Enhancing magneto-optic effects in two-dimensional magnets by thin-film interference
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Enhancing magneto-optic effects in two-dimensional magnets by thin-filminterference
F. Hendriks a) and M.H.D. Guimarães b) Zernike Institute for Advanced Materials, University of Groningen,The Netherlands (Dated: 4 February 2021)
The magneto-optic Kerr effect is a powerful tool for measuring magnetism in thin films atmicroscopic scales, as was recently demonstrated by the major role it played in the discov-ery of two-dimensional (2D) ferromagnetism in monolayer CrI and Cr Ge Te . These 2Dmagnets are often stacked with other 2D materials in van der Waals heterostructures on aSiO /Si substrate, giving rise to thin-film interference. This can strongly affect magneto-optical measurements, but is often not taken into account in experiments. Here, we showthat thin-film interference can be used to engineer the magneto-optical signals of 2D mag-netic materials and optimize them for a given experiment or setup. Using the transfer ma-trix method, we analyze the magneto-optical signals from realistic systems composed ofvan der Waals heterostructures on SiO /Si substrates, using CrI as a prototypical 2D mag-net, and hexagonal boron nitride (hBN) to encapsulate this air-sensitive layer. We observea strong modulation of the Kerr rotation and ellipticity, reaching several tens to hundredsof milliradians, as a function of the illumination wavelength, and the thickness of the SiO and layers composing the van der Waals heterostructure. Similar results are obtained inheterostructures composed by other 2D magnets, such as CrCl , CrBr and Cr Ge Te .Designing samples for the optimal trade-off between magnitude of the magneto-opticalsignals and intensity of the reflected light should result in a higher sensitivity and shortermeasurement times. Therefore, we expect that careful sample engineering, taking intoaccount thin-film interference effects, will further the knowledge of magnetization in low-dimensional structures. a) [email protected] b) [email protected] In these effects, a change of the re-flected or transmitted light intensity and polarization is (often linearly) related to the change ofmagnetization of the illuminated area. When used in combination with microscopy techniques,magneto-optical signals can be used to image the magnetization of systems at the sub-micrometerscale, and when combined with ultrafast lasers, they give access to the magnetization dynamicsat femtosecond timescales.
The magneto-optic Kerr effect (MOKE) was instrumental for thediscovery of two-dimensional (2D) ferromagnetism in monolayer CrI and Cr Ge Te . Dueto its non-destructive nature and easy implementation, MOKE and related magneto-optic effects,such as reflected magnetic circular dichroism, are one of the standard tools for the magnetic char-acterization of 2D van der Waals magnets.
For those measurements, 2D magnets are oftenstacked with other van der Waals materials on a substrate, such as hexagonal boron nitride (hBN)on SiO /Si substrates. These layered systems can display strong thin-film interference effectswhich in turn affect their magneto-optical response. At the start of the 2D materials revolution,it was discovered that exploiting these interference effects allowed for optical identification ofgraphene flakes, providing a way for easily identifying graphene mono- or few-layers. Later,the same techniques were used for identifying thin layers of other van der Waals meterials, suchas transition metal dichalcogenides. Also, the effects of thin-film interference on magneto-optical signals, and how to use these effects to enhance them, have been studied extensively in thecontext of metallic thin-films, oriented moleculecular films , ellipsometry , and manyother fields. However, thin-film interference effects are often not taken fully into account forthe magneto-optical experiments on van der Waals magnets. This could lead to a suboptimalsignal-to-noise ratio, resulting in a lower sensitivity and / or longer measurement times. Therefore,it becomes more difficult and more time-consuming to measure small changes in magnetizationof 2D magnets, caused by for example chiral spin textures in a homogeneously magnetized lat-tice, and to measure under low-light conditions to avoid sample degradation. While some worksdo take into account the effect of the oxide substrate, hBN, or a polymer layer on the magneto-optical signals, a comprehensive study of thin-film interference effects for the magneto-opticsin realistic samples is still lacking.Here, we show that not only the substrate, but also other materials in a van der Waals stackcan greatly affect the MOKE signals, and that these signals can be significantly enhanced bycarefully choosing the illumination wavelength and through heterostructure engineering (Fig. 1),2s is well known from other studies on thin-film interference enhancements of MOKE signalsfrom e.g. metallic thin films. Using a transfer matrix approach for thin-film interference, wedemonstrate that the MOKE signals can reach values of tens to hundreds of milliradians at sizeablereflected light intensities. In particular, we explore this effect on three systems based on the 2Dvan der Waals magnet CrI on a SiO /Si substrate: monolayer CrI , bulk CrI , and monolayerCrI encapsulated in hBN (we also consider other 2D magnets, see supplementary material). Ourresults show that the often disregarded hBN encapsulation used to protect the air-sensitive 2Dmagnet films can strongly affect the magnitude of the MOKE signals, such that subtler magnetictextures in 2D magnets can be measured. FIG. 1. a) A typical 2D ferromagnet sample displaying MOKE in the presence of thin-film interference. b),c) The calculated Kerr rotation and ellipticity depend heavily on the oxide thickness and wavelength. Themaximum signal occurs when the reflectivity is close to its minimum.
