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Modeling layer-by-layer growth in ion beam assisted deposition of thin films
Nuclear Instruments and Methods in Physics Research B 171 (2000) 314±324
www.elsevier.nl/locate/nimb
Modeling layer-by-layer growth in ion beam assisted deposition of thin ®lms Ismo T. Koponen
*
Department of Physics, University of Helsinki, P.O. Box 9, SF-00014 Helsinki, Finland Received 17 February 2000; received in revised form 25 May 2000
Abstract A simple rate equation model for layer-by-layer growth in ion beam assisted deposition is developed. In terms of the rate coecients, the model describes how adatom diusion, island diusion, detachment and breakup, coalescence of large islands and interlayer transitions of adatoms aect the growth. Results based on the model show that enhanced detachment of adatoms due to ion bombardment is required to promote layer-by-layer growth but without suciently fast downward transition path, the ¯ux of adatoms to underlying layers may be too slow for the onset of layer-by-layer growth. In order to obtain optimum conditions, both requirements must be met. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 81.15.Jj; 68.35.Bs; 81.65.)b; 68.55.)a Keywords: Ion beam assisted deposition; Thin ®lms; Surface growth; Modeling
1. Introduction There are many examples on how low-energy ion beams can be utilized in improving the properties and growth of thin ®lm surfaces in deposition [1±6]. The objective of ion beam assisted deposition (IBAD) is to obtain smooth layer-bylayer growth in as low temperatures as possible. This can happen either directly by the destruction of growing 3D structures or by making the 2D growth kinetically a more favorable growth mode
*
Tel.: +358-9-191-8357; fax: +358-9-191-8680. E-mail address: ismo.koponen@helsinki.® (I.T. Koponen).
through promoting enhanced interlayer transitions [1±4,7]. In IBAD, there is an optimal range of ion energies where useful eects can be found, which extends typically from 20 to 600 eV depending on the properties of the growing surface [4±6]. The eects of IBAD on growth have been studied experimentally over two decades (see e.g. [4] and references therein), but only recently there has been growing interest towards the theoretical understanding of the microscopic phenomena aecting the growth in IBAD [5±8]. However, the ®rst sophisticated kinetic Monte Carlo (KMC) simulations date back to the beginning of the 1990s [9]. Quite recently, a detailed and informative study of the microscopic processes in IBAD based on
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molecular dynamics (MD) simulations has been carried out by Jacobsen et al. [7]. This study is of particular interest, because it is the ®rst report, where detailed description of microscopic processes due to ion bombardment is combined with KMC simulation of macroscopically observed growth phenomena. It also raises a renewed interest towards the rate equation modeling, which has suered from serious lack of microscopic evidence to assess the form of the phenomenological rate coecients [10,11]. The knowledge of the underlying microscopic processes is the starting point for the modeling of growth in IBAD. In principle, KMC simulations can be used to obtain very detailed information of the dynamics of growth but it is still too slow a tool for exploring the vast range of experimentally accessible conditions. Therefore, it is of interest to develop a more coarse-grained model. We have recently studied the eects of dierent surface processes in IBAD on submonolayer growth by using simple rate equation model [10,11]. Here, a similar approach is adopted for modeling layer-bylayer growth and it is demonstrated how dierent intra- and interlayer processes aect the growth. The microscopic processes aecting the growth can be divided to processes related to ion bombardment and to thermally activated rate processes, which are essentially similar as in MBE growth. The ion bombardment causes target adatom and defect production (which are, however, omitted here for reasons explained later on), detachment and breakup. All these processes occur in much shorter timescale than thermally activated rate processes. The eects of all these processes on growth are described with reaction rates and the growth process itself is modeled by rate equations governed by the reaction rates. We ®rst describe in detail each microscopic process and identify it with the pertinent reaction rate. This kind of modeling lacks the quantitative accuracy of more detailed KMC±MD models, but has the advantage of being capable to provide rapidly the qualitative information of the growth in dierent regions of model parameters. Combined with the available more detailed knowledge of the microscopic processes, the rate equation models are an invaluable tool for planning experiments and giving guidance for
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pinning down the regions of interest for more detailed studies. The features of growth, whether layer-by-layer or a precursor to 3D growth, are deduced within the framework of rate equation model by examining the model predictions for adatom and island densities, average size of islands and coverage. For comparisons with experiments [12±14] as well as with Jacobsen's results [7] we also calculate antiphase intensity or anti-phase Bragg intensity, which is measured by anti-phase scattering of He, or electron diraction methods such as RHEED and LEED [12±14]. The anti-phase intensity is a useful quantity in pointing out the dierent growth modes and in monitoring the smoothness of growth. Using these quantities the favorable conditions and necessary requirements for layer-bylayer growth are recognized.
2. Model of layer-by-layer growth In modeling layer-by-layer growth in IBAD, one should ®rst describe the processes of interest already in the submonolayer stage of growth. To review brie¯y, the processes are (1) target adatom and defect production, but for simplicity, this is ignored here, (2) adatom diusion, (3) island diffusion and (4) detachment and breakup [10,11,15]. When growth proceeds beyond the submonolayer stage, it is necessary to consider also the role of (5) coalescence of large islands [16] and (6) interlayer transitions of adatoms [7]. Modeling of growth in the level of rate equations requires that it is possible to ®nd the rate coecients and connect them to the microscopic processes. This has been accomplished successfully in the case of growth with immobile island and adatom detachment [17,18] and recently a self-consistent formulation of this problem has been given [19]. However, more detailed comparisons with e.g. KMC or MD simulations show that such self-consistent rate coecients are at best qualitatively but not quantitatively correct [12,17,20]. Therefore, in this report the goal is to formulate qualitatively and phenomenologically satisfactory rate coecients by considering their microscopic origins at depth,
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which are required to obtain correct generic behavior of the growth. Target adatom production has been discussed in [4±6] and it has been shown that target adatom production enhances nucleation of small islands thus aecting the growth. The role of target adatoms in growth was also discussed in our previous contribution [10] and could be included also as a part of the present model. In the present study, we omit target adatom production because we are only interested in the qualitative features of the growth. In layer-by-layer growth, the target adatom production also generates an upward mass current, because the atoms dislodged near steps get piled up on the islands (pileup eect) [7]. However, the eect of piling up on the net mass current is small in comparison to enhanced downward mass current due to thermally activated interlayer transition through kink sites, to be discussed later on. Only in such low temperatures, where thermally activated interlayer transitions are practically frozen, the pileups are expected to aect the growth, but this temperature region is not of interest in the present study. The diusion of adatoms on metallic fcc(0 0 1) surface is a simple hopping over a twofold bridge site [21]. Calculations utilizing ®nite temperature MD and based on EMT-potentials yield an activation energy ED , which is 0.37 eV for diusion of adatoms on open surface on Ag(0 0 1) and 0.40 eV for Cu(0 0 1) [21]. More complex intralayer diusion processes, such as edge diusion, are ignored here. Of the other jump processes of interest here are the interlayer transitions which allow adatoms to move between layers. The interlayer transition may occur in many dierent ways, and the simplest possible transition is from a straight edge down to an underlying layer. In Ag(0 0 1), this process occurs with an activation energy of 0.48 eV and of 0.56 eV in Cu(0 0 1). However, transitions where kink sites are involved are more favorable, with activation energy of only 0.40 eV for Ag(0 0 1) and 0.44 eV for Cu(0 0 1). Exchange mechanism involving transition through a kink site provides a path for transition with an even lower activation energy EIL which is 0.32 eV in Ag(0 0 1) and 0.38 eV in Cu(0 0 1) [21] (see also [22]). This latter process is of importance for interlayer transitions
