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Growth processes in amorphous metallic films: A computer simulation
Thin Solid Films, 158 (1988) 299 312
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PREPARATION AND CHARACTERIZATION
G R O W T H PROCESSES IN AMORPHOUS M E T A L L I C FILMS: A C O M P U T E R SIMULATION R. MANAILA AND A. DEVENYI
Institute of Physics and Technology of Materials, Bucharest (Romania) P. B. BARNA AND G. RADNOCZI
Research Institutefor TechnicalPhysics, Budapest (Hungary) (Received October 27, 1987; accepted December 1, 1987)
A polytetrahedral, three-dimensional computer model was developed under a variety of physical assumptions in order to simulate an amorphous metallic film in its initial (granulated) growth stage. It was shown that the growth conditions imposed and the diffusion length of the adatoms influence the shape of the cluster. Topological parameters (degree of local tetrahedral perfection, average coordination number, atomic pair distribution ) were also derived. The frustrated character of the structure is suggested by the energy-per-adatom evolution during growth. In model alloy clusters, the built-in degree of frustration could be varied, leading to different degrees of chemical short-range order.
1. INTRODUCTION
The processes by which amorphous films grow when deposited in an atom-byatom way (vapour deposition, sputtering etc.) are still poorly understood. In the first growth stage, the film consists of clusters of a more or less regular shape deposited on the substrate, often forming a fractal-like system 1. A large amount of work has been dedicated of late to Monte Carlo simulations of nucleation and cluster growth on a mainly statistical basis 2-7. The influence of the bonding strength between atoms was also taken into account in ref. 2. However, most of this work is restricted to essentially two-dimensional considerations, the clusters being constructed on a two-dimensional lattice. In the frame of this model, the amorphous structure can be simulated only by drastically restricting the adatom mobility on the cluster surface. Thus the growth process is reduced to a geometrical problem of maximal capture section. Simulations of columnar growth 6's are also two dimensional. In this case, the finite adatom mobilities on the cluster and on the substrate are also neglected, the geometrical aspect of"self-shadowing" being again overstressed. In contrast, models for amorphous metals 9-11 have been rather successful in simulating their structural (radial distribution function) and thermodynamic (energy, entropy) features, but very little was predicted about the real process by which an amorphous metallic cluster is formed. 0040-6090/88/$3.50
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We will concentrate in the following on the mechanisms by which an average individual three-dimensional cluster can grow under a variety of physical (experimental) conditions. It is of particular interest to know how these growth conditions can affect the cluster structure and energy. It will be shown that, in the limit of very low a d a t o m mobility(T, ubs,~,.~0 K, low a d a t o m flux), the simulation reduces to the geometrical maximal capture section approach, which can explain dendritic 5 or c o l u m n a r growth ~''s. However, more significant results can be expected in the more realistic range of finite mobilities. The average dimension and shape of clusters in a granulated film land their time evolution) depend in a complex way on experimental parameters: temperature ~ of the substrate, energy E, of impinging adatoms, deposition rate l@ strength of the atomic interaction etc. In this complex situation, c o m p u t e r simulations can shed some light on the effect of one or more of these parameters. 2. GRO~&'TH ALGORITHM
We have constructed a sequential growth algorithm leading to a polytetrahedral dense r a n d o m packing of atoms. Sequentially built models I 2 14- with different construction recipes, have been proposed for a m o r p h o u s metals since that of Bennett 12. Ichikawa 15 built polytetrahedral models, with various degrees of tetrahedral perfection in an attempt to simulate the structure of a m o r p h o u s iron. O u r p r o g r a m starts with a seed of several atoms (e.g. 7 atoms; see Fig. I i lying on the substrate and forming a n u m b e r of irregular triangular "nests". These atoms simulate a nucleation centre with a r a n d o m configuration on the surface of the substrate. An a d a t o m impinging upon the cluster surface is allowed to migrate over a restricted distance until it reaches a nest which best satisfies a given condition where it is arrested in the position of the nest's tetrahedral apex. Three new lateral nests are created thereby, which are added to the nest list, while the occupied nest is deleted from the list. Before inclusion in the list, the apexes of new nests are checked for overlap with already-deposited atoms. The process is continued until the desired n u m b e r of atoms has been added. The occupation of lateral nests on a tetrahedron (tetrahedral faces) precludes the formation of h.c.p, or f.c.c, close packings 15 which would lead to a '~crystalline" structure, and favours, on the contrary, local icosahedral order in our model. A nest is considered as such only if its edge lengths R i ~< R M = bR~, where b embodies the degree of disorder allowed in the build-up of the model (in our simulation, b = 1.2 or 1.4). Consequently, each nest can be assigned a perfection parameter k = (ZiRi)/3R E. The energy of the assembly is calculated as a sum of pair potentials of the Lennard Jones 6-12 type, an equilibrium a t o m diameter RE being assumed. All distances in the model were calculated from a t o m lot nest apex) spatial coordinates and expressed in units of RE/0.031 62. Clusters of 500 and 800 atoms were grown by this algorithm under various conditions in an attempt to simulate those of the actual growth process.
C O M P U T E R S I M U L A T I O N OF A M O R P H O U S METAL FILM G R O W T H
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G R O W T H SITUATIONS
Which conditions should be required for the choice of a particular nest? We could impose occupancy of the nest of(A) minimal energy, (B) maximal coordination number N, (defined as the number of neighbouring atoms at R ~< Ru), (C) maximal surface section, normal to the direction of the impinging atom beam, (D) maximal number ofheteropairs AB (or homopairs AA,BB), in the case of an alloy AxB1 x, or (E) minimal distance to the geometrical centre (most regular shape of the cluster). Conditions (A) and (B) are the most appealing from a physical point of view, especially (B), which should be less sensitive than (A) to the detailed shape of the interatomic potential. Condition (B) will be considered first (variant (A) yields very similar results in each of the following approximations). 4.
A P P R O X I M A T I O N STEPS T O W A R D S A REALISTIC G R O W T H PROCESS
Having established a criterion for nest occupancy, we can conduct the growth process of the cluster in different ways, designed to approach gradually the complexity of the real process.
