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Nuclear Instruments and Methods in Physics Research B 121 (1997) 44-48

MINI B BeamInteractions

with Materials 8 Atoms

Computer simulation of crystal surface modification by accelerated cluster ion impacts

Z. Insepov * , I. Yamada

Ion Beam Engineering Experimental Laboratory, Kyoto University. loshida-Honmachi, Sakyo, Kyoto 606-01, Japan

Abstract Gas cluster ion impacts on a solid suface lead to a modification of the surface microscopic flatness in the case of normal

ion impacts. In this work we have studied the surface smoothing effect under irradiation with cluster beams. Langevin

Dynamics, based on a KPZ-type equation for discrete surface heights, allows us to confirm the experimental findings of surface modifications by gas cluster irradiation. We supposed that a normal cluster impact creates a hemi-spherical crater. with a diameter defined by cluster energy, and use of the Monte Carlo method for the crater formation process. Different

sputtering angles could easily be incorporated into the model. The probabilities of sputtering were taken from our new

hybrid MD method, which has an advantage over the conventional MD method for later impact stages. We have obtained better agreement for the angular distribution of sputtered target material for normal cluster impact, calculated by the new MD method and experiment. In the case of oblique cluster impacts the shape of the crater has been chosen to be a shallower and a wider hollow. Compared with normal impact, an essential part of the cluster energy is reflected back into the vacuum at oblique impact. The surface temperature can be lower, and this effect will reduce the intensity of surface diffusion. We obtained that significant smoothing occurs after irradiation by normal cluster impacts on a surface area which data has been supported by experiment. The rate of the smoothing process depends on the value of the surface diffusion coefficient, and can be significantly accelerated if the lateral sputtering phenomenon is taken into account.

1. Introduction

Materials of high surface quality are needed and ap- plied widely in advanced technologies. Examples include smooth surfaces required for quantum well structures, surfaces with changed electrical and optical properties, surfaces with increased hardness and wear resistance in optoelectronic and microelectronic applications [ 11.

Cluster ion irradiation of solid surfaces is a unique

method for surface modification, distinguished from other techniques due to its ability to deliver high total energy, determined by cluster size and acceleration voltage, to-

gether with soft impact on a surface. The latter feature of cluster impact is explained by the sharing of the total

cluster energy with the constituent cluster and collisional surface atoms on impact. As a consequence of the dual nature of cluster energy, new physical effects occur at cluster impact on a solid surface. The interesting effect of cluster ion irradiation on a solid substrate appears to be the surface smoothing effect. As was established experimen- tally [2], the surface roughness of various substrate materi-

’ Corresponding author. Fax: + 81-757-516-774; email: inse-

pov@kuee.kyoto-u.ac.jp.

als, measured by atomic force microscope, has been re-

duced considerable after irradiation with CO, cluster ions accelerated to IO kV. Gas cluster ions accelerated to high kinetic energies can be used succesfully for surface modifi- cation. These results have been obtained for Pt, Cu, poly-Si, SiO,, Si,N, films and glass substrates.

Molecular Dynamics calculations of cluster impact on a solid surface have revealed that a hot and highly com- pressed region, with a transient temperature of up to IO5 K and pressure up to 1 Mbar, arises in a short time interval which generate strong atomic scale shock waves [3-51. In our MD calculations we demonstrated the lateral sputtering effect, when most of ejected surface atoms acquire mo-

mentum with a lateral orientation, which allows them to diffuse much more widely, noticeably smoothing rough surfaces [3].

