Monte Carlo simulation of atomic processes on the solid surfaces

Surface Science 122 (1982) 99-118 North-Holland Publishing Company

99

MONTE CARLO SIMULATION SOLID SURFACES

OF ATOMIC PROCESSES

ON THE

T.T. TSONG Physics Department, Received

9 March

Pennsylvania 1982; accepted

Rate

University,

for publication

University Park, Pennsylvania

16802, USA

2 June 1982

Atomic processes on solid surfaces can be directly observed in the field ion microscope. We present here a Monte Carlo study of these atomic processes, aiming to facilitate analysis of the experimental data and to derive useful information from them. Problems considered include the effect of elastic and inelastic reflective boundaries on the observable mean square displacements, the frequency of a diffusing atom to encounter a pIane boundary, the site occupation probabilities, and the displacement dist~bution. The discrete random walks are assumed to have an exponential jump length distribution, and the nearest neighbor random walk is treated as a limiting case. The adsorption layer superstructure formation is also simulated based on pair energy data we have measured recently.

1. Introduction The field ion microscope has been successfully applied to study various atomic processes on metal surfaces, yielding detailed information on kinetics and dynamics of jumps of individual adsorbed atoms. These processes include diffusion of single atoms and simple atomic clusters [l-5], interactions between two or more adsorbed atoms [6], and adsorbed layer superstructure formation [7]. Although in a few cases, analysis of experimental data can be done analytically, this is not possible in many cases. Approximate relations need to be confirmed before they can be confidently applied for extracting quantitative information from the experimental data. Where analytical expressions do not exist, the expected observable result of a proposed mechanism has to be compared with the experimental data. For these purposes, Monte Carlo simulations are most useful. Monte Carlo methods have been widely used in studying hundreds of different problems in various branches of science [S]. Some applications of the methods to problems in single atom diffusion on solid surfaces have been reported by Reed and Ehrlich [9], and by Cowan [IO], and extensive work has been done on adsorption layer superstructure formation [ 1 I]. Two crystal planes of particular interest are the W{ 112) and the W{ 1 lo} 0039-6028/82/0000-0000/$02.75

0 1982 North-Holland

100

T. T. Tsong / Monte Carlo simulation of atomic processes

since comprehensive FIM studies of atomic processes are mostly done on these planes. Atomic hoppings on these planes are respectively one- and two-dimensional. We report here a Monte Carlo study of atomic processes on one-dimensional planes and on two-dimensional planes of the bee { 1 lo} structure. The results presented for the bee { 1 lo} plane are directly applicable to other crystal planes of similar atomic structures. Topics to be considered include the effect of plane boundaries on the mean square displacement of a diffusing atom, the frequency of encountering a plane boundary by a diffusing atom, diffusion of single atoms, and displacement distributions. All these problems are considered for random walks with an exponential jump length distribution. We also simulate adsorption layer superstructure formation based on pair energies we measured recently. Our simulations are intended to provide some guidance for future FIM measurements, and also to help extracting physical parameters of interest from existing FIM data.

2. Surface structures In discrete random walks, information on the sites of adsorption and the surface structure is needed to properly interpret the experimental data, especially where more than one adsorption site may be present within a primitive cell as in the case with surface site adsorption on the W{ 1 lo} plane. Most experimental evidence indicates that the sites of adsorption on the W{ 1 lo} and the W{ 112) planes are the lattice sites [ 12,131. A few exceptions reported have not been confirmed. For our purpose here, we will consider only the lattice site adsorption. Each primitive cell contains only one adsorption site. For planes where atomic hoppings are one-dimensional, the jump length can only be an integral multiple of the nearest-neighbor distance 1. When equivalent surface channels intersect, atomic hoppings are two-dimensional. For the W{ 1 lo} plane, the primitive cell has the size of oa/2

Fig. 1. Structure of the W( 1 IO} plane and the adatom-adatom twice the lattice parameter.

bonds with bond lengths less than

T. T. Tsong / Monte Carlo simulation of atomic procxxwe~

101

x $?4,/2 with the lattice vectors pointing along the [Iii] and the [ ITI] surface channel directions; a is the lattice constant of the substrate. The structure of this plane is shown in fig. 1.

