Opto-Electronic Advances  2019, Vol. 2 Issue (11): 190026      DOI: 10.29026/oea.2019.190026     
Imaging the crystal orientation of 2D transition metal dichalcogenides using polarization-resolved second-harmonic generation
George Miltos Maragkakis1,2, Sotiris Psilodimitrakopoulos1, Leonidas Mouchliadis1, Ioannis Paradisanos1,2, Andreas Lemonis1, George Kioseoglou1,3, Emmanuel Stratakis1,2,3     
1. Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, Heraklion Crete 71110, Greece;
2. Department of Physics, University of Crete, Heraklion Crete 71003, Greece;
3. Department of Materials Science and Technology, University of Crete, Heraklion Crete 71003, Greece

Introduction

2D TMDs are atomically thin crystals of the type MX2, where M is a transition metal atom (Mo, W), and X is a chalcogen atom (S, Se, or Te). As part of the family of 2D materials, established by the breakthrough creation of single-layer graphene1, 2, 2D TMDs share the reduced dimensionality and similar crystal structure. Unlike graphene, however, they are direct bandgap semiconductors, exhibiting a variety of remarkable properties, such as strong photoluminescence3, 4, optical valley polarization5-8, high transistor on-off ratio9, and large exciton binding energies10, 11. These exciting characteristics of 2D TMDs offer an ideal field for fundamental studies, as well as numerous possible applications in various research fields12, including electronics and optoelectronics13, 14, energy harvesting15, valleytronics16, and biomedicine17, 18.

The realization of technology and devices based on 2D atomic crystals presupposes the ability to create large-area films of good quality and minimum imperfections, given that defects unavoidably affect the material behavior19-21. These polycrystalline films consist of single-crystalline areas of varying orientation, and grain boundaries22, i.e. the interface regions between crystallites. In such materials, types of defects include the poor domain connectivity between grain boundaries, and the absence of homogeneity in crystalline orientation, material thickness and layer stacking. Unfortunately, state-of-the-art large-area crystal growth techniques, such as CVD, often fail in producing defect-free materials, and moreover, there is currently no easily applicable, non-invasive and fast characterization method of the quality of 2D crystals.

Recently, the potential for nonlinear optical processes in TMDs has been attracting significant scientific interest. Several groups have explored methodologies to acquire information about the TMD crystals by analysing their second-harmonic generation (SHG) signal23-31, which is very sensitive to crystal symmetry. The capability of mapping the armchair orientation distribution over large areas of 2D materials with high resolution, could provide a unique tool towards the evaluation of their crystal quality. Here, we acquire pixel-by-pixel information about the armchair orientation by measuring the SHG intensity, while rotating the linear polarization of the laser beam. It is revealed that crystal imperfections are creating sharp contrast in the PSHG image among domains of different crystallographic orientations, e.g. grain boundaries or defected regions. Such sensitivity enables detailed mapping of the various crystallographic orientations over large areas, providing invaluable information on crystal structure, which is shown to be unattainable with traditional, intensity-only SHG imaging.

PSHG as a powerful tool for characterizing 2D materials

The present work further supports the unique capabilities of PSHG as a method for determining the crystalline integrity of 2D TMDs23. First, the crystal characterization can be performed rapidly in an all-optical manner via a single measurement process. Moreover, it can be applied to both forward and epi detection geometries, allowing the study of samples in their original (even opaque) substrates. Therefore, it is minimally invasive and does not require sample preparation, in contrast to TEM microscopy, which necessitates the transfer of the sample to an electron-permeable TEM-supporting membrane, which is a time-consuming and invasive process.

More importantly, unlike the SHG intensity-only method used for the determination of the main crystallographic axis by rotation of the sample, in the PSHG approach, the change of the polarization of the fundamental field allows pixel-by-pixel imaging with ultrahigh resolution that is determined by the pixel size. As a result, the contrast in PSHG analysis offers a mechanism that provides resolution beyond the optical one, in the 'polarization space', enabling to detect changes in the armchair direction with accuracy ~0.5°.

Another advantage of raster-scanning PSHG over traditional SHG imaging is that it allows the mapping of a large area, and therefore is a unique tool for the characterization of grain boundaries and other extended defects in polycrystalline structures. Finally, the incorporation of a PSHG setup into a CVD chamber could potentially lead to an in-situ, real-time evaluation of crystal quality (in analogy with RHEED in molecular beam epitaxy), which would signify an important advance towards the production of defect-free 2D materials. A comparison between PSHG with traditional SHG is also shown in the Table 1.

