# Resonant soft X-ray reflectivity of ultrathin polymer films at the C-edge: A direct approach

Alain Gibaud, Jayanta Kumar Bal, Eric M Gullikson, Cheng Wang,
Guillaume Vignaud

► To cite this version:

Alain Gibaud, Jayanta Kumar Bal, Eric M Gullikson, Cheng Wang, Guillaume Vignaud. Resonant soft X-ray reflectivity of ultrathin polymer films at the C-edge: A direct approach. AIP Advances, 2016, 6 (9), pp.095016. 10.1063/1.4963295 . hal-05492589

HAL Id: hal-05492589

https://hal.science/hal-05492589v1

Submitted on 4 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Resonant soft X-ray reflectivity of ultrathin polymer films at the C-edge: A direct approach

Alain Gibaud, Jayanta Kumar Bal, Eric M. Gullikson, Cheng Wang, and Guillaume Vignaud

Citation: AIP Advances **6**, 095016 (2016); doi: 10.1063/1.4963295

View online: http://dx.doi.org/10.1063/1.4963295

View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/6/9?ver=pdfcov

Published by the AIP Publishing

## Articles you may be interested in

* Probing porosity at buried interfaces using soft x-ray resonant reflectivity
  J. Appl. Phys. **107**, 023529 (2010); 10.1063/1.3295915

* Resonant soft x-ray reflectivity of organic thin films
  J. Vac. Sci. Technol. A **25**, 575 (2007); 10.1116/1.2731352

* Soft x-ray resonant reflectivity of low-Z material thin films
  Appl. Phys. Lett. **87**, 214109 (2005); 10.1063/1.2136353

* Energy dispersive x-ray reflectivity technique to study thermal properties of polymer films
  J. Appl. Phys. **94**, 2882 (2003); 10.1063/1.1596717

* Analysis of x-ray reflectivity data from low-contrast polymer bilayer systems using a Fourier method
  Appl. Phys. Lett. **76**, 2713 (2000); 10.1063/1.126452

<figure><img src="image_2.png" /></figure>

NEW Special Topic Sections

NOW ONLINE

Lithium Niobate Properties and Applications:
Reviews of Emerging Trends

AIP | Applied Physics
Reviews
# Resonant soft X-ray reflectivity of ultrathin polymer films at the C-edge: A direct approach

Alain Gibaud,<fnref n="1" />,ª Jayanta Kumar Bal,<fnref n="1" />,<fnref n="2" />,ª Eric M. Gullikson,<fnref n="3" /> Cheng Wang,<fnref n="3" /> and Guillaume Vignaud<fnref n="4" />

(Received 27 March 2016; accepted 9 September 2016; published online 19 September 2016)

The use of resonant soft X-ray reflectivity (RSXRR) in s-polarization is presented with the aim to show how far it is possible to go in the understanding the evolution of the refractive index $n(E) = 1 - \delta(E) - i\beta(E)$ of a ultrathin polystyrene film when the RSXRR is measured through the C-edge. We evidence that a direct fit to the data provides a very good estimation of $\delta(E)$ and $\beta(E)$ in a large range of energies. Nevertheless, at some specific energy close to C-edge we observe that it is not possible to obtain a satisfactory fit to the data though the same formalism is applied to calculate the reflectivity. We show that even though we take into account the energy resolution of the incident beam, we still end up with a poor fit at these energies. Incorporating the strong contribution of 2^nd^ order photons appeared near C-edge we could not eliminate the discrepancy. Probably the data normalisations have some impacts on such discrepancies at some specific energies. © 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/10.1063/1.4963295]

