Introduction

The Faraday effect is crucial for numerous scientific and technological advancements in astronomy, biology, chemistry, physics, and materials science. For instance, it is used for investigating the magnetic domain structure in solids1,2, nuclear magnetic resonance in fluids via optical detection3,4, paramagnetic gas molecule detection5, determination of magnetic fields6 and electron-density distribution in outer space and celestial objects7, probing spin coherence in cold atoms8, quantum spin fluctuation measurements9, biochemical and biomolecular detection10, stabilization of laser frequency11, optical current sensing12, optical Hall effect13, and optical isolators14.

In recent years, extraordinary progress has been made in exploring the unique physical phenomena in atomically thin transition metal dichalcogenide (TMDC) semiconductors15,16,17,18,19,20,21. In two-dimensional TMDCs, Coulomb-bound electron–hole composite quasiparticles such as neutral and charged excitons possess large binding energies and giant oscillator strengths19,21, when compared to the traditional group III–V or II–VI quantum wells22,23. These exceptional properties enable investigating 2D quantum effects even at room temperature. Furthermore, the corners of the Brillouin zone i.e. the K± valleys in TMDCs selectively couple to circularly polarized light21,24. The magnetic moments associated with the neighboring K+ and K− valleys are opposite to each other. This property leads to effects unique to atomically thin TMDCs, such as valley polarization24, valley coherence25, valley Zeeman splitting26,27, the valley Hall effect28, the valley-selective optical Stark effect29, and magnetic-field-induced valley polarization30,31,32,33,34,35,36,37. From the magneto-optics perspective, TMDCs have been extensively studied using magneto-photoluminescence, magneto-reflectance and magneto-transmittance21. However, the classic magneto-optical phenomena such as the Faraday and Kerr effect still remain to be explored experimentally due to major challenges involved in these measurements on the microscopic level38,39,40.

For 2D materials, the Faraday and Kerr effects can provide crucial information on the valley-related processes, such as valley Zeeman splitting38 and magnetic-field-induced valley polarization38. While Faraday rotation is generally accompanied with ellipticity when light passes through an absorbing medium (Fig. 1), a measurement of only Faraday rotation is sufficient for obtaining all the magneto-optical information about the sample1.

Fig. 1: Faraday effect in a 2D semiconductor.
figure 1

Schematic drawing depicting how linearly polarized light passes through an atomically thin semiconductor under a magnetic field and acquires Faraday rotation and ellipticity.

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Here, we show that the giant oscillator strength and large valley Zeeman splitting of excitons in WSe2 and MoSe2 monolayers result in the highest known Verdet constant (Faraday rotation per unit length per unit magnetic field) in a material in the visible regime31,32,33,34,35,36,41,42,43,44,45,46,47,48. We measure giant Faraday rotations on the order of degrees under a small magnetic field of around 1 T.

Faraday rotation arises from the non-zero off-diagonal term of the complex dielectric tensor of a material. For light incident along the z-direction perpendicular to the sample plane with B||z, (i.e. Faraday geometry), the complex dielectric tensor for the xy plane can be written as1,49,50,51

$$\mathop{\epsilon }\limits^{\leftrightarrow}=\left(\begin{array}{cc}{\widetilde{\epsilon }}_{{xx}} & {\widetilde{\epsilon }}_{{xy}}\\ -{\widetilde{\epsilon }}_{{xy}} & {\widetilde{\epsilon }}_{{yy}}\end{array}\right)$$
(1)

