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

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Part II

Interfaces

7 Fluctuations and Growth Phenomena

in Surface Diﬀusion

Michael C. Tringides and Myron Hupalo

7.1 Introduction

The development of new and innovative techniques over the last 40 years

to study in a controlled way the adsorption of atoms on surfaces has led

to increased interest in the problem of surface diﬀusion and to explosive

publication growth in the ﬁeld [1, 2]. The interest in the problem is driven

both by questions of fundamental science and by the role of surface diﬀu-

sion in controlling the growth of atomic scale structures, which can have

potential technological applications. Atoms adsorbed on surfaces (at least in

the submonolayer regime) provide a true realization of two-dimensional (2D)

systems, so it is possible to test many theoretical predictions of dynamic phe-

nomena in 2D statistical mechanics. At the same time, surface diﬀusion is the

key process that controls the formation of nanostructures which can be used

as building blocks in electronic nanodevices (i.e. lasers, switches, transistors

etc.) of increasing speed and storage capacity. However within the limited

space of this chapter it is not possible to give justice to the wide-range ac-

tivity in the ﬁeld and only selected topics covering diﬀerent aspects of the

problem will be presented. Still we hope even this limited view of surface

diﬀusion will convey the tremendous progress and excitement in the ﬁeld.

The chapter is organized as follows. In Sect. 7.2 a broad overview of

surface diﬀusion is presented, with a short discussion of measurement meth-

ods and unique features of the diﬀusion problem related to properties of

surfaces (i. e. geometry of surfaces, adatom interactions, equilibrium vs non-

equilibrium methods). Next, four experiments are discussed in detail in order

to illustrate the general issues presented in the overview. Section 7.3.1 de-

scribes equilibrium diﬀusion measurements in reciprocal space on stepped

Si(100) from diﬀraction ﬂuctuations. Section 7.3.2 describes equilibrium dif-

fusion measurements in real space on H/Si(111) with scanning tunnelling

microscopy (STM) current ﬂuctuations. Next, non-equilibrium methods are

discussed, viz in Sect. 7.4.1 non-equilibrium growth studies on Pb/Si(111) at

low temperatures and the observation of self-organized uniform height islands

as a result of quantum size eﬀects (QSE). Section 7.4.2 presents measure-

ments and theoretical analysis of interlayer diﬀusion in Ag/Ag(111) which

is responsible for the 3D growth observed at all temperatures. Section 7.5

lists some general remarks about equilibrium vs non-equilibrium diﬀusion.

286 Michael C. Tringides and Myron Hupalo

Hopefully the reader can appreciate the wide-ranging concepts and diversity

in the ﬁeld of surface diﬀusion despite the subjective and limited choice of

experiments discussed.

7.2 Surface Diﬀusion Beyond a Random Walk

7.2.1 The Role of Structure and Geometry of the Substrate

The traditional paradigm used to describe surface diﬀusion is the random

walk of an atom diﬀusing on a rigid 2D lattice. The atom jumps to nearest

neighbour sites with constant probability and its mean square displacement

(R

−R

)

 =4D

t grows linearly vs time with the constant D

, the so-called

tracer diﬀusion coeﬃcient. The lattice is described in an abstract way, as a

periodic arrangement of mathematical points which is far from the structure

and geometry of a real surface. Surface atoms have ﬁnite size and varying co-

ordination with their neighbors depending on the symmetry of the underlying

lattice and the atom position on the lattice. Surface atoms are connected with

their neighbours and the diﬀusing atom with ﬁnite forces. As a result of the

motion of the diﬀusing atom, they can move out of their initial equilibrium

to new relaxed positions to minimize the total energy of the system. The sub-

strate is therefore not passive and rigid, as in the picture of a mathematical

random walk, but actively participates in determining the optimal pathway

of the lowest barrier. Although the relaxation of the surface atoms aﬀects the

jumping probability to the nearest site and the energetic barrier, it does not

change the basic stochastic nature of the process as a random walk.

