Computational Protocol for Material Optical Properties During Pump-Probe Experiments and in Nanostructures Irradiated with Ultrashort Pulses

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WP9 JRA4 – Research on Time-resolved ultrafast probes on nanosystems D9.7 Computational protocol for material optical properties during pump-probe experiments and in nanostructures irradiated with ultrashort pulses

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NFFA-Europe has received funding from the EU's H2020 framework programme for research and innovation under grant agreement n. 654360

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PROJECT ACRONYM PROJECT TITLE NFFA-Europe NANOSCIENCE FOUNDRIES AND FINE ANALYSIS - EUROPE GRANT AGREEMENT NO: FUNDING SCHEME 654360 RIA - Research and Innovation action START DATE 01/09/2015

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WORK PACKAGE ID WORK PACKAGE TITLE WP9 JRA4 – Research on Time-resolved ultrafast probes on nanosystems

WORK PACKAGE LEADER Emmanuel Stratakis (FORTH)

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DELIVERABLE ID DELIVERABLE TITLE D9.7 Computational protocol for material optical properties during pump- probe experiments and in nanostructures irradiated with ultrashort pulses DELIVERABLE DESCRIPTION A short description of the theoretical framework that was developed to describe laser matter interaction with nanosctructures that can be used to assist fundamental mechanism understanding as well as appropriate design protocols in pump-probe experimental set ups. EXPECTED DATE ESTIMATED INDICATIVE PERSONMONTHS 31/12/2018 16 AUTHORS Emmanuel Stratakis (FORTH), George Tsibidis (FORTH), Leonidas Mouchliadis (FORTH), Andrea Marini (CNR), Paolo Umari (CNR) PERSON RESPONSIBLE FOR THE DELIVERABLE Emmanuel Stratakis (FORTH) NATURE R - Report DISSEMINATION LEVEL ☒ P - Public ☐ PP - Restricted to other programme participants & EC: (Specify) ☐ RE - Restricted to a group (Specify) ☐ CO - Confidential, only for members of the consortium

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ACTUAL SUBMISSION DATE NUMBER OF PAGES 21/12/2018 14 FOR MORE INFO PLEASE CONTACT Emmanuel Stratakis (FORTH) Tel. +30-2810-391274 Email: [email protected]

Description / Reason for Version Date Author(s) Status modification 1 18/12/2018 Emmanuel Stratakis, Final George Tsibidis, Leonidas Mouchliadis, Andrea Marini, Paolo Umari Choose an item.

Contents

Executive Summary 4 1. Concept –Theoretical framework 4 1.1 DFT and BSE+GW calculations 5 GW calculations 5 Bethe-Salpeter equations 6 1.2 Relaxation Processes 7 2. Design specification 7 2.1 Yambo Code/Quantum Espresso 9 Simulation 9 Initialization 9 Input file generation and command line options 9 Hartree-Fock and GW calculation 9 2.2 Two Temperature Model module/Phase transition Module 10 3. Simulation Results for 6H-SiC 10 4. Conclusions and perspectives 13 Publications in which NFFA is acknowledged 14

Executive Summary

To design appropriate efficient pump-probe experiments and evaluate the response of nanostructured materials to ultrashort-pulsed laser irradiation, it is important to understand the underlying physical mechanisms that characterise laser-matter interaction. To this end, a computational protocol is developed to describe the multiscale analysis of all physical processes that are involved following irradiation with very short pulses. To efficiently quantify the optical response of the material in the first stages after the irradiation, a quantum mechanical approach is followed through the synergy of Quantum Espresso and Yambo Code that are coupled with revised versions of the Two Temperature Model (TTM) to eventually describe relaxation processes and/or morphological changes. To test the computational code, 6H-SiC is used as a test material for which the damage threshold fluence (i.e. minimum temperature at which the material melts) is being calculated. The reason for which this material is used is because there are existing experimental protocols designed by groups in the consortium.

