A MONTE CARLO SIMULATION OF THE PHYSICAL VAPOR … · Department of Materials Science and...

Pergamon Acta mafer. Vol. 45, No. 4, pp. 1455-1468, 1997

0 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd

PII: S1359-6454(96)00256-X Printed m Great Britain. All rights reserved 1359-6454197 $17.00 + 0.00

A MONTE CARLO SIMULATION OF THE PHYSICAL VAPOR DEPOSITION OF NICKEL

Y. G. YANG, R. A. JOHNSON and H. N. G. WADLEY Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903.

U.S.A.

(Received 4 March 1996; accepted 15 July 1996)

Abstract-A two-step Monte Carlo method for atomistically simulating low energy physical vapor deposition processes is developed and used to model the two-dimensional physical vapor deposition of nickel. The method consists of an impact approximation for the initial adatom adsorption on a surface and a multipath diffusion analysis to simulate subsequent surface morphology and interior atomic structure evolution. An embedded atom method is used to determine the activation energies for each of the many available diffusional paths. The method has been used to predict the morphology/structure evolution of nickel films over the length and time scales encountered in practical deposition processes. The modeling approach has enabled determination of the effect of vapor processing variables such as flux orientation, deposition rate and substrate temperature on deposit morphology/microstructure as defined by packing density, surface roughness and growth column width (which appears closely related to grain size). Several aspects of the empirical MovchanDemchishin structure zone model are well predicted by this approach. fi 1997 Acta Metallurgica Inc.

1. INTRODUCTION

Physical vapor deposition (PVD) via either electron beam evaporation [l], resistive heating [2] or

sputtering [3] is widely used for depositing metal films at low to intermediate rates (< 1 pm/min). Interest is growing in the development of potentially very high rate (l-100 pm/min) physical vapor deposition processes such as jet vapor deposition (JVDTM) [4-S] and directed vapor deposition (DVD) [9, lo] for synthesizing metal multilayers with minimal inter- layer diffusion. In all these processes, the atomic flux (or equivalently the deposition rate), the flux incidence angle, the incident atom kinetic energy and the substrate temperature can all be independently varied. Since these parameters govern the kinetic phenomena involved in the atomic assembly and reconstruction of surfaces during film growth, many options are available for controlling the morphology/ microstructure of a deposit during vapor phase manufacturing. The many variables of the processes make it difficult to identify the conditions that result in acceptable morphologies/microstructures.

The prediction of these morphologies/microstructures, and their dependence on material system properties and process parameters has proven to be very difficult. The structures predicted by classical arguments [l l] (e.g. layer-by-layer or Frank-van der Merwe [12], island or Volmer-Weber [13], and the layer-plus-island or StranskiLKrastanov [ 141 growth modes) are normally applicable only to the early stages of the nucleation and growth of epitaxy. However, the empirical Structure Zone Model (SZM)

of Movchan, Demchishin and Thornton [15. 161 indicates that, depending upon the deposition temperature, the actual microstructures consist of either tapered crystals with domed tops separated by voided growth boundaries (zone I), columnar grains separated by metallurgical grain boundaries (zone II), or equiaxed grains corresponding to a fully annealed structure (zone III). These microstructures may also be twinned, contain stacking-fault structures [17, 181 or other types of defected structures [19, 201.

This research contributes to the development of an atomistic-based modeling approach that might eventually be used to identify optimal deposition conditions for any material system of interest. Thus, we seek a model that, when given a substrate temperature, a deposition rate, an incidence angle, the adatom kinetic energy, a substrate geometry and a material system as inputs, is capable of predicting the morphology/microstructure of the resulting deposit. As a start, this work concentrates on low incident energy processes such as evaporation in which incident atoms have only thermal energy.

Both Molecular Dynamics (MD) [21] and Monte Carlo (MC) [22] methods are able to simulate some aspects of the microstructure of a vapor deposition process, Each method has its advantages and disadvantages [21-231. In MD calculations, the atomic configuration of the deposit is represented by the coordinates and velocities of the atoms and the dynamics are completely determined by these coordinates and the interatomic potential function that is used to represent the interaction between the atoms [21]. The model permits simulations of

1455

1456 YANG et ~1.: MONTE CARLO SIMULATION OF PHYSICAL VAPOR DEPOSITION

granular structure [24], epitaxial growth [25], twin formation [24], and stress development [26, 271 and their sensitivity to the diffusion processes [28], energetic bombardment modification [27729]. How- ever, since the lattice atoms vibrate in this approach, the forces on atoms must be calculated several times per lattice vibration period (i.e. every femtosecond or so). Because of this, computations in reasonable times (less than several hours of workstation computation) can only be conducted for problems with a limited physical elapsed time and a small system size. Generally, only systems with less than several thousand atoms deposited in a period of less than a nanosecond or two can be simulated. This results in the simulation of a vapor deposition process with unrealistically high deposition rates of l-10 m/s [24, 281.