The diagonal components are responsible for the conventional optical response of materials in reflectance, transmittance, and absorption. For materials with a high degree of in-plane symmetry, such as monolayer MX2 (M = Mo, W; X = S, Se), the diagonal components are equal i.e. \({\widetilde{\epsilon }}_{{xx}}={\widetilde{\epsilon }}_{{yy}}\). Normally, in semiconductors, off-diagonal terms \({\widetilde{\epsilon }}_{{xy}}\) are zero in the absence of a magnetic field49. But these components reach very large values around the exciton resonances under magnetic fields, due to the exciton Zeeman splitting39. We note that in two-dimensional electron/hole gases in the quantum Hall regime, \({\widetilde{\epsilon }}_{{xy}}\) also rises sharply around inter-Landau level transition energies52. However, usually, the magnetic field to reach the quantum Hall regime is much higher compared to the one required for large Faraday effects around Zeeman-split exciton energies1,53. Far from the exciton lines, \({\widetilde{\epsilon }}_{{xy}}\) are smaller by many orders of magnitude. \({\widetilde{\epsilon }}_{{xy}}\) is given as49,54

$${\widetilde{\epsilon }}_{{xy}}(B)={\epsilon }_{{xy}1}+i{\epsilon }_{{xy}2}=\frac{\widetilde{V}{\lambda }_{0}{\widetilde{n}}_{{xx}}B}{\pi }$$
(2)

where \({\widetilde{n}}_{{xx}}=n+{ik}=\sqrt{{\widetilde{\epsilon }}_{{xx}}}=\sqrt{{\epsilon }_{{xx}1}+{i\epsilon }_{{xx}2}}\) is the complex refractive index, and \(\widetilde{V}={V}_{{FR}}+i{V}_{{FE}}\) is the complex Verdet constant with its components \({V}_{{FR}}\) and \({V}_{{FE}}\). The complex Verdet constant is related to the complex Faraday rotation \({\widetilde{\phi }}_{F}={\phi }_{F}+i{\eta }_{F}\) as \(\widetilde{V}={\widetilde{\phi }}_{F}/({d\; B})\) where \({\phi }_{F}\) and \({\eta }_{F}\) are Faraday rotation and Faraday ellipticity, respectively, d is the sample thickness, B is the magnetic field. We note that \({\widetilde{\epsilon }}_{{xy}}\) is related to the popularly defined “Voigt constant” Q as \(Q=i{\widetilde{\epsilon }}_{{xy}}/{\widetilde{\epsilon }}_{{xx}}\)50.

Results

Excitonic Faraday rotation in MoSe2 and WSe2 monolayers

We measure the Faraday rotation (the real component of \({\widetilde{\phi }}_{F}\), i.e., \({\phi }_{F}\)) of light around the neutral and charged A exciton transitions in a hBN-encapsulated MoSe2 monolayer, and the neutral A exciton transition in a hBN-encapsulated WSe2 monolayer. The substrates are c-cut double-side polished sapphire, which enable optical transmission measurements. Details about our experimental setup are described in ref. 38 and are briefly summarized in the supporting information. Here, we first discuss the case of the MoSe2 sample, and extend our conclusions to the WSe2 sample afterwards. Figure 2a shows the measured optical transmittance spectrum of an hBN-encapsulated MoSe2 monolayer in the spectral region of the neutral and charged exciton (trion) at a temperature of T = 10 K. The measured spectrum (solid spheres) is modeled using the transfer-matrix method (lines) to incorporate the effects of optical interference due to multiple layers of the sample on the spectral line shape55,56. The excitonic contribution to the dielectric function is described as a complex Lorentzian

$${\widetilde{\epsilon }}_{{xx}}\left(E\right)={\epsilon }_{{xx}1}+i\,{\epsilon }_{{xx}2}={\left({n}_{b}+i{k}_{b}\right)}^{2}+{\sum }_{j}\frac{{A}_{j}}{{E}_{0j}^{2}-{E}^{2}-i{\gamma }_{j}E}$$
(3)

where, \({A}_{j},{E}_{0j},\) and \({\gamma }_{j}\) are the oscillator strength, transition energy and full-width at half-maximum (FWHM) linewidth of the \({j}^{th}\) resonance. \({n}_{b}+i{k}_{b}\) is the complex background dielectric function of monolayer MoSe2 without excitonic contributions57. The exciton and trion resonances A and T are at 1.634 eV and 1.607 eV, respectively (Fig. 2a). Narrow FWHM linewidths (2.7 meV and 4.0 meV, respectively) approaching the homogeneous linewidth limit indicate the excellent quality of our samples58,59,60.

