Subgridding in the finite-difference time-domain method for simulating ...

A. Kern and M. Walther

Vol. 25, No. 3 / March 2008 / J. Opt. Soc. Am. B

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Subgridding in the finite-difference time-domain method for simulating the interaction of terahertz radiation with metal Andreas Kern* and Markus Walther Department of Molecular and Optical Physics, University of Freiburg, Stefan-Meier-Str. 19, D-79104 Freiburg, Germany *Corresponding author: [email protected] Received September 4, 2007; accepted November 17, 2007; posted January 3, 2008 (Doc. ID 87222); published February 8, 2008 Simulating the interaction of electromagnetic terahertz radiation with metals poses difficulties not encountered in the optical regime. Owing to a penetration depth small compared to the wavelength, such simulations in the terahertz frequency range require large discretization volumes with very small grid spacings. We present a unique subgridding scheme that accurately describes this interaction while keeping computational costs minimal. Bidirectional coupling between grids allows for the complete integration of subdomains into the simulation volume. Implementation in one and two dimensions is demonstrated, and a comparison with theoretical and experimental results [Opt. Express 15, 6552 (2007)] [Phys. Rev. B 69, 155427 (2004)] is given. Using our technique, we are able to accurately simulate plasmonic effects in terahertz experiments for the first time. © 2008 Optical Society of America OCIS codes: 000.4430, 240.0310, 240.5420, 240.6680, 300.6495, 310.1210.

1. INTRODUCTION Simulation has become a valuable tool in system design and provides insight into processes that cannot easily be measured. Often, trial and error can be replaced by simulation, and complex real-world phenomena not accessible by mathematical approximation can be accurately modeled. The finite-difference time-domain (FDTD) method [1,2] is today’s most commonly used technique for simulating time-dependent electromagnetic fields, such as the propagation of broadband pulses. It uses a centered difference form of the time-dependent Maxwell equations to calculate the temporal development of initial field distributions. This method’s stability, however, imposes an upper limit for the electric field gradient, in both space and time. In situations involving media with dielectric numbers ⑀r on the order of 1 and structure sizes comparable to the studied wavelength, a stable discretization grid is usually easily found. Simulating the interaction of electromagnetic terahertz (THz) radiation with real metals, however, poses a problem to the standard technique: Free-space THz pulses are of the spatial dimensions of few hundred micrometers, whereas their interaction length in metals is only ␦ ⬇ 100 nm. This large difference in spatial extent requires a large simulation volume with a finely spaced spatial grid and very fine time steps for system stability. For example, using constant grid spacing to simulate the interaction of a THz pulse with a 1 mm ⫻ 1 mm metal structure over a typical time period of 25 ps would lead to a huge computational cost: 3 ⫻ 109 spatial grid points would have to be calculated over 5 ⫻ 105 time steps. This would require computer memory in 0740-3224/08/030279-7/$15.00

the tens of gigabytes and computational times on the order of months using today’s conventional technology. In the best case, such calculations will become very costly, in the worst case impossible as memory limits are exceeded. One way to reduce these high demands is to allow for variable step sizes. If a fine grid is chosen where needed— e.g., at metal–vacuum interfaces—and spacing is coarsened over large areas of vacuum, the dimensions of the spatial grid can be reduced greatly. The time steps, however, must still be chosen so that even the finest grid points obey the FDTD’s stability condition. To further reduce computational costs by also varying the temporal grid, one can apply subgridding. Using this technique, different spatial grid sizes are stored in separate volumes, each with its own temporal grid. Thus, the optimal time step can be determined for each volume, ensuring stability without wasting processing time on unnecessary precision in the regions with coarse spatial grids. One must, however, establish accurate and efficient coupling between the separate volumes.