While MD methods are deterministic, the many variants of the Monte Carlo method are based on probabilities. Among those addressing low energy processes, all use a form of Henderson’s model [30] to deduce the initial adatom configuration. Some strictly follow the Metropolis procedure in which the atoms are moved in accordance with Boltzmann statistics but the kinetic path of evolution is physically meaningless [31, 321. Others follow a possible path taken by the system and generate a more correct, usually nonequilibrium structure [20, 3040]. Regardless of approach, the common feature is that the detailed interatomic forces need not be evaluated and the movements of atoms are determined by a set of statistical rules established beforehand. The benefits of this method are that simulations can be carried out rapidly and a very large number of atoms can be simulated in relatively short periods of time depending upon the type of algorithm used. MC methods have been applied to studies of condensation phenomena [37], the occurrence of anisotropy [30], the scaling properties of vapor deposition [35], step coverage in metallization processes [40,41] and the reproduction of the columnar structure in vapor deposition processes [20]. However, the MC method also suffers limitations; in particular, it imposes severe con- straints on the crystal structure. In most simulations, an Ising model [42] is used in which a crystal lattice is selected in advance, and the allowed atomic configurations are described by specifying the occupancy of each crystal lattice site. In this approximation it is assumed that crystal growth only involves a very small subset of all the possible atomic configurations. Because of this, it cannot then model amorphous systems, dislocations, twins, grain orientations and other problems (e.g. stress) involving atoms that are not on bulk lattice sites [22, 231. Another limitation of the model as it is normally used arises from the sometimes extreme approximations made to simplify diffusional processes. This has resulted in the absence of a connection between the simulation and the physical

time scale and real temperature of the growth processes [21].

Although MD and MC applications can be used to explore many important features of vapor deposition processes, their use to identify the best practical process parameters is still difficult. A start toward this end was made by Mi_iller [34], who sought to introduce meaningful temperature and time scales into a MC approach. Although his approach was capable of identifying a transition between zone I and zone II structures defined in the SZM model, it suffered from a number of deficiencies. For example, the bond counting method used to assess the jump’s activation energy was too simple and could not take the Schwoebel barrier [43] into account. Also, atomic jump decisions were made by comparing the thermal energy with the bond counting activation barrier and always making a possible jump into the site with the highest coordination. In addition, a proper link between the real deposition time and the diffusive process time was not made. Another effort toward realistically modeling higher energy sputtering processes has been made by Fang and Prasad [41] in which a hybrid numerical scheme was used to account for the entire deposition process. In the first step, a MC algorithm was used to calculate the transport of sputtered atoms in a deposition chamber. A simplified MD approach along with a nuclear scattering theory was adopted to approximate the collision between the incident atom and the substrate/deposited-film. Finally, a mobility parameter was calculated in order to determine the extent of hopping before an atom reached the lowest energy neighbor site. The method has now been successfully used to predict step coverage of sub-micron contact holes. Although the kinetic energy was appropriately treated, an explicit treat- ment of the substrate temperature and the deposition rate was not developed.

Two phenomena must be incorporated in a MC simulation of physical vapor deposition. First, it is necessary to characterize the outcome of the adatom-substrate collision process that results in an initial adatom configuration. Secondly, the subsequent diffusion of atoms both on the growing surface and within the previously deposited bulk of the film must be modeled. For low incident energy processes, the Henderson, Brodsky and Chaudhari model [30], demonstrated in Fig. 1, has been widely used in the past for the first step. Atoms (represented by solid discs) are randomly dropped from above the substrate. They are assumed to travel in a straight line (at a defined angle to the substrate normal) until coming in contact with an already deposited disc on the substrate. The incident disc is assumed to remain in contact with the disc with which it first makes contact, but is allowed to relax to the nearest cradle formed by the first contacted disc and any other previously deposited disc in the surface. After this

YANG et al.: MONTE CARLO SIMULATION OF PHYSICAL VAPOR DEPOSITION 1457

relaxation process is completed, the next incident hard disc is introduced and so on.

Henderson et al. initially intended to use their model to simulate the entire atomic arrangement process in random networks or dense random packed arrays to demonstrate the natural occurrence of anisotropy and voids in film growth. However, they and others subsequently found that the packing densities of the simulated films were much smaller than those of real films [30, 33, 44,451. Furthermore, the angle of inclination of the column structures they predicted was only slightly less than the flux incidence angle [30]. This disagreed with experimental observations [46]. In an effort to improve this model, Kim et al. [33] allowed the incident particle to “bounce” according to a pre-assigned probability after first impact. In the bouncing process, atoms then had a chance to move to sites with a higher coordination. They found that the film density increased as the probability of bouncing was increased. Gau et al. [44] introduced a migration parameter i (defined as the ratio of the average distance traveled by the atom after impingement to the diameter D of an atom) to allow incident atoms to migrate across the surface until a stable location was reached. By choosing different values for d, they crudely simulated different adatom mobility conditions. In a similar approach, Dew et al. [40] approximated long range surface diffusion by introducing an average diffusion length and using a calibration factor to make the simulated column size comparable to that in a real growth case. All these models had the common feature that after an adatom eventually came to rest, it never again moved. Thus the structure was frozen after the initial adsorption/relaxation event, and further diffusional relaxation as deposition progressed was precluded.

Henderson Model

Momentum

Fig. I. Schematic illustration and comparison of the Henderson model and the momentum scheme for the initial relaxation in the two-step Monte Carlo simulation. The shadowed disk represents an atom in flight and the dashed disk an atom hitting a previously deposited atom. Path B stands for the Henderson model and path A the momentum

scheme. a is the incidence angle.