Figure 7.1 illustrates the complexities related to the structure of real

surfaces by using the ball model for an fcc(111) surface with two exposed

levels separated by a step [3]. Diﬀerent types of atoms are seen with increasing

bonding coordination: adatom detached from the step to the terrace, kink

atoms, step atoms and terrace atoms. Depending on the speciﬁc structure and

the growth conditions used, diﬀerent diﬀusion processes can be the controlling

factors of the observed structures. In general, terrace diﬀusion of adatoms

controls mass transport across the surface. Step shape, whether regular or

ramiﬁed, is controlled by 1D diﬀusion of the kink atoms at the island edges.

Even this 1D diﬀusion has additional complexity as seen from the diﬀerence

in the structure of the two steps shown, denoted by A- (Fig. 7.1(a)) or B-type

(Fig. 7.1(b)), dictated by the stacking sequence of the fcc(111) lattice. Atoms

at A-type edges form square (100)-type unit cells with the row of substrate

atoms below; while atoms at B-type steps occupy the in-between sites of the

row of atoms below and form triangular (111)-type unit cells. (If the islands

were of multiple heights, A-type steps develop into (100) and B-type steps

develop into (111) facets).

We have brieﬂy described one major aspect of the surface diﬀusion prob-

lem, the role of the energetic barriers and kinetic pathways on the formation

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 287

Fig. 7.1. Ball models for the two types of fcc(111) steps: A- (at the top) and B-type

(at the bottom). In each case a straight step and a step with kinks is shown.

of the diﬀerent atomic scale structures. This is a major research area within

the surface diﬀusion community, driven by close interaction between theory

and experiment: the potential energy surfaces can be reliably mapped out

with an arsenal of theoretical techniques and directly compared to experi-

ments to identify the corresponding kinetic pathways. The theoretical meth-

ods range from ab initio methods [4] (i.e. the force ﬁelds of the adsorbed

atom to the substrate are calculated from Quantum Mechanics) and approx-

imate methods [5] (where the adatom-substrate interactions are parametrized

with eﬀective classical potentials and the time evolution of the system follows

Newton’s equations of motion). Correspondingly the atomic pathways in real

systems are inferred by comparing initial to ﬁnal state images obtained with

atom resolving techniques (i.e. scanning tunneling microscopy (STM), ﬁeld

ion microscopy (FIM)). Although this aspect of the problem is a very active

research area in surface diﬀusion, it will not be covered in this article and

interested readers can consult the literature [1,2].

However, even after the pathway of individual atoms is fully determined,

it is still an open question to show how this atomistic information can be used

to account for the results of ﬁnite coverage experiments, when many atoms

cooperate to form complicated conﬁgurations . As will be seen in this chapter,

which emphasizes the many-body aspect and cooperativity in the problem

of surface diﬀusion, this is another active research area within the commu-

nity. The cooperativity is particularly important under equilibrium condi-

tions when thermal averaging over all the microscopic processes is required.

288 Michael C. Tringides and Myron Hupalo

However, atomistic information can be suﬃcient under non-equilibrium con-

ditions. A single microscopic process can fully determine the outcome of the

experiment if this process has a time constant comparable to the duration

of the experiment and all other processes “freeze out” because they have

larger time constants. In epitaxy, for example, single atom terrace diﬀusion

can fully determine the nucleated island density during irreversible growth

at suﬃciently low temperatures [6].

7.2.2 The Role of Adsorbate-Adsorbate Interactions

The second complexity of the problem of surface diﬀusion, stated in the pre-

vious section, is related to the cooperativity and interactions between the

diﬀusing atoms at ﬁnite coverage [7]. As seen from the formation of ordered

phases of diﬀerent symmetry formed at low temperatures, the atoms expe-

rience strong mutual interactions to generate the ordering. When interac-

tions are present diﬀusion is not a simple random walk, because the jumping

probability to the nearest neighbor sites depends on the local environment

of the diﬀusing atom rather than being constant. For example, in a simple

model with nearest-neighbour repulsive interaction J the jumping probabil-

ity e

zJ/k

depends on the number of nearest neighbours z of the diﬀusing

atom, as implemented in Monte Carlo algorithms.