We model the thin layered systems as a series of stacked parallel homogeneous layers, wherethe first and last layer (being air and Si), are semi-infinite. An example of this geometry is il-lustrated in Fig. 1a, where a single 2D magnetic layer of thickness t is on top of a SiO /Sisubstrate with oxide thickness t ox . The interfaces are assumed to be smooth, such that there areonly specular reflections. Furthermore, we assume that the illumination intensity is low enough,such that the optical properties of the materials can be described by a linear dielectric permittiv-ity tensor ε and magnetic permeability tensor µ . The intensity and polarization of the light thatreflects off this stratified linear system are calculated using the transfer matrix method. We use amethod similar to the one in, which is explained in full detail in the supplementary material.The transfer matrix relates the components of the electric ( ~ E ) and magnetic ( ~ H ) field parallel tothe layers, called ~ E k and ~ H k , at one interface of a medium to the other one. To construct a transfermatrix, we start by describing plane waves in a single layer. We begin from the Maxwell equations3n isotropic homogeneous media, and consider plane waves with a frequency ω and wave vector ~ k , of the form ~ E = ~ E exp ( i ( ~ k · ~ r − ω t )) , where t is time and ~ r is the position in space. We can thenderive the following wave equation: ε − (cid:16) ~ k × (cid:16) µ − (cid:16) ~ k × ~ E (cid:17)(cid:17)(cid:17) = − ω ~ E . (1)Solving the above equation yields four values for the z -component of ~ k , k z , i , and four correspondingpolarization eigenmodes, ~ E , i , where i labels the polarization mode. These solutions describetwo plane waves traveling in the + z direction, and two in the − z direction. The transfer matrixis the diagonal matrix diag ( exp ( ik z , i t layer )) , which propagates the eigenmodes with wave vectorcomponents k z , i from one interface to the other one over a distance t layer , after it is transformedfrom the basis of the eigenmode amplitudes to the the basis of the amplitudes of the ~ E k and ~ H k components. The transfer matrix of the whole system is simply the product of the transfer matricesof the individual layers, since ~ E k and ~ H k are continuous across the interfaces. This matrix is usedto calculate the amplitudes of the eigenmodes of the reflected and transmitted light, and from thisthe reflected intensity and polarization.We apply the above method to the system illustrated in Fig. 1a, where the 2D ferromagnetis monolayer CrI with a thickness of t DM = 0.7 nm. The dielectric tensor of ferromagneticmonolayer CrI is taken from Wu et al. , where it is calculated from first-principles methodstaking excitonic effects into account. The dielectric constants of Si and thermally grown SiO are experimental values from Herzinger et al. The magnetic permeability of all materials isapproximated by the scalar vacuum permeability µ . Using these parameters, we calculate theKerr angle θ K , Kerr ellipticity ε K , and reflected intensity of linearly polarized light hitting thesample at normal incidence and polar configuration. The results are shown in Fig. 1b and 1c as afunction of t ox and wavelength respectively.Fig 1b shows a clear periodic behavior of the MOKE signals as a function of t ox , with a periodof 216 nm, corresponding to half a wavelength in SiO . It also shows that the Kerr angle andellipticity attain their maximum values when the reflected intensity is close to a minimum, andvice-versa. In Fig. 1c, the largest MOKE signals are found in the wavelength range from 400nm to 750 nm, where the wavelength dependence of θ K and ε K is caused primarily by the wave-length dependence of the dielectric tensor of CrI . Again, θ K and ε K attain their maximum valueswhen the reflectivity is close to a minimum. These results show that the oxide thickness and thewavelength of the light have a strong impact on the sign and magnitude of the MOKE signals. By4ptimizing t ox or the wavelength, the signals can already change by as much as 20 mrad in thisexample, while still having a sizable reflectivity of more than 6%.In order to get a complete picture of the impact of each parameter on the signals, we explorethe full parameter space, varying both the wavelength and oxide thickness for a CrI monolayer ona SiO /Si substrate (Fig. 2). Besides the reflectivity, θ K , and ε K , we also calculate the contrast forthe CrI layer. This can be used to locate the target flake, usually a few µ m in size and thereforehard to find on a large substrate, using a microscope, or using a reflectivity scan in a laser basedexperiment. The contrast is defined as C = ( I − I ) / I , where I and I are the reflected intensityof the system with and without CrI respectively. The reflectivity in Fig. 2 shows a clear fanpattern. The periodicity in t ox in our simulated reflectivity corresponds to half a wavelength inthe SiO , which strongly suggests that this fan