Surface Diffusion

Abstract

Surface diffusion is a key step in several phenomena occurring on crystal surfaces, for example in thin-film growth and in chemical reactions. Here, we review the basic concepts of surface diffusion. The elementary diffusion mechanisms on flat and stepped surfaces, such as jumps and exchanges are treated first. Then, the long-time result of the combination of such elementary diffusion moves is considered, both from the point of view of single-particle (or tracer) diffusion and of collective diffusion. Finally, an overview of the experimental techniques to measure surface diffusion is given, with special attention being paid to the helium spin-echo technique.

Adatom mobility is extremely important in a series of processes of technological relevance, such as thin-film growth and heterogeneous catalysis. In fact, smooth thin films can be grown only if the deposited atoms are able to move around on terraces and between different terraces. On the other hand, surface chemical reactions are possible only if the reactants, which usually impinge on the surface in random positions, can move around to meet each other. In this chapter, we review the basic aspects of adatom mobility on surfaces, first focusing on the elementary atomic moves and then on the combination of these moves in determining the long-time stochastic motion of the adatoms. Literature on this subject is vast, and, therefore, we refer to a narrower domain, metal-on-metal diffusion, as regards the specific examples. The interested reader may refer to books and review articles for more complete overviews [2.1, 2.2, 2.3, 2.4, 2.5, 2.6]. Moreover, we focus on single-adatom mobility, recalling, however, that mobility on surfaces may involve also atomic aggregates [2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20, 2.21, 2.22, 2.23, 2.24, 2.7, 2.8, 2.9].

1 Elementary Mechanism of Surface Diffusion

Let us consider an atom that is deposited on an otherwise empty crystal surface. The adsorbed atom can exchange energy with the substrate, which acts as a thermostat at temperature T. For simplicity, we assume now that the substrate does not participate actively in mass transport, i. e., that substrate atoms simply oscillate around their lattice positions. This assumption will be relaxed when dealing with exchange diffusion. After deposition, a series of different processes may take place in sequence for the adatom. These processes may be schematically described as follows:

• Transient mobility. When the incoming atom hits the surface, it may have a significant amount of kinetic energy, coming from the kinetic energy of the beam and from the condensation energy on the substrate (which is of the order of few eV for transition metals), or from the dissociation energy in the case in which molecules, such as O₂, are deposited. This excess energy may cause some transient mobility of the adatom before the initial nonthermal energy is dissipated to the substrate [2.25, 2.26]. Dissipation may take place either by exciting phonons or by creating electron–hole pairs in the substrate [2.27]. The former mechanism contributes to the phononic part of the friction, whereas the latter contributes to the electronic part of the friction. Recent estimates for diffusion of Na on Cu(111) [2.28] show that dissipation to phonons and to electron–hole pairs account for ≈ 80% and 20% of the total friction, respectively.

• Thermalization. After the initial excess energy is dissipated, the atom equilibrates with the substrate in some adsorption site, where it stays for some time by making small oscillations of frequency ν_osc at the potential well bottom. This is possible if the potential energy well associated with the adsorption site is much deeper than k_BT, where T is the temperature of the substrate. If adsorption sites are very shallow, the adatom equilibrates on the substrate but keeps on moving in a quasicontinuous way. This type of motion is, however, rather uncommon. An example is the case of adatom diffusion on fcc(111) surfaces, as discussed.

• Activated escape. Even if deeper than k_BT, an adsorption site is not infinitely deep, so that the adatom can escape from it through a saddle point, by surmounting an energy barrier E_b. Here, we consider only diffusion by thermal activation, since quantum tunneling is relevant only at low temperatures (< 100 K for hydrogen and at even lower temperatures for other atoms [2.1, 2.29]). The escape rate ν_j is given by an expression known as the Arrhenius law

  $$\nu_{j}=\nu_{0j}\mathrm{e}^{-\frac{E_{\mathrm{b}}}{k_{\mathrm{B}}T}}\;,$$

  (2.1)

  where ν_0j is the rate prefactor. Since ν_0j ≈ ν_osc [2.5] one has

  $$\nu_{j}\ll\nu_{\mathrm{osc}}\;.$$

  (2.2)

  Energy barriers for diffusion depend on many factors. For a given type of substrate and adatom, the barrier depends on the surface orientation. For example, for Au, Ag, and Cu adatom diffusion on the surfaces of these metals, the barrier for hopping diffusion is of a few tens of meV for diffusion on the (111) surface, and of a few hundred meV on the (100) surface [2.30, 2.31, 2.32, 2.33, 2.34, 2.35]. The very low barriers on the (111) surface cause a quasicontinuous diffusive motion in the network on fcc and hcp adsorption sites, which can take place already at room temperature. As discussed in Sect. 2.1.2, E_b is different in the vicinity of steps and other defects than on the flat, perfect surface. For diffusion of metal atoms on insulating surfaces, E_b may also depend on the charge state of the adatom [2.36]. Diffusion barriers may depend on the direction on anisotropic surfaces, to the extent that diffusion may become essentially one-dimensional in some cases [2.37, 2.38, 2.39, 2.40, 2.41].

• Jump diffusion. After jumping out from an adsorption site, the adatom will finally thermalize in another adsorption site, which may be either a nearest/neighbor site or a more distant site [2.42, 2.5]. In the latter case, a long jump has taken place. For deep potential wells, the time of flight τ_fl between escaping from the first site and thermalizing in the new site is much shorter than the average residence time in the well τ_j

  $$\tau_{\mathrm{fl}}\ll\tau_{j}=\frac{1}{\nu_{j}}\;.$$

  (2.3)

  The residence time is, therefore, sufficiently long to eliminate correlations between subsequent jumps, so that the overall adatom motion will be a sequence of uncorrelated events, which produces a random walk in the lattice of adsorption sites.

A schematic representation of the trajectory of a diffusion event of an adatom is given in Fig. 2.1. The adatom oscillates in an adsorption site, then it jumps to a nearest-neighbor site and, finally, thermalizes there and keeps on oscillating.

