Shape of Heteroepitaxial Island Determined by Asymmetric Detachment
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Shape of Heteroepitaxial Island Determined by Asymmetric Detachment
Yukio Saito ∗ and Ryo Kawasaki Department of Physics, Keio University, Yokohama 223-8522, Japan
Square lattice gas models for heteroepitaxial growth are studied by means of kinetic MonteCarlo simulations, in order to find a possible origin of anisotropic island shape observed in growthexperiments of long organic molecules. When deposited molecules form clusters irreversibly attheir encounter during surface diffusion, islands grow in a ramified dendritic shape, similar toDLA. Introduction of molecular detachment from edges makes islands compact with smooth edges.Tilting of adsorbed long molecules or steps in a vicinal substrate may induce orientation-dependencein the detachment rate of edge molecules from an island. In simulations with orientation-dependentdetachment rates, a clear anisotropy in an island shape is observed. Shape anisotropy on a vicinalsubstrate is enhanced as steps get dense, in agreement to the experimental observation.
PACS numbers: 81.15.Aa, 68.43.Hn, 68.47.Pe
I. INTRODUCTION
There have been increasing interests in heteroepitaxialgrowth of organic semiconductors in relation to the fab-rication of thin and deformable microelectronic or opto-electronic devices [1, 2, 3]. To accomplish a high standardin electrical properties, improvement of the film qualityis indispensable, and fundamental researches on growthmechanisms of adsorbed organic molecules are under-taken [4, 5]. In an experiment of pentacene (Pn, C H )growth on Si(001) [4], authors concluded that an organicthin-film growth is similar to the epitaxial growth of in-organic materials, such as the formation of fractal rami-fied islands like diffusion-limited aggregates (DLA) [6, 7].On the other hand, in an experiment of Pn growth on ahydrogen-terminated Si(111) surface [H-Si(111)] [5], a pe-culiarity is found. On a flat H-Si(111) surface islands arecompact with smooth edges and isotropic, whereas ona vicinal surface islands are dendritic in the step-downorientation while edges in the step-up orientation remaincompact: Each island has a strong anisotropy in its shapeon a vicinal substrate. Island shape anisotropy, however,is not specific to a vicinal surface. It is observed in thevapor growth of another organic molecule, a behenic acid,even on a flat singular substrate, namely on an oxidizedGaAs [8]. In the present paper we analyze some modelsystems which lead to shape anisotropy theoretically.In the heteroepitaxial growth of inorganic materials,island shape is mainly controlled by the competition oftwo processes after an adatom deposition; an adatom dif-fusion on a terrace surface and an incorporation kineticsat island edges [9, 10, 11, 12]. With a fast incorporationat island edges, diffusion controls the growth and edgesundergo a morphological instability to form dendrites. Inthe extreme case of irreversible solidification such that anadatom once solidified never detaches again from the is-land edge, then an island takes a ramified irregular shape ∗ Electronic address: [email protected] with many branches [13]. Its structure is similar to theDLA without a characteristic length, and is called a frac-tal. An island is isotropic or symmetric by reflecting thelattice symmetry.To provide a compact island shape, edge smootheningprocesses are necessary, such as an edge diffusion or a de-tachment from the edge. If the detachment rate is high,adatoms loosely attached to the edge will readily de-tach and they are hardly incorporated in two-dimensional(2D) islands. Then, the slow incorporation kinetics gov-erns the growth to make islands compact with smoothedges [14, 15, 16]. Islands may be round at high temper-atures or polygonal at low temperatures.In the heteroepitaxial growth of organic molecules, onehas to take an additional feature into account, i.e. thesize of an adsorbed organic molecule. It can no longerbe regarded as a point object as in inorganic cases so fardiscussed. Both pentacene and behenic acid moleculesare flat and elongated in shape. When they are adsorbedon a substrate, molecules can be normal or lateral tothe substrate surface [18]. Pn molecules are known togrow a wetting film with the molecular long axis nor-mal to the H-Si(111) surface, namely in a standing-uporientation [17]. Behenic acid is also known to be in astand-up orientation but with a small tilting on a sub-strate. We assume that this molecular orientation affectsthe detachment rate of molecules from island edges, andleads to anisotropy in an island shape.In a previous study[19] we proposed a simple latticegas model, and studied general aspects of the effect ofanisotropic detachment on an island shape. In the next § § § II. 2D ISLANDS ON A FLAT SUBSTRATE