pattern is caused by the interference of the lightreflected from the top and bottom interface of the SiO , similar to graphene-based systems. Thesame pattern appears for C , θ K , and ε K , indicating that the interference in the SiO layer also hasa large effect on the contrast and MOKE signals. Additional features at 420 nm, 500 nm, and 680nm, can also be seen, and originate from the wavelength dependence of the dielectric tensor ofCrI (see supplementary material). By tuning both the wavelength and oxide thickness, θ K and ε K can be tuned over a range of several tens of milliradians while keeping the reflectivity above 5%.Furthermore, when the Kerr rotation and ellipticity are maximized, the contrast is large as well,making it easier to locate the CrI using e.g. a simple reflectivity scan.The above results can be compared to the experimental results from Huang et al. In theirexperimental work, using a laser with a wavelength of 633 nm and t ox = 285 nm, they obtained θ K = ± θ K by morethan a factor of 4, or if the wavelength is changed to 560 nm, the Kerr rotation can increase by afactor of about 3.The 2DM thickness can also strongly affect the MOKE signals. Fig. 3 shows the dependenceof the magneto-optical signals as a function of both wavelength and 2DM thickness, using thedielectric tensor of ferromagnetic bulk CrI taken from Wu et al. While the theoretical values5 rI31LSiSiO2 a bc dRe (cid:1)
Contrast (cid:2) K ε K FIG. 2. Simulation results for a CrI (1L)-SiO -Si stack. The reflectivity (a), contrast (b), Kerr rotation (c)and ellipticity (d) are shown as function of illumination wavelength and oxide thickness. Where the colorscale is saturated, the values exceed the bounds of the scale. of ε CrI used in our calculations differ slightly from the available experimental values, ourmain findings are not altered if we consider the experimental values. We therefore opt for usingthe theoretical values since they span a larger wavelength range. For comparison, we providecalculations using the experimental values in the supplementary material. Interestingly, θ K and ε K have a non-monotonic behavior, showing a strong peak and dip around a wavelength of 600 nmCrI thickness of 14 nm. The extreme values of θ K and ε K approach ± π / ± π / θ K and ε K can still be changed over a range of a few hundred milliradians when tuningthe wavelength and CrI thickness, while keeping the reflectivity above 5% and having a goodcontrast.Due to the air sensitivity of many 2DMs, they are often encapsulated in hBN. The presenceof the hBN layers also leads to thin-film interference effects and thus can be used to engineerthe magneto-optical signals as well. To explore the impact of hBN encapsulation, we study theMOKE signals in monolayer CrI encapsulated by a top and bottom hBN flake with the samethickness t hBN . The refractive index of hBN needed for the simulation is calculated using thesingle oscillator model, n ( λ ) = + A λ / ( λ − λ ) , where λ = 164.4 nm and A = 3.263 aredetermined experimentally by Lee et al. The simulation results for an oxide layer of 285 nm6 iSiO2 fl Contrast θ K ε K d FIG. 3. Simulation results for a CrI (bulk)-SiO (285 nm)-Si stack. The reflectivity (a), contrast (b), Kerrrotation (c) and ellipticity (d) are shown as function of wavelength and CrI thickness. Where the colorscale is saturated, the values exceed the bounds of the scale. are shown in Fig. 4. We have also investigated the effect of the hBN thickness on the signal-to-noise ratio (see supplementary material). A striking result is that an hBN thickness of aboutten nanometers, a typical thickness for hBN flakes used for encapsulation in experimental studies,can already lead to dramatic changes in the reflectivity, contrast, and Kerr signals. Therefore, oneshould take into account the system as a whole when engineering their heterostructures for optimalMOKE signals. The hBN encapsulation is particularly important, since the wavelength and oxidethickness are usually more difficult to vary, while hBN flakes of various thicknesses can be easilyfound in a single exfoliation run. Therefore, in addition to protecting the 2DM against degradation,hBN encapsulation can be used as an active method for magneto-optical signal enhancement.The common feature in the results of the simulation of the three systems above that θ K and ε K are maximized when the reflectivity is close to a minimum, is a general and well-knownphenomenon. It can be explained by the behavior of the reflection coefficients for the elec-tric field of the two circular polarizations, r + and r − , near the reflectivity minimum. In this re-gion, the magnitude of both reflection coefficients are small, and their complex phases changerapidly with wavelength and layer thickness. The exact parameter values around which thesecoefficients have a minimum and change phase are different for r + and r − due to the circularbirefingence and dichroism caused by the magnetic layer. Therefore, both the