Kinetic phase diagram for terrace and step nucleation of CaF/Si(1 1 1) $

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1 Applied Surface Science 219 (2003) Kinetic phase diagram for terrace and step nucleation of CaF/Si(1 1 1) $ Joachim Wollschläger a,*, Markus Bierkandt a, Mats I. Larsson b,1 a Institut für Festkörperphysik, Universität Hannover, Appelstrasse 2, Hannover, Germany b epartment of Materials Science and Engineering, Stanford University, CA , USA Received 21 August 2002; accepted 26 January 2003 Abstract Competing processes during the growth on vicinal surfaces are the nucleation on terraces and the nucleation at steps. From kinetic Monte Carlo (KMC) simulations which are performed on vicinal surfaces of various terrace widths, a kinetic phase diagram (parameters: terrace width and deposition p temperature) has been deduced. The boundary between both phases of terrace and step nucleation is described by w / ffiffiffiffiffiffiffiffiffiffiffiffi ðt c Þ. In addition, the nucleation of CaF on Si(1 1 1) has been investigated by means of scanning tunneling microscopy (STM). A comparison of the experimental data with the kinetic Monte Carlo simulations shows that the diffusion barrier of CaF molecules on Si(1 1 1) is 1.8 ev. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Kinetic Monte Carlo simulation; Scanning tunneling microscopy; iffusion; Nucleation; Hetero epitaxy; Calcium fluoride; Silicon 1. Introduction Thin films of homogeneous thickness are essential for many applications such as, for instance, nanoelectronic devices. Growing films on flat surfaces, however, rough films are generally obtained due to the interplay of various diffusion, growth, and nucleation processes (see for instance [1]). Here, the most difficult process which determines the time scale of the growth process is the nucleation of a new island. This problem can be $ ICSCM, Cleveland, OH, USA, 21st 23rd August * Corresponding author. Present address: Universität Bremen, Institut für Angewandte und Physikalische Chemie Leobener Straße Fachbereich 2, NW II Postfach , 28334, Bremen. Tel.: þ ; fax: þ address: wollschlaeger@fkp.uni-hannover.de (J. Wollschläger). 1 On leave from the epartment of Physics, Karlstad University, Karlstad, Sweden. surpassed by depositing the material on vicinal surfaces instead of singular surfaces as described by Burton et al. for the first time [2]. According to classical nucleation theory, islands that are larger than the critical island size i are stable, while islands with a size below i can decay via detachment of monomers. Assuming that two-dimensional islands are formed and that desorption can be neglected, it has been concluded from rate equations that the maximum island density n sing follows i=ðiþ2þ (1) n sing x / F for deposition on singular surfaces without substrate steps [3 5]. Here, F and denote the flux of the deposited material and the diffusion constant of the deposited monomers on the surface, respectively. The flux F is related to the deposition rate R in units of (ML/s) via F ¼ R=A where A denotes the area of the x /$ see front matter # 2003 Elsevier Science B.V. All rights reserved. doi: /s (03)

2 108 J. Wollschläger et al. / Applied Surface Science 219 (2003) surface unit cell. Eq. (1) has often been used to determine the critical nucleus size i if the flux F is varied. iffusion barrier E of monomers can be obtained from Eq. (1) via ¼ 0 expð E =ktþ as well, if one varies the deposition temperature (see also [6]). Here, however, one has to perform experiments in the range of i ¼ 1, since an additional temperature dependent factor expðe i =ði þ 1Þk B TÞ enters the right side of Eq. (1) for the case of i > 1(E i denotes the binding energy of the critical cluster). The prefactor 0 ¼ 1=4a 2 n depends on both the attempt frequency n of the hopping monomers and the lateral lattice constant a. In addition, kinetic Monte Carlo (KMC) simulations have been performed successfully to check the validity of the classical nucleation theory as described by Eq. (1)) (cf. e.g. [7 11]). Pimpinelli et al. obtained Eq. (1) from scaling arguments for island density at the time of island coalescence, too [12]. They furthermore applied scaling arguments to the case of epitaxy on vicinal surfaces and concluded that island density is not considerably affected by substrate steps if the diffusion length l is smaller than the terrace width w (cf. Fig. 1). In the opposite case, i.e. if the diffusion length l is