## I. INTRODUCTION

Low energy XRR is presently becoming more and more attractive as recent advancement in soft X-rays facilities provides access to outstanding soft X-ray reflectometers and some of them equipped with polarized dichroism set-ups. This development in technology is of particular interest for all the scientists who are working in the field of soft materials such as polymers or biological samples. Indeed for such materials, soft X-rays permit to conduct resonant XRR at the C edge. The paper by M. Mezger et al.<fnref n="1" /> is an excellent introduction to XRR measurements made at low energy. The technique that is used is the so-called RSXRR that employs the absorption dichroism for X-rays polarized in the parallel (p) and perpendicular (s) directions of the incident plane of X-rays. On the contrary to classical XRR carried at high energy ($E \sim 10$ keV), RSXRR is performed at an energy close to 285 eV i.e., in the vicinity of the absorption edge of C to take advantage of the very strong anomalous coefficients $f'(E)$ and $f''(E)$ of carbon at this energy and in addition of the resonance observed in the near edge X-ray absorption fine structure (NEXAFS) region for molecular compounds. Let us recall that XRR of a thin film can be calculated if one knows the index of refraction of the film. Whatever the energy, the index of refraction for X-rays is given by $n(E) = 1 - \delta(E) - i\beta(E)$ where $\delta(E)$ and $\beta(E)$ are directly related to $f'(E)$ and $f''(E)$, respectively (see supplementary material, Eq. S6 and S7). At low

<footnote n="1">¹LUNAM Université, IMMM, Faculté de Sciences, Université du Maine, UMR 6283 CNRS, Le Mans Cedex 9 72000, France</footnote>

<footnote n="2">²Centre for Research in Nanoscience and Nanotechnology, University of Calcutta, Technology Campus, Block JD2, Sector III, Saltlake City, Kolkata 700098, India</footnote>

<footnote n="3">³Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA</footnote>

<footnote n="4">⁴Laboratoire d'Ingénierie des MATériaux de Bretagne, Centre de Recherche, Rue de Saint Maudé, BP 92116, 56321 Lorient Cedex France</footnote>

<footnote>ªAuthor to whom correspondence should be addressed. Electronic mail A. G. Alain.Gibaud@univ-lemans.fr and J. K. B. Jayanta.bal@gmail.com.</footnote>
energy, the wavelength becomes quite large ($\lambda$> 4 nm). Therefore the critical angle of incidence given by

$$ \theta_c^2 = 2\delta(E) = \frac{r_e}{2\pi} \lambda^2 \rho_e(E) \quad (1) $$

where $r_e$ is the classical radius of the electron ($r_e = 2.8 \times 10^{-15}$ m), $\lambda$ the wavelength and $\rho_e(E)$ the energy dependant electron density, $\theta_c$ becomes generally > 5°. In such conditions, the reflectivity which can be measured up to 80° becomes sensitive to the polarization of the X-ray beam. This sensitivity can be exploited to access fundamental information about the chain orientation of polymer molecules. Such information is impossible to get from conventional hard X-ray set-ups for which XRR measurements are polarization independent.

Several key considerations are somehow essential to be understood if one wants to take advantage of the RSXRR results. Soft X-rays have the drawback to provide a small accessible range of q wave vector transfers ($q_{max} < 0.2 \AA^{-1}$) compared to the one measurable for hard X-rays ($q_{max} < 1 \AA^{-1}$). This is a serious disadvantage for accessing information about the interfacial roughness as intensity is falling off as $\exp(-q_z^2\sigma^2)$ and/or even for probing the existence of extremely thin layers of thickness less than 3 nm ($2\pi/q_{max}$). However, recently Capelli et al.² have studied organic layers using resonant soft x-ray reflectivity near C K-edge and reported that RSXRR can be probe down to the single molecular layer of thickness 1.6 nm. However, the index of refraction for materials containing C atoms in the soft X-ray region is not known despite the remarkable richness of spectral features found in the Henke database³⁻¹¹ as mentioned in a recent paper by H. Yan et al.³ It is therefore of paramount importance that an accurate knowledge of the complex index of refraction of molecular materials can be found. This issue is not trivial since the complex index of refraction exhibits rapid changes near the absorption edge that depend strongly on composition (see Figure 1). Indeed the bonding in molecular materials can be excited in the specific range of energy close to the C edge. The corresponding databases and compilations for the dispersion properties are entirely lacking for most materials, other than a few reports on characterizing the dispersion of poly (methyl methacrylate) (PMMA), polyimide and amorphous carbon.¹,³,¹²⁻¹⁴ For PMMA and polystyrene (PS), the complete determination of the index of refraction was recently accessed as a function of energy at the C edge.³ This problem was solved by first measuring the absorption coefficient $\beta(E)$ in transmission through a