ellipticity, given by ε K = tan − ( | r + | − | r − | ) / ( | r + | + | r − | ) , and the Kerr rotation, given by θ K = ( arg ( r + ) − arg ( r − )) / c Re fl Contrast θ K ε K Si h(cid:0)(cid:3)(cid:4)(cid:5)(cid:6)S(cid:7)(cid:8) FIG. 4. Simulation results for a hBN-CrI (1L)-hBN-SiO (285 nm)-Si stack. The reflectivity (a), contrast(b), Kerr rotation (c) and ellipticity (d) are shown as function of wavelength and hBN thickness. Where thecolor scale is saturated, the values exceed the bounds of the scale. can become very large when the total reflectivity is near a minimum, as is explained in more de-tail in the supplementary material. On the other hand, if the reflectivity is large, both r + and r − are large, meaning that their relative difference is small, and that their complex phase changesslowly with wavelength and layer thickness. This will result in a low Kerr ellipticity and rotationrespectively. Therefore, the extreme MOKE signals of e.g. ε K ≈ ± π / θ K ≈ π / (bulk)-SiO (285nm)-Si stack can only occur at a low reflectivity. This reasoning isnot restricted to the samples treated in this paper. A general method to increase the Kerr rotationand ellipticity of a multi-layer sample is to use a combination of wavelength and thickness of thelayers that minimizes the reflectivity. A reduction of the reflectivity, and a corresponding increasethe magneto-optical signals, can also be achieved by adding new layers to the sample. Such anti-reflection coatings have been used for over half a century to enhance Kerr signals from magneticfilms. Here we showed that thin-film interference can be a useful tool for improving magneto-opticalsignals in magnetic van der Waals systems. Through careful sample or heterostructure engineering,one is able to optimize their system for a particular experimental setup, improving the signal-to-noise ratio and measurement speed. The optimization of the signals can be done by choosing aparticular illumination wavelength, substrate, thickness of the van der Waals magnet, or hBN usedfor encapsulation. The signal improvement, reaching several tens of miliradians, could lead to8he identification of weaker signals from more delicate effects, such as chiral magnetic structuresembedded in a homogeneously magnetized lattice.
SUPPLEMENTARY MATERIAL
See supplementary material for the simulation details, graphs of the dielectric tensors used inthe simulations, simulation results for other 2D magnetic monolayers, and an explanation for whythe Kerr rotation and ellipticity are large when the reflectivity is close to a minimum.
ACKNOWLEDGMENTS
We thank Alejandro Molina-Sánchez, for sharing their data on the dielectric tensor for chromiumtrihalides. This work was supported by the Zernike Institute for Advanced Materials, the DutchResearch Council (NWO Start-Up, STU.019.014), and the European Union’s Horizon 2020 re-search and innovation programme under grant agreement No 785219 (Graphene Flagship Core3).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authorupon reasonable request.
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F. Hendriks a) and M.H.D. Guimarães b) Zernike Institute for Advanced Materials, University of Groningen,The Netherlands (Dated: 4 February 2021) a) [email protected] b) [email protected] ONTENTSI. Simulation details II. Dielectric tensors of the materials used in the simulations III. Interference effects in other 2D magnets IV. Bulk CrI with experimental values for the diagonal of its dielectric tensor V. Signal to noise ratio VI. Constant dielectric tensor for 2D magnets VII. Large MOKE signals at low reflectivity References . SIMULATION DETAILS
The thin layered systems we simulate in this work are modelled by a series of stacked parallelhomogeneous layers, where the first and last layers (being air and silicon) are semi-infinite. Anexample of this geometry is depicted in Fig. 1a of the main text. The interfaces are assumed tobe smooth, such that all reflections are perfectly specular. Furthermore, the illuminating intensityis assumed to be low enough, such that the optical properties of the materials are described by alinear dielectric permittivity tensor ( ε ) and a linear magnetic permeability tensor ( µ ). To calculatethe intensity and polarization of the light that reflects off this stratified medium, we use a slightvariation of the transfer matrix method described by P. Yeh et al. , where we assume a moregeneral wave equation and use a different basis for the transfer matrix. This method is explainedin detail below.A transfer matrix relates the components of the electric ( ~ E ) and magnetic ( ~ H ) field parallel tothe layers, ~ E k and ~ H k respectively, of the bottom and top interfaces of a medium. Grouping theseparallel field components into a vector ~ F , the transfer matrix M is defined as ~ F II = M ~ F I , ~ F i = E ix E iy H ix H iy , (1)where the superscript i = I , II , specifies the interface. To construct a transfer matrix, we start byassuming plane waves