second harmonic generation in nanoscale films of transition metal dichalcogenide accounting for multipath interference

Second harmonic generation in nanoscale films of transition metal dichalcogenide: Accounting for multipath interference , A V Kudryavtsev, S D Lavrov, A P Shestakova, L L Kulyuk , and E D Mishina Citation: AIP Advances 6, 095306 (2016); doi: 10.1063/1.4962764 View online: http://dx.doi.org/10.1063/1.4962764 View Table of Contents: http://aip.scitation.org/toc/adv/6/9 Published by the American Institute of Physics AIP ADVANCES 6, 095306 (2016) Second harmonic generation in nanoscale films of transition metal dichalcogenide: Accounting for multipath interference A V Kudryavtsev,1 S D Lavrov,1 A P Shestakova,1 L L Kulyuk,1,2,a and E D Mishina1 Moscow Technological University, 119454 Moscow, Russian Federation of Applied Physics, Academy of Sciences of Moldova, MD-2028 Chisinau, Republic of Moldova Institute (Received 18 May 2016; accepted 29 August 2016; published online September 2016) The transfer matrix method has been widely used to calculate wave propagation through the layered structures consisting entirely of either linear or nonlinear optical materials In the present work, we develop the transfer matrix method for structures consisting of alternating layers of linear and nonlinear optical materials The result is presented in a form that allows one to directly substitute the values of material constants, refractive index and absorption coefficient, into the expressions describing the second harmonic generation (SHG) field The model is applied to the calculation of second harmonic (SH) field generated in nano-thin layers of transition metal dichalcogenides exfoliated on top of silicon oxide/silicon Fabry-Perot cavity These structures are intensively studied both in view of their unique properties and perspective applications A good agreement between experimental and numerical results can be achieved by small modification of optical constants, which may arise in an experiment due to a strong electric field of an incident focused pump laser beam By considering the SHG effect, this paper completes the series of works describing the role of FabryPerot cavity in different optical effects (optical reflection, photoluminescence and Raman scattering) in 2D semiconductors that is extremely important for characterization of these unique materials © 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4962764] I INTRODUCTION Layered structures play important role in optical applications Optical methods in turn play important role in characterization of layered structures Modeling of the optical properties of such structures is based on direct calculations of transmissivity and reflectivity of each layer with interference between all transmitted and reflected waves being taken into account.1 The most elegant technique is based on transfer matrix approach in which multiple reflections and interference are included intrinsically.2 Layered structures of nonlinear optical materials, which generate optical harmonics are also used in optical applications, and thus require modeling of the harmonic intensity In the classical work of Bloembergen,3 which considered a nonlinear slab on a substrate, only several reflections were taken into account Later the importance of multiple reflections of fundamental and SHG wave in a slab was shown These effects become even more important if the system consists of more than one layer In the work of Li5 one-dimensional photonic crystals (PCs) were investigated, which represent multiply repeated pairs of nonlinear slabs with different optical constants and/or nonlinear susceptibilities.6,7 However, absorption of the material was not considered in this work a Author to whom correspondence