4.1. Approximation 1 The adatom is free to select from the nest list the one of maximal Nn (i.e. it can freely migrate on the cluster surface, in search of the optimal site to rest into). This condition corresponds to very high Ts and high adatom initial energy Ea (although very energetic atoms could re-evaporate). In addition, it also implies that Vd is very low, so that each adatom is allowed enough time to migrate before being "quenched" by a subsequent atom layer. In this case, the algorithm explores the actual list of nests, chooses the optimal one and places the new adatom at its apex. This variant generates unphysical, sometimes very elongated shapes. They originate in the inhomogeneous nature of the cluster surface (areas of higher and lower coordination) but cannot simulate actual situations. 4.2. Approximation 2 We have to take into account that, for reasonable values of T~, Ea and Vd, a new adatom can move inside a limited area only on the surface of the cluster. This is due to its gradual "thermalization" by inelastic interactions but also to the limited time at its disposal before it is "quenched" by the next layer. Therefore we allow the adatom to choose the optimal nest from a restricted subset only, comprising those nests whose apexes are nearer than a distance Ro from a randomly selected nest (the site of adatom incidence). With R D = 10R E, the adatoms still tend to concentrate on certain zones of the cluster surface, as can be seen by following the patterns of successive adatom positions. The 500-atom cluster (Fig. 1) has an elongated external shape, which is seen to include voids. Thus at high mobilities the model growth is governed by surface inhomogeneities, whose influence can propagate over several interatomic distances. When RD is limited to 3RE, more compact clusters of irregular shape result. The
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4.3. Apt~roximation 3 A further step t o w a r d s the c o m p l e x i t y of the real process requires the u n d e r s t a n d i n g that the nests at the base of the cluster (up to a limiting height Z of the o r d e r of R~) have a higher p r o b a b i l i t y of being occupied than the m o r e highly placed nests. This is due to the fact that the n u m b e r N~ of inferior nests can be occupied by a t o m s falling from a b o v e and also by those which m o v e a l o n g the s u b s t r a t e t o w a r d s the cluster, having fallen on the free area between the islands. In contrast, the s u p e r i o r nests, the n u m b e r of which is N~, can be occupied, r o u g h l y speaking, only by a t o m s falling directly on them, if we a s s u m e a r a t h e r low a d a t o m m o b i l i t y on the cluster surface. The relative role of surface t r a n s p o r t c o m p a r e d with v a p o u r phase t r a n s p o r t at different coverages was also evidenced by c o m p u t e r s i m u l a t i o n in refs. 2 a n d 3. The ratio of the p r o b a b i l i t i e s for o c c u p a n c y of any nest in the inferior ( p r o b a b i l i t y P,) or s u p e r i o r ( p r o b a b i l i t y P~) ranges is given by the c o v e r a g e degree C of the s u b s t r a t e : Pi/P~ ~ (1 - C ) / C . In our algorithm, c o n v e n i e n t l y simple instructions allowed this ratio to be modified by a chosen factor s. K n o w i n g C at a given m o m e n t of an actual d e p o s i t i o n experiment, we can give s a suitable value, which will c o r r e s p o n d to that precise g r o w t h m o m e n t . Clusters g r o w n in this a p p r o x i m a t i o n of variant (BI (Fig. 3) d i s p l a y a shape similar to that o b t a i n e d by a p p r o x i m a t i o n 2.
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SHAPE ('I1ARACTERIZATION
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At this stage, a more quantitative characterization of the cluster shape seems worth attempting. The cluster, considered at its different growth stages, can be viewed as a collection of islands in a granulated film, possibly forming a fractal system t'~'. We could then take advantage of the fractal dimensionality ~ D, as derived from the dependence of the number N of atoms in the cluster and its gyration radius R~: N -~ RgD. The value D refers in our case to the sequence of growth stages and characterizes the scale-insensitive shape of the clusters in the size range where the system shows fractal properties. Disc-like shapes would give D = 2, while hemispherical islands would correspond to D = 3. Irregular, tree-like or columnar shapes should exhibit D values below these. The Witten Sanders model TM was extended by Meakin'* by the simulation of aggregates formed on a seed particle under diffusioncontrolled conditions. It was shown that D obeys the general law D ~ 5d/6, where d is the euclidean dimensionality of the space. This relation predicts D = 1.7 and D = 2.5 for two-dimensional and three-dimensional systems respectively. We calculated the log Rg vs. log N dependence at each 10-atom stage of growth in the 500-atom clusters (Fig. 4), Rg being referred to a [001] axis, normal to the substrate and passing through the centre of gravity of the cluster. The position of the centre of gravity was calculated at each growth stage.
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The gyration radius is in units of RF'0.031 62. In clusters grown in variants (B) or (E), the dimensionality D was found to range between 2.0 and 2.5 for different values of the growth parameters RD and b. The linear log Rg vs. log N dependence extends over the whole range of cluster sizes investigated (10 ~< N ~< 500, i.e. 1.7 decades), with a correlation coefficient of 0.984 0.999. It can be seen that our growth algorithm leads to D values between those mentioned above, as expected for three-dimensional growth on a plane substrate. This analysis of fractal properties in model clusters seems to be worth attempting, although larger-scale simulations are required for more convincing conclusions.
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6.
T O P O L O G I C A L FEATURES
At this stage, it is relevant to ask how the growth conditions affect the structure of the cluster, as characterized by geometrical and topological features.
6.1. Degree of tetrahedral perfection The tetrahedral perfection parameter k (see Section 2) was determined for all the nests formed during the cluster build-up, including the occupied nests as well as those left unoccupied. It can be taken as a measure of the topological disorder of the polytetrahedral packing. Also, a large value ofk should be related to low densities in these non-relaxed models. Clusters grown under the condition of most regular shape (variant (E)) show for b = 1.2 a rather narrow k distribution, ranging between 1.00 and 1.15 (Fig. 5(a)). If the nearest-neighbour searching condition is relaxed to b = 1.4 (Fig. 5(b)) a considerably higher topological disorder results, k being distributed in a quasigaussian way between 1.03 and 1.34. If, instead of variant (E), we build according to the criterion of maximal local coordination number of each adatom (variant (B)), the topological disorder is greater, k being now concentrated in the range 1.22-1.35 (Fig. 5(c) for b = 1.4, RD = 3RE). Thus the preference for distorted sites, with a higher than average coordination, leads to a more disordered structure.
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Analysis of the k distribution at different growth stages does not reveal significant differences. This fact suggests that the growth process is stationary and could be continued without altering the topological properties of the cluster. 6.2. Coordination m~rnber of the aloms The number N, of nearest neighbours at R ~< RM = hR~ gives insight into the local structure of the model. Atoms of low coordination, if not belonging to the external surface, indicate the presence of local defects in the structure, possibly concentrated onto internal surfaces. Figure 6 shows the distribution of N n values among bulk and surface atoms for a cluster grown according to variant (B). The latter atoms were identified by the criteria of belonging to a surface nest and having N. ~< 6. A striking feature about the N. distribution is that there is a number of bulk atoms of low coordination with N,, ~< 6, which points to a large concentration of local defects. The most probable coordination of the bulk atoms is 7 8, a rather low value, which could be explained by the non-relaxed, low density character of the model.