propagating in a layer: ~ E ( ~ r , t ) = ~ E e i ( ~ k · ~ r − ω t ) , (2)where ~ E is the electric field amplitude, ~ k the wave vector, ~ r the position in space, ω the angularfrequency, and t time. Plugging this into Maxwell’s equations gives ~ k · (cid:16) ε ~ E (cid:17) = ~ k × ~ E = ω ~ B (3b) ~ k · ~ B = ~ k × (cid:16) µ − ~ B (cid:17) = − ω ε ~ E . (3d)Using Eq. 3b to eliminate ~ B from Eq. 3d yields ε − (cid:16) ~ k × (cid:16) µ − (cid:16) ~ k × ~ E (cid:17)(cid:17)(cid:17) = − ω ~ E . (4)3his equation gets a more familiar form when the cross products are written in terms of matrixmultiplications. Defining k c . p . = − k z k y k z − k x − k y k x , (5)Eq. 4 becomes ε − k c . p . µ − k c . p . ~ E = − ω ~ E . (6)Now it takes the form of an eigenvalue problem for a matrix given by ε − k c . p . µ − k c . p . witheigenvalue − ω . Since E k and H k are continuous across all interfaces, ω , k x and k y are constantthroughout the system, and therefore they are equal to ω , k x and k y of the incoming light. Theonly unknowns are k z and ~ E . The solution for this eigenvalue problem gives a relation between ~ k , ω , and the material parameters ε and µ . This yields four values for the z -component of ~ k , k z , i , describing two waves traveling in the + z and two in the − z direction. The correspondingeigenvectors, E , i , are the polarization modes of the electric field. For the magnetic field, thepolarization modes are H , i = µ − ( ~ k × ~ E ) / ω . This describes plane wave propagation in a singlelinear homogeneous medium.The transfer matrix for a single layer can be constructed from the polarization modes and thewave vectors of the four plane waves that are allowed for a given incident wave. Given E k and H k at one interface, their value at the other interface of the medium is calculated by decomposing thefields in the polarization modes, propagating these modes to the other interface using Eq. 2, andthen transforming them back to E k and H k . The transfer matrix relating E k and H k between thetwo interfaces is M = APA − , (7)where the matrices A and P are defined as A = E , x E , x E , x E , x E , y E , y E , y E , y H , x H , x H , x H , x H , y H , y H , y H , y , P = e i ( k z , ∆ z ) e i ( k z , ∆ z ) e i ( k z , ∆ z )
00 0 0 e i ( k z , ∆ z ) . (8)4ince E k and H k are continuous across the interfaces, the transfer matrix for light propagationthrough whole system is described by the product of the transfer matrices of the individual layers: M tot = ∏ j M j . (9)This matrix relates E k and H k in the first layer to E k and H k in the final layer.We now write E k and H k as linear combinations of the polarization modes ( ~ F i ), with a coefficient( a i ): ~ F = ∑ i a i ~ F i . (10)For the final layer, we assume that there are only waves traveling in the + z direction, since thereis no interface to reflect them back. Substituting the above relation in Eq. 1 and rearranging theterms results in: a i M tot ~ F i + a i M tot ~ F i + a r M tot ~ F r + a r M tot ~ F r = a t ~ F t + a t ~ F t , (11)where the subscripts i , r , and t label the incident, reflected, and transmitted modes respectively.For example, ~ F r and ~ F r represent the first and second polarization mode of the reflected beam( − k z ).After regrouping the terms, the (complex) amplitudes of the reflected and transmitted modescan be solved from the following matrix equation: h M tot ~ F r M tot ~ F r ~ F t ~ F t i a r a r a t a t = M tot h ~ F i ~ F i i a i a i . (12)These complex amplitudes describe the intensity and the polarization of the reflected and trans-mitted waves. Since they travel in an isotropic medium, the eigenmodes can be chosen to be the p and s polarization modes, or horizontal and vertical polarization modes in case the incoming waveis perpendicular to the layers. The Jones vector is then calculated as J = [ a , a ] / ( | a | + | a | ) .Considering only the phase difference ( ∆ ) between a and a and writing the Jones vector in theform [ a , b exp ( i ∆ )] , where a and b are positive real numbers, the polarization angle θ and ellipticity ε are calculated by: tan ( θ ) = ab cos ( ∆ ) / ( a − b ) (13)tan ( ε ) = p ( − q ) / ( + q ) , q = q − ( ab sin ∆ ) . (14)5e consider the ellipticity being positive when sin ( ∆ ) <
0. The difference between the θ and ε ofthe reflected (transmitted) and initial wave gives the Kerr (Faraday) rotation and ellipticity. II. DIELECTRIC TENSORS OF THE MATERIALS USED IN THE SIMULATIONS
This section shows the values of the dielectric tensor elements of the materials used in oursimulations. The dielectric tensor elements are plotted versus wavelength for the range we considerin the main text, which is from 414 nm to 900 nm. We refer to the respective original publications(referenced in each figure caption) for the full data.
500 600 700 800 900wavelength (nm)−2024681012 ε CrI monolayer Re ε xx Im ε xx Re ε xy Im ε xy FIG. S1. Dielectric tensor elements of monolayer CrI , calculated by Wu et al.