larger than the terrace width, they obtained i=2 n x / Fw2 (2) for island density assuming that the growth of islands is diffusion-limited. In addition, a crossover scaling function has been derived which shows that the crossover from Eqs. (1) and (2) occurs at n sing x 9=w 2 (for i ¼3). This has been proven by KMC studies [13]. Eq. (2) has to be modified if one considers island density on vicinal surfaces at coverages Y far below the coverage where the islands coalesce. Bales studied the nucleation on vicinal using rate equations and kinetic Monte Carlo simulations for the case of i ¼ 1 [14]. He discovered that the island density follows n x / F K pffiffiffiffiffi 1ð 12 a=wþ w3 pffiffiffiffiffi Y (3) K 0 ð 12 a=wþ where K 0 ðxþ and K 1 ðxþ are modified Bessel functions of the order 0 and 1, respectively. For the range of terrace widths of interest (w=a ¼ ) Eq. (3) can be approximated by n x / F wn Y (4) with n 3 (cf. Fig. 1(a) of [14]). From scaling arguments Pimpinelli et al. obtained the generalized form n x / F i w 2ðiþ1Þ Y (5) where sizes larger than one for the critical nucleus have been considered, as well [12]. Obviously Eq. (5) reproduces the dependence on the diffusion constant of Eq. (4) for the case of i ¼ 1 while a different resultis obtained for the dependence on the terrace width w. Fig. 1. Model of the elementary processes during epitaxy on vicinal surfaces. Monomers are deposited on terraces of width w. They diffuse randomly on the terrace until they are captured, either by other monomers (homogeneous nucleation) or by substrate steps (heterogeneous nucleation). In addition, monomers can be captured by islands on terraces (not shown).

3 J. Wollschläger et al. / Applied Surface Science 219 (2003) An other approach via rate equations has been employed by Myers-Beaghton and Vvedensky to study the transition from growth via nucleation on terraces to the growth via step flow, where most of the material is captured by growing steps (step nucleation) [15 17]. In contrast to the publications cited above which deal with the nucleation of submonolayers, Myers-Beaghton and Vvedensky focused their interest on the steady state limit of island formation. They concluded that the competition between terrace pffiffiffiffiffiffiffiffiffiffiffi and step nucleation starts if the diffusion length t ML is of the order of the terrace width w. Here, t ML denotes the time necessary to deposit 1ML. Therefore, the transition between terrace and step nucleation is expected if the critical temperature T c fulfills w 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi p 0 t ML exp E (6) 2k B T c Myers-Beaghton and Vvedensky report that it is important to consider the number of sites which are visited by the migrating monomers between deposition and incorporation into steps if the growth is governed by agglomeration at steps. This consideration modifies Eq. (6) for w 2ð 0 a 2 t ML Þ 1=4 exp E (7) 4k B T c In the following, we will first present STM studies on the growth of CaF 2 deposited on vicinal Si(1 1 1), from which we evaluate kinetic phase diagrams for the phase of terrace nucleation and the phase of step nucleation. Equivalent kinetic phase diagrams are obtained from KMC simulations as well. The goal of this paper is to demonstrate the diffusion barrier E can be obtained from nucleation experiments on vicinal surfaces, if the critical temperature T c is determined where the island density n x falls below a critical island density n c x. Thus, comparing both kinetic phase diagrams we evaluate the height of the diffusion barrier of CaF monomers on Si(1 1 1). 2. Scanning tunneling microscopy on CaF 2 /Si(1 1 1) Vicinal Si(1 1 1) samples with a miscut angle of 1 have been cleaned by repeated flash annealing up to C by applying direct current through the sample. uring the cooling down of the sample, the temperature is kept for some time at approximately 830 8C to improve the quality of the 7 7 reconstruction. After this cleaning procedure the scanning tunneling microscopy (STM) micrographs of the Si samples show large (1 1 1) terraces which are mostly separated by mono-atomic steps, although depending on the details of the preparation, some multi-atomic steps are observed, too. The surface generally has terraces with regular terrace width and step height if the current flows parallel to the substrate steps. Alternating areas with step bunches (narrow terraces) and de-bunched regions (wide terraces) are obtained if the heating