# Analysis of Resonant Soft X-ray Reflectivity of Anisotropic Layered Materials

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## Abstract

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## 1. Introduction

_{2}[7], and tetracene single crystal [8].

## 2. Materials and Methods

#### 2.1. Evaluation of the Organic Film Dielectric Tensor in Correspndence of Absorption Resonances

_{2}(ℏω) along the 3 axes, which are linked, well outside the edge, to the Henke tabulated values [19]. In doing this, the sum is considered of all contributions to f

_{2}(ℏω) of all the atomic constituents of the molecule (for PTCDA 24 C atoms, 6 O atoms, 8 H atoms), up to ℏω = 30,000 eV. The real part of the molecular scattering factors f

_{1}is then evaluated through Kramers–Krönig (KK) integral transformations [6]. In the present case, this was achieved in practice by first constructing the odd functions f

_{2_odd}(ℏω), where f

_{2_odd}(−ℏω) = −f

_{2}(ℏω) and f

_{2_odd}(ℏω) = f

_{2}(ℏω), and then by calculating the Hilbert transforms of f

_{2_odd}(ℏω) on the domain from −30,000 to 30,000 eV, in steps of 0.1 eV. This procedure can be implemented straightforward in most software for data analysis (including Mathematica [20], Matlab [21], IgorPro [22]). The complex refractive index along the three principal axes is derived from f

_{1}(ℏω) and f

_{2}(ℏω):

_{m},y

_{m},z

_{m}denotes the molecular axis index, r

_{e}is the classical electron radius, λ is the light wavelength, and N is the density of the molecules. For PTCDA, we used a value of N = 1.37 × 10

^{21}molecules/cm

^{3}[6].

_{m}axes and an alternate herringbone arrangement in the x

_{m}y

_{m}plane, with adjacent molecules laying nearly perpendicular to each other [23,24,25,26]. This can be described by a diagonal dielectric tensor of a film in which the xx and yy elements are taken as the azimuthal angular average (in the x

_{m}y

_{m}plane) of ${\tilde{\epsilon}}_{{x}_{m}}$ and ${\tilde{\epsilon}}_{{y}_{m}}$ and the zz element is given by ${\tilde{\epsilon}}_{{z}_{m}}$. This model is further supported by the fact that the beam footprint (higher than 100 μm

^{2}) in a reflectivity experiment is much larger than the domains and the experimental signal integrates over a large number of domains that are typically oriented following the threefold symmetry of the Au (111) substrate. In fact, on the (111) surface of Au, the PTCDA molecules are known to organize in equivalent threefold domains, oriented at 120° with respect to each other [23,24,25,27,28]. This results in an overall in-plane isotropy of the reflectivity, assuming that the scattered intensity from the different domains is added up incoherently. Therefore, the dielectric tensor of a PTCDA layer can be written as

#### 2.2. Calculation of Light Propagation in the Layered Medium

_{l}. Layers are separated by flat planar and parallel interfaces. To describe a possible reorientation of the axes of each layer with respect to the x-, y-, and z-axes of the sample frame of reference (Figure 1a), we applied coordinate rotation matrices R

_{l}in terms of the Euler angles ϕ

_{l}, θ

_{l}, ψ

_{l}(describing a sequence of rotations around the z-x-z-axes) to the dielectric tensor so that the dielectric tensor becomes ${\overleftrightarrow{\epsilon}}_{Rl}={R}_{l}\left({\varphi}_{l},{\theta}_{l},{\psi}_{l}\right){\overleftrightarrow{\epsilon}}_{Rl}{R}_{l}^{-1}\left({\varphi}_{l},{\theta}_{l},{\psi}_{l}\right)$. Since R

_{l}is orthogonal, the dielectric tensor in x-, y-, and z-coordinates must be symmetric (for non-magnetic systems). Eventually, if multiple domains with different orientations are present in each layer, an average over the possible orientation angles is also applied.

#### 2.3. Experimental Methods

^{−9}mbar [6] from an home-made source, constituted by a tantalum pinhole source that was resistively heated at a temperature of 270 °C. An evaporation rate of 0.05–0.10 nm/min was used, which was monitored with a quartz microbalance. In particular, we focus here on a PTCDA film of 2 nm of nominal coverage, the same film studied in [6], to which the reader is addressed for further details. The specular reflectivity experiments were taken at a grazing incidence θ ≈ 8° in s- and p-polarization geometries. The photon energy was scanned across the O K-edge with a resolution of 0.2 eV, keeping fixed θ−2θ scattering conditions. The intensity of the direct (I