00 600 700 800 900wavelength (nm)−20246810 ε CrI bulk Re ε xx Im ε xx Re ε xy Im ε xy FIG. S2. Dielectric tensor elements of bulk CrI , calculated by Wu et al.
500 600 700 800 900wavelength (nm)−5.0−2.50.02.55.07.510.012.5 ε CrI monolayer Re ε xx Im ε xx Re ε xy Im ε xy FIG. S3. Dielectric tensor elements of monolayer CrI , calculated by Molina-Sánchez et al.
00 600 700 800 900wavelength (nm)−101234 ε CrBr monolayer Re ε xx Im ε xx Re ε xy Im ε xy FIG. S4. Dielectric tensor elements of monolayer CrBr , calculated by Molina-Sánchez et al.
500 600 700 800 900wavelength (nm)−0.50.00.51.01.52.02.5 ε CrCl monolayer Re ε xx Im ε xx Re ε xy Im ε xy FIG. S5. Dielectric tensor elements of monolayer CrCl , calculated by Molina-Sánchez et al.
00 600 700 800 900wavelength (nm)0510152025 ε Cr Ge Te monolayer Re ε xx Im ε xx Re ε xy Im ε xy FIG. S6. Dielectric tensor elements of monolayer C Ge Te , obtained from the optical conductivity calcu-lated by Fang et al.
500 600 700 800 900wavelength (nm)4.44.54.64.74.8 ε hBN Re ε xx FIG. S7. Dielectric constant of bulk hexagonal boron nitride (hBN), obtained from the refractive indexmeasured by Lee et al. The refractive index is described by the single oscillator model n = + A λ / ( λ − λ ) , with parameter values A = .
263 and λ = .
00 600 700 800 900wavelength (nm)2.122.132.142.152.162.17 ε SiO , thermal Re ε xx FIG. S8. Dielectric constant of thermally grown SiO , obtained from the refractive index measured byHerzinger et al. It is described by the function ε = n = offset + a λ / ( λ − b ) − c λ , with paramtetervalues offset = . a = . b = . µ m, and c = .
500 600 700 800 900wavelength (nm)−5051015202530 ε Si Re ε xx Im ε xx FIG. S9. Dielectric constant of Si, measured by Herzinger et al. II. INTERFERENCE EFFECTS IN OTHER 2D MAGNETS
In this section we present the simulation results for heterostructures composed of different 2Dmagnetic monolayers, with a similar configuration as the one shown in Fig. 2 of the main text formonolayer CrI . CrI31LSiSiO2 a bc dre fl contrast θ K ε K – e θ K ε K FIG. S10. Simulation results for a CrI (1L)-SiO (285 nm)-Si heterostructure. The dielectric tensor of CrI is taken from Molina-Sánchez et al. (Fig. S3). The reflectivity (a), contrast (b), Kerr rotation (c, e) andellipticity (d, f) are shown as function of wavelength and oxide thickness. Note that the color scale used in(e) and (f) is a symmetric log scale that is cut at ± − mrad. All values between ± − mrad are indicatedby the color white. Where the color scale is saturated, the values exceed the bounds of the scale. rBr3 (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16) a bc dre fl contrast θ K ε K (cid:17) e θ K ε K FIG. S11. Simulation results for a CrBr (1L)-SiO (285 nm)-Si heterostructure. The dielectric tensor ofCrBr is taken from Molina-Sánchez et al. The reflectivity (a), contrast (b), Kerr rotation (c, e) and ellip-ticity (d, f) are shown as function of wavelength and oxide thickness. Note that the color scale used in (e)and (f) is a symmetric log scale that is cut at ± − mrad. All values between ± − mrad are indicatedby the color white. Where the color scale is saturated, the values exceed the bounds of the scale. bc dre fl contrast θ K ε K CrC l3 (cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24) (cid:25) e θ K ε K FIG. S12. Simulation results for a CrCl (1L)-SiO (285 nm)-Si heterostructure. The dielectric tensor ofCrCl is taken from Molina-Sánchez et al. The reflectivity (a), contrast (b), Kerr rotation (c, e) and ellip-ticity (d, f) are shown as function of wavelength and oxide thickness. Note that the color scale used in (e)and (f) is a symmetric log scale that is cut at ± − mrad. All values between ± − mrad are indicatedby the color white. Where the color scale is saturated, the values exceed the bounds of the scale. G T a bc dre fl contrast θ K ε K (cid:26) e θ K ε K FIG. S13. Simulation results for a Cr Ge Te (1L)-SiO (285 nm)-Si heterostructure. The dielectric tensorof Cr Ge Te is taken from Fang et al. The reflectivity (a), contrast (b), Kerr rotation (c, e) and ellipticity(d, f) are shown as function of wavelength and oxide thickness. Note that the color scale used in (e) and (f)is a symmetric log scale that is cut at ± − mrad. All values between ± − mrad are indicated by thecolor white. Where the color scale is saturated, the values exceed the bounds of the scale. V. BULK CrI WITH EXPERIMENTAL VALUES FOR THE DIAGONAL OF ITSDIELECTRIC TENSOR
