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[Yujun Wang](https://orcid.org/0009-0002-3321-9781), [Shunzhen Wang](https://orcid.org/0009-0001-8382-590X), [Masashi Kawaguchi](https://orcid.org/0000-0001-5907-9137), [Jun Uzuhashi](https://orcid.org/0000-0003-2023-8158), [Akhilesh Kumar Patel](https://orcid.org/0000-0001-9718-8860), [Kenji Nawa](https://orcid.org/0000-0003-4535-0920), [Yuya Sakuraba](https://orcid.org/0000-0003-4618-9550), [Tadakatsu Ohkubo](https://orcid.org/0000-0003-3548-1951), [Hiroshi Kohno](https://orcid.org/0000-0003-2860-5290), [Masamitsu Hayashi](https://orcid.org/0000-0003-2134-2563)

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[Significant electron-magnon scattering in layered ferromagnet Cr2Te3](https://mdr.nims.go.jp/datasets/bf5422bc-4bdb-4cea-a8b8-c3e3bd5d637b)

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Significant electron-magnon scattering in layered ferromagnetCr2Te3Yujun Wang,1 Shunzhen Wang,1 Masashi Kawaguchi,1 JunUzuhashi,2 Akhilesh Kumar Patel,2 Kenji Nawa,3, 2, ∗ Yuya Sakuraba,2Tadakatsu Ohkubo,2 Hiroshi Kohno,4 and Masamitsu Hayashi1, 5, †1Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan2National Institute for Materials Science, Tsukuba 305-0047, Japan3Graduate School of Engineering, Mie University, Tsu 514-8507, Japan4Department of Physics, Nagoya University, Nagoya 464-8602, Japan5Trans-scale quantum science institute,The University of Tokyo, Tokyo 113-0033, JapanAbstractA layered ferromagnet Cr2Te3 is attracting growing interest because of its unique electronic andmagnetic properties. Studies have shown that it exhibits sizable anomalous Hall effect (AHE) thatchanges sign with temperature. The origin of the AHE and the sign change, however, remainselusive. Here we show experimentally that electron-magnon scattering significantly contributes tothe AHE in Cr2Te3 through magnon induced skew scattering, and that the sign change is causedby the competition with the Berry-curvature or impurity-induced side-jump contribution. Theelectron-magnon skew scattering is expected to arise from the exchange interaction between theitinerant Te p-electrons and the localized Cr d-electrons modified by the strong spin-orbit couplingon Te. These results suggest that the magnon-induced skew scattering can dominate the AHE inlayered ferromagnets with heavy elements.∗ current affiliation: National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8568,Japan† hayashi@phys.s.u-tokyo.ac.jp1I. INTRODUCTIONThe Cr-Te compound[1] is a material system that has attracted significant interest re-cently owing to its unique structural, transport and magnetic properties. Many of the com-pounds form a layered structure and are stable down to a monolayer. The majority of thecompounds exhibit strong ferromagnetism with some exceptions (e.g. antiferromagnetismin CrTe3[2] and Cr1+δTe2[3]). Studies have shown that ferromagnetism persists down to amonolayer[4], allowing studies on two-dimensional magnetism[5–7]. The Curie temperaturetypically lies in a range of 100 K to 200 K. With proper growth conditions[8, 9], however,recent reports show the Curie temperature can be increased, exceeding room temperatureunder certain circumstances[4, 10, 11]. Owing to the crystalline anisotropy, the magneticeasy axis often points along the film normal, with the perpendicular magnetic anisotropyenergy larger than that of other layered ferromagnets[12, 13].The transport properties of the compounds also show unique characteristics. In particu-lar, the compound exhibits a sizable anomalous Hall effect[14–16]. Studies have shown thatthe anomalous Hall resistance changes its sign as the temperature is varied[9, 15, 17–21].The origin of the anomalous Hall effect as well as its sign change with temperature have beenunder scrutiny. It has been reported that the anomalous Hall effect in the Cr-Te compoundsis caused by the large Berry curvature of the bands near the Fermi level[18, 20–23]. As theBerry curvature induced anomalous Hall conductivity was found to be an odd function ofenergy, it causes a sign change in the anomalous Hall resistance as the temperature is varieddue to population change of the occupied bands. However, other studies[14, 15, 24] haveshown that contribution from the skew scattering plays an essential role in the anomalousHall effect, posing question on its origin.Here we show that the unique characteristics of the