Giant Faraday rotation in atomically thin semiconductors

Faraday rotation is a fundamental effect in the magneto-optical response of solids, liquids and gases. Materials with a large Verdet constant find applications in optical modulators, sensors and non-reciprocal devices, such as optical isolators. Here, we demonstrate that the plane of polarization of light exhibits a giant Faraday rotation of several degrees around the A exciton transition in hBN-encapsulated monolayers of WSe2 and MoSe2 under moderate magnetic fields. This results in the highest known Verdet constant of -1.9 × 107 deg T−1 cm−1 for any material in the visible regime. Additionally, interlayer excitons in hBN-encapsulated bilayer MoS2 exhibit a large Verdet constant (VIL ≈ +2 × 105 deg T−1 cm−2) of opposite sign compared to A excitons in monolayers. The giant Faraday rotation is due to the giant oscillator strength and high g-factor of the excitons in atomically thin semiconducting transition metal dichalcogenides. We deduce the complete in-plane complex dielectric tensor of hBN-encapsulated WSe2 and MoSe2 monolayers, which is vital for the prediction of Kerr, Faraday and magneto-circular dichroism spectra of 2D heterostructures. Our results pose a crucial advance in the potential usage of two-dimensional materials in ultrathin optical polarization devices.


Reviewer #2 (Remarks to the Author):
In the paper "Giant Faraday rotation in atomically thin semiconductors" by Benjamin Carey et al, the authors study the Faraday rotation upon transmittance of visible light through monolayer transition metal dichalcogenides encapsulated in hBN.Using a recently introduced, home-built setup for broadband Faraday rotation spectroscopy, the degree of polarization rotation at moderate external magnetic fields can be investigated.Due to the large exciton oscillators strength and g-factor, record values for the Verdet constant are reported, which surpass the ones obtained for conventional semiconductors by two to three orders of magnitude.Using an exhaustive table summarizing previous literature values of Verdet constants, this key result is further highlighted.The data acquisition and subsequent analysis has carefully been carried out as also evidenced by the authors' previous work  Methods 6, 2200885 (2022)), the authors have reported very similar experiments with almost identical plots.The only difference is the material system.From the data in Fig. 3e-g of this paper, a Faraday rotations of ~0.5 degree at the exciton resonance with an applied magnetic field of 1.2 T can be deduced.In this paper, the Verdet constant was simply not extracted and/or discussed, even though it should be very similar (up to a factor 3 or 4) to the values presented in the current manuscript.
-Could the authors comment on the reasons why WS2 yields a slightly worse Verdet constant?Is the broader linewidth of the exciton resonance the main reason?-Whereas the main insights such as the microscopic origin of the large Verdet constant (oscillator strength and g-factor) are nicely illustrated using MoSe2 as an example, Figure 3 is hardly discussed.The underlying reason is the fact, that the second material system does not contain any new information other than slightly different parameters of the exciton and hence a slightly different Verdet constant.In that sense, this comment is closely related to the first one concerning the novelty and impact.Most likely, the variation among different samples of the same type is even similar to the reported difference between MoSe2 and WSe2.
-I would suggest that the authors provide additional experimental data containing physics beyond the pure monolayer case.For example, this could encompass a 2D ferromagnet/TMDC heterostructure with yet higher Verdet constant as alluded to in the outlook.Another interesting scenario might be interlayer excitons, whose g-factor is yet higher than the monolayer excitons.The much smaller oscillator strength of the interlayer excitons compared to the monolayer might render this approach quite challenging, however.
-The authors argue that the Faraday effect has a plethora of applications.Consequently, a large Verdet constant is desirable.Yet the overall rotation cannot easily be scaled further based on the platform of monolayer TMDCs due to their atomically thin nature.Stacking multiple layers on top of each other would inevitably lead to interlayer hybridization and hence a reduction of the exciton binding energy.Thus, an hBN spacer layer is definitely required for incorporating more than one layer into a sample with an even higher Verdet constant.In short, if the experiments suggested in the previous comment prove to be too challenging to implement, an hBN/TMDC/hBN/TMDC/hBN structure might provide additional novel results beyond Figure 2.
