Surface diffusion is a general process involving the motion of adatoms, molecules, and atomic clusters (adparticles) at solid material surfaces.^[1] The process can generally be thought of in terms of particles jumping between adjacent adsorption sites on a surface, as in figure 1. Just as in bulk diffusion, this motion is typically a thermally promoted process with rates increasing with increasing temperature. Many systems display diffusion behavior that deviates from the conventional model of nearest-neighbor jumps.^[2] Tunneling diffusion is a particularly interesting example of an unconventional mechanism wherein hydrogen has been shown to diffuse on clean metal surfaces via the quantum tunneling effect.

Various analytical tools may be used to elucidate surface diffusion mechanisms and rates, the most important of which are field ion microscopy and scanning tunneling microscopy.^[3] While in principle the process can occur on a variety of materials, most experiments are performed on crystalline metal surfaces. Due to experimental constraints most studies of surface diffusion are limited to well below the melting point of the substrate, and much has yet to be discovered regarding how these processes take place at higher temperatures.^[4]

Surface diffusion rates and mechanisms are affected by a variety of factors including the strength of the surface-adparticle bond, orientation of the surface lattice, attraction and repulsion between surface species and chemical potential gradients. It is an important concept in surface phase formation, epitaxial growth, heterogeneous catalysis, and other topics in surface science.^[5] As such, the principles of surface diffusion are critical for the chemical production and semiconductor industries. Real-world applications relying heavily on these phenomena include catalytic converters, integrated circuits used in electronic devices, and silver halide salts used in photographic film.^[5]

Surface diffusion kinetics can be thought of in terms of adatoms residing at adsorption sites on a 2D lattice, moving between adjacent (nearest-neighbor) adsorption sites by a jumping process.^[1][6] The jump rate is characterized by an attempt frequency and a thermodynamic factor that dictates the probability of an attempt resulting in a successful jump. The attempt frequency ν is typically taken to be simply the vibrational frequency of the adatom, while the thermodynamic factor is a Boltzmann factor dependent on temperature and E_diff, the potential energy barrier to diffusion. Equation 1 describes the relationship:

${\displaystyle \Gamma =\nu e^{-E_{\mathrm {diff} }/k_{B}T}\qquad {\text{(eq. 1)}}}$

Where ν and E_diff are as described above, Γ is the jump or hopping rate, T is temperature, and k_B is the Boltzmann constant. E_diff must be smaller than the energy of desorption for diffusion to occur, otherwise desorption processes would dominate. Importantly, equation 1 tells us how strongly the jump rate varies with temperature. The manner in which diffusion takes place is dependent on the relationship between E_diff and k_BT as is given in the thermodynamic factor: when E_diff < k_BT the thermodynamic factor approaches unity and E_diff ceases to be a meaningful barrier to diffusion. This case, known as mobile diffusion, is relatively uncommon and has only been observed in a few systems.^[7] For the phenomena described throughout this article, it is assumed that E_diff >> k_BT and therefore Γ << ν. In the case of Fickian diffusion it is possible to extract both the ν and E_diff from an Arrhenius plot of the logarithm of the diffusion coefficient, D, versus 1/T. For cases where more than one diffusion mechanism is present (see below), there may be more than one E_diff such that the relative distribution between the different processes would change with temperature.

Random walk statistics describe the mean squared displacement of diffusing species in terms of the number of jumps N and the distance per jump a. The number of successful jumps is simply Γ multiplied by the time allowed for diffusion, t. In the most basic model only nearest-neighbor jumps are considered and a corresponds to the spacing between nearest-neighbor adsorption sites. The root mean squared displacement goes as:

${\displaystyle {\sqrt {\langle \Delta r^{2}\rangle }}=a{\sqrt {\Gamma t}}\qquad {\text{(eq. 2)}}}$

The diffusion coefficient is given as:

${\displaystyle D={\frac {\Gamma a^{2}}{z}}\qquad {\text{(eq. 3)}}}$

where ${\displaystyle z=2}$ for 1D diffusion as would be the case for in-channel diffusion, ${\displaystyle z=4}$ for 2D diffusion, and ${\displaystyle z=6}$ for 3D diffusion.^[8]

