Antiferromagnetic Weyl semimetal Mn_{3}Sn exhibiting strong magneto-optical Kerr effect (MOKE) due to non-zero Berry curvature is attractive for spintronic and photonic device applications. Despite many reports on the anisotropic anomalous Hall effect (AHE), so far, there have been few studies on its anisotropic optical properties. In this work, we experimentally characterized the anisotropic optical and magneto-optical (MO) properties of Mn_{3}Sn(20)/MgO(110) and Mn_{3}Sn(0001)/Al_{2}O_{3}(0001) epitaxial films using ellipsometry in the wavelength range from 300 to 1690 nm. By measuring the Mueller matrix of magnetized Mn_{3}Sn, the anisotropic permittivity tensor is determined using the 4 × 4 transfer matrix method. Temperature dependent MOKE measurement confirmed the origin of the anisotropic MO effect to the non-zero Berry curvature of the chiral magnetic phase. The measured permittivity also agrees well with first-principles calculations. The anisotropic optical and MO properties determined in this work can be useful for Mn_{3}Sn based spintronic device characterization and photonic device development.

## I. INTRODUCTION

The antiferromagnetic (AFM) Weyl semimetal material, Mn_{3}Sn, has garnered considerable research interest recently.^{1–5} Thanks to its inverse triangular spin structure in a kagome lattice^{6–8} and noncollinear antiferromagnets in a cubic lattice,^{9,10} Mn_{3}X shows strong anomalous Hall effect (AHE) and magnetooptical (MO) effect at room temperature, attributing to a non-zero Berry curvature of its band structure despite vanishing magnetic moment. These properties make Mn_{3}X an attractive material for spintronic MO device applications.^{3,11,12} A series of studies has been carried out to study the fabrication, characterization and device application of Mn_{3}Sn films.^{4,5,13,14} In particular, the MO effect of Mn_{3}Sn has attracted research interest. The MO effect of Mn_{3}Sn can be used as an effective probe to image the domain structure of spintronic devices. It can also be used to develop MO devices. The large magneto-optical Kerr effect (MOKE) was first theoretically and experimentally observed,^{11,15–18} analog to those observed in ferromagnetic materials even in the absence of net magnetization. The giant MO effect is found to be an effective probe to visualize the domain structure of Mn_{3}Sn based spintronic devices.^{11,16,19} Later, a large ultrafast-modulated Voigt effect was observed in Mn_{3}Sn by quenching the AFM order with ultrafast laser pulses, paving the way for applications in high-speed spintronic devices.^{20} At terahertz frequencies, large Faraday effects were observed in polycrystalline Mn_{3}Sn films grown on SiO_{2} and oxidized silicon substrates.^{21} Meanwhile, the giant MO effect of Mn_{3}Sn at the terahertz frequencies can also be used to develop spintronic terahertz light sources.^{22,23}

Despite these progresses, a fundamental question remains to be answered: what are the anisotropic optical and MO properties of Mn_{3}Sn? Because Mn_{3}Sn has a hexagonal unit cell, the optical constants would be anisotropic in or out of the (0001) plane. The MO effect, manifested as the off-diagonal component of the permittivity tensor, is also anisotropic for magnetization aligned along the [2$1\u03041\u0304$0] or the [0001] orientations. Because the measured MOKE is related to the full anisotropic permittivity tensor of Mn_{3}Sn, the determination of the permittivity tensor is important for the interpretation of the spin orientation in spintronic device characterization and for the development of Mn_{3}Sn based MO devices.

In this work, we characterized the anisotropic optical and MO properties of Mn_{3}Sn epitaxial thin films by measuring their permittivity tensor using magneto-ellipsometry in the 300–1690 nm wavelength range. By measuring the Mueller matrix of magnetized Mn_{3}Sn, the anisotropic permittivity tensor was determined using the 4 × 4 transfer matrix method. Temperature dependence of the MO effect was measured across the magnetic phase transition temperatures. A dramatic change in *ε*_{yz} was observed in the near-infrared range. The measured permittivity tensor agreed well with theoretical calculation results using density functional theory (DFT) and the Kubo formula. The results may provide useful information for spintronic and photonic device applications, such as magneto-optical storage, optical isolators, optical modulators, spintronic devices, and sensors.