-"Faraday rotation per using length".Most likely the authors are referring to Faraday rotation per uNIT length.

Response to the reviewers' comments on manuscript NCOMMS-23-07643-T
We thank the referees for the detailed assessment of our manuscript.In the following, we address the concerns of the referees in a point-by-point response.All changes made to the manuscript are marked red in the revised text.

Referee 1.
Referee comment.Since the experimental report on the Faraday rotation of monolayer graphene with the magnetic field, the magnetooptical responses of two dimensional materials become the hot topic.In this work, the authors reported the experimental Faraday rotation of monolayer WSe2 and MoSe2 under the application of magnetic field.They attributed the giant Faraday rotation obtained here to the combined effect of a giant exciton oscillator strength and a large exciton g-factor.Their experimental woks in this manuscript are valuable.
Our response.We thank the referee for finding our experimental work valuable.
Referee comment.This work looks like the paper "Giant Faraday rotation in single-and multilayer graphene Nature Phys 7, 48 (2010).In this paper, the giant Faraday rotation of monolayer graphene under the application of magnetic field is attributed to its inter Landaulevel transitions (in the low-doping regime) or cyclotron resonances (high-doping regime) in the terahertz region.In this work, the authors also used the magnetic field on monolayer WSe2 and MoSe2 to induce the Faraday rotation.From our understanding, the Landau levels still should exist in pristine monolayer WSe2 and MoSe2 when the magnetic field is applied [Appl.Phys. Lett. 105, 222411 (2014); Nat.Commun.8, 1938 (); Nat.Nano.12, 144 ()].Why the Faraday rotation here is not from the inter Landau level transition here but the combined effect of a giant exciton oscillator strength and a large exciton g-factor?Since the pristine monolayer WSe2 and MoSe2 have also possessed these kind of properties, but there is no magnetooptical response in the absence of magnetic field.The authors should clarify this point well.
Our response.We thank the referee for this important comment.Our present work is fundamentally different from the work on graphene mentioned by the referee (Nature Phys 7, 48 (2010)).In the case of doped graphene, the 2D electron gas gets quantized to form Landau levels in the presence of a magnetic field.However, in our present work, Landau quantization of the excitonic transitions does not happen in undoped or slightly doped 2D semiconductors until under very high magnetic fields (>30 T).The reason for the relatively easy Landau quantization at low magnetic field in graphene is that the electron gas in graphene is free from excitonic effects.Therefore, the magnetic energy does not have to compete with any binding energy of electrons, such as the Coulomb energy in the case of excitons (see "Physics of semiconductors in high magnetic fields", N. Miura, Oxford University Press, 2008, for the phenomenon).As a result, quantization of the electron gas in graphene is possible under low values of the magnetic fields such as well below 7 T in Nature Phys 7, 48 (2010).The energy of these inter-Landau transitions or cyclotron resonances falls in the Terahertz regime.
Corresponding to these resonances, signatures are observed in the Faraday rotation spectrum in the Terahertz region of the spectrum.
In contrast, in hBN-encapsulated monolayers of WSe 2 and MoSe2, we observe narrow excitonic transitions corresponding to the A excitons in the visible/infrared regime.The binding energy of these excitons is very large (>100 meV).For any Landau quantization effects to appear in the optical spectra, the magnetic energy has to compete with this binding energy.It turns out that one requires at least a few tens of Teslas of magnetic field, before any Landau quantization effects are experimentally observed in these systems (see Arora et al., J. Appl. Phys. 129, 120902 (2021) for a review on 2D materials).
The three papers mentioned by the referee and dealing with TMDCs Appl.Phys.Lett. 105, 222411 (2014), Nat. Commun. 8, 1938(2017) and Nat.Nano.12, 144 (2017) are not applicable to our work on the excitonic Faraday rotation due to the following reasons: 1) Appl.Phys.Lett. 105, 222411 (2014) theoretically discusses the creation of Landau Levels in TMDCs under very large magnetic fields (up to 50 T).Furthermore, it ignores the Coulomb interaction.Our work on excitonic Faraday rotation uses very low magnetic fields (<1.4T), and the Coulomb effects are extremely strong (>100 meV exciton binding energy).Therefore, we cannot compare our work to this paper.