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6.3. Atomic pair distribution The atomic pair distribution functions IPDFs) (Fig. 7) give insight into the degree and character of the local order, generated by the imposed growth conditions. Variant (B), which requires maximal coordination, results in a P D F showing a rather large first maximum and a split second m a x i m u m (Fig. 7(a) for RD = 3R~), as commonly found in metallic glasses and dense-random-packing-of-hard-spheres (DRPHS) models 19. The correlation between atom positions extends up to (4-5)R~ only, which should be again attributed to the non-relaxed character of the model. An increase in the diffusion length to 10RE results in a peculiar sharpening of the first peak, as well as of those at certain further distances, due to external shape-related effects, which alter the bulk PDF. The influence of the b parameter (relaxation of the first neighbour conditionl on the shape of the first P D F peak can be seen in Figs. 7(b) and 7(c) for a cluster grown in variant (E). An increase in b from 1.2 to 1.4 results in a considerable spreading of the distances. Variant (E) (most regular shape) gives rise to a P D F whose features are
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considerably broader than in variant (B). Also, the first subpeak in the second maximum is fainter in variant (E) than predicted by D R P H S models. As this subpeak is mainly due to distances between lateral nests on a tetrahedron, it is related to the presence oficosahedral local order. All these facts recommend variant (B) as the most suitable for growing D R P H S assemblies. 7. ENERGY OF THE CLUSTER
The cluster energy and its evolution during the growth process were also investigated. The energetic stability of small clusters relaxed under Lennard-Jones or Morse potentials has received much attention 2° 22. It was shown that polytetrahedral clusters are most probably among the minimal energy configurations for less than 50 atoms 22. Among them, the icosahedron (13 atoms), which belongs to the polytetrahedral Werfelmeier sequence for small clusters, and possibly also larger icosahedral multiply twinned structures 2°'22 are particularly stable. Generalized valence bond calculations 23 also showed that the lowest energy isomers of small metal clusters belong to the polytetrahedral type.
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At the same time, free fragments of crystal lattices of the same size are of higher specific energy so that, if they appear, they are very short lived, relaxing (if allowed to) towards more stable configurations because the energy barriers for rearrangements are low for small clusters. The total energy of our model cluster was calculated at each stage as E = ~EiEj Vu, where i runs over the N atoms in the cluster,./denotes the members of the nest triplet occupied by a t o m i (thereby forming a tetrahedral unit) and ~i is a L e n n a r d - J o n e s 6 12 potential of arbitrarily chosen magnitude. The dependence of specific energy E, = E/N on N in clusters grown by variant (B) is shown in Fig. 8. For b = 1.4, E, increases steadily with N, although the slope decreases, E~ reaching asymptotically a constant value, which would permit the further growth of the cluster. For more perfect clusters (h = 1.2), E~ increases up to a size of 100 200 atoms, although to a lower value than in the case o f b = 1.4. The range above N ~ 150 is relatively flat, allowing the further growth of the cluster from the point of view of the energy. Shallow local minima can be noticed for some of the runs, as are seen for about 200 and 400 atoms on Fig. 8. This type of E~(N) dependence is typical for frustrated structures, which are not allowed to reach their equilibrium state at any m o m e n t during their sequential formation. O u r polytetrahedral cluster with h = 1.2 is clearly unstable towards growth during the first stages (N ~< 100 200 atoms). However, we did not allow the whole structure to relax towards equilibrium (i.e. to decompose) but permitted only a "surface relaxation" of the uppermost layer of atoms by a d a t o m migration. Thus we have built a./rustrated structure, as could be present in thin a m o r p h o u s films, in
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which kinetic and energetic conditions during build-up prevent the attainment of an equilibrium atomic configuration. Our simulation results show that frustrated structures can reach a critical size, beyond which the addition of new atoms does not increase the specific energy, so that the growth process can be continued. There may be other cluster configurations of the same size which are more stable, but energy barriers towards rearrangements should be quite high for large clusters, implying collective movements of extended groups of atoms. Variant IA) has, as expected, a minimal asymptotic Es (about 0.5 in the range 200-500 atoms, in the units of Fig. 8), being built with precisely this requirement. 8. FACTORS WHICH FAVOUR COLUMNAR GROWTH (VARIANT (C))
In the extreme case when Ts, E, and Vd are very low, adatoms stick at the site where they have touched the surface of the cluster (with only minor relaxation possible). High incidence rates Vd can dislocate the atoms out of these impact sites. In such an ideal case, a probabilistic modelling approach seems suitable. In this approach the nest with the maximal upwards section S, as being the most likely to be filled, is chosen from the nest list for the next occupancy. Here S is the projection on the substrate of the nest surface, defined by its three atoms. It is the "capture crosssection" presented by the nest to an atom arriving at normal incidence. The atom beam was considered to fall normally to the substrate, but oblique incidence is as easy to consider. Figure 9 displays a projection of a cluster fragment possessing a striking columnar shape grown according to variant (C). Columnar microstructure (mainly at oblique incidence) is persistently reported in amorphous metallic films 6'8, being sometimes related to magnetic anisotropy. Recently, artificially grown columnar metallic structures 24 seem to confirm this assumption.
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The columnar microstructure can be given a purely geometrical explanation (%elf-shadowing" of certain low-lying nests by the already-deposited atoms25'2~), although some deposition factors, such as substrate bias, are known to be influential. The columnar cluster grown by our variant {C) tFig. 9) is an illustration of this geometrical effect. Recent work 2~' shows that, provided a basic columnar profile of adequate wavelength is present in the first layer, the diffusion equations can favour this shape, but only under conditions of low T and limited adatom mobility. In contrast, our simulation emphasizes the fact that self-shadowing can be effective only in certain experimental conditions (very low ~, E a and 1~). It is worth noting that, in r.f.sputtered Gd Co amorphous films, the columnar structure is obtained only for low gas pressures (low Va) and low discharge power {small E~,I2s, confirming the suggestions of the computer simulation. More realistic columnar shapes could be simulated by %ybridizing'" the rather restrictive condition (C) with the more physical constraints (Bt or {A). 9.
