Optical detection of the susceptibility tensor in two-dimensional crystals

2,5,10-12 . Here we perform a smoking gun experiment on the optical response of a single-layer two-dimensional crystal that addresses these

problems. We successfully remove the substrate contribution to the optical response of these materials 28 by a step deposition of a monolayer crystal inside a thick polydimethylsiloxane prism. This allows for a 29 reliable determination of both the in-plane and the out-of-plane components of the monolayer surface 30 susceptibility tensor. Our results prescribe one clear theoretical model for these types of material. This 31 work creates opportunities for a precise characterization of the optical properties of two-dimensional 32 crystals in all the optical domains such as the nonlinear response 13 , surface wave phenomena 14 or 33 magneto-optical Kerr effect 15 . Our assay will be relevant to future progresses in photonics and 34 optoelectronics with 2D materials 16,17 . 35 One of the great achievements in materials science is certainly the isolation of individual crystal planes, 36 starting from solids with strong in-plane bonds and weak, van der Waals-like, coupling between layers. In 37 general, when dealing with layered crystals, Maxwell's equations increase in complexity in order to 38 account for anisotropy. The susceptibility of these materials is no longer described by a constant, but by 39 a tensor and, as regards to the optical properties, they are at least uniaxial. Consequently, we expect that 40 the optical spectra of single-layer, two-dimensional (2D) crystals also show out-of-plane anisotropy. In 41 spite of the fact that experimental assessment of the out-of-plane anisotropy for the three-dimensional 42 crystal is manageable, this is not the case for isolated monolayers 1 . As a result, fifteen years after the 43 exfoliation of the first atomically thin crystal, the exact description of its optical response remains an 44 active and debated area of research 2-7 . 45 There are currently two main models in use for the optical description of single-layer crystals, the first is 46 a thin film model that can be either isotropic or anisotropic along the vertical direction 1,6-9,18 . Optical 47 contrast measurements, used to detect single and multiple layers of a 2D crystal, are analysed by choosing 48 an isotropic thin film 6,7,9 . This is justified by the normal incidence configuration: the electric field is parallel 49 to the crystal and the in-plane optical constants dominate the optical response. Surprisingly, the same 50 model applies to spectroscopic ellipsometry, a very sensitive technique that works at any angle of 51 incidence. If the first ellipsometric data for graphene were fitted using a uniaxial thin film model 18 , 52 subsequent analysis concluded that ellipsometry is only sensitive to the in-plane optical constants 1 . This 53 because the sensitivity to anisotropy is dependent on the path length through the film, which is extremely 54 limited for a monolayer 1 . We are aware of only two papers that claim a big difference in between optical 55 constants of single-layer transition metal dichalcogenides (TMDC) extracted from ellipsometric data using 56 an anisotropic and an isotropic thin film model 19,20 . However, the same articles also find large differences 57 in the in-plane optical constants, a result in contradiction with several other works 1,8,18,21,22 . 58 In the second model, the monolayer is treated as a 2D surface current without any thickness. In this case, 59 the system is intrinsically anisotropic with both null out-of-plane surface susceptibility (⊥) and 60 conductivity (⊥) 2,5,10-12 . If on one side the use of two different models is a sign of a physical richness, on 61 the other it poses new conundrums. The two approaches, starting from the measured ellipsometric 62 parameters, provide a different in-plane surface susceptibility (∥) and conductivity (∥) 21 . For ∥, this 63 difference is greater than the experimental error 21 . From a theoretical point of view, the choice of setting 64 ⊥ = 0 m and ⊥ = 0  -1 in the surface current model is arbitrary, yet so is the use of an isotropic thin film 65 model, when we expect the system to be highly anisotropic. Ab-initio many body calculation of the optical 66 spectra of graphite, graphene and bilayer graphene report interesting differences in the simulated out-67 of-plane properties going from the bulk limit, down to a monolayer, but they exclude a null value 23 , this 68 also holds for other 2D crystals like TMDC 24 . 69 One main experimental problem in the analysis of the optical response of a 2D crystal is the role of the 70 substrate, that adds a background signal, hiding the small contribution that comes from the out-of-plane 71 optical constants. We measure ∥, ∥, ⊥ and ⊥ in a two-step experiment ( fig. 1 a, b). First, we extract the 72 ellipsometric data (s, s) from a single-layer 2D crystal deposited on a transparent dielectric substrate, 73 namely polydimethylsiloxane (PDMS). Then, the same crystal is completely immersed in PDMS for a 74 second ellipsometric measurement that provides a new set of data (i, i). By inverting the fundamental 75 equation of ellipsometry tan   = / , where rp and rs are the Fresnel coefficients, it is possible to 76 extract ∥, ∥, ⊥, ⊥ from s, s, i, i.