Fig. 2: Excitonic Faraday rotation in hBN-encapsulated monolayers of MoSe2 and WSe2.
figure 2

a Optical transmission spectrum of hBN/1L MoSe2/hBN (left panel) and f hBN/1L WSe2/hBN (right panel) around the exciton and trion resonances A and T in the MoSe2 sample and A exciton in the WSe2 sample. A strong dip is observed around the neutral A exciton resonance. b, g Experimental (spheres) and modeled (solid lines) Faraday rotation spectra of resonances in magnetic fields ranging from \(B=0.2\,{{{{{\rm{T}}}}}}-1.4\,{{{{{\rm{T}}}}}}\) for the two samples. The spectra are vertically shifted by 0.5° successively for clarity. The shift is mentioned along with the respective plots. Characteristic Faraday rotation spectral line shapes are revealed around the resonance energies. The exciton transition exhibits a large peak Faraday rotation of 2.2° and 1.7°, respectively, for the MoSe2 and WSe2 cases at \(B=1.4\,{{{{{\rm{T}}}}}}\). c, h Real and imaginary parts of the off-diagonal dielectric function of the two materials at B = 1.4 T derived from our measurements, as explained in the main text and the supporting information. d, i Zeeman splittings of the resonances derived from line shape modeling. Linear fits (solid lines) are used to derive the effective g-factors \({g}_{A}\) and \({g}_{T}\). e, j Measured peak Faraday rotation of the resonances as a function of the magnetic field. Peak rotation is fitted linearly for deriving the Verdet constants VA = −(2.3 ± 0.2) × 107 deg T−1 cm−1 and VT = −(0.9 ± 0.2) × 106 deg T−1 cm−1 for the MoSe2 and VA = −(1.9 ± 0.2) × 107 deg T−1 cm−1 for the WSe2 sample.

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Faraday rotation ϕF is measured in this spectral region under moderate out-of-plane applied magnetic fields ranging from B = 0.2–1.4 T. We observe characteristic Faraday rotation line shapes with a dip around the exciton and trion energies. From our data, we determine the real and imaginary components (\({\epsilon }_{{xy}1}\) and \({\epsilon }_{{xy}2}\)) of the off-diagonal dielectric constant \({\widetilde{\epsilon }}_{{xy}}\) of monolayer MoSe2 following Eq. (2). An example of \({\epsilon }_{{xy}1}\) and \({\epsilon }_{{xy}2}\) for \(B=1.4\,{{{{{\rm{T}}}}}}\) is shown in Fig. 2c. As required by Eq. (2), the procedure involves a calculation of Faraday ellipticity using a Kramers–Kronig analysis of our data in Fig. 2b61, as well as the complex diagonal dielectric function \({\widetilde{\epsilon }}_{{xx}}\). Details of the calculation are provided in the supporting information.

The Faraday rotation line shapes in Fig. 2b are modeled using the transfer-matrix method to determine the Zeeman splittings of A and T as a function of magnetic field (Fig. 2d)38,55,56. First, the transmittance spectrum is modeled as described before. The complex transmission Fresnel coefficients for left and right circular polarizations σ± are given as

$${\widetilde{t}}_{\pm }={t}_{\pm }{e}^{i{\phi }_{\pm }}$$
(4)

They are identical in the absence of an external magnetic field. \({\widetilde{t}}_{\pm }\) are obtained from modeling the transmission spectrum. In the presence of a magnetic field, excitons and trions undergo a valley Zeeman splitting, leading to different \({\widetilde{t}}_{\pm }\), and thereby different ϕ±. The Faraday rotation of polarized light under a magnetic field is related to ϕ± as

$${\phi }_{F}=-\frac{1}{2}\left({\phi }_{+}-{\phi }_{-}\right)$$
(5)

