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Abstract
Freeform optics enable improved optical solutions but their fabrication usually requires complicated precision machining processes. We report on an approach for freeform shaping of optical surfaces via a stress-induced viscous deformation of glass plates. We studied the deformation of fused silica substrates covered by specifically laser patterned films of substoichiometric silicon oxide during annealing at about 1100 °C in an oxidizing ambient. The obtained large deformation of the substrates can be understood by a mostly viscous deformation but can be described in analogy to a purely elastic deformation. Our results demonstrate the feasibility of a method for freeform shaping of individual optical substrates that only requires the preparation of a flat surface.
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1. Introduction
Freeform surfaces are surfaces without an axis of rotational invariance [1]. They are applied in optics technology to simplify optical setups and their assembly, enhance their performance and to obtain new possibilities [1]. Freeform surfaces can be produced by forming and finishing processes like turning, milling, grinding, water jet polishing, magnetorheological finishing, ion beam figuring or reactive atom plasma processing [2]. Also direct laser ablation and laser polishing can be applied [3,4]. There are also forming processes, like compression molding [5] and slumping [6,7], that are not directly based on material removal. However, they still require precision machining of a suitable mold.
A freeform surface can also be produced by a stress-induced shaping method [8]. Recently, such methods have been studied in detail for the correction of the surface topography of optical substrates [9–14]. In these methods, a permanent field of plane stress is introduced into a surface-near region of the plate-like substrate. The plane stress field then causes a specific elastic deformation. In general, a state of plane stress can be decomposed into an equibiaxial, a uniaxial and a shear component, in which the latter depends on the orientation of the coordinate frame and hence can be replaced by the orientation of the uniaxial component [15, Sec. 4.3.2][16]. To correct or achieve complex deformations, the values of all three components need to be adjusted [9]. As these stress-induced shaping methods rely on an elastic deformation of the substrate, they are limited to small deflections or thin substrates.
To obtain larger deflections, one could induce a non recoverable, i. e. a plastic or viscous, deformation. Although typical substrate materials like silicate glasses and silicon exhibit elastic (recoverable) behavior at room temperature under usual circumstances, they undergo non-recoverable deformations at increased temperatures [17,18]. For crystalline materials, occasionally, the plastic deformation of substrates by a stressed coating has been reported in the literature [19–23]. However, to the best of our knowledge, similar reports for glass substrates are missing. In [24], it was theoretically demonstrated that for a thin film on a viscoelastic substrate, the curvature of the film-substrate system increases. Thus, by annealing of a film-substrate system, large nonrecoverable deformations of the substrate can be obtained. However, to our knowledge, structured films on viscoelastic substrates have not been studied yet.
For precision structuring of thin films, laser ablation is a well suited tool because it is highly flexible, fast and does not require any hazardous chemicals. Patterning of dielectric thin films by excimer lasers leads to straight and steep film edges [25]. Additionally, due to the small wavelengths in the ultraviolet spectral region, excimer lasers allow high lateral resolution [26]. In a process developed by our group, monolithic fused silica phase masks are produced by deposition of a substoichiometric silicon oxide (silicon suboxide) film on a fused silica substrate, excimer laser structuring of this film by rear side ablation and oxidation of the thus structured film to silicon dioxide [27,28]. In rear side ablation, the laser light is transmitted by the substrate and absorbed in the film close to the film-substrate interface, resulting in a binary ablation process [25]. During oxidation of the silicon suboxide film taking place at high temperatures [27], the film thickness increases [29] and a compressive film stress is induced [30].
We demonstrate here the possibility of stress-induced freeform shaping of fused silica substrates via a viscoelastic deformation. The plane stress field is generated by an excimer laser patterned film of silicon suboxide and the samples are annealed at about 1100 °C to achieve viscous flow of the substrates. This approach might be used for production of custom-made freeform mirror substrates or correction plates.
2. Methods
We followed the process chain sketched in Fig. 1. The substrates were double side polished cylindrical plates of fused silica (Heraeus Suprasil 1 or 2, type-III vitreous silica [31], OH-content $\leq$ 1300 ppm) with a thickness of 1 mm and a diameter of 25 mm. We measured the exact substrate thickness with a micrometer gauge. On one side of the substrates, we deposited thin films (roughly 400 to 1100 nm) of silicon suboxide by physical vapor deposition inside a commercial evaporation chamber (Leybold Univex 350). Pellets of silicon monoxide were thermally evaporated by joule heating of a tungsten baffled box. During deposition, the gas pressure inside the evaporation chamber was of the order of 10-6 mbar, the temperature of the sample holder was 300 °C and it was rotated at 10 rpm. By transmission and reflection measurements of the samples after deposition, we estimated the stoichiometry of the films to be close to the one of silicon monoxide.
