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Two-Dimensional Island Shape Determined by Detachment

Effect of an anisotropic detachment on a heteroepitaxial island shape is studied by means of a kinetic Monte Carlo simulation of a square lattice gas model. Only with molecular deposition followed by surface diffusion, islands grow in a ramified dendritic shape, similar to DLA. Introduction of molecular detachment from edges makes islands compact. To understand an anisotropic island shape observed in the experiment of pentacene growth on a hydrogen-terminated Si(111) vicinal surface, asymmetry in detachment around the substrate step is assumed. Edge molecules detach more to the higher terrace than to the lower terrace. The island edge from which molecules are easy to detach is smooth and the one hard to detach is dendritic. If islands are close to each other, islands tend to align in a line, since detached molecules from the smooth edge of the right island are fed to the dendritic and fast growing edge of the left island.


Introduction
For the production of high quality devices, it becomes more and more important to control atomic structures on clean crystalline surfaces. There are many studies, for example, on a shape and density of two-dimensional (2D) islands which are nucleated and grown on a substrate surface by molecular beam epitaxy. [1][2][3][4] They are mainly determined through the competition between the diffusion of adsorbed molecules on a substrate surface and their incorporation kinetics at the island edge. If the molecule incorporation at edges are very fast and diffusion controls the growth, island edges undergo a morphological instability to form dendrites. In the extreme case of irreversible solidification, an island takes a ramified irregular shape with many branches, 5 similar to a diffusion-limited aggregate (DLA). 6,7 Its structure has no characteristic length and is fractal. On the other hand, if the incorporation kinetics is slow and limits the island growth, islands becomes compact. [8][9][10] On a vicinal surface steps exist and may bring about additional effects on surface structures. For a homoepitaxial case, they lead to surface roughening via various instabilities as meandering and bunching. [11][12][13][14][15][16] For a heteroepitaxial case, substrate steps may affect a shape of epilayer islands. It is in fact observed recently in a heteroepitaxial growth of a pentacene (Pn) submonolayer on a vicinal surface of hydrogen-terminated Si(111) [H-Si(111)] substrate. 18 The experiment is performed in an expectation that vicinal steps give control on the epifilm orientation. Pn wets the substrate surface completely and form 2D flat islands in a submonolayer coverage. When a terrace width is large on a vicinal H-Si(111) surface, Pn islands are isotropic and compact in shape. In contrast, when the slope of the vicinal substrate becomes steep and the step density increases such that an island covers many steps, the shape becomes anisotropic; on the higher side of the vicinal surface an island is compact and rounded, whereas on the lower side it is dendritic. The island shape is found to be independent of the impinging orientation of the Pn molecular beam. As for the origin of this shape anisotropy, no definite conclusion has yet been stated. 18 There is another experiment where anisotropy is observed in the shape of two-dimensional islands; a thin film of organic molecules on oxidized Ga-As substrate. 19 The substrate has no steps, and the shape anisotropy is attributed to the tilting of long organic molecules in a crystalline film. Precise mechanism to determine shape anisotropy for this sytem still awaits for an analysis.
Here we propose and study a possibility that may induce a shape anisotropy of islands; anisotropy in the rate of molecular detachment from an island edge. For example, on a vicinal surface, the rate of detachment from the edge of an island that climbs up a substrate step may differ from that from the edge of a descending island. Even on a flat substrate surface, if long molecules are adsorbed in a tilted conformation, detachment rate of molecules from the edge of an island may depend on a relative directions of tilting and migration.
Whatever the mechanism may be, an anisotropy in the detachment rate should be reflected in the island shape. This is the aim we address ourselves in the present paper. We consider a simple and thus generic model where point molecules are adsorbing on a flat square substrate, and study it by kinetic Monte Carlo simulations. 20 The model is described in §2. In order to understand the effect of detachment on the island shape, we first study the case of an isotropic detachment in §3. In §4 we describe the possible origin of anisotropic detachment referring to the Pn experiment by way of example, and study the island shape in an anisotropic case. The last section §5 summarizes the results.