Fig. 2.1

Trajectory (projected in the surface plane) of an adatom jumping between two adsorption sites (of energy −V at the well bottom) through a saddle point of energy 0 (the saddle point is at the center). The energy barrier E_b is given by the energy difference between the saddle point and the minimum, E_b = V

The occurrence of long jumps indicates a weak energy exchange between the adatom and the substrate. This can be qualitatively discussed in the framework of the Langevin model, in which the energy exchange between adatom and substrate is described in terms of friction and white noise. In the Langevin model, the motion of the adatom obeys a stochastic differential equation. For simplicity, we discuss the one-dimensional case, which is relevant, for example, in diffusion along channels or steps. Let us assume that the adatom is diffusing in a spatially periodic system of period a, which corresponds to the lattice distance between nearest-neighbor adsorption sites of our system. The displacement x(t) of the adatom of mass m is ruled by

$$m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=-m\gamma v+F(x)+\Gamma(t)\;,$$

(2.4)

where v is the velocity of the adatom, $F(x)=-\mathrm{d}V/\mathrm{d}x$ is a spatially periodic force derived from the adiabatic potential, which describes the average interaction with the substrate, and γ is the friction per unit mass; Γ(t) is a δ-correlated white noise, whose intensity is related to the friction by the fluctuation-dissipation theorem [2.43]. While the friction term can only subtract energy to the adatom, the noise term can give and subtract energy to insure thermal equilibrium with the substrate.

In order to discuss the occurrence of long jumps [2.44, 2.45], let us calculate the energy dissipated by the adatom that crosses a lattice cell assuming that the amplitude A = E_b ∕ 2 of V(x) is much larger than k_BT. We can assume that the adatom starts from the top of the barrier with a kinetic energy of k_BT and then dissipates the energy Δ while crossing the cell to reach the next barrier top

$$\Delta=\int_{0}^{a}m\gamma v(x)\mathrm{d}x\;.$$

(2.5)

Since long jumps can occur only when dissipation is small, we can estimate v(x) as if the motion of the adatom would be conservative, i. e., assuming that

$$\frac{1}{2}mv^{2}(x)+V(x)=E_{\mathrm{b}}+k_{\mathrm{B}}T\;,$$

(2.6)

so that

$$\begin{aligned}\displaystyle\Delta&\displaystyle=\gamma\int_{0}^{a}\sqrt{2m[E_{\mathrm{b}}+k_{\mathrm{B}}T-V(x)]}\\ \displaystyle&\displaystyle\simeq\gamma\int_{0}^{a}\sqrt{2m[E_{\mathrm{b}}-V(x)]}\;,\end{aligned}$$

(2.7)

because E_b ≫ k_BT. In this limit, the main contribution to the integral comes from x close to the well bottom, so that the integral itself can be approximated by E_b ∕ ν_osc. The condition for the occurrence of long jumps thus becomes

$$\frac{\Delta}{k_{\mathrm{B}}T}\ll 1\quad\longrightarrow\quad\frac{\gamma}{\nu_{\mathrm{osc}}}\ll\frac{k_{\mathrm{B}}T}{E_{\mathrm{b}}}\;.$$

(2.8)

Long jumps have been observed in different experiments [2.37, 2.38, 2.46, 2.47, 2.48, 2.49, 2.50] and in many simulations and calculations [2.16, 2.31, 2.32, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56].

Mass transport on surfaces can take place by several different mechanisms besides jumps. Below we describe the most important ones.

1.1 Exchange Diffusion

In exchange diffusion, an adatom enters the substrate taking the place of a substrate atom, which is pushed up on the surface and becomes the new adatom. Even though this mechanism implies an active role of the substrate, for homoepitaxial systems it can be treated by the same line of reasoning as the jumps, since the final result of the process is the same as in a jump between the appropriate initial and final sites. For heteroepitaxial systems, after the exchange process, the new diffusing adatom may, of course, be of a different species.

There are several systems in which the exchange mechanism is favored over the simple jump. The first example is the diffusion of adatoms on the (110) surface of fcc metals. This is a channeled surface, with atomic rows running along the [1$\bar{1}$0] direction (Fig. 2.2). Diffusion across these channels, i. e., in the [001] direction, can hardly take place by jumps. In fact, in the adsorption site, the adatom has five nearest neighbors, which reduce to two at the saddle point for the jump process, with a corresponding huge amount of energy which is of the order of 1 eV and more for transition and noble metals. On the contrary, diffusion by exchange requires surmounting much smaller barriers, often well <  0.5 eV.

Exchange diffusion has been observed also on surfaces with square symmetry, such as the (100) surface of fcc crystals. As shown in Fig. 2.2b, the adatom enters the substrate taking the place of one of its four nearest neighbors. This process produces a net displacement in a diagonal direction (such as the [011] direction), i. e., the same displacement that would be obtained by a jump to a second-neighbor site. Therefore, if diffusion on this surface takes place exclusively by exchange, the adsorption sites of a single sublattice can be visited by a series of subsequent exchanges, while the other sublattice is never visited. This sublattice is rotated by 45^∘, like the network of white squares on a checkerboard. On the other hand, if diffusion takes place by jumps or by a combination of jumps and exchanges, all sites can be visited. This allows us to single out pure exchange diffusion in experiments [2.57, 2.58]. There are also systems in which both exchanges and jumps occur at comparable frequencies, for example, in W∕W(100) [2.59].

In simulations, exchange diffusion has been observed for a large variety of metal-on-metal homoepitaxial and heteroepitaxial systems [2.31, 2.51, 2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66].

