Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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7 Fluctuations and Growth Phenomena in Surface Diﬀusion 323

Ω = FAτ

/τ =(FA)

. (7.37)

Since an atom diﬀusing very fast can explore all the sites on the island

top with the same probability, the residence time τ

is the inverse of the

escape probability to the lower layer. Thus the escape probability is simply

the product of 2πRa/πR

(the probability to reach the island edge) times

the probability to hop over the barrier D

p (which is the product of terrace

diﬀusion coeﬃcient times the interlayer probability). Equation (7.37) becomes

Ω(R)=α

p. (7.38)

The two expressions for the rate Ω(R) are very diﬀerent in the two the-

ories. For ﬁxed Ω, (7.36) shows that the interlayer probability depends less

strongly on R (p ∼ R

) than predicted by the result of the revised theory

(cf (7.38), p ∼ R

). This implies that if the same second-layer nucleation

data are analyzed by the two theories, a lower probability (and therefore a

higher barrier) will be deduced in the revised theory than in the mean ﬁeld

theory. Because the mean ﬁeld theory assumes a monomer density c(r)tobe

present, it is easier for second-layer nucleation to occur. Ω is overestimated

and therefore a higher p is suﬃcient to match the experimental f(T,R)curve.

Anomalous Prefactors in Interlayer Diﬀusion of Ag/Ag(111)

In principle, for a given f(T,R), p can be used as a ﬁtting parameter to

obtain the best agreement with the data, but such ﬁts have large uncertainty.

A direct method was proposed in [73] by using at least two temperatures,

to deduce uniquely the two contributions to p, ν

/ν

and ∆E

.Fromthe

theoretical expression of the nucleation rate Ω(R) (either (7.36) or (7.38))

the limiting values Ω

min

(when only one island has a second-layer island

on top) and Ω

max

(when all the islands have second-layer islands on top)

the corresponding values R

min

, (the largest island size with no second layer

occupation) and R

max

(the smallest island size with all islands having second

layer occupation) are plotted vs p in Fig. 7.14 (for (7.38)). From the measured

values of (R

min

, R

max

) in Fig. 7.13b: (2 nm, 4 nm at T = 120 K and 3 nm,

7.5 nm at T = 130 K) these values are best matched on the two values of

ln p at −9.2 and −6.6 shown by the vertical lines. From these two values of

p the parameters ν

/ν

=10

,and∆E

=0.32 eV are uniquely extracted for

the revised theory [72]. If the mean ﬁeld expression (7.36) for Ω(R)isused

the corresponding parameters are ν

/ν

=10

and ∆E

=0.13 eV because

the mean ﬁeld theory overestimates Ω(R) and underestimates the barrier.

Although the exact form of the function f (T,R) depends on the theory,

the result ν

/ν

> 1 is independent of the theory and follows from visual

inspection of the raw data of [68], because a dramatic change of the measured

function f(T,R) is seen even within a 10 K temperature change (from 120 K

to 130 K).

324 Michael C. Tringides and Myron Hupalo

Fig. 7.14. Plots of the min-

imum island size (R

max

)with

all islands occupied (top curve)

and maximum island size (R

min

)

when none of the islands is oc-

cupied (bottom curve) vs ln p

(where p = ν

/ν

−∆E/k

)for

the theory of [72]. The experi-

mentally determined sizes R

min

and R

max

at two temperatures

120 K and 130 K are used to de-

termine the corresponding val-

ues of p (shown by vertical lines)

and to extract ν

/ν

and ∆E

Since the interlayer parameters can depend on the step morphology (i.e.

whether straight or kinked, whether A- or B-type are present etc.) exper-

iments on macroscopic crystals measure an eﬀective barrier averaged over

diﬀerent length scales and step morphologies, with techniques which sam-

ple a larger area and diﬀerent step types. During growth of Ag on Ag(111)

over a wide temperature range (T = 90–600 K) and with diﬀerent scattering

techniques (X-ray, He-scattering, reﬂection high energy electron diﬀraction

(RHEED) [63–65]) monotonic intensity decay is observed (instead of diﬀrac-

tion oscillations), which conﬁrms 3D growth and, hence, the small values of

p. Such diﬀraction experiments can be also used to measure the step edge

barrier at the edges of the ﬁnite terraces on the clean surface. Terraces are

present in all crystals, even under the best preparation conditions.