<figure><img src="image_3.png" /><figcaption>FIG. 1. Energy (E) dependant values of absorption coefficient $\beta$ (a) and scattering factor $\delta$ (b) for polystyrene at the carbon edge¹³ showing their drastic changes close to the carbon edge. Values of $\beta$ are measured directly in transmission while values of $\delta$ are obtained through the Kramers-Kronig relationship. (c) The evolution of direct beam intensity for absolute normalization. Note that the variation of direct beam intensity is related to the carbon contamination on the optics.</figcaption></figure>
very thin free standing film of these materials, $\delta(E)$ was then determined by using the Kramers-Krönig relationship³

$$ \frac{\delta(E)}{(E^2 - E_a^2)(E^2 - E_b^2)} - \frac{\delta(E_a)}{(E^2 - E_a^2)(E_a^2 - E_b^2)} - \frac{\delta(E_b)}{(E_b^2 - E^2)(E_a^2 - E_b^2)} = -\frac{2}{\pi} P_c \int_{E_{\min}}^{E_{\max}} \frac{E' \beta(E')}{(E^2 - E^2)(E^2 - E_a^2)(E^2 - E_b^2)} dE', \quad (2) $$

where $E_a$ and $E_b$ are the two energies at which $\delta$ is known from the direct measurements. However, RSXRR was also used for structural and chemical analysis of thin films and multilayer structures.¹⁵⁻¹⁸ Ksenzov et al.¹⁵ reported that RSXRR can be used to discriminate the information of the upper interface from those of interfaces within the multilayer.

In this work, our aim is to address the point whether it is possible to directly determine $\delta(E)$ and $\beta(E)$ of a polymer thin film by fitting the RSXRR curves measured with a polarized (s) beam or not.

## II. MATERIALS AND METHOD

### A. Substrate treatment

Si(100) substrates were cutted into square pieces of size ~15×15 mm². They were sonicated in the presence of acetone (~10 min) and methyl alcohol (~10 min) separately to remove organic contaminants only, preserving the native-oxide layer (thickness ≈ 2 nm) on the surface. Before spin-coating, these substrates were etched with hydrofluoric (HF) acid to remove the native oxide layer.¹⁹ This treatment consisted in immersing the substrate in a HF acid solution (10% in volume) for 3 min followed by rinsing with deionised water and drying the surface under filtered air.

### B. Film preparation

Spin-coating (Karl Suss) technique was adopted to prepare homogeneous film on HF treated Si(100) wafers. For this, PS ($M_w$ = 136k, purchased from polymer source) was diluted in toluene at a concentration of 17g/L and spin-coated at 2000 rpm for 1 min. The deposited film was annealed in a vacuum oven at 160°C for 24h. The oven was slowly cooled back to room temperature by just turning off the heater.

### C. Characterization

Reflectivity ($\theta - 2\theta$ geometry) data were acquired at the ALS beam line in a ultrahigh vacuum,²⁰ following previously established protocols¹³ that include precautions to avoid radiation damage which causes spectral changes. Simulations of the XRR were performed using a homemade program in which a least-squares algorithm based on the theory shown in the supplementary material (Eq. S1) was implemented. In the calculation, it was assumed that the beam coming from the bending magnet was (s) polarized.

## III. RESULTS AND DISCUSSION

Figure 2a shows the XRR measured for a 90 nm thick film of PS in the energy range going from 250 eV to 350 eV stepping every 5 eV. All the data were normalized by the direct beam intensity (see Figure 1c) at a given energy. The measured RSXRR curves (Figure 2a) beautifully show the drastic changes that occur when passing through the edge. Figure 2c and d shows a comparison between the results obtained from our fits for $\delta(E)$ and $\beta(E)$ with the values provided in the paper by H. Yan et al.³ For $\beta(E)$, we clearly obtain an excellent agreement between the results obtained by the fitting method and the ones deduced from a direct measurement of the absorption. For $\delta(E)$, the
<figure><img src="image_4.png" /><figcaption>FIG. 2. Observed (a) and calculated (b) reflectivity versus both the energy and the incident angle $\theta$ for a 90 nm thick polystyrene thin film based on the calculation presented in the Supporting Information. Values of $\beta(E)$ (c) and $\delta(E)$ (d) obtained from Yan *et al.*'s report³ (open circle) for polystyrene and from the fit to the data (solid line) showing the excellent agreement for $\beta$ and for $\delta$ except at some specific energies near 285 eV.</figcaption></figure>

agreement is almost perfect except at $E = 285$ eV. From Figure 2 one can clearly observe the following features:

- in the range 250 eV to 275 eV, the plateau of total external reflection is clearly visible since $\beta(E)$ is small (see Figure 2c) and $\delta(E)$ is positive (see Figure 2d). The Kiessig fringes are well contrasted and are progressively displaced towards small angles of incidence as a result of the decrease of $\delta(E)$ with increasing energy.

- at about 280 eV the fringes and the plateau suddenly disappear as $\delta(E)$ goes to zero. At this energy the PS film is no more visible as its electron density is now the same as the one of the vacuum. This is a way of achieving a perfect contrast matching between the film and the above medium in which the film is placed.

- at about 285 eV there is a very strong absorption in the film as $\beta(E)$ becomes extremely big. This is the location of the carbon edge. Close to this value, there is no external reflection any more as $\delta(E)$ is negative.

- above 285 eV the absorption is still dominant and the plateau of total external reflection is barely visible due to the conjugated effects of a strong absorption and of $\delta(E)$ being mainly negative.

- above 300 eV the Kiessig fringes reappear since $\delta(E)$ becomes positive and is large enough to produce a sufficient contrast between the film and the surrounding media (i.e. the Si substrate and the vacuum).

An excellent overall agreement between the observed and calculated data can be seen in Figure 2a and b. Fits to the data were based on the assumptions that i) the Si index of refraction
was known from the Hanke database and ii) $\delta(E)$ and $\beta(E)$ for PS were free parameters. The thickness of the film was also initially fitted at 250 eV because at this energy the absorption is negligible and the Kiessig fringes are well contrasted. It was then kept to a constant value throughout the fits. A special routine was written so as to calculate all reflectivity curves in a batch. In this batch only $\delta(E)$ and $\beta(E)$ of PS, a scale factor and the background were allowed to vary while all other parameters were kept fixed. The procedure is particularly suitable for handling a large amount of data sets. In addition, all data are treated the same way without any external manipulation during the fitting procedure. In such a way, it is possible to fully trust the calculated results. This formalism is also successfully implemented on other type of polymer like poly n-butyl methacrylate (PBMA) (see supplementary material, Figure S4).

We now discuss the origin of this discrepancy in the vicinity of 285 eV and we start with the possible impact of the instrumental resolution in energy. Away from this specific energy and given the step size of 5 eV, $\delta(E)$ varies smoothly. The fit can then be carried out without taking too much care of the instrumental resolution in energy as there is not much difference between $R(q,E)$ and

<figure><img src="image_5.png" /><figcaption>FIG. 3. Observed (a) and calculated (b) reflectivity for a 90 nm thick polystyrene thin film based on the calculation presented in the supplementary material S1 together with the values of $\delta(E)$ (c) and $\beta(E)$ (d) extracted from the fits to the data in a narrow range of energy through the carbon edge. These results show that we observe clear discrepancies at certain wavelengths with nevertheless an overall good agreement.</figcaption></figure>
$R(q,E+dE)$. On the contrary, when $\delta(E)$ strongly varies in a narrow energetic bandwidth, it is no more possible to get a good estimation of $\delta(E)$ without taking in account the instrumental resolution in energy. This point was further evidenced by narrowing the step size to 0.2 eV and measuring the XRR from to 280 eV to 287 eV (see Figure 3a and b). In this range, $\delta$ and $\beta$ exhibit rapid changes (see Figure 3c and d) due to the existence of the C edge and to the resonances associated with the $\pi*$ transitions from the K-shell to the empty antibinding molecular orbitals of PS.⁵