anomalous Hall effect in Cr2Te3, oneof the most stable compounds in the Cr-Te family, are defined by electron-magnon scatter-ing. Cr2Te3 has a layered structure in which layers of CrTe2 are connected by intercalated Cratoms. We find the electron-magnon scattering significantly contributes to the longitudinaland anomalous Hall resistances. The scaling relation between the longitudinal and anoma-lous Hall resistivities is used to identify the origin of the anomalous Hall effect. We find twocompeting sources: magnon induced skew scattering and impurity induced side jump/Berrycurvature effect. Model calculations show that the former is caused by the exchange inter-2action between the itinerant Te p-electrons and the localized Cr d-electrons modified by thespin-orbit coupling. We consider Te, which possess significant spin-orbit coupling, plays acritical role in setting the magnon-induced skew scattering.II. EXPERIMENTAL RESULTSA. Structural and magnetic propertiesCr2Te3 films were grown on sapphire or MgO substrates using molecular beam epitaxy(MBE). A Ti or Te layer was used as a capping layer. See ”Methods” and SupplementaryNote 1 for the details of sample preparation and characterization. A cross-sectional high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) imageof a 20 nm-thick Cr2Te3 film is displayed in Fig. 1(a). The bright and dark contrasts of theimage represent grains with different crystal orientations within the film plane. The grainsize is of the order of a few tens of nanometers. The high magnification image, Fig. 1(c), andthe corresponding nanobeam electron diffraction pattern, Fig. 1(d), show highly texturedfilm with growth along the Cr2Te3 (001) direction. Energy dispersive X-ray spectroscopy(EDS) maps of the elements are shown in Fig. 1(b). The images show Cr and Te areuniformly distributed within the film. Profile of the film composition along the film normalis presented in Fig. 1(e). From the profile, we determine the film composition is Cr:Te ∼ 2:3.See Supplementary Figure S1 for the reflection high energy electron diffraction (RHEED)images and the X-ray diffraction (XRD) spectra of the films.First, we study the magnetic properties of Cr2Te3. Figure 2(a) shows the temperaturedependence of the saturation magnetization Ms for films with different thicknesses. TheCurie temperature TC is ∼175 K for all samples except for the 5 nm-thick film, which exhibitsTC of ∼215 K. See Supplementary Note 2 for the details of how TC is extracted. The valueof Ms at 2 K for the thicker films is close to that predicted from first principles calculations,which is ∼465 emu/cm3. We findMs is substantially larger for the thinner films (5 nm and 10nm-thick). Previous studies have reported that the magnetic moments of Cr2Te3 are cantedfrom the film normal but the canting can be suppressed when the film thickness is reduced,thereby causing a difference in Ms against the film thickness[13]. Alternatively, it has beenshown, using scanning tunnel microscopy, that Cr3Te4 forms at the beginning of the growth3with molecular beam epitaxy[25]. Since the saturation magnetization of Cr3Te4 is largerthan that of Cr2Te3[26, 27], the magnetization can be larger for the thinner films given thelarger weight of the Cr3Te4 phase. Note that the transport properties of Cr3Te4[28, 29] arenot significantly different from those of Cr2Te3. In addition, Ms shows an upturn below ∼10K for the thinner films. Such change in Ms at low temperature was reported previously[9].Although the physical mechanism behind the upturn is unclear in Cr2Te3, previous studiesfor other systems (e.g. ultra-fine cobalt ferrite nanoparticles) suggested that it may originatefrom surface magnetic moments[30]. Further study is required to clarify the origin of thethickness dependence of Ms and the upturn below ∼10 K. In Supplementary Figure S2, weshow a few exemplary magnetization hysteresis loops. The loops show that the magneticeasy axis of the films points along the film normal, in agreement with previous studies[24].For later use, we define M0s as Ms obtained at the lowest measurement temperature (∼2 K).B. Longitudinal transport propertiesThe transport properties are studied using the patterned Hall bars: see SupplementaryNote 2 and Figure S3 for the details of the device structure. The temperature dependenceof the longitudinal resistivity ρxx of a 10 nm-thick film is shown in Fig. 2(b). We find thetemperature