1. Concept –Theoretical framework

Laser-matter interaction with intense (ultrashort) pulsed lasers triggers a variety of time-scale dependent processes influenced by the applied conditions (i.e. fluence, pulse duration, number of pulses, etc). Determining the material’s response characteristics changes at longer time scales has been hampered by the absence of a direct correlation between the conditions applied and the interdependence of the intermediate mechanisms at all time-scales. In brief, a consistent theoretical approach to describe the multiscale analysis of the response of the irradiated material should comprise the following components: (i) a term that describes energy absorption through a precise estimate of the optical properties, (ii) a term that describes electron excitation and all relevant interactions (electron-electron and electron-phonon collisions, etc.), (iii) a heat transfer component that accounts for electron-lattice thermalisation through particle dynamics and heat conduction and carrier-phonon coupling, and/or (iv) a hydrodynamics component that describes fluid dynamics followed by a mass removal and re-solidification process in areas where a phase transition occurs. In principle, the processes start after some fs, they continue to mechanisms that complete after some picoseconds (ps) while others require more time and they last up to the nanosecond (ns) regime. One very important aspect that requires a thorough investigation is the absorption of laser energy by the irradiated material. A critical parameter is the computation of the optical properties (reflectivity, absorption coefficient) while a detailed analysis of the propagation of the beam inside the irradiated material allows the computation of the spatio-temporal distribution of the absorbed laser energy. There are various models that have been used to simulate the dielectric constant of the material (and therefore the optical properties) as a function of the laser beam wavelength while solution of Maxwell equations can yield the energy distribution. One classical (approximate) approach that has been used to simulate the dielectric constant of the material as a function of the laser beam wavelength is by means of modified versions of the Lorentz-Drude model the interaction of intense laser pulses with an extensive variety of solids giving rise to surface modifications as well as changes in optical properties, has been the subject of intensive research during recent years. However, many of the expressions used to estimate the optical properties, firstly, are valid at low temperatures while they do not take into account their possible variation during the laser pulse (i.e.

for extremely short pulses). Therefore an imprecise calculation of the optical properties is derived that potentially may lead to an inaccurate determination of the absorbed energy; the latter is, then, expected to produce erroneous estimation of the thermal response of the system. Therefore, a first principle analysis of the laser interaction with matter was based on the fundamental quantum mechanical-based processes. Simulations (DFT Calculations, etc) were performed in various conditions in Tasks 9.2 of JRA4 to provide precisely the reflectivity and absorption coefficient of the irradiated material and results were used as a feedback to compute the thermal response of the material. 1.1 DFT and BSE+GW calculations GW calculations The aim of a GW calculation is to obtain the quasiparticle correction to energy levels using many- body perturbation theory (MBPT). The general non-linear quasiparticle equation reads:

As a first step we evaluate the self-energy Σ entering in the quasiparticle equation. In the GW approach the self-energy can be separated into two components: a static term called the exchange self-energy (Σx), and a dynamical term (energy dependent) called the correlation self-energy (Σc):

We treat these two terms separately and find the most important variables for calculating each term. In practice we compute the quasi-particle corrections to the one particle Kohn-Sham eigenvalues obtained through a DFT calculation.

The steps are the following:

Step 1. The Exchange Self Energy or HF quasi-particle correction: We start by evaluating the exchange Self-Energy and the corresponding Quasiparticle energies (Hartree-Fock energies).

Step 2. The Correlation Self-Energy and Quasiparticle Energies

Once we have calculated the exchange part, we next turn our attention to the more demanding dynamical part. The correlation part of the self-energy in a plane wave representation reads:

In the expression for the correlation self energy, Σc, we have (1) a summation over bands, (2) an integral over the Brillouin Zone, and (3) a sum over the G vectors. In contrast with the case of Σx, the summation over bands extends over all bands (including the unoccupied ones), and so convergence tests are needed. Another important difference is that the Coulomb interaction is now screened so a fundamental ingredient is the evaluation of the dynamical dielectric matrix. In the calculation, we use two ways to take into account the dynamical effects. First, we set the proper parameters to obtain a model dielectric function based on a widely used approximation, which models the energy dependence of each component of the dielectric matrix with a single pole function. Secondly, we perform calculations by evaluating the dielectric matrix on a regular grid of frequencies. Once the correlation part of the self-energy is calculated, we will check the convergence of the different parameters with respect to some final quantity, such as the gap. After computing the frequency dependent self-energy, we observe that in order to solve the quasiparticle equation we need to know its value at the value of the quasiparticle itself. We therefore solve the non-linear

quasi-particle equation at first order, by expanding the self-energy around the Kohn-Sham eigenvalue. In this way the quasiparticle equation reads:

where the normalization factor Z is defined as:

The Plasmon Pole approximation

As stated above, the basic idea of the plasmon-pole approximation is to approximate the frequency dependence of the dielectric matrix with a single pole function of the form:

The two parameters RGG' and ΩGG' are obtained by a fit (for each component), after having calculated the RPA dielectric matrix at two given frequencies. Yambo calculates the dielectric matrix in the static limit (ω=0) and at a user defined frequency called the plasmon-pole frequency (ω=iωp). Such an approximation has the big computational advantage of calculating the dielectric matrix for only two frequencies and leads to an analytical expression for the frequency integral of the correlation self-energy.

Step 3. Interpolating Band Structures

After checking convergence for the gap with respect to several parameters, we calculate the quasiparticle corrections across the Brillouin zone in order to visualize the entire band structure along a path connecting high symmetry points. To do that we start by calculating the QP correction in the plasmon-pole approximation for all the k points of our sampling and for a number of bands around the gap.

Bethe-Salpeter equations To solve the Bethe-Salpeter equations (BSE) the latter is usually rewritten in the space of transitions between valence and conduction states as the (pseudo)eigenvalue problem for a two- particle Hamiltonian. The two-particle Hamiltonian matrix elements are given by:

where vck indicates the pair of quasiparticle states vk and ck. The first term on the RHS is the quasiparticle energy differences (diagonal only). The second term is the kernel which is the sum of the electron-hole exchange part V (which stems from the Hartree potential) and the electron-hole attraction part W (which stems from the screened exchange potential). The kernel both shifts (diagonal contributions) and couples (off-diagonal contributions) the quasiparticle energy differences:

-1 where εGG' is the inverse of the dielectric screening matrix.

1.2 Relaxation Processes

The two-temperature model (TTM) constitutes the standard theoretical method to investigate laser- matter interaction upon ultrashort laser irradiation, which assumes an instantaneous electron excitation during the laser pulse that produces fast electron thermalisation on the femtosecond timescale. Therefore, a revision to the model is required, especially for very short pulses (<100 fs) when there is a strong nonthermal electron distribution that is produced before the end of the pulse. The aforementioned first principle-based analysis is firstly, used to compute precisely both the energy absorption and optical properties temporal variation as well as the carrier density distributions. The equilibration of the excited carrier temperature with the lattice temperature and relaxation processes are then modelled while heat transfer mechanisms are included to describe phase transitions and damage on the material. The latter is also described by Navier-Stokes equations to model surface modification or features.

2. Design specification

The computational module design is based on the multi-scale modelling of processes (illustrated in Fig.1) that occur after irradiation of materials with ultrashort pulses. Although, the toolbox in the test case in this report is applied for semiconductors (i.e. 6H-SiC), an extension to include mutliphysical processes after excitation of other materials (metals or dielectrics) is also possible by incorporating appropriate modules. To link processes together (through a sequential modelling) and deduce the final outcome, a number of components/codes will be produced for each stage, each taking its input parameters from the previous stage and, in turn, its output parameters being used as inputs for the next stage. More specifically, we have divided the tasks in which the input parameters of the carrier excitation module are: the fluence, the pulse duration, polarization state, and/or the number of pulses while the code is expected to produce a detailed description of the transient optical properties of the material and density of states information through the use of DFT calculations/Yambo computational software.

Figure 1: Multiscale computational platform.

Similarly, the energy absorption modality has as input parameters the carrier density distribution and the induced density of state distribution assuming the presence of transient optical properties due to excitation and the resulting quantities will be (due to variation of the electron temperature) the electron-phonon coupling, spatio-temporal distribution of absorbed energy, and the electron temperature dependence of thermophysical properties (i.e. carrier heat capacity and heat conductivity). On the other hand, the code that models the relaxation processes will use the quantitative feedback obtained in the previous stage and yield the lattice temperature profile and thermophysical lattice parameters (i.e. lattice heat capacity and heat conductivity, thermal expansion coefficients). Finally, the last component of the software toolbox, the module which computes systematically structural changes and properties is using the last output parameters and it computes structural changes on the material (damage threshold, surface modification, morphological features). The time scale related to the last component is up to hundreds nanoseconds or some microseconds. To assist researchers across Europe in their efforts to understand the fundamentals of particular mechanisms and effects following exposure of materials to intense heating two installation modules are available on the NFFA project: (i) https://www.nffa.eu/offer/theory-simulation/ (Installation 1) and (ii) https://www.nffa.eu/offer/theory-simulation/installation-3/multiscale-modeling-of- materials-under-extreme-irradiation/ (Installation 3). The modules are open to experimental users with no experience in modelling and simulation, as well as to experimental and computational users with experience in the field. Access to the installation may be remote or on- site, depending on the needs of the proposed project.