Deposition and Diffusion on a Square Substrate
Our model consists of point molecules depositing on a flat substrate surface of a square lattice. They diffuse on the substrate, and when they meet each other during the surface diffusion, they attach each other to form a dimer. When a diffusing molecule touches an already existing cluster, it attaches the cluster to increases its size, and the cluster grows to an island. When a molecule is deposited above an island, it diffuses on the island terrace till it reaches an edge, and steps it down to be incorporated into the island. Evaporation of adsorbed molecule from the surface is excluded, since the growth is performed under a low 2/13 temperature. With only surface diffusion and attachment at island edges, grown islands are ramified 5 and look fractal, similar to DLA, 6, 7 at a low coverage. In order to realize compact islands, detachment or edge diffusion of molecules weakly bonded to the island edge are necessary unless evaporation is allowed. In the present model we allow detachment of those molecules connected to the island edge with only one nearest-neighboring bond, for simplicity.
Still, they can be detached only to the substrate surface, but not allowed to hop up on the epitaxial island: There are three possible orientations for them to detach.
We now review theories on an island density and shapes in terms of three parameters; 2-4 a deposition flux F , a surface diffusion constant D s , and a detachment rate D e from a singlyconnected edge site. As for the average island distance ℓ 2 in a stationary state, we have to consider two-dimensional nucleation on a flat substrate. For a freshly deposited molecule, it takes about a time τ 2 = ℓ 2 2 /D s to migrate a region of a linear dimension ℓ 2 . During this time interval, other molecules are deposited to a density n 1 = F τ 2 , and the initial molecule has a probability n 1 ℓ 2 2 to meet another molecule during its migration. (Logarithmic correction is necessary for more detailed analysis.) By assuming that the encounter causes the instantaneous nucleation, the encounter probability is the same with the nucleation probability per site. On the other hand, the latter is given by (a/ℓ 2 ) 2 with a being the lattice constant, since there is one nucleus within a region of ℓ 2 2 . By equating the two results, one obtains n 1 ℓ 2 2 = (a/ℓ 2 ) 2 or a scaling relation This scaling holds when solidification takes place irreversibly and the critical size of a nucleation is i c = 1. In a more general case with a critical nucleation size i c , the island separation is given by 2, 4 where the lattice constant is set to unity.
An irreversibly grown island with i c = 1 has a ramified profile similar to DLA, and consists of many edge molecules loosely connected only by a single bond. These loose molecules may detach from the island with a rate D e = D s e −J/k B T at a temperature T with an energy penalty When the detachment is possible, the dendritic branches dissolve and the island can be compact in a polygonal shape. There is a theory on a stability limit of an edge length ℓ 1 at 3/13 which the polygonal shape becomes unstable. 4,8,9 On a straight edge of an island of a length ℓ 1 , there should be a kink which incorporates molecules diffusing on the edge. With an edge diffusion constant D e , it takes about a time τ 1 = ℓ 2 1 /D e before a freshly deposited molecule on an edge is incorporated into the kink site. During this period, there are about F ℓ 2 2 τ 1 molecules deposited within the territory of an island, where ℓ 2 is an average separation between islands.
With a fast surface diffusion D s , this number of molecules are deposited on the island edge.
Thus, when it is of order unity, a one-dimensional nucleation takes place on the edge and the polygon side would become rough kinetically. Therefore, the critical size to keep an island in a polygonal shape is given as

Isotropic Detachment
In order to understand the effect of detachment on the island shape, we perform kinetic Monte Carlo simulations 20 of a system with deposition, diffusion and detachment processes.
Before the study of an anisotropy effect, we start from the more fundamental case with an isotropic detachment in this section. To concentrate on the island shape variation, parameter values are so chosen as to allow only a single island in a large system with a linear dimension of L = 500. In this large system, we provide initially a nucleus of a size 2 × 2 at the center of the system; This is the minimum size of an immobile island under a finite detachment rate, If the aggregate is compact, the density is independent of the distance r and one obtains d f = 2. For a fractal DLA, its dimension is well-known to be d f = 1.7, less than a space dimension 2. 1, 6 Our simulation data is consistent with this value, though a deviation is observed around the center which is exposed for a long time under deposition; spaces between dendritic branches will be filled under a perpetual deposition, and the island becomes compact eventually.
We now introduce detachment of molecules from the island edge to the substrate surface.
The grown island at a coverage Θ =0.1 becomes fat and anisotropic, as shown in Figs. 1(b) to (f). Every island consists of N a = 25, 000 molecules. As D e /F increases from 10 3 to 10 5 , shown in Fig. 1(b) to (d), each branch becomes fat. On further increase of detachment rate to D e /F = 10 6 , the island becomes skeletal with a corner instability due to the Berg effect, as shown in Fig. 1(e). At a further large value of D e /F = 10 7 (Fig. 1(f)) and more, the island remains square up to a coverage Θ =0.1.
On looking at island shapes in Fig. 1  This causes the deviation from the scaling law at small D e in Fig. 2.