Table 1 Comparison of PSHG with traditional SHG methods.
SHG PSHG
Detection of armchair orientation
Pixel-by-pixel mapping of the armchair orientation over large areas ×
Application as crystal quality marker over large crystal regions ×
Identification of the boundaries between regions of different crystal orientations ×
Methods and analyses Experimental setup for measuring PSHG in stationary, raster-scanned samples

The experimental setup of our laser-scanning microscope is schematically shown in Fig. 1. It is based on a diode-pumped Yb:KGW fs oscillator (1.2 W, 1030 nm, 70–90 fs, 76 MHz, Pharos-SP, Light Conversion, Vilnius, Lithuania), a custom-built inverted microscope (Axio Observer Ζ1, Carl Zeiss, Jena, Germany), and a pair of silver-coated galvanometric mirrors (6215H, Cambridge Technology, Bedford, MA, USA). First, the beam passes through a zero-order half-wave retardation plate (QWPO-1030-10-2, CVI Laser), with which the orientation of the linear polarization of the excitation beam at the sample plane, can be rotated using a motorized rotation stage (M-060.DG, Physik Instrumente, Karlsruhe, Germany). A pair of achromatic doublet lenses, forming a telescope, suitably expands the laser spot in order to fill the back aperture of the objective lens, while the galvanometric mirrors direct the scanning beam towards the inverted microscope and its motorized turret box, just below the objective (Plan-Apochromat × 40/1.3NA, Carl Zeiss).

Fig. 1 Schematic representation of the experimental setup, also adopted in ref.23, allowing high-resolution PSHG measurements in stationary, raster-scanned samples. Abbreviations, as met by the laser fundamental pulse: HWP: zero-order halfwaveplate, L: lens, GM: galvanometric mirrors, M: mirror, D: dichroic, O: objective, S: sample, C: condenser, F: filters, LP: linear polarizer, PMT: photomultiplier tube. The linear polarization of the excitation electric field E starts horizontal in the sample plane and is rotated clockwise with an angle φ (see also Fig. 3).

Fig. 3 The experimental configuration, showing the laboratory X-Y-Z, and the crystalline x-y-z coordinate systems. Angles φ, θ, and ζ describe, respectively, the orientation of the rotating fundamental linear polarization, crystal armchair, and linear polarizer, with respect to X laboratory axis.

At the microscope turret box, we have the choice of using either a silver-coated mirror (PFR10-P01, ThorLabs, Newton, NJ, USA), or a short-pass dichroic mirror (DMSP805R, ThorLabs), both at 45°, depending on whether we detect the signal in the forward direction (silver-coated), the backwards or both simultaneously (dichroic). PSHG measurements in the forward direction ensure that our setup is insensitive to the different orientations of laser polarization, given the silver coating of all mirrors (PF 10-03-P01, ThorLabs), including the galvanometric. It is important to note, however, that our setup also permits PSHG collection in the backward direction, allowing the study of samples in their original substrates and confirming the minimally invasive character of the technique. It also permits simultaneous imaging of PSHG and (back reflected) two-photon-absorption-induced photoluminescence (TPL), in the forward and epi directions, respectively, by using suitable filters, the same objective, and a second detector.

For the experimental results presented here, we work in the forward detection geometry. The beam, reflected by the silver-coated mirror, is tightly focused by the microscope objective lens to a diffraction-limited spot onto the sample, which produces SHG. This signal is collected by a high numerical aperture condenser lens (achromatic-aplanatic, 1.4NA, Carl Zeiss), and then filtered by a short-pass filter (FF01-720/SP, Semrock, Rochester, NY, USA), to remove residual from the fundamental pulse, as well as a narrow bandpass filter (FF01-514/3, Semrock), centered at the second-harmonic wavelength, to separate it from TPL. Finally, the beam passes through a rotating film polarizer (LPVIS100-MP, ThorLabs), and is detected by a photomultiplier tube (PMT) (H9305-04, Hamamatsu, Hizuoka, Japan).

The galvanometric mirrors and the PMTs are connected to a connector block (BNC-2110, National Instruments, Austin, TX, USA), which is interfaced to a PC through a DAQ (PCI 6259, National Instruments). Coordination of PMT recordings with the galvanometric mirrors for the image formation, as well as the movements of all the microscope motors, is carried out using LabView (National Instruments) software.