should be addressed E-mail: kulyuk@phys.asm.md 2158-3226/2016/6(9)/095306/10 6, 095306-1 © Author(s) 2016 095306-2 Kudryavtsev et al AIP Advances 6, 095306 (2016) Recently optical characterization of the two-layered structures containing two-dimensional transition metal dichalcogenides (TMD) on a substrate was found important due to the unique properties of these materials and their potential for application in electronics and optoelectronics.8–12 Unlike graphene, two-dimensional (2D) TMDs are semiconductors with the band gap that can be tuned by changing the number of atomic layers in a slab For technological reasons one of the best substrates for these structures is an oxidized silicon wafer From the optical point of view, this is a two-layered system (TMD/SiO2 Fabry-Perot cavity) on Si substrate The Fabry-Perot (FP) cavity can be tuned for the used wavelength to enhance different optical effects in a single layer, from visualization13 by color contrast 14 to observation of luminescence1 and Raman scattering.15,16 In order to get quantitative information from such measurements, direct calculations of transmissivity and reflectivity with interference taken into account are required For harmonics generation these effects are also important However, one can find a gap within the transfer matrix technique for such systems Although they are simpler than PC, the results obtained for PCs cannot be automatically transferred to the two-layered system on a substrate (by substitution with zero of one of the second order susceptibilities) In nanoscale TMD layers, including exfoliated W S2 , effective second harmonic generation is observed.9,17,18 SHG is used for determination of crystallographic orientation of the flakes or microcrystallites in deposited TMD films.19–21 In many studies, significant enhancement of SHG is reported in nanoscale TMD layers compared to the bulk TMD crystals This phenomenon is of fundamental interest.9 Generally, the reported values of nonlinear susceptibility of TMD nanoscale layers have big dispersion.22,23 In this paper, we develop the transfer matrix technique to calculate intensity of the SH wave generated in TMD (based on work of Li5 ) The presence of SiO2 FP cavity provides conditions for multipath interference of fundamental and SH waves both in TMD and SiO2 The result is obtained in the form that is convenient for experiment analysis and interpretation, because it allows for direct substitution of material constants for the calculated values of the SH field The model is applied for interpretation of experimental dependence of SHG intensity on the thickness of the TMD layer in the W S2 /SiO2 /Si system II THEORETICAL MODEL Schematic view of the investigated structure is shown in Fig It consists of two layers (1 - W S2 , - SiO2 ) on Si substrate All layers are homogeneous For the parameters of layers, the subscripts denote the number of the layer, the superscripts – the wavelength The thickness of the layers and is d1 and d2 , respectively The refractive indices for fundamental field (m = 1) and second harmonic field (m=2) are n1(m) and n2(m) The nonlinear susceptibility of the layer is χ(2) , and of the layer is zero The difference of our model from the model of Li5 is that we (1) consider only one pair of layers, (2) only the layer possesses nonzero nonlinear susceptibility, the top layer have nonzero absorption FIG Schematic view of the investigated multilayered structure and of propagation directions and amplitudes of the forward and backward fundamental and SH waves 095306-3 Kudryavtsev et al AIP Advances 6, 095306 (2016) coefficient and (3) The Si substrate plays the role of the outer medium on the right side of the sample, which is valid for visible range The outer medium on the left