77
While the reflection coefficients for the anisotropic slab model are reported in ref. 25 , it is much more 78 difficult to find in the literature a complete and correct generalization of the surface current model to 79 include also the orthogonal polarization. We provide here the essential conceptual steps. The detailed 80 calculations of the Fresnel coefficients are in the Methods section. The polarization parallel ∥ ⃗⃗⃗ and 81 perpendicular ⊥ ⃗⃗⃗⃗ to the crystal plane induce two surface currents: ∥ ⃗⃗ and ⊥ ⃗⃗⃗⃗ respectively. The reflected 82 field is the superposition of the reflected fields from these two currents. We thus solve two set of 83 boundary conditions 26 , one for ∥ ⃗⃗ : 84 and one for ⊥ ⃗⃗⃗⃗ : 85 Here ,̂ are the unitary vectors in the z and x direction (Fig. 1), ⃗ ⃗ is the magnetic field, ⃗ the electric 86 field, 0 the vacuum permittivity and the subscripts 1 and 2 refer to the media above and below the 87 monolayer. The first set of boundary conditions are discussed in refs 2,10,11 , for the surface current model 88 with null ⊥ ⃗⃗⃗⃗ . The second set has the following simple explanation 26-28 . In the radiation zone 29 , the 89 electromagnetic field due to an oscillating electric dipole in the  direction is identical to an 90 electromagnetic field due to an oscillating magnetic dipole in the −̂ direction. This last one would cause 91 a jump in the tangential component of ⃗ . 92 Assuming a time dependence , ( is the angular frequency of the light) for a monolayer completely 93 immersed in a dielectric medium of refractive index n we find: 94 where the subscript i denotes that the sample is immersed, p is in place of p-polarized light, k is the wave 96 vector of light in vacuum,  is the impedance of vacuum and  is the angle of incidence. For a monolayer 97 deposited at the interface of vacuum with a dielectric substrate of refractive index n we find: 98 where the subscript s denotes the substrate, and  t is the propagation angle in the dielectric.

102
In our experiment, a 7x7 millimeter-size, polycrystalline, single-layer 2D crystal is deposited on a 1 cm 103 thick (to avoid back reflections) PDMS substrate (2x2 cm square) by chemical vapor deposition ( fig. 1a). 104 The first ellipsometric measurement (VASE ellipsometer, J. A. Wollam) provides s, and s and the 105 confirmation that we are dealing with a monolayer. We then place our sample in a prism-shaped mold 106 (base 6x5 cm, height 3.5 cm), we pour non-polymerized PDMS on it and we wait for complete 107 polymerization. This process successfully produces a 2D crystal immersed in PDMS without any additional 108 interface in between the previous and the newly added material. The prism has two optical quality lateral 109 windows (fig 1b). The light reflected by the sample in this second step is 2 orders of magnitude less than 110 in the previous one, because there is no more the substrate contribution. For this reason, we set up a 111 manual ellipsometer at the wavelength of 633 nm ( fig. 1 c) to measure i and i.

112
We studied both monolayer graphene and MoS2. First, we tested our two steps procedure without 113 deposition of the monolayer to ensure that we do not observe any reflection in between the embedded 114 substrate and the final prism. Ellipsometric parameters s and s are measured at angles of incidence s The study of a 2D crystal deposited on a substrate or immersed in a host material is by far the most 142 common situation in laboratory. Still, measuring a free-standing monolayer 30 will be of great interest. If 143 macroscopically it means a 2D crystal immersed in a medium with n = 1 rather than 1.   a. Insulator single-layer 2D crystal immersed in a dielectric medium of refractive index n.