in IBAD because ion bombardment creates excess number of kink sites and thus facilitates the interlayer transitions [7]. The diusion on open surface occurring with hopping rate CD expÿED =kT and the interlayer transitions with rate CIL expÿEIL =kT provide two important timescales for surface growth processes in IBAD. In addition to these timescales, the ¯ux of deposited adatoms given by U (ML/s) and the breakup rate of island determined by the ion bombardment set up the other two important timescales. Island mobility can be described by diusion coecient Ds [10,16], and for islands with more than s 10 atoms a simple scaling law Ds / sÿa with a constant scaling exponent a holds (see [23± 25] and references therein). The prefactor of island diusion coecient is taken to be proportional CD . The scaling exponent a is related to the diusion mechanism and the pertinent single jump process in metallic fcc(0 0 1) surfaces is the so-called periphery diusion (PD) process, corresponding to the value a 3/2 [23,24]. However, most simulations and also available experiments on metallic fcc(0 0 1) surfaces indicate values 1:75 < a < 2:1 [25], and consequently we will here adopt values a 3=2 and 2 for typical values describing fcc(0 0 1) surfaces. The diusion coecient of islands is then used to derive the rate coecient for the aggregation of moving islands, and it can be shown that it takes the form K i; j K0 iÿa jÿa for islands with sizes i and j [15,26]. The term K 1; 1 2K0 is now related to diusion coecient D of adatoms and adatom capture rate r [20] through relation K0 Dr. The detachment of adatoms on the existing islands occurs with enhanced probability due to ion bombardment, and in addition to this also larger fragments are produced [27,7]. The breakup of island of size s to smaller islands with sizes i and j (mass is conserved s i j) is described with rate F i; j i jb and for IBAD values 1=2 < b < 1 are reasonable choices [10,11]. In these equations, the ®rst term F 1; 1 2F0 is for dimer breakup with F0 as a rate of detachment for single adatoms, determined by the ¯ux of the ion beam [10,11]. At large enough island densities, islands begin to coalesce and formation of larger islands proceeds rapidly. Within the rate equation formalism
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there is no entirely satisfactory way to treat coalescence. However, Biham et al. [16] have managed to describe coalescence with a model, where the coalescence rate of island with sizes i and j is proportional to the size of the islands and thus given by a rate coecient Gi;j G0 i j, where G0 U [16]. This description is adopted here also. The growth within a given layer (intralayer growth) can now be modeled with similar rate equations as previously used for island growth in submonolayer region [10,11,15], the only modi®cation being the addition of term describing coalescence, X X on1 U ÿ K1;1 n21 ÿ K1;j n1 nj ÿ F1;j n1j ; ot jP2 jP1 ons 1 X Ki;j Gi;j ni nj ÿ Fi;j ns ot 2 ijs X Ks;j Gs;j ns nj ÿ Fs;j nsj : ÿ
1
2
jP1
As Eqs. (1) and (2) are formulated in complete analogy with growth with mobile islands and breakup, we can assume that the island densities will also follow the general scaling form ns hsÿ2 g s=s, where s is the P average size of the islands, de®ned by s hÿ1 0 s2 ns as in [15,20] (prime denoting summation over s P 2) and where g x is a scaling function which depends on the form of the rates Ki;j , Gi;j and Fi;j [15]. To obtain the detailed form of the scaling function, the complete solution for island densities would be required. However, for qualitative understanding, which is here of primary interest, it suces to know that the scaling function is of the form g x xd expÿx with d depending on the degree of homogeneity of Ki;j , Gi;j and Fi;j [15,28]. In order to obtain tractable set of equations for layer-by-layer growth with interlayer transitions included, it is neither convenient nor necessary to work with complete island distributions. Instead, it suces to use adatom andPtotal island densities, n t n1 t and N t 0 ns t, respectively. From now on, we take into account the interlayer transitions and denote the averaged densities in layer I 1; 2; . . . ; Nmax by nI and NI . Each layer I
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has a coverage hI , and the total coverage is P h Ut hI , with h0 1 for the substrate. Of the deposited adatoms each layer receives thus only a fraction, which is proportional to the exposed surface area hIÿ1 ÿ hI . The rate of change of adatom density due to interlayer transitions is modeled by a term proportional to CIL nI H 1 ÿ hIÿ1 , where H is the unit step function. There is no dependence on the coverage hIÿ2 of the underlying surface, because the transitions are always down to boundary edge, which is depleted of adatoms. The dominant interlayer transition with the lowest activation energy occurs through kink sites, and thus the total number of transitions should be proportional to the number of kink sites, and thus also to the step density. In IBAD, the step density is only weakly dependent on coverage [7], and for most practical purposes it can be treated as constant. The averaging of Eqs. (1) and (2) is done by summing over all island sizes and then de®ning the appropriately averaged rate coecients, as is done in the case of rate equation models for growth with detachment [19,29,30]. In addition to adatom and island densities, a useful third dynamic equation of growth is written for total mass M sN P 0 sns x; t of islands, which also allows one to obtain the average size of the islands s [29,30]. The dynamic evolution of the resulting model is governed by three timescales: the deposition rate F (ML/s), hopping rate CD for diusion and the interlayer transition rate CIL . Accordingly, in the averaged equations the coverage h Ut is introduced as a time parameter, R DK0 =U as a measure for a ratio of timescales characterizing diusion and deposition (note that usually K0 is not included in this ratio but the present choice simpli®es much the notation) and G CIL =CD expDE=kT , with DE ED ÿ EIL as a measure for the rate of interlayer transition CIL to diusion jumps CD . With these timescales and timescale ratios the averaged equations are obtained in the form dnI hIÿ1 ÿ hI ÿ R 2n2I rNI nI ÿ jMIb NI1ÿb dh 3 GnI1 H 1 ÿ hI ÿ nI H 1 ÿ hIÿ1 ;
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dNI 1 1 b 1ÿb 2 ÿa 2a R nI e r MI NI ÿ e j MI NI ÿc fMI NI ; dh 2 2 4 dMI R2n2I rNI nI ÿ jMIb NI1ÿb : dh
5
In the above equations, the dimensionless rate coecients have been introduced: r and j for adatom processes, e r and e j for island processes and ef for coalescence. Derivation of the detailed expression for these rate coecients is straightforward. The averaging is carried out in principle similarly as e.g. in [29,30] but now the scaling form ns hsÿ2 g s=s is utilized. Then a continuum limit is taken by replacing the summations by integrations [28], and by using the condition R of mass conservation M sN ' hn, with n g x dx the average size can be eliminated from the ®nal rate equations. Following this procedure, the rate of breakup is found to be measured by parameter R j F0 =K0 xb g x dx. An order of magnitude estimate for j is obtained by noting that typically R 1 < xb g x dx < 5 and in physically accessible region 10ÿ7 < F0 =K0 < 10ÿ2 (see [11,15] and references therein), which thus yields 10ÿ7 < j < 10ÿ2 . Similarly, for the aggregation rate due to adatom R diusion one obtains r g x dx, which is in the range 2 < r < 4 for typical scaling functions [15]. The rate coecients e r and e j for aggregation due to island diusion and for breakup, respectively, and ef for coalescence are bit more complicated and proportional to integrals ZZ e r xÿa y ÿa g xg y dx dy; ZZ e xb y b g xg y dx dy j F0 =K0
clear that it would require considering the various microscopic processes in detail going beyond the scope of the present report. 3. Results In this report, we concentrate on some representative cases instead of attempting to systematically explore the entire parameter space. Our objective is to demonstrate the eect of interlayer transitions on growth and show that with simple and easily solvable models reasonable results guiding the insight can be achieved. In our previous contributions, we have studied the eect of island diusion and breakup on submonolayer growth [10,11,15], and based on the previous experience the representative cases are chosen to be b 1=2 and a 3=2, corresponding detachment or breakup related to the size of the boundary edge [10], and island diusion mediated by the periphery or edge diusion mechanism [15,25]. Variations of these parameters within the physically reasonable limits for metallic layers, i.e. 0 < b < 1 and 1 < a < 2, do not essentially change the qualitative features of the results given here. The model describes the growth of the layer in terms of the island density, island size and coverage. In addition to these quantities, the smoothness of growth can be monitored by the so-called anti-phase intensity or anti-phase Bragg intensity, which is measured by anti-phase scattering of He, or diraction methods such as RHEED and LEED [12±14]. The anti-phase intensity used in this report is calculated from expression [7] " #2 1 X i I h ÿ 1 hi ÿ hi1 ; 6 i0
and ef
ZZ x yg xg y dx dy:
The range of values for these parameters can be j < 10ÿ2 and estimated to be 2 < e r < 10, 10ÿ7 < e 1 < ef < 3. More detailed assessment of the values for rate coecients is not pursued here. In principle, it is possible to make this connection but it is
where hi is the coverage of the ith layer. In this case, the new layers start to grow only when underlying layers are nearly completed and the antiphase intensity shows oscillations, like a series of parabolas, and the period of oscillations corresponds to 1 ML [7,12]. When nucleation occurs on top of the existing islands before completion of the layer, the anti-phase intensity is strongly damped and approaches zero at h 1.