2) Nat.Commun.8, 1938 (2017) and Nat.Nano.12, 144 (2017) are experimental works on highly-doped TMDC monolayers under magnetic fields.Under such conditions, the binding energy of the excitons and trions drastically reduces due to screening by the high-density electron gas.In fact, the features due to excitons and trions completely vanish under such conditions and inter-Landau level transitions are possible to be observed magneto-optically under large magnetic fields (from 4 T up to 9 T in Nat.Nano.12, 144 (2017)).Our work does not involve highly-doped systems, and Faraday rotation measurements are performed under very low fields (<1.4 T).Therefore, we are working in a completely different regime and are therefore not able to directly compare our present work to these earlier works.Indeed, the observed Faraday rotation around exciton lines is not due to Landau quantization effects.Instead, the intrinsic large Lande g factor of excitons (~ -4) and a giant exciton oscillator strength leads to a strong Faraday rotation around these transitions, which is explained as following.Due to the Zeeman splitting of the exciton line in a magnetic field, the material offers different dielectric functions to the left and right circularly polarized light around the exciton energy.This effect, when combined with a very narrow exciton transition due to a giant oscillator strength results in a large phase shift between the two circular polarization components of the incident linearly polarized light at the exciton energy.As a result, a giant Faraday rotation is observed due to these excitonic properties.
We have clarified this important point in our revised manuscript (marked in red), where we have also cited the works pointed out by the referee: We note that in the present work, Landau quantization effects 78-81 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.ℏ/2 * ~0.3 meV at  = 1.4 T) at the magnetic fields used 19,21,82 .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 regime 66 .(ii) Our hBN-encapsulated samples are not in the highly-doped regime, as evidenced by low (high) trion (exciton) oscillator strengths (Fig. 2 and 3).Inter-Landau transition effects have only been observed previously under high-doping conditions and large magnetic fields 79 .

Referee comment. The Faraday rotation basically comes from the nondiagonal term of the dielectric constant in the materials. We can see the information about the diagonal term of the dielectric constant in monolayer WSe2 and MoSe2 in the work but seldom information is found for the nondiagonal term of the dielectric constant in monolayer WSe2 and MoSe2. That will help the reader understand it better.
Our response.We agree with the referee that Faraday rotation arises from the off-diagonal term of the complex dielectric tensor of the materials.For light incident along the z-direction perpendicular to the sample plane with  ∥  (i.e.Faraday geometry), the dielectric tensor can be written as (e.g.Introduction of Modern Optics, Grant R. Fowles, Dover Publications, New York, 1975, henceforth referred to as Fowles (1975)): Here, each non-zero component is a complex quantity.In case of excitonic transitions in semiconductors, off-diagonal terms are normally zero in the absence of magnetic field, and are activated in the presence of a magnetic field due to the exciton Zeeman splitting.The complex off-diagonal component ̃  is given as (Fowles (1975)) where  � =   +    is the complex Verdet constant, having its components   and   related to Faraday rotation and Faraday ellipticity,  �  =  +  = �x  = � 1 +   2 is the complex refractive index.The calculation of ̃  () is a non-trivial task.Eq.R2 suggests that for calculating ̃  at a given magnetic field , we require the knowledge of the complex diagonal refractive index  �  and complex Verdet constant  � .
For  �  , we measure reflectance () and transmittance () spectra of our hBNencapsulated monolayer MoSe2 and WSe2 samples at  = 10 .We plot () and 1 − () spectra in Fig. R1 a and e for the two samples, respectively.Since () and () are a function of (, ) or in turn ( 1 ,  2 ), we derive these quantities numerically using a transfermatrix method-based analysis (see Fig. R1 c and g).Therefore,  �  in eq.R2 is derived.
is experimentally measured (Eq.6 of the main text).For   (), we make use of the fact that   and   are related to each other through Kramers-Kronig relationships as follows (P.Kielar, J. Opt.Soc.Am.B 11, 854 (1994)): where  represents the principal value of the integral, and  = ℏ.Using eq.R4,   () is determined.Faraday ellipticity for  = 1.4T is plotted in Fig. R1 b and f (dashed orange line).