3. Surface diffusion of single adatom 3.1. General principles The interaction between an adsorbed atom and an infinite substrate, on the time average, can be described by a periodic potential energy function which satisfies the following translational invariance: UkS-P,)

= u(r),

where r is the position vector, and p,, is a surface lattice vector. The lattice site adsorption indicates that within each primitive cell there is an adatom position of minimum and maximum potentital energy and multiple saddle points. If the potential well is deep enough as in chemisorption, there will be localized vibrational states. A light atom may tunnel from one well to another at very low temperatures. In chemisorption of atoms heavier than He thermal activation occurs much more readily. At a given temperature, an adatom has a finite probability of having energy exceeding that at a saddle point. The atom, once overcoming the conf~ement, will translate along the surface. It will be de-excited by some scattering mechanisms and will be localized to a new potential well. Surface diffusion is therefore discrete random walks. The distance of jump depends on the rate of de-excitation [lo]. One may characterize the de-excitation process by a relaxation time 7’. In one-dimensional symmetric random walks, an adatom accomplishes a jump of distance ii if it is de-excited within (i - t)l and (i + i)l where I is the nearest-neighbor distance and i is an integer. The probability of having a jump length il, p( il), is given by p(U) =I (2r’ir)-i

exp( //2r’V)~~~1~~p( ’ 2

=Ce-lilo

for

i= A-1, -C2,...,

- 1x [/A?)

dx 0)

where Q = l/r%,

C=J(e”-

1);

a is the average translational speed; factor for p(d) in the range from jump length distribution. In general the jump distance. When 7% B 1, or to p(il)

= $S(ik

1).

(2) (27’ti)-’ exp( 1/2 r’s) is the normalization - co to -i/2 and I/2 to co; p(U) is the it is an exponentially decaying function of a z+ 1, the jump length distribution reduces (3)

102

T T. Tsong / Monte Carlo simulation of atomic processes

This corresponds to the nearest-neighbor random walks. If we are concerned with only the magnitude but not the direction of the jumps, then the jump length distribution is p(il)

= C’e-”

for

The probability well, p, , is related

i=

1,2,...,

C=(e”-

of de-excitation to a by

1).

of a translating

(4) adatom

within

a potential

p, = (1 -e-O).

(5)

Eq. (4) can be written p(U)

=p,(l

-p,)‘-l

as for

i=

1,2, 3 ,....

(6)

For such an exponential jump length distribution, ment in unrestricted random walk is given by ((Ax)‘)=

the mean square

displace-

5 p,p(il)* i=l

=Fi2

(2-P,)

*

5712

PI For the special case where p, neighbor discrete random walk bor random walk as a limiting namely the walk with discrete,

l+e-”

(1 - e-“)*

’

(7)

1, or a + co, the walk reduces to nearest[lo]. In this report we treat the nearest-neighcase of a more general class of random walks, exponential jump length distribution. =

3.2. Effect of “elastic” and “inelastic” reflective boundaries FIM experiments show that plane boundaries are mostly reflective to adatom migration [l-4]. In this Monte Carlo study of atomic processes, three effects of a reflective boundary will be considered. They are how the boundary will change the displacement distribution and the mean square displacement, and what is the average number of times a diffusing adatom will encounter a boundary in a heating period. The first topic will be discussed in section 3.3. Our simulations consider the jump length distribution to be exponential. The nearest random walk is treated as a special case where p , - 1 or a -+ cc. At least two different kinds of reflective boundaries can be envisioned. They are the elastic and inelastic boundaries. When a translating adatom encounters an “inelastic” reflective boundary, the adatom is de-excited into the boundary site of the plane. An “elastic” reflective boundary reflects a translating adatom without de-excitation of the adatom by the boundary. On a one-dimensional lattice of M adsorption sites, labeled from 0 to M - 1, the final rest site, m,, of a translating adatom will either be 0 or M - 1 if the boundaries are inelastic and if the adatom encounters either of the two boundaries. If the boundaries

103

T. T. Tsong / Monte Carlo simulation of atomic processes

are elastic, the rest site will be m,=2(M-1)-((mi+Am), if the translating

direction

is positive,

and

m,=Am-mi, if the translating direction is negative; mi is the initial site, and Am is the jump length. Whether a reflective plane boundary is elastic or inelastic depends on the dynamics of adatom-plane edge collision. Without further knowledge of this process, one can only hope that the two kinds of boundaries will give rise to experimentally distinguishable observations. A simple method is used to simulate the direction and the distance of an atomic jump. A random number generator which generates numbers between 0 and 1 is used. Two random numbers R, and R, are generated. If R, G 0.5 then the jump is toward right, otherwise it is toward left. If

i=l

i=l

then

the jump

distance

0 -

is

jl,

Computer

or Am = j. To simulate

the average

number

of

Slmulatlon

Analytical

Fig. 2. Mean square displacement as a function of a in un-restricted one-dimensional random walks with exponential jump lengths. The curve is from eq. (7). Each data point is obtained from 10,000 simulated heating periods of observations. The average number of jumps per heating period, iV, is 1.