Solid lines in Fig. 2b are the fits to the Faraday rotation spectra using this procedure. The modeling provides the valley Zeeman splittings of the excitons and trions as a function of magnetic field, which are plotted in Fig. 2d. The absolute value of the Zeeman splitting increases linearly as a function of the magnetic field. The Zeeman splitting of an exciton resonance X is given as \(\Delta E={g}_{{{{{{\rm{X}}}}}}}{\mu }_{B}B\), where \({g}_{{{{{{\rm{X}}}}}}}\) is the effective exciton g-factor and \({\mu }_{B}=5.788\times {10}^{-5}{{{{{\rm{eV}}}}}}{{{{{{\rm{T}}}}}}}^{-1}\) is the Bohr’s magneton. By fitting the data in Fig. 2d using this relation, we obtain the exciton and trion g-factors as \({g}_{A}=-4.3\pm 0.3\) and \({g}_{T}=-4.8\pm 0.6\). These values are in excellent agreement with literature values21,33,45,46. Figure 2f–j presents the results for an hBN-encapsulated WSe2 monolayer around the A exciton, yielding similar results as for MoSe2.

We find that around the A exciton, the magnitude of the Faraday rotation relative to the flat spectral background (i.e., 0° rotation) is 2.2° for monolayer MoSe2, and 1.7° for monolayer WSe2 at \(B=1.4\,{{{{{\rm{T}}}}}}\). The magnitude of Faraday rotation for the A and T transitions in MoSe2 are plotted as a function of the magnetic field in Fig. 2e, and for the A exciton in WSe2 in Fig. 2j. The rotation increases linearly with the magnetic field for all resonances. The Faraday rotation (real part) around a resonance X at a magnetic field B is given as

$${\phi }_{F}={V}_{{{{{{\rm{X}}}}}}}\,d\,B$$
(6)

where \({V}_{{{{{{\rm{X}}}}}}}\) is the Verdet constant of the resonance, and d is the layer thickness. Fitting the data in Figs. 2e, j using this relation yields Verdet constants of A and T, respectively as VA = −(2.3 ± 0.2) × 107 deg T−1 cm−1 and VT = −(0.9 ± 0.2) × 106 deg T−1 cm−1 for monolayer MoSe2, and VA = −(1.9 ± 0.2) × 107 deg T−1 cm−1 for monolayer WSe2.