## II. EXPERIMENTAL SECTION

### A. Sample preparation and characterization

In collinear spin structure altermagnets such as RuO_{2} and MnTe, anisotropic AHE can be tested by growing epitaxial films with different orientations or by varying the angle at which the magnetic field is applied.^{24,25} In this paper, (0001)- and (2$1\u03041\u0304$0)-oriented Mn_{3}Sn epitaxial thin films were prepared by pulsed laser deposition (PLD) on single crystalline Al_{2}O_{3}(0001) and MgO(110) substrates. Details of the fabrication process can be found in our previous publication.^{18} The crystal structure and orientation of the films were analyzed by x-ray diffraction (XRD) using a high-resolution x-ray diffractometer (Bruker D8). The film compositions were measured using energy dispersive spectra (EDS) in a scanning electron microscope (SEM, JSM-7600F). The thickness of the Mn_{3}Sn film was characterized by SEM. Magnetic hysteresis was measured by a commercial superconducting quantum interference device (SQUID) magnetometer (MPMS3, Quantum Design).

### B. MOKE and magneto-ellipsometry characterization

The MOKE hysteresis was measured from 150 to 300 K using a free space MOKE measurement setup (MOKE system, East Changing Technologies), employing a 633 nm semiconductor laser. For magneto-ellipsometry characterization, we used a spectroscopic ellipsometry equipped with a vertical auto-rotating angle platform. The instrument can perform reflectance measurements on samples within a wavelength range of 300–1690 nm and at angles from 45° to 75°. A permanent magnet is placed beside the sample with the Voigt geometry, providing a 1500 Oe magnetic field along different crystalline orientations of the sample. The Mueller matrix is measured with or without magnetic field application. For variable-temperature Mueller matrix characterization, a custom-built temperature-controlling sample stage capable of heating the sample from 300 to 400 K was used, which is equipped with a commercially available semiconductor heater for temperature regulation.

#### 1. DFT calculations

_{3}Sn is calculated using DFT as well as the Kubo formula. The conductivity tensor of Mn

_{3}Sn assumes the following form:

*x*,

*y*, and

*z*correspond to the [2$1\u03041\u0304$0], [01$1\u0304$0], and [0001] directions, respectively.

*σ*

_{αβ}is calculated using the Kubo formula as follows:

_{c}represents the cell volume,

*N*

_{k}is the number of

*k*-points (Weyl points),

*f*(E) is the Fermi distribution function, and

*η*is the smearing parameter. Through the conversion relationship between conductivity and dielectric constant,

*δ*

_{αβ}is the Kronecker

*δ*function, which equals 1 when

*α*=

*β*and 0 otherwise.

To calculate Eqs. (2) and (3), the Kohn–Sham energy E_{nk} and wavefunction $\psi nk\u232a$ were obtained within the generalized-gradient approximation based on DFT, as implemented in the Quantum ESPRESSO package. A 6 × 6 × 6 *k*-point grid, ultrasoft pseudopotentials, and plane wave basis sets with cutoff energies of 500 eV for wavefunctions and 320 Ry for charge densities were employed. The calculated magnetic moment for each Mn atom was 3.07 *μ*_{B}. The total magnetic moment effectively canceled out within numerical error. We conducted calculations with a constraint on magnetization (*M*) and computed *θ*_{k} for *M* = 0.003 *μ*_{B}/f.u. The summation in Eq. (2) was performed using a Wannier-interpolated band structure with a 50 × 50 × 50 *k*-point grid and a smearing width of *η* = 0.5 eV.

We conducted *ab initio* calculations to explore the antiferromagnetic properties of Mn_{3}Sn, employing the Vienna *Ab initio* Simulation Package (VASP).^{26} For these simulations, we utilized relativistic ultrasoft pseudopotentials and applied the Perdew, Burke, and Ernzerhof version of the generalized gradient approximation (GGA) as our exchange–correlation functional.^{27} In our simulations, we adopted lattice parameters a = 5.67 Å and c = 4.53 Å, alongside the Mn atom’s 6*h* Wyckoff positions at *x* = 0.8388, as determined from experimental observations.^{28,29}