These schemes can therefore be considered one-step models. In such models, it is impossible to precisely connect a migration parameter or an average diffusion length to the deposition temperature and rate. Thus, the microstructures they predict are only valid when diffusion is relatively insignificant (i.e. very high rate or low homologous temperature deposition). Two-step models that allow continued atomic movements after the initial deposition event are needed to more realistically simulate the phenomena that occur under conditions where diffusion is likely to be active.

A significant start in the development of two-step MC models in low energy processes began with work by Miiller [34], who permitted unlimited adjustment of an “average diffusion length” and was therefore able to approximately incorporate the consequences of high homologous temperature or low rate deposition. However, when the homologous deposition temperature was very low and/or the adatoms possessed very limited mobilities, this scheme again simply reduced to the Henderson model which may be an inaccurate approximation of the initial impact process.

A two-step MC simulation model for low energy (< 1.0 eV) deposition is proposed in the present report in which the actual process parameters can be used as inputs. It involves the use of a momentum scheme to identify the initial surface configuration of an adatom and then a multipath analysis of all subsequent diffusion. The results of a two- dimensional (2-D) model are presented first for simplicity; subsequent work will extend the approach to the full three-dimensional case and sputtering process. The results of this work will be shown to provide a deeper insight into many of the different physical processes involved in physical vapor deposition and enables exploration of the effects of process conditions upon morphology and microstructure.

2. ADATOM INCORPORATION

In the present scheme to simulate low energy processes, the interaction at the moment of adatom incidence has been modified from the Henderson model, and the relaxation step is treated in a more general way than that used by Miiller. As shown in Fig. 1, an incident atom 0 travelling along the cd direction inclined at an angle CI from the normal to the substrate will first touch atom F located at positi0n.f. With Henderson’s nearest cradle criterion, the atom was assumed to go to the nearest stable site which is B. This method results in a very low packing density which is not observed in experimental work. We note that the adatoms deposited in even low energy deposition process can still carry considerable energy when they arrive at the substrate. For example, Zhou, Johnson and Wadley [47] using MD simulations found that a nickel atom with thermal

1458 YANG et al.: MONTE CARLO SIMULATION OF PHYSICAL VAPOR DEPOSITION

energy of 0.1 eV can be accelerated to a kinetic energy as high as 1 .O eV in the attractive force field of the surface atoms before being incorporated in the lattice. This same phenomenon has also been observed for silicon deposition by Gilmer [48], again through MD simulation. Thus, conservation of momentum is an important consideration and suggests that the atom 0 will more likely move towards the cradle site A when the landing point is to the left side of line ef(for a head-on collision), even though the distance of path oA is larger than that of oB. This is clearly an approximation to the complicated binary collision processes with a rough surface at low energy situations [38].

This simple modification, which we term the momentum scheme, generates configurations having significant differences with those of the Henderson model. Two sample configurations are shown in Fig. 2, both with an incidence angle a of 45”. In the momentum scheme, the column orientation angle with respect to the normal to the substrate, fl, is significantly less than that with the Henderson model. This trend continues for a range of M as shown by Fig. 3(a). From a plot of packing density (defined as the fraction of atoms occupying the lattice sites in the deposit region, with sufficient number of surface layers and the substrate excluded from the measure- ment; the packing density = 1 if fully packed) as a function of incidence angle, Fig. 3(b), it is seen that structures generated by the momentum scheme have considerable higher packing densities than those generated by the Henderson model when a is less than about 30’. When c( approaches 50’, the density tends to be similar for both models. Thus the momentum scheme provides a better initial relaxation model for prediction of density [30, 33, 44-461.

The decrease of the column orientation angle can be indirectly confirmed by experiments. Nakhodkin and Shaldervan [46] using evaporation process conducted an extensive study on the effect of condensation condition on the profile of condensed films. One of their results was that the column orientation angle decreased with the increase of the substrate temperature, which is in essence related to the increase of the surface mobility of adatoms. Since adatoms can relax to a distant cradle site rather than to a near one, the momentum scheme essentially increases the surface mobility that leads to the result of a smaller angle. Therefore, in simulating vapor deposition, atomic kinetic energy must be taken into account in an appropriate way even in very low energy processes.

3. DIFFUSION MODEL

3.1. Activated difSusion process

As shown in Section 2, the structures generated by both the Henderson model [30] and the momentum scheme do not account for atomic diffusion and have low densities. They can only represent the situations

a

4 P

a = 45”

(a) Henderson Model

(b) Momentum Scheme

5nm ,

Fig. 2. Representative configurations simulated using (a) the Henderson model and (b) the momentum scheme. An incidence angle IX of 45” is used for both cases and B is

indicated for column orientation angle.

where diffusion is unimportant, i.e. at very low homologous temperatures and/or very high deposition rates. In practice, most deposition processes are conducted under conditions where atomic diffusion occurs simultaneously with deposition and diffusion must therefore be taken into account in the simulation. The methodology should address the many diffusional pathways available (e.g. due to the atomic coordination number dependence of the


jumping activation energy). It should also connect with the temperature and deposition rate since these control the available time for rapid surface diffusional processes before the surface becomes covered by a new adatom layer.

If Boltzmann statistics are assumed to govern the diffusional processes, the probability per unit time for a possible jump, i, to take place is given by

where 1’” is the effective vibration frequency (taken to be 5 x lO”/s for all the cases in this work), E, is the activation energy for the ith type of jump, k is Boltzmann’s constant and T the absolute temperature. If the different diffusional pathways each have different E, values, they must also have different p, values.