104

T. T. Tsong / Monte Carlo simulation of atomic processes

\

\

\

\

\

\\

inelastic

0’

20

40

60

80

A 100

Ii4 Fig. 3. (( Am)2) ,, as a function of fl for p, ranging from 0.5 to 1 for inelastic boundaries. The size of the one-dimensional plane is 12. Each data point is obtained from 20,000 to 70,000 simulated atomic jumps. The dashed curve is from eq. (8).

M -12 elastic

Fig. 4. Similar curves as shown in fig. 3 for elastic boundaries. All the curves, p, =0.5 to 1.0 with increments of 0.05 each and most of them are not shown, appear to approach the same asymptotic value - M2/6 for large a

T. T. Tsong / Monte Carlo simuhtion

of atomic processes

105

jumps P in a heating period, N’G IP(2N+ 10) trials are performed. If a random number R, generated is smaller than or equal to m//N’ then a jump is performed. Otherwise a new trial is attempted. For a finite size plane, the jumping atom is considered to have encountered a boundary if and only if the final site is outside the plane, i.e., if m, < 0 or m, > M - 1. In such a case, the final site is readjusted according to the definitions of the reflective boundaries discussed in the last paragraph. For nonrestricted one-dimensional random walks, the Monte Carlo simulated mean square displacements agree with eq. (7) for all values of p,, as shown in fig. 2. When boundaries are present, no analytical expression is available. Figs. 3 and 4 show how ((AM)~),, depends on the average number of jumps per heating period F, and on the probability of de-excitation within a well p, for a one-dimensional lattice of M= 12 with inelastic and elastic boundaries. In the case with elastic boundaries, the asymptotic value of ((APz)~) b agrees with the expected value - &C2/6 for large N for all values of p,. This is not the case with inelastic boundaries. The asymptotic value depends on the value of pi as can be expected. It is not difficult to argue that as p, approaches 0, ((ALM)~)~ will approach (M - 1)2/2 = 60.5 for inelastic boundaries since the rest sites will always be either site 0 or N - I. The factor l/2 comes from the fact that only half of the attempts will result in a net displacement. Simulated results for M= 16 are plotted as functions of p, and are shown in fig. 5. The boundaries are assumed to be inelastic. For p, = 1, or the nearest-neighbor random walks, an approximate equation

Inelastic

M:t6

Inelastic

.5

.6

.7

Mz16

.a .9 1.0 S Fig. 5. (a), (b) Similar results as shown in figs. 3, 4 and 6 for a linear plane of M = 16, but now plotted as a function of pl. The boundaries are assumed to be inelastic.

T. T. Tsong / Manre Carlo simulation of atomic processes

106

given by Ehrlich [l],

(8) agrees well with simulated result for both the “elastic” and “inelastic” reflective boundaries as long as ((Am)2), SM2/9. Another question of interest is the average number of times, Nb, a diffusing adatom encounters a plane boundary within a heating period. For the nearestneighbor random walks on a one-dimensional lattice of M adsorption sites, Cowan [IO] has shown that N7,= F//M.

(9)

For an elliptical plane of major and minor axes a and b, Tsong gives [ 141 +--

Ivz -+1 7l i a

1 b’1

(10)

for a square surface lattice structure. However if I<<:a and 1K 6, then the lattice structure is less important. Eq. (10) is equally applicable to the bee { 110) structure. Our Monte Carlo simulations find that eqs. (9) and (10) are valid only for “inelastic” boundaries. Results of these simulations are shown in figs. 6a and 6b. Eqs. (9) and (10) do not agree with the simulation result for planes with elastic boundaries as can be seen in fig. 7. However the linear dependence between gb and g is maintained for all values of p, for both the elastic and the inelastic boundaries.