Nonlinear optical response of 2D TMDs

The induced nonlinear polarization that leads to SHG in crystals, ${P_i}(2\omega ) = {\varepsilon _0}\sum\nolimits_{j, k} {\chi _{ijk}^{(2)}{E_j}(\omega ){E_k}(\omega )} $, is governed by the second-order susceptibility tensor $\chi _{ijk}^{(2)}$, a third-rank tensor that describes the crystal symmetry, and is nonzero for non-centrosymmetric media32.

The structure of monolayer MX2, shown in Fig. 2, comprises three sublattices: an atomic plane of metal atoms, with threefold coordinate symmetry, is hexagonally packed between two trigonal planes of chalgogen atoms. WS2 crystals with 2H stacking order belong to D6h symmetry group and are inversion symmetric, for an even number of layers. However, for odd layer number, the symmetry is broken and the crystal belongs to the D3h space group. Under this symmetry, χ(2) has four nonzero elements, namely $\chi _{xxx}^{(2)} = - \chi _{xyy}^{(2)} = - \chi _{yyx}^{(2)} = - \chi _{yxy}^{(2)}$, where x, y, z denote the crystalline coordinates, with x being the mirror symmetry axis (the armchair direction), and y the axis along which the mirror symmetry is broken (the zigzag direction). The finite second-order optical susceptibility, along with the atomic thickness of 2D TMDs which ensures phase matching, suggest strong optical SHG, which, indeed, has been observed and studied23-31. For the case of TMDs with D3h point symmetry, including the monolayers, the SHG equation can be written in matrix form as

Fig. 2 Schematic representation of the structure of 2D TMDs, containing three sublattices, with a plane of metal atoms being hexagonally packed between two planes of chalgogen atoms.
$\left( {\begin{array}{*{20}{c}} {P_x^{2\omega }} \\ {P_y^{2\omega }} \\ {P_z^{2\omega }} \end{array}} \right) = {\varepsilon _0}\chi _{xxx}^{(2)}\left( {\begin{array}{*{20}{c}} 1&{ - 1}&0&0&0&0 \\ 0&0&0&0&{ - 1}&{ - 1} \\ 0&0&0&0&0&0 \end{array}} \right)\;\left( {\begin{array}{*{20}{c}} {E_x^\omega E_x^\omega } \\ {E_y^\omega E_y^\omega } \\ {E_z^\omega E_z^\omega } \\ {2E_y^\omega E_z^\omega } \\ {2E_x^\omega E_z^\omega } \\ {2E_x^\omega E_y^\omega } \end{array}} \right)$. (1)
Methodology for measuring 2D TMD crystal orientation using PSHG

The adopted experimental configuration, presented in Fig. 3, consists of two coordinate systems: the laboratory X-Y-Z, and the crystalline x-y-z, with Zz, and x parallel to the armchair orientation and at angle θ from X. The fundamental field is propagating along Z axis, normally incident on the crystal, and it is linearly polarized along the sample plane, at an angle φ with respect to X laboratory axis. By rotating the zero-order half-waveplate, we rotate the orientation of the excitation linear polarization at the sample plane, and we record the sample-produced SHG as function of φ. Before reaching the detector, SHG is passing through a linear polarizer, at constant angle ζ from X.

In order to address the laser propagation, we employ the Jones formalism. The excitation polarization after the retarder plate may be expressed as the Jones vector $\left( {\begin{array}{*{20}{c}} {\cos \varphi } \\ {\sin \varphi } \end{array}} \right)$, where the amplitude of the electric field is normalized to unity. Expression of the Jones vector in crystalline coordinates is achieved by multiplication with the rotation matrix$\left( {\begin{array}{*{20}{c}} {\cos \theta }&{\sin \theta } \\ { - \sin \theta }&{\cos \theta } \end{array}} \right)$, containing the armchair angle θ. The result for the nonlinear polarization in crystalline coordinates is $\left( {\begin{array}{*{20}{c}} {P_X^{2\omega }} \\ {P_Y^{2\omega }} \end{array}} \right) = $${\varepsilon _0}\chi _{xxx}^{(2)}\left( {\begin{array}{*{20}{c}} {\cos (2\theta - 2\varphi )} \\ {\sin (2\theta - 2\varphi )} \end{array}} \right)$, or by rotating back to lab coordinates, $\left( {\begin{array}{*{20}{c}} {P_X^{2\omega }} \\ {P_Y^{2\omega }} \end{array}} \right) = $ ${\varepsilon _0}\chi _{xxx}^{\left( 2 \right)}\left( {\begin{array}{*{20}{c}} {\cos (3\theta - 2\varphi )} \\ {\sin (3\theta - 2\varphi )} \end{array}} \right)$. Finally, in order to account for the polarizer, the polarization vector of the detected SHG signal is found after multiplying with the Jones matrix $\left( {\begin{array}{*{20}{c}} {{{\cos }^2}\zeta }&{\cos \zeta \cdot \sin \zeta } \\ {\cos \zeta \cdot \sin \zeta }&{{{\sin }^2}\zeta } \end{array}} \right)$.