Fig. 2.2

(a) Exchange process on a (100) surface of an fcc metal. This channeled surface, with atom rows along the [1$\bar{1}$0] direction (x-direction) is shown in a top view. The atom rows are very difficult to surmount by a jump process to reach the adjacent channel. In the middle panel the typical symmetric dumbbell configuration is shown. This is either the saddle point or a short-lifetime metastable configuration corresponding to a very shallow local minimum. (b) Exchange process on a (100) surface of an fcc metal. The adatom enters the substrate and substitutes an atom that becomes the new adatom. The process causes mass transport along the diagonal, in the [011] direction. A single jump process would cause mass transport either along the x or the y-direction

Exchange processes may also take complex pathways involving the displacement of more than two atoms. Evidence in favor of these multiatom processes essentially comes from simulations [2.51, 2.67, 2.68, 2.69, 2.70, 2.71, 2.72, 2.73, 2.74, 2.75]. In some cases, also correlated processes involving a jump and an exchange part have been seen [2.31, 2.51]. Schematic representations of a correlated jump-exchange process and of a three-atom exchange are given in Fig. 2.3.

Fig. 2.3

(a) A correlated jump-exchange process [2.51] on an fcc(110) surface. The original adatom (blue color) makes a jump along the a channel in the $[1\bar{1}0]$ direction, but, instead of stopping in the next well, it pushes a substrate atom of an adjacent row (red color) to the next channel and takes its place in the row. (b) A multiple exchange process on an fcc(100) surface [2.67] in which the adatom is incorporated into the substrate by pushing two atoms. One of them (violet color) becomes the new adatom

1.2 Diffusion Across and Along Steps

Surface diffusion in the presence of defects is also an important research topic. Steps are the most common extended defects on a surface, and both diffusion along and across steps play an important role in the evolution of the surface during thin-film growth [2.76, 2.77].

Adatom diffusion across a step (Fig. 2.4) is an asymmetric process. In metal-on-metal diffusion, descending from the upper to the lower terrace is usually easier than the reverse process (upward diffusion). This happens because the site reached by the adatom on the lower terrace is more coordinated than sites on the upper terrace, so that there is an energy gain in the downward move. For other systems, the behavior may be different—for example, simulations of the diffusion of oxygen on Cu(001) by reactive force fields [2.78] have shown that in this case, upward diffusion is easier than downward diffusion. In the following, we focus on metal-on-metal diffusion, in which downward diffusion between terraces is the easier process.

Fig. 2.4

(a) An adatom jumps down at a step. (b) The adatom is incorporated in the step riser after an exchange, whereas a former step atom is the new adatom that can diffuse either along the step or on the bottom terrace. (c) The adatom is incorporated in an inner position of the terrace by a multiple-exchange process

Diffusion across steps can take place by jumps and by exchange, the latter being possible also from sites that are not in the close vicinity of the terrace border, as is shown in Fig. 2.4. Even though there is a final energy gain in the downward diffusion processes, the intermediate configurations of these processes often contain severely undercoordinated atoms, as is schematically shown in Fig. 2.4. For this reason, the activation barrier for downward diffusion can be higher than the barrier for diffusion on the upper terrace. The additional barrier for crossing the step is known as the Ehrlich–Schwoebel (ES) barrier [2.79, 2.80]. In Fig. 2.6, we sketch a typical energy profile of the diffusion pathway across a step. If the diffusion on the upper terrace, far away from the step, has a barrier E_ut, and the downward diffusion across the step has a barrier E ^⟂_step , then the ES barrier is defined by $E_{\mathrm{step}}^{\perp}=E_{\mathrm{ut}}+E_{\mathrm{ES}}$. The ES barrier is a crucial factor in determining the thin-film growth mode. Small or nonexistent ES barriers favor layer-by-layer growth, while large ES barrier favor three-dimensional growth because they hinder the mobility between terraces at different levels [2.76]. There are even cases in which the barrier for downward step crossing may be lower than the barrier on the flat terrace, as was found in calculations for exchange crossing on Ag(100) [2.81]. In these cases, the ES barrier is negative. Zero or even negative ES barriers may be found also for downward crossing at step kinks [2.7, 2.82, 2.83].

Fig. 2.5

An adatom on the border between a (111) and a (100) facet on an fcc truncated octahedral cluster pushes a row of atoms of the (100) facet, so that finally an adatom appears on the opposite (111) facet

Also the diffusion of adatoms between adjacent facets on a nanoparticle [2.84, 2.85] takes place by similar types of mechanisms to the diffusion between terraces of high-index crystal surfaces. Interfacet diffusion processes, therefore, have their own ES barriers [2.68, 2.86]. In nanoparticles, there is also the possibility of chain diffusion mechanisms that allow the displacement of atoms between opposite sides of the nanoparticle [2.68, 2.87], as shown in Fig. 2.5.

Fig. 2.6

Potential energy profile along the diffusion pathway of an atom crossing a step to descend to the lower terrace. The additional ES barrier E_ES for crossing the step is indicated

The diffusion of adatoms across steps has been studied in many cases, especially by simulations. For metal-on-metal diffusion, these studies considered both homoepitaxial [2.88, 2.89, 2.90, 2.91, 2.92] and heteroepitaxial systems [2.92, 2.93, 2.94]

Diffusion along steps may take place on the lower and on the upper side of the step. Since the sites on the lower side have higher coordination, they act as traps of adatoms. On the other hand, the adsorption energy in sites on the upper side of the step is usually very close to that of inner terrace sites. For these reasons, most studies have focused on diffusion along the lower sides of steps. Diffusion along steps is quite difficult to measure directly in experiments, but it has important experimental consequences as far as the shape of growing islands is concerned [2.76, 2.95, 2.96], especially if diffusion around corners is also considered. The presence of steps is also important because it can alter the interaction between adsorbed molecules on the terraces, thus influencing their diffusion coefficient [2.97].

Due to the difficulty of measuring the diffusion of adatoms along steps, most studies in the literature are simulation studies. For metal-on-metal diffusion, one may refer to, for example,  [2.100, 2.101, 2.102, 2.93, 2.98, 2.99].

The diffusion barriers for adatoms diffusing along steps strongly depend on the symmetry of the surface and on the type of step. This can be exemplified by considering straight steps on two high-index surfaces of fcc crystals, the (111) and the (100) surfaces.