dependence of ρxx for T < TC can be fitted with the following function:ρxx = ρ0xx + ρmxxT2 (T < TC). (1)The fitting result, shown by the red solid line in Fig. 2(b), is in good agreement with theexperimental results. The change in ρxx for T > TC is almost linear and its slope is small:see also Supplementary Figure S4.The quadratic temperature dependence of ρxx below TC can be attributed to electron-magnon scattering[31, 32]. (As is often the case in metals[32], we neglect electron-electronscattering, which also scales with T 2.) The temperature independent resistivity that isdominant at the lowest temperature is likely associated with impurity induced scattering.We therefore assign ρ0xx and ρmxx as the impurity and electron-magnon scattering coefficients,respectively. The film thickness dependences of ρ0xx and ρmxx are shown in Figs. 2(c) and2(d), respectively. ρ0xx tends to decrease with film thickness, suggesting that the film qualityimproves for thicker films. In contrast, ρmxx increases with the film thickness until it saturates4at ∼50 nm.To show that electron-magnon scattering indeed contributes to the transport properties,the out of plane magnetic field (Hz) dependence of the longitudinal resistivity ∆ρxx ≡ρxx(H) − ρxx(H = 0) is studied. Figure 3(a) shows representative results from a 10 nm-thick film. There are at least two major effects known to contribute to ∆ρxx in magneticmaterials[33]: the Lorentz magnetoresistance and the magnon-induced magnetoresistance.Whereas the resistance quadratically increases with Hz for the former, it linearly decreaseswith Hz for the latter. As is evident, ∆ρxx decreases almost linearly with increasing Hz whenthe temperature is lower than Tc, suggesting that the magnon-induced magnetoresistance isdominant[33]. We fit the data with a linear function near zero field (from ∼0 to 10 kOe) toobtain the slope of ∆ρxx vs. Hz, which is defined as ∂∆ρxx∂Hz. ∂∆ρxx∂Hzis plotted as a function oftemperature for all films in Fig. 3(b). Clearly, ∂∆ρxx∂Hzincreases as the temperature approachesTc, suggesting that magnon-scattering plays a larger role at higher temperatures.In Fig. 3(c-e), we plot ∂∆ρxx∂Hzvs. ρmxx to study if they are correlated. At the lowest temper-ature [Fig. 3(c)], there is no significant correlation between the two quantities, as magnon-scattering is suppressed in this temperature range. As the temperature is raised[Fig. 3(d,e)],we observe a positive correlation between the two, indicating that ∂∆ρxx∂Hzis dependent onmagnon-scattering.C. Anomalous Hall resistanceNext, we study the transverse resistivity ρyx of Cr2Te3. The Hz dependence of ρyx ofa 10 nm-thick film, measured at different temperatures, are plotted in Fig. 4(a). A clearhysteresis loop is observed at temperatures below TC. The small hump near the magne-tization switching fields may be due to the topological Hall effect or the superposition ofcompeting anomalous Hall effects with opposite signs. Although similar features have beenobserved in other systems and were attributed to the topological Hall effect, e.g. Cr5Te6[34],CrTe2/Bi2Te3[35] and Cr2Te3/Cr2Se3[36], we cannot identify its origin in Cr2Te3 single layerfilm from the current data set. The anomalous Hall resistivity ∆ρyx is obtained by sub-tracting the linear background found at high magnetic field and taking half the difference ofthe background subtracted ρyx at positive and negative Hz. The linear background is pre-dominantly caused by the ordinary Hall effect: see Supplementary Figure S5 for the carrier5density and mobility estimated from the background signal. We normalize ∆ρyx with MsM0stoexclude the temperature dependence of Ms from ∆ρyx[37]. The normalized anomalous Hallresistivity ∆ρ̃yx ≡ ∆ρyx/MsM0sis plotted as a function of temperature in Fig. 4(b). (See Sup-plementary Figure S4 for the temperature dependence of ∆ρyx.) As is evident, ∆ρ̃yx changesits sign at T ∼ 100 K, a trend that has been reported in previous studies[9, 15, 17, 20]. Sim-ilar to ρxx, ∆ρ̃yx exhibits a T 2 scaling. The red solid line in Fig. 4(b) shows a parabolicfitting, which agrees well with the data. Later, we show that such T 2 scaling is one of thecharacteristics of magnon-induced skew scattering.The scaling relation between the anomalous Hall and the longitudinal resistivities isstudied to identify the origin of the anomalous Hall effect[38–40]. ∆ρ̃yx is plotted againstρxx in Fig. 4(c). |∆ρ̃yx| increases with increasing ρxx, with the temperature as an implicitparameter, exhibiting a predominantly linear scaling. In general, the scattering sources thatcause the temperature dependence of the anomalous Hall resistivity can be classified intotwo categories, static and dynamic disorders. The former is caused by impurities and scaleswith ρ0xx, whereas the latter can be induced by magnons and phonons and is proportional toρTxx ≡ ρxx − ρ0xx. From multi-variable scaling derived by Hou et al.