Revealing all states of dewetting of a thin gold layer on a silicon surface by nanosecond laser conditioning

Dewetting is a ubiquitous phenomenon which can be applied to the laser synthesis of nanoparticles. A classical spinodal dewetting process takes place in four successive states, which differ from each other in their morphology. In this study all states are revealed by interaction of pulsed nanosecond UV laser light with thin gold layers with thicknesses between 1 nm and 10 nm on (100) silicon wafers. The specific morphologies of the dewetting states are discussed with particular emphasis on the state boundaries. The main parameter determining which state is formed is not the duration for which the gold remains liquid, but rather the input energy provided by the laser. This shows that each state transition has a separate measurable activation energy. The temperature during the nanosecond pulses and the duration during which the gold remains liquid was determined by simulation using the COMSOL Multiphysics® software package. Using these calculations, an accurate local temperature profile and its development over time was simulated. An analytical study of the morphologies and formed structures was performed using Minkowski measures. With aid of this tool, the laser induced structures were compared with thermally annealed samples, with perfectly ordered structures and with perfectly random structures. The results show that both, structures of the laser induced and the annealed samples, strongly resemble the perfectly ordered structures. This reveals a close relationship between these structures and suggests that the phenomenon under investigation is indeed a spinodal dewetting generated by an internal material wave function. The purposeful generation of these structures and the elucidation of the underlying mechanism of dewetting by ultrashort pulse lasers may assist the realisation of various technical elements such as nanowires in science and industry. O. C. Ernst, D. Uebel, S. Kayser, F. Lange, Th. Teubner, T. Boeck 2

. States of spinodal dewetting. Spinodal dewetting takes place in four different states. The initial state represents the as-grown gold layer. In phase 1 the layer starts to oscillate spatially. The oscillations reach the ground in phase 2. This is represented by the red areas. The red, dry areas expend till droplets are formed in the final state.
The initial layer state represents a homogeneous, undisturbed layer. In phase 1 the layer experiences heat and consequentially increased Brownian motion. The effective interfacial potential can possibly intensify this movement and create a spatial oscillation or warping in the layer thickness. These spatial oscillations are stable in phase 1, but become unstable in phase 2. Here, the motion gets amplified even more, which leads to valleys of the material waves hitting the underlying substrate. In this way a hole is formed in the layer and this hole grows in the same way as they grow in the induced mechanism. The main difference is that in spinodal dewetting, the holes are not created randomly, but in the order dictated by heat-induced warping amplified by the effective interface potential.
First, these states of dewetting were not identified individually, but successfully isolated from each other, whereby the correctness of the postulates and energy calculations could be proven [9][10] [11]. This was done by thermally induced dewetting [12] and also by experiments in which heat was applied to the thin metal layers by a laser [13] [14] [15]. In this work, this earlier results will be not only confirmed. The focus lies predominatly on the transitions between the different states of dewetting. For this purpose, a precise adjustment of the laser parameters and a detailed description of the heat influence of the laser in the gold-silicon system were carried out. The COMSOL Multiphysics® software package provides a detailed model of the dynamic temperatures and heat flows in the system [16]. Numerical simulations such as these provide a reproducible method for solving complex systems [17]. This allows new and more specific insights into the system than it was previously possible with classical analytical calculations, which was done in great detail by other scientist for heat flows in laser-induced dewetting [18].
Additionally, Minkowski measures were computed to estimate the quality of the transition states of the laser-treated layers compared to the thermally treated ones. Minkowski functionals are an effective tool to compare structures with each other and to find relations between structures which are not visible to the eye. The Minkowski functionalism offers a statistically stable method of integral geometry analysis, in which integration is performed over the entirety of all boundaries of a structure in all domains [19]. It is remarkable that the topology and geometry can be investigated in great detail for very diverse structures and morphologies [20]. This makes them suitable both for largescale systems whose size is unknown or of which only a very small part can be observed, and for micro-and nanoscopic systems that are difficult to study, image or reproduce. For this reason, Minkowski functionalism is a highly valued tool in theoretical and statistical physics to describe a DOI 10.20347/WIAS.PREPRINT.2777 Berlin 2020 variety of widely different systems, such as galaxies in astronomy [21][22] [23], chemical reactions in physical chemistry [24][25] [26] or, as also demonstrated here, the motion of ensembles of atoms to gain a deeper insight into dewetting mechanisms [27]. In this work, Minkowski measures are used to determine whether the dewetting process is random or whether there is an internal ordering mechanism. By determining whether the structures created by dewetting are randomly arranged or ordered, it is possible to distinguish between induced dewtting and spinodal dewetting.
For this study, a thin gold layer in the thickness range of 1 nm to 10 nm on a (100) silicon wafer is treated with a 266 nm nanosecond laser. The system of gold on silicon was chosen for two reasons: Firstly, early qualitative descriptions were made for this system, which promotes a quantitative experiment [28]. And secondly, gold on silicon is a promising material system for many applications -