At high temperatures no single microscopic process of barrier E

(i.e. ter-

race diﬀusion, edge diﬀusion, dimer diﬀusion, atom detachment from islands

etc.) controls the dynamics of the system because all rates (∼ e

)ofthe

diﬀerent microscopic processes become comparable and an averaging over all

the processes is necessary (cf. Chap. 18).

The magnitude of the interactions between the adatoms depends on the

strength of the bonding to the substrate. For example, in chemisorbed systems

the bonding to the substrate is strong and the adatom interactions larger than

about 0.1 eV, because they involve charge transfer or are mediated via the

conducting electrons of the substrate [8]. Physisorbed atoms (e. g. noble gases)

are weakly bound to the substrate, with van der Waals type interactions

between the adatoms that are less than about 0.1 eV, i. e. comparable to

the potential corrugation of the substrate. In some systems, strong long-

range adatom interactions mediated through the anisotropy of the intrinsic

substrate stress can be present (i. e. stepped Si(100), Au(23 ×

√

3). They

result in the spontaneous formation of periodically arranged domains of large

size determined by the competition between the energy increase (because of

missing bonds at the domain boundary) and the energy decrease due to the

long-range stress mediated interactions [9].

Adatom interactions modify the energy minima at the binding site of the

adsorbed atom and the corresponding diﬀusion barrier. Qualitatively, the ef-

fect of the interactions on diﬀusion is summarized as: repulsive interactions

lower the diﬀusion barrier while attractive interactions raise it. It is not a pri-

ori clear how competing interactions will aﬀect the barrier. Since the number

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 289

of atoms within the local environment of the diﬀusing atom increases with

coverage, one expects the diﬀusion coeﬃcient to be coverage dependent. This

in turn changes the linear diﬀusion equation into a non-linear one

∂θ

∂t

= ∇D

(θ)∇θ (7.1)

where θ(r, t) is the coverage at position r and time t and D

the collective

diﬀusion coeﬃcient. The coverage is deﬁned as the ratio θ = c(r, t)/c

with

c(r, t) the concentration at r, t and c

the saturation concentration, i. e. when

all binding sites on the surface are ﬁlled. However, as will be discussed later,

for systems truly at equilibrium the coverage dependence is not a severe prob-

lem and the diﬀusion equation can be still treated as linear if the variation of

θ is small (a few % from its average value

θ), which justiﬁes the assumption

that D

(θ) is constant.

Extensive theoretical work has already been carried out to answer quan-

titatively the question how adsorbate-adsorbate interactions aﬀect diﬀusion

[1, 2]. Commonly a lattice gas model is used, with an eﬀective Hamiltonian

which includes the adatom interactions, to deduce the concentration and tem-

perature dependence of the diﬀusion coeﬃcient D(θ, T )(whereD refers to

either D

and D

). The Hamiltonian is written in the form

H =



n





n



ijk

+ ... (7.2)

where n

= 1, 0 denotes the occupation of site i, J

denotes the pairwise

interaction energy between sites i and j, J

ijk

a trio interaction between a

triplet of sites, etc.. In principle, the Hamiltonian can include more complex

interactions than the pairwise and trio interactions shown above, but (7.2)

can account for the majority of the ordered phases observed experimentally.

The calculation of D(θ, T ) for a given Hamiltonian and set of interaction

energies {J’s} is carried out either analytically (i.e. mean ﬁeld theory of var-

ious levels of sophistication, renormalization group techniques), numerically

(series expansion methods), or, far more commonly, with Monte Carlo simula-

tions. The family of calculated curves D(θ,T ) from (7.2) are compared to the

corresponding measured curves to obtain the best agreement with the exper-

imental results. Since the temperature dependence of the diﬀusion coeﬃcient

is commonly parametrized in Arrhenius form D = D

(θ)exp−(E(θ)/k

T ),

in practice, the extracted parameters from the data (i.e. the prefactor D

(θ)

vs θ and activation energy E(θ)vsθ) are compared to the calculated ones.