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The numerical solution of the ®rst-order coupled nonlinear rate equations has been performed by using MATHEMATICA [31], which is a convenient and accurate tool for such initial value problems as encountered here. Checks of accuracy were done by varying maximum stepsize and error bounds, and by checking some of the results against solutions obtained by NAG-library routines. The set of equations was also tested against the existing results for similar rate equations used in submonolayer thermal growth [16] and growth with breakup [10]. The ®rst example to be studied is growth without interlayer transitions, described by G 0. The timescale ratio of diusion to deposition was chosen to be R 105 , favoring diusion driven aggregation. Breakup of islands was varied from moderate with j 1:5 10ÿ6 to enhanced breakup with j 1:5 10ÿ4 (compare with [15]). In accordance with previous results for submonolayer growth, it was found that only the order of magnitude of the fragmentation is important [15], and therefore one can set j e j without loss of generality. Aggregation is described by rate coecients r 1 and e r 4, and in order to mimic the real growth after submonolayer region the eect of coalescence is included in all cases and rate of coalescence is chosen to be ef 1, which is in accordance with [16]. This choice of parameters is consistent with low temperature growth in case of Cu(0 0 1) and Ag(0 0 1). For Cu(0 0 1) surface, at temperatures lower than 100 K, aggregation and island growth dominate and with suppressed downward transitions there is no possibility for layer-by-layer growth. In the case of Cu(0 0 1) layer-by-layer growth without ion beams is expected to occur only at temperatures 200 K or higher [12]. Fig. 1 shows the scaled island density N =Nmax , scaled island size s=smax and coverage hI for each growing layer in the case for moderate breakup with j 1:5 10ÿ6 . The results resemble the case of thermal growth, but are however strongly affected by the breakup. In the case shown in Fig. 1, the maximum average size of islands is smax 250, without breakup average size would be 1950, diminishing to 70 with j 1:5 10ÿ5 , and eventually to 30 with j 1:5 10ÿ4 . The maximum
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Fig. 1. Scaled island density N =Nmax , scaled average size s=smax and coverage for growing layers in deposition with moderate breakup and suppressed downward transitions. The total coverage is 6 ML corresponding to 6 completed layers shown. The parameters are G 0:0 and R 105 . Rates for aggregation and detachment are r 1 and e r 4, and j j 1:5 10ÿ6 , respectively, and for coalescence f 1:0. Maximum island density is Nmax 8:6 10ÿ3 and maximum island size is smax 250.
island density changes accordingly from Nmax 8:6 10ÿ3 for j 1:5 10ÿ6 to Nmax 3:2 10ÿ2 for j 1:5 10ÿ4 . In the case of suppressed interlayer transitions, the eect of breakup is mainly
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to moderate unbounded growth of average size, which would now otherwise increase rapidly due to coalescence. From the results for coverages of each layer it is seen, how all layers grow at the same time and the completion of layers occurs very slowly. Even after deposition of 3±4 layers, the ®rst layer is neither completed, nor has the average size reached the stationary value. The growth is, however, retarded, because underlying layers acquire mass at low and lower rates due to shading eects of islands formed on top of them. Therefore, all growing layers compete eectively only of the deposited adatoms, in proportion to the exposed surface. The corresponding antiphase intensities measured e.g. RHEED (and calculated now from Eq. (6)) are in these cases simply decaying functions reaching zero rapidly at coverage of order unity. All the above discussed cases are favorable for 3D growth but clearly undesired if 2D layer-by-layer growth is pursued. The downward transitions have an essential role in promoting the layer-by-layer growth, and e.g. in the case of Cu(0 0 1) interlayer transitions begin to aect the growth at temperatures of 200 K or higher, which leads to promotion of layer-bylayer growth [13,14]. The importance of downward transitions at low temperatures in IBAD is clearly demonstrated by Jacobsen et al. [7] in their KMC± MD simulations, where growth on Ag(1 1 1) and Pt(1 1 1) in IBAD is studied in detail. Jacobsen identi®es the eect of ion bombardment with increased rate of downward transitions, and ®nds an optimum window for this around 25 eV deposition. Within the framework of our model, this effect is mimicked by increasing the rate of downward transitions. In the present model, the eects of IBAD on enhanced interlayer transitions enter through timescale ratio G, which contains information on both the increased step density and temperature and thus G nStep expDE=kT , where nStep is the relative increase in step density. With typical activation barriers in fcc(0 0 1) metals DE 70 MeV, and for relative increase in step density a conservative estimate 2 < nStep < 5 is used. Thus, reasonable values are estimated to be in the range 50 < G < 500 under typical conditions in IBAD.
The eect of the downward transitions is demonstrated in Fig. 2, where the results for G 100 are shown. In the case of fcc(0 0 1) metallic surfaces, this corresponds to the situation encountered at room temperature. The other parameters
Fig. 2. Scaled island density N =Nmax , scaled average size s=smax and coverage for growing layers in deposition with enhanced breakup and moderate downward transitions. The parameters are G 100 and R 5 103 and j j 1:5 10ÿ4 , otherwise as in the case shown in Fig. 1. Maximum island density is now Nmax 4:1 10ÿ2 and maximum island size is smax 26.
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are comparable to the case shown in Fig. 1. The eect of downward transitions is to accelerate the completion of layers, due to additional ¯ux of adatoms from upper layers to lower ones. Now each layer is nearly completed before the growth of other layers starts, and as seen from the results for island densities, each layer grows essentially in a similar way. At the same time, the average size of islands is greatly reduced and island density increases. However, the average size does not reach the stationary value but continues to increase, which reveals that aggregation is still strong enough to allow slow but steady growth of islands. In the case shown in Fig. 2, the maximum island density is Nmax 4:1 10ÿ2 and maximum island size is smax 26. The case displayed in Fig. 2 is the optimum case, where the increase in breakup does not lead to more pronounced layer-by-layer growth because the rate of downward transitions acts as a rate limiting process. The corresponding anti-phase intensity calculated from Eq. (6) is shown in Fig. 3 with result for more moderate breakup. In the optimum case, the oscillations in anti-phase intensity are pronounced, but they diminish substantially with reduced breakup rate and for j 1:5 10ÿ6 the oscillations are essentially similar to those found for thermal deposition at room temperature (compare with results in [12]). These results point out clearly the important
Fig. 3. Anti-phase intensities calculated from Eq. (6) for enhanced breakup and moderate downward transitions with G 100. The cases shown are for j j 1:5 10ÿ4 , 1:5 10ÿ5 , 1:5 10ÿ6 , from top to bottom. Other parameters are as in the case shown in Fig. 2.