## III. RESULTS AND DISCUSSION

We first characterized the crystal structure, magnetic, and transport properties of the Mn_{3}Sn films. As shown in Fig. 1(a), the 2*θ* − *ω* XRD pattern for Mn_{3}Sn(0001) films deposited on Al_{2}O_{3}(0001) substrates. Figure 1(b) shows the 2*θ* − *ω* XRD pattern of (2$1\u03041\u0304$0) Mn_{3}Sn thin film deposited on the MgO(110) substrates, indicating that the prepared Mn_{3}Sn film with [2$1\u03041\u0304$0] and [0001] orientations is of high quality. More details of the crystal structure characterization can be found in our previous publication.^{18} Energy-dispersive spectroscopy (EDS) reveals that the stoichiometric ratio of the film is Mn_{3.12}Sn_{0.88}.

Figure 1(c) illustrates the magnetic hysteresis loop for magnetic field B//[2$1\u03041\u0304$0] and [0001] of Mn_{3}Sn. The magnetization (*M*) vs magnetic field (*B*) at 300 K shows a weak ferromagnetic moment of 27.4 and 2.3 emu/cm^{3}, and a coercivity of 200 and 250 Oe, respectively. The longitudinal magneto-optical Kerr (LMOKE) hysteresis with applied magnetic field along the [2$1\u03041\u0304$0] and [0001] orientations of the (0001) and (2$1\u03041\u0304$0) oriented Mn_{3}Sn epitaxial films are shown in Fig. 1(d). When the magnetic moments aligned along the [2$1\u03041\u0304$0] direction, a Kerr rotation angle |*θ*_{k}| of 16.2 mdeg was observed, accompanied by a square-shaped hysteresis loop. When the magnetic field is oriented parallel to the [0001] direction of the (2$1\u03041\u0304$0) oriented film, the LMOKE is nearly negligible. These results agree well with Mn_{3}Sn single crystals,^{1,11,12,16} indicating that the Mn_{3}Sn epitaxial films we have grown possess good crystal quality.

_{3}Sn using spectroscopic ellipsometry. In anisotropic samples, cross-polarization may occur. Thus, Mueller matrix analysis is required to address this issue. Figure 2 shows a top-down view of a single atomic layer Mn

_{3}Sn lattice. In each a–b plane, there is a kagome lattice made of Mn moments, each ∼3

*μ*

_{B}. This lattice has geometrical frustration, creating an inverse triangular spin arrangement. Hexagonal Mn

_{3}Sn is uniaxial anisotropic. We define the Cartesian coordinate axis to align with the crystal orientations of Mn

_{3}Sn. In particular, the [2$1\u03041\u0304$0] orientation is designated as the

*x*-direction [01$1\u0304$0], as the

*y*-direction, and [0001] as the

*z*-direction. The tensor expression for the dielectric constant of the Mn

_{3}Sn is then as follows:

Mn_{3}Sn is a uniaxial anisotropic material; when its optical axis ([0001] axis) is perpendicular to the plane of the incident light, it can minimize the off-diagonal elements of the Mueller matrix. This simplifies the steps of the standard ellipsometry method. Therefore, an Mn_{3}Sn epitaxial thin film with an out-of-plane orientation parallel to the *x*-direction was chosen. The Mueller matrix was measured on the 60 nm thick Mn_{3}Sn film covering 300–1690 nm wavelength range with a step size of 1 nm. Measurements were taken at multiple incident angles ranging from 50° to 70°, with a step size of 10°. Measurements were performed with in-plane rotation of the sample stage, covering angles from 0° to 360° with a step size set of 45°. The detailed results of the variable-angle Mueller matrix testing are shown in supplementary material S1. Utilizing these data, *ɛ*_{xx} = *ɛ*_{yy} and *ɛ*_{zz} can be directly obtained by fitting the data using a uniaxial anisotropic model. For detailed data fitting steps, please refer to supplementary material S3.

The measured diagonal elements of the permittivity tensor are shown in Fig. 3(a). Mn_{3}Sn thin films are semi-metallic, with absorption across the entire measurement wavelength range arising from intraband transitions. The real parts *ɛ*_{xx} and *ɛ*_{zz} are positive for wavelengths shorter than ∼680 nm (1.82 eV) and 510 nm (2.43 eV), respectively. At longer wavelengths, *ɛ*_{xx} and *ɛ*_{zz} become negative. The imaginary parts of *ɛ*_{xx} and *ɛ*_{zz} increase monotonically with the wavelength. These dielectric properties are typically attributed to intraband transitions of free carriers, which also agree well with the Drude model.