T. T. Tsong / Monte Carlo simulation of atomic processes

107

-25 I

-20

-15

D

-10

-

Llnaar

Flta

-5

rqo

2qo

390

3

Fig. 6. (a) Fb versus fl from 10,000 to 100,000 to 1.0. Data points for the bee (110) structure

plots for a linear lattice of 12 adsorption sites. Each data point is obtained simulated atomic jumps. Plane boundaries are inelastic. p, ranges from 0.5 p, = 1 agree with p#, = N/M. (b) q versus F plots for circular planes of with R =51 and 101, and p, = 1.

Since the linear relation between & and &r is maintained for elastic boundaries, we ask the question of whether or not the proportionality constant is independent of the plane size. Or specifically, if we write & = IW,,M,

(11)

what is the value of K, and how will K depend on M. Fig. 8 shows a simulation result obtained by varying the size of the plane M from 10 to 50 with steps of 10 each. Our data show that within the statistical uncertainty shown, K is independent of M for all values of p, studied, i.e. from 0.5 to 1.0. The value of

T. T. Tsong / Monte Carlo simulation

of atomic processes

M=12 elastic

Fig. 7. fib versus flplots for a linear lattice of M = 12. Plane boundaries are elastic. These plots are significantly different from those shown in fig. 6a.

1.4, 1.2..

0.8.. 0.6

o.4 I 1o

06

0.7 p 0.8

0.9

1.0

Of,

0.7

0.9

1.0

’ 0.8

Inelastic

Fig. 8. Values of K as functions of p, for both the elastic and inelastic boundaries. The curves are from eqs. (12) and (13).

T. T. Tsong / Monte Carlo simulation of atomic processes

K for “elastic”

boundaries

can be very well approximated

109

by

Km 0.5~;‘.

(12)

By the same token, approximated by

the value of K for “inelastic”

Kmp,‘.s.

boundaries

can be very well

(13)

3.3. Displacement

distributions

The geometrical aspects of surface diffusion of single atoms can be best studied by measuring the displacement distribution [ 15,161. In one-dimensional diffusion, a displacement distriution can provide information on the jump length distribution of an adatom. In two-dimensional diffusion, a displacement distribution can provide the information on both the jump length distribution and the direction, or the path, of the atomic jumps. For unrestricted nearest-neighbor random walks, an analytical expression exists for the displacement distribution [ 151 W,-(Am)=exp(-N)I,,(N),

(14)

where 00 (t77)(Am+W LP)=

E k=O k! (Am+k)!

(15)

For exponential jump length distribution, the displacement distribution can best be found by Monte Carlo simulation. In fig. 9, a simulation result for one-dimensional unrestricted random walks with exponential jump lengths, with the value of a to range from 0.5 to 5, is shown. The average number of jumps per heating period is taken to be 1. The asymptotic values for large a approach the distribution of the nearest-neighbor random walks given by eq. (14). It is interesting to note that while frequencies of all displacements other than Am = 2 and 3 either increase or decrease monotonically with a, the frequencies of Am = 2 and 3 show a nonmonofohic dependence on a. This behavior changes with a given value of &? On a plane of finite size, the displacement distribution will be drastically changed for large x For small g, the change is also noticeable. In fig. 9 we also include displacement distributions on one-dimensional planes of M = 12, one with elastic reflective boundaries, and one with inelastic reflective boundaries. Although there are noticeable differences, it will not be easy to find out whether the boundaries are elastic or inelastic by measuring the displacement distribution. Two difficulties are the expected large statistical fluctuations of the very limited amount of data collectable under the present experimental procedures, and the possibility of the occurrence of the non-

T. T. Tsong / Monte Carlo simulation of atomic processes

.L 5

.6

.7

.B

.9

4

39

1

0

I

-.-.