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role of enhanced detachment as an agent promoting layer-by-layer growth. The larger the rate of downward transition is, the more easily the layer-by-layer growth occurs if breakup rate is large enough to maintain the ¯ux of adatoms to lower layers. In the region of large but yet reasonable downward transition rates where G 200 the overall appearance of island density, coverage and average size are quite dierent to the cases shown in Figs. 1 and 2. With optimally large breakup rate 2:0 10ÿ4 all these quantities reach their stationary values already within h 1, as shown in Fig. 4. Now there is no considerable overlap between temporal behavior of growth in dierent layers. With increasing rate of downward transitions, the upper layers are rapidly depleted of the adatoms and the large rate of breakup provides an eective means for continuous detachment thus preventing the formation of large islands and eectively suppressing the 3D growth mode. In the case displayed in Fig. 4, the maximum average size is only smax 12, the maximum island and adatom densities are Nmax 8:3 10ÿ2 and nmax 6:0 10ÿ2 , respectively. The values of these quantities do not depend on the rate of downward transitions because they are determined by the intralayer process, by the balance of aggregation and breakup (compare with results in [11,15]). However, the saturation values are reached much earlier in the case of enhanced downward transitions, in the case of G 200 it occurs already at h 1 while in the case G 100 it happens only at h 2. The nearly perfect layer-by-layer growth is now clearly revealed also by the oscillations of the antiphase intensity, shown in Fig. 5 for cases G 100; 150 and 200. In comparison with Fig. 3 the intensity of anti-phase oscillations is nearly doubled, indicating more perfect layer-by-layer growth. Moreover, the anti-phase intensity is comparable to that found by Jacobsen et al. [7] for layer-by-layer growth on Ag(1 1 1) when 25 eV Ag ion bombardment was used. These results demonstrate that with suitable combination of detachment and accompanying enhanced downward transitions it is possible to achieve nearly perfect layer-by-layer growth.
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Fig. 5. Anti-phase intensities calculated from Eq. (6) for enhanced breakup j j 2 10ÿ4 and downward transitions with G 200; 150 and 100, from top to bottom. Other parameters are as in the case shown in Fig. 4.
Fig. 4. Scaled island density N =Nmax , scaled average size s=smax and coverage for growing layers in deposition with enhanced breakup and downward transitions. The parameters are G 200 and R 5 103 and j j 2 10ÿ4 , otherwise as in the case shown in Fig. 1. Maximum island density is now Nmax 8:3 10ÿ2 and maximum island size is smax 12.
4. Discussion In metallic fcc(0 0 1) surfaces, it is possible to obtain layer-by-layer growth at suciently high temperatures, e.g. Cu(0 0 1) at 200±250 K. This is due to the fact that interlayer transitions through
kink sites can eectively compete with intralayer transitions [12,21]. However, even at lower temperatures it is possible to obtain layer-by-layer growth by using ion beams in promoting the interlayer transitions through additional kink sites created by ion bombardment. Here the situation was chosen, so that without IBAD layer-by-layer transitions do not occur. The eects of IBAD on enhanced interlayer transitions enter through timescale ratio G, but contributions of the dierent microscopic processes on parameter G were not separated in detail. Anyway at present only for Ag(1 1 1) there are microscopic data available for such detailed description [7]. The choice of island size independent total transition rate is in agreement with results of Jacobsen et al. [7], which show that there is no noticeable correlation between step density and coverage after the transient stage in the submonolayer region, i.e. step density and transition rate proportional to it are essentially constants. The fact that the total transition rate is eectively independent of the coverage of the lower layer and average edge size of islands can be of course due to the fact that product s1=2 hIÿ1 ÿ hI happens to be constant in the region of growth of interest. The rate equation model developed here is not quantitatively accurate in the limit h ! 1 mainly due to lack of detailed knowledge on how to model coalescence. The model adopted here, based on the results in [16], is expected to be reliable up to
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h 0:5 but at larger coverages the accuracy of the model probably deteriorates rapidly. We expect that the present model underestimates the role of coalescence. However, this is not a serious drawback, because we are mainly interested in region of growth, where detachment or breakup keeps the island size in moderate limits and prevents the formation of large islands. The scaling analysis was needed in the derivation of the rate equations. In all regions of growth, the scaling law sN h holds but the coecient of proportionality may vary. Moreover, the detailed study of the validity of scaling analysis in the present case is a rather demanding task and clearly beyond the scope of present study. Detailed knowledge of the scaling function is needed to get quantitatively accurate rate coecients [15] but ®nding the scaling function amounts up to solving the complete model with all island distributions. Therefore, even within the framework of rate equations we have good reasons to limit the model in the phenomenological and qualitative levels of description. 5. Conclusions In this report, a model describing layer-by-layer growth in IBAD is developed. It is demonstrated how the dierent intra- and interlayer processes aect the growth. The microscopic processes affecting the growth considered are adatom diusion, island diusion, detachment and breakup, coalescence of large islands and interlayer transitions of adatoms. The eects of all the dierent processes on growth are described with reaction rates and the growth process itself is modeled by rate equations governed by these reaction rates. The advantage of the model is its great simplicity, which also allows a rapid numerical solution. This makes it possible to explore eectively the interesting parameter regions and use it as a tool for planning experiments or e.g. more detailed simulations. From results based on the present model, it can be deduced that enhanced detachment of adatoms due to ion bombardment is a prerequisite for promoting layer-by-layer growth but without suciently fast downward transition path adatom
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¯ux to underlying layers may be too moderate and prevent the onset of layer-by-layer growth. In order to obtain optimum conditions, both requirements must be met. In practice, the energy and mass of the bombarding ions can be used to tune the breakup and detachment rate. The ambient temperature plays a role in determining the parameter G, but ion bombardment has a central role in controlling the rate of transfer of adatoms through the kink density, which is re¯ected in the special form of the term chosen to model this effect.
References [1] C.-H. Choi, R. Ai, S.A. Barnett, Phys. Rev. Lett. 67 (1991) 2826. [2] B.K. Kellerman, E. Chason, J.A. Floro, S.T. Picraux, J.M. White, Appl. Phys. Lett. 67 (1995) 1703. [3] G. Rosenfeld, N.N. Lipkin, W. Wulfhekel, J. Kliewer, K. Morgenstern, B. Poelsema, G. Comsa, Appl. Phys. A 61 (1995) 455. [4] W. Ensinger, Nucl. Instr. and Meth. B 127/128 (1997) 796. [5] S. Esch, M. Breeman, M. Morgenstern, T. Michely, G. Comsa, Surf. Sci. 365 (1996) 187. [6] M. Kalf, M. Breeman, M. Morgenstern, T. Michely, G. Comsa, Appl. Phys. Lett. 70 (1997) 182. [7] J. Jacobsen, B.H. Cooper, J.P. Sethna, Phys. Rev. B 58 (1998) 15847. [8] E. Chason, B.K. Kellerman, Nucl. Instr. and Meth. B 127/ 128 (1997) 225. [9] E. Chason, P. Bedrossian, J.E. Houston, J.Y. Tsao, B.W. Dodson, S.T. Picraux, Appl. Phys. Lett. 59 (1991) 3533. [10] J. Sillanp a a, I. Koponen, Nucl. Instr. and Meth. B 142 (1998) 67. [11] M. Rusanen, I. Koponen, J. Heinonen, J. Sillanp a a, Nucl. Instr. and Meth. B 148 (1999) 116. [12] M. Breeman, G.T. Barkema, M.H. Langelaar, D.O. Boerma, Thin Solid Films 272 (1996) 195. [13] H.J. Ernst, F. Fabre, J. Lapojoulade, Surf. Sci. Lett. 275 (1992) L682. [14] W.F. Egelho, I. Jacob, Phys. Rev. Lett. 62 (1989) 921. [15] I. Koponen, M. Rusanen, J. Heinonen, Phys. Rev. E 58 (1998) 4037. [16] O. Biham, G.T. Barkema, M. Breeman, Surf. Sci. 324 (1995) 47. [17] C. Ratsch, A. Zangwill, P. Smilauer, D.D. Vvedensky, Phys. Rev. Lett. 72 (1994) 3194. [18] J.A. Blackman, A. Wilding, Europhys. Lett. 16 (1991) 115. [19] G.S. Bales, A. Zangwill, Phys. Rev. B 55 (1997) R1973. [20] G.S. Bales, D.C. Chrzan, Phys. Rev. B 50 (1994) 6057. [21] J. Merikoski, I. Vattulainen, J. Heinonen, T. Ala-Nissila, Surf. Sci. 387 (1997) 167.