[39], the anomalous Hallresistivity can be expressed as:∆ρ̃yx = ∆ρ̃skewyx +∆ρ̃sjyx +∆ρ̃intyx ,∆ρ̃skewyx = a1ρ0xx + a2ρTxx,∆ρ̃sjyx = b1(ρ0xx)2 + b2(ρTxx)2 + b3ρ0xxρTxx,∆ρ̃intyx = cρ2xx,(2)where ∆ρ̃skewyx , ∆ρ̃sjyx, ∆ρ̃intyx are contributions from the skew scattering, the side jump andthe Berry curvature effect, respectively. a1, a2, b1, b2, b3 and c are the scaling coefficients.In contrast to the note made in Ref. [39], here we have included a dynamic skew scattering(the a2-term).We first study the effect of static disorders on the anomalous Hall resistance. We setρTxx = 0 and rearrange Eq. (2) to obtain∆ρ̃0yxρ0xx= a1 + (b1 + c)ρ0xx (3)where ∆ρ̃0yx is ∆ρ̃yx measured at the lowest temperature (2 K). In Fig. 4(d), we plot∆ρ̃0yxρ0xxas a function of ρ0xx to determine a1 and b1 + c. (As a reference, ∆ρ̃0yx vs. ρ0xx is plotted6in the inset.) Data is fitted with Eq. (3): the fitted curve is shown by the solid line. Wefind a1 ∼ −0.034 and b1 + c ∼ 1.5 × 10−4 (µΩ cm)−1. These results show that films withlarger ρ0xx exhibit positive ∆ρ̃0yx due to the larger contribution from the impurity inducedside-jump/Berry curvature effect, i.e. the b1 + c term. This is the case for the thinner films.For the thicker films, ∆ρ̃0yx is negative since contribution from the impurity induced skewscattering (a1 term) is larger.Next, we look into the influence of dynamic disorders, which cause the temperaturedependent anomalous Hall resistance. We find that most of the data (∆ρ̃yx vs. ρxx) can bedescribed by a linear function, shown by the solid lines in Fig. 4(c), particularly when thefilm thickness is small. The predominant linear dependence indicates that skew scattering(the a2 term in Eq. (2)) contributes to the anomalous Hall effect, consistent with previousreports on Cr-Te systems[14–16, 19]. We fit the data from 2 ∼ 100 K with a linear line.From Eq. (2), the slope of the linear line is equal to a2 + (b3 + 2c)ρ0xx. (Note that the slopeof ∆ρ̃yx vs. ρTxx is the same as that of ∆ρ̃yx vs. ρxx since ρ0xx is a constant.) We thus plotthe slope as a function of ρ0xx in Fig. 4(e) and fit a linear function, which is shown by thesolid line. From the fitting, we find a2 ∼ −0.067 and b3 + 2c ∼ 0.7× 10−4 (µΩ cm)−1. a2 iscomparable in magnitude with the skew scattering coefficient in other systems[41–43].Before discussing the origin of the a2 term, we comment on the other terms in the anoma-lous Hall resistivity. As is evident in Fig. 4(c), the data deviates from the linear fitting forthe thicker films at higher temperatures. The deviation is caused by the side jump and/orBerry curvature contributions, i.e., b2 and c terms in Eq. (2). (For the thinner films, becauseof the lower resistivity at high temperature, influence from the quadratic terms is limited.)A previous study showed that the Berry curvature contribution in Cr2Te3 can vary withtemperature due to thermal broadening of the Fermi surface and may depend on the filmthickness via growth induced strain[20, 23]: see Supplementary Note 3 and Figure S6 for thefirst principles calculations we performed as well. The possible change of the Berry curvaturecontribution with temperature and thickness make it difficult to extract the coefficients b2and c using Eq. (2). We therefore focus on the predominant linear term (a2) and simplynote that the combined contributions from the quadratic terms (b2 and c) take a maximumof ∼ 25% for the thickest film near the Curie temperature, which is estimated from thedeviation of ∆ρ̃yx from the linear fitting.To identify the source of the a2 term, we plot ∆ρ̃Tyx ≡ ∆ρ̃yx − ∆ρ̃0yx as a function of7ρmxx in Fig. 4(f). ∆ρ̃Tyx is positively related to ρmxx, particularly at higher temperatures,suggesting that magnons play a dominant role in the dynamic disorder induced scattering.To corroborate this observation, the Hz dependence of the Hall resistance is measured upto 140 kOe for the 65 nm-thick film. The results obtained at measurement