Faraday rotation around interlayer excitons in a MoS2 bilayer

As an example of a material with a positive Verdet constant, we perform Faraday rotation spectroscopy of interlayer excitons (IL) in a hBN-encapsulated MoS2 bilayer (Fig. 3). Interlayer excitons in bilayer and bulk TMDCs are known to have a positive g-factor which is opposite in sign to the intralayer excitons21,62,63. However, interlayer excitons have a much smaller oscillator strength and a larger line width compared to the intralayer excitons62,63,64. Therefore, their Verdet constant is expected to be smaller. The measured transmittance (for B = 0) and the Faraday rotation spectra (under \(B=0.4-1.4\,{{{{{\rm{T}}}}}}\)) of the hBN-encapsulated MoS2 bilayer sample are shown in Fig. 3a. Clear signatures corresponding to the intralayer exciton (A) at 1.930 eV, intralayer trion (T) at 1.909 eV, and a split interlayer exciton (IL1 at 1.994 eV and IL2 at 2.004 eV) are visible in the Faraday rotation spectra. The assignment of the features are performed on the following grounds: A and T resonances have similar g-factors, nearly equal to −4, suggesting their intralayer character21,63. T polarizes strongly under magnetic field, with its valley polarization approaching ≈ +14% (Fig. 3c). Furthermore, A polarizes only weakly (valley polarization ≈ −2%). The large polarization of T with an opposite sign compared to A is characteristic for the appearance of a trion–exciton pair65. We notice that the binding energy of the trion in the MoS2 bilayer is about 21 meV. In comparison, the reported value in a non-encapsulated bilayer is 27 meV66. A smaller value in our work signifies the effect of an increased dielectric constant around the trion, due to hBN encapsulation. The split IL exciton lines are identified due to their positive g-factors (\({g}_{{{{{{\rm{IL}}}}}}1}=+ 6.6\pm 0.3\) and \({g}_{{{{{{\rm{IL}}}}}}2}=+ 7.2\pm 0.3\))21,62,63 in Fig. 3d. In previous works, one IL resonance has been observed in optical reflectance spectra21,63. In our transmittance spectra, we also notice one (broad) IL line (Fig. 3a), while Faraday rotation spectroscopy is able to resolve two close-lying IL features due to the high sensitivity of the technique (Fig. 3b). We believe that the reason for the appearance of the two IL features is the Stark effect splitting of the IL exciton due to a static electric field67,68, which can be created by charge transfer from impurities in the substrate69. The Verdet constant of the interlayer excitons are VIL1 = +(1.9 ± 0.5) × 105 deg T−1 cm−1 and VIL2 = +(2.2 ± 0.5) × 105 deg T−1 cm−1 (Fig. 3e). In contrast, the intralayer exciton in this sample has a Verdet constant of VA = −(1.5 ± 0.2) × 106 deg T−1 cm−1.

Fig. 3: Faraday rotation of intralayer and interlayer excitons in hBN-encapsulated bilayer MoS2.
figure 3

a Optical transmission spectrum of hBN/2 L MoS2/hBN displaying trion T, intralayer exciton A, and a pair of interlayer excitons IL1 and IL2. b Experimental (spheres) and modeled (solid lines) Faraday rotation spectra of resonances in magnetic fields ranging from \(B=0.4\,{{{{{\rm{T}}}}}}-1.4\,{{{{{\rm{T}}}}}}\). The spectra are vertically shifted by 0.1° successively for clarity. The shift is mentioned along with the respective plots. The characteristic line shapes of the T, A, and IL resonances are used for their assignment explained in the main text. c Magnetic-field-induced valley polarization of A and T derived from modeling in (b). d Valley Zeeman splittings deduced for the four resonances from the line shape modeling in (b). Linear fits (solid lines) are used to derive the effective g-factors \({g}_{A}\) and \({g}_{T}\). e Measured peak Faraday rotation of the resonances as a function of the magnetic field. Peak rotation is fitted linearly for deriving the Verdet constants VA = −(1.5 ± 0.2) × 106 deg T−1 cm−1, VIL1 = +(1.9 ± 0.5) × 105 deg T−1 cm−1 and VIL2 = +(2.2 ± 0.5) × 105 deg T−1 cm−1.

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Discussion

In Table 1, we compare the derived Verdet constants for our non-magnetic atomically thin semiconductors with conventional semiconductors, diluted magnetic semiconductors (DMSs), and magneto-optical elements used in optical isolators, covering the visible to infrared wavelength range14,70. Most importantly, we find that the Verdet constants for monolayer MoSe2 and WSe2 are larger by many orders of magnitude compared to other materials in this wavelength region14,70. A comparable magnitude is only known in the far-infrared (Terahertz) region, where graphene has a Verdet constant of 2.7 × 107 deg T−1 cm−1 (see ref. 52), which is due to inter-Landau-level transitions (in the low-doping regime) or cyclotron resonances (high-doping regime) in the Terahertz region52.