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[22] U. K urpick, T.S. Rahman, Phys. Rev. B 57 (1998) 2482. [23] S.V. Khare, T.L. Einstein, Phys. Rev. B 54 (1996) 11752. [24] S.V. Khare, N.C. Bartelt, T.L. Einstein, Phys. Rev. Lett. 75 (1995) 2148. [25] J. Heinonen, I. Koponen, J. Merikoski, T. Ala-Nissila, Phys. Rev. Lett. 82 (1999) 2733. [26] P.L. Krapivsky, J.F.F. Mendes, S. Redner, Phys. Rev. B 59 (1999) 15950. [27] H. Feil, Mater. Res. Soc. Symp. Proc. 317 (1994) 609.
[28] C.M. Sorensen, H.X. Zhang, T.W. Taylor, Phys. Rev. Lett. 59 (1987) 363. [29] H.T. Dobbs, A. Zangwill, D.D. Vvedensky, in: M.T. Tringides (Ed.), Surface Diusion: Atomistic and Collective Processes, Plenum, New York, 1997, pp. 263±275. [30] H.T. Dobbs, A. Zangwill, D.D. Vvedensky, J. Johansson, N. Carlsson, W. Seifert, Phys. Rev. Lett. 79 (1997) 897. [31] S. Wolfram, The Mathematica Book, fourth ed., Wolfram Media/Cambridge University Press, Cambridge, 1999.
www.elsevier.nl/locate/nimb
Modeling layer-by-layer growth in ion beam assisted deposition of thin ®lms Ismo T. Koponen
*
Department of Physics, University of Helsinki, P.O. Box 9, SF-00014 Helsinki, Finland Received 17 February 2000; received in revised form 25 May 2000
Abstract A simple rate equation model for layer-by-layer growth in ion beam assisted deposition is developed. In terms of the rate coecients, the model describes how adatom diusion, island diusion, detachment and breakup, coalescence of large islands and interlayer transitions of adatoms aect the growth. Results based on the model show that enhanced detachment of adatoms due to ion bombardment is required to promote layer-by-layer growth but without suciently fast downward transition path, the ¯ux of adatoms to underlying layers may be too slow for the onset of layer-by-layer growth. In order to obtain optimum conditions, both requirements must be met. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 81.15.Jj; 68.35.Bs; 81.65.)b; 68.55.)a Keywords: Ion beam assisted deposition; Thin ®lms; Surface growth; Modeling
1. Introduction There are many examples on how low-energy ion beams can be utilized in improving the properties and growth of thin ®lm surfaces in deposition [1±6]. The objective of ion beam assisted deposition (IBAD) is to obtain smooth layer-bylayer growth in as low temperatures as possible. This can happen either directly by the destruction of growing 3D structures or by making the 2D growth kinetically a more favorable growth mode
*
Tel.: +358-9-191-8357; fax: +358-9-191-8680. E-mail address: ismo.koponen@helsinki.® (I.T. Koponen).
through promoting enhanced interlayer transitions [1±4,7]. In IBAD, there is an optimal range of ion energies where useful eects can be found, which extends typically from 20 to 600 eV depending on the properties of the growing surface [4±6]. The eects of IBAD on growth have been studied experimentally over two decades (see e.g. [4] and references therein), but only recently there has been growing interest towards the theoretical understanding of the microscopic phenomena aecting the growth in IBAD [5±8]. However, the ®rst sophisticated kinetic Monte Carlo (KMC) simulations date back to the beginning of the 1990s [9]. Quite recently, a detailed and informative study of the microscopic processes in IBAD based on
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 0 ) 0 0 2 8 7 - 1
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molecular dynamics (MD) simulations has been carried out by Jacobsen et al. [7]. This study is of particular interest, because it is the ®rst report, where detailed description of microscopic processes due to ion bombardment is combined with KMC simulation of macroscopically observed growth phenomena. It also raises a renewed interest towards the rate equation modeling, which has suered from serious lack of microscopic evidence to assess the form of the phenomenological rate coecients [10,11]. The knowledge of the underlying microscopic processes is the starting point for the modeling of growth in IBAD. In principle, KMC simulations can be used to obtain very detailed information of the dynamics of growth but it is still too slow a tool for exploring the vast range of experimentally accessible conditions. Therefore, it is of interest to develop a more coarse-grained model. We have recently studied the eects of dierent surface processes in IBAD on submonolayer growth by using simple rate equation model [10,11]. Here, a similar approach is adopted for modeling layer-bylayer growth and it is demonstrated how dierent intra- and interlayer processes aect the growth. The microscopic processes aecting the growth can be divided to processes related to ion bombardment and to thermally activated rate processes, which are essentially similar as in MBE growth. The ion bombardment causes target adatom and defect production (which are, however, omitted here for reasons explained later on), detachment and breakup. All these processes occur in much shorter timescale than thermally activated rate processes. The eects of all these processes on growth are described with reaction rates and the growth process itself is modeled by rate equations governed by the reaction rates. We ®rst describe in detail each microscopic process and identify it with the pertinent reaction rate. This kind of modeling lacks the quantitative accuracy of more detailed KMC±MD models, but has the advantage of being capable to provide rapidly the qualitative information of the growth in dierent regions of model parameters. Combined with the available more detailed knowledge of the microscopic processes, the rate equation models are an invaluable tool for planning experiments and giving guidance for
315
pinning down the regions of interest for more detailed studies. The features of growth, whether layer-by-layer or a precursor to 3D growth, are deduced within the framework of rate equation model by examining the model predictions for adatom and island densities, average size of islands and coverage. For comparisons with experiments [12±14] as well as with Jacobsen's results [7] we also calculate antiphase intensity or anti-phase Bragg intensity, which is measured by anti-phase scattering of He, or electron diraction methods such as RHEED and LEED [12±14]. The anti-phase intensity is a useful quantity in pointing out the dierent growth modes and in monitoring the smoothness of growth. Using these quantities the favorable conditions and necessary requirements for layer-bylayer growth are recognized.