The reciprocal of an atomic jump probability per unit time (equation (1)) is a residence time for an atom that moves by that specific type of jump. Since the jump probabilities of all the different types of jumps are independent, the overall probability per unit time for the system to change its state by any type of jump step is just the sum of all the possible specific jump type probabilities, and so the residence time for the system in a specific configuration is the reciprocal of this overall jump probability. The next diffusional step is determined by randomly choosing

80 I I I / /

(a) Column orientation angle 1’

60 - & 1 I I

Henderson model 7 ,A’ p A” /d

I I

0.9 / I

(b) Packing density

0.80 I3 c - n 0 / Momentum scheme

b I I /

20 40 60 80

Incidence angle CL (deg)

Fig. 3. Comparison of the difference between the Henderson model and the momentum scheme in terms of (a) the relationship between column orientation angle /II and incidence angle r and (b) the relationship between packing

density and incidence angle, a.

from among all the possible jumps weighted by their relative probability of occurrence. By following the ensuing discrete jump path for the system, accumulat- ing the residence time of the system along the path, and linking the duration of this history to the adatom arrival interval (or the deposition rate), the diffusion process can be realistically simulated.

The calculation begins by determining a time interval between adatom arrivals based on the deposition rate. The average time interval between the arrival of two atoms in a 2-D lattice model is

At=& where R is the deposition rate (deposited thickness per unit time), a is the nearest neighbor distance and n is the number of atoms comprising a monolayer in a close-packed simulation system. Clearly, the higher the deposition rate, the smaller the time interval between two consecutive deposition events in the model. Time periods are then related to the diffusion process through a net residence time, t,,, given by

where N is the number of different types ofjumps (i.e. different diffusion pathways). In this model, a single jump is allowed only to vacant nearest-neighbor sites or over a ledge at the surface, i.e. a Schwoebel jump [43]. The two time definitions (equations (2) and (3)) are then linked for the simulation.

An atom is next dropped from a random position above the surface. It travels to the surface in a straight line and impacts the substrate at an angle r* to the normal. It is then initially relaxed using the momentum scheme. The set of atoms in the simulation system are then monitored for diffusional modification prior to the arrival of the next atom, i.e. in the time At. If the system has a net jump probability greater than 1 in the At time period, a jump is made and a time equal to the net residence time as calculated using equation (3) is subtracted from At so that there is less time remaining for further jumps in the allotted time period. This process is iterated until the probability of making any jump in the remaining time is less than one. Whether or not any jump is made in this time period is then determined by random choice based on the remaining time and the net time for that particular state of the system. Whenever a jump is to be made, the specific one is determined by random choice based on the relative probabilities of all potential jumps. When the remaining time reaches zero, the clock is turned ahead by At and another atom is then deposited.

3.2. Calculation of the activation energies ,for migration

Each of the probabilities, p,, for the possible migration steps are required as input for the


calculation of diffusion by this MC method. To obtain physically reasonable values for the 2-D problem being studied, atomistic calculations for the activation energy for migration for different diffusion paths (defined by the initial and final configurations) have been carried out.

A 3-D EAM model [49] with parameters for nickel has been modified for use with a close packed 2-D array constrained to remain in a plane. The energy of the 2-D array was minimized to find a 2-D equilibrium lattice constant and cohesive energy. No adjustments were made to the 3-D EAM parameters, and static calculations were then carried out to track the energy barrier for a number of possible jump configurations. Since at this stage the calculations are designed to indicate overall behavior and not provide exact results for each jump type, the static calculations used an approximate relaxation scheme that resulted in activation energies that were accurate within several hundredths of an eV.

The basic equations of the EAM [50] in the notation used by Johnson (511 are

E=&%)f~KG P6= cf(y,) (4) I ,ir i”

where E is the total internal energy, 4(r,,) is the two-body potential between atoms i and j, P,, is the distance between atoms i and j, F(p,) is the embedding energy of atom i, p, is the electron density at atom i from all other atoms, and f(~,,) is the contribution to the electron density at atom i from atom ,j. Although the physical interpretations are different, these equations have the same form as those developed by Finnis and Sinclair [52]. The model used above has been “normalized”, so that independent equilibrium is attained from the two-body interaction and from the embedding function [53].

Analytic forms for the two-body potential, the electron density, the embedding function and the parameters for a nearest-neighbor EAM model for nickel are used in the present calculations [50]. When applied to an infinite 2-D close-packed plane of atoms, the equilibrium spacing decreases from 2.49 to 2.42 8, and the cohesive energy decreases from 4.45 to 3.515 eV. The 2-D bulk vacancy formation and migration energies are 1.14 and 0.83 eV, respectively, and the 2-D surface energy, assuming the 2-D sheets are stacked as 3-D {ll l}planes, is 1.4 J/m’.

