Pattern formation during the oscillatory photoelectrodissolution of n-type silicon: Turbulence, clusters and chimeras

We report and classify the rich variety of patterns forming spontaneously in the oxide layer during the oscillatory photoelectrodissolution of n-type doped silicon electrodes under limited illumination. Remarkably, these patterns are often comprised of several dynamical states coexisting on the electrode, such as subharmonic phase clusters and spatio-temporal chaos, and include so-called 'chimera states'. The experiments suggest that the subharmonic phase clusters emerge from a period doubling bifurcation which, upon further parameter changes, evolve into classical phase clusters. Experimentally the occurrence of the patterns is controlled via two coupling mechanisms: A linear global coupling by an external resistor and a nonlinear coupling imposed on the system by the limitation of the illumination.


Introduction
Oscillatory media are known to exhibit a variety of spatial patterns [1,2]. Best known are spiral waves and chemical turbulence, as most famously encountered in the Belousov-Zhabotinsky (BZ) reaction or the oxidation of CO on Pt in UHV [3,4,5,6]. These systems can be understood as ensembles of diffusively coupled individual oscillators. Additionally, there are many physical mechanisms leading to a long range or even global coupling of the oscillators. This applies, e.g., to electrically controlled systems, such as semiconductor devices [7,8], gas discharge tubes [9] or electrochemical systems [10,11,12,13]. The most prominent patterns emerging from a global coupling are scale free cluster patterns where the medium separates into a few domains, oscillating with a certain, constant phase lag [14,15,16,17,18]. Properties of these cluster patterns have been intensely studied theoretically employing phase oscillator models or the complex Ginzburg-Landau equation with linear global coupling [19,20]. Yet, the exploration of the phenomena occurring and the understanding of the dynamics of these patterns is far from being complete. In this article, we present a collection of peculiar patterns and unusual synchronization phenomena observed in an electrochemical experiment, the oscillatory photoelectrodissolution of n-type silicon, and their classification.
During the electrodissolution the silicon is first oxidized electrochemically according to the following reaction [21]: where νVB denotes the number of charge transfer processes via the valence band of the silicon, i.e., processes involving minority charge carriers in n-type silicon. This oxide is subsequently etched away by fluoride species present in the electrolyte in a purely chemical process. For n-type doped silicon, the dominant spatial coupling mechanism between the oscillators involves the diffusion of holes. The holes drive the initial charge transfer step in equation 1, i.e., the inequality νVB ≥ 1 holds [22]. A limited illumination and, therefore, limited generation rate of holes in the valence band of the working electrode thus introduces a cut-off for the total current. Clearly, this mechanism has a nonlinear coupling characteristic. An additional global coupling mechanism is introduced by connecting a resistance Rext in series to the working electrode in the potentiostatically controlled experiments. This external resistance thus couples the voltage drop ∆φint across the electrode|oxide|electrolyte interface at each individual point of the surface of the working electrode to the total current through the working electrode and consequently has a linear characteristic.
Here, U is the externally applied voltage, j the spatial average of the current density and A the electrode area.
In order to get a comprehensive picture of the dependence of the patterns on the experimental parameters, four of these parameters are varied over a wide range where current oscillations occur. Two of these parameters characterize the electrolyte, the first one being the total concentration of fluoride species cF and the second one the pH value which determines the distribution of the fluoride on the three species F − , HF and HF − 2 according to the law of mass action. Both parameters together determine the total silicon oxide etching rate as the different fluoride species follow different etching pathways with distinct rates [23,24]. The other two parameters represent the strengths of the coupling mechanisms given by the illumination intensity I ill for the nonlinear coupling and RextA for the purely linear global coupling, respectively.