2. Model of layer-by-layer growth In modeling layer-by-layer growth in IBAD, one should ®rst describe the processes of interest already in the submonolayer stage of growth. To review brie¯y, the processes are (1) target adatom and defect production, but for simplicity, this is ignored here, (2) adatom diusion, (3) island diffusion and (4) detachment and breakup [10,11,15]. When growth proceeds beyond the submonolayer stage, it is necessary to consider also the role of (5) coalescence of large islands [16] and (6) interlayer transitions of adatoms [7]. Modeling of growth in the level of rate equations requires that it is possible to ®nd the rate coecients and connect them to the microscopic processes. This has been accomplished successfully in the case of growth with immobile island and adatom detachment [17,18] and recently a self-consistent formulation of this problem has been given [19]. However, more detailed comparisons with e.g. KMC or MD simulations show that such self-consistent rate coecients are at best qualitatively but not quantitatively correct [12,17,20]. Therefore, in this report the goal is to formulate qualitatively and phenomenologically satisfactory rate coecients by considering their microscopic origins at depth,
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which are required to obtain correct generic behavior of the growth. Target adatom production has been discussed in [4±6] and it has been shown that target adatom production enhances nucleation of small islands thus aecting the growth. The role of target adatoms in growth was also discussed in our previous contribution [10] and could be included also as a part of the present model. In the present study, we omit target adatom production because we are only interested in the qualitative features of the growth. In layer-by-layer growth, the target adatom production also generates an upward mass current, because the atoms dislodged near steps get piled up on the islands (pileup eect) [7]. However, the eect of piling up on the net mass current is small in comparison to enhanced downward mass current due to thermally activated interlayer transition through kink sites, to be discussed later on. Only in such low temperatures, where thermally activated interlayer transitions are practically frozen, the pileups are expected to aect the growth, but this temperature region is not of interest in the present study. The diusion of adatoms on metallic fcc(0 0 1) surface is a simple hopping over a twofold bridge site [21]. Calculations utilizing ®nite temperature MD and based on EMT-potentials yield an activation energy ED , which is 0.37 eV for diusion of adatoms on open surface on Ag(0 0 1) and 0.40 eV for Cu(0 0 1) [21]. More complex intralayer diusion processes, such as edge diusion, are ignored here. Of the other jump processes of interest here are the interlayer transitions which allow adatoms to move between layers. The interlayer transition may occur in many dierent ways, and the simplest possible transition is from a straight edge down to an underlying layer. In Ag(0 0 1), this process occurs with an activation energy of 0.48 eV and of 0.56 eV in Cu(0 0 1). However, transitions where kink sites are involved are more favorable, with activation energy of only 0.40 eV for Ag(0 0 1) and 0.44 eV for Cu(0 0 1). Exchange mechanism involving transition through a kink site provides a path for transition with an even lower activation energy EIL which is 0.32 eV in Ag(0 0 1) and 0.38 eV in Cu(0 0 1) [21] (see also [22]). This latter process is of importance for interlayer transitions
in IBAD because ion bombardment creates excess number of kink sites and thus facilitates the interlayer transitions [7]. The diusion on open surface occurring with hopping rate CD expÿED =kT and the interlayer transitions with rate CIL expÿEIL =kT provide two important timescales for surface growth processes in IBAD. In addition to these timescales, the ¯ux of deposited adatoms given by U (ML/s) and the breakup rate of island determined by the ion bombardment set up the other two important timescales. Island mobility can be described by diusion coecient Ds [10,16], and for islands with more than s 10 atoms a simple scaling law Ds / sÿa with a constant scaling exponent a holds (see [23± 25] and references therein). The prefactor of island diusion coecient is taken to be proportional CD . The scaling exponent a is related to the diusion mechanism and the pertinent single jump process in metallic fcc(0 0 1) surfaces is the so-called periphery diusion (PD) process, corresponding to the value a 3/2 [23,24]. However, most simulations and also available experiments on metallic fcc(0 0 1) surfaces indicate values 1:75 < a < 2:1 [25], and consequently we will here adopt values a 3=2 and 2 for typical values describing fcc(0 0 1) surfaces. The diusion coecient of islands is then used to derive the rate coecient for the aggregation of moving islands, and it can be shown that it takes the form K i; j K0 iÿa jÿa for islands with sizes i and j [15,26]. The term K 1; 1 2K0 is now related to diusion coecient D of adatoms and adatom capture rate r [20] through relation K0 Dr. The detachment of adatoms on the existing islands occurs with enhanced probability due to ion bombardment, and in addition to this also larger fragments are produced [27,7]. The breakup of island of size s to smaller islands with sizes i and j (mass is conserved s i j) is described with rate F i; j i jb and for IBAD values 1=2 < b < 1 are reasonable choices [10,11]. In these equations, the ®rst term F 1; 1 2F0 is for dimer breakup with F0 as a rate of detachment for single adatoms, determined by the ¯ux of the ion beam [10,11]. At large enough island densities, islands begin to coalesce and formation of larger islands proceeds rapidly. Within the rate equation formalism
I.T. Koponen / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 314±324
there is no entirely satisfactory way to treat coalescence. However, Biham et al. [16] have managed to describe coalescence with a model, where the coalescence rate of island with sizes i and j is proportional to the size of the islands and thus given by a rate coecient Gi;j G0 i j, where G0 U [16]. This description is adopted here also. The growth within a given layer (intralayer growth) can now be modeled with similar rate equations as previously used for island growth in submonolayer region [10,11,15], the only modi®cation being the addition of term describing coalescence, X X on1 U ÿ K1;1 n21 ÿ K1;j n1 nj ÿ F1;j n1j ; ot jP2 jP1 ons 1 X Ki;j Gi;j ni nj ÿ Fi;j ns ot 2 ijs X Ks;j Gs;j ns nj ÿ Fs;j nsj : ÿ
1
2
jP1
As Eqs. (1) and (2) are formulated in complete analogy with growth with mobile islands and breakup, we can assume that the island densities will also follow the general scaling form ns hsÿ2 g s=s, where s is the P average size of the islands, de®ned by s hÿ1 0 s2 ns as in [15,20] (prime denoting summation over s P 2) and where g x is a scaling function which depends on the form of the rates Ki;j , Gi;j and Fi;j [15]. To obtain the detailed form of the scaling function, the complete solution for island densities would be required. However, for qualitative understanding, which is here of primary interest, it suces to know that the scaling function is of the form g x xd expÿx with d depending on the degree of homogeneity of Ki;j , Gi;j and Fi;j [15,28]. In order to obtain tractable set of equations for layer-by-layer growth with interlayer transitions included, it is neither convenient nor necessary to work with complete island distributions. Instead, it suces to use adatom andPtotal island densities, n t n1 t and N t 0 ns t, respectively. From now on, we take into account the interlayer transitions and denote the averaged densities in layer I 1; 2; . . . ; Nmax by nI and NI . Each layer I
317
has a coverage hI , and the total coverage is P h Ut hI , with h0 1 for the substrate. Of the deposited adatoms each layer receives thus only a fraction, which is proportional to the exposed surface area hIÿ1 ÿ hI . The rate of change of adatom density due to interlayer transitions is modeled by a term proportional to CIL nI H 1 ÿ hIÿ1 , where H is the unit step function. There is no dependence on the coverage hIÿ2 of the underlying surface, because the transitions are always down to boundary edge, which is depleted of adatoms. The dominant interlayer transition with the lowest activation energy occurs through kink sites, and thus the total number of transitions should be proportional to the number of kink sites, and thus also to the step density. In IBAD, the step density is only weakly dependent on coverage [7], and for most practical purposes it can be treated as constant. The averaging of Eqs. (1) and (2) is done by summing over all island sizes and then de®ning the appropriately averaged rate coecients, as is done in the case of rate equation models for growth with detachment [19,29,30]. In addition to adatom and island densities, a useful third dynamic equation of growth is written for total mass M sN P 0 sns x; t of islands, which also allows one to obtain the average size of the islands s [29,30]. The dynamic evolution of the resulting model is governed by three timescales: the deposition rate F (ML/s), hopping rate CD for diusion and the interlayer transition rate CIL . Accordingly, in the averaged equations the coverage h Ut is introduced as a time parameter, R DK0 =U as a measure for a ratio of timescales characterizing diusion and deposition (note that usually K0 is not included in this ratio but the present choice simpli®es much the notation) and G CIL =CD expDE=kT , with DE ED ÿ EIL as a measure for the rate of interlayer transition CIL to diusion jumps CD . With these timescales and timescale ratios the averaged equations are obtained in the form dnI hIÿ1 ÿ hI ÿ R 2n2I rNI nI ÿ jMIb NI1ÿb dh 3 GnI1 H 1 ÿ hI ÿ nI H 1 ÿ hIÿ1 ;