current direction is perpendicular to the substrate steps due to electro migration (cf. e.g. [18]). Both kinds of preparation techniques have been applied to investigate the growth of CaF 2 on variously stepped Si(1 1 1) substrates. CaF 2 was deposited from e-beam evaporators with either graphite or tantalum crucibles. The deposition rate was controlled by quartz microbalances. Typical deposition rates were ML/min. The temperature of the samples was controlled by an infra red pyrometer which was calibrated to the 7 7to1 1 phase transition of the Si(1 1 1) surface at 830 8C [20,19]. The base pressure in the UHV system was Pa. From studies by X-ray photoelectron spectroscopy (XPS) it is well known that CaF 2 dissociates on the Si(1 1 1) surface in the temperature range of C used in our STM experiments [21]. Thus, the first monolayer has CaF stoichiometry instead of the CaF 2 stoichiometry of the non-dissociated molecules. CaF islands with 1 1 structure are formed for submonolayer coverages [21 23]. The formation of islands with 3 1 structure has been reported for higher substrate temperatures [23 26]. The residual Fluorine forms SiF x species on the surface which do not desorb below 600 8C. Fig. 2 shows STM micrographs after deposition of 0.1 ML CaF 2 on stepped Si(1 1 1) substrates at 450 and 560 8C, respectively. The terraces are nm wide. At low deposition temperatures, the morphology is governed by CaF islands which are formed on terraces although a considerable amount of the deposited material is agglomerated at the substrate steps. Some of the islands on the terraces are nucleated at anti-phase boundaries of the 7 7 reconstruction of

4 110 J. Wollschläger et al. / Applied Surface Science 219 (2003) Fig. 3. Kinetic phase diagram for CaF nucleation on terraces and at steps of vicinal Si(1 1 1) substrates. Fig. 2. STM micrographs after deposition of 0.1 ML CaF 2 at 450 and 560 8C, respectively. The terrace width is approximately 300 nm in the upper micrograph. The major part of the CaF layer forms islands on the terraces. The arrows indicate CaF islands which are nucleated at anti-phase boundaries of the 7 7 reconstruction. Terrace widths between 100 and 200 nm are observed in the lower micrograph. Most of the CaF is agglomerated at substrate steps. The residual CaF islands on the terraces are nucleated at anti-phase boundaries. the Si(1 1 1) surface. These anti-phase boundaries are running over the terraces perpendicular to the substrate steps. Therefore these islands form chains (cf. arrows in Fig. 2). For the higher deposition temperature of 560 8C, the major part of the deposited material covers the substrate steps. Islands on terraces are only observed on the right hand side of the micrograph where the terraces are wider than on the left hand side. Probably, these islands are nucleated at anti-phase boundaries. Considering these STM micrographs, we conclude that both the terrace width w and the deposition temperature T are important for the transition from terrace to step nucleation. A comparison of the micrographs shows that the nucleation on terraces reduces if the substrate temperature increases. A comparison of both sides of the high temperature micrograph demonstrates that the nucleation on terraces reduces with decreasing terrace width w as well. We deposited 0.1 ML CaF 2 at various temperatures on substrates with various terrace widths to determine the critical temperature for suppression of the nucleation on terraces where all deposited material is incorporated in the substrate steps. The critical island density n c x ¼ nm 2 has been used here. The kinetic phase diagram obtained from these experiments is presented in Fig. 3. The Arrhenius behavior of the phase boundary between terrace and step nucleation, respectively, demonstrates that the process is thermally activated and that it depends on the diffusion barrier of CaF on Si(1 1 1) terraces. To elucidate the exact relationship between the diffusion barrier and the critical temperature for the transition between terrace and step nucleation, we performed KMC simulations on vicinal surfaces with fcc(1 1 1) symmetry. These simulations will be described in the next sections. 3. Model of kinetic Monte Carlo simulations The KMC simulations are performed on hexagonal lattices of size and sites. Chiral periodic boundary conditions are used in the model to consider the vicinality of the substrate. The substrate steps are A oriented ((1 0 0) like steps, cf. Fig. 4).