We patterned the films via rear side ablation with the light of an ArF excimer laser (Coherent LPX-Pro) with a wavelength of 193 nm and a pulse duration of about 20 ns. To obtain a large quadratic laser spot of edge length of about 1 mm, we applied a nonimaging fly’s eye homogenizer setup (combination of crossed cylinder lens arrays and a spherical plano-convex fused silica lens [32, pp. 133-135]). For the ablation of narrow (10-100 µm) and many millimeters long lines, the image of a slit mask was projected onto the sample via a cylindrical fused silica lens, as described in [33].
Fig. 1. Our approach for stress-induced freeform shaping of fused silica plates. The silicon suboxide film deposited on one side of the plate is patterned by ablation with an ArF excimer laser. Annealing leads to an oxidation of the film, which causes a compressive stress, and a decrease of substrate viscosity, allowing for a significant nonrecoverable deformation.
For annealing, we placed the samples on a flat, ground fused silica plate with the coated side facing upwards. We used a muffle furnace (Nabertherm L5/13/B410) for annealing at 1120(1) °C in ambient air. Annealing in nitrogen flow was achieved inside a tube furnace (Nabertherm R50/250/13) with an inner tube made of fused silica with a volume of about 0.6 L, that was constantly purged by a stream of nitrogen at a rate between 50 to 100 L/h. The annealing temperature, measured by introduction of a type K thermocouple through the gas outlet, was 1060(10) °C.
In between the annealing steps, we measured the height profiles and the height maps of the non coated side of the samples at room temperature by tactile profilometry (Bruker Dektak XT-A) with a diamond stylus of 2 µm radius. During measurement, the samples were placed on three steel balls for defined and reproducible boundary conditions. Reproducible orientation of the measurements was obtained by orientation at a mark on the rim of the sample or at a feature of the film pattern. By tactile profilometry, we also measured the step heights of the silicon suboxide films to determine the film thickness and feature geometry.
To monitor the oxidation progress of the films, we measured transmission spectra of the samples (Perkin Elmer Lambda 19 UV/VIS/NIR) and reflection spectra of the coated sample sides (Filmetrics F20UV) in between the annealing steps at room temperature. By fitting the reflection spectra with the internal program of the F20UV, we obtained values for the thickness of the grown SiO2 films.
For interpretation of the experimental results, we applied linear elastic and linear viscoelastic three dimensional finite element simulations using the FEniCS platform [34,35] and the mesh generator Gmsh 4.3.0 [36] (see also Sec. S1 in Supplement 1). We assumed the material properties to be isotropic. For the linear viscoelastic simulation, we described the deviatoric deformation by a Maxwell element and the dilatational deformation by a purely elastic behavior. The latter assumption is well justified as long as the sample is only constrained in two dimensions [17, Sec. 7.7].
3. Results
3.1 Continuous film
Figure 2 shows results for the deformation of a sample with a continuous film of 500(6) nm thickness for annealing at 1120 °C in ambient air. We measured the height profiles (Fig. 2(a)) and the height maps (Fig. 2(b)) of the non coated sample surface after different total annealing times in the central $20 \times 20$ mm2 or $15 \times 15$ mm2 area, respectively. Initially, the surface is flat. Due to the annealing, it develops a concave shape with a maximum amplitude of about 70 µm after 9 h of total annealing time. Thus, the sample deformation corresponds to a compressive stress inside the film and the sample domes up, as is indicated by the inset in Fig. 2(d). From the height map after 9 h of total annealing time (Fig. 2(b)), it can be seen that the deformation is isotropic.