RHEED experiments were carried out to measure the step edge barrier at

the edge of Ag(111) crystals with small average terraces of 50 nm diameter

[69]. Since the diﬀracted intensity is summed over more than 10

terraces,

the experiment averages over diﬀerent step conﬁgurations and local barriers.

The temperature of the experiment is chosen to be in the “1-island regime”.

In this regime, because diﬀusion is so fast, the deposited atoms will attach to

existing islands and only few (1 or 2) islands nucleate on the terrace. After a

ﬁxed deposition ∆θ =0.1 ML, the deposited atoms end either at ascending

or descending steps at the terrace edges or at the few nucleated islands.

At ascending steps the arriving atoms stick irreversibly. At descending steps,

atoms have only a ﬁnite probability p to descend and attach to the step below.

As the temperature is increased more atoms at descending steps overcome

the barrier at the expense of the atoms which attach to the nucleated islands.

The size of the nucleated island determines the magnitude of the RHEED

intensity drop. By comparing the RHEED intensity drop both at in-phase and

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 325

Fig. 7.15. RHEED inten-

sity decay after deposition in

Ag/Ag(111) for in-phase and

out-of-phase conditions (top two

sets of data also shown in the

inset).Thelowercurveisthe

result of Monte Carlo simula-

tions. The diﬀerence between

the experimental and simulation

data implies higher interlayer

probability p and therefore

/ν

= 100.

out-of-phase condition (see the discussion in Sect. 7.3.1) it is deduced that at

in-phase (k

d = (even)π) the intensity drop is not determined kinematically,

but is related to the change of the step density on the surface. Quantitative

information about the step edge barrier at the terrace edges can be extracted

from the diﬀerence between the measured and the simulated step density.

The experimental intensity drop is essentially determined by the increase in

the step density at the nucleated island (since the step density increase at

the terrace perimeter is much smaller than at the island perimeter).

The process was modeled with Monte Carlo simulations with parameter

/F ≈ 10

(i. e. with terrace diﬀusion to ﬂux rate ratios at T ≈ 150 K

typical of the “1-island” regime for 200 × 200 lattice or 50 nm Ag(111) ter-

race [69]). The additional amount ∆θ =0.1 ML deposited on the lattice is

divided between ascending steps, descending steps and the nucleated island.

By monitoring the increase of the nucleated island perimeter N

1/2

isl

/L, with

N denoting the number of atoms in the island and L the system size (i.e.

increase in the step density and therefore the intensity drop ∆I/I)thesim-

ulation result is compared to the experimental result in Fig. 7.15. The top

curves are for in-phase and out-of-phase conditions and the bottom curve

shows the result of the simulations. From the comparison one can deduce

the temperature dependence of the interlayer probability p(T )andextract

the Arrhenius parameters ν

/ν

∼ 100 and ∆E

=0.13 eV, in good agree-

ment with the second-layer nucleation experiments [68]. The reason for the

larger prefactor ratio can be directly seen from Fig. 7.15. ∆I/I both at in-

phase and out-of-phase conditions is less than the simulation result N

1/2

isl

for ν

/ν

∼ 1and∆E

=0.13 eV (at T = 150 K p is so low that only one

out of 10

atoms encountering a step will descend). For such high-step edge

barriers many atoms will be reﬂected at descending steps, the reﬂected atoms

will attach to the nucleated island, resulting in larger perimeter and larger

326 Michael C. Tringides and Myron Hupalo

intensity drop. Since the experimental drop is well above the expected one,

p is larger than 10

−5

as the ﬁtting result ν

/ν

= 100 suggests.

It is not clear what is the physical reason for the prefactor at steps to be

larger than the terrace prefactor. As discussed in Sect. 7.2.2, the “normal”

value of the prefactor is D

=1/4a

ν ∼ 10

−8

/s where a is the lattice

constant and ν the typical vibrational frequency. Numerous diﬀusion exper-

iments have measured prefactor values diﬀerent from this “normal” result.

Several heuristic reasons have been suggested to explain these “anomalous”

values, but there is still no clear understanding of the underlying physics.