_{0}) and reflected (I) beams was measured with a photodiode (IRD SXUV-100) in separate scans. Possible beam instabilities were taken into account by acquiring the photocurrent drained by the last optical element of the beamline at the same time during I

_{0}and I scans, just before the scattering chamber. The reflectivity was obtained by evaluating the ratio between I and I

_{0}, after each of the two signals was corrected (normalized) for its respective beam instabilities. The degree of linear light polarization was selected of the order of 0.9, obtained by accepting photons only from the central portion of the beam from the bending magnet, with an angular acceptance of ± 0.1 mrad with respect to the plane of the synchrotron orbit. In simulations, we assumed completely linear polarized light, as expressed by Equation (21) above. A Si window filter (thickness 0.1 µm) was placed in the beam before incidence on the sample to reduce diffuse light contribution. X-ray absorption spectra (XAS) were acquired simultaneously, along with reflectivity, in total electron yield mode, by measuring the drain current from the sample. The XAS spectra were normalized to the beam flux, measuring the total electron yield from a carbon- and oxygen-free Au sample in the same energy region of the O K-edge.

## 3. Results and Discussion

## 4. Conclusions

^{−2}, with oscillations in the 10

^{−3}scale. In spite of this, high sub-nanometer sensitivity was obtained. The experimental spectra can be interpreted in terms of films of flat laying molecules, with the effective thickness obtained by simulations nicely corresponding with the nominal coverage. Small differences in the experimental spectra with respect to simulations were interpreted in terms of some level of disordered molecules, showing a more tilted alignment with respect to the rest of the film, and/or due to the effect of the non-complete linear polarization of the incoming photons.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Sketch of sequence of layers, each described by a complex dielectric tensor and a thickness d

_{n}. The layers are delimited by vacuum on top and by a semi-infinite substrate at the bottom, which is typically (but not necessarily) described by an isotropic material. A representation of the possible impinging and reflected electric fields are given, with xz being the scattering plane. E

_{01}and E

_{04}are the components of the fields parallel to the surface plane (perpendicular to the scattering plane—s-polarization for the incoming field), and E

_{02}and E

_{03}are the components of the fields laying in the scattering plane (p-polarization for the incoming field). (

**b**) 3,4,9,10-Perylene-tetracarboxylic dianhydride (PTCDA) molecule with its principal axes.

**Figure 2.**Calculated real (

**a**) and imaginary (

**b**) parts of the molecular film dielectric tensor in correspondence of the O K-edge; experimental near-edge X-ray absorption (

**c**) taken in s- and p-polarization scattering condition, with a grazing incidence of 8° on a film of 2 nm of nominal coverage of PTCDA on Au(111).

**Figure 3.**Experimental reflectivity (solid lines) measured at 8° of grazing incidence in s- and p-scattering geometries (labelled as refs and refp, respectively) compared with simulations (broken lines) obtained for a film of flat laying molecules of 2 nm of coverage.

**Figure 4.**Simulations of the variation of s- and p-reflectivity as a function of (

**a**) thin thickness (in steps of 0.5 nm), with fixed grazing incidence of 8° and flat laying molecules; (

**b**) grazing incidence angle (in steps of 0.1°), with fixed thickness of 2 nm and flat laying molecules; (

**c**) tilt angle of the molecules with respect to flat alignment (in steps of 20°, from 0° to 80°), with fixed thickness of 2 nm and fixed grazing incidence of 8°.

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**MDPI and ACS Style**

Pasquali, L.; Mahne, N.; Giglia, A.; Verna, A.; Sponza, L.; Capelli, R.; Bonfatti, M.; Mezzadri, F.; Galligani, E.; Nannarone, S.
Analysis of Resonant Soft X-ray Reflectivity of Anisotropic Layered Materials. *Surfaces* **2021**, *4*, 18-30.
https://doi.org/10.3390/surfaces4010004

**AMA Style**

Pasquali L, Mahne N, Giglia A, Verna A, Sponza L, Capelli R, Bonfatti M, Mezzadri F, Galligani E, Nannarone S.
Analysis of Resonant Soft X-ray Reflectivity of Anisotropic Layered Materials. *Surfaces*. 2021; 4(1):18-30.
https://doi.org/10.3390/surfaces4010004

**Chicago/Turabian Style**

Pasquali, Luca, Nicola Mahne, Angelo Giglia, Adriano Verna, Lorenzo Sponza, Raffaella Capelli, Matteo Bonfatti, Francesco Mezzadri, Emanuele Galligani, and Stefano Nannarone.
2021. "Analysis of Resonant Soft X-ray Reflectivity of Anisotropic Layered Materials" *Surfaces* 4, no. 1: 18-30.
https://doi.org/10.3390/surfaces4010004