Abstract
We investigate Second Harmonic Generation (SHG) in monolayer WS_{2} both deposited on a SiO_{2}/Si substrate or suspended using transmission electron microscopy grids. We find unusually large second order nonlinear susceptibility, with an estimated value of d_{eff} ~ 4.5 nm/V nearly three orders of magnitude larger than other common nonlinear crystals. In order to quantitatively characterize the nonlinear susceptibility of twodimensional (2D) materials, we have developed a formalism to model SHG based on the Green's function with a 2D nonlinear sheet source. In addition, polarized SHG is demonstrated as a useful method to probe the structural symmetry and crystal orientation of 2D materials. To understand the large second order nonlinear susceptibility of monolayer WS_{2}, density functional theory based calculation is performed. Our analysis suggests the origin of the large nonlinear susceptibility in resonance enhancement and a large joint density of states and yields an estimate of the nonlinear susceptibility value d_{eff} = 0.77 nm/V for monolayer WS_{2}, which shows good orderofmagnitude agreement with the experimental result.
Introduction
Twodimensional (2D) materials, exhibiting extraordinary and novel properties not available in their bulk forms, are at the forefront of nanomaterial technologies^{1,2,3,4,5,6,7,8}. For instance, the transition metal dichalcogenide (TMD) family of monolayer materials (such as MoS_{2}, WS_{2}), unlike their bulk counterparts, possesses direct band gaps^{9,10,11,12} and could thus generate strong photoluminescence^{13,14,15,16,17}. The TMD family of monolayer materials have also garnered interest in nonlinear optics^{18,19}. Characterization of exfoliated WS_{2}^{20} and synthesized MoS_{2} monolayers^{21,22,23} has revealed large secondorder nonlinear susceptibility, as high as nm/V^{22,23}. Here we investigate SHG in monolayer WS_{2} synthesized by chemical vapour deposition. We have developed a formalism to model SHG in 2D materials using the Green's function with a 2D nonlinear sheet source, which also takes into account the focused excitation geometry and the substrate effect as opposed to a simple planewave model. SHG in WS_{2} monolayers both suspended and deposited on SiO_{2}/Si substrates is studied. The estimated second order nonlinear susceptibility is approximately three orders of magnitude higher than that of common nonlinear optical crystals such as BBO^{24}. Polarized SHG in WS_{2} is studied to probe the tensorial nonlinear susceptibility, which is related to the structural symmetry of the 2D crystal. To gain further insight into the large nonlinear susceptibility, density functional theory (DFT) calculations of the secondorder nonlinear susceptibility are also performed.
Results
A WS_{2} monolayer lacks inversion symmetry and has nonvanishing secondorder nonlinear susceptibility. An experimental setup was developed to characterize the SHG in the synthesized WS_{2} samples (Fig. 1a). Fig. 1(b) shows an optical micrograph of a triangular monolayer of WS_{2} grown on a SiO_{2}/Si substrate. The same triangle is also shown as an SHG image in Fig. 1(c), obtained by raster scanning the sample with a step size of 1 μm. As can be observed from the SHG image, the background contribution from the substrate surface in the surrounding area is negligible. To further isolate substrate effects, the same experiments were also performed on a WS_{2} sample that was suspended on a transmission electron microscopy (TEM) grid. Fig. 1(d) shows an optical micrograph of the WS_{2} film on the TEM grid and Fig. 1(e) shows the SHG image of the same triangle. Although the WS_{2} triangle can barely be seen on the TEM grid micrograph due to weak reflection contrast, the triangle is easily apparent in the SHG image. A depth scan was also performed on the WS_{2} samples and bare substrates to measure the SHG signal as a function of sample depth position, by translating the sample axially with a computercontrolled linear stage. The results shown in Fig. 2(a) indicate that the Second Harmonic (SH) signals generated from both the onsubstrate and suspended samples are predominantly from the WS_{2} monolayers, as the bare substrates had no detectable SH signal under the same measurement conditions. To characterize the secondorder nonlinear susceptibility, we measured the average power of the SHG signal as the power of the fundamental beam was varied. Fig. 2(b) shows that the resulting relation is quadratic: a linear fit of the loglog plot reveals a slope of 2.04 for the TEMsuspended WS_{2} sample and 2.12 for the WS_{2}/SiO_{2}/Si substrate sample. The lower inset of Fig. 2(b) shows representative spectra of the fundamental and SHG signal, which also clearly indicate the frequency doubling.