The final SHG intensity recorded by the detector can be expressed as

${I_{{\rm{SHG}}}} = A \cdot {\cos ^2}(\zeta - 3\theta + 2\varphi )$, (2)

where A is a multiplication factor depending on χ(2) and the excitation amplitude.

For ζ=0 and ζ=π/2, i.e. polarizer paraller to X and Y laboratory axes, respectively, the SHG intensity reads23

${I_X} = A \cdot {\cos ^2}(3\theta - 2\varphi )$, (3)

and

${I_Y} = A \cdot {\sin ^2}(3\theta - 2\varphi )$. (4)

These equations summarize the polarization-dependent SHG modulation from 2D TMDs, for the experimental configuration presented here. This modulation is plotted in polar diagrams in Fig. 4 as a function of φ, with φ ϵ[1°, 360°], for fixed polarizer (ζ=0° and ζ=90°). As may be seen, a fourfold pattern (four-leave rose) is obtained, which rotates for different values of crystal armchair orientation, θ. Given that each θ corresponds to a characteristic polar modulation of specific orientation, we can calculate θ for every pixel of an area image, by fitting pixel-by-pixel the PSHG experimental data to Eq. (2). It should be noted that, since there exist three equivalent armchair directions in each hexagon of the crystal lattice (threefold rotational symmetry), one can determine this direction modulo 60°, constraining θ ϵ[0°–60°], while sampling φ in [0°–90°] is adequate to extract all possible armchair orientations. The armchair angle could also be determined by combining Eqs. (3) and (4), as

Fig. 4 Simulated PSHG modulation presented in polar diagrams, as function of the linear polarization orientation φ, with φ ϵ [1°, 360°], for fixed polarizer at angle (a) ζ=0° and (b) ζ=90°. The orientation of the fourfold pattern rotates for different crystal armchair directions θ.
$\theta = \frac{1}{3}\left( {2\varphi + \arctan \sqrt {\frac{{{I_Y}}}{{{I_X}}}} } \right)$. (5)
Extending the consideration for multi-layer structures

The approach we described applies to crystals of D3h symmetry, where the dipoles created by the incident field act as surface array antennas radiating at double frequency and thus producing SHG signals. In the case of multi-layer systems, the second-harmonic fields from each layer interfere before detection, and can be treated as a vector superposition, giving:

${{\boldsymbol{E}}_{\rm{S}}} = \mathop \sum \limits_{i = 1}^N {{\boldsymbol{E}}_i}$, (6)

where N is the total number of layers, and ${{\boldsymbol{E}}_i}$ can be obtained by

${{\boldsymbol{E}}_i} = {A_i} \cdot \cos (\zeta - 3{\theta _i} + 2\varphi ) \cdot (\cos \zeta \cdot {\boldsymbol{\hat X}} + \sin \zeta \cdot {\boldsymbol{\hat Y}})$. (7)

For ζ=0°, we get the simplified form ${{\boldsymbol{E}}_i} = {A_i} \cdot \cos (3{\theta _i} - 2\varphi ) \cdot {\boldsymbol{\hat X}}$.

The total intensity recorded by the detector can then be expressed as

$I_{\rm{S}}^{{\rm{PMT}}} = {\left| {{{\boldsymbol{E}}_{\rm{S}}}} \right|^2} = {\left| {\mathop \sum \limits_{i = 1}^N \;{{\boldsymbol{E}}_i}} \right|^2} = \mathop \sum \limits_{i = 1}^N \;{I_i} + \mathop \sum \limits_{i, j(i \ne j)}^N \sqrt {{I_i}{I_j}} \cdot \cos (3{\delta _{ij}})$, (8)

with ${I_i}$ the intensity of the ith layer, and ${\delta _{ij}}$ the twist angle between layers i and j, ${\delta _{ij}} = {\theta _i} - {\theta _j}$. For N=2, the SHG intensity of the bilayer is given by29