*ω*

_{p}is the plasma frequency, $\gamma =1\tau $ is the damping constant,

*ω*is the angular frequency of the electromagnetic wave, and

*ɛ*

_{∞}is the dielectric constant at high frequencies.

*x*and

*z*directions can be obtained as τ = 0.659 and 0.671 fs. The plasma frequency

*ω*

_{p}is 2.77 × 10

^{15}and 3.69 × 10

^{15}rad/s in the

*x*and

*z*directions, respectively. The plasma frequency can also be expressed as

*n*is estimated to be ∼1.5 × 10

^{22}cm

^{−3}.

^{1,30–32}Thus,

*m**/

*m*

_{e}are 4.17 and 2.35 along the

*x*and

*z*directions, respectively. This indicates that there are strong electronic correlation effects in Mn

_{3}Sn.

^{32}

_{3}Sn thin films, we measured the transverse magneto-optical Kerr effect (TMOKE). The Mueller matrix element difference spectra [Δ

*m*

_{ij}=

*m*

_{ij}(

*H*+) −

*m*

_{ij}(

*H*−),

*k*=

*x*,

*y*,

*z*] normalized to

*m*

_{11}were recorded as a function of wavelength at a 60° angle of incidence; see details in supplementary material S2. These measurements involved applying an external magnetic field of ±1500 Oe by permanent perpendicular to the incident plane. We obtained the Mueller matrix element difference five times under both positive and negative magnetic fields by changing the direction of the permanent magnet. We then calculated the variance and mean of the five test results, ensuring the accuracy of the test outcomes. At the same time, a rotatable magnet holder was designed to ensure that the center point of the magnetic field remains unchanged when the direction of the permanent magnet is altered. In Fig. 3(b), the TMOKE spectra of Mn

_{3}Sn(0001) epitaxial thin films with magnetic field along the [2$1\u03041\u0304$0] crystal orientation are displayed within the wavelength range of 300–1690 nm, showcasing measurements taken at magnetic field strengths of 1500 and 0 Oe, respectively. Combining with supplementary material S3, the calculation formula for TMOKE can be derived as follows:

^{33}

^{,}

_{3}Sn at 600 nm is consistent with the findings reported in the literature regarding LMOKE and in agreement with first-principles calculations.

^{11,16}It is evident that the TMOKE value decreases by half at 0 Oe compared to 1500 Oe. Although the net magnetization is zero, the system lacks a “good” symmetry with spatial symmetry (translational or spatial inversion), which breaks Kramer’s spin degeneracy of its energy bands. In the presence of spin–orbit coupling and Berry curvature, MO effects can occur in Mn

_{3}Sn.

^{15,34–36}

To evaluate the anisotropic MO effect, Mueller matrices were measured on the Mn_{3}Sn(0001) sample with magnetic field along [2$1\u03041\u0304$0] or on the Mn_{3}Sn (2$1\u03041\u0304$0) sample with magnetic field along [0001], respectively. Based on the measured Mueller matrix elements and the diagonal component of Mn_{3}Sn, we computed the off-diagonal dielectric tensor components using Yeh’s 4 × 4 matrix formalism as shown in Figs. 3(c) and 3(d), see details in supplementary material S3. The real [Re(ε_{yz})] and imaginary [Im(ε_{yz})] parts of *ɛ*_{yz} peak at 450 nm (2.76 eV) and 680 nm (1.82 eV), respectively. However, when the magnetic field aligns with the [0001] direction, the MO effect approaches zero, leading to ε_{xyw} ∼ 0 across the entire measured spectrum range, as shown in Fig. 3(c). This result also agrees with previous reports on anisotropic LMOKE characterizations.^{11} In Mn_{3}Sn, the Mn moments are confined within the a–b plane, with no out-of-plane component. In this inverse triangular configuration, only one of the moments in each Mn triangle aligns with the local easy axis. This unique spin magnetic structure and crystal structure are responsible for the anisotropic MOKE observed in Mn_{3}Sn. To ascertain the origin of the MO effect in Mn_{3}Sn, we conducted MO measurements on Mn_{3}Sn epitaxial thin films under zero magnetic field, as shown in Fig. 3(d). In this case, the off-diagonal elements of the dielectric constant value decrease by ∼40%. Nevertheless, the large *ε*_{yz} demonstrates that the magnitude of the MO effect is mostly influenced by the Berry curvature.^{37}