5 J. Wollschläger et al. / Applied Surface Science 219 (2003) Fig. 4. (a) Model for the kinetic Monte Carlo simulations on vicinal fcc(1 1 1) surfaces. Atoms of the lower and upper terrace are presented in black and grey, respectively. The substrate step is a type A step with (1 0 0) symmetry. In addition, a triangular island with A steps is shown on the lower terrace. The white circles show two typical diffusion pathways. (b) iffusion barriers for both pathways depicted in (a). In addition, the center graph shows the diffusion barriers for a pathway similar to path I but with diffusion along a B step with (1 1 1) symmetry. The third positions in the two upper graphs are positions at steps. In the bottom graph, the monomer on the third position is on a transitent site while the fourth position is at the lower side of the A step. Step distances of 16, 32, 64, and 128 lattice sites are investigated in the simulations. The deposition rate of the adatoms is 0.1 ML/s. The adatom hopping rate n i;f ¼ n 0 exp Eðn i; n f Þ (8) k B T on the surface depends on the energy barrier Eðn i ; n f Þ¼n i E 0 ðn i þ n f ÞZ c E b (9) where k B, T and n 0 denote Boltzmann s constant, the temperature, and the effective attempt frequency (n 0 ¼ Hz). n i and n f denote the number of nearest neighbors before and after the hop, respectively. Some fundamental aspects of this model will be presented here, although details of the model are published elsewhere [27 29]. In this study, we used E b =E 0 ¼ 1=300. The configuration dependent factors Z c are Z c ¼ Z T ¼ 1, Z c ¼ Z A ¼ 68:5, and Z c ¼ Z B ¼ 23:5 for the hopping between terrace sites, the hopping between sites at A steps ((1 0 0) like steps), and the hopping between sites at B steps ((1 1 1) like steps), respectively. Therefore, denoting the diffusion barrier on terraces (n i ¼ n f ¼ 3) by E, the diffusion barriers E A and EB at steps (n i ¼ n f ¼ 5) of both types A and B are E A ¼ 0:92E and E B ¼ 1:42E, respectively. The enhanced diffusion along steps of type A (compared to the slow diffusion along B steps) leads to the formation of triangular islands with A steps (cf. Fig. 5) as observed in the STM studies. We used E A < EB to obtain smooth growth at the substrate steps. In addition, we also studied the reverse case (E A > EB ), where rough zig-zag like steps are produced [30]. The characterization of this growth, however, is beyond the scope of this paper. The hopping frequency on terraces ranged from 7 to 10 7 Hz (equivalent to =F ¼ 2 ð Þ in the study presented here. Fig. 4(a) presents typical configurations during growth. On the lower part of the drawing, a substrate step is illustrated. A triangular island with A steps is grown on the lower terrace in front of the step. In addition, two pathways of diffusing monomers are depicted. Pathway I shows the diffusion for a monomer which is deposited on the lower terrace and hops to an A step of the island. It is captured by the island if it is on third position of the pathway, since the barrier to leave the island (e.g. to hop backwards to the second position) is rather high (1:67E, cf. Fig. 4(b)). As mentioned above, the diffusion barrier to hop along the A step is much smaller. Therefore, the monomer most probably travels along the A step until it is captured either by a kink or by a second monomer at the step.