2. SUBGRIDDING IN THE FREQUENCYDEPENDENT FDTD METHOD A. Finite-Difference Time-Domain Method The FDTD method has established itself as the preferred technique for simulating broadband electromagnetic waves and transients, as its time-resolved calculations allow for a single simulation to describe a broad frequency range. In principle, arbitrary geometries and initial conditions can be modeled. Even in its simplest form, inho© 2008 Optical Society of America

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J. Opt. Soc. Am. B / Vol. 25, No. 3 / March 2008


mogeneous, time-dependent dielectrica and perfect metallic conductors can easily be integrated. The FDTD method is based on the time-dependent Maxwell equations

⳵D ⳵t ⳵B ⳵t

= ⵜ ⫻ H,

b0 + b1z−1 + b2z−2 + . . . + bNz−N a0 + a1z−1 + a2z−2 + . . . + aMz−M

,

where N and M are the orders of the power series in numerator and denominator. In z space, the second line of Eq. (2) becomes

= 共a0 + a1z−1 + a2z−2 + . . . + aMz−M兲D共z兲, = − ⵜ ⫻ E,

共1兲

H = ␮−1B, E = ⑀−1D.

共2兲

Here, ␮ = ␮0␮r and ⑀ = ⑀0⑀r describe the magnetic and dielectric properties of the medium and are, in general, a function in space. To obtain the best possible accuracy with a given grid size, centered differences are used, and consequently discrete derivatives are required not on the grid points themselves but exactly between two grid points. This procedure was first described by Yee in his monumental paper [1] and has become the standard technique. Centered differences are used not only for spatial derivatives, but also for those in time. This leads to the fact that electric and magnetic field components are stored not only at different points in space, but also at different times. Electrical fields are stored at whole time steps n⌬t and magnetic fields at 共n + 21 兲⌬t. B. Frequency Dependence in the FDTD Method To realistically simulate metals and other media with frequency-dependent dielectric functions, the FDTD method must be expanded. As calculations with the FDTD method are performed in the space and time domain, all parameters in the simulation such as material properties can be functions only of space and time. Thus the implementation of frequency-dependent dielectric functions is not straightforward. One approach is based on digital filter theory [3,4]. D and E field values are known at discrete, equidistant points in time and are coupled by the dielectric function ⑀. If one assumes ⑀ to be frequency dependent, coupling corresponds to a digital frequency filter imposed on the time-dependent field values. Digital signal processing methods describe the transformation of a frequency-dependent factor to a form that can be applied in the discrete time domain. Given a dielectric function in Laplace space ⑀共s兲 = ⑀共−i␻兲, one first applies the bilinear transform 2 1 − z−1 ⌬t 1 + z−1

,

共4兲

共b0 + b1z−1 + b2z−2 + . . . + bNz−N兲E共z兲

which are coupled by the electric and magnetic permeabilities

s=

⑀共z兲 =

共3兲

to attain ⑀共z兲 in z space. This term is then brought into the form of a quotient of power series in z−1,

共5兲

where D共z兲 = Z兵Dn其 is the z transform of the discrete time series Dn, E共z兲 likewise. Using the identity Z兵fn−k其 = z−kZ兵fn其,

共6兲

meaning that a multiplication with z−k in z space is equivalent to a shift back k steps in time, Eq. (5) can be transformed to En = 关共a0Dn + a1Dn−1 + . . . + aMDn−M兲 − 共b1En−1 + b2En−2 + . . . + bNEn−N兲兴b0−1 .

共7兲

Equation (7) is the standard form of a digital infinite impulse response (IIR) filter in digital signal processing. Implemented in the FDTD method, dielectric functions ⑀共␻兲 as well as other frequency-dependent behavior, expressed by the coefficients ai and bj, can be modeled. The bilinear transform in Eq. (3) is a commonly used method for transforming from Laplace space to z space owing to its simplicity. One flaw, however, should be mentioned: The response of an IIR filter calculated using a bilinear transform displays frequency warping near the Nyquist frequency. Since stability of the FDTD method imposes an upper limit to the time step ⌬t, the Nyquist frequency is often high enough that this effect is not noticeable. Nevertheless, it should be considered, especially when mapping systems with steep frequency responses. C. Nonuniform Grids The stability of the FDTD method depends on the chosen grid spacing. If cmax is the maximum velocity of light in the simulated system, stability is given if the Courant criterion, 关共⌬x兲2 + 共⌬y兲2 + 共⌬z兲2兴1/2 ⬎ cmax⌬t,