Next, we studied the temperature dependence of the permittivity tensor of Mn_{3}Sn. We conducted temperature-dependent TMOKE measurements at a wavelength of 633 nm, as shown in Fig. 4(a). As the temperature decreased to 250 K, the Kerr rotation reached its peak value, followed by a fast decrease with the lowering of the temperature, corresponding to the spin structure change. Within the temperature range where Mn_{3}Sn maintains the kagome lattice, the value of TMOKE gradually decreases from 250 to 390 K and then sharply drops to zero at 400 K, as shown in Fig. 4(b). Mn_{3}Sn single crystal shows a Néel temperature of 420 K. The deviation from the Néel temperature of the bulk single crystal may be attributed to the off-stoichiometry of Mn_{3}Sn thin films, which was measured to be Mn_{3.12}Sn_{0.88} in our films. The spin structure of Mn_{3}Sn manifested a noncollinear inverse-triangular spin configuration between 270 and 420 K, transitioned to a helical spin ordering state within the temperature range of approximately 50–270 K, and eventually exhibited a spin glass phase below 50 K. The measured transition temperature of the helical spin structure is 250 K, which is also due to the Mn-rich stoichiometry of Mn_{3}Sn.^{21,38} These temperatures were clearly depicted in Fig. 4(b), suggesting TMOKE to be a good probe for magnetic phase transition measurements.

As depicted in Figs. 4(c) and 4(d), we acquired the real and imaginary components of *ε*_{yz} within the temperature range of 300–400 K, under a magnetic field of 1500 Oe. It can be seen that as the temperature increases, the values of Re(*ε*_{yz}) and Im(*ε*_{yz}) change significantly. In the shorter wavelength range (300–800 nm), the changes in the curves for different temperatures are relatively small. The amplitude of Re(*ε*_{yz}) and Im(*ε*_{yz}) monotonically decreases with temperature. However, for wavelengths larger than 800 nm, the curves show stronger divergence. $\epsilon yz\u2032$ continued to decrease, reaching more negative values, particularly for higher temperatures. As the wavelength increases beyond 800 nm, $\epsilon yz\u2032\u2032$ becomes positive, indicating increased magnetic circular dichroism at longer wavelengths. At 400 K, the MOKE goes to 0 thanks to the phase transition of the spin structure. This result indicates a significant impact of temperature variation on the MOKE in the near-infrared wavelength range. Future studies on temperature dependent spin and electronic structures may reveal more about the origin of the observed phenomenon.

To elucidate the origin of the MO effect in Mn_{3}Sn, we calculated the permittivity tensor using DFT, as shown in Fig. 5. The analysis is grounded on a minimal model that characterizes a single pair of Weyl points near the Fermi level (E_{F}) resulting from the crossing of linearized bands. The derived fitting formula for *σ*_{yz} involves three essential parameters: the strength of Berry curvature, $\Omega \u0303$, in the vicinity of E_{F}, the separation in *k*-space, Δ*k*, between the two Weyl points, and the energy of the Weyl point, E, relative to E_{F}. In the case of systems with multiple pairs of Weyl points, the E value provides an estimate of the lowest possible Weyl point energy. For Mn_{3}Sn, this model effectively describes the experimental *σ*_{yz}, and the E value of Weyl points obtained from the best fit aligns reasonably well with the predicted energy from band structure calculations, as shown in Fig. 5(a). The Mn_{3}Sn thin films often contain an excess of Mn, as discussed previously.^{38} Given that the electronic states at the E_{F} are primarily occupied by 3*d* electrons, even a minor amount of Mn doping can induce a significant shift in the chemical potential relative to the E_{F} in the stoichiometric case. For instance, a 1% Mn doping in Mn_{3}Sn may lead to a chemical potential shift of ∼6 meV above E_{F}.^{39} As a result, in the case of Mn composition with Mn_{3.12}Sn_{0.88}, the chemical potential would be E = E_{F} + 72 meV. Next, utilizing the Kubo formula, we calculated the permittivity tensor of Mn_{3}Sn, as described in formula (3).