F R U S T R A Y E I ) C H E M I C A L ORDER IN ALI,OY I'II.MS
The chemical short-range order (CSRO) in A,.B 1 .~ alloys very often favours heteropairs AB over homopairs {wrong bonds) AA or BB on the grounds of lower enthalpy of formation. It has long been recognized that in amorphous films the CSRO is often frustrated in comparison with glassy materials {generally prepared from the melt), which are closer to the maximal CSRO degree (see ref. 29 for the AsxS1 x system). In our simulation, variant (D) {Section 3} should maximize the CSRO in amorphous films, under a given degree of structural frustration, controlled by the diffusion length RD. Thus the effect of CSRO frustration in amorphous alloy films could be put on a quantitative basis. Clusters of 500 atoms were grown for alloy compositions x = 0.1, 0.3 and 0.5 with restricted diffusion over distances RD -- 2 R ~ , 3 R E and 10R L. The nature of each new adatom was determined by a random number generator which imposed the condition of a given composition .v. Also, to each nest was attributed a number, related to the composition of the triplet, which allowed an adatom of a given nature to settle into the nest in such a way as to maximize the number of AB pairs. As was noticed above (Section 4), the shape of the resulting clusters depends on RD, but what interests us most is the degree n of CSRO in these models. As discussed by Cargill 3°. for amorphous alloys hAlt = { Z A B , , Z ; B t - - 1, whcrc ZAl~ is the coordination number of the species A by B and its statistically determined value is Z*aB = C ~ Z A Z R / ( C A Z A + c B Z B ) (where c is the concentration and Z the total coordination number). For complete disorder, n = 0 because ZA~ = ZAB, while n > 0 signals the presence of CSRO. A maximal n value can only be reached at a given composition x: n ~ x = xZa/"l I - x ) Z ~ < 1. This formula yields reliable values only for n < 1/3 in close-packed structures 3°. Also, the topology of the network can further limit the maximal CSRO that can be attained. As both A and B spheres have the same RE value in our model, we expect ZA = ZR. With this simplifying assumption, n can be derived from the total number
COMPUTER SIMULATION OF AMORPHOUS METAL FILM GROWTH
31 1
of AB pairs and the actual composition Xeff of the model: n = ( N A B ~ N A B ) - l, where NAB = 2Xeff(1 -- x e f f ) N and N -- NAA -b NAB -t- NBB. Table I shows the n values found in models of different frustration degrees (different R D values) together with the n max values calculated for the same models as discussed above. The slight variations in n max for the same intended x are due to statistical fluctuations in the composition Xeff that are still present at this size of the model. Table I shows, for compositions x = 0.1 and 0.3, that limitation of the adatom mobility on the cluster surface (small R D values) strongly frustrates the degree of CSRO. An increase in RD from 2R E to 10RE for x = 0.3 results in an n value almost twice as large but still far from the maximal value. For x = 0.1, the large statistical fluctuations in NAB preclude a meaningful comparison between the actual n value in the model and n max.At equimolar composition (x = 0.5), n is rather insensitive to Rt~. NAa
TABLE I CHEMICAL SHORT-RANGE ORDER r/ /~S. n max (CALCULATED FOR THE EFFECTIVE COMPOSITION Xeff) IN MODEL CLUSTERS GROWN ACCORDING TO VARIANT (D) WITH DIFFERENT DIFFUSION LENGTHS R D (RE, ATOM DIAMETER)
X
RD/R E
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3 10 2 3 10 3 10
0.138 0.142 0.384 0.348 0.350 0.534 0.532
0.160 0.165 0.623 0.534 0.538 0.873 a 0.880 a
0.259 0.342 0.183 0.208 0.326 0.303 0.314
a Unreliable values: see text.
Relaxation of one of these alloy models towards a larger degree of CSRO would probably encounter very large activation barriers and could be achieved only by extensive collective rearrangements. However, in a real film CSRO relaxation could proceed much more easily on annealing as a result of the presence of structural defects, which are not embodied in our models. 10. CONCLUSIONS A model of polytetrahedral, sequential growth under a variety of physical assumptions was devised which is particularly suitable for understanding structural particularities in amorphous discontinuous films in comparison with the more equilibrated glassy systems. These features originate in the frustrated character of the film structure as a result of the physical conditions which control the growth process, and they can be identified as follows. (1) The amorphous model clusters are energetically unstable in the first growth stages (specific energy E/N increases with N). Beyond a critical size, however, E/N displays a rather constant plateau. This suggests the possibility that the cluster can continue to grow, even if its configuration is not that of absolute minimal energy for
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the given N, because collective rearrangements would encounter high energy barriers. (21 Various quantities can be used to characterize the degree of structural disorder in the model cluster: nest perfection parameter k, distribution of coordination numbers and PDFs. They are controlled by the growth conditions imposed as well as by the nearest-neighbour disorder parameter h assumed in the simulation (related, in physical systems, to the detailed shape of the interatomic potentiall. (3) The degree of built-in CSRO (for alloy films) is lower than the upper limit set by composition and decreases with increasing frustration degree. The model allows visualization and quantification of the frustration degree by the limited number of accessible sites (inside the diffusion distance RDt which can be occupied by each newcoming adatom. REFERENCES I 2 3 4 5 6 7 8 9 10 II 12 13 I4 15 I6 17 18 19 20 21 22 23 24 25 26 27 28 29 30
R. Messier and J. E. Yehoda, J, ,41~pl. Phys,, 5~' { 1985) 373tL R. Messier, A. P. Giri and R. A. Roy, J. 1"ac. Sci. li,chmd..I, 2 (1984) 51)0, J, Salik, J. Appl. Phy.s., 57(1985) 5017: Phv,~. Rev. B, 32(1985) 1824. R.A. Outlaw and J. H. Heinbockel, Thin Solid Fibns, 123 ( 19851 159. P. Meakin, Phy.f. Rcr. B, 31(19851564:3O(1984)4207:Phy.~'.Rec, A,27(1983) 1495,261o. A. Barna, P. B. Barna, G. Radnoczi, |1. Sugawara and P. Thomas, Thin Solid Film,~, 4S { 1978) 163 K . H . Mfiller, ,/. Appl. Ph.v.~., 58 (1985) 2573. S.H. Garofalini. T. ttalicioglu and G. M. Pound, SuUI &i.. 114(1982) 161 | I. J. Leamy and A, G. Dirks, .I. Appl. Phy.~., 49 (1978) 3430. M.R. ltoare, J. Non-('r.t'.~l. 5"olid~, 31 (19781157. L. yon tleimendahl. J. Phv.~. F, 511975) LI41. R. Harris and L. J. Lewis, Phys. Rer. B, 25 (1982)4997. C. It. Bennett, J. Appl. Phys., 43 (I972) 2727. D. J, Adams and A. J. Matheson,,l. ('hem. Phys.. 56 11972) 1989. J.F. Sadoc, J. Dixmier and A. Guinier. J. Not>()'vst. Solids'. 12 (1973) 46. T. Ichikawa, Physica Status Solidi A, 29 (1975) 293. B.B. Mandelbrot, Fractals, Form, Cham'e and Dimension, Freeman, San Francisco, CA, 1977 H . E . Stanley, J. Phys..4, 10 (1977) [,21 I. T.A. Winen and k. M. Sanders. Phys. Rec. Lelt., 47(1981) 1400. J.L. Finney, B. J. Gellally and J. Wallace, in C. Hargittai, 1. Bakonyi and T. Kemeny leds. ), Metallic Ghtssev' Sciem'e and 7~'chmdo&v, Akadelniai Kiado, Budapest, 1981, p. 55. C.L. Briant and J. J. Burton, Phy,sica Status Solidi B, ,~¢5( 19781393. M . R . H o a r e a n d P. Pal, 4de. Phys., 2011971) 161. M.R. Hoare and P. Pal, J. Cry,s/. Growth, 1711972) 77. M . H . McAdon and W. A. Goddard II1, Phys. Rev. Lt'tI., 55 (1985) 2563: J. Non-Crvst. Solid~, 75 (1985) 149. M. Abe, M. Gomi and F. Yokoyama, J. Appl. Phy,s., 57 (1985) 3909. 1t. J. Leamy, G. H. Gilmer and A. G. Dirks, in E. Kaldis (ed.), ('urrent Topics in Materials Science, Vol. 6, North-Holland, Amsterdam. 1980. S. Kim and D. J. Henderson, Thin SolMFilms, 47(1977) 155. S. Lichter and J. (7hen, Phys. Rer. Lett., 56 (1986) 1396. V. Florescu and D. Serbanescu, personal communication, 1986. A.J. Apling, A. J. Leadbetter and A. C. Wright, J. Non-('rysl. Solid~, 23 (1977) 369. G.S. Cargill Ilk J. Non-('O'st. Solids', 43 (1981) 91.