2.1 Yambo Code/Quantum Espresso Simulation A typical Yambo calculation comprises three steps: (i) a DFT calculation with Quantum Espresso (ii) conversion to Yambo format (iii) adjustment of the various parameters and convergence checks. As a paradigm material we have used hBN and first calculated the ground state properties by using the relevant databases downloaded from the Yambo website. In the simulation we considered bulk hBN with a HCP lattice, four atoms per unit cell B and N (16 electrons), plane wave cutoff energy of 40 Ry and a shifted 6x6x2 grid (14 k-points) with 8 bands. The QE SCF calculation was performed with pw.x and the ground state charge density, occupations and Fermi level were obtained and saved. After this run a new SAVE directory was created and the PWscf output was converted into the Yambo format using the p2y executable (pwscf to yambo), found in the yambo bin directory. By entering the SAVE directory and launching p2y an output with information about the system is produced and a hBN.SAVE directory is generated. The Yambo databases for bulk hBN are contained in the hBN.SAVE directory and contain information about the G-vector shells and k-point meshes as defined by the DFT calculation. Initialization Every Yambo run commences with an initialization step. Launching the code yambo from inside the folder containing the hBN-bulk SAVE directory generates a report file. This file contains information about the ground state properties as defined by the DFT run, but also information about the bandgaps, occupations, shell of G-vectors, BZ grids, the CPU structure etc. Yambo recalculates again the Fermi level and from there on, however, the Fermi level is set to zero, and other eigenvalues are shifted accordingly. The system is insulating (8 filled, 92 empty) with an indirect band gap of 3.87 eV. The minimum and maximum direct and indirect gaps are indicated. There are 72 k-points in the full BZ, generated using symmetry from the 14 k-points in our user-defined grid. Input file generation and command line options After initialization, yambo generates its own input files. Various quantities can be calculated by launching Yambo with one or more lowercase options. Any time yambo runs with a lowercase option the appropriate input file is generated (yambo.in) and is automatically opened in a vi editor. Multiple options can be used together to activate various tasks or runlevels. For instance, the following command generates an input file for optical spectra including local field effects:

yambo –o c –k hartree Hartree-Fock and GW calculation We performed a Hartree-Fock and GW calculation using a plasmon-pole approximation by launching:

yambo –x –p p –F

This command built up the input file for a GW/PPA calculation, including the exchange self-energy.

Similar to the HF calculation, we have focused on the direct gap of the system and therefore have selected the last occupied (8) and first unoccupied (9) bands. The value of cutoff energy was kept to its converged value of 40 Ry and the plasmon pole energy to its default value of 1 Hartree. Finally, the direction of the electric field was also fixed for the evaluation of the dielectric matrix to a non- specific value. By inspection of the output we identified the different steps of the calculation: computation of the screening matrix, the exchange self-energy and finally the correlation self-energy and quasiparticle energies.This file was subsequently modified in order to check the convergence of the gap as a function of the screening parameters: the bands employed to build the RPA response

function and the dimension of the microscopic inverse dielectric matrix. By adjusting these two parameters we have built a series of input files differing by the values of bands and block sizes of the dielectric matrix. In order to generate the files automatically we have used the appropriate bash script provided in the Yambo website.