temperaturesof 5 K and 150 K are shown in Fig. 5(a,b). At low temperature [Fig. 5(a)], ρyx scaleslinearly with Hz for fields outside the hysteresis loop, whereas it varies in a non-linearfashion for higher temperature [Fig. 5(b)]. (ρyx that scales with Hz at large field is mostlycaused by the ordinary Hall effect.) To display the effect more clearly, we fit the data witha linear function in the range of Hz ∼ 130 − 140 kOe and subtracted it from the data.The subtracted data, defined as ρ′yx, are shown in Fig. 5(c,d). ρ′yx tends to decrease as |Hz|increases when the temperature is high, whereas it is nearly constant in the entire field rangefor lower temperature. Previous studies have shown that large magnetic field suppressesexcitation of magnons[33, 44]. The reduction of ∆ρyx at large Hz can therefore be attributedto decrease in magnon population. These results thus support the notion that electron-magnon scattering contributes to the anomalous Hall resistance at higher temperatures. Wenote that the saturation of ρ′yx in Fig. 5(d) is caused by the linear background subtractionprocess. Measurements at even larger magnetic field are needed to determine the saturationfield. Indeed, previous studies showed that suppression of magnon induced effects requiresmagnetic field of the order of a few hundreds of kOe[33]. Suppression of electron-magnonscattering by magnetic field can also be found in the magnetoresistance measured at highermagnetic field. Figure 5(e) shows the magnetoresistance ∆ρxx measured up to 140 kOefor the 65 nm-thick film. The slope of ∆ρxx vs. Hz clearly changes with Hz at highertemperatures. Note that the slope represents the strength of magnon scattering: see thediscussion pertaining to Fig. 3. We plot the temperature dependence of the slope ∆ρxx/Hzat lower magnetic field (0-10 kOe) and higher magnetic field (130 -140 kOe) in Fig. 5(f).The former is significantly larger than the latter when the temperature is high, indicatingthat magnon excitation is suppressed at larger magnetic field. These results strongly suggestthat the source of the a2 term is magnon-induced skew scattering.Based on these results, we discuss the reason behind the sign change of the anomalousHall resistance with temperature in Cr2Te3. At the lowest temperature, ∆ρ̃0yx is governedby static disorder (impurity) ρ0xx. The linear (a1) and quadratic (b1+ c) terms have oppositesign. For the thinner films (with larger ρ0xx), the net contribution is positive since the8quadratic term dominates. In contrast, ∆ρ̃0yx is negative for the thicker films (with smallerρ0xx) as the linear term is dominant. With increasing temperature, contribution from themagnon induced skew scattering (a2), which is negative, increases and takes over, resultingin a sign change of ∆ρ̃yx only for the thinner films.III. MODEL CALCULATIONSFinally, we discuss the microscopic origin of the magnon induced skew scattering inCr2Te3. The anomalous Hall resistivity ∆ρ̃yx that originates from magnon induced skewscattering is obtained by including the effect of the spin-orbit coupling in a p-d exchangeinteraction. The Hamiltonian that describes the electron-magnon skew scattering has beenproposed as[45]:Hpd = iλJpda20∑k,k′(k × k′) · (δS)k−k′ c†kck′ , (4)where λ is a dimensionless parameter that characterizes the spin-orbit coupling, Jpd rep-resents the p-d exchange interaction between the Te 5p conduction electrons and the Cr3d localized moments, a30 is the volume per localized spin, k and ck are the electron wavevector and annihilation operator, S represents the localized spin (|S| = 3/2 for Cr) and δSis the deviation from its equilibrium direction due to magnon excitation. Assuming a freeelectron like band with exchange splitting, the anomalous Hall resistivity ∆ρ̃calyx is calculatedby considering the process shown in Supplementary Figure S7 , as∆ρ̃calyx = −Ξyxλa20J3pdS8ℏ2e2v3F(kBTAex)2(5)where vF is the Fermi velocity and Aex is the exchange stiffness parameter. (ℏ is the reducedPlanck constant, e is the elementary charge and kB is the Boltzmann constant.) Note thatthis process was not considered in Ref. [39]. The longitudinal resistivity ρcalxx due to electron-magnon scattering is given by[45]ρcalxx = Ξxxm2J2pdS(2π)3ℏ3e2n2a30(kBTAex)2(6)where Ξyx [in Eq. (5)] and Ξxx [in Eq. (6)] are constants of order unity and weakly dependenton temperature (see Supplementary Figure S8). m is the effective electron