Experimental
The working electrodes are the (111) faces of monocrystalline, n-type doped silicon samples with a resistivity of 3 − 5 Ωcm. They are ohmically back-contacted with a thermally evaporated 200 nm aluminum layer annealed at 250 • C for 15 minutes. The silicon is attached to a custom made PTFE holder via silicone paste (Scrintex 901, Ralicks GmbH, Rees-Haldern, Germany) which also serves as a sealing for the back-contact against the electrolyte. The typical electrode size is A ≈ 4 × 6 mm 2 . As a counter electrode, a circular shaped platinum wire is placed symmetrically in front of the working electrode and a Hg|Hg2SO4 reference electrode is used which is placed behind the working electrode at a distance of several centimeters. During the experiments the electrolyte is constantly stirred with a magnetic stirrer rotating at about 10 Hz. The sample illumination is realized by a He-Ne laser and the illumination intensity is adjusted by a polarizer plate. The laser beam is optically expanded to a diameter of ca. 1.5 cm and passes an iris diaphragm so that only its central part is incident on the sample to guarantee a spatially uniform illumination.
For the ellipsometric imaging of the optical path through the oxide, light from a blue LED (λ = 470 nm) is first elliptically polarized by passing a linear polarizer and a zeroth order λ/4-plate. It then hits the sample surface at an angle close to the Brewster angle of the silicon|electrolyte system (≈ 70 • ). The reflected light passes another polarizer which converts changes in the polarization upon reflection caused by the optical path through the oxide into an intensity signal. Using a suitable lens, an image of this ellipsometric intensity distribution across the surface is then created on the CCD chip of a camera (JAI CV-A50). The data is then digitized using a frame grabber card (PCI-1405, National Instruments) and preprocessed and recorded by a Labview program. A sketch of the setup (with the optical components of the ellipsometric imaging path omitted) is shown in figure 1.
Before the experiments the silicon working electrodes Figure 1: Sketch of the experimental setup showing the arrangement of the three electrodes and the external resistance together with the optical paths for the sample illumination (red) and the spatially resolved ellipsometric imaging (blue). A cross section of the interface and the growth direction of the oxide is shown in the inset. A schematic of the optical components in the light path between the LED and the CCD can be found in [25] are cleaned by first rubbing them carefully with an acetone soaked tissue and subsequently immersing them acetone (p.a.), ethanol (p.a.), methanol (p.a.) and ultrapure water for 5 min each. For the analysis of the patterns the spatial distribution of the ellipsometric intensity at specified points in time is considered as well as the ellipsometric signal of one dimensional subsets of the electrode over the entire time window. Furthermore, the time series of the ellipsometric intensity at each individual point is Fourier transformed (FFT, MATLAB) and the spatial distribution of absolute value and phase of relevant Fourier coefficients is then used to gain insight into phase clustering behavior for these Fourier modes. Lastly, the data from a selected area on the electrode is Hilbert transformed (MATLAB) and the resulting analytical signal is used to visualize the different dynamics on the electrode.

Parameter variation
The occurrence of oscillations and the accompanying pattern formation for a given electrolyte depends on the respective strengths of the two coupling mechanisms, i.e., on the values of the external resistance RextA and the illumination intensity I ill . An examplary 2d parameter space is shown in figure 2. If the total coupling is too weak, i.e., at high I ill and low Rext, the system settles to a stable focus. Conversely, if the total coupling becomes too strong, the system relaxes monotonically to a stable node. If the nonlinear coupling is weak, i.e., for relatively high illumination intensities, the entire electrode oscillates uniformly reflecting a strong synchronizing spatial coupling. An example for such a uniform oscillation is shown in figure 3.
Uniform oscillations are also found in all oscillatory measurements at non-illuminated p-type silicon samples as discussed in detailed studies of such samples conducted previously in our group [25,26]. In addition to the uniformity, the shape of the oscillations for highly illuminated n-type doped silicon and p-type doped silicon samples are identical at otherwise matched experimental parameters and in both cases a minimal external resistance has to be introduced. This similarity in behavior can be fully attributed to the sufficiencyof the amount of holes in the valence band to maintain the current present in both cases, strongly corroborating the notion that the nonlinear coupling introduced by the restriction of holes is solely responsible for the emergence of patterns. Consequently, in the parameter space shown in figure 2 only at illumination intensities below an electrolyte specific threshold value of I ill = 2.5 mW/cm 2 pattern formation can be observed. If the linear global coupling is chosen too strong, the patterns vanish and uniform oscillations are observed although the current is still restricted by the illumination and an effect of the nonlinear coupling is thus still to be expected. This behavior is in accordance with literature where the global coupling through an external resistor tends to synchronize the oscillations, at least when the electrode potential acts as the activator [27,28]. As a consequence patterns are only found in the parameter region where the nonlinear coupling dominates the linear global coupling, i.e., in the lower left hand corner of figure 2. Most notably, here, sustained oscillations are observed even without an external resistance.