6 112 J. Wollschläger et al. / Applied Surface Science 219 (2003) temperature and deposition rate were studied. Comparing the simulated island densities with Eq. (1), we obtained the critical nucleus size i ¼ 1 from both kinds of simulations. This can be explained by the large barrier of 1:33E for dissociation of dimers. 4. Results of kinetic Monte Carlo simulations Fig. 5. Kinetic Monte Carlo simulations of the growth of 0.1 ML at various temperatures. The field has size lattice sites. The terraces are 128 sites wide. Therefore, one substrate step and two terraces are presented. The position of the substrate step is marked by the dashed line. eposition temperatures: (a) k B T=E ¼ 0:0549, (b) k B T=E ¼ 0:0587, (c) k B T=E ¼ 0:0664, and (d) k B T=E ¼ 0:0702. For higher temperatures, the island density decreases and an increasing amount of material grows at the steps. Assuming that the monomer is captured by a B step, the situation for backward hopping is quite similar due to the high diffusion barrier. On the other hand, the diffusion along the B step is very improbable as well (cf. Fig. 4(b)). Therefore, the monomer remains on the site where it is attached to the island. For a monomer deposited on the upper terrace, a further pathway (pathway II) is presented in Fig. 4(a). The diffusion pathway crosses the substrate step. Here, the monomer occupies a transient state on the third position, since it only has two next neighbors at this position. There is an equal probability for the monomer to hop from the third position backward on the upper terrace or to hop to the fourth position on the lower terrace at the substrate. The diffusion barriers for both hops are 0:67E. The soundness of our KMC model was proven by performing simulations on singular surfaces (without substrate steps). The dependence on both substrate Fig. 5 shows simulated surface images after deposition of 0.1 ML at various temperatures. The higher the temperature, the smaller the number of islands on terraces and the more material is incorporated in substrate steps. The steps are very smooth with few kinks since the diffusion along the A steps is very fast. enuded zones with lower island densities are formed close to substrate steps because the steps consume deposited material. This can clearly be seen in Fig. 5(b) and (c). Furthermore, the few islands formed in the simulation presented in Fig. 5(d) are far away from the substrate steps. Simulations at various temperatures were also performed for larger coverages than 0.1 ML to determine the maximum island density, since many theoretical result are strictly valid only for maximum island density. The maximum island densities, however, do not significantly disagree with the island densities at 0.1 ML coverage. Nevertheless, Fig. 6 depicts the temperature dependence of the maximum island density for a terrace width w=a ¼ 128 sites, where a denotes the surface lattice constant. The solid line is the island density we obtained for nucleation on singular surfaces n sing x / 1=3 ¼ 1=3 0 exp E 3k B T (10) (cf. Eq. (1) with the critical cluster size i ¼ 1). The island density on the vicinal surface starts to deviate from the island density on singular surfaces for temperatures higher than k B T=E ¼ 0:053 (or E =k B T < 19 or =F > 10 5 ). There is a small temperature range (0:053 k B T= E 0:059 or 17 E =k B T 19 or 10 5 < =F < 10 6 ) which is well described by n x / 1=2. This proportionality has been concluded for island density on vicinal surfaces assuming that the islands start to coalesce and that the diffusion length is larger than terrace width w. This assumption, however, is not

7 J. Wollschläger et al. / Applied Surface Science 219 (2003) Fig. 6. Temperature dependence of the maximum island density for deposition on a vicinal surface with terrace width w=a ¼ 128. The solid and the dashed lines are fits of the data to Eqs. (10) and (11), respectively. fulfilled as demonstrated in Fig. 5(a) and (b). A better agreement over a larger temperature range is obtained if n x / 1 ¼ 1 0 exp E (11) k B T is used to describe the temperature dependence of the island density in this temperature range. Eq. (11) was derived for deposition of less material than would be necessary to obtain island coalescence. However, the island densities deviate from Eq. (11) for higher temperatures than k B T=E ¼ 0:071 (or E =k B T < 14 or =F ¼ 10 7 ). At such high temperatures almost all material travels to the substrate steps and islands are rarely formed. Since island densities at these deposition conditions are very small, we performed several simulations to