First, as stated in Sect. 7.2.2, even for single-atom diﬀusion non-Arrhenius

behavior (and “anomalous” prefactors) can result when the residence time

of an atom in the potential well becomes comparable to the phonon vibra-

tional time of the substrate atoms, which invalidates the assumption of a

rigid potential energy surface. Second, ﬁnite coverage experiments on in-

teractive systems can lead to a coverage-dependent activation energy E(θ),

non-Arrhenius temperature dependence and “anomalous” prefactors. An al-

most complete understanding of such eﬀects has been attained for O/W(110)

where prefactor changes from 10

−11

/s to 10

−8

/s and activation energy

increase from 0.6 eV to 1 eV have been observed with increasing θ.Thesein-

creases have been correlated to a phase transition from a disordered phase at

low θ to an ordered p(2 × 1) phase at higher θ [74]. “Anomalous” prefactors

have been also related to the presence of impurities, to long-range interactions

and to concerted collective cluster motion, mediated through substrate elastic

interactions [2]. However, a better understanding of diﬀusion prefactors and

reasons why prefactors can vary so widely is one of the major open questions

in the ﬁeld.

Adatom vs Vacancy Island Decay Experiments

to Measure Interlayer Diﬀusion on Ag/Ag(111)

Experiments to measure the interlayer parameters for Ag/Ag(111) were

also carried out at higher temperatures (T ≈ 300 K) under assumed quasi-

equilibrium conditions during the decay of adatom and vacancy islands of

radius r inside larger vacancy islands of radius R [75]. The geometry of the

vacancy island decay experiment is shown schematically in Fig. 7.16. Adatom

islands decay because the atom detachment rate exceeds the rate at which

atoms attach to the island from the terrace; vacancy islands decay because

the reﬁlling rate by atoms hopping over its edge exceeds the detachment rate

from its boundary. Vacancy islands decay over longer times than adatom

islands because in the analysis of [75] incoming atoms experience the step

edge barrier and hop into the island with probability p<1. In [75] the decay

rates for an r = 7 nm adatom (and vacancy) island located at the center

of an R = 70 nm vacancy island were studied as a function of temperature

(T = 280–360 K). For example, at 300 K the decay rate for the adatom island

is 25 times faster than the decay rate of the vacancy island.

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 327

Fig. 7.16. Schematic presenta-

tion of the experiment of [75]

for adatom-island decay of ra-

dius r = 7 nm in the middle

of a larger vacancy island R =

70 nm. Dark areas are at the

same height and the region be-

tween them is one step lower.

The bottom curve shows the

atom arrangement at the island

corner: #1 denotes kink atoms,

#2 corner atoms and #3 straight

segment atoms. If the indepen-

dent detachment model is used

the prefactor ratio ν

/ν

 25 is

deduced at T = 300 K (in agree-

ment with the low-temperature

experiments of [68, 69]).

The experiment was analyzed under the assumption of steady state. For

the adatom island this means that the 2D atom ﬂux (i.e. number of atoms

per unit length per time) at r is equal to the ﬂux diﬀusing away from r and

also equal to the atom ﬂux incorporated at R. Similar steady-state condition

holds for the vacancy-island decay. The steady-state condition implies that

the concentration c(r



) (within the annular disc R ≥ r



≥ r) is described

by the same functional form at all times, which is a solution of the diﬀusion

equation in cylindrical coordinates. Atoms are evaporated from the island

perimeter because of diﬀerences between the real monomer density outside

the island from the expected equilibrium monomer density c

(r)whichis

given by the Gibbs Thomson relation c

(r)=c

∞

exp(γ/k

Tnr)whereγ is

the 1D “surface” tension, n =13.6atoms/nm

the two-dimensional Ag(111)

density and c

∞

the monomer density in front of a straight step. The decay

of the adatom island of radius r obeys the relation

d(πr

)

= −κ



+ln(R/r)+a/R

−1

×[exp(γ/k

Tnr) − exp(−γ/k

TnR)] (7.39)

which is derived from the steady-state condition with κ =2πD

∞

/n de-

noting the eﬀective detachment rate. The three terms in the ﬁrst bracket

of the right hand side represent the contribution of the three microscopic

processes which are needed to transport atoms from r to R,i.e.evaporation

at r because c(r) <c

(r), diﬀusion in the annular region R ≥ r



≥ r,and

incorporation at R because c(R) >c

(R). A similar equation holds for the

328 Michael C. Tringides and Myron Hupalo

decay of the vacancy island except in this case the curvature in the Gibss

Thomson relation is negative, because it is easier to evaporate atoms from

the boundary of a larger than a smaller vacancy island. In addition, the ﬁrst

term (a/r) in the ﬁrst bracket in the right hand side is replaced by (pa/r)

since the incorporation of atoms at r has probability p because of the step

edge barrier, which leads to

d(πr

)