Pattern classification
Compared to patterns classically found in oscillatory media the silicon oscillator exhibits two characteristic and peculiar phenomena: First, in a large part of the pattern forming parameter region the global quantities, i.e., the total current j and the spatially averaged ellipsometric intensity ξ oscillate in a simple periodic fashion although local time series exhibit complex oscillation forms up to completely irregular ones. Besides, ξ may oscillate with a period-2 or irregularly. Second, typically different dynamic behaviors as for example cluster patterns and spatio-temporal chaos coexist on the electrode. We call a part of the surface with one distinct dynamics a region.
In the following we classify the patterns, in a first step, according to the spatially averaged ellipsometric intensity ξ into three categories: • Simple periodic oscillations in ξ, 2d simulations using the suitably modified complex Ginzburg-Landau equation with a nonlinear global coupling show striking similarities with the patterns observed in the experiments [29,30,31]. This opens a pathway to a possible deeper understanding of both the origin of the pattern formation and the coupling mechanism. Consequently, in the following the emphasis is first put on the analysis of the first class of patterns and the second and third class are treated as modifications of this first class and are discussed later.

Simple periodic oscillations in ξ
In a first step we now consider the patterns forming during simple periodic ξ oscillations depicted in the top of figure 4. The regularity of the spatially averaged signal implies that the spatial variations of the patterns cancel out on average. There are two dominant types of patterns forming. First, subharmonic phase clusters are found, where the ellipsometric intensity at each point shows oscillations with a strong frequency component at an integer fraction of the spatially averaged oscillation frequency. These points are arranged in phase domains where all points within a domain are synchronized and the distinct domains differ in their phase [29]. Typically but not exclusively, 2-phase clusters are found. The second type of dynamical behavior found are desynchronized oscillations with a broad distribution of oscillation frequencies.
Hence, at any given point the oscillation amplitude is irregular.
As already mentioned, typically the electrode forms different regions. In a region either one of the two patterns mentioned or a synchronized oscillation with the frequency of the spatially averaged signal exists. The occurrence of only one type of dynamical behavior on the entire electrode surface and any pairwise coexistence have been found in this study and are subsequently presented.

One region
First, we take a look at the patterns with only one region covering the entire electrode. We start by discussing 2-phase clusters. An example is shown in figure 5. As apparent from More insight into the details of the cluster dynamics can lytical signal of the points in the two areas indicated in figure  5 are arranged on a bar (blue and green points, respectively) whose center follows the analytical signal of ξ (black solid line). The end points of the bar, i.e., the points from the individual phase domains, trace a Moebius strip projected on the complex plane. This suggests that the subharmonic clusters result from a period doubling bifurcation of the uniform oscillation. The four snapshots in figure 7 are taken at consecutive extrema of ξ and the sequence of pictures thus shows that a full rotation of the bar is accompanied by two full rotations of its center.
The second type of dynamical pattern covering the entire active area observed is a desynchronized oscillation spreading across the entire electrode as shown in figure 8. Here, the remarkable case is shown where the ensemble of pointwise time series of the ellipsometric intensity across the surface is oscillating in a desynchronized manner, while in the spatially averaged signal these differences cancel out almost perfectly leaving a simple periodic oscillation in ξ and j. The corresponding full analytical signal of two sets of points on the electrode further illustrates this behavior. It is shown in figure  9. The trajectories of the individual points are uncorrelated, both within the two areas shown and between them, leading to broad point clouds circling the origin.