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dNI 1 1 b 1ÿb 2 ÿa 2a R nI e r MI NI ÿ e j MI NI ÿc fMI NI ; dh 2 2 4 dMI R2n2I rNI nI ÿ jMIb NI1ÿb : dh
5
In the above equations, the dimensionless rate coecients have been introduced: r and j for adatom processes, e r and e j for island processes and ef for coalescence. Derivation of the detailed expression for these rate coecients is straightforward. The averaging is carried out in principle similarly as e.g. in [29,30] but now the scaling form ns hsÿ2 g s=s is utilized. Then a continuum limit is taken by replacing the summations by integrations [28], and by using the condition R of mass conservation M sN ' hn, with n g x dx the average size can be eliminated from the ®nal rate equations. Following this procedure, the rate of breakup is found to be measured by parameter R j F0 =K0 xb g x dx. An order of magnitude estimate for j is obtained by noting that typically R 1 < xb g x dx < 5 and in physically accessible region 10ÿ7 < F0 =K0 < 10ÿ2 (see [11,15] and references therein), which thus yields 10ÿ7 < j < 10ÿ2 . Similarly, for the aggregation rate due to adatom R diusion one obtains r g x dx, which is in the range 2 < r < 4 for typical scaling functions [15]. The rate coecients e r and e j for aggregation due to island diusion and for breakup, respectively, and ef for coalescence are bit more complicated and proportional to integrals ZZ e r xÿa y ÿa g xg y dx dy; ZZ e xb y b g xg y dx dy j F0 =K0
clear that it would require considering the various microscopic processes in detail going beyond the scope of the present report. 3. Results In this report, we concentrate on some representative cases instead of attempting to systematically explore the entire parameter space. Our objective is to demonstrate the eect of interlayer transitions on growth and show that with simple and easily solvable models reasonable results guiding the insight can be achieved. In our previous contributions, we have studied the eect of island diusion and breakup on submonolayer growth [10,11,15], and based on the previous experience the representative cases are chosen to be b 1=2 and a 3=2, corresponding detachment or breakup related to the size of the boundary edge [10], and island diusion mediated by the periphery or edge diusion mechanism [15,25]. Variations of these parameters within the physically reasonable limits for metallic layers, i.e. 0 < b < 1 and 1 < a < 2, do not essentially change the qualitative features of the results given here. The model describes the growth of the layer in terms of the island density, island size and coverage. In addition to these quantities, the smoothness of growth can be monitored by the so-called anti-phase intensity or anti-phase Bragg intensity, which is measured by anti-phase scattering of He, or diraction methods such as RHEED and LEED [12±14]. The anti-phase intensity used in this report is calculated from expression [7] " #2 1 X i I h ÿ 1 hi ÿ hi1 ; 6 i0
and ef
ZZ x yg xg y dx dy:
The range of values for these parameters can be j < 10ÿ2 and estimated to be 2 < e r < 10, 10ÿ7 < e 1 < ef < 3. More detailed assessment of the values for rate coecients is not pursued here. In principle, it is possible to make this connection but it is
where hi is the coverage of the ith layer. In this case, the new layers start to grow only when underlying layers are nearly completed and the antiphase intensity shows oscillations, like a series of parabolas, and the period of oscillations corresponds to 1 ML [7,12]. When nucleation occurs on top of the existing islands before completion of the layer, the anti-phase intensity is strongly damped and approaches zero at h 1.
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The numerical solution of the ®rst-order coupled nonlinear rate equations has been performed by using MATHEMATICA [31], which is a convenient and accurate tool for such initial value problems as encountered here. Checks of accuracy were done by varying maximum stepsize and error bounds, and by checking some of the results against solutions obtained by NAG-library routines. The set of equations was also tested against the existing results for similar rate equations used in submonolayer thermal growth [16] and growth with breakup [10]. The ®rst example to be studied is growth without interlayer transitions, described by G 0. The timescale ratio of diusion to deposition was chosen to be R 105 , favoring diusion driven aggregation. Breakup of islands was varied from moderate with j 1:5 10ÿ6 to enhanced breakup with j 1:5 10ÿ4 (compare with [15]). In accordance with previous results for submonolayer growth, it was found that only the order of magnitude of the fragmentation is important [15], and therefore one can set j e j without loss of generality. Aggregation is described by rate coecients r 1 and e r 4, and in order to mimic the real growth after submonolayer region the eect of coalescence is included in all cases and rate of coalescence is chosen to be ef 1, which is in accordance with [16]. This choice of parameters is consistent with low temperature growth in case of Cu(0 0 1) and Ag(0 0 1). For Cu(0 0 1) surface, at temperatures lower than 100 K, aggregation and island growth dominate and with suppressed downward transitions there is no possibility for layer-by-layer growth. In the case of Cu(0 0 1) layer-by-layer growth without ion beams is expected to occur only at temperatures 200 K or higher [12]. Fig. 1 shows the scaled island density N =Nmax , scaled island size s=smax and coverage hI for each growing layer in the case for moderate breakup with j 1:5 10ÿ6 . The results resemble the case of thermal growth, but are however strongly affected by the breakup. In the case shown in Fig. 1, the maximum average size of islands is smax 250, without breakup average size would be 1950, diminishing to 70 with j 1:5 10ÿ5 , and eventually to 30 with j 1:5 10ÿ4 . The maximum
319
Fig. 1. Scaled island density N =Nmax , scaled average size s=smax and coverage for growing layers in deposition with moderate breakup and suppressed downward transitions. The total coverage is 6 ML corresponding to 6 completed layers shown. The parameters are G 0:0 and R 105 . Rates for aggregation and detachment are r 1 and e r 4, and j j 1:5 10ÿ6 , respectively, and for coalescence f 1:0. Maximum island density is Nmax 8:6 10ÿ3 and maximum island size is smax 250.
island density changes accordingly from Nmax 8:6 10ÿ3 for j 1:5 10ÿ6 to Nmax 3:2 10ÿ2 for j 1:5 10ÿ4 . In the case of suppressed interlayer transitions, the eect of breakup is mainly
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to moderate unbounded growth of average size, which would now otherwise increase rapidly due to coalescence. From the results for coverages of each layer it is seen, how all layers grow at the same time and the completion of layers occurs very slowly. Even after deposition of 3±4 layers, the ®rst layer is neither completed, nor has the average size reached the stationary value. The growth is, however, retarded, because underlying layers acquire mass at low and lower rates due to shading eects of islands formed on top of them. Therefore, all growing layers compete eectively only of the deposited adatoms, in proportion to the exposed surface. The corresponding antiphase intensities measured e.g. RHEED (and calculated now from Eq. (6)) are in these cases simply decaying functions reaching zero rapidly at coverage of order unity. All the above discussed cases are favorable for 3D growth but clearly undesired if 2D layer-by-layer growth is pursued. The downward transitions have an essential role in promoting the layer-by-layer growth, and e.g. in the case of Cu(0 0 1) interlayer transitions begin to aect the growth at temperatures of 200 K or higher, which leads to promotion of layer-bylayer growth [13,14]. The importance of downward transitions at low temperatures in IBAD is clearly demonstrated by Jacobsen et al. [7] in their KMC± MD simulations, where growth on Ag(1 1 1) and Pt(1 1 1) in IBAD is studied in detail. Jacobsen identi®es the eect of ion bombardment with increased rate of downward transitions, and ®nds an optimum window for this around 25 eV deposition. Within the framework of our model, this effect is mimicked by increasing the rate of downward transitions. In the present model, the eects of IBAD on enhanced interlayer transitions enter through timescale ratio G, which contains information on both the increased step density and temperature and thus G nStep expDE=kT , where nStep is the relative increase in step density. With typical activation barriers in fcc(0 0 1) metals DE 70 MeV, and for relative increase in step density a conservative estimate 2 < nStep < 5 is used. Thus, reasonable values are estimated to be in the range 50 < G < 500 under typical conditions in IBAD.