Table 1 Comparison of the Verdet constant of the excitons in our work with typical Verdet constants of excitons in semiconductors, diluted magnetic semiconductors (DMSs), and Faraday rotators used in optical isolators
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In general, for the case of an ultrathin dielectric material with the thickness of our monolayer samples, Faraday rotation is expected to be negligibly small for energies far away from excitonic resonances. For example, the Faraday rotation of a 0.65 nm (typical thickness of a TMDC monolayer) thick InSb crystal in the sub-bandgap region at 0.23 eV under a magnetic field of B = 1.4 T (maximum applied field in this work) would be of the order of 10−7 deg71. However, close to an excitonic resonance, Faraday rotation is enhanced by many orders of magnitude. For instance, Faraday rotation for InAs at the exciton resonance is expected to be about 10−3 deg at the abovementioned conditions, which is four orders of magnitude larger71. In the case of TMDC monolayers, this value is enhanced even further by another 2–3 orders of magnitude leading to a giant Faraday effect. This is explained as follows. In general, Faraday rotation around exciton resonances occurs if both oscillator strength and Zeeman splitting of the exciton transition are non-zero. A large oscillator strength implies strong dips in transmission amplitudes t± for the σ± transitions (Eq. (4)). This results in an appreciable phase difference \({\phi }_{+}-{\phi }_{-}\) and thereby a strong Faraday effect around the exciton transition. This enhancement of up to four orders of magnitude in the Verdet constant around the exciton transitions, compared to the sub-bandgap region, has been confirmed earlier in III–V and II–VI semiconductors1. The observed giant excitonic Faraday rotation in TMDC monolayers is however due to a combined effect of (i) a giant exciton oscillator strength21,72,73,74,75, and (ii) a large exciton g-factor (≈ −421,27,44,76), when compared to conventional III–V and II–VI semiconductors1,56,77. The giant exciton oscillator strength in TMDCs is both due to the small excitonic spatial extent (≈1 nm) and the character of the electron and hole wavefunctions (localized in d-orbitals of the transition metals)72,73,74,75,78. Furthermore, considering the Wannier model, TMDCs are expected to have a large joint density of states at the van-Hove singularity at the K point due to a large exciton reduced mass21,79,80, which is about 5–10 times larger compared to a typical III–V semiconductor such as GaAs77,81. This results in a giant exciton oscillator strength. The exciton g-factor of ≈ −4 in TMDC monolayers is mainly due to the contributions of d-orbitals at the top of the valence bands at the K point, with modifications due to electron–hole interactions21,27,76,78. This is a large value compared to exciton g-factors in typical III–V 2D semiconductors i.e. quantum wells. For instance, the heavy-hole exciton g-factor in a GaAs/AlGaAs quantum well varies from −2 to +1 for well widths increasing from 2 to 25 nm56,82. The large exciton g-factor in a TMDC monolayer results in an appreciable lifting of the energetic degeneracy of the σ± polarized exciton transitions under a magnetic field. As a result, a strong Faraday effect is observed around the exciton resonance due to a large phase difference ϕ± under a magnetic field (Eq. 5)83. We note that in the present work, Landau quantization effects84,85,86,87 can be neglected due to the following reasons: (i) the large exciton binding energy (>150 meV) in hBN-encapsulated monolayers strongly dominates the magnetic quantization energy scale (i.e., \({\hslash }{{{{{\rm{eB}}}}}}/{{2}{{{{{\rm{m}}}}}}}^{*}\) ≈ 0.3 meV at \(B=1.4\,{{{{{\rm{T}}}}}}\)) at the magnetic fields used19,21,53. This situation is unlike the case of graphene, where such Coulomb interactions are absent, and low magnetic fields (<5 T) are sufficient to observe Landau quantization in the THz regime52. (ii) Our hBN-encapsulated samples are not in the highly-doped regime, as evidenced by low (high) trion (exciton) oscillator strengths (Figs. 2 and 3). Inter-Landau transition effects in TMDCs have only been observed previously under high-doping conditions and large magnetic fields85.

In conclusion, we have measured giant Faraday rotation around the A exciton transitions in hBN-encapsulated MoSe2 and WSe2 monolayers, as well as interlayer excitons in a MoS2 bilayer. The Verdet constants are many orders of magnitude larger than those observed in conventional III–V or II–VI semiconductors and the well-known Faraday rotators used in optical isolators.