With a normalized nearest-neighbor EAM model, the 2-body potential yields equilibrium in a 2-D model at the 3-D lattice spacing and will produce no forces for relaxation at a surface. The embedding energy favors the bulk 3-D electron density at each site, so the decrease in the lattice constant for 2-D equilibrium is caused by the system adjusting to increase the electron density at each atom, and is

Table I. Calculated activation energies for possible jumps usmg a 2-D nickel EAM

Configurational Bonds Energy transition (fromto) (eV)

I

2

3

4

6

I

8

9

10

11

12

13

14

2-2 0.44

Z-3 0.38 3-2 0.91

3-3 0.85

24 0.3 I 4-2 1.34

444 0.96

3-4 4-3

0.71 1.21

4-s 5-4

3-5 5-3

5-5

0.48 0.93

0.20 I .02

0.70

2-s Spontaneous s-2 Unstable, > I .30

bulk 3-3 bulk 0.80

bulk 5-5 bulk 0.83

2-3 via 24 and 4-3

2-3 via 2-l-3

0.57, Reverse I .06

0.66, Reverse 1. I5

counterbalanced by the consequent increase in energy in the two-body bonds as the system contracts. This same effect leads to an inward displacement of atoms at a surface.

The basic jump configurations were identified by the number of nearest neighbors to the jumping atom before and after the jump. The arrangement closest to the surface was chosen with two bulk cases included for comparison. One of the bulk cases was just vacancy migration; the other was the most bulk-like 3 to 3 atom coordination configuration. The numerical results are summarized in Table 1 where each run number corresponds to a different diffusion path. Run 9, which in bond number is like bulk vacancy migration, involved a vacancy in the second row moving parallel to the surface. Runs 13 and 14 have the same initial and final configuration: the configuration changes from there being one atom on top but at the edge of a one-atom high terrace (i.e. at the terrace ledge) to the terrace being extended by one atom. In run 13, the ledge “crumbles” with two


atoms moving concurrently so that the atom which had been on top of the terrace replaces the atom below it at the edge of the terrace, with atom moves to extend the terrace. In run 14, the single atom goes over the top of the ledge. The activation energy for crumbling is less than that for over the top, but for purposes of the diffusion calculation, this is taken as a two-bond to one-bond case followed by spontaneous decay from the unstable one-bond configuration. Both jumps represented by runs 13 and 14 are considered as Schwoebel jumps [43]. The Schwoebel barrier is defined [54] as the activation energy difference between a Schwoebel jump (either run 13 or run 14) and a two-bond to two-bond jump on smooth surface (run 1) and so that the barrier energy is either 0.13 or 0.22 eV depending upon mechanism.

4. RESULTS

In this two-step MC model, the momentum scheme, an approximation for low energy processes, is used for the initial adsorption of the impinging atom and the diffusion model is used for simul- taneous annealing during the deposition. Desorption is ignored because simulations are carried out at relatively low temperatures. All calculations are based on the parameters for the 2-D nickel model and lateral periodic boundary conditions are employed to account for the limited system size. The substrate consisted of a perfect array of 200 close-packed atoms. A representative incidence angle, c( = 38’, was used for all the calculations, and 8000 atoms were deposited during each run. Five runs were carried out for each data point to determine the statistical spread in the data. Since this is a 2-D model, the actual temperatures used for deposition are not directly related to those of a 3-D system. To compensate, we note that the 2-D vacancy formation energy is about 2/3 of the corresponding 3-D value [55]. Thus, a homologous temperature is also given, based on a melting temperature, T,,,, of 1150 K, which is 2/3 of the 3-D nickel melting temperature of 1726 K.

Two series of computer experiments are reported. In one, the deposition rate was held fixed and the substrate temperature was varied. In the other, the substrate temperature was held fixed and the deposition rate was varied. Since this model incorporates a fixed (Ising) lattice of sites which are either occupied or not, dislocations and grain boundaries cannot occur. However, mounding of the surface profile yields a characteristic width that is used as a measure of the column size. The column width was approximated by dividing the width of the system by the number of mounds. The surface roughness W is defined as the standard deviation of the surface height [35, 541, UR = CN-‘(h, - @, with k the average height of the surface layer, h, the height of ith site and N the total number of surface sites. Since a hexagonal rather than the usual square grid is chosen in the simulation, when a part of the

surface is perfectly smooth, the heights of neighbor- ing sites are adjusted to be equal to that of the higher one for proper averaging. Finally, the packing density is measured in the same way as that used in Section 2.

4.1. Eflect of temperature on microstructure/mor-

phology

The effect of the substrate temperature on the thin film structure has been simulated by changing the temperature systematically at three considerably different deposition rates. The basic physical properties reported are packing density, surface roughness and column width. For some simulations, tracer atoms were used to reveal the evolution of the internal structure of the deposit and to indicate possible mechanisms by which the evolution occurs. The tracer atoms are a set of marked adatoms deposited during a specific time period. Their position is observed throughout the remainder of the simulation run. They are shown as filled circles while the remainder of the atoms are open circles in the configuration displays. When an adatom arrives at the growing surface, it is first adsorbed and then diffuses to some extent depending upon its local configuration and the computer experimental conditions. The pattern of development can be visualized by motion of the tracer atoms.