To test if the derived Faraday ellipticity is correct, we used eq.(R3) to derive the Faraday rotation using the dashed orange line as the input.The calculated spectrum (dashed blue line) is in agreement with the experimental data within 10% error, giving us confidence about our method.Hence,  � () in eq.(R2) is obtained.
Finally, eq. ( R2) is used to calculate the complex off-diagonal dielectric constant ̃  .Real and imaginary parts of the ̃  i.e.  1 and  2 are plotted in Fig. R1 d and h for  = 1.4 T.
In the main text, we have introduced the complex dielectric tensor as well as complex Verdet constant on page 2. We have modified Fig. 2 and Fig. 3 to include the off-diagonal dielectric constant as Figs.2c and 3c.We include the following text in the main manuscript file on page 3: "From our data, we determine the real and imaginary components ( 1 and  2 ) of the offdiagonal dielectric constant ̃  of monolayer WSe2 as well following eq.2.An example of  1 and  2 for  = 1.4 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. 2b 59 , as well as the complex diagonal dielectric function ̃  .The details of the calculation are provided in the supporting information." The detailed process to extract ̃  has been included in the supporting information of the revised manuscript.We thank the referee for this comment since it has improved its quality and also underlines the novelty of our manuscript.There are some works working on this issue.Therefore, my impression is that the discussion about the MO effect in monolayer TMDC in the present manuscript is not quite new and it would be good for the authors to clarify their achievements in their work.

Referee comment. The MO effects have been already discussed in monolayer TMDCs
Our response.We thank the referee for suggesting these important references.We notice that all of the references pointed out by the referee describe magneto-optical effects in 2D semiconductors from a theoretical point of view.Phys.Rev. B 100, 045411 (2019) discusses the theoretically expected Verdet constants of excitons in monolayer MoS2 and WSe2 with values reaching up to 5 × 10 6 rad/T per monolayer thickness, i.e. 4.4 × 10 3 deg T −1 cm −1 .Our work experimentally measures the Verdet constant of excitons in monolayer TMDCs for the first time.Furthermore, we find that the values which we actually measure are about 4 orders of magnitude higher than the theoretically predicted values in Phys.Rev. B 100, 045411 (2019).We thank the referee for pointing out these papers which we have now cited in the revised version of the manuscript.
We notice that in the absence of a magnetic field in a bare semiconductor, the valley Zeeman splitting and valley polarization are zero.However, one can create a heterostructure of a 2D semiconductor such as WSe2 with a 2D magnet such as CrI3 and induce a magnetic-exchange interaction from the 2D magnet to the 2D semiconductor (e.g.Zhong et al., Sci.Adv. 3, e1603113 (2017).).In this case, magneto-optical effects such as Faraday/Kerr effects are expected even in the absence of an applied external magnetic field.However, experimentally realizing such a heterostructure system of high quality is extremely challenging and part of our future works.We have included this in the last paragraph of the main text of our manuscript: "Furthermore, a heterostructure of a TMDC with a 2D ferromagnet could further raise the Verdet constant 20,86,87 .In such a heterojunction, strong magnetic exchange interaction effects between the ferromagnetic layer and the excitons in the TMDC are expected 88 ."Referee 2.  3e-g of this paper, a Faraday rotations of ~0.5 degree at the exciton resonance with an applied magnetic field of 1.2 T can be deduced.In this paper, the Verdet constant was simply not extracted and/or discussed, even though it should be very similar (up to a factor 3 or 4) to the values presented in the current manuscript.

Referee comment. In the paper "Giant Faraday rotation in atomically thin semiconductors" by Benjamin Carey et al, the authors study the Faraday rotation upon transmittance of visible light through monolayer transition metal dichalcogenides encapsulated in hBN. Using a recently introduced, home-built setup for broadband
Our response.We thank the referee for the comment.Our earlier work (Carey et al., Small Methods 2022) involved a detailed description of a new Faraday rotation spectroscopy (FRS) technique, which we developed for performing spatially-resolved temperature-dependent FRS measurements.We demonstrated the functionality of the method by performing exemplary FRS measurements on an hBN-encapsulated WS2 monolayer and a TbFe0 ferrimagnetic alloy.