The smoothing effect requires an enormously long time interval when compared to the typical MD calculation time for single cluster impact. The MD method is a detailed method, which provides data for single and many cluster impacts, however it is limited to the study of small surface pieces with typical sizes in the order of 100 x 100 i2. The computational cell size for this new surface smoothing phenomenon should be at least in the order of 1000 X 1000 A2.‘, which imposes extreme requirements for hardware. In

0168-583X/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved

P/I SO1 68-583X(96)00450-8

Z. Insepw, 1. Yamadu/ Nucl. Instr. and Mefh. in Phys. Rex. B 121 (1997) 44-48 4s

our previous paper [6] we introduced a new numerical method of surface modification with many cluster impacts

based on the Kardar-Parisi-Zhang (KPZ) equation of

motion. The dynamics of the surface geometry was defined in the KPZ equation by the competition of surface contrac-

tion, which reduces the surface energy, and surface expan- sion (or roughening) with crater formation process, and did

not consider a surface diffusion process. The aim of this work is to develop further the numeri-

cal model of solid surface modification by cluster irradia-

tion, based on a phenemenological Langevin Dynamics

equation of motion. To use this equation, we need to consider the probabilities of surface sputtering with cluster

irradiation. A new hybrid MD technique, represented in this paper, allow us to find the sputtering flux for the later time instants of cluster impact.

2. Hybrid molecular dynamics model

Conventional MD simulations of cluster-surface im-

pacts use the periodic boundary conditions (PBC), with typical system sizes - 100 A, and some velocity resealing techniques to keep a substrate temperature. If we are interested in the dynamics of the sputtering of a substrate material at cluster impacts, we need to improve the MD method in order to avoid an unphysical reflection of elastic waves from the system’s boundaries, which may signifi- cantly alter the dynamics of sputtering. Particularly for the

later time instants, where the sputtering yield may contain atoms evaporated from hot surface, we need another tech- nique than the periodic boundary conditions. The velocity resealing techique, in turn, may induce an artificial surface stiffness which also will change the dynamics of the target’s atoms.

As obtained before [7], the central collisional zone, undergoing a significant disturbance from cluster impact, becomes thermodynamically nonequilibrated. We can esti-

mate the size L of a central non-linear zone, where the atomic dynamics should be treated by conventional MD, by the use of the shock wave theory [8] as L - d . IO”“, where d - 30 A is the cluster diameter, which gives L - 100 A. Parameters of cluster shock wave become thermally equilibrated for distances larger than this L. The

shock front reaches the boundaries at a time r - L/u,, - I ps, where u,, is an initial cluster velocity. Displacements for the atoms in a region outside the central zone should be

small, and can be treated by the use of continuum mechan- ics and thermodynamics. Here we used thermodynamics to introduce thermal boundary conditions (TBC) and the fi- nite-elements method for continuum mechanics in order to obtain the displacements of atoms in outer target region. According to the heat transfer theory, the macroscopic body with a typical size L needs a time to cool down, estimated as T- L’/,y, where x is the target thermal

diffusivity. By the use of x- I cm’/s for Si at room

temperature [9], we can get T- I ps. This means that deformation and thermal processes have comparable char-

acteristic time intervals, and they can be treated together. The new thermal boundary conditions allow us to extend

our computation to a longer time instants because they can efficiently suppress a nonphysical reflection of elastic

waves from the system’s boundaries. The other advantage

is that they significantly reduce the computation time, because we do not need to calculate inessential degrees of

freedom for the atoms in outer substrate region by MD.

We devided the outer substrate region into symmetrical

cells, with positions r, at a time I, and used the equations

for heat transfer and a continuum mechanics equations for the cell’s displacements [lo]:

d*u au,, L=_

‘dr2 ’ ax,

a,,= -Ku73,k+Ku,,S,k+2p(u;k-u,,~)

+&t~;k+z+,k-&y.

