Anisotropic optical and magneto-optical properties of antiferromagnetic Weyl semimetal Mn₃Sn epitaxial thin films

The antiferromagnetic (AFM) Weyl semimetal material, Mn₃Sn, 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 Mn₃X 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 Mn₃X 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 Mn₃Sn films.^4,5,13,14 In particular, the MO effect of Mn₃Sn has attracted research interest. The MO effect of Mn₃Sn 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 Mn₃Sn based spintronic devices.^11,16,19 Later, a large ultrafast-modulated Voigt effect was observed in Mn₃Sn 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 Mn₃Sn films grown on SiO₂ and oxidized silicon substrates.²¹ Meanwhile, the giant MO effect of Mn₃Sn 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 Mn₃Sn? Because Mn₃Sn 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$1̄1̄$0] or the [0001] orientations. Because the measured MOKE is related to the full anisotropic permittivity tensor of Mn₃Sn, the determination of the permittivity tensor is important for the interpretation of the spin orientation in spintronic device characterization and for the development of Mn₃Sn based MO devices.

In this work, we characterized the anisotropic optical and MO properties of Mn₃Sn 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 Mn₃Sn, 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.

In collinear spin structure altermagnets such as RuO₂ 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$1̄1̄$0)-oriented Mn₃Sn epitaxial thin films were prepared by pulsed laser deposition (PLD) on single crystalline Al₂O₃(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 Mn₃Sn film was characterized by SEM. Magnetic hysteresis was measured by a commercial superconducting quantum interference device (SQUID) magnetometer (MPMS3, Quantum Design).

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.

To calculate Eqs. (2) and (3), the Kohn–Sham energy E_nk and wavefunction $ψnk〉$ 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 Mn₃Sn, 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 6h Wyckoff positions at x = 0.8388, as determined from experimental observations.^28,29

We first characterized the crystal structure, magnetic, and transport properties of the Mn₃Sn films. As shown in Fig. 1(a), the 2θ − ω XRD pattern for Mn₃Sn(0001) films deposited on Al₂O₃(0001) substrates. Figure 1(b) shows the 2θ − ω XRD pattern of (2$1̄1̄$0) Mn₃Sn thin film deposited on the MgO(110) substrates, indicating that the prepared Mn₃Sn film with [2$1̄1̄$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_3.12Sn_0.88.

Figure 1(c) illustrates the magnetic hysteresis loop for magnetic field B//[2$1̄1̄$0] and [0001] of Mn₃Sn. The magnetization (M) vs magnetic field (B) at 300 K shows a weak ferromagnetic moment of 27.4 and 2.3 emu/cm³, and a coercivity of 200 and 250 Oe, respectively. The longitudinal magneto-optical Kerr  (LMOKE) hysteresis with applied magnetic field along the [2$1̄1̄$0] and [0001] orientations of the (0001) and (2$1̄1̄$0) oriented Mn₃Sn epitaxial films are shown in Fig. 1(d). When the magnetic moments aligned along the [2$1̄1̄$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$1̄1̄$0) oriented film, the LMOKE is nearly negligible. These results agree well with Mn₃Sn single crystals,^1,11,12,16 indicating that the Mn₃Sn epitaxial films we have grown possess good crystal quality.

Mn₃Sn 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 Mn₃Sn 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 Mn₃Sn 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). Mn₃Sn 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.

To evaluate the anisotropic MO effect, Mueller matrices were measured on the Mn₃Sn(0001) sample with magnetic field along [2$1̄1̄$0] or on the Mn₃Sn (2$1̄1̄$0) sample with magnetic field along [0001], respectively. Based on the measured Mueller matrix elements and the diagonal component of Mn₃Sn, 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 Mn₃Sn, 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 Mn₃Sn. To ascertain the origin of the MO effect in Mn₃Sn, we conducted MO measurements on Mn₃Sn 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 Mn₃Sn. 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 Mn₃Sn 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). Mn₃Sn 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 Mn₃Sn thin films, which was measured to be Mn_3.12Sn_0.88 in our films. The spin structure of Mn₃Sn 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 Mn₃Sn.^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. $εyz′$ continued to decrease, reaching more negative values, particularly for higher temperatures. As the wavelength increases beyond 800 nm, $εyz′′$ 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 Mn₃Sn, 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, $Ω̃$⁠, in the vicinity of E_F, the separation in k-space, Δk, 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 Mn₃Sn, 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 Mn₃Sn 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 3d 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 Mn₃Sn 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_3.12Sn_0.88, the chemical potential would be E = E_F + 72 meV. Next, utilizing the Kubo formula, we calculated the permittivity tensor of Mn₃Sn, 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 3d 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 Mn₃Sn is a key factor in the MO effect of Mn₃Sn.

In summary, we determined the anisotropic permittivity tensor of Mn₃Sn 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 Mn₃Sn 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 Mn₃Sn. The results are potentially useful for the development of spintronic and MO devices based on such materials.

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.

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).

The authors have no conflicts to disclose.

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).

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