Supplementary material
Dynamic competition between island growth and coalescence in
metal-on-insulator deposition
B. Lü, V. Elofsson, E.P. Münger, and K. Sarakinos
Department of Physics, Chemistry and Biology, Linköping University, SE 581 83, Linköping, Sweden
1. Additional simulation details
The islands are forced to maintain a hemispherical shape throughout the simulation. Therefore, an
island refers strictly to an object which has the shape of a hemisphere. During growth, an atom
attaching to an island is immediately relocated to a position on the island surface that best retains the
hemispherical shape. During coalescence between two islands, their individual hemispherical shapes
are maintained until the coalescence clock runs out. In other words, the processes of neck formation
and shape equilibration commonly associated with coalescence are not treated atomistically.
While the coalescence clock is running, atomistic growth of the islands is allowed to progress as
normal. This means that the total coalescence time of each pair of islands is updated as the sizes of
these islands are increased (strictly, as the size of the smaller island is increased). Several islands may
impinge on each other at the same time to form a single cluster, but coalescence clocks are only started
for each unique pair of islands.
In general, a cluster refers to an agglomeration of islands linked to each other through impingement,
which means a cluster may have a randomly ramified shape. However for the sake of simplifying
calculations, all single, isolated islands are also counted as clusters of size one. In this way, the
average cluster size can be calculated by taking the total number of clusters divided by the total
number of individual islands. This guarantees that when the average cluster size crosses the value two,
the surface is predominantly covered by impinging islands (as opposed to isolated islands), and the
elongation transition is said to be reached (see Fig. S1).
FIG S1. Average cluster size, calculated by dividing the island density by the cluster density, as a function of material
coverage. The dashed lines indicate the average cluster size of 2 and the corresponding material coverage, which
signifies the coverage at the elongation transition.
2. Additional experimental details
Film growth
All experiments were conducted in a vacuum chamber with a minimum base pressure of 1.3×10-6 Pa at
an Ar buffer gas (99.9997 % purity) pressure of 0.67 Pa. Films were grown on electrically floating Si
(100) substrates covered by a 3000 Å thermally grown SiO2 layer from a circular Ag target (99.99 %
purity) with a diameter of 75 mm. Radiative heating from the target region was minimized by placing
the substrates at a distance of 12.5 cm away from the target, as well as by keeping the power supplied
to the target at low values. For each growth temperature (300 and 330 K), films were grown at various
deposition rates F using continuous vapor fluxes generated by direct current magnetron sputtering
(MDX 1K, Advanced Energy Industries Inc.).
In situ growth monitoring
The effects of temperature and deposition rate on film growth were monitored in situ using
spectroscopic ellipsometry (M-88, J.A. Woollam Inc.) with a rotating analyzer. Ellipsometric data
were acquired every ∼0.12 ML (nominal thickness) for 67 wavelengths in the spectral range 1.63.2
eV at an incidence angle of 65°. The substrates were modeled as 1 mm thick Si slabs with a top layer
of SiO2 using optical data from Ref 1.1 The thickness of the latter was determined by recording a
spectrum of the substrate prior to deposition.
Continuous and close to continuous films were modeled as perfect metals using the Drude free
electron theory where the complex dielectric function, 𝜀̃, as a function of the energy of the incident
light, 𝜔, is given as2
𝜀̃(𝜔) = 𝜀∞ −
𝜔𝑝2
𝜔 2 + 𝑖Γ𝜔
Here, 𝜀∞ is a constant that accounts for interband transitions occurring at higher energies that
otherwise are not accounted for in the model, 𝜔𝑝 is the plasma frequency and Γ represents damping of
the electron oscillations. Using the parameters that globally minimizes the mean square error when
fitting the model to experimental data it is possible to calculate the film resistivity, 𝜌, as3
𝜌=
Γ
𝜀0 𝜔𝑝2
where 𝜀0 is the permittivity of free space. 𝜌 was then plotted as a function of the nominal thickness,
FIG S2. Nominal thickness as a function of resistivity ρ. The point used to demark the formation of a continuous
film 𝜃𝑐𝑜𝑛𝑡 , is depicted in the figure.
see Fig. S2, where 𝜌 is seen to decrease quickly as the nominal thickness increases before reaching a
steady-state value. The nominal thickness at which a continuous film is formed, θcont , was taken as the
nominal thickness where the steady-state behavior first was obtained.4 The nominal thickness was
calculated from a continuous film by assuming the same growth rate throughout all growth stages.