4.1.1. Structural conjiguration, packing density and surface roughness. The final configurations at four deposition temperatures are shown in Fig. 4 for a deposition rate of 10 pm/min. At a fairly low temperature of 250 K (T/T,,, = 0.22) Fig. 4(a), a low density structure (packing density = 0.79) with voided growth boundaries, typical of zone I in the SZM model, is found. This structure forms because of self-shadowing [16]. With an increase of temperature to 300 K (T/T,,, = 0.26) Fig. 4(b), the voided growth boundaries are found to have almost disappeared (packing density = 0.97) and the internal columnar boundaries have become better defined, as demonstrated by the surface contour in relation to the tracer atom contour, although the surface is still quite rough. With a further increase of temperature to 400 K (T/T, = 0.35) a fully dense columnar structure (packing density = 1.0) and a facet-like surface typical of zone II in the SZM model was observed, Fig. 4(c). The column boundaries within the bulk can be seen with the aid of the tracer atoms: the columns nucleate at the beginning of deposition, are tilted toward the flux, and retain their approximate width throughout the run. At a still higher temperature of 550 K (T/T, = 0.48), Fig. 4(d), the surface becomes fairly flat and the tracer atoms show a rapid increase of the column width. Tracer atoms also show that only limited bulk diffusion occurs at this temperature.

The packing density as a function of temperature for this deposition rate (10 pm/min) is represented by the filled circles in Fig. 5. The fitted curve exhibits


a L R= lOpm/min

I(a) T = 250 K (T/T, = 0.22)

(b) T = 300 K (T/T, = 0.26)

(d) T = 550 K (TIT, = 0.48)

12.5 nm

Fig. 4. Representative 2-D configurations of Ni growth at various substrate temperatures at a deposition rate of 10 pm/min and an incidence angle c( of 38”. The momentum scheme is used to treat the effect of low kinetic energy (< 1 .O eV) in the deposition process. Atoms, tagged as tracers to show sample atom

movements, are shown as solid circles.

Homologous temperature T/T, Homologous temperature T/T,,,

0.00 0.10 0.20 0.30 0.40 0.50 1 0.10 0.20 0.30 0.40 0.t 50 I I I I

I Dewxition rate R= O.k5 f i4 a=38” a=38'

i lb 1 I 2;O pm / min

1 I I I I I

0.6 / 100 200 300 400 500 6C IO

I I I I I 0 100 200 300 400 500 60( 1 Substrate temperature T (K)

Deposition temperature T (K) Fig. 6. Surface roughness vs substrate temperature at

Fig. 5. Packing density vs substrate temperature at various various deposition rates for s( = 38’. The momentum

deposition rates for d( = 38-. The momentum scheme is used scheme is used to treat the effect of low kinetic energy

to treat the effect of low kinetic energy (< 1 .O eV) in the (< 1.0 eV) in the deposition process.

deposition process.

The surface morphology (or roughness) also varied three regions. When the temperature is below about with the processing parameters and is well known to 150 K, the porous structure has a fairly constant exhibit transition phenomena [16]. The effect of density of about 0.68, whereas a fully dense structure temperature on surface roughness predicted using the is obtained when the temperature is above about MC model is shown in Fig. 6. It can be seen that in 350 K. A transition occurs in the region between each case the roughness is approximately constant at these two temperatures and marks the change from low temperature, decreases with an increase of a porous columnar structure to a fully dense temperature and approaches a final saturation value columnar structure. The transition temperature, T,, is of about 6 A. The roughness transition is coincident defined as the onset of full structural densification, in with the transition in the density plot: for a conformity with the characteristic columnar bound- deposition rate of 10 pm/min, the roughness is ary difference of zone I and zone II [16]. Using this definition, r, = 350 K (T/T, = 0.3) for a deposition Homologous temperature T/T,,, rate of 10 pm/min. This temperature is defined by 0.20 0.25 0.30 0.35 0.40

both the mobility of the adatoms and the deposition 200 I I / I rate. It delineates a temperature where adatom mobility is sufficient to fill shadowed regions. The

Deposition rate R = 0.06 Km I min

shadowed regions grow at a rate proportional to 160 a=38’ T

R and so T, corresponds to the temperature at which z

a balance between mobility and deposition rate is E

achieved, and so it must be deposition rate ‘ci 120

dependent. E’

The results of simulations conducted for both a 3

8 i. - much higher (250 pm/min) and a much lower g 80

deposition rate (0.05 pm/min) are also plotted in S

Fig. 5. The faster deposition is seen to shift the P a

transition region to a higher temperature (Tr/ 40

T,, = 0.35) while the slower deposition shifts it to a lower temperature (7’,/T, = 0.24). The transition also occurs over a greater temperature spread as the 0 b I I I / 1 I

100 200 300 400 500 600

deposition rate increases. This zone I-II transition was seen by Miiller [34] as well from an analysis of

Substrate temperature T (K)

packing density in his 2-D calculations, although at Fig. 7. Average column size vs substrate temperature at the

a significantly higher temperature partly because of same deposition rate of 0.06 pm/min as that used in the

his use of the Henderson model and the bond experiments [55]. The momentum scheme is used to treat the

counting activation barrier energy. effect of low kinetic energy (< 1.0 eV) in the deposition

process.


1464 YANG et al.: MONTE CARLO SlMULATION OF PHYSICAL VAPOR DEPOSITION

\ Minimum observed grain size

01 I , I I I I I I 1100 0 2 4 6 8 10 12 14 16 18

T, / T

Fig. 8. (a) Plot of measured grain size variation with substrate temperature for Ni films deposited at a deposition rate of 0.06 pm/min using an e-beam evaporation process (ref. [55] with permission) compared with the simulated column size plotted in Fig. 7. (b) Comparisons with other

metals.

greatest below about 150 K (where densification starts) and becomes lowest at and above 350 K when densification is complete. Surface roughness thus provides an alternative way to measure the zone I-II

transition. 4.1.2. Comparison of simulated column size and

experimentally observed grain size. Since a compre- hensive experimental study of the relationship between grain size of Ni thin films and processing parameters has been carried out [55], simulations were run at the same deposition rate of 0.06 pm/min used in the experiments. The relationship between average column size and substrate temperature is plotted in Fig. 7. The column size does not experience any significant change when the temperature is at or below about 200 K, but increases rapidly above about 250 K.