It would have been far more eﬃcient if it were possible to do the analysis

inversely i. e. to start from the experimental quantities D

(θ)andE(θ)and

deduce the appropriate Hamiltonian (7.2) and the interaction parameters

{J’s}. In practice, the Hamiltonian is based on information obtained inde-

pendently in other experiments (for example low energy electron diﬀraction

(LEED) measurements of I vs θ or I vs T to determine the diﬀerent or-

dered regions of the system and to map out the phase diagram [10]). But

290 Michael C. Tringides and Myron Hupalo

Fig. 7.2. Schematic energy dia-

gram of hopping from an initial

energy state E

to a ﬁnal state

, through the saddle point of

energy E

. ∆E denotes the en-

ergy change because of adatom

interactions.

even if such inversion in the data analysis were possible, there would still be

the question whether the Arrhenius pametrization of D(θ,T )overthewhole

temperature range is justiﬁed or whether non-Arrhenius behaviour should

be expected. Arrhenius expressions are empirically used by experimentalists

over limited temperature range (∼ 100 K) but it is not clear what is the full

theoretical justiﬁcation of this practice.

In a simple-minded way the Arrhenius expression is explained in terms

of the 1D energy diagram of Fig. 7.2. At temperature T and in the har-

monic approximation the adsorbed adatom oscillates at the well minimum

because of its thermal energy. It can overcome the energy barrier with a

temperature-dependent probability which leads to the Arrhenius-like expres-

sion D = D

(θ)exp(− E(θ)/k

T ). In the θ → 0 limit E(θ) is the energy

diﬀerence between the minimum and the saddle point in the potential energy

surface of a single adatom and the prefactor is simply D

(θ → 0) = 1/4a

where a is the lattice constant and ν the typical adatom vibrational frequency.

Such expressions are better justiﬁed at low temperatures or for strong bar-

riers, when the typical residence time of the adatom in the potential well is

much longer than the vibrational time of the substrate phonons, so the pic-

ture of an adatom adsorbed at the minima of a rigid potential energy surface

as shown in Fig. 7.2 is better justiﬁed.

In addition to the coverage and temperature dependence of the diﬀu-

sion coeﬃcient, important questions related to how sensitive D(θ, T )isto

the phase boundaries separating diﬀerent ordered regions and whether crit-

ical eﬀects resulting from phase transitions can be observed in the θ or T

dependence of D, have been actively pursued and will be discussed later.

7.2.3 Diﬀusion in Equilibrium and Non-Equilibrium

Concentration Gradients

Fluctuation Measurements

Diﬀusion is a result of the presence of non-zero concentration gradients in

the system. These gradients exist either because they are imposed as initial

conditions, i.e. at t = 0 the coverage θ(r, t) has an externally imposed value

(r, 0) (diﬀerent from its average equilibrium value θ) or they are generated

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 291

because chemical potential gradients ∇µ(r) exist in the system. Such chemical

potential gradients are present even at equilibrium (i.e. when the average

coverage

θ is constant both in time and space) because the energy cost of

adsorption at a site depends on the type of site occupied. It is more general

to write the diﬀusion equation in terms of the chemical potential gradient

(cf.,e.g.,Chap.5)

J = −L∇µ (7.3)

where J is the adatom ﬂux generated by the gradient ∇µ and L a phenom-

enological transport coeﬃcient. Using the continuity equation

∂θ

∂t

+ ∇J =0 (7.4)

we derive (7.1) with D

related to L by the expression D

=(∂µ/∂θ)L.When

the system is at equilibrium, the derivative ∂µ/∂θ is determined thermody-

namically from the µ

(θ) isotherm of the system, otherwise the chemical

potential µ(θ) under non-equilibrium conditions depends on the speciﬁc con-

dition θ

(r, 0) applied initially. For example, in non-equilibrium step proﬁle

experiments when the coverage has a value θ

for r<0 and a diﬀerent value

for r>0, as will be discussed below, µ(θ) if diﬀerent from µ

(θ)andthe

extracted D

depends explicitly on ∂θ/∂r.