Similar experimental results are also encountered in case of confined PS film of thickness ≈ 7 nm (see Figure 4). At energies below 284 eV the Kiessig fringes are still visible indicating a high contrast at the film/air and film/substrate interfaces. The former one decreases and vanishes at 284.5 eV where no fringes are visible indicating the passage by $\delta = 0$. In other words, near the C edge as the absorption is getting very big the penetration depth inside the film gets smaller (see Figure 5b). This can hinder

<figure><img src="image_6.png" /><figcaption>FIG. 4. Observed (a), (c) and calculated (b), (d) reflectivity versus both the energy and the incident angle $\theta$ for a 7 nm thick polystyrene thin film.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>FIG. 5. Calculated penetration depth for polystyrene film at an energy (a) E = 250 eV and (b) E = 285.2 eV. A large decay in penetration depth from 250 eV to 285.2 eV. (c) Evolution of simulated penetration depth (Z, in nm) as a function of energy and incident angle (θ).</figcaption></figure>

the observation of the Kiessig fringes which is evident from the simulation of penetration depth as a function of energy (see Figure 5c). At energy E = 250 eV, a large variation (from 6 to 1000 nm) of penetration depth with incident angle θ is observed whereas at E = 285.2 eV a small variation (from 5 to 12 nm) of penetration depth is encountered. Due to less adsorption, soft X-ray can penetrate a large distance within PS film away from the C edge as illustrated in Figure 5c. By restricting the range to these energies one can see the real problem occurring during the fitting procedure. As soon as the change in the sign of the derivative of δ is big, one cannot get a good fit to the data and additionally the agreement with the values obtained by applying the Kramers Kronig relationship and the direct fitting becomes rather poor.

For achieving a good fit to the data in the specific range of energy, we assumed that it would be necessary to consider the instrumental energy resolution. We have thus introduced a normalized energy distribution p(E) in a Gaussian form having a half width at half maximum ΔE = 0.06 eV. The reflectivity is thus given by

$$R(q) = \int_{E-\Delta E}^{E+\Delta E} p(E)R(E, q)dE \quad (3)$$

As it was not possible to fit the index of refraction for different energies in this method of calculation, we have just assumed that they could be taken from Ref. 3. We carried out the calculation, assuming the given values of δ(E) and β(E), calculating at each energy the reflectivity and weighting it with a Gaussian probability. Nevertheless we were unable to fit properly our results at the same specific values of the energy. This was very surprising as we would have expected that the discrepancy between
the calculated and observed values was due to the instrumental resolution in energy. Scholze et al.<fnref n="21" /> reported that 2nd order harmonics is quite strong at the C-edge. This means that when calculating the reflectivity, one must keep in mind that part of the reflected intensity comes from 2nd order or 3rd order harmonics for which the reflectivity curve can differ strongly than the one measured at the given energy. We have also considered this issue as a possible source of error in our calculation. We have calculated the reflectivity as the weighted sum of the reflectivities at energy *E* and energy 2*E*. Yet we found that any contribution of higher energy yielded a worse adjustment of the calculated curve. Therefore this assumption was also ruled out. A possible reason to explain why at some energies it is difficult to fit the X-ray reflectivity curve could arise from an improper evaluation of the direct beam intensity used to normalize the data. Indeed this value was measured at 2θ = 2° at all energies on a PS film itself. This is quite amazing as the fits are amazingly good at most energies even though the fits are automatically carried out in a batch file.

## IV. CONCLUSION

In conclusion we have shown in this paper that it is possible to extract in a direct way the optical index of refraction for X-rays of a thin film of PS when measurements are carried out close to the C edge. However we have failed to get a perfect agreement between the observed and calculated XRR at some specific energies. This problem was the most prominent when the derivative of δ(*E*) was changing in sign. Introducing the energy resolution in the calculation did not produce a better output. The underlying reason has been attributed to Considering the presence of significant contribution from 2nd order harmonics in the vicinity of C-edge we could not obtain a better fitting. Normalisation of the direct beam intensity could be the problem. However the underlying reason remain uncleared.

## SUPPLEMENTARY MATERIAL

See supplementary material for additional experimental details and figures.