The referee is correct that we did not focus on the Verdet constants of the A exciton i.e. −(4.6 ± 0.2) × 10 6 of the WS2 layer in this earlier manuscript.As discussed in the next section, this value is found to be a factor of 4 to 6 smaller than the WSe2 and MoSe2 monolayer samples in the present work due to the relatively poor sample quality of the hBN-encapsulated WS2 monolayer used in Carey et al., Small Methods, 2022.
We believe that the topic of giant Verdet constants in the visible and infrared region of the spectrum in high-quality monolayers deserves a focused attention due to the fundamental scientific importance and its potential in opening a new route to ultrathin magneto-optical device-based applications.Only a separate manuscript with high visibility would justify this.It also requires the best sample qualities.Therefore, for the present work, we tried hard and succeeded in producing high-quality hBN-encapsulated WSe2 and MoSe2 monolayers on sapphire substrates, with A exciton line widths approaching the homogeneous limit.
We further address the comments of the referee concerning the novelty of our work as follows: 1) We perform additional optical spectroscopy measurements such as reflectance and transmittance on our samples simultaneously.Using these in conjunction with our measured Faraday rotation spectra, we calculate the complete complex dielectric tensor of the two representative materials monolayer WSe2 and MoSe2 from the van der Waals semiconductors family for the first time.We include the real and imaginary parts of the diagonal and offdiagonal dielectric elements of the tensor in the revised version (see above for details, answer to referee 1).Knowledge of the dielectric tensor is important for designing and predicting the magneto-optical response of heterostructures of 2D semiconductors.
2) We perform additional Faraday rotation spectroscopy measurements on interlayer excitons in a hBN-encapsulated MoS2 bilayer for the first time, which have a positive and higher absolute g-factor than intralayer excitons, and include them in our revised manuscript (see further below).
Following these further improvements in the manuscript, we are confident that our work deserves publication in a high-quality journal such as Nature Communications.
Referee comment.-Could the authors comment on the reasons why WS2 yields a slightly worse Verdet constant?Is the broader linewidth of the exciton resonance the main reason?
-Whereas the main insights such as the microscopic origin of the large Verdet constant (oscillator strength and g-factor) are nicely illustrated using MoSe2 as an example, Figure 3 is hardly discussed.The underlying reason is the fact, that the second material system does not contain any new information other than slightly different parameters of the exciton and hence a slightly different Verdet constant.In that sense, this comment is closely related to the first one concerning the novelty and impact.Most likely, the variation among different samples of the same type is even similar to the reported difference between MoSe2 and WSe2.
Our response.These two points are related, therefore we are addressing them together.The referee is correct.All the three materials WS 2, WSe2 and MoSe2 are qualitatively similar concerning their exciton Verdet constants.In Fig. R2 a and b, we compare the optical transmittance spectra of the three hBN-encapsulated monolayer samples.An overall difference in the transmission between the three materials is due to the different extent of interference effects due to encapsulation with hBN of dissimilar thicknesses.There are quantitative differences in the A exciton oscillator strengths of the three materials (Table 1).We find that the A exciton oscillator strength is the largest in WS2, followed by WSe2 and MoSe2 respectively.However, WS2 yields a slightly worse Verdet constant compared to WSe2 and MoSe2 in our samples.The derived A exciton full-width at half-maximum (FWHM) line widths are compared in Table 1.We find that the FWHM line width for the WS2 case is about a factor of 2 larger than WSe2 and MoSe2, even though the oscillator strength parameters are comparable.Therefore, the poor quality of the WS2 layer results in a lower Verdet constant.We also notice that although the A exciton oscillator strength in the MoSe2 sample is the smallest among the three samples, its Verdet constant is the largest due to the narrowest line width in this sample.Additionally, we agree with the referee that there will be a variation found in the samples of the same type depending on A-exciton line widths.Referee comments.I would suggest that the authors provide additional experimental data containing physics beyond the pure monolayer case.For example, this could encompass a 2D ferromagnet/TMDC heterostructure with yet higher Verdet constant as alluded to in the outlook.Another interesting scenario might be interlayer excitons, whose g-factor is yet higher than the monolayer excitons.The much smaller oscillator strength of the interlayer excitons compared to the monolayer might render this approach quite challenging, however.