Fig. 2. Experimental results for a sample with a continuous film of 500(6) nm of silicon suboxide on a 1 mm thick substrate of fused silica. The sample was annealed at 1120 °C in ambient air. (a) Height profiles of the non coated surface after different total annealing times. (b) A height map of the non coated surface after 9 h of annealing. The vertical stripes are the individual height profiles obtained by the measurement procedure. (c) Curvature profiles of the height profiles in (a). (d) The average curvature values in the interval of 2.5 to 17.5 mm for two orthogonal measurement directions depending on total annealing time. The error bars of the curvature values have been magnified by a factor of ten for better visibility. Approximate values for the rates of curvature change have been obtained by linear fits to the data. The inset shows a sketch of the sample cross section after deformation to illustrate that it domes up due to annealing.
From the measured height profiles, smoothed by convolution with a Gaussian function, we calculated the curvature profiles, which are shown in Fig. 2(c). They are approximately homogeneous but develop a slightly parabolic shape with ongoing annealing. The diverging curvature at the end points of the profiles is an artifact of the smoothing algorithm, while the waviness of the profiles presumably is some low frequency noise of the instrument. For a more quantitative analysis, we calculated the average curvature values of the profiles in between 2.5 and 17.5 mm for two orthogonal measurement directions. Fig. 2(d) shows the obtained average curvature depending on the total annealing time. Please note that the error bars only reflect the uncertainty in curvature measurement and do not consider a statistical variation in between different samples. The average curvature increases with increasing annealing time until, at 9 h of total annealing time, oxidation of the film is complete, which can be deduced from transmission spectra of the sample (Fig. S1 in Supplement 1). For longer annealing times, the curvature decreases again.
For a similar sample as in Fig. 2, we removed the whole film via excimer laser ablation before the oxidation was complete. After decoating, nearly the whole deformation remained, which indicates a major contribution by viscous flow. If an annealing temperature of 1025 °C is applied instead of 1120 °C as in Fig. 2, the rate of deformation is roughly reduced by an order of magnitude. This shows that the deformation is rather sensitive to the annealing temperature and is another argument for the large contribution of viscous flow because, according to the manufacturer, the viscosity of the substrate material increases by roughly an order of magnitude if the temperature is reduced from 1120 (annealing point) to 1025 °C (strain point).
3.2 Macroscopic film pattern
The above results indicate the possibility of a forming process of fused silica plates by a silicon suboxide film. Actually, after deposition of a reflective coating, Fig. 2(b) would already represent a concave spherical mirror with a focal length of roughly 360 mm. To evaluate the possibility of obtaining more complex surface topographies, we applied different patterns to the silicon suboxide film via laser ablation.
Figure 3 shows results for a sample where the film has been removed on one half of the coated sample surface such that a semicircle of coated area remains (Fig. 3(a)). The film thickness was 431(8) nm and the sample was annealed in ambient atmosphere at 1120(1) °C. We measured the height profiles on the non coated side of the sample in direction across the film edge and along the film edge. These profiles are plotted in Fig. 3(b) after 5 h of total annealing time. The corresponding curvature profiles are plotted in Fig. 3(c). In direction across the film edge, a large positive value of curvature is obtained in the coated area and a small negative value of curvature is obtained in the decoated area. In contrast, in direction along the film edge, a uniform curvature is observed. The values of the average curvature (Fig. 3(d)), calculated as described in the preceding subsection, exhibit a similar behavior as the ones for the continuous film in Fig. 2(d).
Fig. 3. Experimental and simulation results for a sample with a macroscopic film pattern. The film thickness was 431(8) nm and the substrate thickness 1 mm. The sample was annealed at 1120(1) °C in ambient air. (a) Sketch of the view on the non coated side of the sample. The measurement directions are indicated. (b) Height profiles after 5 h of total annealing time in both measurement directions. The results of a linear elastic FE simulation are also shown (see text). (c) The corresponding curvature profiles. (d) The average curvature values for the two measurement directions depending on the total annealing time. The y error has been magnified by factor of ten for better visibility.