Unrcslricted

-

Indostic

Hi

----

Elastic

M: 12

2

3

I2

5

a Fig. 9. Displacement distributions for non-restricted ID random walks (solid line), restricted ID random walks with elastic reflective boundaries (dotted line), and with inelastic reflective boundaries (dashed line). M = 12 and fl= 1. These curves are obtained from 400,000 simulated atomic jumps. Accuracy of the frequencies is about -C 1%. The jump length distributions are exponential.

nearest-neighbor jumps. The latter difficulty is especially severe if one intends to distinguish the real effect of non-nearest-neighbor jumps and the reflective boundaries. This difficulty can be alleviated by a measurement of site occupational probability as will be discussed in the next section. On two-dimensional planes similar behavior prevails. Since it is rather difficult to present 2D displacement distributions in graphical forms, we list only two sets of the simulation results in table 1. For noncrossing terms, the frequencies are essentially the same as the 1D case. The frequencies for cross terms are new information. All frequencies depend very sensitively on the atomic jump direction. In fact from a measurement of the displacement distribution, it is concluded that the jump directions of single W adatoms on the W{ 1 lo} are along the (111) surface channels [ 161. Thus the geometrical aspect of atomic jumps in 2D diffusion can be most effectively studied by measuring the displacement distributions.

III

T. T. Tsong /- iUonre Carlo simulation of atomic processes Table 1 Two-dimensional

displacement

AZ

Al

For a = I.0 0 *I -i2 23 -t4 r5 -6 27 For a = 8.0 0 -t-l *2 23 24

distribution

(W)

0

rl

*2

r3

24

2s

+6

27

37.8 13.2 5.9 2.8 1.2 0.8 0.1 0

13.5 4.5 2.0 0.9 0.4 0.2 0 0

6.2 2.0 0.7 0.4 0.2 0.1 0 0

2.7 0.7 0.4 0.2 0.1 0 0 0

1.3 0.5 0.1 0.1 0 0 0 0

0.6 0.2 0.1 0.1 0 0 0 0

0.2 0.1 0 0 0 0 0 0

0.1 0 0 0 0 0 0 0

40.2 20.2 2.6 0.2 0

21.4 10.4 1.2 0 0

2.3 1.2 0.1 0 0

0.2 0.1 0 0 0

4. ‘Ihe site occupation probability distribution When the average number of jumps per heating period is small and the number of heating periods is also small then the sites in the vicinity of the starting position of a diffusing adatom tend to be more frequently occupied than sites far away even if the plane boundaries are reflective. This rather obvious fact is reflected in our many Monte Carlo simulations. A more interesting question is that under a sufficiently large number of atomic jumps where a near uniform site occupation probability distribution can be expected, will the inelastic reflective and the elastic reflective boundaries give rise to distinguishable features in the site ~stributions, and whether these distributions depend on atomic jump length ~st~butions. To answer these questions, we have made a Monte Carlo study on one-dimensional plane of 12 adsorption sites. Our result indicates that for a plane with elastic reflective boundaries, the site occupation probability distribution depends only very slightly with the jump length distribution. For a large value of a, or the nearest-neighbor random walk, sites near the plane boundaries have a slightly larger occupation probability than sites near the center of the plane. This slight nonuniformity is undetectable for a small value of a. The boundary sites, however, have an

112

T. T Tsong / Monte Carlo simulation of atomic processes

occupation probability only - l/2 of other sites irrespective of the value of a. This is shown in fig. 10. For a plane with inelastic reflective boundaries, the site occupation proba-

C .C L% : s.

Elastic

O"'"',"'. 01234567891011 SITE Fig. 10. The site occupation probabilities for a plane with 12 adsorption sites. The boundaries are elastically reflective. Simulation data: a =0.5 to 4.5, N = 10,000, r= 1. For elastic boundaries, the occupation probabilities are, within statistical uncertainties, independent of the values of p,. This is not so for inelastic boundaries as can be easily seen.

bility distribution differs slightly from a uniform distribution for a small value of a. The boundary sites have a larger occupation probability than the other sites as can be seen in fig. 10. For nearest-neighbor random walks, the distribution is nearly uniform. The behavior on this plane is therefore very different from the other plane. This fact suggests that the reflective property of a boundary can be studied by a measurement of the site occupation probability distribution. However, one may encounter another uncertainty of whether the potential well depth is uniform over the entire plane. Nonuniformity of potential well depth will produce a similar distribution. At the present time we are unaware of any method to discriminate the reflective property of the boundary and a possible potential barrier variation across the plane.