side is air The calculation of the SHG intensity implies two stages:(1) calculation of the complex amplitudes of the fundamental electromagnetic waves in both layers; (2) - calculation of the complex amplitudes of the free SHG waves generated in nonlinear layer and then SHG intensity We consider the Si layer as a semi-infinite medium because the intensity of the transmitted fundamental and the SHG waves on the right side of the structure is zero due to large thickness of the layer and high absorption coefficient in the visible range A plane electromagnetic wave with an angular frequency ω and amplitude E0+ incident onto the multilayered structure and propagates along the z axis Polarization of the electric field is along the x axis and polarization of the magnetic field – along the y axis As a result of nonlinear interaction, two SH waves propagate in air (out of the structure): “reflected” SH wave with the amplitude E0(2)− and “transmitted” SH wave with the amplitude E0(2)+ In each layer the electric field of fundamental wave can be written as (z − zi−1 ) − ωt E1i (z, t) = Ω+i exp i k(1) i (z − zi−1 ) − ωt + Ω−i exp −i k(1) i , (1) where z0 is set to 0, zi = zi−1 + di , ki(1) = ni(1) k10 , and k10 = ωc , c is the light speed in vacuum Ω±1 are the complex amplitudes of the electric field at the interface between the layers, and the sign “+” or “-” denotes the forward- or backward-propagating wave It can be shown that the continuity condition of the E and H fields at the interface between the layers and can be written in the matrix form as n1(1) + n2(1) n1(1) − n2(1) Ω+1 = (1) Ω−1 2n1 Ω+2 Ω+2 = T12 − Ω−2 Ω2 n1(1) − n2(1) n1(1) + n2(1) (2) 1 1 (1) (1) (1) and D2 = n2 −n2(1) n1 −n1 The “propagation” matrix, which describes the phase changes of the electric field under the wave, propagating from one side of the layer to the other, can be written as We can define the matrix T12 as T12 = D1−1 D2 , where D1 = Pj = exp iki(1) di 0 exp −iki(1) di (3) The overall matrix, which describes the propagation of the fundamental forward and backward waves through the entire structure, can be written as −1 T = DSi D2 P2 D2−1 D1 P1 D1−1 Dair , (4) 1 1 (1) (1) (1) (1) and DSi = nSi −nSi nair −nair The relation between the fundamental fields in the air and in the Si substrate can be expressed by the equation where Dair = Ω+t =T Ω+0 Ω−0 (5) If the value of the amplitude Ω+0 of the pump wave is known, this equation can be solved for the variables Ω−0 and Ω+t Then, the relative complex amplitudes for both of the layers can be further determined from the analogous equations: Ω+1 Ω−1 Ω+2 Ω−2 = D1−1 Dair Ω+0 Ω−0 = D2−1 D1 P1 D1−1 Dair , Ω+0 Ω−0 (6) (7) The nonlinear polarization induced by the fundamental wave in the W S2 layer can be written as PNL (z, t) = ε χ(2) E1(1) (z) exp (−i2ωt) (8) 095306-4 Kudryavtsev et al AIP Advances 6, 095306 (2016) The propagation equation for the second harmonic field in the nonlinear media is ∂ E2ω (z) ∂ PNL (z, t) (z) + k E = µ 2ω ∂z2 ∂t (9) Solving this equation and taking into account the ratio between the fields H (2) and E (2) H1(2) (z) = ∇ × E1(2) (z) , ik20 (10) we can write the relation for the H (2) and E (2) in W S2 layer in matrix form as E1(2) (z) H1(2) (z) = 1 E1(2)+ (z) n1(2) −n1(2) E1(2)− (z) + 1 2n1(1) k10 k20 2n(1) k10 − k1 20 A1 Ω+1 A1 Ω−1 (z) + (z) C1 Ω+1 Ω−1 , (11) where A1 = −4ε (2)2 χ (2) ω −4k1(1)2 k1 χ(2) ω ;C1 = −4ε0(2)2 k1 and E1(2)+ (z) = E1(2)+ exp ik1(2) (z − z0 ) ; (12a) E1(2)− (z) = E1(2)− exp −ik1(2) (z − z0 ) ; (12b) Ω+1 Ω−1 (z) = Ω+1 (z) = Ω−1 2 exp 2ik1(1) (z − z0 ) ; (12c) exp −2ik1(1) (z − z0 ) (12d) The first part on the right side of the equation (11) denotes the free-wave amplitudes of the SH electromagnetic fields, the second part denotes the bound-wave amplitudes of the SH