We found that the obviously viscoelastic sample deformation due to the film pattern can be formally well described by a linear elastic deformation. However, unrealistic large values of the film stress need to be assumed. To demonstrate this analogy between the viscoelastic experimental results and the elastic simulation, we calculated a hypothetical film stress of $\sigma _f=-33\,\text {GPa}$ by the Stoney equation [37]
and the average curvature value $k_{av}$ in both measurement directions. Note that we introduced a factor of $-2$ to account for the fact that half of the film has been removed and that we measured the curvature on the non-coated side. We assumed a film thickness of $t_f=431\,\text {nm}$, a substrate thickness of $t_s=1.078\,\text {mm}$, a Young’s modulus of $E_s=72.9\,\text {GPa}$ and a Poisson’s number of $\nu _s=0.165$ for the fused silica substrate at room temperature [38, p. 16]. We then applied the obtained value of the film stress in a linear elastic finite element simulation for a plate that is covered by a film of same geometry as in the experimental case and the same properties as mentioned above. The resulting height profiles and curvature profiles are plotted in Fig. 3(b) and Fig. 3(c), respectively. A good agreement with the experimental profiles is obtained. This indicates that similar film patterns can be applied in the viscoelastic case as in the elastic case to obtain a qualitatively similar deformation.3.3 Line pattern
To study the sample behavior in case of uniaxial plane stress components, we patterned silicon suboxide films of about 1.1 µm thickness into line patterns with a duty cycle (linewidth divided by line period) of approximately 0.5, but different line periods (Fig. 4). Thereby, we achieved different aspect ratios (height divided by width) of the lines and correspondingly a differently pronounced anisotropy of the average film stress in lateral directions [39]. However, during oxidation of a silicon suboxide film, the oxidation front propagates from the film-air interface to the film-substrate interface [40]. Thereby, the film stress distribution is changed during oxidation. For a continuous film or a film with a macroscopic pattern, this effect can be neglected because the curvature only depends on the integrated stress. However, for a line patterned film, it has been shown that the distribution in direction normal to the surface of the film stress has a major influence on the anisotropy of the thickness integrated stresses in lateral directions [8]. Additionally, due to the incorporation of oxygen atoms, oxidation causes a swelling of the oxidized structures [29]. Therefore, in order to avoid too rapid oxidation, which would cause a change in stress distribution and structure height, we annealed the samples in flowing nitrogen. This measure significantly reduced the oxidation rate but, presumably due to residual oxidant molecules, a slow growth of a SiO2 layer, which seems to be essential for the stress generation, could still be detected. The annealing temperature was 1060(10) °C.
Fig. 4. For studying the influence of uniaxial plane stress components, we patterned films of silicon suboxide into narrow lines of different period and a duty cycle of about 0.5. Here, the surface of a sample with a line-period of 20 µm is shown. (a) A bright-field reflected light microscope image. The bright and dark lines correspond to the residual film and the uncovered substrate surface, respectively. (b) A surface profile measured by tactile profilometry in direction across the lines.
Figures 5(a) and 5(b) show the curvature evolution of the non-coated side of three samples with different line periods and one sample with a continuous film. For the continuous film, no turning point in curvature evolution, as in Figs. 2(d) and 3(d), is observed. This is because after 34 h of annealing, a SiO2 thickness of only 129(3) nm is reached. Thus, oxidation still proceeds. For the patterned films, in direction across the lines, the curvature values split up for long annealing times and become ordered depending on the line period. This trend is also shown in Fig. 5(c) for the curvature ratio $k_s/k_p$ of average curvature values $k_s$ and $k_p$ in direction across and along the lines, respectively, depending on the annealing time. For line periods of 100 and 200 µm, the curvature ratio approximately becomes constant for long annealing times but for a line period of 20 µm it still decreases after 34 h of total annealing time and would, according to the curvature evolution in Figs. 5(a) and 5(b), eventually approach the zero line. Figure 5(d) shows a height map of the sample with a line period of 20 µm after 34 h of annealing. A toroid surface segment has been obtained.
Fig. 5. Experimental results for three samples with a line patterned film and one sample with a continuous film on fused silica. The film thickness was about 1.1 µm and the substrate thickness about 1 mm. Annealing was done at 1060(10) °C in flowing nitrogen. The period of the line patterns is specified in the figures. (a),(b) Curvature evolution of the non coated surface in directions along and across the lines, respectively. The errorbars of the curvature values have been magnified by a factor of ten for better visibility. (c) The evolution of the curvature ratio (across/along). (d) Height map of the sample with 20 µm period after 34 h of annealing.