共8兲

is fulfilled, meaning that in one time step, field propagation is at most one spatial step. If the spatial grid is chosen too coarse, numerical dispersion causes the simulation to display unphysical attenuation and anisotropy. In general, however, the effects of numerical dispersion can be neglected if grid spacing is chosen finer than one tenth of the shortest occurring wavelength. Also, it is not necessary to keep step sizing constant over the whole grid. This allows to refine the spatial grid where needed, e.g., in media with large indeces of refraction, while choosing a coarse grid in vacuum to reduce storage and computational costs. However, as the FDTD method uses grid neighbors to calculate the field development at a given point, special care must be taken when applying nonuniform grids [5]. Although nonorthogonal grids are in principle possible, they make deriving the discrete derivatives in Eq. (1) nontrivial, and stability must be verified. By introducing subgrids—domains in which grid spacing has


been uniformly reduced—one can assume stability within the single domains, having only to pay attention to the coupling between grids, which is often done using spatial interpolation. Time steps, on the other hand, must be chosen so that Eq. (8) is satisfied even for the smallest occurring spatial step sizes. This leads to unneccessarily small time steps and needlessly exact calculation in areas with coarse spatial grids, wasting computing time. D. Multipass Subgridding To further reduce computational costs, especially when grid sizes vary considerably between domains, multipass subgridding can be utilized. This technique allows each subdomain to run with time steps optimized for its specific spatial grid. In cases involving domains with very different spatial step sizes, using separate temporal grids can speed up processing by many orders of magnitude. Kunz and Simpson proposed a two-pass approach [6] using a subgrid to simulate the field behind a small aperture. In the first pass, one time step was calculated in the coarse grid. Refining the subgrid required it to be calculated with 25-fold shorter time steps, so 25 time steps had to be run to reach the same point in time as the coarse grid. Coupling to the subgrid was done by imposing a temporal interpolation of the coarse grid to the outer boundary of the subgrid. This method allows for accurate modeling of the field behind the aperture. However, coupling is essentially one-way, so that structures in the subgrid will not effect the field in the coarse grid. This, however, is of crucial importance in cases where the field in the subdomain effects the behavior outside, like when simulating plasmonic effects at a dielectric–metal interface.

3. SIMULATING THE INTERACTION OF TERAHERTZ RADIATION WITH METALS Using the frequency-dependent FDTD method to simulate the interaction of electromagnetic waves with metals has been done for optical frequencies [7]. In this regime, wavelengths 共␭ ⬃ 600 nm兲 and skin depths 共␦ ⬃ 30 nm兲 are of similar orders of magnitude. Thus a simple, equidistant square mesh with grid spacing of a few nanometers can resolve skin effects while being able to store the whole free-space waveform without excessive memory usage. In the THz regime, however, wavelengths 共␭ ⬃ 300 ␮m兲 and skin depths 共␦ ⬃ 60 nm兲 differ by almost 4 orders of magnitude. To resolve the skin depth while being able to store the entire terahertz pulse in the simulation volume, the grid would have to be huge, and many time steps would have to be calculated. Subgridding is an ideal method for overcoming this problem. Assuming a flat metal surface, the domain in which the skin depth must be resolved is very thin, thus a subgrid containing the metal’s “skin” could be kept small. In this case, the tiny time steps needed for the fine grid to run stably are calculated only within the subgrid and can be computed in reasonable time. In the following, we present a subgridding technique that will prove especially useful for simulating the interaction of THz radiation with metals. Our concept unites nonuniform meshing, multipass subgridding, and bidirec-


281

tional coupling to fully integrate the subgrid into the discretization volume. An outline of our method is given in Fig. 1. In areas where high field gradients are expected, refined subgrids are introduced, overlapping the coarse grid by one full step on each side. The calculation procedure can be described in four steps. (a) For a given time step, assume the field values to be correct at all points except for the components of the coarse grid within the subgrid’s fine mesh (black cross). Using the standard FDTD scheme, one time step is evaluated in the coarse grid, calculating first magnetic, then electric field components. Since the field development is calculated from neighboring field values, the erroneous field propagates one half cell in both directions (b). Assuming the subgrid to be refined at most N-fold, N time steps are evaluated in the subgrid at ⌬t / N each, imposing the temporal interpolation of the corresponding coarse grid electric field values on both subgrid boundaries (c). By replacing the erroneous field values in the coarse grid with the correct values from the subgrid (d), the initial situation is established (a). In our example, the spatial grid in the subdomain is nonuniform, which allows mesh refinement to be limited to a small fraction of a cell, e.g., to simulate a thin metal film or skin of a metal surface. Also, it allows the subgrid’s overlap of only one step to span a whole cell in the coarse grid, enabling bidirectional coupling of the two grids.

