Experiment and Monte Carlo simulation of Ni(977)
| Isolated single step | Frustrated dead end |
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| Figure 1 | |
In order to make effective use of the extreme density of nanoscale elements that form spontaneously in self-assembling architectures, one must address the associated issue of minimizing defect creation during the formation of such structures. In this paper we examine the competing roles that nucleation kinetics and two-dimensional growth processes play in nanostructure formation and defect minimization. We employ oxygen-induced step doubling of vicinal Ni(977) surfaces as our physical system, using elevated temperature scanning tunnelling microscopy and Monte Carlo simulations to extract the desired details of interface evolution.
The oxygen-driven reconstruction dynamics and step merging mechanism of Ni(977) have been previously studied using both HAS and STM. It was found that steps commence coalescence at a point contact (i.e., a step-edge bulge connects to its downstairs neighbor step-edge); doubling then proceeds via bi-directional zippering. Under optimized oxygen coverage (step-edge saturation) at 465 K, the step doubling linear rate is ~3.7 Å s-1 corresponding to an areal sweep rate of ~60 Å2 s-1 (the width of the propagating step terrace is ~16.5 Å).
With the dynamics and the mechanism of the reconstruction understood, we focussed on several features that remained on the surface after the oxygen-driven step doubling reached its asymptotic limit. There are two possible defects that hinder perfect doubling: frustrated "dead ends" and isolated "2-1-2" single steps. Assume that there are initially three single steps: 1, 2 and 3. When the oxygen-induced doubling begins, steps 1 and 2 start to merge and begin growing from top to bottom. Meanwhile, steps 2 and 3 start to merge and begin growing from bottom to top. Their intersection forms a frustrated dead-end. Experimentally, such frustrated ends are observed to remain stationary once formed, hence why they are used as internal references to monitor surface morphology evolution. For the isolated single steps, assume that initially there are five single steps: 1, 2, 3, 4 and 5. When steps 1-2 and 4-5 are doubled, 3 remains as an isolated single step having no neighbor with which it can merge.
Fig. 1 contains schematic pictures of the frustrated end and isolated single step anomalies. These two features are important since, ideally, they are the only possible defects that can inhibit the formation of a perfectly doubled surface. Studying the relationship between their formation and the experimental conditions helps further our understanding of step doubling and facetting phenomena, enabling the production of more perfect surface patterns for use as nano-scale templates.
Fig. 2 contains two final-state images that resulted from two different initial oxygen exposures. The frustrated ends are marked with circles. Small oxygen exposure (0.0375 L) results in only one frustrated end, while larger oxygen exposure (0.15 L) leads to five frustrated ends. The ultimate number of frustrated ends depends on both the nucleation rate (the rate for generating the initial contact points) and the step zippering rate. It is clear that the initial oxygen exposure is proportional to the ultimate number of frustrated ends. Even before the oxygen exposure reaches its optimal value (0.15 L), additional oxygen increases the probability of forming contact points between two neighboring steps, and therefore increases the probability of forming frustrated ends. When the oxygen exposure is beyond the optimal value, the nucleation rate remains virtually unchanged, but the extra adsorbed oxygen hinders step zippering. In effect, this too increases the probability of forming frustrated ends.
| Figure 2 |
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| O2 exposure: 0.0375 L Total Number of frustrated ends: 1 |
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| O2 exposure: 0.15 L Total Number of frustrated ends: 5 |
In addition to dead-ends, isolated single steps also remain on the surface. The ultimate isolated single step populations for experimental images (100 nm x 100 nm) range from 11% to 17% with an average of 15% and do not show an obvious dependence on the initial oxygen exposure. Under ideal conditions (i.e., no frustrated ends), the best result will be a perfectly doubled surface with 0% isolated single steps and the worst result will be 20% isolated single steps. A simple statistical model using various (1 x N) matrices, with N ranging from 101 to 105, was used to determine the expected single step population excluding the effect of frustrated ends. Statistically, one expects approximately 13.5% isolated single steps, agreeing well with the observed population.
Monte Carlo simulation of the step doubling phenomenon was performed using a simple physical model based on extant experimental understanding for oxygen-driven Ni(977) step doubling. Our simulation study focusses on the final state of the surface, in particular defect content, in order to understand the relationship between experimental conditions and surface structure upon reaching an asymptotic limit.
An (N x N) matrix was used to mimic the Ni(977) surface, where N is a positive integer. Every column of the matrix is regarded as a single surface step. Two experimental conditions were set as parameters: nucleation rate (the rate for generating the initial contact points) and step zippering rate after the first point contact was made. Since the nucleation rate and the zippering rate are both related to the local oxygen coverage at the step-edges and the temperature, it is possible to connect the simulation results to the experimental findings to further our understanding and guide future experiments.