The eect of the downward transitions is demonstrated in Fig. 2, where the results for G 100 are shown. In the case of fcc(0 0 1) metallic surfaces, this corresponds to the situation encountered at room temperature. The other parameters
Fig. 2. Scaled island density N =Nmax , scaled average size s=smax and coverage for growing layers in deposition with enhanced breakup and moderate downward transitions. The parameters are G 100 and R 5 103 and j j 1:5 10ÿ4 , otherwise as in the case shown in Fig. 1. Maximum island density is now Nmax 4:1 10ÿ2 and maximum island size is smax 26.
I.T. Koponen / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 314±324
are comparable to the case shown in Fig. 1. The eect of downward transitions is to accelerate the completion of layers, due to additional ¯ux of adatoms from upper layers to lower ones. Now each layer is nearly completed before the growth of other layers starts, and as seen from the results for island densities, each layer grows essentially in a similar way. At the same time, the average size of islands is greatly reduced and island density increases. However, the average size does not reach the stationary value but continues to increase, which reveals that aggregation is still strong enough to allow slow but steady growth of islands. In the case shown in Fig. 2, the maximum island density is Nmax 4:1 10ÿ2 and maximum island size is smax 26. The case displayed in Fig. 2 is the optimum case, where the increase in breakup does not lead to more pronounced layer-by-layer growth because the rate of downward transitions acts as a rate limiting process. The corresponding anti-phase intensity calculated from Eq. (6) is shown in Fig. 3 with result for more moderate breakup. In the optimum case, the oscillations in anti-phase intensity are pronounced, but they diminish substantially with reduced breakup rate and for j 1:5 10ÿ6 the oscillations are essentially similar to those found for thermal deposition at room temperature (compare with results in [12]). These results point out clearly the important
Fig. 3. Anti-phase intensities calculated from Eq. (6) for enhanced breakup and moderate downward transitions with G 100. The cases shown are for j j 1:5 10ÿ4 , 1:5 10ÿ5 , 1:5 10ÿ6 , from top to bottom. Other parameters are as in the case shown in Fig. 2.
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role of enhanced detachment as an agent promoting layer-by-layer growth. The larger the rate of downward transition is, the more easily the layer-by-layer growth occurs if breakup rate is large enough to maintain the ¯ux of adatoms to lower layers. In the region of large but yet reasonable downward transition rates where G 200 the overall appearance of island density, coverage and average size are quite dierent to the cases shown in Figs. 1 and 2. With optimally large breakup rate 2:0 10ÿ4 all these quantities reach their stationary values already within h 1, as shown in Fig. 4. Now there is no considerable overlap between temporal behavior of growth in dierent layers. With increasing rate of downward transitions, the upper layers are rapidly depleted of the adatoms and the large rate of breakup provides an eective means for continuous detachment thus preventing the formation of large islands and eectively suppressing the 3D growth mode. In the case displayed in Fig. 4, the maximum average size is only smax 12, the maximum island and adatom densities are Nmax 8:3 10ÿ2 and nmax 6:0 10ÿ2 , respectively. The values of these quantities do not depend on the rate of downward transitions because they are determined by the intralayer process, by the balance of aggregation and breakup (compare with results in [11,15]). However, the saturation values are reached much earlier in the case of enhanced downward transitions, in the case of G 200 it occurs already at h 1 while in the case G 100 it happens only at h 2. The nearly perfect layer-by-layer growth is now clearly revealed also by the oscillations of the antiphase intensity, shown in Fig. 5 for cases G 100; 150 and 200. In comparison with Fig. 3 the intensity of anti-phase oscillations is nearly doubled, indicating more perfect layer-by-layer growth. Moreover, the anti-phase intensity is comparable to that found by Jacobsen et al. [7] for layer-by-layer growth on Ag(1 1 1) when 25 eV Ag ion bombardment was used. These results demonstrate that with suitable combination of detachment and accompanying enhanced downward transitions it is possible to achieve nearly perfect layer-by-layer growth.
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Fig. 5. Anti-phase intensities calculated from Eq. (6) for enhanced breakup j j 2 10ÿ4 and downward transitions with G 200; 150 and 100, from top to bottom. Other parameters are as in the case shown in Fig. 4.
Fig. 4. Scaled island density N =Nmax , scaled average size s=smax and coverage for growing layers in deposition with enhanced breakup and downward transitions. The parameters are G 200 and R 5 103 and j j 2 10ÿ4 , otherwise as in the case shown in Fig. 1. Maximum island density is now Nmax 8:3 10ÿ2 and maximum island size is smax 12.
4. Discussion In metallic fcc(0 0 1) surfaces, it is possible to obtain layer-by-layer growth at suciently high temperatures, e.g. Cu(0 0 1) at 200±250 K. This is due to the fact that interlayer transitions through
kink sites can eectively compete with intralayer transitions [12,21]. However, even at lower temperatures it is possible to obtain layer-by-layer growth by using ion beams in promoting the interlayer transitions through additional kink sites created by ion bombardment. Here the situation was chosen, so that without IBAD layer-by-layer transitions do not occur. The eects of IBAD on enhanced interlayer transitions enter through timescale ratio G, but contributions of the dierent microscopic processes on parameter G were not separated in detail. Anyway at present only for Ag(1 1 1) there are microscopic data available for such detailed description [7]. The choice of island size independent total transition rate is in agreement with results of Jacobsen et al. [7], which show that there is no noticeable correlation between step density and coverage after the transient stage in the submonolayer region, i.e. step density and transition rate proportional to it are essentially constants. The fact that the total transition rate is eectively independent of the coverage of the lower layer and average edge size of islands can be of course due to the fact that product s1=2 hIÿ1 ÿ hI happens to be constant in the region of growth of interest. The rate equation model developed here is not quantitatively accurate in the limit h ! 1 mainly due to lack of detailed knowledge on how to model coalescence. The model adopted here, based on the results in [16], is expected to be reliable up to
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h 0:5 but at larger coverages the accuracy of the model probably deteriorates rapidly. We expect that the present model underestimates the role of coalescence. However, this is not a serious drawback, because we are mainly interested in region of growth, where detachment or breakup keeps the island size in moderate limits and prevents the formation of large islands. The scaling analysis was needed in the derivation of the rate equations. In all regions of growth, the scaling law sN h holds but the coecient of proportionality may vary. Moreover, the detailed study of the validity of scaling analysis in the present case is a rather demanding task and clearly beyond the scope of present study. Detailed knowledge of the scaling function is needed to get quantitatively accurate rate coecients [15] but ®nding the scaling function amounts up to solving the complete model with all island distributions. Therefore, even within the framework of rate equations we have good reasons to limit the model in the phenomenological and qualitative levels of description. 5. Conclusions In this report, a model describing layer-by-layer growth in IBAD is developed. It is demonstrated how the dierent intra- and interlayer processes aect the growth. The microscopic processes affecting the growth considered are adatom diusion, island diusion, detachment and breakup, coalescence of large islands and interlayer transitions of adatoms. The eects of all the dierent processes on growth are described with reaction rates and the growth process itself is modeled by rate equations governed by these reaction rates. The advantage of the model is its great simplicity, which also allows a rapid numerical solution. This makes it possible to explore eectively the interesting parameter regions and use it as a tool for planning experiments or e.g. more detailed simulations. From results based on the present model, it can be deduced that enhanced detachment of adatoms due to ion bombardment is a prerequisite for promoting layer-by-layer growth but without suciently fast downward transition path adatom
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¯ux to underlying layers may be too moderate and prevent the onset of layer-by-layer growth. In order to obtain optimum conditions, both requirements must be met. In practice, the energy and mass of the bombarding ions can be used to tune the breakup and detachment rate. The ambient temperature plays a role in determining the parameter G, but ion bombardment has a central role in controlling the rate of transfer of adatoms through the kink density, which is re¯ected in the special form of the term chosen to model this effect.
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