299
PREPARATION AND CHARACTERIZATION
G R O W T H PROCESSES IN AMORPHOUS M E T A L L I C FILMS: A C O M P U T E R SIMULATION R. MANAILA AND A. DEVENYI
Institute of Physics and Technology of Materials, Bucharest (Romania) P. B. BARNA AND G. RADNOCZI
Research Institutefor TechnicalPhysics, Budapest (Hungary) (Received October 27, 1987; accepted December 1, 1987)
A polytetrahedral, three-dimensional computer model was developed under a variety of physical assumptions in order to simulate an amorphous metallic film in its initial (granulated) growth stage. It was shown that the growth conditions imposed and the diffusion length of the adatoms influence the shape of the cluster. Topological parameters (degree of local tetrahedral perfection, average coordination number, atomic pair distribution ) were also derived. The frustrated character of the structure is suggested by the energy-per-adatom evolution during growth. In model alloy clusters, the built-in degree of frustration could be varied, leading to different degrees of chemical short-range order.
1. INTRODUCTION
The processes by which amorphous films grow when deposited in an atom-byatom way (vapour deposition, sputtering etc.) are still poorly understood. In the first growth stage, the film consists of clusters of a more or less regular shape deposited on the substrate, often forming a fractal-like system 1. A large amount of work has been dedicated of late to Monte Carlo simulations of nucleation and cluster growth on a mainly statistical basis 2-7. The influence of the bonding strength between atoms was also taken into account in ref. 2. However, most of this work is restricted to essentially two-dimensional considerations, the clusters being constructed on a two-dimensional lattice. In the frame of this model, the amorphous structure can be simulated only by drastically restricting the adatom mobility on the cluster surface. Thus the growth process is reduced to a geometrical problem of maximal capture section. Simulations of columnar growth 6's are also two dimensional. In this case, the finite adatom mobilities on the cluster and on the substrate are also neglected, the geometrical aspect of"self-shadowing" being again overstressed. In contrast, models for amorphous metals 9-11 have been rather successful in simulating their structural (radial distribution function) and thermodynamic (energy, entropy) features, but very little was predicted about the real process by which an amorphous metallic cluster is formed. 0040-6090/88/$3.50
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We will concentrate in the following on the mechanisms by which an average individual three-dimensional cluster can grow under a variety of physical (experimental) conditions. It is of particular interest to know how these growth conditions can affect the cluster structure and energy. It will be shown that, in the limit of very low a d a t o m mobility(T, ubs,~,.~0 K, low a d a t o m flux), the simulation reduces to the geometrical maximal capture section approach, which can explain dendritic 5 or c o l u m n a r growth ~''s. However, more significant results can be expected in the more realistic range of finite mobilities. The average dimension and shape of clusters in a granulated film land their time evolution) depend in a complex way on experimental parameters: temperature ~ of the substrate, energy E, of impinging adatoms, deposition rate l@ strength of the atomic interaction etc. In this complex situation, c o m p u t e r simulations can shed some light on the effect of one or more of these parameters. 2. GRO~&'TH ALGORITHM
We have constructed a sequential growth algorithm leading to a polytetrahedral dense r a n d o m packing of atoms. Sequentially built models I 2 14- with different construction recipes, have been proposed for a m o r p h o u s metals since that of Bennett 12. Ichikawa 15 built polytetrahedral models, with various degrees of tetrahedral perfection in an attempt to simulate the structure of a m o r p h o u s iron. O u r p r o g r a m starts with a seed of several atoms (e.g. 7 atoms; see Fig. I i lying on the substrate and forming a n u m b e r of irregular triangular "nests". These atoms simulate a nucleation centre with a r a n d o m configuration on the surface of the substrate. An a d a t o m impinging upon the cluster surface is allowed to migrate over a restricted distance until it reaches a nest which best satisfies a given condition where it is arrested in the position of the nest's tetrahedral apex. Three new lateral nests are created thereby, which are added to the nest list, while the occupied nest is deleted from the list. Before inclusion in the list, the apexes of new nests are checked for overlap with already-deposited atoms. The process is continued until the desired n u m b e r of atoms has been added. The occupation of lateral nests on a tetrahedron (tetrahedral faces) precludes the formation of h.c.p, or f.c.c, close packings 15 which would lead to a '~crystalline" structure, and favours, on the contrary, local icosahedral order in our model. A nest is considered as such only if its edge lengths R i ~< R M = bR~, where b embodies the degree of disorder allowed in the build-up of the model (in our simulation, b = 1.2 or 1.4). Consequently, each nest can be assigned a perfection parameter k = (ZiRi)/3R E. The energy of the assembly is calculated as a sum of pair potentials of the Lennard Jones 6-12 type, an equilibrium a t o m diameter RE being assumed. All distances in the model were calculated from a t o m lot nest apex) spatial coordinates and expressed in units of RE/0.031 62. Clusters of 500 and 800 atoms were grown by this algorithm under various conditions in an attempt to simulate those of the actual growth process.
C O M P U T E R S I M U L A T I O N OF A M O R P H O U S METAL FILM G R O W T H
3.