fields, and the third part denotes the influence of interference between the forward and backward fundamental fields on the SH fields Let us define new matrices for simplicity: 1 (2) nair (2) −nair 1 n1(2) −n1(2) Gair = G1 = B1 = Q1 = F1 = 2n1(1) k10 k20 − 0 exp −ik1(2) d1 2n1(1) k10 k20 ; G2 = ; B2 = Q2 = ; exp GSi = exp ik1(2) d1 exp i2k1(1) d1 ; −i2k1(1) d1 ; F2 = 1 (2) nSi (2) −nSi 1 n2(2) −n2(2) 2n2(1) k10 k20 ; (13a,b) ; (13c,d) − 2n2(1) k10 k20 ; (13e,f) exp ik2(2) d2 0 exp −ik2(2) d2 ; exp i2k2(1) d2 0 exp −i2k2(1) d2 (13g,h) (13i,j) Then we write the boundary conditions for the fields H (2) and E (2) between different layers: E0(2) (z) H0(2) (z) = air/W S2 E1(2) (z) H1(2) (z) ; air/W S2 E1(2) (z) H1(2) (z) = W S2 /SiO2 E2(2) (z) H2(2) (z) ; W S2 /SiO2 (14a,b) 095306-5 Kudryavtsev et al AIP Advances 6, 095306 (2016) E2(2) (z) H2(2) (z) = SiO2 /Si (2) (z) ESi HSi(2) (z) (14c) SiO2 /Si Taking into account the expressions (13, 14), one can write these boundary conditions more detailed: Gair E0(2)− = G1 E1(2)+ E1(2)− + B1 A1 Ω+1 C Ω+ Ω− 1 (15a) E2(2)+ + − + C Ω Ω = G2 (2)− 1 E2 (15b) A1 Ω−1 + - at the air/W S2 interface; G1 Q1 E1(2)+ E1(2)− + B1 F1 A1 Ω+1 A1 Ω−1 - at the W S2 / SiO2 interface; G2 Q2 E2(2)+ E2(2)− = GSi Et(2)+ (15c) - at the SiO2 /Si interface Getting rid of unknown variables E1(2)+ , E1(2)− , E2(2)+ , E2(2)− and using the values of the amplitudes Ω−1 , which can be obtained by solving equation (6), we can write the matrix equation for the amplitudes Et(2)+ and E0(2)− for free SHG waves To this, we need to consistently perform the E (2)+ E (2)+ following steps: 1) express 1(2)− from (15a); 2) express 2(2)− from (15b) using the expression E2 E1 (2)+ (2)+ E1 E2(2)+ Et for ; 3) express from (15c) using the expression for Thus the expression E1(2)− E2(2)− Ω+1 , for Et(2)+ is 0 A1 Ω+1 Et(2)+ −1 = G−1 (2)− + GSi (N2 B1 F1 − SB1 ) Si SGair E0 A1 Ω−1 + G−1 Si (N2 − S) 2 C1 Ω+1 Ω−1 , (16) −1 −1 where S = G2 Q2 G−1 G1 Q1 G1 , N2 = G2 Q2 G2 From this equation one can calculate the SH fields E0(2)− and Et(2)+ radiated by the structure into air and into the Si substrate, respectively The final expression for E0(2)− is E0(2)− = − Ω+0 2 K T22 22 (1) (1) × A1 L21 T11 T22 − T12 T21 (1) (1) + M21 C1 T21 T22 − T22 T21 (1) (1) + L21 T21 T22 − T22 T21 (1) (1) T11 T22 − T12 T21 , (17) −1 D P D−1 D P D−1 D ,T (1) = D−1 D , where T = DSi 2 1 air air −1 −1 K = G−1 Si SGair , L = GSi (N2 B1 F1 − SB1 ) , M = GSi (N2 − S) Figure shows the calculated dependences of the SHG intensity on the thicknesses of the SiO2 layer We used the following parameters for the calculations: the fundamental and SH wavelengths are 800 nm and 400 nm respectively, complex refraction indices on the fundamental wavelength are 095306-6 Kudryavtsev et al AIP Advances 6, 095306 (2016) FIG Calculated dependences of the SHG intensity on the thickness of the oxide from the multilayered W S2 /SiO2 /Si structure with different thicknesses of the W S2 (monolayer, and 20 layers, left axis, solid lines) and linear reflection contrast of monolayer at wavelengths 800 nm and 400 nm (right axis, dash and dotted lines) (1) 15 (1) (1) nW S2 = 3.79 + 0.36i; nSiO2 = 1.453; nSi = 3.688 + 0.006i; complex refraction indices on the SH (2) 15 (2) (2) wavelength are nW S2 = 3.68 + 2.28i; nSiO2 = 1.47; nSi = 5.57 + 0.387 The dependences of the SHG intensity on the thickness of SiO2 layer I 2ω dSiO2 are the sequences of double asymmetric bands The most effective second harmonic generation corresponds to the W S2 thickness of 20 monolayers (Fig 2) The I 2ω dSiO2 dependence for monolayer was calculated using optical constants for monolayer, (2) (1) 24 which are different from parameters for the bulk W S2 : nW S2 = 5.27 + 0.36i; nW S2 = 5.22 + 2.28i The SHG intensity in its maximum for