T. 27 Tsong / Mome Carlo sim~la~ionof aiomicprocesses

113

5. Adsorbed layer superstructure formation When a fraction of a monolayer of foreign atoms adsorbs on a surface, the adsorbed atoms almost always form a superlattice. This was already discovered by Davison and Germer in 1927 and reported in their well known paper of confirming the wave nature of electrons [ 171. Adlayer superstructure formation has since then been extensively investigated by LEED technique [ 18,191. It is generally believed that adlayer superstructures are formed by the lateral interaction between adsorbed atoms. In particular, Einstein and Schrieffer [20] proposed that the adlayer superstructures are formed by the oscillatory nature of the electronic indirect interaction between adsorbed atoms. A quantitative evidence of a close correlation between adatom-adatom interaction and adlayer superstructure formation was reported only recently by Tsong and Casanova [7]. From the relative pair energies measured for Si adatoms on the W{ 1 IO} they were able to calculate the two-dimensional binding energy per adatom in an adlayer of different structures. They found the p(2 X 1) structure to have the lowest energy. This prediction was confirmed in a later field ion microscope observation. We report here a Monte Carlo simulation of superstructure formation based on the pair energies measured by them. For the convenience of discussions here, the data presented by them are plotted in graphic form and is shown in fig. 11. In thermodynamic equilibrium, adatoms assume a layer structure such that the entropy of the system is maximized and the free energy ~ni~zed [11,21]. In this study we assume that the adatom-substrate interaction is not affected

St -SI

on W {llok

r. 5 2 2 : i

0

1

2

Dletance

5

2 L G P

t

-SC t

Fig. 11. Relative pair energies for Si-Si interaction on the W{ 110) measured in FIM studies. These data are from ref. [7]. Dashed line is an educated guess from the data points shown.

114

T. T. Tsong / Monte Cnrfo simulation of atomic processes

by the adatom-adatom interaction. Thus the adatoms are treated as an independent two-dimensional system confined to certain rest sites imposed by the substrate lattice structure. This same assumption is made in all FIM derivations of pair energies. The assumption is thus consistent in both the Monte Carlo simulation and the experimental data analysis. The entropy of the two-dimensional adlayer can be maximized and the free energy minimized if the total number of pair bonds of i configuration in the adlayer is given by [Zl] e--E
n; =

xj e-E,/kT3

(16)

where Xj is over all possible bond configurations on the surface. A Monte Carlo simulation of adlayer superstructure formation can be done by allowing adatoms to jump on the surface according to a given probability distribution so that the adsorbed layer atoms will eventially satisfy the relations given by eq. (16) after a sufficiently large number of jumps. To conserve the computation time, we assume the pair interaction to be negligibly small beyond twice the substrate lattice parameters, or 6.32A. The bonds considered are designated as a, b, c, d, e and f bonds (see fig. 1). A specified number of adatoms is randomly deposited on a plane of the W{ 1 IO} structure. Relative probabilities of jumping in different surface channel directions of all the adatoms are then computed. If an atomic jump is breaking a bond of energy E,, the relative probability of the jump is reduced by a factor of exp( - E,/kT). If the atomic jump will result in formation of a bond of energy E, then the relative probability is increased by a factor of exp( - Ej/kT). The adatom to perform a jump and the jump direction are randomly chosen based on the relative probabilities of all the adatoms to jump in all four possible surface channel directions. After an atomic jump, the probability distribution is recomputed since the bond configuration of the atoms has been changed. This program does not provide information on the time rate of the adatoms to approach an equilibrium adlayer structure. It will, however, give a correct adlayer structure with a configuration consistent with the bond number distribution given by eq. (16). The program thus can give the structure of an adlayer to be expected from a given set of pair energies of the six bonds. The pair energies can either be experimental data or hypothetical data. The program is not intended to study structural phase transition since the number of adatoms has to be small to avoid excess computation time. The small systems in FIM experiments will not give singularities observed in large systems. Fig. 12 gives an example of Si adlayer superstructure formation using the experimentally measured pair energies. Thirty-five adatoms are deposited on a plane of 150 adsorption sites. Within about 100 atomic jumps, on average three jumps per adatom, a recognizable superstructure is usually formed. To reach a near perfection of the adlayer structure, more than one thousand jumps are

T. T. Tsong / Monte Carlo sjmula~ion of atomic processes

Fig. 12. A Monte Carlo simulation pair energies shown in fig. 11.

of silicon adsorption

layer superstructure

formation

using the

needed. The number however fluctuates widely, depending on the u~for~ty of the initial deposition. In fig. 13 we show near equilibrium structures with different degrees of adatom coverages. The shape of islands at low degrees of coverages and various two-dimensional lattice imperfections such as intersti-

Fig. 13. Island and layer structures in Si adsorption layers of varying degrees of coverage. The total number of adsorption sites is 150.