mass and n isthe electron density. See Supplementary Note 4 for the outline of the derivation of Eqs. (5)9and (6), details of Ξyx and Ξxx and the Feyman diagram (Supplementary Figure S7) usedto calculate the anomalous Hall conductivity due to magnon scattering.Both ∆ρ̃calyx and ρcalxx scale with the temperature T quadratically, in agreement with theexperiments: see Figs. 2(b) and 4(b). Substituting the parameters listed in SupplementaryTable 1, suitable for Cr2Te3, we obtain ∆ρ̃calyx ∼ −22 µΩ·cm and ρcalxx ∼ 329 µΩ·cm at T = 200K from Eqs. (5) and (6). Here we adjusted λ, the dimensionless parameter that characterizesthe spin-orbit interaction, to match the value of a2 = −0.067 obtained in the experimentswith ∆ρ̃calyx/(ρcalxx). Note that Jpd = −0.1 eV is estimated from the band structure of Cr2Te3using first principles calculations. Jpd is normally negative when the exchange coupling isbetween conduction electrons and localized moments that belong to different elements. FromEq. (5), we find λ must be negative in order to account for the negative ∆ρ̃yx found in theexperiments (see e.g. Fig. 4(b)). The strength of the p−d exchange interaction, characterizedby λJpd, is ∼ 0.084 eV, which is similar in magnitude with the atomic spin-orbit coupling ofTe[46–48]. We thus infer that the predominant magnon-induced skew scattering in Cr2Te3is primarily induced by the large spin-orbit coupling of Te. A microscopic model that takesinto account the band structure of host material (here Cr2Te3) is required to clarify therelation between the atomic spin orbit coupling and λ.IV. CONCLUSIONWe have studied the longitudinal and anomalous Hall resistivities of Cr2Te3 thin films.We find a quadratic temperature dependence of the longitudinal resistivity below the Curietemperature, suggesting that electron-magnon scattering is one of the major sources ofthe resistivity. This is corroborated by a linear magnetoresistance found against the out-of-plane magnetic field. The anomalous Hall resistivity includes two major contributions:temperature dependent and temperature independent terms. The former exhibits a pre-dominant linear dependence with the longitudinal resistivity and is positively correlatedwith the electron-magnon scattering coefficient of the longitudinal resistivity, suggestingthat magnon induced skew scattering significantly contributes to the anomalous Hall effectin Cr2Te3. We also find the anomalous Hall resistivity at higher temperature is suppressedby large magnetic field, supporting the scenario that the skew scattering originates fromcollision with magnons. For the latter (i.e. the temperature independent term), we find that10the contribution from the impurity induced side-jump and/or the Berry curvature effectpossess the opposite sign with that of magnon-induced skew scattering. The sign change ofthe anomalous Hall resistivity with temperature is thus accounted for by the competitionbetween the two effects. Using model calculations, we show that the p-d type exchangeinteraction modified by spin-orbit coupling can account for the significant electron-magnonscattering contribution to the anomalous Hall effect in Cr2Te3. These results suggest thatmagnon-induced skew scattering plays a significant role in the anomalous Hall effect in lay-ered ferromagnets with heavy elements, a class of 2D materials that are of current interest.ACKNOWLEDGMENTSThis work was partly supported by JSPS KAKENHI (Grant Numbers 21H01608,22K14290, 22K18935, 23H00176, 23H05463), JST CREST (JPMJCR19T3), MEXT Initia-tive to Establish Next-generation Novel Integrated Circuits Centers (X-NICS) and Coopera-tive Research Project Program of RIEC, Tohoku University. Computations were performedon a Numerical Materials Simulator at NIMS. Y.W. is supported by the GSGC program ofUniversity of Tokyo, S.W. thanks JST SPRING GX, Grant Number JPMJSP2108.Author contributionsY.W. deposited the films, fabricated the samples and performed the magnetic and transportmeasurements, with the help of M.K. S.W. helped the film growth and the RHEED obser-vation. A.K.P. and Y.S. carried out the high field transport measurements. J.U. and T.O.fabricated the specimen and conducted the HAADF-STEM observation. K.N. performedthe first principles calculations and H.K. carried out the model calculations. Y.W. and M.H.wrote the manuscript with substantial inputs from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information: The online version contains supplementary material availableat XX.11V. METHODSA. Sample preparationCr2Te3 films were grown on sapphire (0001) or MgO (001) substrates in a commercialmolecular beam epitaxy (MBE) system. The substrates were pre-annealed in vacuum at650 oC for 150 min for degassing. After degassing, the substrate temperature Ts was set to∼ 380 oC. Tellurium and chromium were co-evaporated using a Knudsen cell for Te and anelectron beam gun for Cr. To form Cr2Te3, the flux ratio of Te over Cr was kept to ∼20or higher. The growth rate of Cr2Te3 was monitored using a quartz crystal spectrometerand its typical value is ∼0.01 nm/s. Cr2Te3 thin films with different thicknesses (5, 10,26, 49 and 65 nm) were grown. The thickness of the films were determined using X-rayreflectivity measurements. A 5 nm-thick Te or 3 nm-thick Ti capping layer was deposited atroom temperature to protect the Cr2Te3 film from oxidation. We find the type of substrate(sapphire vs. MgO) and the capping layer material (Te vs. Ti) have little influence on themagnetic and transport properties of the films.B. Sample characterizationMagnetic properties of the films were examined by superconducting quantum interferencediffractometer (SQUID). The specimen for cross-sectional transmission electron microscopy(TEM) studies was prepared by using a focused ion beam (FIB). High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) observation, nanobeamelectron diffraction, and energy dispersive x-ray spectroscopy (EDS) analysis were carriedout using a commercial system.To study the transport properties, the films were patterned into Hall bars using opticallithography and Ar ion milling. Contact electrodes, made of ∼50 nm thick conductingmaterials, were patterned on the Hall bars using standard liftoff technique. The electrodeswere deposited via RF magnetron sputtering. The width w and length L of the Hall barchannel are 10 µm and 25 µm, respectively. Measurements were performed using a physical12property measurement system (PPMS). A current Ix of 100 µA was applied to the sam-ple and the longitudinal voltage Vxx and the transverse voltage Vyx were measured. Thelongitudinal and transverse resistivities are obtained from the relations ρxx = VxxIxwtLandρyx = VyxtIx, respectively, where t is the thickness of the film. See Supplementary Note 2and Figure S3 for the details of the configuration used to measure the longitudinal andtransverse resistivities.Data availabilityExperimental data presented in the main text are stored in the Supplementary Data. Allrelevant data are available from the authors upon request.13References[1] H. Ipser, K. L. Komarek, and K. O. Klepp, “Transition metal-chalcogen systems viii: Thecr-te phase diagram,” Journal of the Less Common Metals 92, 265 (1983).[2] M. A. McGuire, V. O. Garlea, K. C. Santosh, V. R. Cooper, J. Q. Yan, H. B. Cao, and B. C.Sales, “Antiferromagnetism in the van der waals layered spin-lozenge semiconductor crte3,”Phys. Rev. B 95, 144421 (2017).[3] Y. Fujisawa, M. 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Irkhin, “Electronic structure, correlation effects and physical prop-erties of d- and f-metals and their compounds,” (2021), arXiv:cond-mat/9812072 [cond-mat].[46] T. Doi, K. Nakao, and H. Kamimura, “Valence band structure of tellurium .1. k.p perturbationmethod,” J. Phys. Soc. Jpn. 28, 36 (1970).[47] T. Furukawa, Y. Shimokawa, K. Kobayashi, and T. Itou, “Observation of current-inducedbulk magnetization in elemental tellurium,” Nat. Commun. 8, 954 (2017).[48] M. Sakano, M. Hirayama, T. Takahashi, S. Akebi, M. Nakayama, K. Kuroda, K. Taguchi,T. Yoshikawa, K. Miyamoto, T. Okuda, K. Ono, H. Kumigashira, T. Ideue, Y. Iwasa, N. Mit-suishi, K. Ishizaka, S. Shin, T. Miyake, S. Murakami, T. Sasagawa, and T. Kondo, “Radialspin texture in elemental tellurium with chiral crystal structure,” Phys. Rev. Lett. 124, 136404(2020).18Figure captions50 nmCr Te Ti O Al10 nm(e)(a)(b)2 nm00"4004116000"1"12112"1"16(c) (d)AlOCrTeTiAuFigures_092424a.pptxFig. 1FIG. 1. Structural characterization. (a) Cross-sectional high-angle annular dark-field scanningtransmission electron microscopy (HAADF-STEM) image and (b) EDS maps of a 20 nm-thickCr2Te3 film. (c) High magnification image of the Cr2Te3 film shown in (a). (d) Nanobeam electrondiffraction pattern of the image shown in (c). Indices of Cr2Te3 are labeled. (e) Depth profile ofthe elements using energy dispersive X-ray spectroscopy (EDS) mapping.19(a) (b)𝜌 !! (µΩ⋅cm)𝑀" (emu cm#$ )(c)𝜌 !!