In our case we are interested in calculating the optics in G-space at the independent particle level and this simply done by executing: yambo –o c

For the optical properties it is sufficient to consider the long wavelength limit which corresponds to the first q-point. In order to select just the first q-point we changed the variables for transferred momenta and total energy steps in the input file, then saved it and launched again the above command witout the lowercase options. In the optics runlevel yambo first computes the nonlocal commutator term due to the pseudopotential, then reads the wavefunctions from disc, calculates the dipole matrix elements and saves the result as databases. In the calculation all the G-vectors are used in expanding the wavefunctions. This corresponds to a cutoff energy of 40 Ry used in the DFT calculation. We have also tried using a smaller value of G-vectors and the two spectra were identical, highlighting an important advantage of excited state properties calculations: fewer G- vectors are required than in the case of DFT calculations. 2.2 Two Temperature Model module/Phase transition Module

The relaxation process module takes feedback from the above module, the temporal evolution of the carrier densities and the optical properties of the irradiated material. In other words, results from the first-principle-based approach provide the initial conditions for the relaxation process. For the case of semiconductors, an Auger recombination assisted mechanism is used to ‘force’ the decrease of the carriers following excitation while an equilibration mechanism leads to the exchange of energies between the carrier and phonon subsystems. Conditions for isothermal lines exceeding the melting temperature of the material are used to determine damage thresholds while Navier Stokes equations or thermoelastic models are employed to describe surface modification. State of the art finite difference or finite element methods were developed to quantify the temporal scales of the occurrence of a series of physical phenomena and analyse response of the irradiated material. The advantage of the in-house codes is that they are capable to provide a multiscale analysis which is available in commercial software.

3. Simulation Results for 6H-SiC

As a test case, simulations have been performed for 6H-SiC. The fundamental structural unit of the crystal comprises a covalently bonded tetrahedron of four C atoms with a single atom Si at the center and each C atom is likewise surrounded by four Si atoms. The input file for bulk SiC has been downloaded from GitHub and a norm-conserving type pseudopotential was chosen for Si and C atoms. Some of the changes in the input file included changing the symmetry or the number of atoms. For the cases of 3C, 4H and 6H SiC we have downloaded the crystallographic information files (cif) from the Bilbao crystallographic database and used the Avogadro interface to edit them. The resulting input files were displayed by using xCrySDen. Figure 2 shows the relevant unit cells for the cases of 3C and 6H SiC as they are read from the scf.in file using xCrySDen.

Figure 2: Crystal structure of 6H (left) and 3C (right) silicon carbide

In our case we calculated the optics in G-space at the independent particle level. The dielectric function was calculated using the following formula:

The real and imaginary part of the dielectric function are presented in Figure 3 for three different time delays with respect to the excitation pulse. The simulation under ultrafast laser irradiation was performed for the following conditions: laser 2 wavelength λL= 401 nm, pulse duration τp=50 fs, and laser (peak) fluence Ep=266 mJ/cm . Firstly, a combination of DFT calculations and Yambo Code were used to derive the evolution of the optical properties (Fig.4) and the carrier density evolution (Fig.5a). It is noted that after the end of the pulse (Fig.5a) the carrier density reaches a plateau which is due to the fact that the first-principle- based code does not include the Auger recombination which is essential to allow further decrease of the carrier density Ne. Therefore, the employment of the final Ne (time=end of pulse) as the initial parameter in the TTM can provide the carrier density evolution till relaxation as well as the carrier and lattice temperatures, Te and TL, respectively (Fig.5b).

Figure 3: Real (upper panel) and imaginary (lower panel) part of the dielectric function for 6H SiC.

Knowledge of the absorption coefficient can also provide an estimation of the heat propagation inside the material and computation of Ne and TL (Fig.5b). The estimated maximum lattice temperature is ~900 K for the above conditions that is well below the melting temperature of the material (~3100 K). Further calculations are under way to compute the damage threshold (~1 J/cm2) which appears to agree with early experimental results following pump-probe experiments conducted by experimental partners in the NFFA consortium. Finally, after the estimation of the damage threshold of the 6H-SiC, the computational code can be further used to describe morphological characteristics on the surface of the material following

0.25 1.6

0.245 1.4

0.24 1.2

0.235 1 m]  [

0.23 -1 0.8  Reflectivity 0.225 0.6

0.22 (a) 0.4 (b) 0.215 0.2 50 100 150 200 250 300 350 50 100 150 200 250 300 350 Time [fs] Time [fs] Figure 4: (a) Reflectivity and (b) Optical penetration depth as a function of time. Simulation results are derived 2 through the Yambo Code (Irradiation of 6H‐SiC with λL=401 nm, τp=50 fs, Ep=266 mJ/cm ). repetitive irradiation. Based on the fact that the material exhibits a ‘metallic’ behaviour after exposure 2 to laser fluences Ep=1 J/cm , surface plasmon excitation-related effects are expected to lead to periodic structure formation that has been confirmed by experimentalists a NFFA partner.