Let us consider first the (111) surface. In Fig. 2.7, a schematic representation of an island on a (111) surface is shown. This island is limited by four straight steps, which, however, are not equivalent, because two of them present square facets on the step riser (steps of type A), and two of them present triangular facets on the step riser (steps of type B). If there is an A step on one side of the island, on the opposite side there will be a B step. Rhombic and hexagonal islands must, therefore, present both step types, while triangular islands (of orientations differing by 60^∘) present a single type of step [2.76]. Diffusion of adatoms along these steps may be characterized by different energy barriers [2.98]. Also the diffusion pathway along A and B steps may be somewhat different. In fact, calculations for Au(111) and Ag(111) showed that the preferential diffusion pathway is a straight line along step B, whereas it has a cosine-like shape along step A [2.98]. However, the most important point is that on this surface, diffusion along steps has much higher barriers than diffusion on the flat terrace. In Ag and Au(111), the former has barriers of a few tenths of eV, while the latter has barriers of ≤  0.1 eV [2.30, 2.98]. This means that there is a temperature range in which terrace diffusion is activated while diffusion along steps is not, allowing the growth of fractal-like islands [2.76].

Fig. 2.7

An island on a (111) surface of an fcc crystal. The island is limited by straight steps, and there are four adatoms diffusing on the lower edges of the steps. There are two different types of steps. The steps of type A (those of the adatoms marked by A) have square facets on the step riser. The steps of type B (those of the adatoms marked by B) have triangular facets on the step riser

On the (100) surface, the situation is quite different. In fact, referring again to Ag and Au as examples, diffusion on the flat terrace is characterized by barriers of a few tenths of eV [2.31]. In the jump mechanism, terrace diffusion takes place by passing through a saddle point, where coordination is decreased from four to two. For diffusion along a straight step, the decrease in coordination at the saddle point is only from five to four, as shown in Fig. 2.8. This indicates that the barrier for diffusion along straight steps may be lower than that on the terrace, as is, indeed, verified by several calculations ([2.101] and references therein). Therefore, on these (100) surfaces, there is no temperature interval in which terrace diffusion is activated while diffusion along steps is not. This causes the growth of islands limited by straight steps, which are much more compact than the island grown on the on the (111) surface of the same metals.

Fig. 2.8

Diffusion of an adatom along a straight step on an fcc(100) surface. From the initial position (I), the adatom passes through the saddle point (S) and then moves to the final site (F). In (I) and (F), the adatom has five first neighbors, while in (S) it has four first neighbors

2 Single-Particle and Collective Diffusion Coefficients

In the previous section, we focused on the elementary moves by which an adatom can displace from one site to another on a crystal surface. Now we focus on the long-time result of the combination of such elementary moves, assuming that they are statistically independent of each other. This amounts to assuming that each time the diffusing atom reaches a new adsorption site on the surface, it stays there for a sufficiently long time to equilibrate with the substrate.

As we will show in the following, the combination of statistically independent elementary moves leads to a long-time behavior in which the mean square displacement of the adatom increases (apart from a numerical constant related to the dimensionality of the system) linearly as D_st, where D_s is the single-particle (or tracer) diffusion coefficient; D_s is, thus, related to the average behavior of each individual diffusing particle. However, this is not the only diffusion coefficient that can be defined. In fact, there is also the collective diffusion coefficient D_c, which describes how density gradients relax to equilibrium in the system; D_s and D_c may both depend on the density of the adsorbate, but, as we will see in the following, in quite different ways.

2.1 Single-Particle (Tracer) Diffusion

Let us consider an isotropic surface and a single diffusing adatom whose position as a function of time t is r(t). The diffusing adatom is in contact with the substrate, which exchanges energy with the adatom, thus acting as a thermostat at a temperature T. For sufficiently long times, the motion of the diffusing adatom can be described as a random walk. The net mean displacement of the adatom is

$$\langle\Updelta\boldsymbol{r}(t)\rangle=\left\langle\left[\boldsymbol{r}(t)-\boldsymbol{r}(0)\right]\right\rangle=0$$

(2.9)

due to left-right symmetry. On the contrary, the mean square displacement

$$\langle\Updelta\boldsymbol{r}^{2}(t)\rangle=\langle\left[\boldsymbol{r}(t)-\boldsymbol{r}(0)\right]^{2}\rangle$$

(2.10)

is different from zero and tends to increase as t^α, with the exponent α > 0, for sufficiently long times. This allows us to define the single-particle (or tracer) diffusion coefficient D_s

$$\begin{aligned}D_{\mathrm{s}}=\lim_{t\to\infty}\frac{\langle\Updelta\boldsymbol{r}^{2}(t)\rangle}{2dt}\;,\end{aligned}$$

(2.11)

where d is the dimensionality of the system, d = 2. The limit in (2.11) converges to a finite value if asymptotically ⟨Δr²(t)⟩ ∝ t, i. e., α = 1. If this does not happen, i. e., if the asymptotic behavior is proportional to t^α with α ≠ 1, one has the so-called anomalous diffusion, of which there are some examples also in the case of surfaces [2.103, 2.104]. It can be easily shown that (2.11) is equivalent to [2.43]

$$D_{\mathrm{s}}=\frac{1}{d}\int_{0}^{\infty}\langle\boldsymbol{v}(t)\cdot\boldsymbol{v}(0)\rangle\mathrm{d}t\;,$$

(2.12)

where v is the velocity of the adatom.