There are four different general schemes in which diffusion may take place.^[9] Tracer diffusion and chemical diffusion differ in the level of adsorbate coverage at the surface, while intrinsic diffusion and mass transfer diffusion differ in the nature of the diffusion environment. Tracer diffusion and intrinsic diffusion both refer to systems where adparticles experience a relatively homogeneous environment, whereas in chemical and mass transfer diffusion adparticles are more strongly affected by their surroundings.

• Tracer diffusion describes the motion of individual adparticles on a surface at relatively low coverage levels. At these low levels (< 0.01 monolayer), particle interaction is low and each particle can be considered to move independently of the others. The single atom diffusing in figure 1 is a nice example of tracer diffusion.
• Chemical diffusion describes the process at higher level of coverage where the effects of attraction or repulsion between adatoms becomes important. These interactions serve to alter the mobility of adatoms. In a crude way, figure 3 serves to show how adatoms may interact at higher coverage levels. The adatoms have no "choice" but to move to the right at first, and adjacent adatoms may block adsorption sites from one another.
• Intrinsic diffusion occurs on a uniform surface (e.g. lacking steps or vacancies) such as a single terrace, where no adatom traps or sources are present. This regime is often studied using field ion microscopy, wherein the terrace is a sharp sample tip on which an adparticle diffuses. Even in the case of a clean terrace the process may be influenced by non-uniformity near the edges of the terrace.
• Mass transfer diffusion takes place in the case where adparticle sources and traps such as kinks, steps, and vacancies are present. Instead of being dependent only on the jump potential barrier E_diff, diffusion in this regime is now also dependent on the formation energy of mobile adparticles. The exact nature of the diffusion environment therefore plays a role in dictating the diffusion rate, since the formation energy of an adparticle is different for each type of surface feature as is described in the Terrace Ledge Kink model.

Orientational anisotropy takes the form of a difference in both diffusion rates and mechanisms at the various surface orientations of a given material. For a given crystalline material each Miller Index plane may display unique diffusion phenomena. Close packed surfaces such as the fcc (111) tend to have higher diffusion rates than the correspondingly more "open" faces of the same material such as fcc (100).^[10][11]

Directional anisotropy refers to a difference in diffusion mechanism or rate in a particular direction on a given crystallographic plane. These differences may be a result of either anisotropy in the surface lattice (e.g. a rectangular lattice) or the presence of steps on a surface. One of the more dramatic examples of directional anisotropy is the diffusion of adatoms on channeled surfaces such as fcc (110), where diffusion along the channel is much faster than diffusion across the channel.

Surface diffusion is a critically important concept in heterogeneous catalysis, as reaction rates are often dictated by the ability of reactants to "find" each other at a catalyst surface. With increased temperature adsorbed molecules, molecular fragments, atoms, and clusters tend to have much greater mobility (see equation 1). However, with increased temperature the lifetime of adsorption decreases as the factor k_BT becomes large enough for the adsorbed species to overcome the barrier to desorption, Q (see figure 2). Reaction thermodynamics aside because of the interplay between increased rates of diffusion and decreased lifetime of adsorption, increased temperature may in some cases decrease the overall rate of the reaction.

Surface diffusion may be studied by a variety of techniques, including both direct and indirect observations. Two experimental techniques that have proved very useful in this area of study are field ion microscopy and scanning tunneling microscopy.^[3] By visualizing the displacement of atoms or clusters over time, it is possible to extract useful information regarding the manner in which the relevant species diffuse-both mechanistic and rate-related information. In order to study surface diffusion on the atomistic scale it is unfortunately necessary to perform studies on rigorously clean surfaces and in ultra high vacuum (UHV) conditions or in the presence of small amounts of inert gas, as is the case when using He or Ne as imaging gas in field-ion microscopy experiments.

• Surface engineering
• Surface science
• False diffusion