= κ



+ln(R/r)+a/R

−1

×[exp(−γ/k

Tnr) − exp(−γ/k

TnR)] . (7.40)

Depending on which of the three processes dominates, (7.39) and (7.40)

can be simpliﬁed. Diﬀerent regimes are deﬁned for the vacancy-island decay

(i.e. diﬀusion limited when ln(R/r)  pa/r or interface limited when pa/r 

ln(R/r)), while for adatom-island decay the decay is always diﬀusion limited

(since ln(R/r) >a/r, a/R)). For each adatom-island decay curve at a given

temperature T two parameters γ and κ(T ) and for the vacancy-island decay,

in addition to these two, the interlayer probability p(T ) are extracted by

ﬁtting the decreasing island area vs time according to (7.39) and (7.40). From

this analysis the step edge barrier parameters ∆E

=0.13 eV, ν

/ν

≈ 1, the

“surface” tension γ =0.22 eV/atom and the detachment activation energy

κ =0.71 eV were extracted.

It is puzzling why the prefactor ratio deduced in this experiment (ν

/ν

≈

1) is lower than the one deduced in the growth experiments [68, 69], which

implies diﬀerent interlayer probability p. However, a single Arrhenius form

might fail over a temperature range of 150 K between the two experiments.

In addition, if the step edge barrier depends on site type (i.e. whether it is

kink vs straight step site, A- or B-step site type, etc.) the measured step edge

barrier will be an eﬀective average over the distribution of diﬀerent sites. The

quasi-equilibrium experiments were carried out on well-equilibrated islands

with a larger fraction of straight steps, while the low temperature experiments

were carried out on partially equilibrated island shapes, with a larger fraction

of kink sites. The interlayer probability p should be lower in the high- than

in the low-temperature experiments, because interlayer diﬀusion is higher

at kink than straight step sites. However, it is not clear why the extracted

barrier ∆E

is the same in the two experiments and only the prefactor is

diﬀerent.

Independent Atom Detachment Model for Island Decay

Experiments

As another possible explanation for the diﬀerence in the prefactor ratios the

assumption of steady-state conditions might not be applicable to Ag/Ag(111)

at 300 K. The validity of this assumption depends on the time scales of the

diﬀerent microscopic processes in the system [76]. If τ(= κ

−1

)istheav-

erage detachment time and t

the time for an atom to diﬀuse across the

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 329

region R ≥ r



≥ r,thent

must be larger than a minimum value (which

depends on τ). Otherwise, for very fast diﬀusion, the detached atoms are

transferred “instantaneously” to the boundaries, the concentration on the

terrace falls to zero and steady-state conditions cannot be established. For

Ag(111), terrace diﬀusion measured in a diﬀerent experiment is extremely

high (2×10

−0.1eV/k

hops/s [68]). At 300 K, using the measured value of κ

(obtained in [75]) and D

corresponding to ∼ 10

hops/s, the equilibrium den-

sity in front of a straight step is c

∞

=(κn/2πD

)=0.25 × 10

−9

atoms/nm

(with n =13.8atoms/nm

as the Ag(111) density). At 300 K, for the mea-

sured “surface” energy γ =0.22 eV/atom and r =7nm,R =70nmweobtain

(r)=0.33×10

−9

atoms/nm

and c

(R)=0.24×10

−9

atoms/nm

, respec-

tively. This implies diﬀerences c

(r) − c

(R)oflessthan10

−9

atoms/nm

indicating that steady-state condition might be violated.

An alternative way to view the island decay is to assume that it is a result

of independent atom detachment, followed by single-atom random walk until

the atom is adsorbed at the island boundaries. In this analysis the steady-

state condition is not satisﬁed [76].

One way to decide which analysis is applicable in a given experiment, is to

relate the detachment rates for adatom and vacancy islands to the expected

rates from the experimentally determined island shapes [77]. As found in [77]

the island shape consists of two types of segments: six straight segments

connected with six kinked segments, shown schematically in Fig. 7.16 for the

small adatom island in the middle of the large vacancy island. Straight and

kinked segments have comparable free energies at T ∼ 300 K; so the ratio

of their lengths is ∼ 1. Atoms can detach easier from kinked than straight

segments simply because they have lower coordination (i. e. 3 vs 4 neighbors).

We can determine the average detachment rate E(r) for an adatom island of

radius r and E(−R) for a vacancy island of radius R based on their shapes, if

we use a simple bond counting estimate of the corresponding energy barriers.