4. Discussion
Figure 2(d) shows the experimental curvature evolution for a sample with a continuous film. Our understanding of the sample behavior is as follows. Oxidation of the silicon suboxide film proceeds from the surface and thereby leads to the formation of a silicon suboxide/silicon dioxide bilayer [40]. Due to incorporation of oxygen atoms, a compressive stress is generated inside the freshly grown silicon dioxide layer [30]. At such a high temperature, fused silica behaves like a viscoelastic fluid [41]. Therefore, the film stress causes a small elastic but a large viscous deformation. As long as oxidation proceeds, an overall compressive stress of the film is obtained. However, after complete oxidation of the film, the stress inside the silicon dioxide film fully relaxes and the integrated film stress becomes zero. Now, the gravitational influence dominates and tends to flatten the sample again, which causes a decrease in curvature. Further, gravity causes the evolution of an inhomogeneous curvature profile, as can be observed in Fig. 2(c).
It might be argued that a stress inside the silicon suboxide film itself, e. g. a thermal stress, causes the viscous deformation of the samples. However, we did not observe any dependence of the rate of curvature change on the initial film thickness (Sec. S3 in Supplement 1). Therefore, we conclude that there is a fast relaxation of any stress inside the silicon suboxide film. Considering that silicon suboxide undergoes a phase separation into silicon nanoclusters inside a matrix of stoichiometry close to the one of silicon dioxide and crystallization of the Si nanoclusters [42], our interpretation is supported by a fast stress relaxation on the order of seconds reported for silicon nanocrystals inside a silicon dioxide matrix at even smaller temperatures than applied in this work [43]. It might also be argued that the deformation only happens during heating or cooling of the sample. However, we found that the curvature depends on the annealing duration of single annealing steps. This can be observed in Fig. 3(d), where annealing steps of different duration have been applied but the curvature still follows a clear trend.
The above hypothesis for the gravitational influence is supported by a different curvature evolution after the oxidation is complete if the sample is oriented vertically during annealing (not shown). Plus, the inhomogeneous curvature profiles can be qualitatively reproduced by a linear viscoelastic finite element simulation for the gravitational influence on a flat plate under simply supported boundary conditions. Note that a linear curvature decrease is observed in Fig. 2(d) after the oxidation is complete. By the simulation, we obtained a similar rate of curvature change for a Newtonian shear viscosity of the material of roughly $\eta = 2.2 \times 10^{11}\,\text {Pa}\,\text {s}$. From [41], a viscosity value of $2.9 \times 10^{11}\,\text {Pa}\,\text {s}$ can be estimated for a vitreous silica of about 1200 ppm OH-content at 1120 °C. The excellent agreement with the estimated value supports our interpretation.
According to the viscoelastic Stoney equation
As we used non thermally stabilized glass substrates for this study, one would expect a change in fictive temperature, and thereby a change in viscosity during annealing. However, from the data of [44] and the results in [45], we estimated the structural relaxation to take place during the first annealing step only. Nevertheless, for improved precision and reproducibility, thermal stabilization of the substrates before coating should be applied.
In Sec. 3.2, we demonstrated an analogy between the experimentally observed mostly viscous deformation and the results of a linear elastic simulation. Such an analogy can be understood in light of the correspondence principle of linear viscoelasticity, which states that under certain circumstances the solution of a viscoelastic problem can be obtained from the elastic solution by replacing the elastic modulus by a time-dependent modulus [15, Sec. 9.3.3]. From a mathematical point of view, however, it does not matter if the modulus or the stress value is changed, as was the case in our simulations.