Discussion
In order to quantitatively characterize the secondorder nonlinear susceptibility, we have developed a formalism to model SHG from 2D materials on a semiinfinite substrate. A planewave fundamental beam uniformly filling the aperture of a lens (focal length F, aperture radius a, numerical aperture NA = a/F) is focused onto the 2D material and the generated SH signal is epicollected. The fundamental field at the sample can be represented by the Debye integral^{25}, where E_{10} represents the incident field at the lens aperture, n_{1} is the refractive index of the substrate at the fundamental wavelength λ_{1} and circ() is the circular aperture function. The paraxial approximation is used and the Fresnel coefficient for normal incidence is also taken into account. The SHG is governed by the wave equation , where E_{2} is the complex SH field, and represent the polarization of the fundamental and SH beam, ω_{2} is SH angular frequency, c is the speed of light in vacuum, ε_{0} and μ_{0} are the vacuum permittivity and permeability and χ^{(2)} is the secondorder nonlinear susceptibility tensor^{19}. Due to its atomic thickness, the nonlinear susceptibility of the 2D material can be modelled by a Dirac delta function with an effective surface nonlinear susceptibility, i.e. . Additionally, n(z) = 1 (z < 0) or n_{2}(z > 0), the refractive index of the substrate at the SH frequency. The SHG from the substrate surface is neglected, as justified by the experimental observation of no signal for the substrate background measurements of Fig. 1(c) and Fig. 2(a).
The relationship between the fundamental excitation power P_{1} and the SH signal power P_{2} is obtained by solving for the Green's function with the nonlinear sheet source:
with .
From Eq. (1) we can express the surface nonlinear susceptibility in terms of the average power, the pulse width and the pulse repetition rate:
where R is the repetition rate, t_{i} is the pulse width and P_{avi} is the average power (i = 1: fundamental, 2: SH). Since the WS_{2} monolayer has atomic thickness much shorter than the pulse length involved, the continuous wave approximation is still valid even though chirped femtosecond pulses are used in our experiment. In our calculation, the instantaneous power of the fundamental beam is approximated by the pulse energy divided by the pulse duration time. The calculated nonlinear susceptibility represents an averaged value over the pulse bandwidth. Based on data taken similar to that of Fig. 2(b) (from 30 triangles on 3 separate substrates) and Eq. (2), the effective bulklike secondorder susceptibility of the grown WS_{2} monolayers, with where T = 0.65 nm is the thickness of a WS_{2} monolayer, was estimated to be 4.46 nm/V for the suspended film with a standard deviation of 0.09 nm/V across all samples and 4.51 nm/V for the WS_{2} triangle on a SiO_{2}/Si substrate with a standard deviation of 0.55 nm/V. Note that the standard deviations should be interpreted as the consistency of the measured SHG power rather than the accuracy of the susceptibility value due to the assumptions that were made in our calculation. These estimated values are nearly three orders of magnitude greater than that of typical nonlinear crystals^{24}. It should also be noted that the SiO_{2}/Si substrate may not be simply treated as a semiinfinite homogeneous medium due to the presence of the SiO_{2}/Si interface, as it is well known such substrate has contrast enhancement^{26}. The calculated nonlinear susceptibility thus needs to be adjusted by an enhancement factor, i.e., where η_{1} and η_{2} are the field enhancement factors for the fundamental and the epidetected SH field, respectively, due to the FabryPerot cavity formed by air/SiO_{2} and SiO_{2}/Si interfaces. is the calculated value using Eq. (2) and is the actual nonlinear susceptibility. For a substrate with 300 nm of SiO_{2} above silicon, the susceptibility value is estimated to be 15% more when the cavity enhancement factors are accounted. This analysis of the enhancement factor assumes a normally incident plane wave. A more rigorous approach would require angular spectrum decomposition of the incident wave. For this reason, the suspended sample is likely to yield a better estimate of the nonlinear susceptibility.