It is more relevant to discuss diﬀerences between equilibrium vs non-

equilibrium conﬁgurations in systems when ordered phases are present, i. e. in

interactive systems at suﬃciently low temperatures. As an illustration assume

the ordered state is c(2×2). At equilibrium, atoms can diﬀuse into neighboring

sites even when there is no externally imposed gradient, because within the

ordered domains neighboring sites are vacant or at the domain walls the sites

are energetically costly, i. e. they have diﬀerent chemical potentials. If one

monitors the coverage at position r and time t, θ(r, t), its occupation will

ﬂuctuate in time around its average value

θ. As a result of diﬀusion in or out

of position r, a random initial ﬂuctuation δθ(r, 0) = (θ(r, 0) −

θ) will spread

out. At distance r



away and at later time t, we expect to see the “remnant”

ﬂuctuation δθ(r



,t)=(θ(r



,t) − θ) if the time of the ﬂuctuation at r to

diﬀuse to site r



is larger than t. Information about the diﬀusion coeﬃcient

is obtained from the time dependence of the autocorrelation function S(r



−

r, t)=δθ(r, 0)δθ(r



,t) where the bracket average includes an average over

all sites in the system, since at equilibrium all sites are equivalent. Since the

diﬀusion equation (7.1) is linear for systems at equilibrium (even when D

(θ)

is coverage dependent) it is also obeyed by S(r



− r, t). Truly equilibrium

systems have low amplitude concentration ﬂuctuations (when compared to

the average concentration

θ) |δθ

(0, 0)

1/2

|/θ  1 and as discussed earlier a

constant diﬀusion coeﬃcient can be assumed. Since S(r



− r, t) depends on

the diﬀerence (r



− r) it can be simpliﬁed to S(r, t)andisgivenby

S(r, t)=S



exp −(r

/(4D

t))



(7.5)

292 Michael C. Tringides and Myron Hupalo

with S

= δθ

(0, 0). As will be discussed in Sect. 7.3.2, more commonly

theinitialﬂuctuationisprobedwithinanareawithseveralsitesinorder

to increase the measurable signal. The expression for S(r, t) can be written

equivalently in terms of its Fourier transforms for the conjugate variables of

r and t, i.e. wavevector k and frequency ω. The Fourier components S(k, t)

follow exponential decay with k-dependent initial amplitudes. For the k → 0

component the decay is given by

S(k, t)=





· exp(−D

t) (7.6)

where a

is the unit cell area and S(k, ω) has a Lorentzian dependence on ω

S(k, ω)=2S



(ω

+(k

)



. (7.7)

For the measurement of the diﬀusion coeﬃcient D

diﬀerent experimental

methods based on the three equivalent expressions (7.5), (7.6), (7.7) have been

developed.

The extraction of D

from (7.5) requires the measurement of time corre-

lations between ﬂuctuations at two sites separated by distance r



− r. Such

measurements are diﬃcult because (i) precise control of the separation (r



−r)

is necessary and (ii) the measurable signal is low if the correlation between

only two single sites is probed. However, it is not necessary to restrict the

experiment to the correlation of single sites [11]. The signal can be increased

considerably if instead ﬂuctuations within a ﬁnite area, which includes sev-

eral sites, are monitored in time. For example for a circular area A of radius

, shown schematically in the upper part of Fig. 7.3, it can be easily shown

that the autocorrelation function is given by the integral

∆N(0)∆N (t) =



r d



δθ(r, 0)δθ(r



,t) =



rS(r



− r, t)

(7.8)

where N(t) is the number of adatoms within the area A at time t and c

the

saturation concentration. The magnitude of the initial ﬂuctuation at t =0is

given by ∆N

(0) = Ac

S(k =0, 0). Since the signal increases linearly with

the probe area A, it is easier to be detected in the experiment. D

depends

on the time constant τ of the decay of the correlation function and the probe

area A via D

= A/(4τ).

Equation (7.6) has been implemented in simulations where a periodic

long-wavelength (λ =2π/k) modulation of the coverage is generated by

imposing at t = 0 a small-amplitude variation of the chemical potential

µ(r)=µ

+ δµ cos(kr)whithδµ/µ

 1. The system is still close to equilib-

rium but the coverage diﬀers in a sinusoidal fashion from the average value

θ (determined from µ

= µ

(θ)byasmallamountδθ proportional to δµ).

For times t>0thevariationδµ cos(kr) is removed and the system relaxes

back to equilibrium. This is shown schematically in the lower part of Fig. 7.3

where the amplitude of the coverage modulation decays with times t

, t