When molecules in a 2D crystal stack almost normalto a substrate surface but with a finite tilting, as to the a -axis in the case of a behenic acid [8], kinetics at islandedges may depend on relative orientations of the edge andof the molecular tilting. To an edge where the molecu-lar inclination restricts the incorporation space, diffusingmolecules may be difficult to be attached to the edge,but once attached they may be difficult to be detachedfrom this edge. Depression of the attachment rate slowsdown the incorporation kinetics, and the edge becomessmooth. On the contrary, suppression of detachment in-creases incorporation and leads to a diffusional instabilityof a smooth edge. So far the relation between the molec-ular tilting and the orientation of dendritic edge is notidentified experimentally [8], we here assume simply thata molecular tilting affects detachment process such thatthe detachment rate of a molecule depends on the edgeorientation. In our theoretical treatment, we forget aboutthe finite size and shape of an adsorbing organic molecule,but consider it implicitly in the orientation-dependenceof the detachment rate.We perform kinetic Monte Carlo simulations [19, 20]of a lattice gas models of point molecules depositing on asquare substrate with a deposition rate F per area. Ad-sorbed molecules diffuse on a surface with a diffusion con-stant D s . When two diffusing molecules meet, they makea bond to lower an energy and form a cluster. Clustersenlarge their sizes by incorporating diffusing moleculesat edges. On the other hand, molecules at edges candetach from the islands when they are loosely attached:Those particles with a single bond to the island can breakthe bond to migrate out on the substrate surface with arate D e , but molecules with more than two bonds areassumed immobile. When a particle is deposited abovean island, it diffuses on the island terrace till it reachesan edge, and steps it down to be incorporated into theisland. Evaporation of adsorbed molecules from the sur-face is excluded, since the growth is assumed to takeplace at a low temperature. When there is no detach-ment, grown islands take a ramified dendritic form withirregular fine branches, similar to the DLA [6, 7]. With asmall detachment, one obtains dendritic aggregates withfat branches, as shown in Fig. 1(a). Here, parametersare D s /F = 10 and D e /F = 10 with a system size L = 500 and the coverage Θ = 0 .
1. Although islandsinterfere with each other in a diffusion field, they have al-most a symmetric shape. By applying the box-countingmethod, one obtains a rough estimate of the fractal di-mension as d f ≈ . R L .Island morphology simulated by means of kinetic MonteCarlo simulation [20] becomes anisotropic, as shown inFig. 1(b) and 1(c). When the detachment from theleft edge is reduced by a factor 10 as R L = 0 . FIG. 1: Islands on a substrate with a size 500 with a surfacediffusion D s /F = 10 at a coverage Θ = 0 .
1. A detach-ment rate is D e /F = 10 , and (a) isotropic, and (b) and (c)anisotropic. Detachment from the left edge is suppressed bya factor (b) R L = 0 .
1, and (c) R L = 0 . d f ≈ .
7. On further reducing the detachmentrate from the left edges to R L = 0 .
01 (Fig. 1(c)), is-lands point sharply to the left with the left-to-right ratioranging about 3 to 6, depending on environment. Pri-mary branches extending to the left are fine and haveless side-branches, compared to Fig. 1(a) and 1(b), andprimary branches to other orientations have many sec-ondary branches extending long to the left. Pointingstructure leads to the small fractal dimension of d f ≈ . y direction, and the obtainedshape looks different to what is observed in the exper-iment [8]. On increasing the detachment rate D e /F ,islands become more compact with smooth edges as askeletal shape or even to a complete square. The effectof anisotropic detachment in these cases have been dis-cussed previously [19]. III. ISLANDS ON A VICINAL SURFACE
We know consider the case of a heteroepitaxial growthon a vicinal surface, which might be relevant to Pngrowth on H-Si(111) [5]. On a flat terrace, Pn moleculesform 2D islands, and they are compact with an isotropicshape. Therefore, Pn molecules might often detach fromisland edges. On a vicinal H-Si(111) surface, Pn islandsshow different morphology in the upward and downwarddirections relative to the substrate steps. Island edgesextending to the step-down direction are pointed in adendritic form, while those to the step-up directions re-main to be smooth and round.Since Pn is known to grow in a standing-up orientation D e R L D e D e D e R H L s FIG. 2: Detachment rates of adsorbed long molecule fromislands on a vicinal surface. [17], an interaction between Pn molecules is stronger thanthat between Pn and the substrate. Let us now imaginethat a Pn crystalline film is growing on a vicinal surfacewith a positive slope, as shown in Fig. 2. When a filmgrows to the left in a step-down direction and reaches anupper side