Preparation of substrates
〈100〉 oriented 3 inch Si wafers with an initial surface roughness (RMS) of 0.1 nm were used. To dissolve the native silicon oxide and to passivate the free surface bonds by hydrogen, the substrates were immersed for 10 s in 2 % HF solution and rinsed in deionized water.
Au deposition was carried out in a molecular beam epitaxy (MBE) chamber at a base pressure of 2 × 10 -10 mbar. To reduce the amount of water inside the MBE chamber, the substrate was annealed DOI 10.20347/WIAS.PREPRINT.2777 Berlin 2020 at 870 K for 1 h in a separate preparation chamber. Before Au deposition, the substrates were annealed at 1170 K for 5 min to guarantee an oxide-free and reconstructed 2 × 1 Si(100) surface.
Then, 1 nm Au was evaporated from an effusion cell onto the substrate with a nominal deposition rate of 0.01 nm s -1 at room temperature. The absence of characteristic RHEED reflections after the deposition process indicates an amorphous Au layer on the crystalline substrate.

Thermal treatment
A couple of samples were annealed at 800 K or 1400 K for 10 min by a wafer heater in a high vacuum chamber (10 -6 mbar) as reference to the laser experiments.

Implementation of COMSOL simulation
The heat transfer in the silicon-gold system was modelled with the heat transfer module of the COMSOL Multiphysics® 5.4 software package in 2.5 dimensions using a finite element approach (FEM) and MUMPS solver [17].
The model as used for COMSOL calculations is composed of a cylindrical stack of a 200 nm thick silicon wafer, of an infinitesimal small interface between silicon and gold, and of a 5 nm thick gold layer (Fig. 3a). The backside is set to constant room temperature. The whole system will cool down to this temperature in less than 170 µs.
The radius of each cylindrical stack is diminutive compared to the assumed laser pulse. Due to its modest size, it is assumed that no heat leaves the cylindrical stack through the side walls. Fig. 3b pictures a larger cylinder representing the entire laser pulse function. Due to the low thickness of the thin gold layers, it is assumed no heat is generated within the gold layer. The assumption heat originates exclusively in silicon is underlined by the fact that the absorption of UV light in gold generates ballistic electrons [40] [41]. These electrons are most likely to be diverted into the silicon, where they lead to additional heating [42].
Accordingly, the laser pulse has a lateral extension of 100 µm and lasts 1.4 ns (FWHM in each case).
To calculate the change in temperature over time, the heat equation was used A detailed derivation of this formula can be found in [16]. Table 1 contains all important material parameters used in the simulation.

Integral geometry analysis
Integral geometry analyses such as the computation of Minkowski measures are elegant methods for studying the connectivity of different structures [28]. Minkowski measures are used to compare structures with each other and to identify related structures [19]. These structures can be of natural or simulated origin. In this study, the mathematical tool will be used to determine whether structures are of random or ordered origin, respectively induced or spinodal dewetting. A structure may be based on a complex ordering mechanism causing it to appear random to the human eye although it is not. Integral geometry analysis as it has been applied here can only predict that an ordering mechanism exists, but not which one. A complete integral geometry analysis can be eased by a simple, graphical evaluation of the different primary quantities such as the total area, the boundary length, and the Euler-Poincaré characteristic [24] and secondary quantities such as the number of closed areas and the formal radii [50]. For this study, the geometry of the gold film after laser conditioning will be compared with perfectly random or perfectly ordered models to determine geometric relationships.