These simple estimates of the barriers have been also conﬁrmed with more

sophisticated calculations [66].

For the adatom island we assume that all the atoms at the kinked seg-

ments evaporate ﬁrst within τ

since they have lower number (3) of bonds.

Since the corner atoms also have 3 bonds they detach next, followed by the

“unzipping” of the atoms at the straight segment in time (n

/2)τ

(where

is the number of atoms on the straight segment). There are two corner

atoms to initiate the “unzipping” which accounts for the factor of 2. For the

adatom island of [75] with r = 7 nm we have n

=12andn

=7,sothe

average detachment time is τ

=(1+n

/2)τ

=7τ

The time for an atom to detach from the vacancy island of radius R

determines the decay of the smaller vacancy island, since these detached

atoms diﬀuse “instantaneously” to r and reﬁll the small vacancy island. As

argued before, ﬁrst the atoms at the kinked segments detach within time

. However, a major diﬀerence between the convex adatom vs the concave

330 Michael C. Tringides and Myron Hupalo

vacancy island is that once the atoms in the kinked segment are evaporated,

the corner atom has 5 bonds (instead of 3 for the adatom island) so it does

not detach next. Atoms need to detach from the straight segment (where

an atom has 4 bonds) to generate kink atoms (with 3 bonds). The time

for a kink atom to be generated on a straight step of the vacancy island

will be by a factor e

ε/k

longer (with ε denoting the nearest neighbor at-

tractive energy), followed by the “unzipping” process as before. The kink

atom can be generated anywhere on the straight segment, so averaging over

this position we ﬁnd the average time to “unzip” (3/4n



)τ

. Since these

processes are sequential, the total time is τ

= τ

+ e

ε/k

+(3/4n



)

corresponding respectively to detachment from kink segment, straight seg-

ment and “unzipping” of the straight segment. For R = 70 nm one has



= 120, which results in τ

= 5053τ

using ε =0.22 eV/atom [75]. Since

E(r)andE(−R) are rates per detached atom and the number of atoms

that can detach is proportional to the island perimeter, the ratio of de-

tachment rates is E(r)/E(−R)=(τ

/τ

)(2πr/2πR)=72.1. This ratio is

much larger than the ratio E(r)/E(−R)=D

c(r)/D

c(−R) expected for the

steady-state analysis, since using the Gibbs Thompson expression one has

c(r)/c(−R)=exp(γ/k

T (1/r +1/R)) ≈ 1.4. Such a low ratio is not compat-

ible with the diﬀerence in the island shape discussed before, i. e. convex for

the adatom vs concave for the vacancy island.

The decay rates for the adatom and vacancy islands can now be estimated

in an independent atom model. For the adatom island

/dt =(2πr)aE(r)P

/n (7.41)

where P

is the probability of an atom detached from the island at r,toreach

the boundary at R. The other possible outcome for the atom is to return back

to r with probability (1 − P

). The contribution of the atoms detached from

the boundary of the vacancy island at R is neglected because E(−R)  E(r).

has been shown to be a/r ln(R/r) [75].

Similarly, the vacancy island decay rate is given by

/dt =(2πR)aE(−R)P

/n (7.42)

where P

is the probability for an atom detached at R to be adsorbed

at r after overcoming the step edge barrier. The probability P

for the

atom detached at R to be adsorbed at r, if there was no reﬂecting bound-

ary at r (i.e. p =1),isP

= a/R ln(R/r). P

is easily found to be

p/(1 −(1 −p)(1 −P

)) by summing the probabilities p(1 −P

)

(1 −p)

of all possible paths to reach the small vacancy island before the atom over-

comes the barrier. n denotes number of reﬂections at r.

We can express the ratio of adatom- vs vacancy-island decay rates in terms

of the detachment rates, (dA

/dt)/(dA

/dt)=(r)(P

E(r))/(RP

E(−R)) =

E(r)(1 − (1 − p)(1 − P

))/E(−R). Using the data for island vs vacancy de-

cay of Ag/Ag(111) at T = 300 K [r =7nm,R =70nm,E

=0.13 eV (for

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 331

p = ν

/ν

exp(−∆E

T )), E(r)/E(−R)=72.1, (dA

/dt)/(dA

/dt) = 25]

we obtain a ratio ν

/ν

 25. We have a strong inequality in ν

/ν

(instead

of an exact value) because from the measured ratio of the decay rates it fol-

lows that p>P

, which implies that p is suﬃciently large so that it does

not control the atom attachment to the small vacancy island at r.Anyvalue

of p>P

is allowed and therefore (for a ﬁxed value ∆E

=0.13 eV of the

barrier) the experiment is consistent with any ratio ν

/ν

 25.