Anisotropic Detachment
We now consider a shape anisotropy induced by an anisotropic detachment rate. The problem may be relevant to the shape anisotropy observed in the heteroepitaxial growth of Pn islands on a vicinal surface of H-Si(111) substrate 18 or of an organic epifilm on a flat oxidized GaAs substrate. 19 In the following a Pn system is chosen by way of an example to explain the possible origin of a detachment anisotropy, but the applicability of our generic model is by no means restricted to this specific system.
Pn is a flat, elongated molecule, and on H-Si(111) surface it is known to grow a wetting film with the molecular long axis perpendicular to the surface, namely in a standing-up orientation. 21 It means that Pn interaction with the substrate surface is weaker than the Pn-Pn interaction. On a flat terrace, deposited Pn molecules make surface diffusion and they are incorporated into a 2D island when they encounter. Island are compact and their shape is isotropic. If the incorporation were instantaneous and irreversible, the island should be ramified and fractal, in contradiction to the experiment. A detachment from the island edge may accomplish an observed compact island shape on a flat terrace, as discussed in the previous section. We here study an asymmetry observed in island shape when Pn grows on a vicinal H-Si(111) surface. As the step density in the vicinal surface increases and an island spans over many steps, island edges at the higher level are round whereas lower edges become dendritic.
Steps may affect molecular diffusion on a substrate surface. If the step enhances an energy barrier against surface diffusion, the effect is symmetric around the step. The rate of step-up diffusion slows down by the same amount as that of step-down diffusion. Since no net flow is expected in this case, islands should be symmetric. If a step edge makes an extra bonding with a diffusing molecule, it will reside preferentially in front of the step. Then, nucleation rate will be enhanced in front of the step, and consequently islands will be nucleated along it, as often observed in various experiments. In the Pn experiment, 18 however, this kind of island alignment along steps is not observed. Therefore, these kind of step effects on surface diffusion may be negligible for the Pn experiment.

7/13
Instead, we consider a step effect on molecular detachment from island edges. Imagine that a Pn crystalline film grows in a step-down direction and reaches an upper side of the descending step. Then, further crystal growth takes place by incorporating a Pn molecule from the lower terrace. In this case, the whole length of a newly attached molecule interacts with the island edge and the step ledge, and it is hard to be detached back to the lower terrace. The detachment rate is diminished by a factor R L < 1. On the contrary, when the film grows in the step-up direction and reaches the lower side of the ascending step, the next Pn molecule to be incorporated is located on the higher terrace, and its upper part does not contribute to the Pn-Pn bonding. Since the substrate step has a height of about one fourth of the length of a Pn molecule, a loss in the bonding energy may be significant. Also, the free upper part of the long molecule may be susceptible to thermal fluctuation, and there may be a substantial entropy contribution which reduces the bonding free energy. Accordingly, those molecules incorporated into an island edge from the higher terrace are easily detached back with an enhancement factor R H > 1. We now study the effect of this anisotropic detachment with R H and R L on the island morphology. As we want to find out the essential factors of shape determination, the model here is chosen to be as simple as possible: Instead of providing a vicinal surface tilted down in +x direction, we simulate a model of heteroepitaxial growth on a flat substrate surface, but with anisotropic detachment rate. The molecule that resides on the left edge of an island and connected to the island only by one bond has a higher detachment rate D e R H with R H > 1, whereas that on the right edge detaches from the island less frequently with the rate D e R L

