Campus Extens Introducció

The difficulty in creating a realistic microscopic configuration from a few coarse observables is strongly dependent on the nature of the system under consideration. By definition, the problem is under-determined; the key is to lift to a micro-configuration in which, at most, only the ‘unimportant’ (fast-relaxing) aspects of the configuration are incorrect.

In addition to these generic issues, the evolution of amorphous Ge islands on an amorphous SiO2 substrate poses unique challenges. To illustrate these challenges,

consider a situation in which the macroscopic observable at some time t, X t( ), is the

complete island size distribution (ISD), i.e., the number of islands at every size. On one hand, this is a detailed collection of coarse observables, and in fact, it would be quite difficult to estimate derivatives in time for the number of each of the island sizes from microscopic simulations because of stochasticity in these quantities. On the other hand, this information, in many respects, is still insufficient for lifting because of the amorphous nature of the system. First, the morphology of each island at each size is not known, and placing configurationally unrealistic islands on the surface would underestimate their stability. Second, we have shown previously [111] that the a-SiO2

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Again, placing islands at locations that are not energetically favorable will tend to produce unstable islands and an inability to maintain consistency with the slow manifold.

Next, we present a protocol to lift a macroscopic configuration, defined by a given ISD, to an atomistic one. As noted above, a crucial aspect of the lifting procedure for the present situation is the ability to construct realistic amorphous Ge islands. Samples of islands from direct MD simulations of deposition exhibit a rather wide range of island shapes, which alter the island capture zones and may therefore be important in the evolution of the system. To address this, we implemented a database approach in which a library of Ge island configurations was compiled from the direct MD deposition simulations reported in Chapter 3. The MD deposition simulations used to construct the library of Ge island configurations were performed at deposition fluxes of 4.14×1024

atoms/cm2s, 1.38×1023 atoms/cm2s, 2.76×1022 atoms/cm2s, and at temperatures ranging

from 2000K to 2200K. At each deposition condition, about 250 snapshots of the system were used to inform the library of island configurations. The procedure was as follows. First, individual Ge islands were identified in a given system snapshot using the Stillinger criterion. For each island, the particle with the lowest z-coordinate was set as the origin, and the positions of all other particles were adjusted accordingly to maintain their relative positions. The particle coordinates after adjustment, along with the island size, were recorded in the library of cluster configurations. Overall, the island morphology library contains O(105) configurations with island sizes ranging from 2 to 153. Some example

configurations taken from the configuration library are shown in Figure 4.2. The islands exhibit a wide range of morphologies, particularly at smaller sizes, and include both compact clusters and long, extended structures. The shapes tend to become more hemi- spherical at larger sizes.

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Figure 4.2. Example Ge island morphologies from the library of Ge island configurations collected from direct simulations of Ge deposition on a-SiO2. Particle rendering (here

and in the remainder of this document) performed using the OVITO visualization package [128].

Given a ‘target’ ISD, the lifting procedure was executed by first randomly selecting Ge island configurations from the library of configurations according to the sizes required by the ISD. For every selected island configuration, the particle with the lowest z-coordinate was placed at a randomly generated xy-position, (xi, yi) over the a-

SiO2 surface. The island size was characterized by computing its maximum lengths

along the x- and y-directions, L_x and L_y, respectively. A rectangular region with area (L_x10) ( L_y10) Å2 was then centered at the island center-of-mass. The z-coordinate

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of the lowest atom in the island then was initialized to be 5 Å above the maximum height of the existing atoms inside this rectangular region. The atoms in the island were then assigned a downward velocity corresponding to the system temperature.

The island initialization procedure was repeated for all islands needed to fulfill the target ISD. Island overlapping was prevented by first defining a circular area around each island with diameter equal to the maximum distance between any two atoms in the island. Inter-island spacing was enforced by requiring that each of circular domains be at least 6Å away from any other circular domain. As each island was created sequentially, any conflict with existing islands resulted in a destruction of the island and a resample of the ISD to generate a new one.

The system was subsequently evolved with MD in the NVT ensemble for 0.3 ps, allowing the islands to establish interactions with atoms on the surface. During this procedure, all surface atoms (Si and O particles) remained fixed. Once all islands had made contact with the surface, all surface atoms (except for the ones in the bottom 1 nm layer of the a-SiO2 substrate) were released. The system then was further relaxed with

energy minimization. Finally, the system was subjected to an NVT-MD anneal at the deposition temperature for an additional 2 ps to stabilize the islands on the surface. Due to the strongly heterogeneous binding environment of the a-SiO2 surface [111], some Ge

islands were found to quickly desorb from the surface, moving the ISD away from the target value. If this was the case, additional islands were introduced to replace any islands that desorbed. In this procedure, the newly introduced islands were brought down to the surface using another 0.3 ps-long NVT anneal. Here, only atoms in the new candidate islands were free to move while all other atoms, including those in existing (stable) islands, were kept fixed. The energy minimization and finite temperature

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annealing procedure described above was then reapplied to the entire system. The entire process was repeated until no islands desorbed during the relaxation protocol and the ISD matched the target distribution.