The re-examination of the island decay experiments in terms of the in-

dependent atom model brings the value of ν

/ν

in good agreement with the

low-temperature experiments. It is also consistent with the fact that the ra-

tio of atom-detachment rates E(r)/E(−R), expected from the experimentally

known island shapes and the contribution of the diﬀerent segments (kink vs

straight), is larger than the ratio found under the steady-state assumption.

7.5 Conclusion

We have reviewed recent developments in the fast growing ﬁeld of surface

diﬀusion by presenting results from four diﬀerent experiments to illustrate

diﬀerent techniques and objectives in the ﬁeld, especially the important dis-

tinction between equilibrium and non-equilibrium experiments. We would

like to stress again that the issues and problems discussed in this chapter can

only be a small subset of the wide-ranging activities in the ﬁeld, focused on

the authors’ expertise. Despite this limitation, the reader is expected to gain

a good idea about the breadth and diversity of the ﬁeld.

The basic dichotomy in the ﬁeld, i. e. equilibrium vs non-equilibrium

methods, is characteristic of surface diﬀusion. Work in the ﬁrst area is more

intimately related to topics covered in other chapters of this volume. Non-

equilibrium diﬀusion especially in the context of epitaxy, growth and in de-

veloping the ability to control nanostructures, as discussed in Sects. 7.4.1 and

7.4.2, is unique to surface diﬀusion with no clear analogue to bulk diﬀusion.

After recognizing this essential fact, which deﬁnes two subcommunities in

surface diﬀusion, it is worth commenting on pragmatic limitations in the rela-

tion between theory and experiment in each subcommunity. Equilibrium sur-

face diﬀusion is highly developed theoretically with mature methods of high

sophistication to deduce the coverage and temperature dependence of the

diﬀusion coeﬃcient D(θ, T ) and its relation to phase transitions and thermo-

dynamic parameters, for a given set of adatom interactions. This connection

is universally amenable to general and global predictions. Unfortunately, this

area is diﬃcult to implement experimentally because the measurable signal

is small, originating from few percent concentration ﬂuctuations. The meth-

ods presented in Sects. 7.3.1 and 7.3.2, based on ﬂuctuation measurements

in reciprocal and real space, have shown that such methods are now possi-

ble, but further tests are necessary, to carry ﬂuctuation experiments to other

systems and compare them with the results from other techniques.

332 Michael C. Tringides and Myron Hupalo

On the other hand, non-equilibrium experiments such as proﬁle evolu-

tion methods, island growth in epitaxy, second layer nucleation, island decay

etc. are easier to implement experimentally because the measurable signal is

large and easy to detect. However, such methods are either less developed

theoretically or they easily lead to ambiguities in their interpretation. The

implicit expectation, i. e. to apply the results universally to other physical

phenomena, is not easily achieved in non-equilibrium experiments, since the

questions posed are speciﬁc to the situation at hand, and the controlling fac-

tors in non-equilibrium experiments are diﬀerent microscopic processes. The

main issue is the identiﬁcation of the kinetically limiting key processes which

control the evolution of the system. This was clear both in the uniform-height

island in Pb/Si(111) and in the diﬀerent interpretations of the interlayer diﬀu-

sion experiments on Ag/Ag(111). However, the ultimate justiﬁcation of non-

equilibrium experiments is the potential to fabricate well-controlled atomic-

scale structures which are technologically useful. Success even in one system

may therefore have a larger payoﬀ than ﬁnding, if possible, a universal de-

scription of all non-equilibrium phenomena.

Notation

a lattice constant

c(r, t) local concentration at r, t

d single step height

step edge diﬀusion coeﬃcient

collective Diﬀusion coeﬃcient

tracer Diﬀusion coeﬃcient

E(r) detachment rate from island of radius r

∆E

step edge barrier

F ﬂux rate

f atomic scattering factor

I diﬀraction Intensity

k wavevector

Boltzmann constant

L terrace length of stepped surfaces

n Ag(111) atom density in atom/nm

p interlayer probability

probability to diﬀuse from r to R

probability to diﬀuse from R to r

probability to diﬀuse from R to r in the presence of a step edge

barrier at r

S(k, t) structure factor

t time

T temperature

critical temperature