monolayer is about 500 times smaller than the one for 20 monolayers For comparison, Fig shows dependences of linear reflection contrast for W S2 monolayer on SiO2 /Si substrate at the wavelengths of 800 nm and 400 nm Fig shows clearly that conditions for the best observation (visualization) of TMDs on SiO2 /Si in linear and SHG images are very different However, since in any experiment firstly one should find the single-layer flake, it is more practical to use SiO2 thickness, which is optimal for linear optical contrast (around 300 nm for white light) III EXPERIMENT Thin W S2 microcrystals were obtained by multiple mechanical exfoliation25 from the bulk W S2 crystals onto Si/SiO2 substrate The sample obtained by this method contained a variety of differently oriented single crystalline flakes with thicknesses ranging from to 130 nm For the visualization and characterization of the flakes, we used laser scanning confocal microscope (Alpha300S+, WITec) combined with atomic force microscope (AFM) For the SHG experiments, we used Ti:Sap laser (“Avesta-Project”) with a wavelength of 800 nm and repetition rate of 100 MHz The average power was 3mW (at the sample surface) and the pulse duration was 100 fs Focusing of the pumping beam and detection of the reflection SH beam were performed by ×100 objective (NA = 0.75) in confocal geometry The diameter of the laser beam on the sample surface was µm, which provided the pulse energy density of mJ/cm2 and the pulse power density of 50 GW/cm2 The SH signal was detected by a photomultiplier tube of the WITec detection system Polarization of the SHG wave was chosen by the Glan prism to be parallel to the polarization of the fundamental wave For each flake, the sample was rotated over its normal to find the orientation of the sample, which corresponded to the maximum of the SHG azimuthal dependence.26 The thicknesses of different regions of the flakes were measured by AFM 095306-7 Kudryavtsev et al AIP Advances 6, 095306 (2016) IV RESULTS AND DISCUSSION In Fig.3, the AFM and SHG images of several flakes are presented By comparison of these two different types of the images, the experimental SHG dependence on the thickness of the flakes I 2ω dW S2 was obtained (Fig.4, points) This dependence (Fig 4) possesses two maxima: big one around 10 nm and much lower at about 100 nm Figure 4, lines, shows the SHG intensity dependence on the TMD flake thickness I 2ω (dTMD ) (with the oxide thickness fixed) calculated in the frame of our model (Eq 17) The theoretical I 2ω (dTMD ) dependence is largely determined by optical constants First of all, we calculated the I 2ω (dTMD ) dependence for WS2 using the bulk and the monolayer values of optical constants (see Section II) The dependences are shown in Fig by black and blue lines, respectively It gives for the bulk values the first two maxima at 17 nm and 138 nm (for oxide thickness of 300nm), and for the monolayer values of the optical constants the first two maxima at 10 nm and 86 nm, both of which not match the experimental dependence The maxima positions that match the experimental values can be obtained for the different values of the optical constants The best fit (1) (2) gives nW S2 = 4.8 + 0.36i, nW S2 = 4.6 + 2.28i, the values, which fall between the values for the bulk and monolayer of W S2 Several reasons may result in modification of optical constants in 2D semiconductors: (1) dependence of refractive index on the thickness within few monolayers, and (2) modification of the optical constants by light-matter interaction (1) The optical parameters of the layered TMD strongly depend on the number of monolayers.15,24,27 Increasing the thickness up to eight monolayers, changes the electronic structure of W S2 Based on that, we introduce dependence