8/13
and R L < 1. Those molecules residing on the upper and the lower edge of the island detach as before with the rate D e . The diffusion constants are set to D s /F = 10 15 , D e /F = 10 6 to allow a single island in our system of a size L 2 = 500 2 . By initially providing an embryo of a size 2 by 2 at the center of the system, an island is nucleated and grows to a shape as shown in Fig. 3 at the coverage Θ = 0.1. The detachment enhancement factor to the higher terrace R H is chosen to be inversely proportional to the suppression factor to the lower terrace R L in order to decrease the number of free parameters, and they are varied as (a) (R H , R L )=(2, 1/2), (b) (5, 1/5), and (c) (10, 1/10). In the isotropic case with R H = R L = 1, the island has a skeletal shape shown in Fig.1(e). Even though enhancement and the suppression factors, R H and R L , are applicable everywhere in the system and there is no local inhomogeneity, the left and the right halves of the island show a drastic contrast: The left-hand side (lhs) is polygonal and the right-hand side (rhs) is dendritic.
To quantify the shape difference in the lhs and the rhs of an island, one may think to utilize fractal dimensions in both sides. In Fig. 3(c), for instance, the lhs of an island is very narrow, and a box counting in a limited width yields only a rough evaluation as d f ≈ 1.9 ∼ 2.
On the rhs, dendrites are clearly observable but they are still fat and not long enough. A fractal dimension depends on the narrow fitting range as d f ≈ 1.7 ∼ 1.8. For a more accurate determination of d f larger systems are necessary, but it is out of scope of the present study.
As for the upper and lower edges in Fig. 3, the lhs and the rhs of an island also show a contrast. In fact, the detachment rates from them are the same D e as in Fig. 1 (e). The lhs of islands in Fig. 3 are similar to that in Fig. 1(e), whereas the upper and lower edges in the rhs of an island in Fig. 3(c) are quite flat compared to the rhs of an island in Fig. 1(e).
It may be because branching of the upper or lower edges to the left ( or to −x direction) is suppressed. Branching takes place only to the right, and these dendrites grow to the right ( or +x direction ) fast. This suppression of left branching decreases fluctuation of upper and lower edges of an island.
Furthermore, on looking through Fig. 3(a) to (c), one may get an impression that an island seems to be shifted to the right as R L decreases. This apparent shift is due to the difference in the growth rate in ±x-directions. The dendritic right edge grows faster than the facetted left edge. Difference in the two edge speeds is quantified in a previous experiment 18 by a ratio of growth distances of side edges from the nucleation center. Since the initial nucleus is provided at the center of the system x c = y c = L/2, the degree of the shape anisotropy is measured by the maximum and the minimum distance in the x direction of the island edges from the initial nucleation center as The simulation result is summarized in Fig. 4. As the reduction factor R L of detachment to the lower terrace decreases, the shape anisotropy increases. they seem to be distributed homogeneously. In Fig. 5(b), the surface diffusion is decreased to D s /F = 10 9 , and the island density increases. Then one may detect some correlation in the alignment of islands in x-direction. In Fig. 5(c) the surface diffusion is kept the same with the case in Fig. 5(a) but, instead, the detachment rate at the upper and the lower edges of an island is increased by a factor of ten: Molecules are easily detached from the three edges except the right one, and accordingly the right dendritic edge grows faster. Enhanced molecule detachment in y direction enlarges the longitudinal island separation and weakens the correlation in this direction. In contrast, one has an impression that islands align laterally in x direction.
In order to quantify structural correlation, structure function or the Fourier transform of the density correlation is usually employed. In the present small system, however, the structure function is noisy and it is difficult to identify the anisotropy of the island alignment. Instead, we use a simple measure to indicate an anisotropy in x and y directions; number densities of empty rows and columns which are free of adsorbed molecules. For example, in Fig. 5(a) 4.8% of the vertical columns are free of adsorbed molecules whereas 9.2% of the lateral rows are free. Though each island has a clear anisotropic shape, islands arrangement in space is 10/13 rather isotropic. In Fig. 5(b), the empty row increases to the density of 16.6%, whereas there are no empty columns; there are at lease one adsorbed molecule on every column. In Fig. 5(c) alignment in x direction tends stronger; 51% of empty rows but no empty columns. Thus, the measure of empty rows and columns seems very clear and helpful to indicate the degree of anisotropic arrangement of islands.

Summary and Discussions
An effect of an anisotropic detachment from island edges on the determination of a twodimensional island shape is studied by means of a kinetic Monte Carlo simulation of a square lattice gas model of heteroepitaxial growth. The study may be relevant to the formation of anisotropic islands observed in heteroepitaxial growth experiments of Pn molecules on a vicinal surface 18 and/or of tilted organic molecules on a flat substrate. 19 Only with a molecular deposition with flux F followed by a surface diffusion D s , islands take a ramified dendritic shape, similar to DLA. Addition of detachment from the edge with a rate D e lets dendritic branches fat, and with a large D e compact square islands are formed.
By assuming that a detachment rate depends on an edge orientation, island shape becomes anisotropic. For example, the left edge with an enhanced detachment with a rate R H D e with R H > 1 is found smooth and compact, whereas the right edge with a small detachment rate R L D e with R L < 1 is ramified and dendritic. If islands are close to each other, one notices a tendency that islands align linearly, since detached molecules from the smooth edge of the right island are fed to the dendritic and fast growing edge of the left island.
We comment on the anisotropy factors, R H and R L , in Pn experiments. 18 Even without steps, Pn islands have compact shape, and thus the detachment process takes place often it is conceivable that the upper part is most susceptible to thermal fluctuation. When substrate steps are present, a Pn molecule incorporated from the higher terrace misses bonding at its upper part, but if thermal fluctuation already breaks the bonding there, the effect will be small. Then, R H can be close to unity. For a Pn molecule attaching from the lower terrace, on the other hand, its upper part may make a strong bond with the pre-existing island edge which lies higher. Then, the detachment may be strongly supressed, yielding a very small R L . Estimation of precise values of these factors requires microscopic information of chemical bondings, molecular rigidities, and so on, and out of scope of the present study.
Above explanation is refering to the Pn epilayer on a vicinal surface just by way of example, but it is not meant to be specific to that system. We expect our model is universal and is applicable to other cases, such as heteroepitaxial growth of tilted molecules on a flat substrate surface. 19