Fig. 3 plots the total number of frustrated ends as a function of the nucleation and zippering rates. The simulation results exhibit a clear effect: the higher the nucleation rate, the more dead ends; the higher the zippering rate, the fewer dead ends; and if the ratio of the nucleation rate and the zippering rate is kept constant, the larger their absolute magnitudes, the more frustrated ends. Also, the ultimate number of frustrated ends is not as sensitive to the zippering rate as it is to the nucleation rate. The relation between the nucleation rate or the zippering rate and the total number of frustrated ends is as expected. The nucleation rate increases as the step mobility increases, that is, the higher the possibility for two steps to stick together and to possibly form a frustrated end. A faster zippering rate will allow the step doubling to be completed in a shorter time, thereby minimizing the number of nucleation sites and decreasing the ultimate number of frustrated ends. However, the result that the total number of frustrated ends depends on the absolute magnitudes of these two rates (with their ratio constant) is somewhat unintuitive. The explanation lies in the physical difference between these two rates. The nucleation rate can be understood as a two-dimensional parameter, while the zippering rate is limited to only the step-parallel direction. These dimensional differences lead to different scaling behavior. Simulation results for the final percentage of isolated steps fluctuate between 13.0% and 14.5% and do not show a strong dependence on either nucleation or zippering rate. This result agrees well with the population predicted by the one-dimensional statistical model described above.
| Figure 3 |
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By comparing experimental data with Monte Carlo simulation results, a physical calibration can be applied to the simulation parameters. With this correspondence in hand, it will be possible to design future experiments that incorporate optimized substrate conditions. It is known that when the initial oxygen exposure is 0.15 L, the step zippering rate is 3.7 Å s-1 at 465 K. When the step doubling reaches an asymptotic limit, the total number of frustrated ends is 5~6 on a 100 nm by 100 nm surface sample, and the final percentage of isolated single steps is ~15%.
Since the final percentage of isolated single steps is not sensitive to either the nucleation or the zippering rate, we use the total number of frustrated ends as the comparison criterion. To achieve 5~6 frustrated ends on a 100 nm x 100 nm area, the corresponding simulation parameters for a 60 x 60 matrix are a nucleation rate of 1 and a zippering rate of 43. Assuming each element of the matrix is a square, the length of each element will be 16.5 Å, which is equal to the Ni(977) terrace width. By comparing to the experimental zippering rate, one assigns every program cycle to 194 seconds at 465 K, the experimental temperature. Hence, on a 100 nm x 100 nm Ni(977) surface, we would expect to see about 19 nucleation sites (point contacts) within one hour under these conditions. Given the timescales of imaging, it is unfeasible to catch the formation of every individual nucleus. If multiple nuclei form on a certain step, the exact number of them often cannot be determined by examining discrete STM images. The simulation, therefore, enhances our understanding of this phenomenon by providing insight into the true nucleation rate.
By analyzing the Monte Carlo simulation, the effects of both the nucleation rate and the zippering rate on step doubling are better understood. If the goal is to make a perfectly doubled Ni(977) surface, the key factor is to minimize the total number of frustrated ends while enabling the most doubling to occur. In order to prevent the formation of these defects, a low nucleation rate and a high zippering rate are required. For a doubling event to occur, the steps need to be mobile enough to have a significant probability of overcoming entropic repulsions. Nucleation and subsequent zippering then occur provided that both steps are decorated with oxygen. In order to minimize the number of frustrated ends, the nucleation events need to be controlled. By using a small initial oxygen exposure, the nucleation rate could be reduced. However, this will result in incomplete step doubling since the total amount of oxygen would not be enough to cover all the step-edges. Mastering this balance is key to generating a more perfectly doubled surface.
Since the simulation results show that the final number of frustrated ends is not as sensitive to the zippering rate as it is to the nucleation rate, lowering the temperature will likely be effective in enhancing the extent of perfection of the asymptotic surface. The steps will zipper slightly slower than at the upper end of the temperature regime, but the nucleation rate will be affected far more dramatically, thereby minimizing the number of frustrated ends. Preliminary experiments testing this hypothesis drawn from our simulations have substantiated this methodology: One can gain conversion with only mild increases in defects up to a temperature of approximately 425 K. At higher temperatures, there is little further net conversion, yet many more defects are introduced.
In all, we have critically examined, combining experiment and numerical simulation, the key control parameters which
govern the formation of relatively defect-free modified-structure vicinal surfaces. We have employed oxygen-induced
step doubling of Ni(977) surfaces as our physical system using elevated temperature scanning tunnelling microscopy
and Monte Carlo simulations to extract the desired details of interface evolution. These studies have elucidated
the competing roles that nucleation kinetics and two-dimensional growth processes play in nanostructure formation
and defect minimization. Elevated temperature STM has been used to probe the oxygen-induced reconstruction behavior
of Ni(977). Special attention was paid to characterizing the morphological features which remained on the surface
after the step doubling process reached its asymptotic limit.
93. "In search of nano-perfection: Experiment and Monte Carlo simulation of nucleation-controlled step doubling"
