MORPHOLOGICAL CHANGES TO InAs ISLANDS CAUSED BY RIPENING
After the initial material deposition to form islands, if the surface is kept at an elevated temperature after growth, the material on the surface remains dynamic. After island formation and growth has occurred, islands are at a high-energy state. To lower this energy, islands can redistribute their material into fewer islands so the energy cost associated with the island edges and surfaces is lowered. This process is called ripening. While we observed two different shapes for InAs islands on GaAs we also observed a change in the distribution of these two shapes with annealing. Based on the correlation between shape and volume, and the increase in island volume with ripening, this might well be expected.
Island Volume
The histogram charts showing the number of islands per volume for each
shape are displayed in Figure 7. It should be noted that the
number of islands for each chart is unimportant since there were a different
number of samples taken for each anneal time. This means for some anneal
times, more islands were available to be counted and measured. The charts
show that for each anneal time up to 3 minutes, the smaller islands are
type-I shaped and the larger islands are type-II shaped. The hybrid-shaped
islands lie between the type-I and type-II islands. However, there is
not one critical volume, Vcr, at which the islands transform from type-I
to type-II. The critical volume increases with increased annealing.
This further explains why only type-I islands are observed at 6 minutes.
If Vcr is increasing with increased annealing, then at 6 minutes the critical
volume increases past the largest volume in the population of islands.
Therefore, all islands would have volumes less than the Vcr and would
not exist in the type-II variety. This idea agrees with the intermixing
that is observed with increasing anneal time. The intermixing is a strain
relief mechanism that causes the critical volume to increase. With increased
intermixing, strain does not have to be relieved as quickly by transitioning
from type-I islands to type-II islands. At 1 minute of annealing, a bimodal
distribution begins to appear, with the lower peak being composed of type-I
islands and the upper peak being composed of type-II and hybrid islands. At
3 minutes of annealing, the bimodal distribution is even more pronounced
with the smallest peak consisting of type-I islands and the largest peak
consisting of mainly type-II islands. These results are similar to the
study mentioned in section by Ross et al. which reported a bimodal distribution
caused by a shape transition in the ripening of Ge islands grown on Si.
Island Height 
Figure 8 shows histogram charts for the number of islands per height for each island shape. The shorter islands are type-I and the taller islands are type-II, while hybrid islands again rest in the middle of the distribution. The average height of the islands increases from 0 to 3 minutes, with the critical height at which type-I islands transform to type-II islands also increasing. Just as with the island volumes, there is also a bimodal distribution of island height. At 3 minutes of annealing, the smaller peak in the distribution consists of type-I shaped islands while the larger peak consists mainly of type-II and hybrid shaped.
Island Aspect Ratio
The island aspect ratio (height/width) is shown for each island in the
histogram charts in Figure 9. The trend is similar to that of the
volume and height of the islands. The smaller islands have a smaller
aspect ratio while the larger islands have a larger aspect ratio. At
3 minutes there is again a bimodal distribution where the lower peak is
composed of type-I islands and the upper peak is composed of type-II islands.
A difference in the aspect ratio and the volume and height of the islands
is that the critical aspect ratio seems to remain constant at around 0.36
with each anneal time. So, whereas the critical volume and height at
which type-I islands become type-II islands increase with annealing, the
critical aspect ratio seems to remain constant. At 6 minutes, notice
how all the islands have aspect ratios below the estimated critical aspect
ratio of 0.36. Since they are below the critical value, they are all
type-I shaped. In this way, annealing has provided a means of selecting
Type I over Type II shaped 3-D islands.
Ripening Models
There are at least two very different kinds of ripening, Ostwald Ripening (OR) and Smoluchowski Ripening (or sometimes referred to as cluster diffusion). The main difference between Ostwald ripening and Smoluchowski ripening is that only for the former process, mass transport between islands is governed by a thermodynamic driving force (i.e., the chemical potential difference between islands of different size).
Ostwald ripening theory describes a ripening process in which the islands do not move on the surface, but atoms move to and from islands. The mass transport between islands occurs between a 2D gas of atoms. For SK growth of islands, it is thought this 2D gas of atoms is the wetting layer on which the islands rest. The physical mechanism for the occurrence of mass transport in Ostwald ripening is the energy associated with the curvature of the islands’ surfaces. Islands with small curvature (lower kink density) are at a lower energy configuration than islands with large curvature (higher kink density), so atoms migrate from the large curvature islands to the small curvature islands. This means migrating material favors a larger island since the larger island has less curvature.