116

T. T. Tsong / Monte Carlo simulation of atomic processes

tials, antiphase domain boundaries, etc. can be readily visualized especially before a near equilibrium structure is reached [ 11. In the case of Si adatoms, the low coverage adlayer superstructure can be easily figured out without the aid of a computer simulation because of the two relative minima in the pair interaction. The simulation, however, can give results not readily anticipated. To give an example, pair interaction between Ir adatoms show relatively strong binding at twice the nearest-neighbor distance, or the d bond [22]. Away from this bond, the interaction is repulsive. There seems to exist a small oscillatory tail beyond 8 A, but the amplitude is much smaller than kT at the temperature where Ir adatoms can diffuse that the tail is not expected to affect the adlayer superstructure formation. Bonds in the vicinity of the d bond such as the a, b, c, e, and f bonds are so much less attractive than the d bond, that it is impossible to determine the bond energies by the field ion microscope measurement. A possible method of extracting information on pair energies is by assuming a set of pair energies and then compare the simulation adlayer structures with direct FIM observations. In fig. 14, a simulation based on a set of assumed pair energies is shown. The pair

__

.,“.,“.“..’

150 siln.35atoms As drposiled

100 jumps

:

1..” _,-a_” 655 jumps

_’

Fig. 14. Using the pair energies shown, which are intended to approximate the Ir-Ir interaction on the W( 1 IO) as given in ref. [22], an adlayer superstructure with a well matched phase boundary is formed after 655 jumps.

T. T. Tsong / Monte Carlo simulation of atomic processes

117

energies are based on an educated guess from the FIM data shown in fig. 11 of ref. [22]. From this set of pair energies, it is almost impossible to visualize what kind of adlayer structures one can expect. This simulation shows that two structure phases can coexist with well matched phase boundaries. It is plausible that for a certain set of pair energies, a slight change of temperature will completely change the adlayer structure from one form to another if the plane size is sufficiently large. T’his will then be a phase transition. No such sharp transition seems to occur for the very small number of adatoms used in our simulation. When a given number of atoms are randomly deposited on a finite size plane, there is a certain statistical distribution of the numbers of various bonds of the adsorbed layer. By allowing the adatoms to jump according to the method described above the numbers of these bonds will change. One will expect the attractive bonds to increase in numbers while the repulsive bond to decrease in numbers monotonic to the total number of atomic jumps performed. This is found, however, not to be the case. In the initial stages of the adsorbed layer superstructure formation, a transient effect is found: both the number of attractive and the repulsive bonds increase temporarily. The number of repulsive bonds reaches a maximum value and then decreases steadily whereas that of attractive bonds increases steadily after the transient effect is over. This transient effect is shown in fig. 15. We believe that this transient effect is real and should be observable in macroscopic systems. 150 sites,

35 atoms

c-bond ___+___JLA----&

d-bond

Bond energy: b: -4OmeV c: +60 d: +lO kT: 26 Total

no. of atomic

600 jumps

800

Fig. 15. A transient effect in which both the number of attractive and repulsive bonds increases temporarily as the adsorption layer gradually approaches an equilibrium structure.

6. Summary Atomic processes on solid surfaces can be directly observed and studied in the field ion microscope in atomic details. A Monte Carlo simulation can be used to generate data according to a hypothetical mechanism. By comparing the simulation data with the FIM data a correct mechanism can be found. In many cases, analytical relations are too complicated to derive. Monte Carlo simulations can provide proper relations for data analysis. These relations are valid within the statistical uncertainties specified. Problems considered in this report include the effect of plane boundaries on observable mean square displacements (for random walks with an exponential jump length distribution), the average number of times a diffusion adatom encounters the plane boundary, the site occupation probabilities on plane with either the elastic or the inelastic boundaries, the displacement distribution, and the adsorption layer superstructure formation. From this study we also conclude that an adlayer superstructure is formed by the total lateral interaction of adatoms, not by the electronic indirect interaction alone.

References [I] [2] [3] [4] [S] [6] [7] [8] [9]

[IO] [II]

[ 121 [ 131 [ 141 [15] [16] [ 171 [18] [ 191 [20] [21] [22]

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