4. APPLICATION OF THE PROPOSED SCHEME A. 1-D Simulation To demonstrate the effectiveness of the proposed subgridding scheme, we simulated a system that has been experimentally characterized in a recent study of the antireflection behavior of a thick silicon substrate covered with a thin chromium film [8]. It has been shown that if the chromium film has the correct thickness, internal reflection of the pulse exiting the substrate can be suppressed owing to wave-impedance matching at the surface. When attempting to simulate this system with the FDTD method, difficulties arise, as the required film thickness of about 10 nm is much smaller than the free-space THz pulse and the silicon substrate. We show that using our subgridding scheme, the metal film can be resolved and the behavior of the system can be modeled without any further approximations to be made. The simulation volume in this case could be reduced to one dimension, minimizing memory and processing requirements. Perfectly matched layers [9] were set on both ends of the grid to absorb outgoing waves, producing an effectively unbounded simulation volume. The coarse grid was set to 400 cells at ⌬xc = 2 ␮m step size, sufficient to include the 500 ␮m thick substrate and the absorbing boundaries. The experiment was conducted using films between 0 and 10 nm thickness [8]. By choosing a refinement factor of N = 1000, our simulation achieves a resolution of 2 nm in the subgrid. A subgrid size of only 12 cells including overlap is sufficient to simulate films up to 18 nm thickness. Chromium was modeled using the Drude model: ⑀r共␻兲 = 1 − ␻p2共␻2 + i␻␯c兲−1. Plasma frequency

282



Fig. 1. Concept of the proposed subgridding scheme with bidirectional coupling in one dimension. Circles represent grid points containing electric field components, vertical dashes are magnetic field components. ⍀1 is the coarse grid, and ⍀2 is the subgrid. The large gray rectangle represents a spatial structure that demands subgridding.

␻p and collision frequency ␯c were obtained by a fit to data from [10], resulting in ␯c = 1.25⫻ 1013 Hz, ␻p = 1.06 ⫻ 1015 s−1. In the experiment, the coated side of the silicon substrate was illuminated by a THz pulse, and the field behind the substrate was measured. We chose the same geometry as in the experiment, illuminating the substrate with a pulse sampled from our own THz time-domain spectrometer. Figure 2 shows the simulated electric field amplitude behind the substrate for different chromium thicknesses. The first pulse at t = 5 ps is the pulse transmitted through the substrate. Without coating, this pulse is followed by the first internal reflection about 11 ps later, which corresponds to the propagation time through

Fig. 2. Simulation of the system described in [8]. (a) shows the field amplitude of a THz pulse after passing through the substrate without (upper line) and with (lower line) 8 nm chromium coating. (b) shows the effect of chromium films thinner 共4 nm兲 and thicker 共12 nm兲 than the optimal thickness. Here, the field amplitudes are magnified 4⫻ to the right of the dashed line. A time-resolved animation of the simulation can be viewed on our homepage [11].

1 mm of silicon 共n = 3.42兲. With an 8 nm thick chromium coating, the internal reflection almost vanishes. Our simulation shows remarkable agreement with the experimental results reported in [8]. In particular, the characteristic antireflection behavior is correctly reproduced. By varying the film thickness in our simulation, we obtained an optimal suppression of the internal reflection for a thickness of 8 nm, in perfect agreement with the experiment. Figure 2(b) shows simulations of the substrate with 4 nm and 12 nm thick chromium coatings. As expected, we find that if the film is chosen too thin, reflection without phase shift occurs, which corresponds to a transition to an optically less dense medium. If the film is too thick, the reflected pulse shows a phase shift of ␲, as occurs when reflecting on an optically denser medium.