In the general case, the surface can be anisotropic, so that one has to consider the diffusion tensor D_s

$$\textbf{D}_{\mathrm{s}}=\begin{pmatrix}D_{\mathrm{s}}^{xx}&D_{\mathrm{s}}^{xy}\hfil\\[0.5mm] D_{\mathrm{s}}^{yx}&D_{\mathrm{s}}^{yy}\\ \end{pmatrix}\;,$$

(2.13)

where

$$\begin{aligned}\displaystyle D_{\mathrm{s}}^{xx}&\displaystyle=\lim_{t\to\infty}\frac{\langle[x(t)-x(0)]^{2}\rangle}{2t}\;,\\ \displaystyle D_{\mathrm{s}}^{yy}&\displaystyle=\lim_{t\to\infty}\frac{\langle[y(t)-y(0)]^{2}\rangle}{2t}\;,\\ \displaystyle D_{\mathrm{s}}^{xy}&\displaystyle=D_{\mathrm{s}}^{yx}=\lim_{t\to\infty}\frac{\langle[x(t)-x(0)][y(t)-y(0)]\rangle}{2t}\;;\end{aligned}$$

(2.14)

D_s is, thus, symmetric by definition, so that it can be diagonalized. This allows us to find two principal diffusion directions x_p and y_p for which

$$\begin{aligned}\displaystyle D_{\mathrm{s}}^{x}&\displaystyle=\lim_{t\to\infty}\frac{\langle[x_{\mathrm{p}}(t)-x_{\mathrm{p}}(0)]^{2}\rangle}{2t}\;,\\ \displaystyle D_{\mathrm{s}}^{y}&\displaystyle=\lim_{t\to\infty}\frac{\langle[y_{\mathrm{p}}(t)-y_{\mathrm{p}}(0)]^{2}\rangle}{2t}\;.\end{aligned}$$

(2.15)

If x_p and y_p are equivalent, one recovers the isotropic case, i. e., D ^x_s  = D ^y_s .

Let us consider jump diffusion. We first restrict ourselves to the case of isotropic lattices in which only jumps to nearest neighbor sites are allowed. There are n_pv nearest neighbors (n_pv = 4, 6, and 3 for the square, the triangular, and the honeycomb lattices, respectively). The adatom starts from the origin of the coordinates and jumps on average at intervals δt, which means that its total jump rate (or frequency) is ν_j = 1 ∕ δt. At the i-th jump, the adatom displacement from its previous position is

$$\delta\boldsymbol{r}_{i}=(\delta x_{i},\delta y_{i})\;,$$

(2.16)

while its total displacement after n jumps is

$$\boldsymbol{r}^{(n)}(t)=\sum_{i=1}^{n}\delta\boldsymbol{r}_{i}\;,$$

(2.17)

with t = nδt. Taking into account that for i ≠ j jumps are statistically independent and that all jumps are equivalent, the mean square displacement is given by

$$\begin{aligned}\displaystyle\langle\Updelta\boldsymbol{r}^{2}(t)\rangle&\displaystyle=\langle\boldsymbol{r}^{(n)}(t)\cdot\boldsymbol{r}^{(n)}(t)\rangle\\ \displaystyle&\displaystyle=\left\langle\sum_{i=1}^{n}\delta\boldsymbol{r}_{i}\cdot\sum_{j=1}^{n}\delta\boldsymbol{r}_{j}\right\rangle\\ \displaystyle&\displaystyle=\left\langle\sum_{i=1}^{n}\delta\boldsymbol{r}_{i}\cdot\delta\boldsymbol{r}_{i}\right\rangle=n\langle\delta\boldsymbol{r}_{1}^{2}\rangle\;.\end{aligned}$$

(2.18)

The tracer diffusion coefficient is, thus, given by

$$D_{\mathrm{s}}=\lim_{t\to\infty}\frac{\langle\Updelta\boldsymbol{r}^{2}(t)\rangle}{4t}=\lim_{t\to\infty}\frac{n\langle\delta\boldsymbol{r}_{1}^{2}\rangle}{4n\delta t}=\frac{1}{4}\nu_{j}\langle\delta\boldsymbol{r}_{1}^{2}\rangle\;.$$

(2.19)

If the distance of first neighbors is a, whatever direction is taken for the jump, the square displacement is a², so that

$$\langle\delta\boldsymbol{r}_{1}^{2}\rangle=a^{2}\;,$$

(2.20)

which leads to the final expression of the tracer diffusion coefficient in a jump model with nearest-neighbor jumps

$$D_{\mathrm{s}}=\frac{1}{4}\nu_{j}a^{2}\;.$$

(2.21)

For anisotropic lattices, we have

$$D_{\mathrm{s}}^{x}=\frac{1}{2}\nu_{j}^{x}a^{2}\quad D_{\mathrm{s}}^{y}=\frac{1}{2}\nu_{j}^{y}b^{2}\;,$$

(2.22)

where ν ^x_j  , ν ^y_j are the total jump rates in x and y directions, respectively, and a , b are the nearest-neighbor distances in the directions x and y.

As we saw in the previous section, there are several examples of surface diffusion in which jumps to more distant neighbors than the first ones are possible. Let us consider the contribution of these long jumps to diffusion. We consider the isotropic case and calculate ⟨δr ²₁ ⟩. From the site of departure (in r = 0 for simplicity), the adatom can jump to any other site l of the lattice with probability π_l, with the normalization condition

$$\sum_{\boldsymbol{l}\neq 0}\pi_{\boldsymbol{l}}=1\;.$$

(2.23)

We have that ⟨δr ²₁ ⟩ equals the mean-square jump length ⟨l²⟩

$$\langle\delta\boldsymbol{r}_{1}^{2}\rangle=\sum_{\boldsymbol{l}\neq 0}\pi_{\boldsymbol{l}}(\boldsymbol{l}\cdot\boldsymbol{l})=\langle l^{2}\rangle\;,$$

(2.24)

giving

$$D_{\mathrm{s}}=\frac{1}{4}\nu_{j}\langle l^{2}\rangle\;.$$

(2.25)

If long jumps occur in one dimension, for example, on channeled surfaces such as the (1 × 2) missing-row reconstructed surface, the expression for ⟨l²⟩ is

$$\langle l^{2}\rangle=\sum_{l\neq 0}\pi_{l}(la)^{2}\;,$$

(2.26)

with l an integer. Since π_l = π_−l, we can define p_l = 2π_l for l > 0 and write

$$\langle l^{2}\rangle=\sum_{l=1}^{\infty}p_{l}l^{2}a^{2}\;.$$

(2.27)

The diffusion coefficient is

$$D_{\mathrm{s}}=\frac{1}{2}\nu_{j}a^{2}\sum_{l=1}^{\infty}p_{l}l^{2}\;.$$

(2.28)

