0127}{\mathrm{\omega}}_{\text{ph}}^{3}}({\mathrm{\chi}}_{\mathit{u}}+{\mathrm{\chi}}_{\mathit{d}})\text{Im}\left[{\mathit{F}}_{2}\right(\frac{\mathit{M}}{{\mathrm{\omega}}_{\text{ph}}},\frac{{\mathit{V}}_{0}}{{\mathrm{\omega}}_{\text{ph}}}\left)\right]$$

(14)

*(χ*_{u}*) is the thermal average of the scalar spin chirality*_{d}*S**· (*_{i}*S**×*_{j}*S**) for the upward (downward) triangles. Here, the sites*_{k}*i*,*j*, and*k*are in counterclockwise order. The result implies that a finite Hall current forms when χ*≠ − χ*_{u}*. The imbalance of chirality generally occurs when the upward and downward triangles are inequivalent, such as in the trimerized case.*_{d}## Acknowledgments

We acknowledge fruitful discussions with Y. Tokura, Y. Taguchi, N. Kanazawa, M. Hirschberger, K. Ishizaka, M. Sakano, S. Yoshida, and Y. Ohigashi. Unpolarized/polarized neutron scattering experiments were performed at Materials and Life Science Experimental Facility (MLF) in Japan Proton Accelerator Research Complex (J-PARC) (proposal nos. 2017L0701 and 2019C0006).

**Funding:**This work was supported by JST PRESTO grant no. JPMJPR18L2 and CREST grant no. JPMJCR16F1, Japan; by Grant-in-Aids for Scientific Research (B) nos. JP18H01866, JP19H01856, JP17H02815, and JP21H01804 from MEXT, Japan; and by Inamori Foundation, Japan.

**Author contributions:**M.U. and M.Ka. designed the experiments. M.U., S.S., and Y.N. grew films and performed transport measurements with R.K. and M.T., and magnetization measurements with M.Kr. and A.M. T.N. performed neutron scattering experiments with K.O., T.M., and T.-h.A. H.I. and N.N. performed model calculation of the anomalous Hall conductivity. M.S.B. performed first principles band structure calculations. M.U. and S.S. analyzed the data and wrote the manuscript with input from all authors. T.-h.A. and N.N. jointly discussed the results. M.U. and M.Ka. conceived the project. All authors have approved the final version of the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.

## Supplementary Materials

### This PDF file includes:

Notes S1 and S2

Figs. S1 to S8

References

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#### Published In

Science Advances

Volume 7 | Issue 52

December 2021

December 2021

#### Copyright

Copyright © 2021 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is

**not**for commercial advantage and provided the original work is properly cited.#### Submission history

**Received**: 20 July 2021

**Accepted**: 3 November 2021

#### Acknowledgments

We acknowledge fruitful discussions with Y. Tokura, Y. Taguchi, N. Kanazawa, M. Hirschberger, K. Ishizaka, M. Sakano, S. Yoshida, and Y. Ohigashi. Unpolarized/polarized neutron scattering experiments were performed at Materials and Life Science Experimental Facility (MLF) in Japan Proton Accelerator Research Complex (J-PARC) (proposal nos. 2017L0701 and 2019C0006).

**Funding:**This work was supported by JST PRESTO grant no. JPMJPR18L2 and CREST grant no. JPMJCR16F1, Japan; by Grant-in-Aids for Scientific Research (B) nos. JP18H01866, JP19H01856, JP17H02815, and JP21H01804 from MEXT, Japan; and by Inamori Foundation, Japan.

**Author contributions:**M.U. and M.Ka. designed the experiments. M.U., S.S., and Y.N. grew films and performed transport measurements with R.K. and M.T., and magnetization measurements with M.Kr. and A.M. T.N. performed neutron scattering experiments with K.O., T.M., and T.-h.A. H.I. and N.N. performed model calculation of the anomalous Hall conductivity. M.S.B. performed first principles band structure calculations. M.U. and S.S. analyzed the data and wrote the manuscript with input from all authors. T.-h.A. and N.N. jointly discussed the results. M.U. and M.Ka. conceived the project. All authors have approved the final version of the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.

### Authors

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JST PRESTO: JPMJPR18L2

JST CREST: JPMJCR16F1

Grant-in-Aids for Scientific Research (B): JP18H01866

Grant-in-Aids for Scientific Research (B): JP19H01856

Grant-in-Aids for Scientific Research (B): JP17H02815

Grant-in-Aids for Scientific Research (B): JP21H01804

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- Maximizing intrinsic anomalous Hall effect by controlling the Fermi level in simple Weyl semimetal films, Physical Review B,
**105**, 20, (2022).https://doi.org/10.1103/PhysRevB.105.L201101

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