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G R O W T H SITUATIONS
Which conditions should be required for the choice of a particular nest? We could impose occupancy of the nest of(A) minimal energy, (B) maximal coordination number N, (defined as the number of neighbouring atoms at R ~< Ru), (C) maximal surface section, normal to the direction of the impinging atom beam, (D) maximal number ofheteropairs AB (or homopairs AA,BB), in the case of an alloy AxB1 x, or (E) minimal distance to the geometrical centre (most regular shape of the cluster). Conditions (A) and (B) are the most appealing from a physical point of view, especially (B), which should be less sensitive than (A) to the detailed shape of the interatomic potential. Condition (B) will be considered first (variant (A) yields very similar results in each of the following approximations). 4.
A P P R O X I M A T I O N STEPS T O W A R D S A REALISTIC G R O W T H PROCESS
Having established a criterion for nest occupancy, we can conduct the growth process of the cluster in different ways, designed to approach gradually the complexity of the real process.
4.1. Approximation 1 The adatom is free to select from the nest list the one of maximal Nn (i.e. it can freely migrate on the cluster surface, in search of the optimal site to rest into). This condition corresponds to very high Ts and high adatom initial energy Ea (although very energetic atoms could re-evaporate). In addition, it also implies that Vd is very low, so that each adatom is allowed enough time to migrate before being "quenched" by a subsequent atom layer. In this case, the algorithm explores the actual list of nests, chooses the optimal one and places the new adatom at its apex. This variant generates unphysical, sometimes very elongated shapes. They originate in the inhomogeneous nature of the cluster surface (areas of higher and lower coordination) but cannot simulate actual situations. 4.2. Approximation 2 We have to take into account that, for reasonable values of T~, Ea and Vd, a new adatom can move inside a limited area only on the surface of the cluster. This is due to its gradual "thermalization" by inelastic interactions but also to the limited time at its disposal before it is "quenched" by the next layer. Therefore we allow the adatom to choose the optimal nest from a restricted subset only, comprising those nests whose apexes are nearer than a distance Ro from a randomly selected nest (the site of adatom incidence). With R D = 10R E, the adatoms still tend to concentrate on certain zones of the cluster surface, as can be seen by following the patterns of successive adatom positions. The 500-atom cluster (Fig. 1) has an elongated external shape, which is seen to include voids. Thus at high mobilities the model growth is governed by surface inhomogeneities, whose influence can propagate over several interatomic distances. When RD is limited to 3RE, more compact clusters of irregular shape result. The
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4.3. Apt~roximation 3 A further step t o w a r d s the c o m p l e x i t y of the real process requires the u n d e r s t a n d i n g that the nests at the base of the cluster (up to a limiting height Z of the o r d e r of R~) have a higher p r o b a b i l i t y of being occupied than the m o r e highly placed nests. This is due to the fact that the n u m b e r N~ of inferior nests can be occupied by a t o m s falling from a b o v e and also by those which m o v e a l o n g the s u b s t r a t e t o w a r d s the cluster, having fallen on the free area between the islands. In contrast, the s u p e r i o r nests, the n u m b e r of which is N~, can be occupied, r o u g h l y speaking, only by a t o m s falling directly on them, if we a s s u m e a r a t h e r low a d a t o m m o b i l i t y on the cluster surface. The relative role of surface t r a n s p o r t c o m p a r e d with v a p o u r phase t r a n s p o r t at different coverages was also evidenced by c o m p u t e r s i m u l a t i o n in refs. 2 a n d 3. The ratio of the p r o b a b i l i t i e s for o c c u p a n c y of any nest in the inferior ( p r o b a b i l i t y P,) or s u p e r i o r ( p r o b a b i l i t y P~) ranges is given by the c o v e r a g e degree C of the s u b s t r a t e : Pi/P~ ~ (1 - C ) / C . In our algorithm, c o n v e n i e n t l y simple instructions allowed this ratio to be modified by a chosen factor s. K n o w i n g C at a given m o m e n t of an actual d e p o s i t i o n experiment, we can give s a suitable value, which will c o r r e s p o n d to that precise g r o w t h m o m e n t . Clusters g r o w n in this a p p r o x i m a t i o n of variant (BI (Fig. 3) d i s p l a y a shape similar to that o b t a i n e d by a p p r o x i m a t i o n 2.
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SHAPE ('I1ARACTERIZATION
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At this stage, a more quantitative characterization of the cluster shape seems worth attempting. The cluster, considered at its different growth stages, can be viewed as a collection of islands in a granulated film, possibly forming a fractal system t'~'. We could then take advantage of the fractal dimensionality ~ D, as derived from the dependence of the number N of atoms in the cluster and its gyration radius R~: N -~ RgD. The value D refers in our case to the sequence of growth stages and characterizes the scale-insensitive shape of the clusters in the size range where the system shows fractal properties. Disc-like shapes would give D = 2, while hemispherical islands would correspond to D = 3. Irregular, tree-like or columnar shapes should exhibit D values below these. The Witten Sanders model TM was extended by Meakin'* by the simulation of aggregates formed on a seed particle under diffusioncontrolled conditions. It was shown that D obeys the general law D ~ 5d/6, where d is the euclidean dimensionality of the space. This relation predicts D = 1.7 and D = 2.5 for two-dimensional and three-dimensional systems respectively. We calculated the log Rg vs. log N dependence at each 10-atom stage of growth in the 500-atom clusters (Fig. 4), Rg being referred to a [001] axis, normal to the substrate and passing through the centre of gravity of the cluster. The position of the centre of gravity was calculated at each growth stage.
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The gyration radius is in units of RF'0.031 62. In clusters grown in variants (B) or (E), the dimensionality D was found to range between 2.0 and 2.5 for different values of the growth parameters RD and b. The linear log Rg vs. log N dependence extends over the whole range of cluster sizes investigated (10 ~< N ~< 500, i.e. 1.7 decades), with a correlation coefficient of 0.984 0.999. It can be seen that our growth algorithm leads to D values between those mentioned above, as expected for three-dimensional growth on a plane substrate. This analysis of fractal properties in model clusters seems to be worth attempting, although larger-scale simulations are required for more convincing conclusions.
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6.
T O P O L O G I C A L FEATURES
At this stage, it is relevant to ask how the growth conditions affect the structure of the cluster, as characterized by geometrical and topological features.