Single-particle diffusion is not limited to isolated adatoms. In fact, in general, D_s can be defined for a dense adsorbate. For N diffusing adatoms (which are here assumed to be identical), D_s is defined exactly in the same way as for an isolated adatom, with the advantage (quite important in simulations) of averaging on all adatoms. The definition for the isotropic case is

$$\begin{aligned}D_{\mathrm{s}} & =\lim_{t\to\infty}\frac{1}{2dt}\sum_{i=1}^{N}\langle\Updelta\boldsymbol{r}_{i}^{2}(t)\rangle\;.\end{aligned}$$

(2.29)

Lattice models are very often used to treat diffusion in dense adsorbates, because they allow a simplified treatment. Here, we consider a specific two-dimensional lattice model, which is known as the Langmuir gas. This model allows us to easily understand the main differences between tracer and collective diffusion. Our discrete lattice contains M equivalent adsorption sites, and it is filled by N adatoms. Each site can be occupied by no more than one adatom, so that the fraction of occupied sites, usually called coverage, is θ = N ∕ M; D_s will depend on θ, generally in a quite complex way. The adatoms diffuse by jumping between nearest-neighbor sites on the lattice and are assumed to interact only by site blocking. Site blocking mimics the short-distance hard-sphere repulsion between the adatoms forbidding multiple occupancy of sites. Even for this simple model, the exact analytical expression for D_s(θ) is not known. In the limit θ → 0, D_s must tend to the expression of (2.21). Then, due to site blocking, D_s(θ) decreases with θ. As a first approximation, we note that, on average, the jump rate is decreased proportionally to the average fraction of occupied nearest-neighbor sites, which is given by θ is this simple model. Therefore, a mean-field expression for D_s(θ) is

$$D_{\mathrm{s}}(\theta)=\frac{1}{4}\nu_{j}a^{2}(1-\theta)\;.$$

(2.30)

This expression is exact for both limits θ = 0 and θ = 1 (for θ = 1, no adatom can move), but it overestimates the true D_s for intermediate θ. This is due to neglecting memory effects, which correlate sequences of subsequent jumps. To understand why memory effects lower D_s compared to the mean-field expression of (2.30), one may note that after a jump, the adatom leaves a free site behind, so that a jump back is more likely than a jump to another site, whose probability of being empty is 1 − θ. In order to take into account memory effects, a tracer correlation factor f_t(θ) is introduced, so that

$$D_{\mathrm{s}}(\theta)=\frac{1}{4}\nu_{j}a^{2}(1-\theta)f_{\mathrm{t}}(\theta)\;.$$

(2.31)

Evaluating f_t(θ) by analytical means is quite complicated, so that numerical evaluations by Monte Carlo simulations are often used [2.105]. Anyway, very accurate analytical formulas have been produced [2.106, 2.107, 2.108], mostly by using projection-operator techniques. For example, for a square lattice, one may use

$$f_{\mathrm{t}}(\theta)=1-\frac{2\theta}{6-\theta-\xi(2-\theta)}\;,$$

(2.32)

with ξ = 10440 ∕ 9443. The behavior of D_s(θ) according to the different approximations is reported in Fig. 2.9.

Fig. 2.9

Normalized diffusion coefficients D_{s,c} ∕ D_s(0) as functions of the coverage θ in a Langmuir gas: a mean-field approximation for D_s (2.30), b D_s with memory effects taken into account (2.31) and (2.32), c D_c (Sect. 2.2.2), which, in the Langmuir gas, does not depend on coverage

Further interactions between the adatoms besides site blocking are usually called lateral interactions. In general, attractive lateral interactions slow down tracer diffusion, because each adatom has to break the bonds with its neighbors to make a jump. On the other hand, repulsive lateral interactions may enhance diffusion, unless they are so strong (compared to k_BT) as to cause the formation of ordered phases, whose rigid framework leads to a decrease in adatom mobility.

2.2 Collective Diffusion

While tracer diffusion describes the long-time motion of each adatom individually, collective diffusion describes the time decay of long-wavelength density fluctuations of the whole adsorbate [2.1, 2.109, 2.2, 2.5]. Let us consider an isotropic adsorbate with average density $\overline{\rho}$. At a given point r on the surface and time t, the density is ρ(r , t), so that the density fluctuation δρ is

$$\delta\rho(\boldsymbol{r},t)=\rho(\boldsymbol{r},t)-\overline{\rho}\;.$$

(2.33)

We assume that the flux of matter in the adsorbate is proportional to the density gradient according to Fick's law

$$\boldsymbol{J}(\boldsymbol{r},t)=-D_{\mathrm{c}}\nabla\rho(\boldsymbol{r},t)=-D_{\mathrm{c}}\nabla\delta\rho(\boldsymbol{r},t)\;,$$

(2.34)

where J is the current density, and D_c is the collective diffusion coefficient. Combining this equation with the continuity equation we obtain the diffusion equation

$$\frac{\partial\delta\rho}{\partial t}=D_{\mathrm{c}}\nabla^{2}\delta\rho\;.$$

(2.35)

Let us consider a periodic density fluctuation and calculate how it decays with time according to (2.35). The fluctuation at time t = 0 is

$$\delta\rho(\boldsymbol{r},0)=\delta\rho_{0}\cos(\boldsymbol{q}_{0}\cdot\boldsymbol{r})=\delta\rho_{0}\frac{\mathrm{e}^{\mathrm{i}\boldsymbol{q}_{0}\cdot\boldsymbol{r}}+\mathrm{e}^{-\mathrm{i}\boldsymbol{q}_{0}\cdot\boldsymbol{r}}}{2}\;.$$

(2.36)

We define

$$S(\boldsymbol{q},t)=\int\mathrm{d}\boldsymbol{r}\mathrm{e}^{\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r}}\delta\rho(\boldsymbol{r},t)\;,$$