Does this correspondence still hold in case of line patterns? For a precise application of uniaxial stress components, the relation between line geometry and stress anisotropy or curvature ratio must be known. Such an analytical relation was derived by Wikström et al. for the case of thermal stresses and a purely elastic deformation of the line patterned film and the substrate [46]. According to their theory, the curvature ratio $k_s/k_p$ is given by the expression [46]
with the Poisson’s ratios $\nu _f$ and $\nu _s$ of film and substrate, respectively, and the function $\chi (\rho )$ given in [46], which depends on the aspect ratio $\rho$ of the lines. Equation (4) is plotted in Fig. 6 for the curvature ratio depending on the aspect ratio $\rho$ for a Poisson’s ratio of the film in the reasonable range of $\nu _f = 0$ to $0.5$ and for $\nu _s = 0.165$, the Poisson’s ratio of fused silica at room temperature (elastic case) [38, p. 16]. In case of a mostly viscous deformation of the substrate, the volume of the substrate should be conserved and therefore the effective Poisson’s ratio should become $\nu _s = 0.5$ for long annealing times (see Sec. S5 in Supplement 1 for a more sophisticated argument). Inserting this value into Eq. (4), the orange area in Fig. 6 is obtained (viscous case). It can be seen that a smaller value of the curvature ratio is expected for a viscous deformation of the substrate than for an elastic deformation. Thus, the analogy between the elastic and viscoelastic deformation demonstrated in Sec. 3.2 does not hold for the curvature ratio in case of uniaxial plane stress components. Actually, a more pronounced anisotropy can be obtained by a viscoelastic deformation. This extends the space of accessible deformations compared to the entirely elastic case.Fig. 6. The ratio $k_s/k_p$ of curvature $k_s$ across the lines divided by curvature $k_p$ along the lines depending on the aspect ratio of the lines. The colored areas reflect the expected values according to Eq. (4) and a Poisson’s ratio of the film in the range of $\nu _f = 0$ to 0.5. The values of $\nu _f$ corresponding to the boundaries of the shaded areas are given on the right side of the figure. In the blue area, a Poisson’s ratio of the substrate of $\nu _s=0.165$ has been assumed, as is the case for fused silica at room temperature. In case of a viscous deformation, the effective Poisson’s ratio should become $\nu _s=0.5$ (orange area). The curvature ratio of the samples in Fig. 5 after 34 h is plotted as well. The error bars of the curvature ratio have been magnified by a factor of ten for better visibility. Besides, for the point of highest aspect ratio, the error bars of the aspect ratio are covered by the dots.
In Fig. 6, also the experimentally obtained values after 34 h of annealing from Fig. 5 are plotted. The aspect ratio has been calculated by division of the measured film thickness after deposition of the film by the measured line width after patterning. The experimental values for the curvature ratio exhibit a better match to the theoretical estimate for the mostly viscous deformation than for the elastic deformation. However, a precise comparison is hindered by ignorance of the stress distribution inside the cross section of the lines, which was recently shown to have a significant influence on the curvature ratio [8], and a non uniformity of the experimental curvature profiles (see Sec. S6 in Supplement 1), which presumably is caused by the gravitational influence.
For the evolution of the curvature ratio with annealing time, one might expect a decrease from the value for a purely elastic deformation to the value for a purely viscous deformation (cf. Eqs. (S24) and (S25) in Supplement 1). However, in Fig. 5(c), a more complicated behavior is observed in the experiment. After 3 h of annealing, three of the four samples exhibit a curvature ratio close to one, while the sample with 100 µm line period exhibits a curvature ratio of about 0.1. From a mathematical point of view, this discrepancy is caused by rather small average curvature values in direction across the lines for the sample with 100 µm line width for small total annealing times (Fig. 5(b)). One should keep in mind that in the experiment, the elastic deformation is determined by the stress and material properties at room temperature, while the viscous deformation is determined by the stress and material properties at the annealing temperature. Therefore, the evolution of the curvature ratio with annealing time could be more complicated. Additionally, especially at small annealing times, there are influences like delayed elasticity, structural relaxation, the flatness of the underlay and structural changes of the silicon suboxide film, which should also be considered.
5. Conclusion
Recently, stress-induced shaping was studied for correction of figure errors of optical substrates. In this work, we proposed a method for forming of optical substrates by a similar approach and a viscoelastic deformation. To reach a specific deformation, similar film patterns as in the elastic case can be applied, but much larger deformations can be reached. The analogy to the elastic deformation indicates that also complex surface topographies should be obtainable by appropriate film patterns. To obtain transparent freeform components, annealing could be stopped before the oxidation of the film is complete. The remaining layer of UV absorbing silicon suboxide could then be used as absorber for complete removal of the film via excimer laser ablation. We recently demonstrated that via rear side ablation, a roughness of the uncovered surface as small as 3 nm can be obtained [28]. Thus, after decoating, the surface could still be used for optical applications. The advantage of our approach is that only a flat polished surface and no mold is required. Due to the latter, the process is highly flexible. However, precision of the process should be studied in detail and possibly improved.
Funding
European Regional Development Fund (ZW 6-85005827); Bundesministerium für Wirtschaft und Energie (03THW05K09); Niedersächsisches Ministerium für Wissenschaft und Kultur (03THW05K09).
Acknowledgments
We would like to thank Jörg Meinertz (IFNano) for helpful discussions and support.
Disclosures
The underlying idea of this work has been patented by the authors institution.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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