In addition to the value of d_{eff}, the tensorial property of the secondorder nonlinear susceptibility of WS_{2} monolayers can also be probed by polarized SHG. The WS_{2} monolayer has P6_{3}/mmc crystal symmetry, resulting in a secondorder nonlinear susceptibility tensor with nonzero elements d_{yyy} = −d_{yxx} = −d_{xxy} = −d_{xyx} where x and y represent the crystal axes^{19}. The polarized SH signal power P_{2} ∝ cos^{2} 3θ where θ is the angle between the x crystal axis and the incidence polarization^{27}. This relationship is shown compared to experimental results in Fig. (3). Two WS_{2} monolayer triangles on the same SiO_{2}/Si substrate are shown by SHG raster scan. The two inserts are polar plots where the circles are measured data points and the solid line is the cos^{2} 3θ fit. The two polar plots are rotated by 18 degrees with respect to each other and the left edge of the two triangle SHG images appear to be rotated by a similar angle. The experimental data and theoretical curve agree well and clearly demonstrate sixfold symmetry. As the tensor structure of χ^{(2)} is closely related to the symmetry of the material structure, polarized SHG can be a useful tool for probing the structural symmetry of 2D materials.
To understand the origin of the giant experimentally observed secondorder nonlinearity we have also calculated the secondorder nonlinear optical susceptibility of monolayer WS_{2} at the level of density functional theory (DFT), extending beyond existing literature results in the tight binding approximation^{28}. Note that the band gap measured by photoluminescence agrees with the DFT calculation surprisingly well, due to the partial cancellation of quasiparticle effects and exciton binding^{29}. On the other hand, calculations at the quasiparticle or BetheSalpeter equation level are computationally expensive, due to the very fine kpoint mesh required. For an accurate description of the secondorder response we follow the formalism of Sipe and Ghahramani^{30,31,32,33} as implemented in the ABINIT package^{34,35}, including both interband transitions and intraband currents. Although completely filled bands produce no intraband current at the linear order, intraband motion does occur at second order^{30,32}. The groundstate properties and response functions were calculated within the local density approximation^{35} using normconserving pseudopotentials with an energy cutoff of 760 eV and a kpoint mesh of 69 × 69 × 1 within ABINIT^{36,37,38,39,40}. The optical spectra achieved convergence with 50 conduction bands and a smearing of 15 meV.
The band structure of monolayer WS_{2} features three nearly parallel bands along ΓK, as shown by the shaded areas near II and IV in Fig. 4. The highest valence band is parallel to both the first and the second conduction bands, yielding two peaks labelled II and IV in j_{ω} the joint density of states (JDOS) of Fig. 5. Another prominent feature in the JDOS, labelled V, originates from transitions near the K point. The secondorder interband transition V could be strongly enhanced by the intermediate transition I at the halfway point in energy, if these matrix elements are nonvanishing. This is confirmed by the calculated nonlinear spectra.
In addition to the calculated χ^{(2)}, Fig. 5 also shows the JDOS, horizontally rescaled to j(2ω) to match resonances arising from the 2ω terms and separated out by colourcoded destination bands following the same colour scheme as in Fig. 4. The main features in the JDOS match well with the calculated χ^{(2)} spectra. Resonance I has the lowest excitation energy, corresponding to half the band gap. Resonances II and IV clearly benefit from a large JDOS. Resonance V arises from transitions near the K point, which lies at almost twice the band gap (also at K). Near transition II (corresponding to a wavelength of 832 nm) we obtain χ^{(2)} = 0.5 nm/V. With the simulation cell height of 2 nm, we obtain a value of d_{eff} = 0.77 nm/V. The calculation confirms the very large nonlinear response observed experimentally and considering the overall magnitude of the effect and the neglect of quasiparticle and excitonic effects, the semiquantitative agreement to experiment is surprisingly good.
In summary, we report strong SHG from synthesized WS_{2} monolayer islands both on SiO_{2}/Si substrate and suspended on a TEM grid, as well as a theoretical calculation of the secondorder susceptibility using DFT. In order to experimentally determine the secondorder susceptibility of 2D materials, we have developed a SHG model based on the Green's function, yielding an estimated bulklike secondorder nonlinear susceptibility d_{eff} approximately 4.5 nm/V for a WS_{2} monolayer. In addition, by using polarized SHG, the crystal symmetry and orientation of the WS_{2} monolayer can be revealed, demonstrating SHG as a useful method to probe structural information in 2D materials. Further analysis through DFT calculations indicate that the large secondorder susceptibility is due to resonance enhancement as well as the large joint density of states. Despite the approximations involved, the calculated nonlinear susceptibility (d_{eff} = 0.77 nm/V) using DFT shows reasonable orderofmagnitude agreement with the experimental result. With giant χ^{(2)} and subnanometre thickness, the integration of 2D TMD materials with photonic circuits to realize new nonlinear optical devices could be a fertile ground worth further exploration.