(2.37)

with the inversion formula

$$\delta\rho(\boldsymbol{r},t)=\frac{1}{(2\uppi)^{d}}\int\mathrm{d}\boldsymbol{q}\mathrm{e}^{-\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r}}S(\boldsymbol{q},t)\;.$$

(2.38)

Substituting in (2.35) we find an equation for the evolution of S

$$\frac{\partial S}{\partial t}=-D_{\mathrm{c}}q^{2}S(\boldsymbol{q},t)\;,$$

(2.39)

with solution

$$S(\boldsymbol{q},t)=S(\boldsymbol{q},0)\mathrm{e}^{-D_{\mathrm{c}}q^{2}t}\;,$$

(2.40)

with

$$S(\boldsymbol{q},0)=(2\uppi)^{d}\frac{\delta\rho_{0}}{2}\left[\delta(\boldsymbol{q}-\boldsymbol{q}_{0})+\delta(\boldsymbol{q}+\boldsymbol{q}_{0})\right].$$

(2.41)

Substituting in (2.38)

$$\delta\rho(\boldsymbol{r},t)=\delta\rho_{0}\cos(\boldsymbol{q}_{0}\cdot\boldsymbol{r})\mathrm{e}^{-D_{\mathrm{c}}q_{0}^{2}t}\;,$$

(2.42)

which gives that the fluctuation decays exponentially with time constant

$$\tau_{\mathrm{c}}(\boldsymbol{q}_{0})=\frac{1}{D_{\mathrm{c}}q_{0}^{2}}\;.$$

(2.43)

Now, we treat collective diffusion in a Langmuir gas. We consider fluctuations in the coverage, by considering the evolution of θ_l(t), which represents the probability that at time t site l is occupied, from an initial periodic configuration. This evolution is described by a master equation

$$\frac{\mathrm{d}\theta_{\boldsymbol{l}}}{\mathrm{d}t}=\alpha\sum_{\boldsymbol{a}}(\theta_{\boldsymbol{l}+\boldsymbol{a}}-\theta_{\boldsymbol{l}})\;,$$

(2.44)

where l + a are the positions of the first neighbors of site l of the two-dimensional Bravais lattice (noting that the only isotropic two-dimensional Bravais lattices are the square and the hexagonal lattices). The coefficient α = ν_j ∕ n_pv is the directional jump rate from l to l + a. The equilibrium probability is the average coverage θ, so that the fluctuation of the coverage is given by

$$\delta\theta_{\boldsymbol{l}}(t)=\theta_{\boldsymbol{l}}(t)-\theta\;.$$

(2.45)

The master equation for δθ_l(t) is, thus,

$$\frac{\mathrm{d}\delta\theta_{\boldsymbol{l}}}{\mathrm{d}t}=\alpha\sum_{\boldsymbol{a}}(\delta\theta_{\boldsymbol{l}+\boldsymbol{a}}-\delta\theta_{\boldsymbol{l}})\;.$$

(2.46)

which has to be solved with the initial condition

$$\delta\theta_{\boldsymbol{l}}(0)=\delta\theta_{0}\cos(\boldsymbol{q}_{0}\cdot\boldsymbol{l})\;;$$

(2.47)

S(q , t), in this case, is defined as

$$S(\boldsymbol{q},t)=\sum_{\boldsymbol{l}}\mathrm{e}^{\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{l}}\delta\theta_{\boldsymbol{l}}(t)\;,$$

(2.48)

with the inversion formula

$$\delta\theta_{\boldsymbol{l}}(t)=A_{\mathrm{c}}\int\mathrm{d}\boldsymbol{q}\mathrm{e}^{-\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{l}}S(\boldsymbol{q},t)\;,$$

(2.49)

where A_c is the area of the primitive unit cell of the Bravais lattice. Inserting (2.49) in (2.46), one obtains

$$\frac{\mathrm{d}S}{\mathrm{d}t}=-f(\boldsymbol{q})S(\boldsymbol{q},t)\;,$$

(2.50)

with

$$f(\boldsymbol{q})=\alpha\sum_{\boldsymbol{a}}(1-\mathrm{e}^{-\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{a}})=\alpha\sum_{\boldsymbol{a}}[1-\cos(\boldsymbol{q}\cdot\boldsymbol{a})]\;.$$

(2.51)

For long-wavelength fluctuations q ≪ 1 ∕ a,

$$f(\boldsymbol{q})=\alpha\sum_{\boldsymbol{a}}\frac{1}{2}(\boldsymbol{q}\cdot\boldsymbol{a})^{2}=\frac{\nu_{j}}{4}a^{2}q^{2}\;,$$

(2.52)

recalling that α = ν_j ∕ n_pv and verifying the equation for the square and hexagonal lattices. However, from (2.30) ν_ja² ∕ 4 = D_s(0), so that

$$f(\boldsymbol{q})=D_{\mathrm{s}}(0)q^{2}\;.$$

(2.53)

Therefore,

$$S(\boldsymbol{q},t)=S(\boldsymbol{q},0)\mathrm{e}^{-D_{\mathrm{s}}(0)q^{2}t}\;,$$

(2.54)

with

$$S(\boldsymbol{q},0)=\frac{\delta\theta_{0}}{2A_{\mathrm{c}}}[\delta(\boldsymbol{q}-\boldsymbol{q_{0}})+\delta(\boldsymbol{q}+\boldsymbol{q_{0}})]$$

(2.55)

and, finally,

$$\delta\theta_{\boldsymbol{l}}(t)=\delta\theta_{0}\cos(\boldsymbol{q}_{0}\cdot\boldsymbol{l})\mathrm{e}^{-D_{\mathrm{s}}(0)q_{0}^{2}t}\;,$$

(2.56)

which shows that the fluctuation decays with a time constant

$$\tau_{\mathrm{c}}(\boldsymbol{q}_{0})=\frac{1}{D_{\mathrm{s}}(0)q_{0}^{2}}\;.$$

(2.57)

Comparing this with (2.43), one obtains

$$D_{\mathrm{c}}(\theta)=D_{\mathrm{s}}(0)\;,$$

(2.58)

which means that in a Langmuir gas, the collective diffusion coefficient does not depend on coverage, and its value is equal to the zero-coverage tracer diffusion coefficient, as shown in Fig. 2.9. Therefore, collective diffusion in the Langmuir gas is as fast in the limit of full coverage as in the limit of zero coverage. This seemingly counterintuitive result can be explained by recalling that D_c always measures the mobility of density fluctuations. In the limit of θ → 1, density fluctuations are a few isolated atomic vacancies in the adsorbate, which diffuse as fast as isolated adatoms in the limit θ → 0.