3 40 T e

] 35

-3 T 2 L 30 K] cm 3 21 1 25 20

0 15 N (from DFT) e 10 N (from TTM) Temperature [10 -1 e Carrier Density [10 Density Carrier (a) Laser Intensity 5 (b)

-2 -2 -1 0 1 0 -1 0 1 10 10 10 10 10 10 10 Time [ps] Time [ps]

0 0 1.2 8 ] -3 0.2 0.2 K] 2 1 7 cm 21

m] 0.4 0.8 m] 0.4   6 0.6 0.6 0.6

Depth [ Depth [ 5 0.4 0.8 0.8

Carrier Density [10 Density Carrier 4 0.2 Lattice Temperature[10 (c) (d) 1 0 1 3 2 4 6 8 2 4 6 8 Time [ps] Time [ps] Figure 5: (a) Carrier density evolution by using the Yambo Code and TTM approaches (blue line indicates the laser intensity which is in arbitrary units). (b) Evolution of electron and lattice temperatures. Spatio‐temporal dependence of Carrier Density (c) and Lattice Temperature (d). (Irradiation of 6H‐SiC with λL=401 nm, τp=50 2 fs, Ep=266 mJ/cm ).

4. Conclusions and perspectives

Efficient linking of the codes developed to compute (i) optical response of the irradiated material and carrier densities by the end of the laser pulse, and (ii) carrier density decrease (through Auger recombination) and relaxation processes following exposure to intense pulses, have been developed to describe ultrafast dynamics and thermal response of 6H-SiC. Design of experimental protocols are under way to test the computational platform which indicates a good agreement between theoretical results and preliminary experimental observations. Next, the computation platform is expected to be used to describe multi-pulse irradiation.

Publications in which NFFA is acknowledged

1 Margiolakis A., Tsibidis G.D., Dani K.M. and Tsironis G.P, ‘Ultrafast dynamics and sub- wavelength periodic structure formation following irradiation of GaAs with femtosecond laser pulses’ Physical Review B 98, 224103 (2018). (arXiv:1807.11422)

2 Tsibidis G.D., ‘The influence of dynamical change of optical properties on the thermomechanical response and damage threshold of noble metals under femtosecond laser irradiation’, Journal of Applied Physics 123, 085903 (2018). (arXiv:1812.02554)

3 Tsibidis G.D., ‘Ultrafast dynamics of non-equilibrium electrons and strain generation under femtosecond laser irradiation of Nickel’, Applied Physics A, 124,311 (2018). (arXiv:1611.03207)

4 Papadopoulos A., Skoulas E., Tsibidis G.D., and Emmanuel Stratakis E., ‘Formation of periodic surface structures on dielectrics after irradiation with laser beams of spatially variant polarisation: a comparative study’, Applied Physics A 124, 146 (2018).( arXiv:1812.02673)

5 Tsibidis G.D., Mimidis A, Skoulas E., Kirner S.V, Krüger J, Bonse J and Stratakis E., ‘Modelling periodic structure formation on 100Cr6 steel after irradiation with femtosecond-pulsed laser beams’, Applied Physics A 124, 27 (2018).( arXiv:1812.02668 )

6 Gaković B., Tsibidis G.D, Skoulas E., Petrović S.,Vasić B. and Stratakis E., ‘Selective ablation of Ti/Al nano-layer thin film by single femtosecond laser pulse’, Journal of Applied Physics 122, 223106 (2017).

7 Tsibidis G.D., and Stratakis E., ‘Ripple formation on silver after irradiation with radially polarized ultrashort-pulsed lasers’, Journal of Applied Physics 121, 163106 (2017).( arXiv:1605.06630 )

8 S. Psilodimitrakopoulos, L. Mouchliadis, I. Paradisanos, A. Lemonis, G. Kioseoglou and E. Stratakis, ‘Ultrahigh-resolution nonlinear imagin optical imaging of armchair orientation in 2D transition metal dichalcogenides’, Light: Science & Applications 7 18005 (2018).