${I_{{\rm{BL}}}} = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cdot \cos (3\delta )$. (9)

Furthermore, for layers of equal intensity (${I_1} = {I_2} = {I_{{\rm{ML}}}}$) at zero twist angle ($\delta = 0$), we obtain, ${I_{{\rm{BL}}}} = 4{I_{{\rm{ML}}}}$, i.e. the well-known result that SHG intensity scales quadratically with layer number, while for$\delta = {\rm{ \mathsf{ π} }}/6$, we have ${I_{{\rm{BL}}}} = 2{I_{{\rm{ML}}}}$. If the intensities are measured, the twist angle could also be estimated from Eq. (9), as 29

$\delta = \frac{1}{3} \cdot \arccos \left( {\frac{{{I_{{\rm{BL}}}} - {I_1} - {I_2}}}{{2\sqrt {{I_1}{I_2}} }}} \right)$, (10)

or

$\delta = \frac{1}{3} \cdot \arccos \left( {\frac{{{I_{{\rm{BL}}}}}}{{2{I_{{\rm{ML}}}}}} - 1} \right)$, (11)

for the case of${I_1} = {I_2} = {I_{{\rm{ML}}}}$.

Samples

The WS2 sample was grown by the low-pressure CVD method on a c-cut (0001) sapphire substrate (2D semiconductors). It was characterized using micro-Raman spectroscopy with a 473 nm excitation wavelength.

Results and discussion

Typical experimental configurations investigating the main crystallographic orientation of 2D TMDs are based on the rotation of the sample (e.g. Refs.24-27). In such experiments, the parallel and perpendicular polarization component of the second-harmonic field is measured, with dependence ${I_X} \sim {\cos ^2}(3\gamma )$ and ${I_Y} \sim {\sin ^2}(3\gamma )$, respectively, where γ is the angle between the armchair direction and the polarization orientation of the incident field. This results in a six-leave rose polar diagram.

In contrast, here, we rotate the orientation of the linear polarization of the excitation, measuring PSHG in stationary raster-scanned crystals and allowing pixel-wise mapping of the armchair orientation. For this purpose, we define angles θ and φ, which characterize the armchair direction and fundamental field polarization, respectively, both with respect to the X laboratory axis (see Fig. 3). In our approach, high angular accuracy was realized by using a step of only 1° for the excitation polarization orientation φ. In Fig. 5, we present PSHG images of raster-scanned WS2, for φ ϵ[0°–360°] with step 40° for each consecutive image, and the linear polarizer at constant angle, ζ=0°. Rotation of the fundamental field is found to switch on and off the PSHG signal from the triangular flake according to its relative armchair crystal orientation θ (see movie in supplementary material, for φ ϵ[0°–360°] with step 1°).

Fig. 5 Snapshots of experimental PSHG images of a WS2 flake, CVD-grown on a sapphire substrate. The white double arrow shows the constant angle, ζ=0°, of the linear polarizer, while the orange double arrow shows the rotating angle, φ, of the excitation linear polarization. Here, the rotation of φ ϵ[0°–360°] with step 40°, clearly shows the switching on and off of the SHG signal. The supplementary material includes a movie capturing the complete experimental PSHG modulation, for φ ϵ[0°–360°] with step 1°.

By summing up the PSHG images for φ ϵ[0°–90°] with step 1°, we obtain Fig. 6(a), showing the total SHG signal collected. In contrast to Fig. 5, where the PSHG signal is changing with respect to the orientation φ of the excitation linear polarization, the sum of the PSHG images is no longer polarization-dependent. Since the dependence on the armchair crystal angle θ is lost, any differences in intensity observed in Fig. 6(a) can now be attributed to material variations such as material thickness or layer stacking. For example, it is known that larger number of 3R-stacked monolayers generates SHG of higher intensity31, which depends quadratically on the number of layers.

Fig. 6 (a) Integration of the experimentally detected PSHG intensity from the WS2 island, for φ ϵ[0°–90°] with step 1°, presented upon marking three POIs and two LOIs for further analysis. The POIs are actually single pixels of the 1200×1200 original image, magnified here, for illustration purposes. (b) Intensity profile of the experimental PSHG modulation presented in (a), along the LOIs shown there. As may be seen for LOI i, the intensity in the central, brighter area is magnified by a factor of ~4, which suggests the presence of second layer29. (c) Polar diagrams of the experimental PSHG modulation for φ ϵ[0°–360°] with step 1°, for the POIs illustrated in (a). We show with red color the raw data, and with blue the fitting using Eq. (3). We also present the retrieved values of the armchair orientation θ and the quality of fitting R2.