The calculation results are shown in Figs. 5(b) and 5(c). This property is primarily associated with the density of states (DOS) and the probability of interband transitions, shown as typical Drude characteristics. The calculated results agreed well with the measurement spectrum shown in Fig. 3. The off-diagonal part of the permittivity tensor is shown in Fig. 5(c). Strong anisotropy is observed for the MO effect. For *ε*_{yz}, two peaks were observed at 3.1 and 1.6 eV, respectively. These two peak transitions agree well with the measurement results in Fig. 3(c), which indicate the peaks of the magnetic circular dichroism. The origin of these two peaks can be inferred from the DOS calculations. The DOS results in this case are depicted in Fig. 5(d), in agreement with previous reports.^{40–42} As shown in Figs. 3(c) and 5(c), a peak at 3.1 eV in the imaginary part of the *ε*_{yz} is observed. This corresponds to the peaks of the DOS of the Mn 3*d* orbitals shown in Fig. 5(d). Whereas the broad peak at around 1–2 eV in the imaginary part of *ε*_{yz} may be attributed to the transition of the electrons on the occupied states near the Weyl point. These results show a close relation of MO effects to the electron transition near the Weyl points, which further proves that the non-zero Berry curvature caused by the octupole magnetic moment of Mn_{3}Sn is a key factor in the MO effect of Mn_{3}Sn.

## IV. CONCLUSIONS

In summary, we determined the anisotropic permittivity tensor of Mn_{3}Sn epitaxial thin films at visible to near infrared wavelength range using magneto-ellipsometry measurements. The film showed clear uniaxial anisotropy of the optical constants, whereas its anisotropic MO effect is related to the noncollinear spin structure. Temperature dependent magneto-ellipsometry measurements of Mn_{3}Sn indicate that the non-zero Berry curvature induced by the octupole magnetic moment is the primary source of the MO effect. The measured permittivity tensor shows good agreement with first-principles calculation results. Our results provide an understanding of the anisotropic optical and MO effects of Mn_{3}Sn. The results are potentially useful for the development of spintronic and MO devices based on such materials.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional information, including the complete Mueller matrix element spectrum in spectroscopic ellipsometry dependent on the incident angle and sample rotation angle, Mueller matrix element differences Δm33 and Δm34 at 1500 and 0 Oe, and ellipsometry data fitting and solving the diagonal and off-diagonal components of the permittivity tensor.

## ACKNOWLEDGMENTS

The authors are grateful for the support of the Ministry of Science and Technology of the People’s Republic of China (MOST) (Grant No. 2021YFB2801600), the National Natural Science Foundation of China (NSFC) (Grant Nos. U22A20148, 51972044, 52021001, 52102357, and 51902033), the Sichuan Provincial Science and Technology Department (Grant No. 99203070), the China Postdoctoral Science Foundation (Grant No. M2020683282), and the Scientific Research Foundation of Chengdu University of Information Technology (Grant Nos. KYTZ202006 and KYQN202312).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

D.G. and L.B. conceived the research idea; D.G. designed experiments, fabricated the sample, and wrote the manuscript; D.G., W.Y., and J.S. conducted the sample characterization and analyzed the data; T.Y. and X.L. conducted DFT calculations. X.L. and L.B. supervised the study. All authors contributed to the technical discussions and writing of the manuscript.

**Dong Gao**: Data curation (equal); Formal analysis (equal); Writing – original draft (equal). **Ting Yang**: Formal analysis (equal). **Fu Tang**: Formal analysis (equal). **Jiejun Su**: Formal analysis (equal). **Weihao Yang**: Formal analysis (equal). **Dengfu Deng**: Formal analysis (equal). **Yunfei Xie**: Formal analysis (equal). **Jun Qin**: Formal analysis (equal). **Xiao Liang**: Data curation (equal); Formal analysis (equal). **Lei Bi**: Funding acquisition (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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