These results are compared with the experimental results from the literature [55], Fig. 8. Figure 8(a) shows that at all substrate temperatures a range of grain sizes was observed and the difference between the maximum and the minimum observed grain sizes increased with the temperature. A reasonable agreement between the calculation and the minimum observed grain sizes can be seen. This suggests that a simulation approach that uses the activated processes of atomic diffusion and a relatively large system size can predict the trends in microstructure/ morphology surprisingly well. Figure 8(b) suggests that when the deposition temperature is scaled by the melting temperature, the grain sizes of many metals collapse on to a master curve and all can be fitted by a nickel model prediction.

Grovenor et al. [55] suggested that the universal nature of their graph of grain size vs homologous temperature is due to the scaling of the activation energy for bulk diffusion with T,. Since there is little bulk diffusion in the present calculations, the agreement of these results with their data indicates that surface diffusion activation energies, which also scale with T,, [46], are controlling the microstructural development at lower temperatures.

4.2. Eff;ct of deposition rate on microstructure

Some insight into the effect of the deposition rate on thin film structures has already been given. A more detailed view has been obtained by using a range of deposition rates and fixing the substrate temperature and the system size. Examples of representative structural configurations for deposition at 350 K are given in Fig. 9. As shown in Fig. 9(a), a fairly porous columnar structure (packing density = 0.96) results when a rapid deposition rate of 250 pm/min is used. The rate of 10 pm/min, typical of the high rate JVD”“‘/DVD processes, results in a fully dense columnar structure with a facet-like surface morphology, Fig. 9(b). As indicated in Fig. 9(c), further decreasing the rate to 0.5 pm/min yields a structure with an increased column size and a facet-like surface. At a still lower rate of 0.05 pm/min (typical deposition rate for a sputtering process), a structure with a further enlarged column size is generated, Fig. 9(d).

The corresponding variations of the packing density, the surface roughness and the column size with deposition rate are plotted in Fig. 10. A transition from a porous columnar structure to a fully dense columnar structure occurs at a deposition rate of about 50 pm/min, Fig. 10(a). This is similar to the transition due to the change of substrate temperature observed earlier.

The roughness values show a plateau over a considerable range of deposition rates, as illustrated in Fig, 10(b). This plateau can be ascribed to the presence of a facet-like surface morphology, Fig. 9. That is, although the column size changes with the rate, the standard deviation of the surface height, which is used as the measure of surface roughness in this work, remains fairly constant. The variation of the column size with deposition rate is illustrated in Fig. 10(c). It increases approximately as the logarithm of the deposition rate at rates less than about 200 pm/min, and is fairly constant at higher rates.

5. DISCUSSION

The computer experiments above describe the atomistic evolution processes during low energy deposition over a wide range of substrate temperatures and deposition rates. A dependence of the density and roughness with deposition rate, flux angle or substrate temperature, a coarsening phenomenon


at relatively high temperatures and low deposition microstructural evolution are relatively insignificant rates as well as structural transitions are all exhibited for the range of deposition rates and temperatures by this model. These observations can be explained investigated here. physically as the effect of the initial impact followed Although the activation energy parameters used in by thermal activation leading to atomistic surface the present work are for nickel and the model results diffusion. Bulk diffusional contributions to the apply strictly only to this metal, the study indicates

T = 350 K

(a) R = 250 pnhnin

(b) R = 10 pm/min

(c) R = 0.5 pmhnin

(d)R= 0.05 pn/min

12.5 nm

Fig. 9. Representative 2-D configurations of Ni film growth at various deposition rates at a substrate temperature of 350 K and an incidence angle a of 38 The momentum scheme is used to treat the effect

of low kinetic energy (< 1 .O eV) in the deposition process.

1466 YANG et a/.: MONTE CARLO SIMULATION OF PHYSICAL VAPOR DEPOSITION

Substrate temperature T = 350K, c( = 38”

A c (cj Average column size ‘B 150

h 01 ,,,(, I .I,.,, I .,.,,, I ,.,,, (.,,I, t .-I 102 10' 100 IO' 102 103 104

Deposition rate R (pm / min)

Fig. 10. Deposition rate vs (a) packing density, (b) surface roughness and (c) column size. A substrate temperature of 350 K and an incidence angle ix of 38- are used for all cases. The momentum scheme is used to treat the effect of low

kinetic energy (< 1.0 eV) in the deposition process.

the observed patterns of behavior for physical vapor deposition are applicable to a broad range of metals. Experimental results for vapor deposition demonstrate such a general pattern regardless of the material and the crystal structure used [55]. As shown in Fig. 8(b), the variation of grain size as a function of T,,,/T, (where T,,, is melting point and T, substrate temperature) is very similar for 10 elements including a variety of b.c.c., hexagonal and f.c.c. metal films and motivated the development of the SZM model. The results of the present calculation are also plotted on this graph as the solid squares and show the same pattern.