minimize numerical errors. All these simulations underline that the island density is less than predicted by Eq. (11), as shown in Fig. 6 for E =k B T ¼ 14. For practical reasons, it is much easier to investigate island densities at a certain coverage than at the coverage of maximum island density. Since 0.1 ML CaF 2 have been deposited in the STM experiments we studied the nucleation and growth of 0.1 ML on vicinal surfaces, as well. Fig. 7(a) presents the Fig. 7. Analysis of the Monte Carlo simulations for R ¼ 0:1 ML/s and 0.1 ML deposited coverage on vicinal surfaces of various terrace widths. (a) Island density. The solid line describes the island density for deposition on a singular surface. (b) Material incorporated into steps. temperature dependence of island density on surfaces with different terrace widths. It is obvious that the temperature dependence of the island density for the terrace width w=a ¼ 128 is almost identical with the temperature dependence of the maximum island density (cf. Fig. 6). The solid line in Fig. 7 represents the island density for singular surfaces. As demonstrated before, the island density on vicinal surfaces is smaller than the island density on singular substrates for high deposition temperatures. The onset of deviation from the behavior described by Eq. (10), however, depends on the terrace width w. The smaller the terrace width w,

8 114 J. Wollschläger et al. / Applied Surface Science 219 (2003) Fig. 8. ependence of island density on terrace width for fixed values of =F (coverage Y ¼ 0:1 ML). For low temperatures (small =F) the island density is almost independent of the terrace width. For high temperatures (large =F) the island density decreases with decreasing terrace width w. No clear power law is observed for the case of large =F. Fig. 9. Kinetic phase diagrams as obtained from the Monte Carlo simulations with terrace nucleation in the upper right part and step nucleation in the lower left part. While the position of the phase boundary depends on the transition criterion n c x, the slope of the phase boundary is described by Eq. (12). the lower the temperature at which less islands are formed on vicinal surfaces. The smaller island density on vicinal surfaces is accompanied by substantial growth of deposited material at substrate steps. This effect is obvious in Fig. 7(b), where the material coverage grown at substrate steps is shown. The coverage at steps cannot exceed 0.1 ML (solid line in Fig. 7(b)) because this amount of material is deposited in the KMC simulations. Fig. 8 presents the dependence of island density n x on terrace width w for fixed =F. For small values of =F the island density is almost constant since nucleation on terraces dominates. Increasing =F the island density decreases more drastically for vicinal surfaces with short terraces than for surfaces with large terraces. However, we do not obtain clear power laws n x / w n even for the highest =F. Thus, we are not able to decide unambiguously whether Eq. (4) or (5) is valid although it seems that Eq. (5) with n x / w 4 describes the behavior better than Eq. (4) with n x / w 3. The goal of this combined STM and KMC study is to determine the kinetic phase diagram of terrace and step nucleation. Therefore, we needed a criterion for the transition which can be applied to both the STM experiments and the KMC simulations. A very simple criterion which can easily be applied is to define a critical temperature T c as the temperature where island density decreases below a critical number n c x (that can be introduced quite arbitrarily). We tested several critical island densities and the results are presented in Fig. 9 as an Arrhenius plot of the terrace width w versus the inverse critical temperature. The line of critical temperature shows Arrhenius behaviour: w / exp E (12) 2k B T c (solid lines in Fig. 9). To the left (high temperature side) of a particular n c x line it is n x < n c x and to the right (low temperature side) of the line it is n x > n c x. The higher the critical island density is, the stronger the lines of critical temperature are shifted toward lower temperatures. eviations from Arrhenius behavior for large terrace widths are observed if the selected critical island density is very high. This can clearly be seen in Fig. 9 for n c x ¼ 10 3 and w=a ¼ 128. The reason for using an arbitrary but fixed value n c x w 2, as we do in this paper lies in the analysis of STM micrographs. As a different criterion, n c x ¼ w 2 might be applied, since w is the natural length scale introduced by the vicinality of the surface. We will