% (µΩ⋅cm)𝜌 !!& (10#$  µΩ⋅cm K#' ) (d)Figures_092424a.pptxFig. 2FIG. 2. Saturation magnetization and longitudinal resistivity. (a) Temperature T depen-dence of the saturation magnetization Ms for films with different thicknesses. (b) Longitudinalresistivity ρxx of a 10 nm-thick Cr2Te3 film plotted against T . The solid red line shows fit to thedata using Eq. (1) in the appropriate temperature range. (c,d) Film thickness dependence of theimpurity scattering coefficient ρ0xx (c) and the electron-magnon scattering coefficient ρmxx (d). Fora given film thickness, results from a few devices are presented using different symbols. Definitionof the symbols are the same for panels (c,d): see the legend shown in (c).20(a) (b)(c) 2 K (d) 100 K (e) 150 KΔ𝜌!! (µΩ⋅cm)𝜕Δ𝜌 !!/𝜕𝐻 ( (µΩ⋅cm kOe#) )𝜕Δ𝜌 !!/𝜕𝐻 ( (µΩ⋅cm kOe#) )𝐻( (kOe)𝜌!!&  (µΩ ⋅ cm K#')Figures_092424a.pptxFig. 3Temperature (K)FIG. 3. Longitudinal magnetoresistance. (a) Magnetoresistance ∆ρxx as a function of outof plane magnetic field Hz obtained for a 10 nm-thick film. Different symbols indicate differentmeasurement temperatures. The solid lines show linear fits to the data when Hz is in the rangeof 0 to 10 kOe. (b) Slope of fitted linear line, ∂∆ρxx∂Hz, plotted as a function of temperature. (c-e)Electron-magnon scattering coefficient ρmxx dependence of ∂∆ρxx∂Hzobtained at 2 K (c), 100 K (d) and150 K (e). (b-e) For a given film thickness, results from a few devices are presented using differentsymbols. Definition of the symbols are the same for panels (c-e): see the legend shown in (c).21Δ6𝜌 *! (µΩ⋅cm)Δ6𝜌 *!%/𝜌!!%×10#$𝜌!!%  (µΩ ⋅ cm)𝜌!!  (µΩ ⋅ cm)Slope(c)(d)(e)(a)Δ6𝜌 *! (µΩ⋅cm)𝜌 *! (µΩ⋅cm)𝐻( (kOe)(b)𝜌!!&  (µΩ ⋅ cm K#')(f)Δ6𝜌 *!+µΩ⋅cmFig. 4!𝜌 !"#µΩ⋅cm𝜌""#  (µΩ ⋅ cm)PPMS-Rxy_Yujun Wang_100924.opju𝜌!!%  (µΩ ⋅ cm)FIG. 4. Anomalous Hall resistivity. (a) Transverse resistivity ρyx, measured at differenttemperatures, is plotted against the out of plane magnetic field Hz for a 10 nm-thick Cr2Te3 film.Data are shifted vertically for clarity. (b) Normalized anomalous Hall resistivity ∆ρ̃yx = ∆ρyx/MsM0s(black circles) of a 10 nm-thick Cr2Te3 film plotted against the temperature T . Red solid linesshow a parabolic fitting to the data. (c) The symbols indicate the longitudinal resistivity ρxxdependence of ∆ρ̃yx. Data displayed are from a temperature range of 2 K to 175 K. Fit to the datawith a linear function is shown by the solid lines. (d)∆ρ̃0yxρ0xxplotted against the impurity scatteringcoefficient ρ0xx (symbols). The solid line shows fit to the data using Eq. (3). The inset shows∆ρ̃0yx, i.e. ∆ρ̃yx obtained at the lowest temperature, plotted as a function of ρ0xx. Definition of thesymbols are the same as in (c): see the legend shown in (c). (e) The slope of the linear functionused to fit the data shown in (c) plotted against ρ0xx (symbols). The solid line is a linear fit tothe data. (f) ∆ρ̃Tyx ≡ ∆ρ̃yx −∆ρ̃0yx plotted as a function of electron-magnon scattering coefficientρmxx. The open, dot center and solid symbols show data obtained at temperatures of 50, 100, 150K, respectively. The color of the symbols represents the film thickness whereas the symbol shapeindicates results from different devices: see the legend shown in (e). The solid linear lines are guideto the eye. (c-f) For a given film thickness, results from a few devices are presented using differentsymbols.22𝜌 *! (µΩ⋅cm)𝐻( (kOe) 𝐻( (kOe)𝜌 *! (µΩ⋅cm)𝜌 *!, (µΩ⋅cm)𝜌 *!, (µΩ⋅cm)(a) (b)(c) (d)5 K 150 KΔ𝜌!!/𝐻( (µΩ⋅cm T#) )Δ𝜌!! (µΩ⋅cm)𝐻( (kOe)(e) (f)Fig. 5 Figures_092424a.pptxTemperature (K)PPMS-Rxy_Yujun Wang_100924.opjuFIG. 5. High field longitudinal and transverse resistivities. (a,b) Transverse resistivity ρyxvs. out of plane magnetic field Hz measured at 2 K (a) and 150 K (b) for a 65 nm-thick film. (c,d)Hz dependence of the linear background subtracted transverse resistivity ρ′yx. The backgroundis determined by fitting the data in the field range of 130-140 kOe with a linear function. Thehorizontal dotted line in (d) is a guide to the eye. (e) Hz dependence of the magnetoresistance∆ρxx of a 65 nm sample measured up to 140 kOe. (f) The slope of the linear line fitted to ∆ρxx vs.Hz in field ranges 0-10 kOe (black squares) and 130-140 kOe (red circles) are plotted as a functionof temperature.23 Article File - Production Ready Figure 1 Figure 2 Figure 3 Figure 4 Figure 5