The authors argue that the Faraday effect has a plethora of applications.Consequently, a large Verdet constant is desirable.Yet the overall rotation cannot easily be scaled further based on the platform of monolayer TMDCs due to their atomically thin nature.Stacking multiple layers on top of each other would inevitably lead to interlayer hybridization and hence a reduction of the exciton binding energy.Thus, an hBN spacer layer is definitely required for incorporating more than one layer into a sample with an even higher Verdet constant.In short, if the experiments suggested in the previous comment prove to be too challenging to implement, an hBN/TMDC/hBN/TMDC/hBN structure might provide additional novel results beyond Figure 2.
Our response.We thank the referee for the suggestions and giving us further ideas to incorporate in the manuscript concerning novelty.The referee provides us three interesting experiments, which are the following: 1) FR spectroscopy of 2D ferromagnet/TMDC heterostructures 2) FR spectroscopy of interlayer excitons whose g-factor is larger, but the oscillator strength is smaller than the intralayer excitons 3) FR spectroscopy of an hBN/TMDC/hBN/TMDC/hBN heterostructure Out of these ideas, while trying 1) is extremely challenging, we have tried 2) and 3).Here we summarize our results:

Idea 2). FR spectroscopy of interlayer excitons in a MoS 2 bilayer.
We performed Faraday rotation spectroscopy on hBN-encapsulated bilayer MoS2 and included our results in the revised main manuscript as Fig. 3, reproduced as Fig. R3 (a -d the features are performed on the following grounds: A and T resonances have similar gfactors, nearly equal to −4 suggesting their intralayer character 21,61 .T polarizes strongly under magnetic field, with its valley polarization approaching ~+ 14% (Fig. S3 of the supporting information).Furthermore, A polarizes only weakly (valley polarization ~− 2%).Large polarization of T with an opposite sign compared to A is characteristic for the appearance of a trion-exciton pair 63 .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 meV 64 .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 gfactors ( IL1 = +6.6 ± 0.3 and  IL2 = +7.2± 0.3) 21,60,61 .In previous works, one IL resonance has been observed in optical reflectance spectra 21,61 .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 field 65,66 , which can be created by charge transfer from impurities in the substrate 67 .The Verdet constant of the interlayer excitons are  IL1 = +(1.9± 0.5) × 10 5 deg T −1 cm −1 and  IL2 = +(2.2± 0.5) × 10 5 deg T −1 cm −1 .In contrast, the intralayer exciton in this sample has a Verdet constant of  A = −(1.5 ± 0.2) × 10 6 deg T −1 cm −1 .
In the revised supporting information, p.3.: "Assignment of the trion-exciton pair in the hBN-encapsulated MoS2 bilayer.The Faraday rotation spectra are modelled using a transfer-matrix-based method for B = 0.4 T -1.4 T [1].The modelled spectra are shown as solid lines in Fig. 3(b) of the main text.The magnetic-field-induced valley polarization of the neutral and charged exciton (A and T, respectively) is plotted in Fig. S3.The opposite degree of the polarization with magnetic field is a well-established evidence for the occurrence of a trion-exciton pair in the literature of semiconductor quantum wells [5].Therefore, we assigned the first two resonances in our transmittance/FR spectra to a neutral and a charged exciton.Our assignment also agrees with an earlier report [6]." Idea 3).FR spectroscopy of a hBN/TMDC/hBN/TMDC/hBN heterostructure.For this experiment, we fabricated a few hBN/WSe2/hBN/WSe2/hBN samples.The goal was to obtain a sample where the A excitons have homogeneous line widths, and are at the same time energetically degenerate in the two individual WSe2 monolayers.However, this experiment has proven extremely challenging.It is because, normally, in creating hBN-encapsulated heterostructures using the standard mechanical exfoliation process, one finds only extremely small regions (~1 − 2 μm across) with homogeneous line widths.The remaining interfacial areas are either mostly covered with bubble-like regions, which collect dirt between the interfaces, or have a broad exciton line width due to imperfect encapsulation.In an hBN/TMDC/hBN/TMDC/hBN heterostructure, there are four interfacial junctions.The probability that there will be a clean interface with an identical quality for all of the four junctions of the heterostructure, resulting in two overlapping homogeneously broadened excitons is extremely small.In our trials of a few hBN/WSe2/hBN/WSe2/hBN samples which we created and tested, we find identical results as follows: The exciton lines are not very narrow.The best exciton line widths are ~10 meV compared to a homogeneous line width of ~3 − 4 meV in a single hBN-encapsulated monolayer.Furthermore, although the optical transmission shows two spectrally overlapping A exciton lines, the FR spectra always show two close-lying but separate exciton lines, corresponding to each of the two WSe2 monolayers.We have not found a location on the samples where the two excitons exactly spectrally overlap.Due to these reasons, the Faraday rotation is not found to be as large as our hBN-encapsulated monolayers.Figure R4 shows one set of results we obtained.The deduced Verdet constant is  1 = −1.9× 10 6 degT −1 cm −1 .