of n and k on the number of layers Since the position of band gap decreases approximately exponentially15,28,29 with the increase of thickness, we use the FIG AFM (middle column) and related SHG (left column) images of the different W S2 flakes Right column – height and SHG signal profiles along the marked lines 095306-8 Kudryavtsev et al AIP Advances 6, 095306 (2016) FIG Dependence of the SHG intensity from the thickness of the flake Points: experimental values Lines: theoretical dependences approximation: u(d) = ubulk + u1ML − ubulk · exp(−γd), where u = nω , k ω , n2ω Such approximation broadens the first maximum and it closer to the experimental one, but does not help the second maximum (2) Strong light-matter interaction was shown in Refs 30,31 for atomically thin TMDs and earlier for bulk TMDs.32 It results in various effects due to electron-hole generation Another effect may arise due to Kerr nonlinearity, associated with nonlinearity of bound rather than free electrons Additionally laser light causes heating of the sample All these effects may result in changes of optical constants Further, we present the estimation of the probable contribution of these effects to optical constants in typical SHG experiment Free carrier generation Free carrier generation is implemented by optical absorption The W S2 monolayer is a direct band semiconductor with a bandgap of about 2.1 eV Bulk W S2 is an indirect band semiconductor with a bandgap of about 1.3 eV.8 For the used wavelength of 800 nm (1.55 eV) for low power, absorption is entirely due a single photon process However, for high laser power density used in our experiments, a two-photon absorption is quite significant as well The total concentration of photocarrier generation is33 N = (1 − R)feff β (1 − R) feff α + , ω ωτ (18) where R = 0.35 is the average reflectivity, τ = 100 fs is the pulse width, feff = mJ/cm2 is pulse energy density, ω is the photon energy, α ∼ × 105 cm−1 24 is the linear absorption coefficient and β = × 10−7 cm/W is the two-photon absorption coefficient.34 It gives the value of N 1.3 × 1021 cm−3 The changes in dielectric function and, consequently, in n, due to the presence of free carriers is described with the Drude model:35 ∆nFC = − e2 N ∗ (T ) , 2εn0 ω2 mopt e (19) −1 ∗ (T ) = 1/m∗ + 1/m∗ where ω is the angular laser frequency, e is the electron charge, and mopt e e h is the carrier effective optical mass For our estimation, we took me∗ = mh∗ and used the value of me∗ = 0.83m0 for bulk W S2 36 It gives the value of ∆n ≈ 0.25 095306-9 Kudryavtsev et al AIP Advances 6, 095306 (2016) As we see, the order of magnitude of ∆n associated with the free carriers’ the same as required for interpretation of our experimental dependencies Thus, free carriers generation for the used laser power may be the reason of the deviation of refractive index from the reported bulk value All other effects associated with free carrier generation (plasma screening effects, band-filling effects and band-gap renormalization) not play noticeable role because they are all important in the vicinity of band edge (and exciton resonances),31,37,38 which is far from the photon energy used in our experiment Kerr effect We can expect that in our case more significant changes in refraction index for each thicknesses are associated with optical Kerr effect The effective refractive index in this case can be described by the following expression: n = n0 + n2 I,37 where n0 and n2 are the effective linear and nonlinear refractive indices, respectively, and I is the incident laser intensity (pulse power density), which in our (1) caseis n0 = nW S2 = 3.79 Regarding to TMDs, the effect was considered in Refs 39,40 The third-order nonlinear susceptibility of W S2 monolayer was estimated as