Generation of droplets
Silicon-gold samples were prepared as described and treated with a 266 nm nanosecond-laser, whereby the thin gold layer changes its shape and morphology. The resulting morphologies are strongly reminiscent of the different states of dewetting.  fig. 5d and 5f. This laser-induced substructure consists of parallel lines with a distance of the laser wavelength (266 nm). Remarkably, these substructures are not visible in laser treated gold layers with an initial thickness of less than 5 nm but are more refined with growing gold layer thickness. Droplet size also change with initial gold thickness and number of pulses ( Table 2). As the initial gold layer thickness increases, the droplets become larger. In general, the morphology alterations at greater thicknesses appear to occur only at higher pulse numbers. This means, the distances of the phase boundaries from the centre of the laser spot increase with higher number of pulses. At low thicknesses or high number of pulse, there are also areas near the centre where no gold, neither as a layer nor as drops, can be detected. With more than 100'000 pulses, no gold was detected in any sample in the entire laser spot, regardless of the initial gold thickness. Melting of silicon was never detected.  (Fig. 6a). Structures similar to phase 2 and droplet like bodies are observable at 5 nm (Fig. 6b). Gold layers with a 10 nm thick gold layer (Fig. 6c) show early phase 2 and late phase 1 behaviour. In any case, the gold structures appear larger than in the laser-treated samples. At 1400 K samples of all considered gold thicknesses form droplets.    190 µJ laser pulse. Since the laser pulse is described as a Gaussian function, the resulting temperature also resembles a bell-shaped distribution with a peak height of 1720 K at 2.4 ns in the centre of the laser spot. Afterwards, the temperature decreases according to heat equation, except at 1337 K and 1687 K. These are the melting points of gold and silicon, respectively, at which solidification heat is released into the system, causing unsteady behaviour. The colour map in Fig. 7a corresponds to the distance from the centre of the laser spot. According to this simulation, the silicon melts from the laser centre up to a distance of 30 µm from the centre of the laser spot. However, no melting was observed in the practical experiments. Fig. 7b

Minkowski measures
Minkowski measures are used to compare structures with each other and to identify related structures. Fig. 8 shows this kind of Minkowski measures for the obtained data in a graphical version. Five different structures respectively networks of holes are compared: Two structures are perfectly ordered (Fig. 8a); one structure is perfectly random (Poisson distributed, Fig. 8b); and two are derived from experimental data (Fig. 8c).

Discussion
The the heat is exchanged into the gold or silicon bulk faster than a melting process of the silicon surface can take place. Since gold cannot give off its heat to the bulk material, but can only cool down by heat radiation into the void, the simulation seems to fit much better here than for the silicon substrate. In contrast, the distance of 34 µm for liquid gold matches the practical results very closely, as the transition from the phase 1 to phase 2 lies at 35 µm. This indicates that only phase 2 and the final state occur as classical fluid dewetting, but phase 1 as solid state dewetting. To initiate a dewetting process, the material does not necessarily have to be liquid. In general, the state of aggregation is a macroscopic effect with limited significance on the nanometre scale. For dewetting it is especially important that the mobility of the particles, in this case the gold atoms, is high enough for the dewetting process. This can be done in a formal solid state -also known as solid state dewetting [1]. Fig. 7b shows how long the gold remains in liquid state at various distances from the laser centre. Since it is known from the simulations that the gold layer cools down completely between the pulses, it can be assumed that the effect of one pulse multiplied by 1'000 is the same as 1'000 consecutive pulses. The same applies, of course, to 10'000 and 100'000 pulses. In this context, it is obvious that it is not only how long the gold remains liquid that matters, but also how much energy is introduced during each heat-up. For example, a 10 nm gold layer at 10 µm from the centre  Table 3 shows the maximum energies required to activate the state transition calculated from the experimental data and the lateral decline of the laser energy. Once a state is formed, the time the gold remains liquid comes into play more strongly: as soon as the activation energy is reached, the morphology changes and the structures grow. This is particularly evident in the size of the droplets in the final state (Figs. 5g and 5h). Table 2 shows here the dependence of droplet size on the layer thickness h and the number of pulses n.
Empirical data in the literature show a dependence of h 5/3 for droplet size in thermally treated layers.
As an approximation it can be assumed that the droplet size for short ranges increases linearly with the number of pulses. Fig. 9 shows that these correlations apply not only to thermally treated gold layers but also to nanosecond laser treated layers. Consequently, the mechanism of dewetting in the thermal process and in laser conditioning is not only theoretically the same, but also leads to the same empirical observations. Figure 9. Correlation between droplet size, number of pulses (N) and initial height of the gold layer (h) for droplets generated by nanosecond laser conditioning.
The major advantage of laser treatment is its better controllability. With finely adjustable input energies, laser profiles in time and space and the number of pulses, many more parameters can be controlled than with thermal treatment. Thus the structures can be controlled in more detail and better adapted to specific applications. If, for example, the gold droplets are used as a catalyst for the growth of nanowires, the diameter of the nanowire depends strongly on the droplet size, which can be adjusted more easily with a laser treatment than with thermal exposure. Moreover, the synthesis DOI 10.20347/WIAS.PREPRINT.2777 Berlin 2020 of all dewetting states on one single substrate was only possible by the laser-matter interaction carried out here and not by thermal treatment. Nevertheless, it must be mentioned that the laser interaction creates at least one additional substructure. Due to the high plasmonic resonance of the gold, a substructure is formed which generates parallel lines with a distance of the laser wavelength (266 nm). It is known that gold on silicon forms a thin wetting layer consisting of only a few monolayers. Presumably, the plasmonic resonance can build up an oscillation within the wetting layer and thus form the observed substructure. The existence of a wetting layer of gold on silicon has already been proven elsewhere in the literature [52] [54]. The results there also indicate the option of an energy-induced subsequent deformation of the wetting layer. In this study, such a warping seems to occur due to the oscillation of the free electron gas density.
The thermally treated samples form nothing but droplets when heated above the melting temperature of gold. Nevertheless, the morphologies of phase 1 and phase 2 could be isolated at much lower temperatures. Here all states occur by solid state dewetting, in which the material may be formally solid and not liquid. This is particularly true for gold on silicon because of the high chemical affinity between silicon and gold: at temperatures above 550 K, silicon can dissolve inside the gold and vice versa [55]. In this way, the system is able to form alloys or even eutectics locally, which increases local mobility many times over. Fortunately, the increase in mobility is not high enough to form droplets immediately in all cases, but only to isolate phase 1 morphologies in 10 nm gold samples and phase 2 morphologies in 5 nm samples.
The Minkowski measures in Fig. 8 show that there is no significant difference between laser-induced and thermally induced structures. It is clearly shown that perfectly ordered systems behave differently in respect to the Minkowski measures than perfectly random systems. This is also true for systems where it is not possible to recognize an order or a randomness with the eyes. By the measures the graphs show that both laser-and thermally treated morphologies behave rather like well-ordered systems. This is particularly evident with regard to the boundary length (Fig. 8e). Only the experimental data of the laser experiment for the normalized area in Fig. 8d