6.1. Degree of tetrahedral perfection The tetrahedral perfection parameter k (see Section 2) was determined for all the nests formed during the cluster build-up, including the occupied nests as well as those left unoccupied. It can be taken as a measure of the topological disorder of the polytetrahedral packing. Also, a large value ofk should be related to low densities in these non-relaxed models. Clusters grown under the condition of most regular shape (variant (E)) show for b = 1.2 a rather narrow k distribution, ranging between 1.00 and 1.15 (Fig. 5(a)). If the nearest-neighbour searching condition is relaxed to b = 1.4 (Fig. 5(b)) a considerably higher topological disorder results, k being distributed in a quasigaussian way between 1.03 and 1.34. If, instead of variant (E), we build according to the criterion of maximal local coordination number of each adatom (variant (B)), the topological disorder is greater, k being now concentrated in the range 1.22-1.35 (Fig. 5(c) for b = 1.4, RD = 3RE). Thus the preference for distorted sites, with a higher than average coordination, leads to a more disordered structure.
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Analysis of the k distribution at different growth stages does not reveal significant differences. This fact suggests that the growth process is stationary and could be continued without altering the topological properties of the cluster. 6.2. Coordination m~rnber of the aloms The number N, of nearest neighbours at R ~< RM = hR~ gives insight into the local structure of the model. Atoms of low coordination, if not belonging to the external surface, indicate the presence of local defects in the structure, possibly concentrated onto internal surfaces. Figure 6 shows the distribution of N n values among bulk and surface atoms for a cluster grown according to variant (B). The latter atoms were identified by the criteria of belonging to a surface nest and having N. ~< 6. A striking feature about the N. distribution is that there is a number of bulk atoms of low coordination with N,, ~< 6, which points to a large concentration of local defects. The most probable coordination of the bulk atoms is 7 8, a rather low value, which could be explained by the non-relaxed, low density character of the model.
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6.3. Atomic pair distribution The atomic pair distribution functions IPDFs) (Fig. 7) give insight into the degree and character of the local order, generated by the imposed growth conditions. Variant (B), which requires maximal coordination, results in a P D F showing a rather large first maximum and a split second m a x i m u m (Fig. 7(a) for RD = 3R~), as commonly found in metallic glasses and dense-random-packing-of-hard-spheres (DRPHS) models 19. The correlation between atom positions extends up to (4-5)R~ only, which should be again attributed to the non-relaxed character of the model. An increase in the diffusion length to 10RE results in a peculiar sharpening of the first peak, as well as of those at certain further distances, due to external shape-related effects, which alter the bulk PDF. The influence of the b parameter (relaxation of the first neighbour conditionl on the shape of the first P D F peak can be seen in Figs. 7(b) and 7(c) for a cluster grown in variant (E). An increase in b from 1.2 to 1.4 results in a considerable spreading of the distances. Variant (E) (most regular shape) gives rise to a P D F whose features are
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considerably broader than in variant (B). Also, the first subpeak in the second maximum is fainter in variant (E) than predicted by D R P H S models. As this subpeak is mainly due to distances between lateral nests on a tetrahedron, it is related to the presence oficosahedral local order. All these facts recommend variant (B) as the most suitable for growing D R P H S assemblies. 7. ENERGY OF THE CLUSTER
The cluster energy and its evolution during the growth process were also investigated. The energetic stability of small clusters relaxed under Lennard-Jones or Morse potentials has received much attention 2° 22. It was shown that polytetrahedral clusters are most probably among the minimal energy configurations for less than 50 atoms 22. Among them, the icosahedron (13 atoms), which belongs to the polytetrahedral Werfelmeier sequence for small clusters, and possibly also larger icosahedral multiply twinned structures 2°'22 are particularly stable. Generalized valence bond calculations 23 also showed that the lowest energy isomers of small metal clusters belong to the polytetrahedral type.
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At the same time, free fragments of crystal lattices of the same size are of higher specific energy so that, if they appear, they are very short lived, relaxing (if allowed to) towards more stable configurations because the energy barriers for rearrangements are low for small clusters. The total energy of our model cluster was calculated at each stage as E = ~EiEj Vu, where i runs over the N atoms in the cluster,./denotes the members of the nest triplet occupied by a t o m i (thereby forming a tetrahedral unit) and ~i is a L e n n a r d - J o n e s 6 12 potential of arbitrarily chosen magnitude. The dependence of specific energy E, = E/N on N in clusters grown by variant (B) is shown in Fig. 8. For b = 1.4, E, increases steadily with N, although the slope decreases, E~ reaching asymptotically a constant value, which would permit the further growth of the cluster. For more perfect clusters (h = 1.2), E~ increases up to a size of 100 200 atoms, although to a lower value than in the case o f b = 1.4. The range above N ~ 150 is relatively flat, allowing the further growth of the cluster from the point of view of the energy. Shallow local minima can be noticed for some of the runs, as are seen for about 200 and 400 atoms on Fig. 8. This type of E~(N) dependence is typical for frustrated structures, which are not allowed to reach their equilibrium state at any m o m e n t during their sequential formation. O u r polytetrahedral cluster with h = 1.2 is clearly unstable towards growth during the first stages (N ~< 100 200 atoms). However, we did not allow the whole structure to relax towards equilibrium (i.e. to decompose) but permitted only a "surface relaxation" of the uppermost layer of atoms by a d a t o m migration. Thus we have built a./rustrated structure, as could be present in thin a m o r p h o u s films, in
150
/ 1 00
x ~ ' ~ 50
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oe
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200
i
400
600
Is0
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Fig. 8. Specitic energy E~ (in a r b i t r a r y units) t:s. N for four clusters grown a c c o r d i n g to variant [B) with RI~ = 3R n : points 1, h = 1.4: points 2, t~ = 1.2 (O, O , /X, different initial inputs of the r a n d o m n u m b e r generatorl.
COMPUTER SIMULATION OF AMORPHOUS METAL FILM G R O W T H
309
which kinetic and energetic conditions during build-up prevent the attainment of an equilibrium atomic configuration. Our simulation results show that frustrated structures can reach a critical size, beyond which the addition of new atoms does not increase the specific energy, so that the growth process can be continued. There may be other cluster configurations of the same size which are more stable, but energy barriers towards rearrangements should be quite high for large clusters, implying collective movements of extended groups of atoms. Variant IA) has, as expected, a minimal asymptotic Es (about 0.5 in the range 200-500 atoms, in the units of Fig. 8), being built with precisely this requirement. 8. FACTORS WHICH FAVOUR COLUMNAR GROWTH (VARIANT (C))
In the extreme case when Ts, E, and Vd are very low, adatoms stick at the site where they have touched the surface of the cluster (with only minor relaxation possible). High incidence rates Vd can dislocate the atoms out of these impact sites. In such an ideal case, a probabilistic modelling approach seems suitable. In this approach the nest with the maximal upwards section S, as being the most likely to be filled, is chosen from the nest list for the next occupancy. Here S is the projection on the substrate of the nest surface, defined by its three atoms. It is the "capture crosssection" presented by the nest to an atom arriving at normal incidence. The atom beam was considered to fall normally to the substrate, but oblique incidence is as easy to consider. Figure 9 displays a projection of a cluster fragment possessing a striking columnar shape grown according to variant (C). Columnar microstructure (mainly at oblique incidence) is persistently reported in amorphous metallic films 6'8, being sometimes related to magnetic anisotropy. Recently, artificially grown columnar metallic structures 24 seem to confirm this assumption.