(Ia)

( lb)

where T(r, I) is the temperature for the finite elemental

cell with a position r, at a time t, x is the target thermal diffusivity (cm’/s), p is the average crystal density, u, is

the displacement vector of a ith cell, u,~ and uik are the stress and strain tensors. The parameters entering the stress tensor are as follows: (Y is the thermal expansion coeffi- cient, K and .$ are the bulk modulus and viscosity, p and q are the shear modulus and viscosity. Eqs. (1) were already used in [I I] to define analytically a geometry of a single heavy ion track in an amrphous target for one-di- mensional symmetry. Here we solved them numerically in two dimensions. By supposing a cylindrical geometry of the problem, we get:

1 dT(r,z;r) tj’T(r,z;r) 1 l+T(r,z;r) - f- X dr = ?tr2 r ilr

aV(r,z;r) +

a? ’ @a)

ii7 1 au, Pu, a2u. pii,= --Km; fp -“+;-&+--++* r2

3:

P)

II. ION-SOLID INTERACTION /DEFECT FORMATION

46 2. In%pou, I. Yamudu /Nucl. Instr. and Meth. in Phys. Res. B 121 (1997) 44-48

8T pii,= -Kaz

1 au, a*l4, a2uz ;T+q+Z az

( I airz a2ici a2ti, +71 ;z+ar2+2. az 1

Eqs. (2) for the outer region can be solved numerically, with the MD procedure for the central zone, which means that MD parameters, that we obtained like displacements and temperature, were used as boundary conditions for computation of Eqs. (2). This is an important detail of the proposed method. Eqs. (2) are stable due to the presence of the coefficients. We can use explicit numerical methods, with a common time unit and a time increment, which significantly simplifies a hybrid calculation. As far as we know, this is the first attempt to combine the MD code with continuum mechanics or with thermodynamics in a cluster-surface impact study. Similar ideas were used for studying complex fluids flows in [ 121.

The Buckingham potential is used to model two-body forces between two Ar and between Ar and Si atoms, and the Stillinger and Weber potential was used to define forces between three Si atoms in the central MD zone. The case of an argon cluster of a hundred atoms impacting a silicon target with the energy of a few keV has been considered. The MD calculations were performed for a cylindrical substrate sample consisting of about lOOOO-- 25000 atoms, and the continuum mechanics calculations have extended our results to sizes about ten times larger.

3. Surface modification model

To study surface modification with cluster ion irradia- tion, we use in this paper a new type of equation of motion, in a form of the noisy Kuramoto-Sivashinsky (KS) equation [ 131:

dh(r,t) - = vV’h(r,t) - tcV4h(r,t) + AV2(Vh(r,r))2

dt

+.&(Vh) + v(r,r), (3)

Here, Y is the term which accounts for the evaporation and deposition processes, K is the coefficient which is related to the surface diffusion coefficient, A is the coeffi- cient of nonlinearity, f&Vh) is the Monte Carlo crater formation term, which we included additionally, and 7) is the noise term which reflects a random redeposition pro- cess. This equation represents the nonlinear dynamics of growing surface profiles in terms of the coarse-grained interface heights h(r,t) in a d-dimensional space where r is the radius vector in a (d - 1) dimensional plane at time t, and accurately describes behavior in later stages, or scaling properties, of a growing interface. The nonlinear term conserves the surface current, and the last term was chosen as the Gaussian white noise 1141.

The typical irradiation parameters used for surface smoothing are as follows: cluster ion doses are in the range of 10’2-10’5 ion/cm2, average cluster sizes are in the order of lo3 atoms, total cluster energies are - 20-200 keV. A single hill, having a typical area of order 106-lo7 A2, was placed in the center of the computational cell. The model cluster dose was in the order of 103-lo4 cluster/hill. We assumed also that clusters hit the surface normally at a random position. Displacements of surface particles after the cluster impacts were modeled in accor- dance with a probability, obtained in our MD simulation of single cluster ion impact on a flat surface. To find the result of many cluster impacts, we assumed that a given amount of surface material is sputtered independently for each cluster impact. We have defined the surface rough- ness from the variance:

5=

where hi is the height of ith column, L is the length of computational cell, and h,, is the mean surface height:

Chi h si

a” L2 .