Conclusion
Most important aspects for an industrial process are reliability and versatility. To be recognised, a scientific method must be reproducible. In this study, a laser treatment process proved to be more controllable than the classical thermal treatment in selective dewetting processes aimed at forming nanoparticles such as nanodroplets, since the laser process includes more adjustable parameters such as pulse energy, number of pulses and pulse duration. Gold droplets are created by the interaction of a 266 nm nanosecond laser with a thin gold layer on silicon. These droplets show a very similar behaviour to the thermally treated droplets. In fact, both processes are induced by heat introduced into the gold layer. Since the laser energy is predominantly absorbed inside the silicon, the gold layer heat up indirectly. With the laser process it is possible to warm the gold very selectively. In this way, not only droplets are obtained, but also the different states of dewetting, such as the morphologies of phase 1 and phase 2. To our knowledge, it has never been possible to separate each dewetting state on a single thermally treated sample. Due to the high adjustability of the laser, this was possible here with laser-treated samples. This allows a more in-depth clarification of the dewetting mechanisms and the transition between the states. Here it is shown that there is no smooth transition between the states, but sharp boundaries. Taking temperature simulations with the software package COMSOL Multiphysics® into account, it has been shown that the duration of how long the gold remains liquid is not the critical parameter for determining which state is formed.
Rather, it is the energy applied that is crucial: the transitions appear to have separate activation energies quantified here. The duration of how long the gold remains liquid only influences how intense the state is formed, e.g. how large the droplets grow.
Minkowsi measures were used to determine which of the both following mechanisms effectively happens: induced dewetting, which occurs randomly or by mechanical impact, or spinodal dewetting, which occurs by internal self-amplified material warping. The holes in the gold layer, which form during phase 1 and phase 2 as the primary state of the droplets, were taken into account. The holes in the experimental samples, laser treated and thermally treated, were compared with perfectly ordered and perfectly (Poisson) random models. It was found that both types of experimental structures show more similarities with the ordered models. This clearly indicates that spinodal dewetting is the main process. Nevertheless, other secondary phenomena also come into play. with a distance of the laser wavelength and are most likely formed as a result of the plasmonic resonance of the wetting layer between the droplets with the laser.
In summary, it was shown that the high adjustability of the interaction between a nanosecond laser and a thin metal layer can be exploited to gain a deeper insight into the mechanisms of dewetting and the energetic requirements needed for the different states of dewetting. Since the morphology of the system and the size and shape of the resulting droplets can be controlled in great detail, this laser interaction is very promising for further research with high reproducibility needs and for industrial processes requiring reliability and versatility. An example of both is the growth of silicon nanowires, whose diameter and length are crucial for their application. Since gold droplets are used as catalysts for nanowire growth and their size and shape determine the properties of the nanowires, laser-induced dewetting can help to bring silicon nanowire technology to a new, more industrial standard.