For bulk CrI , experimental data is available for the diagonal elements of its dielectric tensor .Fig. S14 shows the simulation results for the same stack as in Fig. 3 of the main text, but here theexperimental values are used for the diagonal of the dielectric tensor of CrI . Using the experimen-tal instead of theoretical values changes the wavelength and CrI thickness for which θ K and ε K are maximized, and it slightly changes the shape of the patterns seen in the plots of the reflectivity,contrast, Kerr rotation and ellipticity. However, the Kerr signals still have a similar non-monotonicbehavior, showing a strong peak and a dip where the reflectivity is close to a minimum. SiSiO2 a bc re fl contrast θ K ε K d FIG. S14. Simulation results for a stack of CrI (bulk)-SiO (285 nm)-Si stack. Experimental values are usedfor the diagonal of the dielectric tensor of CrI , and theoretical vales for the off-diagonal elements. Thereflectivity (a), contrast (b), Kerr rotation (c) and ellipticity (d) are shown as function of wavelength andoxide thickness. Where the color scale is saturated, the values exceed the bounds of the scale. . SIGNAL TO NOISE RATIO To find the best signal to noise ratio (SNR), we take R Θ def = R q θ K + ε K as a measure for the sig-nal strength, and consider two different noise regimes. In the case where the noise is independentof illumination intensity, the SNR is proportional to the signal strength (Fig. S14e). If the noise isdominated by shot noise, which grows as √ I , the SNR is proportional to R Θ / √ R = √ R Θ , whichis shown in Fig. S14f. In both cases, the highest SNR is attained at specific wavelength intervals.In the shot noise dominated regime, the best SNR is obtained when the MOKE signals are (almost)maximized, while for the intensity independent noise regime, the best SNR is obtained when theMOKE signals are further away from their maximum, where the reflectivity is higher.16 i hBNhBNSiO2285nmCrI31L a bc Re fl Contrast θ K ε K d e f2.5×R ϴ √ R ϴ √ FIG. S15. Simulation results for a hBN-CrI (1L)-hBN-SiO -Si(285nm) stack, including an estimationfor the signal to noise ratio (SNR) in two different noise regimes, as a function of wavelength and hBNthickness. The reflectivity (a), contrast (b), Kerr rotation (c) and ellipticity (d) are also shown in the maintext in Fig. 4. The signal strength R Θ shown in (e) is proportional to the SNR for the case where the noiseis independent of the intensity of the light. If the noise is dominated by shot noise, the SNR is proportionalto √ R Θ , shown in (f). Where the color scale is saturated, the values exceed the bounds of the scale. I. CONSTANT DIELECTRIC TENSOR FOR 2D MAGNETS
To determine the effect of the wavelength dependence of the dielectic tensor of CrI on ourresults, we perform the same simulation as done for Fig. 2 in the main text, but with a dielectrictensor of CrI that is fixed to its value at a wavelength of 680 nm. The results displayed in Fig.S16, unlike the results for the wavelength-dependent dielectric tensor of CrI displayed in Fig. 2 ofthe main text, do not show any additional features on top of the fan pattern, which is the dominantpattern for the reflectivity. This indicates that the additional features seen in the figures of the maintext and in sections III and IV, are caused by the wavelength dependence of the 2D magnet. CrI31LSiSiO2 a bc dre fl contrast θ K ε K FIG. S16. Simulation results for a CrI (1L)-SiO (285 nm)-Si heterostructure where the dielectric tensor ofCrI is fixed to is value at 680 nm. The reflectivity (a), contrast (b), Kerr rotation (c) and ellipticity (d) areshown as function of wavelength and CrI thickness. Where the color scale is saturated, the values exceedthe bounds of the scale. II. LARGE MOKE SIGNALS AT LOW REFLECTIVITY
A common feature in all our simulations is that the magnitudes of θ K and ε K are maximizedwhen the reflectivity is close to a minimum. This can be explained by the behaviour of the re-flection coefficients for the electric field of the two circular polarizations, r + and r − . Since thecircular polarizations are eigenmodes for all materials we used, reflection off the heterostructuredoes not mix them in the polar geometry. Therefore the two circular polarizations can be treatedindependently.Due to the small circular birefringence and dichroism induced by the 2DM, the reflectivity forthe two circular polarizations are only slightly different. Therefore, when the reflectivity (definedas | r + | + | r − | ) has