χ(3) (esu) = (5.15 ± 0.12) × 10−9 39 To estimate the value of n2 in this case we used this value of χ(3) in well-known formula37 n2 cm2 /W = 0.0395 χ(3) (esu)/n02 Substituting the values in the expression, we get n2 = 1.4 × 10−11 cm2 /W Taking into account the value of pulse power density of 50GW/cm2 , we get ∆n = n2 I ≈ 0.7 The order of magnitude of this estimation corresponds to the changes ofn in our experiment The local changes in n can also be caused by laser heating However, it was shown in Ref 40 that due to experimental geometry and high thermal conductivity influence of heating is minor In order to check an extension of our model for other systems, we perform calculations of reflectivity for MoS2 /SiO2 (300 nm)/Si structure, the calculated I 2ω dMoS2 is shown in Fig by dashed line The result is identical to the one reported in work 40 that was obtained by different method V CONCLUSIONS In this work, we report on calculation of SHG intensity in a nonlinear film placed on top of the Fabry-Perot cavity The system is of particular importance, because it represents wide and intensively studied 2D semiconductors exfoliated on top of SiO2 /Si substrate Calculations of SHG intensity are performed using optical matrix technique and result is presented in an elegant and clear expression, which allows direct substitution of optical constants; we believe that it will be definitely appreciated by experimentalists Since nonlinear optical processes requires quite high laser power density, strong light-matter interaction may change optical constants Our estimations based on the dependence of the SHG intensity on the thickness of W S2 layers show that if the photon energy falls away from the band edge, optical Kerr effects play the dominant role in the optical constant modifications If the pump energy approaches the band edge and excitonic resonances, then additional effects of free carrier generation become important One more important result is the proof of different conditions for the best observation (visualization) of TMD attached to SiO2 /Si substrate in linear and SHG images The obtained result is also important for calculations of nonlinear optical susceptibility of TMDs attached to SiO2 Fabry-Perot cavity Overall, we believe that our results, both experimental and theoretical, will be used as convenient tool for optical diagnostics of TMD nanolayers and devices based on these materials ACKNOWLEDGEMENTS The work is supported by Russian Science Foundation (Grant #14-12-01080) The part of the work devoted to calculations of Kerr effect contribution was supported by Ministry of Education and Science of RF (State task “Organization of Scientific Research,” No 11.144.2014/K) 095306-10 H Kudryavtsev et al AIP Advances 6, 095306 (2016) Zhang, Y Ma, Y Wan, X Rong, Z Xie, W Wang, and L Dai, Appl Phys Lett 10, 101904 (2015) D Palik, Handbook of Optical Constants of Solids (Academic, San Diego, 1985) N Bloembergen, Nonlinear optics (Benjamin, New York, 1965) W N Nerman and L M Hayden, J Opt Soc Am 12, 416 (1995) J.-J Li, Z.-Y Li, and D.-Z Zhang, Phys Rev E 75, 056606 (2007) P Szczepa´ nski, T Osuch, and Z Jaroszewicz, Appl Opt 48, 5401 (2009) S M Hamidi, T Parvini, and M M Tehranchi, Appl Phys A 11, 525 (2013) Q H Wang, K Kalantar-Zadeh, A Kis, J N Coleman, and M S Strano, Nature Nanotech 7, 699 (2012) K L Seyler, J R Schaibley, P Gong, P Rivera, A M Jones, S Wu, J Yan, D G Mandrus, W Yao, and X Xu, Nature Nanotech 10, 407 (2015) 10 M M Glazov, T Amand, X Marie, D Lagarde, L Bouet, and B Urbaszek, Phys Rev B 89, 201302 (2014) 11 Z Ye, T Cao, K O’Brien, H Zhu, X Yin, Y Wang, S G Louie, and X Zhang, Nature (London) 513, 214 (2014) 12 M Koperski, K Nogajewski, A Arora, V Cherkez, P Mallet, J.