Two regions
A region of one distinct dynamical behavior is typically not covering the entire electrode. More often two such regions are observed to coexist on the electrode. In figure 10 an example for the coexistence of a 2-phase cluster with an approximately synchronous oscillation with the frequency of the spatially averaged signal is depicted. The two regions are clearly distinguishable in figure 10 b) with the 2-phase cluster showing a remarkably fine structured border between its two phase domains. Note that the spatially averaged signals still oscillate in a simple periodic manner.
The Fourier analysis of this pattern is shown in figure  11. Outside the clustered domain the subharmonic frequency is inactive as can be seen in figure 11 b). As in the experiment where the 2-phase cluster covers the entire electrode shown in figures 5-7, the domain walls are Ising-type walls. The analytical signal of points from three areas on the electrode is shown in figure 12. Two main results can be read from the analytical signal in figure 12. First, the center of the rotating bar formed by the two domains of the 2-phase cluster and their separating wall is now approximately stationary in the origin. This means that the underlying harmonic frequency of the spatially averaged signal is strongly suppressed in the region of the 2-phase cluster. Second, the trajectory of the points taken from the approximately synchronized region does not follow the trace of the spatially averaged analytical signal perfectly. This can also be seen in the one dimensional cut in figure 10 b) where the active phase seems to propagate across the surface. We treat this, however, as a minor variation of a perfectly uniform oscillating region. Actually, the region oscillating with the frequency of the spatially averaged signal in a synchronous manner tends to this wave-like disturbance easily.
An interesting point arises comparing the measurement where the 2-phase cluster covers the entire electrode shown in figures 5-7 to the measurement showing coexistence of the 2-phase cluster and synchronized, harmonic oscillation shown in figures 10-12. The transition from the former to the latter dynamical state is seemingly achieved by spatially decomposing the two active frequencies. This is best seen by the corresponding deconvolution of the movement of the bar in the analytical signal shown in figure 7 into a pure rotation of the bar around the origin and a pure center of mass rotation around the origin in the different regions on the electrode as shown in figure 12. The frequency ratio between the center of mass movement and the rotation of the bar remains unaffectedly 1:0.5. A further investigation of this possible deconvolution process would be desirable.
The next case frequently found in the experiments is the coexistence of a synchronized and a desynchronized region on the electrode. This dynamical state is typically called 'chimera state' and has attracted a lot of research interest recently [32,33,34,35,36,37]. While in most experiments in literature these states have to be initialized in a special way, their spontaneous occurrence in our system is especially interesting [38]. An exemplary 'chimera state' is shown in figure 13. Note that while a region of desynchronized local oscillations can easily be seen in figure 13 b), the spatial average is again unaffected. Furthermore, the region of synchronized oscillations seems to be less perturbed by the presence of the region of desynchronized oscillations than by the presence of the subharmonic 2-phase cluster as discussed for figure 10. The coexistence of the two dynamical regions is further emphasized in figure 14 where the analytical signal of points from three areas on the surface shows the qualitative difference between the synchronized and the desynchronized oscillations. Another type of coexistence of synchronized and desynchronized patches on the electrode arises in the case of the coexistence of a 2-phase cluster with a desynchronized region as shown in figure 15. Here, the individual domains in the 2-phase cluster are again synchronized, show a phase difference of π and are separated by Ising-type walls. This is further elucidated in the analytical signal obtained from points spreading across the Ising-type wall in figure 16 (blue and green points). Comparable to figure 7 these points form a bar that rotates at half the angular frequency of the ξ oscillation. In contrast, the points from the desynchronized region, displayed in red, spread out in a cloud. As this pattern state is again showing the coexistence of synchronized and desynchronized regions one might argue that it could also be called a 'chimera state'. It has to be noted that a slight period-2 perturbation is visible in the oscillation of ξ in figures 15 and 16. This point will be further discussed in section 3.3. The coexistence of 2-phase clusters with a synchronous simple periodic oscillation as well as chimera states discussed above have also been observed in two other contexts. Tinsley et al. observed them experimentally in a network of coupled chemical oscillators and reproduced the dynamics with a reduced two group model [35]. Here the two groups were obtained by the introduction of a non-local feedback. Furthermore, the uniform oscillation was always stable and thus special initial conditions had to be prepared. This is in contrast to the work described here as well as to the predictions of the modified complex Ginzburg-Landau equation where the patterns occur spontaneously in a purely globally coupled system [31].