of the descending step, as a left island in Fig.2, further crystal growth takes place by incorporating aPn molecule from the lower terrace. In this case, thewhole length of a newly attached molecule interacts withthe island edge and the step ledge, and it is hard to bedetached back to the lower terrace. Also, the free upperpart of the long molecule may in general be susceptibleto thermal fluctuation, and there may be a substantialentropy contribution which reduces the bonding free en-ergy. Attachment from the lower terrace may lessen thisentropy contribution and enhances the Pn-Pn bonding.Thus, the detachment rate from the lower terrace is di-minished from the normal rate D e on a flat terrace bya factor R L < D e R L . On the contrary, whenthe island grows to the right in the step-up direction andreaches the lower side of the ascending step, as a rightisland in Fig. 2, the next Pn molecule to be incorporatedis located on the higher terrace, and its upper part doesnot contribute to the Pn-Pn bonding. Since the substratestep has a height of about one fourth of the length of a Pnmolecule, a loss in the bonding energy may be significant.Accordingly, those molecules incorporated into an islandedge from the higher terrace are easily detached back tothe higher terrace with an enhancement factor R H > R H and R L on the islandmorphology.In kinetic Monte Carlo simulation of heteroepitaxialgrowth on a vicinal surface, steps are separated with adistance L s . After a deposition, molecules perform D s /F times of diffusive migrations until a monolayer is covered,but during this period molecules collide to form clusters.Steps are assumed to have no effect on the surface dif-fusion, for simplicity. A large value of surface diffusionconstant D s /F = 10 is chosen in order to grow onlya single island in our system of a size L = 500 . Fromedges of an island on a flat terrace, molecules with a singlebond detach with a rate D e /F = 10 . Above a descend-ing step, detachment is enhanced by a factor R H = 10,but in front of the step detachment is assumed completelyforbidden, R L = 0, corresponding to an extreme case.By initially providing an embryo of a size 2 by 2 at thecenter of the system, an island is nucleated and grows to FIG. 3: An island on a vicinal substrate surface. The surfacediffusion is D s /F = 10 and the detachment rate on a flatterrace is D e /F = 10 . Detachment to the higher terrace (tothe right) is enhanced by a factor R H = 10 and to the lowerterrace (to the left) is completely suppressed: R L = 0. Stepseparations are L s =(a) 5, (b) 4, (c) 3 and (d) 2. The coverageis Θ = 0 . . the coverage Θ = 0 .
1. As the step separation decreasesfrom L s = 5 to 2, the island shape alters as shown in Fig.3(a) to 3(d). Even with a very large surface diffusion, asecond island is nucleated accidentally in Fig. 3(c). At alarge step separations L s = 5 and 4, the island has onlya weak left-right anisotropy, as in Fig. 3(a) and 3(b).When steps come closer as L s = 3 and 2, anisotropybecomes obvious, as in Fig. 3(c) and 3(d). V L / V H L s FIG. 4: Ratio of the edge velocities to the lower side V L tothe higher side V H as a function of the step separation L s . Acurve is a guide for eyes. The surface diffusion is D s /F = 10 and the detachment rate is D e /F = 10 with an enhancementfactor to the uphill R H = 10 and a complete suppression tothe downhill R L = 0. In order to quantify the shape anisotropy, we measurethe ratio V L /V H of the front velocities to the lower side V L and to the higher side V H as a function of the step sep-aration. The result shown in Fig. 4 clearly demonstratesthat an anisotropy increases as the step separation de-creases, in a qualitative agreement to the experimentalresult obtained in Pn system [5].So far, we discussed the shape of a single island. Ina large area of the vicinal surface many islands are nu-cleated and they affect their mutual growth and shapethrough the diffusion field. In the Pn experiment onemay identify some order in spatial arrangement of islands[5]: Islands look to align in queues parallel to the tiltingdirection of a vicinal substrate, with a dendritic left edgeof one island being contiguous to the smooth right edgeof a left one. ( See, for example, Fig.2 (c) and (d) inRef.[5]. ) FIG. 5: Island distribution on a vicinal surface with a size500 at a coverage Θ = 0 .
3. Step separation is L s = 2 andsurface diffusion constants are (a) D s /F = 10 , (b) 10 , and(c) 10 . Other parameters are D e /F = 10 , R H = 5 , R L =0. In our system with a limited size L = 500 , island den-sity is controlled by varying the surface diffusion constant D s . In some combinations of parameter values, simula-tion can produce island arrangements with some spatialordering, as shown in Fig. 5. The height of the vicinalsurface is increasing to the right with ascending steps atthe separation L s = 2. Molecules are deposited to thecoverage Θ = 0 .