Thin film growth is generally considered to occur in a series of steps [56]: transport of coating atoms to the substrate, adsorption of these atoms on the growing surface, their diffusion over the surface, eventual incorporation into the coating or release from the surface by thermal desorption or sputtering, and finally movement of atoms to lower energy positions within the lattice by bulk diffusion and solid state reactions. In an actual process, however, the simulations show that the surface morphology of a film and its internal structure may be dominated by just one of these steps.

The simulation confirms that in the SZM model, the zone I structure represents the result of limited atomic rearrangement in the vapor deposition

process. When the deposition temperature is suffi- ciently low, relative to the deposition rate, the residence times for the adatoms are relatively large compared to the adatom arrival interval so that there is limited relaxation/diffusion on the surface before it is covered with additional atoms. Since every adatom essentially sticks close to where it arrives, the growth is dominated by the transport of the depositing atoms to the substrate and in particular by the set of directions from which these atoms arrive at the substrate. As a result, the self-shadowing effect is very important and void networks develop. Thus, the structure in zone I is generally characterized by low density poorly aligned crystals, a very rough domed top surface, and small column sizes. In each column, there are essentially several smaller sub-columns aligned in a similar direction, as reflected both in the experimental observations [55-591 and the computer simulations. We find that this type of porosity (due to the limited rearrangement of adatoms) is difficult to eliminate [45, 601. Even when the films are subjected to post-deposition annealing, these voids tend only to change shape since the overall structural framework has been previously established.

The residence times of adatoms at a surface site decrease with an increase in temperature. When the temperature is increased to the point where the residence is comparable to the adatom arrival interval, At, the porous columnar structure begins to change to a fully dense columnar structure separated by distinct, dense and intercrystalline boundaries. This is the transition reflected in the density plot of Fig. 5 and in the roughness plot of Fig. 6.

The zone II structure is controlled by surface diffusion, as indicated by the smooth contour of the tracer atoms in Fig. 4(c). Activation energies for surface diffusion are significantly less than those for bulk diffusion [56]. Accordingly, at a specific deposition rate, a temperature range normally exists which is high enough so that surface diffusion dominates over arrival rates, and the coating atoms lose memory of their arrival directions. This temperature is still low enough (TJTm < 0.5) that bulk diffusion rates remain orders of magnitude less than surface diffusion rates and are consequently negligible. The direct effect is that the columns grow bigger with increasing temperature, Figs 4(b)-(d), and facet-like faces are usually present in this zone. Alternatively, a zone II structure can be obtained by fixing the temperature and increasing the arrival interval, as shown in Fig. 9. The roughness is nearly constant in zone II, Fig. 10(b), and can therefore be used as a delimiter of this range.

It is found that the formation of facet-like mounds in zone II structures is quite universal and was present in every simulation carried out in this work. Experimentally, this also appears to be a fairly general phenomenon, occurring in a number of different systems [57, 58, 61-661. We attribute this phenomenon to both self-shadowing and the presence


of Schwoebel [43] or step edge barriers in the material. They make it more difficult for adatoms to move from one terrace to a lower one. For the two possible Schwoebel jumps considered in this work, Table 1, the Schwoebel barriers are 0.13 and 0.22 eV, respectively as indicated earlier.

Zone III conditions occur at higher temperatures where bulk diffusion dominates over all other processes so that atoms lose all memory of the initial events associated with their condensation. Zone III conditions are not modeled in this work, in part because the activated character of the diffusion rate intrinsically demands a high computational cost. Zone III conditions are less likely in many advanced applications due to concerns about thermal stress and the stability of the substrate [57, 58, 671.

6. CONCLUSIONS

A two-step Monte Carlo method has been proposed and shown to generate reasonably realistic low energy deposition simulations over a wide range of deposition conditions. The model incorporates a momentum scheme in which the effect of low atomic kinetic energy (< 1.0 eV) at the instant of adatom impact with the substrate is approximated. This yields initial packing densities and column orientation angles that are closer to experimentally reported values than those yielded from the Henderson model used in prior calculations. Using basic kinetic considerations of solid-state diffusion, a multipath diffusion model has been developed to provide a fundamental link between the deposition rate and atomic diffusive process on, and within, the sample. An embedded atom method is employed to calculate the activation energies for diffusion of a variety of atomic configurations.

This two-step approach makes possible the simulation of low energy deposition over a broad range of physically realistic deposition parameters. The approach provides a practical method to simulate several aspects of the vapor deposition processes. In particular, it enables determination of the effect of vapor processing variables such as deposition flux density (deposition rate), flux angle and substrate temperature upon deposit microstructure/morphology parameters such as packing density, surface roughness and column size.

Although a two dimensional model is used in this work, the simulation results demonstrate the transition from a porous columnar structure to a fully dense columnar structure in a way that agrees well with the SZM model. The transition is found to occur at a higher temperature as the deposition rate increases. The width of the columns in the simulated microstructures appears to correlate well with grain size, and the temperature dependence of the width correlates closely with experimental grain size data for many metals. This suggests that a Monte Carlo simulation that uses the activated processes of atomic

surface diffusion can predict well the trends in microstructure/morphology evolution.

Acknowledgements-We are grateful for the support of this work by the Advanced Research Projects Agency (A. Tsao, Program Manager) and the National Aeronautics and Space Administration (D. Brewer, Program Monitor) through grant NAGW1692.

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