## ACKNOWLEDGMENTS

We thankfully acknowledge CEFIPRA/IFCPAR program for the financial support. J.K.B thankfully acknowledged to Department of Science and Technology (DST), Government of India, for providing research grant through INSPIRE Faculty Award (IFA13-PH-79).

<footnote n="1">¹ M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, Phys. Rev. B **83**, 155406 (2011).</footnote>

<footnote n="2">² R. Capelli, N. Mahne, K. Koshmak, A. Giglia, B. P. Doyle, S. Mukherjee, S. Nannarone, and L. Pasquali, J. Chem. Phys. **145**, 024201 (2016).</footnote>

<footnote n="3">³ H. Yan, C. Wang, A. R. McCarn, and H. Ade, Phys. Rev. Lett **110**, 177401 (2013).</footnote>

<footnote n="4">⁴ B. L. Henke, E. M. Gullikson, and J. C. Davis, At. Data Nucl. Data Tables **54**, 181 (1993).</footnote>

<footnote n="5">⁵ O. Dhez, H. Ade, and S. G. Urquhart, J. Elec. Spec. Relat. Phenom. **128**, 85 (2003).</footnote>

<footnote n="6">⁶ J. Kikuma and B. P. Tonner, J. Elec. Spec. Relat. Phenom. **82**, 53 (1996).</footnote>

<footnote n="7">⁷ B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, J. Chem. Phys. **134**, 024702 (2011).</footnote>

<footnote n="8">⁸ S. G. Urquhart, A. P. Hitchcock, A. P. Smith, H. Ade, W. Lidy, E. G. Rightor, and G. E. Mitchell, J. Elec. Spec. Relat. Phenom. **100**, 119 (1999).</footnote>

<footnote n="9">⁹ S. G. Urquhart and H. Ade, J. Phys. Chem. B **106**, 8531 (2002).</footnote>

<footnote n="10">¹⁰ P. L. Cook, X. Liu, W. Yang, and F. J. Himpsel, J. Chem. Phys. **131**, 194701 (2009).</footnote>

<footnote n="11">¹¹ K. Kaznacheyev, A. Osanna, C. Jacobsen, O. Plashkevych, O. Vahtras, H. Agren, V. Carravetta, and A. P. Hitchcock, J. Phys. Chem. **106**, 3153 (2002).</footnote>

<footnote n="12">¹² C. Jacobsen, S. Y. Wang, W. Yun, S. Frigo, R. Soufli, and J. F. Seely, Proc. SPIE Int. Soc. Opt. Eng. **23**, 5538 (2004).</footnote>

<footnote n="13">¹³ C. Wang, T. Araki, and H. Ade, Appl. Phys. Lett. **87**, 214109 (2005).</footnote>

<footnote n="14">¹⁴ S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, Nano Lett. **10**, 2863 (2010).</footnote>

<footnote n="15">¹⁵ D. Ksenzov, C. Schlemper, and U. Pietsch, Appl. Opt. **49**, 4767 (2010).</footnote>

<footnote n="16">¹⁶ M. Nayak and G. S. Lodha, J. Appl. Cryst. **46**, 1569 (2013).</footnote>

<footnote n="17">¹⁷ I. A. Makhtokin, E. Zoethout, E. Louis, A. M. Yakunin, S. Müllender, and F. Bijkerk, Opt. Express **20**, 11778 (2012).</footnote>

<footnote n="18">¹⁸ M. Nayak, P. C. Pradhan, G. S. Lodha, A. Sokolov, and F. Schäfers, Scientific Reports **5**, 8618 (2015).</footnote>

<footnote n="19">¹⁹ J. K. Bal, T. Beuvier, A. B. Unni, E. A. C. Panduro, G. Vignaud, N. Delorme, M. S. Chebil, Y. Grohens, and A. Gibaud, ACS Nano **9**, 8184 (2015).</footnote>

<footnote n="20">²⁰ J. H. Underwood and E. M. Gullikson, J. Elec. Spec. Relat. Phenom. **92**, 265 (1998).</footnote>

<footnote n="21">²¹ F. Scholze, M. Krumrey, P. Miiller, and D. Fuchs, Rev. Sci. Instrum. **65**, 3229 (1994).</footnote>