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The columnar microstructure can be given a purely geometrical explanation (%elf-shadowing" of certain low-lying nests by the already-deposited atoms25'2~), although some deposition factors, such as substrate bias, are known to be influential. The columnar cluster grown by our variant {C) tFig. 9) is an illustration of this geometrical effect. Recent work 2~' shows that, provided a basic columnar profile of adequate wavelength is present in the first layer, the diffusion equations can favour this shape, but only under conditions of low T and limited adatom mobility. In contrast, our simulation emphasizes the fact that self-shadowing can be effective only in certain experimental conditions (very low ~, E a and 1~). It is worth noting that, in r.f.sputtered Gd Co amorphous films, the columnar structure is obtained only for low gas pressures (low Va) and low discharge power {small E~,I2s, confirming the suggestions of the computer simulation. More realistic columnar shapes could be simulated by %ybridizing'" the rather restrictive condition (C) with the more physical constraints (Bt or {A). 9.
F R U S T R A Y E I ) C H E M I C A L ORDER IN ALI,OY I'II.MS
The chemical short-range order (CSRO) in A,.B 1 .~ alloys very often favours heteropairs AB over homopairs {wrong bonds) AA or BB on the grounds of lower enthalpy of formation. It has long been recognized that in amorphous films the CSRO is often frustrated in comparison with glassy materials {generally prepared from the melt), which are closer to the maximal CSRO degree (see ref. 29 for the AsxS1 x system). In our simulation, variant (D) {Section 3} should maximize the CSRO in amorphous films, under a given degree of structural frustration, controlled by the diffusion length RD. Thus the effect of CSRO frustration in amorphous alloy films could be put on a quantitative basis. Clusters of 500 atoms were grown for alloy compositions x = 0.1, 0.3 and 0.5 with restricted diffusion over distances RD -- 2 R ~ , 3 R E and 10R L. The nature of each new adatom was determined by a random number generator which imposed the condition of a given composition .v. Also, to each nest was attributed a number, related to the composition of the triplet, which allowed an adatom of a given nature to settle into the nest in such a way as to maximize the number of AB pairs. As was noticed above (Section 4), the shape of the resulting clusters depends on RD, but what interests us most is the degree n of CSRO in these models. As discussed by Cargill 3°. for amorphous alloys hAlt = { Z A B , , Z ; B t - - 1, whcrc ZAl~ is the coordination number of the species A by B and its statistically determined value is Z*aB = C ~ Z A Z R / ( C A Z A + c B Z B ) (where c is the concentration and Z the total coordination number). For complete disorder, n = 0 because ZA~ = ZAB, while n > 0 signals the presence of CSRO. A maximal n value can only be reached at a given composition x: n ~ x = xZa/"l I - x ) Z ~ < 1. This formula yields reliable values only for n < 1/3 in close-packed structures 3°. Also, the topology of the network can further limit the maximal CSRO that can be attained. As both A and B spheres have the same RE value in our model, we expect ZA = ZR. With this simplifying assumption, n can be derived from the total number
COMPUTER SIMULATION OF AMORPHOUS METAL FILM GROWTH
31 1
of AB pairs and the actual composition Xeff of the model: n = ( N A B ~ N A B ) - l, where NAB = 2Xeff(1 -- x e f f ) N and N -- NAA -b NAB -t- NBB. Table I shows the n values found in models of different frustration degrees (different R D values) together with the n max values calculated for the same models as discussed above. The slight variations in n max for the same intended x are due to statistical fluctuations in the composition Xeff that are still present at this size of the model. Table I shows, for compositions x = 0.1 and 0.3, that limitation of the adatom mobility on the cluster surface (small R D values) strongly frustrates the degree of CSRO. An increase in RD from 2R E to 10RE for x = 0.3 results in an n value almost twice as large but still far from the maximal value. For x = 0.1, the large statistical fluctuations in NAB preclude a meaningful comparison between the actual n value in the model and n max.At equimolar composition (x = 0.5), n is rather insensitive to Rt~. NAa
TABLE I CHEMICAL SHORT-RANGE ORDER r/ /~S. n max (CALCULATED FOR THE EFFECTIVE COMPOSITION Xeff) IN MODEL CLUSTERS GROWN ACCORDING TO VARIANT (D) WITH DIFFERENT DIFFUSION LENGTHS R D (RE, ATOM DIAMETER)
X
RD/R E
Xeff
g/max
n
0.1 0.1 0.3 0.3 0.3 0.5 0.5
3 10 2 3 10 3 10
0.138 0.142 0.384 0.348 0.350 0.534 0.532
0.160 0.165 0.623 0.534 0.538 0.873 a 0.880 a
0.259 0.342 0.183 0.208 0.326 0.303 0.314
a Unreliable values: see text.
Relaxation of one of these alloy models towards a larger degree of CSRO would probably encounter very large activation barriers and could be achieved only by extensive collective rearrangements. However, in a real film CSRO relaxation could proceed much more easily on annealing as a result of the presence of structural defects, which are not embodied in our models. 10. CONCLUSIONS A model of polytetrahedral, sequential growth under a variety of physical assumptions was devised which is particularly suitable for understanding structural particularities in amorphous discontinuous films in comparison with the more equilibrated glassy systems. These features originate in the frustrated character of the film structure as a result of the physical conditions which control the growth process, and they can be identified as follows. (1) The amorphous model clusters are energetically unstable in the first growth stages (specific energy E/N increases with N). Beyond a critical size, however, E/N displays a rather constant plateau. This suggests the possibility that the cluster can continue to grow, even if its configuration is not that of absolute minimal energy for
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the given N, because collective rearrangements would encounter high energy barriers. (21 Various quantities can be used to characterize the degree of structural disorder in the model cluster: nest perfection parameter k, distribution of coordination numbers and PDFs. They are controlled by the growth conditions imposed as well as by the nearest-neighbour disorder parameter h assumed in the simulation (related, in physical systems, to the detailed shape of the interatomic potentiall. (3) The degree of built-in CSRO (for alloy films) is lower than the upper limit set by composition and decreases with increasing frustration degree. The model allows visualization and quantification of the frustration degree by the limited number of accessible sites (inside the diffusion distance RDt which can be occupied by each newcoming adatom. REFERENCES I 2 3 4 5 6 7 8 9 10 II 12 13 I4 15 I6 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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