Data Analysis

1.4.1.1 Unrestricted random walk

In the way of getting the general picture we need to ask two questions; how do atoms move on a crystal surface, and, at what rate are they moving with. In order to answer both questions we need to have some sort of a formula so we can deal with the ideas of surface transportation. Let’s look at the question of the rate of movement first.

From a single atom diffusion system, for an isolated atom migrating on a semi-infinite crystal terrace, we define a relevant diffusion coefficient, D, as the tracer diffusion coefficient, by

>

−

=< ( ) (0)²

2mDτ rτ r , (1.1)

where m is the dimensionality of the diffusion (either 1 or 2 for surface diffusion)*,τ is the time interval of the observation, and r is the vector position of the adatom. The brackets indicate a time average over repeated observation periods of durationτ . By defining r(0) in Eq.(1.1) as the origin, the diffusion coefficient becomes simply

mτ

D r 2

2 >

=< , (1.2)

where <r² >is the adatom’s average mean-square displacement during the time intervalτ . In one dimension,<r² >=< x² >and in two dimensions<r² >= . From random-walk theory, the mean-square displacement is given by the number of jumps, N, made by the adatom multiplied by the square of the jump distance, l, or

>

For migration exclusively by thermal activation across a surface with a periodic potential energy distribution, the average number of jumps in a time interval,τ , can be written as the product of an attempt frequency,ν₀, and a Boltzmann factor or

)

where is the difference in the Gibbs free energy between the maximum(saddle point) and the minimum(equilibrium site) of the potential energy curve (also called the potential energy well), is the Boltzmann’s constant, and T is the temperature.

ΔG

kB

*From Eq. (1.2), for diffusion in two dimensions the equation will become< r² > 4τ, or< r² > 2τ for the diffusion in one dimensions. We will be referring to two dimensions throughout this text.

The Gibbs free energy is usually expressed as a sum of an activation energy of surface

From Eqs.(1.3) and (1.5) the mean-square displacement is given by ),

and from Eqs. (1.2) and (1.6) we can then derive an expression for the diffusion coefficient as

whereD₀is all the prefactor terms combined, and is expressed as

)

In most cases the entropy difference between the saddle point and the equilibrium site is negligible and the prefactor is simply the product of the attempt frequency and the square of the jump distance. If one assumes that the attempt frequency is of the same order as atomic vibrational frequencies (~10¹²s ) and that the jump distance corresponds to the typical nearest-neighbor lattice separations (~3Å), then should be roughly .

−1

D0 10⁻³cm /² s

The diffusion coefficient D has often being referred to as the hopping rate (or the jumping rate), as this will also be referred in the remainder of the text.

1.4.1.2 Arrhenius relations

The procedure used to obtain the diffusion parameters is to measure the mean-square displacement at several temperatures and make an Arrhenius plot from Eq. (1.7) of the hopping rate (diffusion coefficient) D versus the inverse temperature 1/kT. This equation is known as the Arrhenius relation. From a system at equilibrium, an atom spends most of its time vibrating around an equilibrium site in the potential energy well. If we consider an equilibrium site as state A and a next neighbor site as state B, for the atom to move from state A to state B requires the atom to overcome an energy barrier, the so called activation barrier (or the diffusion barrier). And so, this activation energy is the energy needed for the atom to overcome the barrier and causes the system to move from A to B. See Fig 1.11.

Fig. 1.11 A diagram showing the activation barrier with an atom vibrating at an attempt frequency

ν0 inside a quantum well. The atom requires an energy E to overcome the barrier and move from state A to state B.

The hopping rate is the rate at which the atom “jumps” to the neighboring sites (from A to B) at an unit time, usually in seconds. If we take the logarithm of Eq. (1.7), we will obtain an expression as

ln 0

lnD E k T D

B

d +

=− , (1.9)

so slope of the plot from the relation in Eq. (1.9) will results in a straight line as , the activation (diffusion) energy, and the frequency prefactor is the intercept ln . Figure 1.12 shows an example of the Arrhenius plot in a case of a Pt atom diffusing on Pt(100) surface [4]. With rates taken at different temperatures it is plotted in a least square fit. The slope and the intercept of the plotted line then yield an activation energy and a prefactor of and , of the surface diffusion, respectively.

Ed

Fig. 1.12 An Arrhenius plot for the diffusions of Pt atoms on Pt(100) surface. The slope and the intercept yield an activation energy and a prefactor of and respectively.

(From Ref. [4])

eV 47 .

0 1.3×10⁻³cm /² s

1.4.1.3 Jump length distributions

Until now, in all previous discussions on diffusion analysis, assumes the displacements take place in single jumps, that is, jumps between adjacent binding sites. It is possible, however, that an atom once promoted to a transition state, can move at distances more than one lattice spacing before retrapping. For diffusions in one dimension, if say, the rate of single jumps isα, and the rate of double jumps isβ , then the probability p(x,t) of a particle being found at a distance x from the origin after time t is given by

∑

where the summation extends over all integer values of j and is the modified Bessel function of order x. Calculations of this probability in comparison to the experimental measured displacement distributions have been done in several adsorbate-substrate systems The results have indicated that the relative fraction of long jumps is system-specific. Wang et al. [5] find that only nearest-neighbor jumps occur for Re and Mo atoms diffusing on the W(211) surface over a range of temperature studied, whereas for Ir and Rh atoms, a small fraction of the displacements (<5%) are due to long jumps on the same surface. Reports by Lovisa and Ehrlich [6] suggested that long jumps on the W(110) surface occur significantly more often, presumably due to its smaller corrugation. Fr example, the fraction of long jumps in the diffusion of Ir on the W(110) surface may be as high as 20% [6]. The distribution of the jump lengths is found to be temperature-dependent suggesting that the transition state varies as a function of temperature. It is known that long jumps are more likely to occur at high temperatures where distribution of is comparable to the activation barrier for diffusion. At lower temperatures, long jumps are thought to occur if the energy dissipation to the substrate lattice is weak [7]. If one ignores the presence of long jumps, for most systems, it may introduce a source of error in measuring the activation energy. Long jumps have a strong contribution to the determination of the activation barrier from measurements of diffusing atoms [8].

Ix

T k_B