The percolation transition was monitored using a graphical method developed by Arwin and Aspnes. 5
It utilizes an absorption feature of the substrate in the measured range to concurrently determine film
thickness and 𝜀̃. This is accomplished by fitting 𝜀̃ to the experimental data using an assumed film
thickness. If the thickness is incorrect the substrate feature shows up as an artifact in 𝜀̃. By minimizing
or eliminating the artifact both the correct film thickness and 𝜀̃ are determined simultaneously.
Following this procedure for each acquired set of data the evolution of 𝜀̃ with increasing thickness was
obtained. As the real part of 𝜀̃ first becomes negative in the infrared (low energy) part of the spectrum,
the film starts to show metallic behavior.6 This point was used to demark the nominal percolation
transition thickness, θperc.
3. The jamming limit and surface coverage at the elongation transition
When sequentially placing circles at random positions on a square without overlapping (random
sequential absorption, RSA), the jamming limit is reached when there is no longer any space available
to place a circle without overlapping older ones. This type of problem was solved during the 1970’s
and 1980’s, when it was found that the surface (or areal) coverage converged to a characteristic value
of 0.547 at the jamming limit.7–10 By fixing the density and position of a collection of circles at this
limit and increasing the size of all circles by ≈ 20%, percolation was found to occur in ref. 7. Since the
elongation transition is considered to be the precursor to percolation, it is reasonable to believe that the
elongation transition occurred sometime earlier during this size increase.
The concept of the jamming limit coincides with a stage in the growth of water droplets condensed on
a cold surface. After initial formation of small, well separated droplets, these grow to enter a time
regime in which instantaneous coalescence upon contact between droplets becomes dominant. This
regime has the attribute of maintaining a constant surface coverage of 0.55, as material is redistributed
among the droplets, and the droplet pattern remains self-similar in time.11,12 Once the amount of
surface revealed by coalesced droplets becomes significant, secondary nucleation occurs, and the
surface coverage begins to increase again.
The model of thin film growth and nucleation described in the main article differs from the droplet
condensation case in one key aspect: coalescence is not instantaneous. However the two cases can be
considered identical in the pre-coalescence growth regime. In fact, in deriving the rate equations for
3D growth, the surface coverage at the saturation island density is taken as roughly 0.5.13
From the droplet condensation case, it can be understood that coalescence should begin to dominate at
a surface coverage of 0.55, which is slightly larger than the coverage at island density saturation.
Figure S3a) shows simulated island densities at the elongation transition (symbols) from this work
compared to calculated saturation island densities (lines) based on the atomistic nucleation theory. It
can be seen that the symbols and lines only coincide in the coalescence-free regime. If no coalescence
events are allowed to complete, it is reasonable to believe that a small expansion of the islands will
cause the elongation transition. Based on this line of argument, it is reasonable to assume that the
surface coverage will always converge to 0.577 in the impingement-controlled regime, as shown in
Fig. S3b).
FIG S3. Simulation data (symbols) taken at the elongation transition for simulation set 2 (constant B), plotted against
deposition rate. a) The island density compared to calculated saturation island densities (lines) b) The surface or areal
coverage. The dashed line indicates the characteristic surface coverage at the elongation transition in the impingementcontrolled growth regime. The inset in b) shows the elongation transition thickness, and the solid line highlights the
coalescence-controlled regime (θelong ∝ F-1/3).
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