Meanwhile, mass transport between islands during Smoluchowski ripening is a result of stochastic island diffusion. Only the fact that two accidentally colliding islands irreversibly combine into one large island directs the process towards a state of lower energy.
An important aspect of ripening is the distribution of island sizes. From island distribution, information about the growth can be known—such as what physical process limits the ripening. For Ostwald ripening, the distribution can be calculated from the continuity equation as follows:
(1)
where F(r,t) is the island size distribution function, r is the island radius, and v(r)=dr/dt. It is found that at long times, t, the critical island radius, rc, at which an island either loses or gains material obeys a scaling relation rc~tß where ß is called the scaling exponent. For a rate-limited step of detachment at the island edge, ß is equal to 1/2. Solving this partial differential equation with the proper boundary conditions leads to the distribution function for detachment limited Ostwald ripening given as
(2)
where rs is the standardized island radius, rs=r/ravg. Figure 10 shows the plot of this distribution function as well as the others to be discussed below.
For the case where the ripening process is limited by the diffusion of atoms in the 2D gas phase (diffusion limited), ß is equal to 1/3 and the island size distribution function becomes:
(3)
where again rs is the standardized radius and C is a scaling constant. Again, refer to Figure 10 to see what this distribution looks like. From this figure, it is easy to see that the diffusion-limited Ostwald ripening process leads to a distribution that is more narrow than detachment limited.
Smoluchowski ripening, or cluster diffusion (CD), is similar to Ostwald ripening in dynamic properties in that both processes slow down as the average island size increases. The main difference between Ostwald ripening and CD is the mechanism of mass transport. As mentioned, the mass transport in Ostwald ripening occurs through a background 2D gas of atoms. During CD ripening, mass transport occurs through island diffusion and combination. Unlike Ostwald ripening where islands remain in a fixed location, in CD islands move during the process. Islands get larger as they collide and combine with each other. Because of this, CD is a purely accidental and random process.
Just as the island size distribution function was calculated for Ostwald ripening, it can also be calculated for CD ripening. To do this, the Smoluchowski equation can be used and is given as follows:
(4)
where cN is the concentration of islands with size N, and the kernel Ki,j is the collision rate constant between islands of mass i and j. The first term is due to the creation of islands of size N=i+j, while the second term is due to coalescence of islands of size N with islands of size i. The diffusion kernel, Ki,j, is often chosen as the Brownian diffusion kernel and is given by
(5)
where d=2 is the spatial dimension of flat islands on a surface and Di is the size and temperature dependent diffusion coefficient for an island with size i. It is possible that the diffusion coefficient shows power law scaling such that DN=D0N-a for large N. The mean island radius then scales as ravg~tß where ß =1/d(a+1). The distribution function can be shown to be given by
(6)
where W=G(a+1+1/d)/G(a+1) and G is the gamma function. The single parameter a determines the CD coarsening characteristics and is related to the atomic mechanism of island diffusion. For periphery diffusion a=3/2 giving ß=1/5. Terrace diffusion (or correlated evaporation-condensation diffusion) gives a=1 and ß=1/4. Uncorrelated evaporation-condensation diffusion gives a=1/2 and ß=1/3, the same value as the diffusion limited Ostwald ripening case. Figure 10 shows the distribution functions for each ripening theory discussed. The distribution function for cluster diffusion in the plot is for the terrace diffusion (a=1) case.
Determination of Ripening Model
One interesting question centers on which ripening model would more closely
fit the ripening of InAs islands on GaAs. To determine
this, a set of histogram charts was created for V1/3, which is proportional
to the island radius. The charts have the type-I islands grouped together
and separated from the hybrid and type-II islands that are also grouped
together. The theoretical ripening distributions were compared with the
experimental data to determine which mode was more closely being followed
for each group. The charts are shown in Figure 11.
During all four annealing times, the Ostwald ripening process seems to be taking place. Again, this means that ripening occurs as stationary islands exchange material with a background 2D gas of atoms. The exchange occurs from islands with larger curvature (more energy, smaller islands) to islands with lower curvature (less energy, larger islands). However, the limiting step in each anneal time seems to be different depending on which island shape is being examined.