The crater geometry was obtained in the previous paper [7], and it was introduced by the use of Monte Carlo procedure [7]. For better understanding of the geometry effect of ejected surface atoms due to cluster impact, we modeled different occurences for the redeposition event. Particles can be. reflected from the first and redeposited at another position. Thus, we have taken into consideration a lateral sputtering effect. The shape of the rim around the crater was chosen depending on the local hill slope. Peri- odical boundary conditions were used in two directions along the x and y axis. Eq. (3) containes two model parameters v and k. The first coefficient is linearly pro- portional to the equilibrium pressure of Si atoms in a close vicinity of a Si surface, and it can usually be neglected, in a comparison with the second term. The KS-equation contains the second surface diffusion term, which meets the process of our interest more adequately [15]. To esti- mate this coefficient, we used the formula [16]: K =

DsyR2n,/ksT, where D, is the surface diffusion coeffi- cient, y is the surface tension, D is the atomic volume, n, is the number of atoms per unit area, and k,T has the meaning of surface temperature.. A choice of these con- stants allows one to model surface modification by differ- ent sources and for different substrate materials.

4. The results and discussion

Fig. 1 shows the dependences of substrate temperature, T,, defined as the mean kinetic energy per substrate atom,

Z. Insepov, I. Yamada/Nucl. Insrr. and Meth. in Phys. Reu. B I21 (1997) 44-48 47

‘edge

0.00 0.75 1.50 225

wps

Fig. 1. The dependences of substrate temperatures T,, T,,,,, and

T edgc, defined as the mean kinetic energy per atom for the whole

substrate, and for two neighboring edge layers, on time. The

circles correspond to the first mesh cell’s temperature which was

taken to be equal to the edge temperature. All these three depen-

dences were averaged over z. The inset in the figure shows

schematically our hybrid MD model geometry. The MD atoms are

depicted by the gray region, and a finite element region is shown

as squared.

on time, together with two layer temperatures, T,,, and

T edge. The first curve, T,,,, gives the temperature for the last MD layer before the edge layer. The second, cdee, gives the temperature for the edge MD layer, and is shown

by a dashed line. The edge temperature was used as an initial temperature for the mesh region. The circles corre- spond to the first mesh cell’s temperature. All these three

dependences were averaged over z. The inset in this figure shows schematecally our hybrid MD model geometry.

Fig. 2 shows our MD result for the angular dependence of sputtering yield of a Si(100) substrate bombarded by a Ar,, cluster, with energy of 50 eV per cluster atom, obtained by the new hybrid MD method. While for the

earlier time instant of 308 fs after the impact (thin line) we found the lateral spreading, as before [3], for the time interval after 8.1 ps (thick line) we have a much higher sputtering yield with an orientation of atoms between 30 and 80”. This figure was drawn from the velocity orienta-

tions of sputtered surface atoms, and it agrees much better with the experimental observations [2]. The other boundary conditions like the velocity resealing technique or Langevin dynamics technique have efficiently suppressed a thermal movement of target atoms, and we could not find a high sputtering yield for the latest time instants.

The nonlinear differential equation, Eq. (31, have been transformed to a difference equation and computed numer-

ically. We have supposed in this work that the sputtered

surface atoms walk with different diffusion coefficients.

The calculations were performed for the cell sizes L2 = 2601, 10000 and 40000 atomic positions, and the periodic

boundary conditions were applied along the x and y axis.

Each cluster impact was modeled by the creation of a single impact crater, with a rim around it, having a shape

which depends on the local surface slope. After the cre-

ation of a crater at a random position, we computed Eq. (31 for 200 time steps, so the total number of time steps for

one curve is 200 thousands. For the case where no diffu-

sion was allowed, we checked our results for 2 millions

time steps. It should be noted that if one uses a typical

surface diffusion coefficient, of the order of IO- I8 cm2/s

[ 161 for Si adatoms on a Si surface at room temperature, no modification could be achieved. Therefore, we have sup-

posed that modification of Si surface with cluster irradia-

tion would be possible if laterally sputtered substrate atoms are overheated due to their significant momenta along the surface, and, hence, they are highly nonequilibrated with the surface. This is a new, non-equilibrium mechanism of

surface modification which does not need so much energy compared with conventional surface diffusion or melting.