Aiming to experimentally explore these considerations, we present the intensity profile of detected SHG intensity (Fig. 6(b)), along two lines of interest (LOIs) shown in Fig. 6(a). As can be seen, in the case of LOI i, the intensity in the central, brighter area of the triangle is magnified by a factor of ~4, which suggests the presence of a second layer at twist angle 0°. Let us now examine this assumption based on our PSHG analysis. We focus on the specific points of interest (POIs) 1 and 2, shown in Fig. 6(a), which belong to different intensity regions. By plotting the experimental PSHG data in a polar diagram, for φ ϵ[0°–360°] with step 1°, and by fitting using Eq. (3), we can determine the armchair angle θ, for each pixel. Indeed, in Fig. 6(c), we present the raw data (in red) and fitted line (in blue), that correspond to the three POIs, along with the retrieved armchair angles and quality of fitting R2. As can be seen, POIs 1–3, correspond to almost identical values of θ, and thus the PSHG analysis further supports the presence of a second layer, at the central, brighter area, vertically stacked at twist angle 0°, as was suggested by intensity-only SHG measurements.

For the case, however, of LOI ii, the intensity-only SHG measurements, considered alone, could give misleading results. More specifically, the intensity profile of LOI ii (Fig. 6(b)), shows a signal change of ~1.3. By using Eq. (11), this corresponds to a twist angle of δ=37°. Nevertheless, by performing PSHG analysis, we find a similar θ for the POIs 2 and 3, and therefore δ~0°. A possible explanation for the intensity variation between POIs 2 and 3 might be the change in stacking sequence31. The above example indicates that SHG intensity-only measurements are insufficient for an all-optical determination of the twist angle between layers of different armchair orientations, and therefore a polarization-dependent analysis is necessary.

By repeating this analysis for every pixel of the PSHG image, for φ ϵ[0°–360°] with step 1°, we obtain the color map presented in Fig. 7(a), showing the distribution of armchair directions across the WS2 flake, and the corresponding histogram (Fig. 7(b)). Color variation in such a map, quantified by the standard deviation of the respective histogram, denotes absence of homogeneity in crystalline properties, supporting the application-suitability of the presented technique as a crystal quality marker.

Fig. 7 Mapping in (a) 2D diagram and (b) histogram of the armchair orientation distribution of the WS2 flake (with < θ > being the mean vale), based on the pixel-by- pixel fitting (R2≥0.88) of Eq. (3) on the experimental PSHG modulation for φ ϵ[0°–360°] with step 1°. Significant color changes in the map, or equivalently, large standard deviation (∆θ) of the histogram, denote inhomogeneity in either crystalline orientation, material thickness, or layer stacking.
Conclusions

In conclusion, we have demonstrated an all-optical, fast and minimally-invasive method to accurately image the armchair orientation in atomically thin 2D crystals, via probing of their PSHG properties. It is shown that different crystal orientations provide different PSHG modulations and subsequent contrast in the images obtained. The presented method comprises the measurement of the PSHG signal anisotropy, produced by a stationary raster-scanned 2D crystal, as a response to the rotating linear polarization of the fs excitation field. By fitting, pixel-by-pixel, this polarization-dependent modulation into a generalized nonlinear model, we are able to extract and map, with high resolution, the distribution of armchair crystal orientations over large areas of the 2D lattice. This approach allows us to obtain valuable information of crystal homogeneity, and therefore can provide a unique tool for the evaluation of crystal quality and for emerging 2D material applications. Given that such capabilities cannot be attained via traditional, intensity-only SHG imaging, we envisage that this work can establish PSHG as a state-of-the-art 2D material characterization tool.

Acknowledgements

GMM acknowledges financial support from the Stavros Niarchos Foundation within the framework of the project ARCHERS ('Advancing Young Researchers' Human Capital in Cutting Edge Technologies in the Preservation of Cultural Heritage and the Tackling of Societal Challenges'). This research has been co-financed by the European Union and Greek national funds through the Operational Program Competitiveness, Entrepreneurship and Innovation, under the call European R & T Cooperation-Grant Act of Hellenic Institutions that have successfully participated in Joint Calls for Proposals of European Networks ERA NETS (National project code: GRAPH-EYE T8ΕΡΑ2-00009 and European code: 26632, FLAGERA).

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information for this paper is available at

https://doi.org/10.29026/oea.2019.190026

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