9 J. Wollschläger et al. / Applied Surface Science 219 (2003) emphasize the importance of the correct criterion in a different paper [31]. Eq. (12) can also be described via w / 1=2 ðt c Þ. Therefore, the ratio between p terrace width w and diffusion length l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ðt c Þt is constant. Setting t ¼ 1 s which is the time needed to deposit 0.1 ML in our KMC simulation, values of 4.5 and 8.3 for the ratios w=l for n c x ¼ 10 4 and 10 5, respectively, pffiffiffiffiffi are obtained. Setting t ¼ t ML these ratios are 10 times larger. 5. iscussion and conclusion The objective of this report is to obtain a kinetic phase diagram from the KMC simulations and to apply this phase diagram to the STM studies on the system CaF 2 /Si(1 1 1) to determine the height of the diffusion barrier of CaF molecules which diffuse on the Si(1 1 1) substrate. Therefore, we used the criterion of critical island density to describe the transition from terrace to step nucleation. Myers-Beaghton and Vvedensky used a different criterion in their study [15]. They defined the critical temperature as the temperature at which 99% of the growth of a substrate step areis due to incorporation of monomers and 1% is due to incorporation of islands which are formed on terraces during deposition. The critical temperature T c determined via this criterion follows Eq. (7). Here, it is important that the phase boundary between terrace and step nucleation is described by w / 1=4 ðt c Þ while we obtain w / 1=2 ðt c Þ from our criterion of critical island density. It is not surprising that both criteria lead to different kinetic phase diagrams. Unfortunately, it is not possible to apply the criterion of Myers-Beaghton and Vvedensky to STM studies, since it cannot be determined from STM micrographs how much material at the substrate step is due to incorporation of monomers and how much material is due to incorporation of islands. Therefore, it is much easier to work with the criterion of critical island density introduced here. This latter criterion can also be applied to island density on vicinal surfaces as evaluated from scaling arguments (cf. Eqs. (4) and (5)). This leads to w / 1=3 ðt c Þ and w / 1=4 ðt c Þ, respectively, if a size of i ¼ 1 of the critical island is adopted. A fundamental assumption used to derive Eq. (5) is that the growth of islands is diffusion-limited. This assumption may not be valid for the temperature range which is of interest for the transition from terrace to step nucleation in our study. Probably, the diffusion of monomers is so fast that the growth has to be described as ballistic transport and the continuum equations on which the scaling arguments are based are not valid in this temperature range. Therefore, the motion of monomers is well described by a random walk of a single monomer which finally is most probably incorporated into a growing step while it can occasionally meet a second monomer to form a nucleus of an island [30,31]. Finally, we lke to mention that the clear power law dependence of Eq. (12) is very surprising since the island density does not depend on =F nor on terrace width w via a power law. This emphasizes that the important time scale during nucleation stage is the diffusion time w 2 which has to be compared with the deposition time t ML. Applying Eq. (12) to the phase diagram of CaF/ Si(1 1 1) as obtained from our STM studies, we conclude that the height of the diffusion barrier of CaF monomers on Si(1 1 1) is 1.8 (0.2) ev. It has to be emphasized that this barrier is an effective barrier, since besides CaF monomers, there are SiF x species on the surface which can block some sites. Since all islands observed in the STM micrographs have the same height, they are nucleated on the bare Si substrate while no islands are formed on the F-covered parts of the surface. This effect of site blocking is confirmed by X-ray photoemission spectroscopy [21]. Therefore, CaF 2 molecules which are deposited on SiF x occupied areas, travel until they reach the bare Si substrate where they are dissociated. Therefore, comparing STM results and KMC studies it is clear that our KMC model simplifies the experimental situation. However, the main effect of dissociation CaF 2 molecules is that more than one site is occupied after dissociation. Therefore, number of sites which can be occupied by CaF molecules decreases with increasing coverage. Since we restrict our study on the very early stages of nucleation with coverage Y ¼ 0:1 ML we believe that the dissociation process does not influence substantially our analysis. An additional complication in the analysis of nucleation and growth of CaF islands may be caused by the underlying 7 7 reconstruction while a 1 1