B. 2-D Simulation In the following, we applied our subgridding scheme to a problem requiring a 2-D simulation volume. To be able to compare results, we simulated a system similar to an experimental setup used to measure the coupling of a broadband THz pulse to surface plasmon polaritons (SPPs) on a flat gold sheet [12]. In the experiment, aperture coupling of the THz pulse to surface plasmons occurs at a gap between a razor blade and the gold surface. In our simulation, a Hz-polarized THz pulse is focused on a 100 ␮m aperture above a gold sheet at an angle of 67 deg to the gold surface’s normal. The z axis is perpendicular to the 2-D simulation volume, resulting in the conservation of the magnetic polarization throughout the simulation. Again, the metal was described by the Drude model, the parameters for gold obtained from [13]: ␯c = 4.76⫻ 1013 Hz, ␻p = 1.37⫻ 1016 s−1. In the coarse grid, a step size of 10 ␮m was chosen, allowing us to store a 10 mm⫻ 4 mm volume in less than 5 MBytes of memory. The subgrid was only refined normal to the gold surface, as a high field gradient is only expected in this axis. A refinement factor of N = 250 resulted in a grid spacing of 40 nm, fine enough to resolve the skin depth in gold. Though the gold sheet in the simulation was slightly thicker than in [12], this had little effect on the system’s behavior, as the skin depth is considerably thinner than the sheet in either case. The razor blade’s function is only to block the field above the surface and create an aperture, thus it can be modeled as a perfect metallic conductor. This simple model forces the electric field components parallel to the conductor’s surface to zero and does not require a fine grid. Figure 3



283

Fig. 3. (Color online) Simulation of THz pulse passing through a 100 ␮m aperture above a gold sheet. The figure shows superimposed snapshots of the simulated field in 4 ps steps, Hz component shown. (a), the coarse grid; (b), the refined subgrid. Horizontal lines represent gold surfaces; the vertical line is a perfect metallic conductor. Step sizes in (a) are ⌬x = ⌬y = 10 ␮m; in (b), ⌬x = 10 ␮m, ⌬y = 40 nm. An animation of the simulation can be viewed on our homepage [11].

shows snapshots of the simulated pulse before, while, and after passing through the aperture. Field penetration of the gold surface is as expected [14]: Hz, parallel to the surface, penetrates the metal with a decay length of the skin depth ␦. An exponential fit to the Hz field amplitude yields a skin depth of 共99± 3兲 nm, or about 2.5 steps in the subgrid. This is in good agreement with the Drude model’s prediction for our pulse’s spectral average of 870 GHz. Ex (not shown) is forced to nearly zero at the metal surface by the good conductivity of the gold. Ey (not shown) is present at the gold surface, but immediately falls to zero as it enters the metal. 1. Coupling to Surface Plasmon Polaritons Since, using our technique, the interaction of THz pulses with metals of finite conductivity can be simulated, we should also be able to observe such effects as coupling to SPPs in the aforementioned setup. To investigate this potential, the simulation was repeated replacing the gold, described by the Drude model, with a perfect metallic conductor (PMC). A perfect conductor does not support SPP propagation, and so no plasmonic effects should be visible in this case. In two dimensions, the electric and magnetic fields emitted by a source of finite size, in our case the aperture, will decay with r−1/2 in the far field, r being the distance from a virtual infinitesimal source. Thus, in our simulation, the field amplitude just above the PMC sheet will behave like

Hz,PMC共r兲 = H0*r−1/2 .

共9兲

The parameter H0* as well as the location of the virtual source can be obtained by a fit of Eq. (9) to the field amplitudes above the PMC. In the simulation implementing our subgridding scheme (SGr), on the other hand, coupling to SPPs would modify the field just above the gold surface. In order for this coupling to take place, wave vectors of the plasmons and the wave in vacuum must be matched [15]. This is not possible for waves propagating in free space. However, in the near field of an aperture, the wave is distorted so that coupling can occur. Once induced, the SPP mode will propagate along the metal sheet with very low attenuation [16]. In the far field of the aperture, therefore, the magnetic field amplitude Hz just above the real metal surface is expected to behave like Hz,SGr共r兲 = H0*共Rr−1/2 + A兲,

共10兲

where R is the reflectivity of the gold surface and A is a measure for the coupling strength to the SPP. The amplitude of the SPP can be assumed to stay constant in our simulation, as the size of the simulation volume is much smaller than the calculated decay length, which is on the order of meters in the THz range [16]. Thus, the quotient of Eqs. (10) and (9), comparing the field amplitudes obtained using our subgridding scheme with those assuming a PMC, respectively, can be written as

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have not yet been reported. In fact, in contrast to our simulation, experimental determination of the SPP coupling strength is greatly hindered by the difficutly to discern free-space waves skimming along the surface from true surface-bound plasmon polaritons.