Methods
Synthesis
The WS_{2} triangle monolayer samples were grown on SiO_{2}/Si substrates using a twostep process^{17}. Briefly, onenm films of WO_{3} (99.998%, Alfa Aesar) were thermally evaporated onto a SiO_{2}/Si substrate at 10^{−6} Torr. The films were placed into a quartz reaction tube next to a boat containing sulphur powder (99.5%, Alfa Aesar). The sulphur zone was heated up to 250°C and the furnace where the WO_{3} sample was located was heated to 800°C. The synthesis was carried out in an inert environment under an Ar flow using 100 sccm and atmospheric pressure at the outlet. To verify the structural quality and integrity of the grown WS_{2} monolayers, Raman spectroscopy using a Renishaw inVia microRaman was carried out at 514.5 nm laser excitation. The 2LA(M) phonon mode (352 cm^{−1}) had an intensity twice as high as that of the A1g mode (418 cm^{−1}), which, according to reference 41, is a characteristic of monolayer WS_{2}, thus confirming that the synthesized WS_{2} samples are indeed monolayers.
Characterization
The output from a modelocked Ti:sapphire laser (from KM Labs with centre wavelength of 832 nm and repetition rate of 88 MHz) was filtered, attenuated and focused onto a sample by a longworkingdistance objective lens (Mitutoyo 50×, NA = 0.55, spot size ~ 1.8 μm at fundamental wavelength). The generated SH signal was then back collected by the same lens, separated using a dichroic mirror and filtered by a 405 nm bandpass filter before entering a spectrometer (PI Acton 2500i with a liquid nitrogen cooled charge coupled device − CCD camera). The signal value of the spectrometer was calibrated using an attenuated (by neutral density filters) SH signal separately generated from a BBO crystal with known power measured by a power meter (Newport 1830C). In order to assist the alignment and ensure that the signal was collected from a WS_{2} monolayer island, an imaging system was coset up so that a removable mirror could direct the image signal towards an imaging CCD camera. In addition, in order to calculate the susceptibility, the laser pulse width needs be known in situ, i.e. directly on the sample at the focal point of the objective lens. Therefore, we performed a collinear frequency resolved optical gating (cFROG) measurement^{42} by using the same WS_{2} sample to determine the pulse width (106 fs, chirped). Although there exists dispersion in the χ^{(2)} of WS_{2} (c.f. fig. 5), with our laser's relatively narrow bandwidth of about 18 nm, this was determined not to be an issue and was confirmed by the agreement of the retrieved spectrum with the measured laser spectrum. WS_{2} monolayer may also enable the characterization of complex pulses with broad bandwidth if its susceptibility is precalibrated. To measure the polarization dependence, the fundamental beam was first prepared in a circular polarization state using a quarterwave plate. A broadband polarizer was inserted between the objective lens and the dichroic mirror to serve as both the polarizer for projecting the input fundamental beam into a linear polarization state and the analyzer for the generated SHG signal. The polarizer was mounted on a computercontrolled rotational stage to continuously rotate its polarization axis.
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Acknowledgements
The authors acknowledge the support from NSF MRSEC program under DMR 0820404. ZL acknowledges support from NSF award ECCS 0925591. ALE, NPL and MT acknowledge support by the U. S. Army Research Office under MURI ALNOS project, contract/grant number W911NF1110362.
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Contributions
C.J., D.M. and N.M. performed SHG characterization and analysis. A.L.E. and N.P.L. synthesized samples. Y.W. performed DFT calculation. Z.L., M.T. and V.C. supervised the project.
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Janisch, C., Wang, Y., Ma, D. et al. Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers. Sci Rep 4, 5530 (2014). https://doi.org/10.1038/srep05530
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DOI: https://doi.org/10.1038/srep05530
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