Fig. 3 represents our results for the surface roughness calculations for the cell size L* = 51 X 51 in the case of

normal cluster impacts on a surface. To get these results we supposed that all target atoms ejected at the impact were redeposited on the surface. The calculations were

made with a time increment of about 10e6 s, the computa- tion time for each cluster impact was 1O-3 s, and the total computation time for one curve was of about 1 s. The coefficient Y in Eq. (3) is usually very small, and we have set it to zero. We have chosen the coefficient K = 2.84 X

0 10 20 30 40 so

N,siaB

Fig. 2. The angular dependence of the sputtering yield from a

Si( 100) substrate bombarded by a Ar,, cluster, with energy of 50

eV per cluster atom. The thii solid line corresponds to a time

instant of 308 fs, and the thick solid line gives the sputtering yield

for 8.1 ps after impact.

II. ION-SOLID INTERACTION/DEFECT FORMATION

48 Z. Insepou. I. Yumuda / Nucl. Instr. and Meth. in Phys. Res. B 121 (1997) 44-48

0.00

0 10 20 30 40

ION DOSE, in lo’* ion/cm*

Fig. 3. The dose dependence of surface roughness for the cell size

L2 = 51 X 51 in the case of normal Ar,,,, cluster impacts, with

energy of 50 eV/atom, on a Si(100) surface. The numbers on the

curves correspond to different surface diffusion coefficients: (1)

no diffusion was allowed, (2) 10e9 cm2/s, (3) 5X 10m9 cm*/%

(4) lo-* cm’/s.

1O-23 cm4/s for the surface diffusion D, = IO-’ cm*/s

for TL 1000 K [16]. Similar results were obtained for all cell sizes and for

cluster energies higher than 50 eV/atom when surface

sputtering is noticeable. The technique that was used can also easily take into account the case when only a fraction of the surface atoms ejected by impact can be really sputtered from the surface, and, thus, be detected experi- mentally. As we can see, the intensive smoothing effect occurs at a diffusion constant of about lo-’ cm2/s after irradiation with a dose 4 X 1013 ion/cm2 [ 171.

As we obtained before [7], the oblique cluster impacts cannot smooth a surface efficiently. The reason is that they cannot deliver a significant energy because they can more efficiently be reflected from the surface. The experiment [2] showed that oblique cluster impacts, with an polar angle of about 60”, gives an increase of a surface rough- ness. Unfortunately we still could not find a proper expla- nation of this roughening phenomenon.

5. Summary

A new hybrid molecular dynamics (MD) method, com- bining the conventional MD technique for the highly non- linear central substrate region, with a continuum mechan- ics finite-elements equations for the linear boundary re- gions, was proposed in this paper. This method can reduce greatly the computation time and allowed us to obtain the sputtering yield for Ar,, cluster impact, with an energy of 50 eV/atom, on a Si(100) surface. Our new boundary conditions revealed sputtering at a higher rate for the later time interval 5 10 ps, for larger sputtering angles between

30-80”, after cluster impacts on a Si surface.

A new numerical model for surface smoothing, by accelerated cluster impacts based on the Kuramoto- Sivashinsky equation, was developed. This model allows to take more adequately into account the different mecha-

nisms of surface modification: evaporation and redeposi- tion, crater formation and surface diffusion. A crater and a

rim around the crater, formed after each cluster impact on

a substrate surface, have been modeled by the use of the Monte Carlo method. The probabilities of sputtering were

extracted from our MD calculations of cluster-surface im-

pacts. The calculations were performed for the cell sizes

L2 = 2601, 10000 and 40000 atomic positions. If no lateral

diffusion was included in the equation of motion, surface

roughness increases with cluster impacts. A significant smoothing effect was obtained for all cell sizes, due to the

enchancement of surface diffusion coefficient, after cluster bombardment. The shape of the rim around a crater effects the surface roughness very little.

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