255
We compute the reflected field due to ∥ ⃗⃗ by solving the boundary equations (1). In this case, the electric 256 field parallel to the crystal plane is continuous across the crystal itself giving: ∥ ⃗⃗⃗ = 0  ∥ ∥ ⃗⃗⃗ (5). Choosing 2 257 ⃗ ⃗ along −, equations (1) plus (5) run: 258 Where the subscript i, r, t denote the incident, reflected and transmitted fields. We compute the reflected 262 field due to ⊥ ⃗⃗⃗ by solving the boundary equations (2). In this case, the electric field orthogonal to the 263 crystal plane is continuous across the crystal itself giving: ⊥ ⃗⃗⃗⃗ = 0  ⊥ ⊥ ⃗⃗⃗⃗ (6) . Choosing ⃗ ⃗ along −̂, 264 equations (2) plus (6) run: 265 In accordance to the superposition principle, the total reflected field is: = 1 + 2 , the total 269 transmitted field is: = 1 + 2 − = − 1 + 2 . The reflection and the transmission 270 coefficients are respectively: = ; = . We verify that: b. Insulator single-layer 2D crystal at the vacuumdielectric medium interface. 272 The electromagnetic fields in vacuum satisfy:  ⃗ ⃗ =̂  ⃗⃗⃗ . For ∥ ⃗⃗ equations (1) plus (5) run: 273 For ⊥ ⃗⃗⃗ equations (2) plus (6) run: 277 In accordance to the superposition principle, the total reflected field is: = 1 + 2 − , the total 281 transmitted field is: We have prepared one large-area (up to millimeters), polycrystalline, continuous, single-layer MoS2 with 308 chemical vapor deposition (CVD). Monolayer MoS2 grows on soda-lime glass using a three-zone tube 309 furnace under low-pressure atmosphere. A piece of Mo foil (Alfa Aesar, 9.95%; 0.025mm thick) was 310 folded as a "bridge" and placed on top of the glass substrate with a gap of 10 mm. S power(Alfa Aesar, 311 purity 99.5%) was located at the upstream of the furnace. Before heating, the system was purged 312 with Ar (80 sccm) for 10 min to get out of the air. Then, Ar (50 sccm) and O2 (6 sccm) mixed gas flows were 313 introduced into the system to create a stable growth atmosphere. The temperature of the 314 S powder and the glass substrate were set at 100 °C and 720 °C, respectively. The growth time was set at 315 1-3 min. After growth, the furnace was naturally cooled to room temperature. The as-synthesized MoS2 316 monolayer/soda-lime samples were firstly spin-coated with PMMA at 1000 rpm for 1 min, followed by 317 baking at 80 °C for 20 min. The PMMA-supported samples were then inclined into the pure water, and the 318 PMMA/MoS2 complex was naturally peeled off under the surface tension effect. The PMMA/MoS2 film 319 was collected by PDMS. Finally, the PMMA film was removed via acetone. 320 c. PDMS substrate preparation and immersion of the 2D crystal in PDMS. 321 The PDMS (Sylgard 184, by Dow Corning) substrate is prepared by mixing the base elastomer and the 322 curing agent in ratio 10:1, and by 72 hours room temperature (RT) polymerization. Then, after the 323 deposition of the monolayer 2D crystal (see above), new pre-polymerized PDMS is poured on the 324 structure, using a 3D printed prism shape structure (ABS filament) as mold. We again wait for RT 325 polymerization. Since the 3D printed structure presents a high roughness that would compromise the 326 optical measurements, glass slides are glued on them. In this way, we obtain a PDMS prism with optical 327 quality lateral windows. To prevent the PDMS bonding to the glass slides during the polymerization 328 process, a fluorinated antiadhesive coating (Trichloro(1H,1H,2H,2H-perfluorooctyl)silane, by Sigma-329 Aldrich) is applied.  (Table 1). Although small, this last difference shows the importance of a careful 340 substrate characterization when dealing with monolayers. For MoS2 these discrepancies are even smaller 341 than for graphene and equal respectively to 0.02 nm, 710 -7  -1 , 0.03 nm and 910 -6  -1 . 342 For graphene, we used a 12 x 12 x 12 regular mesh of k-points for sampling the Brillouin zone (relative to 380 the super-cell) which was shifted in order not to contain the  point and we included 12 occupied and 48 381 unoccupied orbitals. A stretching factor of 1.67 was applied to the DFT valence and conduction bands in 382 order to match GW energies in the proximity of the band closure at the K-point. These GW values were 383 obtained extrapolating, to the bulk limit, results for models of 12 and 32 atoms. 384 For MoS2, we used a 6 x 6 x 6 regular mesh of k-points for sampling the Brillouin zone (relative to the 385 super-cell) together with a fully relativistic treatment of the spin-orbit coupling. We included 56 occupied 386 and 96 unoccupied orbitals. A scissor factor of 1.30 eV was applied to the DFT band structure in order that 387 the DFT gap matches the bulk limit of the GW one. The latter was evaluated at the scalar relativistic level. 388 Moreover, for MoS2 we added a constant term of 2.13 to ∥ and one of 1.33 to ⊥ in order to account for 389 the upper unoccupied orbitals which were not included in the BSE calculation. These factors were chosen 390 in order to match the corresponding components of the dielectric tensor calculated from density 391 functional perturbation theory. 392 It is worth noting that the calculated BSE susceptibilities exhibit only a mild dependence on the strategy 393 chosen for updating the DFT bands in order to account for GW effects. In particular, the out-of-plane 394 responses are particularly insensitive as it is shown, for MoS2 in Figs. 5 and 6 of the extended data section 395 where we report results obtained using scissor factors of 1.0, 1.3 eV and a scissor factor of 1.0 396 accompanied by a stretching one of 1.2 for both manifolds as in ref. 42 . 397 The in-plane C∥ and out-of-plane C⊥ complex susceptibilities are rigorously defined as the ratio between 398 the induced surface dipole densities with respect to the transmitted electric field. In the case of C∥ the 399 transmitted field is parallel to the 2D layer and is conserved along the simulation cell. It is worth noting 43 400 that in first-principles simulations with periodic boundary conditions (PBC) the directly accessible quantity 401 is the response with respect to the total or internal (electric) field. This is due to incommensurate 402 extension of the field with respect to the length period of the PBC. This makes it possible to obtain C∥ where L is the periodic length of the simulation cell in the direction perpendicular to the 2D layer. This 406 length is big from a microscopic point of view in order to null the interaction between the atomic layers 407 but small from a macroscopic point of view (for instance it is much smaller than the wavelength of the 408 incident light). 409 For C⊥ the transmitted electric field Et⊥, which is now perpendicular to the 2D layer, can be found 43 from   See manuscript for full gure caption.