Fig. 4. Simulated field amplitude above Drude metal calculated with subgridding relative to a perfect metallic conductor (circles). The line is a fit according to Eq. (11). Fit parameters R and A correspond to the reflectivity of the gold and the SPP coupling strength.

Hz,SGr共r兲 Hz,PMC共r兲

= R + Ar1/2 .

共11兲

Figure 4 shows this quotient determined from our simulations. A fit of Eq. (11) to the simulated data results in a reflectivity of R = 99.4% and a value for A = 0.17 m−1/2. The Drude model for gold predicts a field amplitude reflectivity of RDr = 99.6% for the spectral average of our initial THz pulse and the simulation’s geometry. The value that we deduce from our simulations is slightly smaller, which we attribute to distortion of the fields near the aperture. The conducting surfaces of the aperture cause the field lines to skew vertically, causing an effective incident angle larger than that of the incoming THz pulse. This leads to a slight reduction in reflectivity, consistent with our observation. The factor A cannot be directly compared to an available model. However, the field amplitude Hz,SPP = H0*A of the surface plasmon can be calculated and compared to the incident THz amplitude to determine a coupling efficiency. In our simulation, a fraction of 0.24% of the incident THz pulse is coupled to the SPP mode. Unfortunately, experimental coupling efficiencies for this system

2. Other Methods for Simulating SPPs Recently, FDTD simulations of THz photons coupling to SPPs in metal have been reported [17]. The method used in this work also incorporates the Drude model to describe the metal. However, grid steps larger than the skin depth in the metal were used. In this case, the field in the metal cannot be resolved, and the vacuum field couples to the average over a whole cell in the metal. To compare the result of this approach with our subgridding method, we repeated our 2-D simulation using a Drude model for the gold but without SGr, instead using equidistant 10 ␮m step sizes. All other parameters were kept the same. The resulting field propagation can be seen in Fig. 5. One can see that the field is concentrated more strongly in the vicinity of the metal surface than in the simulation incorporating our subgridding method. This would represent a stronger coupling to the SPP mode. However, as the pulse propagates further, unphysical, emissive ringing occurs in the metal. This instability is caused by numerical dispersion of the field in the metal. Also, one can see that the velocity of the pulse along the metal surface is approximately 1% lower than that of the pulse in vacuum, whereas the calculated velocity of a SPP mode in gold is only less than the vacuum light velocity by a factor of approximately 10−6 [15]. These inconsistencies with physical expectations lead us to the conclusion that besides causing system instability, an approach based on a grid insufficient for accounting for the skin depth, as in [17], causes the metal to display an effective conductivity below that dictated by the Drude parameters of the model, thereby overestimating the coupling of THz radiation to SPPs. It should be mentioned that the unphysical effects originated by this method become increasingly visible as interaction distances become larger and might not be rec-

Fig. 5. (Color online) 2-D simulated system without subgridding. Hz field component shown. The horizontal line represents Drude modeled gold surface; the vertical line represents a perfect metallic conductor.



ognizable for systems with small dimensions. Also, the observed inaccuracy of the metal’s conductivity is likely to abate as step sizes approach the skin depth. However, the true interaction can correctly be portrayed only if the field in the metal can accurately be resolved.

3.

5. CONCLUSION

5.

We present a subgridding scheme that enables a subgrid of arbitrary dimensions and refinement to be inserted into a FDTD simulation’s discretization volume. Bidirectional coupling accurately conveys influences on the simulated fields from one grid to another. Successful implementation has been demonstrated in one and two dimensions but can easily be expanded to three, the biggest tasks being bookkeeping and appropriate spatial interpolation. Using our subgridding scheme, accurate simulation of a broadband THz pulse coupling to SPPs in metal could be performed for the first time. Our method provides the computational means to investigate the importance of plasmonic effects in the THz regime quantitatively. This opens the way to explore exciting applications such as waveguiding or enhanced field transmission through subwavelength apertures.

4.

6.

7. 8. 9. 10. 11. 12.

ACKNOWLEDGMENTS The authors acknowledge ongoing support from and fruitful discussions with Hanspeter Helm. We thank our colleague Andreas Thoman for sharing his experience on thin metal films.

13.

14.

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