-Y Veuillen, J Marcus, P Kossacki, and M Potemski, Nature Nanotech 10, 503 (2015) 13 M M Benameur, B Radisavljevic, J S Heron, S Sahoo, H Berger, and A Kis, Nanotech 22, 125706 (2011) 14 M R Muller, A Gumprich, E Ecik, K T Kallis, F Winkler, B Kardynal, I Petrov, U Kunze, and J Knoch, J Appl Phys 118, 145305 (2015) 15 S.-L Li, H Miyazaki, H Song, H Kuramochi, S Nakaharai, and K Tsukagoshi, ACS Nano 6, 7381 (2012) 16 J.-H Fan, P Gao, A.-M Zhang, B.-R Zhu, H.-L Zeng, X.-D Cui, R He, and Q.-M Zhang, J Appl Phys 115, 053527 (2014) 17 C Janisch, Y Wang, D Ma, N Mehta, A L Elıas, N.-P Lopez, M Terrones, V Crespi, and Z Liu, Sci Rep 4, 5530 (2014) 18 G Wang, X Marie, I Gerber, T Amand, D Lagarde, L Bouet, M Vidal, A Balocchi, and B Urbaszek, Phys Rev Lett 114, 097403 (2015) 19 D Dumcenco, D Ovchinnikov, K Marinov, P Lazi, M Gibertini, N Marzari, O Lopez Sanchez, Y C Kung, D Krasnozhon, M.-W Chen, S Bertolazzi, P Gillet, A Fontcuberta I Morral, A Radenovic, and A Kis, ACS Nano 9, 4611 (2015) 20 Y Li, Y Rao, K F Mak, Y You, S Wang, C R Dean, and T F Heinz, Nano Lett 13, 3329 (2013) 21 D J Clark, V Senthilkumar, C T Le, D L Weerawarne, B Shim, J I Jang, J H Shim, J Cho, Y Sim, M.-J Seong, S H Rhim, A J Freeman, K.-H Chung, and Y S Kim, Phys Rev B 90, 121409 (2014) 22 W T Hsu, Z A Zhao, L J Li, C H Chen, M H Chiu, P S Chang, Y C Chou, and W H Chang, ACS Nano 8, 2951 (2014) 23 L M Malard, T V Alencar, A P M Barboza, K F Mak, and A M de Paula, Phys Rev B 87, 201401 (2014) 24 H.-L Liu, C.-C Shen, S.-H Su, C.-L Hsu, M.-Y Li, and L.-J Li, Appl Phys Lett 105, 201905 (2014) 25 K S Novoselov, A K Geim, S V Morozov, D Jiang, Y Zhang, S V Dubonos, I V Grigorieva, and A A Firsov, Science 306, 666 (2004) 26 J J Dean and H M van Driel, Appl Phys Lett 95, 261910 (2014) 27 H.-C Kim, H Kim, J.-U Lee, H.-B Lee, D.-H Choi, J.-H Lee, W H Lee, S.-H Jhang, B H Park, H Cheong, S.-W Lee, and H.-J Chung, ACS Nano 9, 6854 (2015) 28 W Zhao, Z Ghorannevis, L Chu, M Toh, C Kloc, P.-H Tan, and G Eda, ACS Nano 7, 791 (2013) 29 L Britnell, R M Ribeiro, A Eckmann, R Jalil, B D Belle, A Mishchenko, Y.-J Kim, R V Gorbachev, T Georgiou, S V Morozov, A N Grigorenko, A K Geim, C Casiraghi, A H Castro Neto, and K S Novoselov, Science 340, 1311 (2013) 30 A Chernikov, C Ruppert, H M Hill, A F Rigosi, and T F Heinz, Nature Phot 9, 466 (2015) 31 C Ballif, M Regula, and F Levy, Solar Energy Materials and Solar Cells 57, 189 (1999) 32 A J Sabbah and D M Riffe, Phys Rev B 66, 165217 (2002) 33 S Zhang, N Dong, N McEvoy, M O’Brien, S Winters, N C Berner, C Yim, Y Li, X Zhang, Z Chen, L Zhang, G S Duesberg, and J Wang, ACS Nano 9, 7142 (2015) 34 R A Soref and B R Bennet, IEEE J Quantum Electron 23, 123 (1987) 35 D Wickramaratne, F Zahid, and R K Lake, J Chem Phys 140, 124710 (2014) 36 R W Boyd, Nonlinear Optics (Academic Press, San Diego, 2003) 37 B R Bennett, R A Soref, and J A Del Alamo, IEEE J Quantum Electron 26, 113 (1990) 38 G Wang, S Zhang, X Zhang, L Zhang, Y Cheng, D Fox, H Zhang, J N Coleman, W J Blau, and J Wang, Photon Res 3, A51 (2015) 39 Y Wu, Q Wu, F Sun, C Cheng, S Meng, and J Zhao, Proc Nat Acad Sci USA 112, 11800 (2015) 40 S D Lavrov, A V Kudryavtsev, A P Shestakova, L Kulyuk, and E D Mishina, Optics and Spectroscopy 120, 808 (2016) E  ...AIP ADVANCES 6, 095306 (2016) Second harmonic generation in nanoscale films of transition metal dichalcogenide: Accounting for multipath interference A V Kudryavtsev,1 S D Lavrov,1... zero of one of the second order susceptibilities) In nanoscale TMD layers, including exfoliated W S2 , effective second harmonic generation is observed.9,17,18 SHG is used for determination of. .. calculate intensity of the SH wave generated in TMD (based on work of Li5 ) The presence of SiO2 FP cavity provides conditions for multipath interference of fundamental and SH waves both in TMD