Referee comment: "Faraday rotation per using length".Most likely the authors are referring to Faraday rotation per uNIT length.
Our response.We thank the referee for pointing out this typo error.We have corrected this in the revised version.The Verdet constant obtained from the linear fit is  1 = −1.9× 10 6 degT −1 cm −1 .
introducing their experimental method.While the results are interesting and the record values for the Verdet constant are indeed impressive, I cannot support publication of the manuscript in Nature Communications in its current form.Please find my individual concerns listed below: -The most important point concerns the degree of novelty of the work.In their previous paper (ref.38, Carey, B. et al.High-Performance Broadband Faraday Rotation Spectroscopy of 2D Materials and Thin Magnetic Films.Small

Fig. R1 .
Fig. R1.Calculation of the dielectric tensor.See main text for details.

Fig. R2 .
Fig. R2.Comparison of the transmittance spectra of the three materials.(a) Comparison of transmittance of hBN-encapsulated WSe2 and (b) hBN-encapsulated MoSe2 with hBN-encapsulated WS2 (from Carey et al., Small Methods 2022) on the sapphire substrates.We notice that WSe2 and MoSe2 samples have a deeper transmission dip compared to WS2.The reason is a larger exciton line width in our WS2 sample (seeTable 1 of this document).
Figure R3.Faraday rotation of intralayer and interlayer excitons in hBN-encapsulated bilayer MoS2.a) Optical transmission spectrum of hBN/2L MoS2/hBN displaying a trion T, an intralayer exciton A, and a pair of interlayer excitons IL1 and IL2.b) Experimental (spheres) and modelled (solid lines) Faraday rotation spectra of the resonances in magnetic fields ranging from  = 0.4 T − 1.4 T. The spectra are vertically shifted successively by 0.1 ∘ 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) Valley Zeeman splittings deduced for the four resonances from the line shape modelling in (b).Linear fits (solid lines) are used to derive the effective g-factors   and   .d) Measured peak Faraday rotation of the resonances as a function of the magnetic field.The peak rotation is fitted linearly for deriving the Verdet constants  A = −(1.5 ± 0.2) × 10 6 deg T −1 cm −1 ,  IL1 = +(1.9± 0.5) × 10 5 deg T −1 cm −1 and  IL2 = +(2.2± 0.5) × 10 5 deg T −1 cm −1 .(e) Magnetic-fieldinduced valley polarization of the trion-exciton pair in the MoS2 bilayer.The inverted valley polarization of the trion-exciton pair supports their observation in optical transmittance and Faraday rotation spectra.
We thank the referee for the detailed report on our manuscript, and appreciating the careful data acquisition and analysis in our work.
Referee comment.-Themostimportant point concerns the degree of novelty of the work.In their previous paper (ref.38,Carey,B. et al.High-Performance Broadband Faraday Rotation Spectroscopy of 2D Materials and Thin Magnetic Films.SmallMethods 6, 2200885 (2022)), the authors have reported very similar experiments with almost identical plots.The only difference is the material system.From the data in Fig.

Table 1
of this document).

Table 1
Comparison of the FWHM line widths and the oscillator strength parameters of the three samples