Period-2 oscillations in ξ
A first deviation from the simple periodic oscillations in ξ is the occurrence of period-2 oscillations in ξ as shown in the middle of figure 4. In all experiments where this behavior is found it is accompanied by the formation of a 2-phase cluster region where one phase domain covers a larger area or has a higher amplitude in the subharmonic mode than the other. An example is

Irregular oscillations in ξ
The last type of patterns discussed here occurs accompanied by an irregular oscillation of ξ as shown in the bottom of   figure 19, can again be seen as a deviation from the case of simple periodic spatial average oscillations. Here the irregular oscillation of the spatial average is imposed on the entire electrode and the relative intensity of the maxima and minima of ξ in figure 19 a) can be traced in the synchronized region in the lower part of the one dimensional cut in figure 19 b). A striking similarity between this type of pattern and the patterns forming accompanied by a simple periodic oscillation in ξ is that the regions on the electrode retain their respective dynamical behaviors and sizes until the measurement is finished. Spatially uniform oscillations with an irregular amplitude are also encountered with p-type [25,26] and highly illuminated n-type silicon samples, i.e., in the absence of the nonlinear coupling. For this reason the measurement shown in figure 19 can be seen as a deviation from the 'chimera state' shown in figure 13. The former is caused by the nonlinear coupling imposed on a spatially uniform oscillation with irregular, most likely chaotic, dynamics, while the latter is caused by the nonlinear coupling acting on a simple periodic, spatially uniform oscillation.
The other two categories of patterns forming during os- cillations of ξ with an irregular amplitude are qualitatively different from all other types of patterns considered so far. The most important difference is that the regions spontaneously forming on the electrode, while still well distinguishable, show dynamical behavior varying with time. A first example of such spatio-temporal pattern formation is shown in figure 20. One important feature the spatio-temporal pattern shown in figure 20 shares with all spatio-temporal patterns shown so far, is that the regions with distinct dynamical behavior do not change their shape as is well visible in the temporal evolution of the cross section in figure 20 b). Conversely, the dynamical behavior in the region in the middle of the cross section changes significantly twice on a long time scale from a completely desynchronized oscillation pattern to a nearly synchronized oscillation pattern with a significantly lower amplitude and back. These profound changes in the overall behavior are also visible in the time series of ξ.
Another qualitatively different behavior where the walls of the regions themselves are moving is shown in figure 21. In this example the respective sizes of the two regions distinguishable on the electrode surface change as one region first grows at the expense of the other to completely vanish once it fills up the entire electrode. It is remarkable in this example how long the memory of the surface seems to be as the period of the growing process is about 30 times larger than the base oscillation period.

Summary and Conclusions
In this work we presented a variety of self-organized spatiotemporal patterns forming in the ellipsometric intensity during the oscillatory photoelectrodissolution of n-type silicon under limited illumination.
Remarkably, complementary regions of distinct dynamical behavior formed spontaneously under completely uniform experimental parameters and without any external feedback. We classified these patterns by the behavior of the spatially averaged oscillation of the ellipsometric intensity ξ, the dominant case being a simple periodic oscillation of ξ. In this case, the patterns included synchronized oscillations, subharmonic 2-phase clusters emerging from the uniform oscillation by a period doubling bifurcation and desynchronized oscillations. Any pairwise coexistence of such dynamical regions was found including so-called 'chimera states'. The conservation of the simple periodic oscillation of ξ and j suggest the presence of a nonlinear global coupling. Indeed, the patterns only occur when the total coupling was found to be dominated by the nonlinear coupling imposed by the limited illumination. However, whether the coupling induced by the limited illumination of the sample is indeed global, remains an open question at this point. In addition to the case of simple periodic oscillations in ξ, patterns accompanied by period-2 and irregular oscillations in ξ have been presented. In the former case the corresponding pattern forming could always be linked to the occurrence of an unbalanced subharmonic 2-phase cluster. The latter case opens a wide variety of possible patterns which were classified by the stability of the region walls and their long term behavior but not yet studied in detail.
The study reveals important open questions concerning the dynamics of silicon electrodissolution as well as universal laws governing pattern formation in oscillatory media. Among the former is the physical mechanism which leads to the conserved oscillation of the spatially averaged quantities. This should be closely related to the dynamics of holes under illumination limited reaction conditions. As for the latter, the conserved oscillation of the spatially averaged quantities seems to strongly promote the spontaneous emergence of the coexistence of different dynamical states on the electrode. In a wider sense, this can be seen as diversification of the dynamical behavior of a system with uniform parameters. The role of self-organization for the emergence of differentiation in uniform systems is a central problem in non-linear dynamics, having implications in diverse fields such as biology or sociology, e.g., in decision making processes.