The Faraday rotation in monolayer TMDCs could be further enhanced by optimizing the multilayer structure of hBN and the TMDC monolayer by enhancing the dip in the transmittance in Figs. 2a and  3a. Furthermore, a heterostructure of a TMDC with a 2D ferromagnet could further raise the Verdet constant20,88,89. In such a heterojunction, strong magnetic exchange interaction effects between the ferromagnetic layer and the excitons in the TMDC are expected90. Finally, our work paves the way for a new generation of ultrathin optical polarization devices based on 2D materials.

Methods

Experimental setup for Faraday rotation spectroscopy on 2D materials

Broadband light from a Xe-arc lamp is collimated, and is linearly polarized, as shown in Fig. S1 of the Supplementary Information. It is focused on the sample mounted on the cold finger of a continuous-flow cryostat. The cold finger hangs between the pole pieces of an electromagnet, with a maximum magnetic field of B = 1.4 T applied perpendicular to the sample surface. The light undergoes a Faraday rotation \({\phi }_{F}\), and is reflected from a mirror. It is collimated after passing through a 10× long-working-distance infinity-corrected objective lens. The collimated light is focused on a 20 μm diameter pinhole, which selects light from a spot of about 4 μm diameter on the sample surface. The light is again collimated and passes through a beam displacer, which spatially separates the linear polarization components. The two components are focused on the input slit of a 0.3 m focal length monochromator, are wavelength dispersed, and are collected simultaneously using a Peltier-cooled CCD detector. A Jones matrix analysis of the setup is performed to obtain the Faraday rotation as discussed in ref. 38.

Determination of the complex dielectric tensor of hBN-encapsulated atomically thin semiconductors

The complex magneto-optical dielectric tensor is

$$\mathop{\epsilon }\limits^{\leftrightarrow}=\left(\begin{array}{cc}{\epsilon }_{{xx}1}+i{\epsilon }_{{xx}2} & {\epsilon }_{{xy}1}+i{\epsilon }_{{xy}2}\\ -({\epsilon }_{{xy}1}+i{\epsilon }_{{xy}2}) & {\epsilon }_{{xx}1}+i{\epsilon }_{{xx}2}\end{array}\right)$$
(7)

Diagonal elements of the dielectric tensor

There are four unknowns in this matrix i.e. \({\epsilon }_{{xx}1},{\epsilon }_{{xx}2},{\epsilon }_{{xy}1}\) and \({\epsilon }_{{xy}2}\). Out of these, \({\epsilon }_{{xx}1}\) and \({\epsilon }_{{xx}2}\) can be determined by measuring reflectance \(R({\widetilde{\epsilon }}_{{xx}})\) and transmittance \(T({\widetilde{\epsilon }}_{{xx}})\) of the materials. The measured \(R({\widetilde{\epsilon }}_{{xx}})\) and \(T({\widetilde{\epsilon }}_{{xx}})\) spectra are shown in Fig. S2a, e for hBN-encapsulated MoSe2 and WSe2 monolayers in the Supplementary Information. Essentially, the experimentally measured \(R({\widetilde{\epsilon }}_{{xx}})\) and \(T({\widetilde{\epsilon }}_{{xx}})\) are simultaneously a function of the two unknowns \({\epsilon }_{{xx}1}\) and \({\epsilon }_{{xx}2}\)55. A transfer-matrix-based approach is used to numerically calculate \({\epsilon }_{{xx}1}\) and \({\epsilon }_{{xx}2}\) with \(R({\widetilde{\epsilon }}_{{xx}})\) and \(T({\widetilde{\epsilon }}_{{xx}})\) spectra as inputs to the calculation55. The results of the calculation are presented in Fig. S2c, g of the Supplementary Information, respectively for the two materials.