3. The surface diffusion constant is set at(a) D s /F = 10 , (b) 10 , and (c) 10 , by keeping thedetachment rate fixed to D e /F = 10 , and the enhance-ment factor R H = 5 and no detachment to the step-downdirection R L = 0. At the smallest surface diffusion D s /F in Fig. 5(a), there are many islands distributed almostrandomly. The stochastic noise associated to random de-position is frozen in during the nucleation process. InFig. 5(b), as the surface diffusion increases, island den-sity decreases and the diffusion process seems to averageout the deposition noise and to induce some correlationin islands arrangement. Islands do not seem to align per-pendicular to the step orientation. One might notify sim-ilar tilting in islands arrangement in the Pn experiment[5]. For larger D s in Fig. 5(c), island density becomes solow that it is difficult to identify spatial correlation forcertain. FIG. 6: Spatial correlation function g ( r ) corresponding toFig. 5. ”H” marks the high hill-top, and ”L” marks the lowvalley. In order to evaluate spatial correlation explicitly, wecalculated the correlation function g ( r ) = h X i n ( r + r i ) n ( r i ) i /N (1) where the occupation variable n ( r i ) is unity when the lat-tice site i is occupied by an admolecule, and zero whenthe site i is empty. Since the system is periodic bothin x and y directions with a periodicity L , the correla-tion function has the symmetry g ( x, y ) = g ( L − x, y ) = g ( x, L − y ) = g ( L − x, L − y ). Thus, we obtained the heightcontour of g ( x, y ) for 0 ≤ x, y ≤ L/
2, as shown in Fig.6. A mark ”H” represents the high hill-top with a strongcorrelation, and a mark ”L” represents the low valleywith a weak correlation. From Fig.6(b) one clearly ob-serves that the pattern has a strong periodicity with a pe-riod p = ( L/ , L/
8) at a surface diffusion D s /F = 10 .With a smaller diffusion D s /F = 10 , average islanddensity gets higher as in Fig. 5(a), and the periodicityis not commensurate to the system size. Therefore, g ( r )has a rather broad first peak around r = (100 , D s /F = 10 , thereare a few islands as in Fig. 5(c), and g ( r ) has a broadpeak at around r = (150 , L s = 2 in order to realize a strong anisotropiceffect in our small system. Since the island shapeanisotropy is expected to be caused by an accumula-tion of many weak anisotropy effect at steps, one hasto pack as many steps in an island as possible. Actually,with L s = 3 we can observe similar arrangement of manyanisotropic islands, but with a larger separation L s ≥ µ m andit spans many steps. It may be interesting if there is acritical size for an island to show a shape anisotropy. IV. SUMMARY AND DISCUSSIONS
Motivated by heteroepitaxial growth experiments oflong organic molecules on a flat [8] or on a vicinal surface[5], effect of detachment anisotropy on the island shapeis studied by means of kinetic Monte Carlo simulationsof a lattice gas model.Only with a molecular deposition followed by a surfacediffusion, islands take a ramified dendritic shape, similarto DLA. Addition of detachment from the edge makes fatdendritic branches, and with a large enough detachmentrate islands become compact in a square form.If the long molecules are tilted in an epitaxial film, therate of detachment from island edges may depend on theedge orientation. Simulation shows that the edge with asmaller rate of detachment shows diffusional instabilityand forms a dendritic tip. Resulting island shape lookssimilar to the one obtained in the experiment [8].On a vicinal surface, an interaction between an edgemolecule and a neighboring one depends on edge position.When an edge has just climbed up the substrate step, amolecular bonding is weak and an edge molecule mayeasily detach from the edge site. On the other hand,when an edge has just stepped down the substrate step,a bonding is strong and the edge molecule is hard todetach. In our simulation we assume an extreme case ofno detachment at a lower edge to stress an anisotropyeffect. Simulations show clear anisotropy in adsorbedisland shape as the step density increases.In addition to the shape anisotropy of a single island,the orientation-dependent detachment rate may inducea spatial correlation in the arrangement of islands withclear peaks in the spatial correlation function. Since therandom deposition process introduces shot noise in thenucleation process, a fast surface diffusion seems neces-sary to average out fluctuation and to induce a spatialorder, but then the island separation becomes large. Forfurther studies on spatial correlation, one needs a largersystem size.As for the reason of island shape anisotropy, one maythink about a diffusional anisotropy, such that a stepprovides an asymmetry in an energy barrier for the ad-molecule diffusion. However, the asymmetric potentialis unable to lead unidirectional motion nor a net drift,as the Feynman’s ratchet and pawl machine cannot pro- voke directional rotation [21]. Another effect that stepsmay provide is a nucleation center by lowering the energybarrier. But in the experiment there is no observation ofalignment of nucleation centers along the step direction.Therefore, the latter effect might be small. Observationof admolecule motion on the substrate, if possible, shouldresolve these points clearly.
Acknowledgments
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