During the 0, 1, and 3 minute anneal times, the type-I islands are detachment limited. The exchange of material is limited by the attachment and detachment of atoms at the perimeter of the type-I islands. This seems counterintuitive since the higher curvature type-I islands have more broken bonds at the surface and the atoms would want to detach from the surface more quickly. However, the perimeter of the type-I islands is small, so there is less room for attachment/detachment to occur, explaining why the type-I islands are detachment limited.
For the same anneal times, the hybrid and type-II islands are diffusion
limited. The exchange of material between hybrid and type-II
islands is limited by the diffusion of atoms in the wetting layer and
not by the attachment/detachment of atoms at the perimeter. Since the
hybrid and type-II islands are larger than the type-I islands, they have
a larger perimeter at which atoms can attach and detach (see Figure 12). Therefore,
the limited process is not in atom detachment. As islands get larger,
the inter-island spacing gets larger as well since the island density
becomes lower. This means that diffusion of atoms from island to island
becomes more difficult since the diffusion must take place over larger
distances. Based on this, one would expect that as islands get larger
and more spread apart they would become more and more diffusion limited.
At 6 minutes of annealing, all of the islands are type-I. For 0, 1 and 3 minutes, the type-I islands were detachment limited. However, it would be expected that as the islands become further and further apart due to ripening, they would also become more diffusion limited since the atoms would have to diffuse at larger distances to reach other islands.
For the data for 6 minute annealing, both the diffusion limited and detachment limited Ostwald ripening distributions are plotted along with the histogram. It seems that the islands are more diffusion limited than detachment limited after 6 minutes—even though they are type-I. Earlier, it was explained that the type-I islands have a smaller perimeter at which to exchange atoms so it made sense that they were detachment limited. However, the type-I islands at 6 minutes have undergone a good deal of intermixing and the perimeter of the islands has spread out and become larger (see Figure 12). Therefore it makes sense that detachment limiting Ostwald ripening would not play as big a role here as at earlier anneal times. On the other hand, it does make sense that since the islands are further apart the diffusion limited process would play a larger role.
In this study, size distributions were between 32.1 percent and 46.8 percent depending on the anneal time. The largest size distribution was due to a bimodal distribution of islands shapes. Therefore, one would want to avoid stopping growth and annealing where a bimodal distribution existed if a small size distribution was required. The smallest size distribution occurred at one minute of annealing. At six minutes the distribution decreased from the one at three minutes, and it is possible that it would have decreased even more in annealing times between six minutes and nine minutes after which the islands disappeared. The disappearance is also a surprise, since the islands are expected to grow as long as material could be exchanged between islands. If the islands had become so far apart that diffusion between them stopped, the islands would reach a self-limiting size. However, in this study neither of these conditions were observed. The islands reached a maximum size, and then started to decrease. Our understanding of this behavior is that the InAs began to intermix with the GaAs substrate.
The evidence for intermixing is strong. At six minutes, STM scans showed 3D islands with material spreading from the base preferentially in the [-110] direction. As InAs intermixed with GaAs, an alloy of InxGa1-xAs is formed that reduces the lattice mismatch between the islands and the substrate. The lattice mismatch reduced the strain between the two materials, and thus removed the requirement for InAs to form 3D islands to relieve strain. The result was that 3D islands began to transform to 2D layers as intermixing took place. Further evidence for intermixing is that after six minutes the type I volume is greater than the Type II volume after zero minutes. That is, the larger Type I volume is consistent with the lesser strain or lower lattice mismatch due to intermixing.
As discussed earlier, in this study, three faceted island shapes were observed. Small islands had a small facet angled shape (type-I) while larger islands had a larger facet angled shape (type-II). Islands in the middle of the size distribution had a shape that was a hybrid of the two others. From the historgrams, it is clear that a shape transition occurred during annealing. The transition occurred as a strain relief mechanism as larger facet angled shapes relieve more strain than smaller facet angled shapes. The critical size at which the shape transition occurred increased with longer anneal times. During all anneal times except six minutes, all island shapes were found on the surface. At six minutes, only type-I islands were found. The reason for this is related to intermixing occurring as a strain relief mechanism. As intermixing occurred more at longer anneal times, it was not required for the once type-II islands to keep that shape in order to reduce energy. Since intermixing relieved the strain in the islands by reducing the lattice mismatch, the type-II shaped islands could become type-I islands once again. The fact that all islands are of one shape at six minutes of annealing is promising. Since electronic properties of QDs are partly dependent on the island shape, a single shape among the island population is desired for uniformity.