Off-diagonal elements of the dielectric tensor

The real and imaginary parts of the off-diagonal element \({\widetilde{\epsilon }}_{{xy}}\) can be determined from the measured Faraday rotation spectrum \({\phi }_{F}(E)\) as follows. \({\phi }_{F}(E)\) is the real part of the complex Faraday rotation spectrum \({\widetilde{\phi }}_{F}(E)={\phi }_{F}+i{\eta }_{F}\) where \({\eta }_{F}\) is the Faraday ellipticity. Both \({\phi }_{F}\) and \({\eta }_{F}\) are functions of complex diagonal and off-diagonal components of the dielectric tensor, i.e., \({\widetilde{\epsilon }}_{{xx}}\) and \({\widetilde{\epsilon }}_{{xy}}\) (see Eq. (2)) which are four unknowns considering their real and imaginary parts. Since \({\widetilde{\epsilon }}_{{xx}}\) is already determined above (Fig. S2c, g of the Supplementary Information) for hBN-encapsulated MoSe2 and WSe2 monolayers, respectively, we are left with two unknowns, i.e., \({\epsilon }_{{xy}1}\) and \({\epsilon }_{{xy}2}\). The knowledge of \({\phi }_{F}\) and \({\eta }_{F}\) enable us now to calculate \({\epsilon }_{{xy}1}\) and \({\epsilon }_{{xy}2}\). Experimentally we measure only \({\phi }_{F}\). To determine \({\eta }_{F}\), we make use of the Kramers–Kronig analysis61. The real and imaginary parts of the complex Faraday effect given by \({\widetilde{\phi }}_{F}(E)={\phi }_{F}+i{\eta }_{F}\) are related to each other through Kramers–Kronig dispersion relations for small values (i.e., \(\tan {\eta }_{F} \sim {\eta }_{F}\)) as follows:

$${\phi }_{F}(\omega )=\frac{2}{{{{{{\rm{\pi }}}}}}}{\omega }^{2}{{{{{\rm{P}}}}}}{\int }_{0}^{\infty }\frac{{{{{{\rm{arctanh}}}}}}{\eta }_{F}\left({\omega }^{{\prime} }\right)\,}{{\omega }^{{\prime} }\left({{\omega }^{{\prime} }}^{2}-{\omega }^{2}\right)}d{\omega }^{{\prime} }$$
(8)
$${\eta }_{F}\left(\omega \right)=\tanh \left[-\frac{2}{{{{{{\rm{\pi }}}}}}}\omega \,{{{{{\bf{P}}}}}}{\int }_{0}^{\infty }\frac{{\phi }_{F}\left({\omega }^{{\prime} }\right)}{{{\omega }^{{\prime} }}^{2}-{\omega }^{2}}d{\omega }^{{\prime} }\right]$$
(9)

Here P is the principal value, \(\omega\) and \(\omega {\prime}\) are the frequencies given as \(E/\hslash\). An application of Eq. (9) on \({\phi }_{F}(E)\) spectra as input (solid lines in Fig. S2b, f of the Supplementary Information) yields Faraday ellipticity spectra \({\eta }_{F}(E)\) (dashed orange lines in Fig. S2b, f) for our samples. To test the applicability of our method, we use these dashed orange lines as input \({\eta }_{F}(E)\) spectra to calculate \({\phi }_{F}(E)\) using Eq. (8) (dashed blue lines) and compare with the experimentally measured \({\phi }_{F}(E)\) spectra (solid lines). The respective dashed blue and solid plots are in good agreement with each other, providing us confidence in our method. Finally, Eq. (2) is used to calculate the real and imaginary parts of the off-diagonal dielectric tensor, \({\epsilon }_{{xy}1}\) and \({\epsilon }_{{xy}2}\). These are shown in Fig. S2d, h of the Supplementary Information and also in Figs. 2c and 3c of the main text for hBN-encapsulated MoSe2 and WSe2 respectively.