a minimum, the reflectivity of both circular polarizations must be very close totheir minimum. This means that the ellipticity, defined as tan − ( | r + | − | r − | ) / ( | r + | + | r − | ) , has itsextrema close to this reflectivity minimum. In Fig. S17 and S18, | r + | , | r − | and ε K are plotted forthe stack used in Fig. 3 of the main text, which consists of bulk CrI on top of a SiO (285nm)/Sisubstrate. The former shows it as a function of CrI thickness at a constant wavelength, and thelatter as a function of wavelength at a constant CrI thickness. These figures indicate that theminima of r + and r − are indeed close to the reflectivity minimum, and that their minima have ingeneral different minimum values and are located at different positions. b |r+||r−| (cid:27) K (cid:28) =617.5nm a |r+||r−| (cid:29) K (cid:30) =600nm FIG. S17. Simulated | r + | , | r − | and ε K for a CrI (bulk)-SiO (285nm)-Si stack, plotted as a function of CrI thickness. Results are shown for a wavelength of 600 nm (a), and 617.5 nm (b). The large Kerr rotations can be explained by the behavior of the complex phase of r ± when theyare approximated to be linear in wavelength and layer thickness close to the reflectivity minimum.Let r ± ( x ) = r ± ( x ) + r ′± ( x ) ∆ x , where x is the wavelength, thickness, or a linear combination of19 ε K |r+||r (cid:31) | b ε K |r+||r | FIG. S18. Simulated | r + | , | r − | and ε K for a CrI (bulk)-SiO (285nm)-Si stack, plotted as a function ofwavelength. Results are shown for a CrI thickness of 13 nm (a), and 14nm (b). the two, r ′± ( x ) is the derivative of r ± ( x ) with respect to x , ∆ x is x − x , and x is the position ofthe reflectivity minimum. Around x , the phase of the complex reflectivity coefficients changesby π (from arg ( − r ′± ∆ x ) to arg ( r ′± ∆ x ) ) at a scale of | r ± ( x ) sin ( ψ ) | , where ψ is the difference incomplex phase of r ± ( ) and r ′± . The change in phase of r + and r − are in general centered atslightly different values of x , and happen at different scales, because the ferromagnetic layer has adifferent refractive index for the two circular polarizations. Therefore, the Kerr rotation, given by θ K = ( arg ( r + ) − arg ( r − )) /
2, increases when | r ± ( x ) sin ( ψ ) | becomes smaller. This happens when | r ± ( ) | is small, i.e. when the reflectivity is close to its minimum. In Fig. S19 and S20, arg ( r + ) ,arg ( r − ) and θ K are plotted for the stack used in Fig. 3 of the main text, which consists of bulk CrI on top of a SiO (285nm)/Si substrate. The former shows it as a function of CrI thickness at aconstant wavelength, and the latter as a function of wavelength at a constant CrI thickness. Thesefigures indicate that arg ( r + ) and arg ( r − ) indeed change rapidly close to the reflectivity minimum,and that this happens in general at different scales and at different positions. The Kerr rotation ismapped to the interval ( − π , π ] . When the phase difference of r + and r − crosses ± π , the Kerrrotation is mapped back into this interval, giving rise to discontinuities in θ K .20 r g(r+)arg(r-) A n g l e ( r a d ) R e fl ! =600nm a arg(r-)arg(r+) " =615nm b arg(r+) arg(r-) c =617.5nm FIG. S19. Simulated arg ( r + ) , arg ( r − ) and θ K for a CrI (bulk)-SiO (285nm)-Si heterostructure, plotted asa function of CrI thickness. Results are shown for a wavelength of (a) 600 nm, (b) 615 nm, and (c) 617.5nm. ar g(r+)arg(r − ) a arg(r+)arg(r − ) b arg(r+)arg(r − ) c FIG. S20. Simulated arg ( r + ) , arg ( r − ) and θ K for a CrI (bulk)-SiO (285nm)-Si heterostructure, plotted as afunction of wavelength. Results are shown for a CrI thickness of (a) 13 nm, (b) 13.5 nm, and (c) 14 nm. REFERENCES P. Yeh, Surface Science , 41 (1980). M. Wu, Z. Li, T. Cao, and S. G. Louie, Nature Communications , 2371 (2019). A. Molina-Sánchez, G. Catarina, D. Sangalli, and J. Fernández-Rossier, Journal of MaterialsChemistry C , 8856 (2020). Y. Fang, S. Wu, Z.-Z. Zhu, and G.-Y. Guo, Physical Review B , 125416 (2018). S.-Y. Lee, T.-Y. Jeong, S. Jung, and K.-J. Yee, Physica Status Solidi B , 1800417 (2019). C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, Journal of AppliedPhysics , 3323 (1998). P. M. Grant and G. B. Street, Bulletin of the American Physical Society , 415 (1968).21 B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, Seyler, D. Zhong,E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu,Nature , 270 (2017). W. A. Challener and T. A. Rinehart, Appl. Opt.26