10 116 J. Wollschläger et al. / Applied Surface Science 219 (2003) structure is used in the KMC simulations. The 7 7 reconstruction, however, is not stable under the influence of CaF 2. It has been demonstrated that the 7 7 reconstruction is very quickly removed probably due to the reaction of CaF 2 with the Si(1 1 1) surface [32]. The barrier of 1.8 ev obtained here has to be compared with barriers of 1.4 and 1.6 ev for the diffusion of CaF 2 molecules on the CaF interface layer and on CaF 2 films, respectively [33]. Therefore, the transition from terrace to step nucleation should occur at lower critical temperatures for the growth of CaF 2 on both the CaF interface layer and thicker CaF 2 films. However, various studies demonstrated that step nucleation is suppressed even at very high temperatures, where the CaF 2 monomers can easily travel to substrate induced steps [34 36]. This effect has been attributed to the B orientation of the CaF 2 layers [37]. The different orientation of the CaF 2 layers lead to too long next neighbor distances so that the CaF 2 monomers cannot form strong bonds with step atoms. In conclusion, we presented a KMC study on the different regimes of terrace and step nucleation on vicinal surfaces. We obtained a kinetic phase diagram for terrace and step nucleation by introducing the concept of critical island density to define the critical temperature for the transition from terrace to step nucleation. The phase boundary follows w / expð E =2k B T c Þ. This analysis has been applied to the growth of CaF layers on Si(1 1 1). The diffusion barrier of 1.8eV for CaF/Si(1 1 1) has been determined from the kinetic phase diagram of CaF/ Si(1 1 1). Acknowledgements We kindly acknowledge financial support by the eutscher Akademischer Austauschdienst (AA) and the Swedish Institute (SI). References [1] I.V. Markov, Crystal Growth for Beginners, World Scientific Publishing, Singapore, [2] W.K. Burton, N. Cabrera, F.C. Frank, Philos. Trans. R. Soc. 243 (1951) 299. [3] J.A. Venables, G.L. Price, in: J.W. Matthews (Ed.), Epitaxial Growth (Part B), Academic Press, New York, 1975, p [4] J.A. Venables, G..T. Spiller, M. Hanbücken, Rep. Prog. Phys. 47 (1984) 399. [5] J.A. Venables, Surf. Sci (1994) 798. [6] H. Brune, Surf. Sci. Rep. 31 (1998) 121. [7] G.S. Bales,.C. Chrzan, Phys. Rev. B 50 (1994) [8] J.W. Evans, M.C. Bartelt, J. Vac. Sci. Technol. A 12 (1994) [9] M.C. Bartelt, L.S. Perkins, J.W. Evans, Surf. Sci. 344 (1995) L1193. [10] J.W. Evans, M.C. Bartelt, Langmuir 12 (1996) 217. [11] J. Wollschläger, Th. Schmidt, M. Henzler, M.I. Larsson, Surf. Sci (2000) 566. [12] A. Pimpinelli, P. Jensen, H. Larralde, P. Peyla, in: Z. Zhang, M.G. Lagally (Eds.), Morphological Organization in Epitaxial Growth and Removal, World Scientific Publishing, Singapore, 1998, p [13] A. Pimpinelli, P. Peyla, Int. J. Modern Phys. B 11 (1997) [14] G.S. Bales, Surf. Sci. 356 (1996) L439. [15] A. Myers-Beaghton,.. Vvedensky, Phys. Rev. B 42 (1990) [16] A. Myers-Beaghton,.. Vvedensky, J. Phys. A: Math. Gen. 24 (1991) L35. [17] A. Myers-Beaghton,.. Vvedensky, Phys. Rev. A 44 (1991) [18] A.L. Aseev, A.V. Latyshev, A.B. Krasilnikov, Surf. Rev. Lett. 4 (1997) 551. [19] A. Ishizaka, T. oi, M. Ichikawa, Appl. Phys. Lett. 58 (1991) 902. [20] J.. enlinger, E. Rotenberg, U. Hessinger, M. Leskovar, M.A. Olmstead, Phys. Rev. B 51 (1995) [21] M.A. Olmstead, Thin Films: Heteroepitaxial Systems, World Scientific Publishing, Singapore, 1999, p [22] T. Nakayama, M. Katayama, G. Selva, M. Aono, Phys. Rev. Lett. 72 (1994) [23] C.A. Lucas,. Loretto, G.C.L. Wong, Phys. Rev. B 50 (1994) [24] T. Sumiya, T. Miura, S. Tanaka, Jpn. J. Appl. Phys. 34 (1995) L1383. [25] T. Sumiya, T. Miura, S. Tanaka, Surf. Sci (1996) 896. [26] T. Sumiya, Appl. Surf. Sci. 156 (2000) 85. [27] M.I. Larsson, H. Frischat, J. Wollschläger, M.C. Tringides, Surf. Sci. 381 (1997) 123. [28] M.I. Larsson, Phys. Rev. B 56 (1997) [29] M.I. Larsson, Phys. Rev. B 64 (2001) [30] M. Bierkandt, Ph.. Thesis, Hannover, [31] M. Bierkandt, J. Wollschläger, M.I. Larsson, in preparation. [32] J. Wollschläger, A. Meier, J. Appl. Phys. 79 (1996) [33] U. Hessinger, M. Leskovar, M.A. Olmstead, Phys. Rev. Lett. 75 (1995) [34] A. Klust, H. Pietsch, J. Wollschläger, Appl. Phys. Lett. 73 (1998) [35] J. Wollschläger, A. Klust, H. Pietsch, Appl. Surf. Sci (1998) 496. [36] J. Wollschläger, H. Pietsch, A. Klust, Appl. Surf. Sci (1998) 29. [37] J. Wollschläger, Appl. Phys. A 75 (2002) 155.