An equivalent definition of D_c can be given in terms of the asymptotic behavior of the displacement of the center of mass of the adsorbate $\boldsymbol{R}=\sum_{i=1}^{N}\boldsymbol{r}_{i}$

$$D_{\mathrm{c}}=\lim_{t\to\infty}\frac{1}{2dNS_{0}t}\langle[\boldsymbol{R}(t)-\boldsymbol{R}(0)]^{2}\rangle\;,$$

(2.59)

where S₀ is the static structure factor, which can be related to the isothermal compressibility of the adsorbate χ_T by

$$S_{0}=\overline{\rho}k_{\mathrm{B}}T\chi_{T}\;,$$

(2.60)

where the average density $\overline{\rho}$ is given by $\overline{\rho}=\theta/a^{2}$. The equivalence of the definitions of D_c by (2.59) and by Fick's law can be demonstrated in a straightforward but lengthy way. The interested reader is referred to [2.111]. From the computational point of view, the comparison of (2.11) and (2.59) shows that D_s is much easier to calculate than D_c, because in a single simulation box, there are N independent samples for calculating D_s, while there is a single sample (the coordinate of the center of mass) for calculating D_c.

The calculation of D_c in interacting systems turns out to be quite difficult. However, at variance with D_s, we note that for D_c memory effects are absent in the Langmuir gas, so that we may assume that they are somewhat less important even in the presence of lateral interactions besides site blocking. If memory effects are neglected, D_c in a lattice gas of coverage θ turns out to be given by the following expression [2.112, 2.113]

$$D_{\mathrm{c}}=\frac{a^{2}\langle\nu_{j}\rangle}{4k_{\mathrm{B}}T\theta\chi_{T}}\;,$$

(2.61)

where ⟨ν_j⟩ is the thermally averaged jump rate.

Results about the dependence of D_c in a triangular lattice gas with repulsive interactions are given in Fig. 2.10. These results [2.110] have been obtained by the approximation of (2.61), in which the thermally averaged quantities ⟨ν_j⟩ and χ_T were calculated by the cluster variation method [2.114]. The results of Fig. 2.10 are representative of the complex dependence of D_c on coverage in systems in which there is the formation of ordered phases. This complex behavior results from the competition between the decrease of both ⟨ν_j⟩ and of χ_T in the ordered phases. Outside the region of the phase diagram corresponding to the ordered phases, D_c shows a tendency to increase with θ because of the repulsive character of the interactions.

However, it must be noted that the approximation of (2.61) may be insufficient in the vicinity of phase transitions in the adsorbate, where memory effects may become important also for collective diffusion [2.115]. More sophisticated analytical approaches have been developed in more recent times [2.116, 2.117], also for heterogeneous and anisotropic lattices [2.118, 2.119, 2.120].

Fig. 2.10

(a) Phase diagram of the repulsive lattice gas on a triangular lattice. Nearest-neighbor interactions are assumed. The inset considers an adatom that jumps from the adsorption site 0 to the adsorption site a, passing through the saddle point Σ. In 0, the adatom feels the repulsion of other adatoms which may occupy sites b, c, d, e, and f. This repulsion has intensity J > 0. At the saddle point, the adatom feels the interaction with other adatoms in sites b and f, with intensity J_Σ, which may be either positive or negative. Putting Δ = J − J_Σ, the energy barrier E_b for this jump is given by $E_{\mathrm{b}}=E_{0}-J(n_{\mathrm{c}}+n_{\mathrm{d}}+n_{\mathrm{e}})-\Delta(n_{\mathrm{b}}+n_{\mathrm{f}})$, where E₀ is the energy barrier for an isolated adatom, and the variables n are either 0 or 1, depending on the corresponding site being free or occupied by an adatom. (b) Behavior of D_c depending on the coverage θ for βJ = 4 and different values of βΔ, with β = 1 ∕ (k_BT) (adapted from [2.110])

3 Experimental Measurements of Diffusion

Various techniques have been established to measure diffusion of atoms and molecules at surfaces experimentally. Again, the literature is vast, and there have been a number of comprehensive reviews on experimental progress [2.1, 2.121, 2.122, 2.123, 2.4]. There is a broad distinction between experimental techniques that either measure the evolution of an initially imposed concentration profile of a surface species, and those that follow the motion of individual particles moving in equilibrium. The former generally measures diffusion over relatively large length scales and interprets the motion within Fick's law and usually provides chemical diffusion constants and their temperature dependence. Once the concentration profile has been established, its evolution can be measured by various methods [2.1], including work function probes, photoelectron microscopy, and scanning Auger microscopy. Similarly, laser-induced thermal desorption has been used to repeatedly desorb species from a particular region of the surface and to subsequently monitor the rate of refilling by diffusion. All these methods rely on being able to generate a suitable initial configuration and, more fundamentally, are only indirectly related to the underlying microscopic characteristics of the system.

Equilibrium methods generally follow the motion of individual particles or fluctuations in the equilibrium configuration of an assembly of particles. Equilibrium methods, therefore, provide more direct microscopic information; specifically, local correlations in individual particle motion can be understood in detail, along with the transition between localized microscopic interactions and dynamics that follow Fick's law in the macroscopic limit. Equilibrium diffusion measurements can be routinely performed using time-lapse imaging, typically based on scanning tunneling microscopy (STM) or field ion microscopy (FIM) [2.121, 2.124]. However, because of the limited framing rate (interframe times are at least milliseconds [2.124]) they can only distinguish the before and after states associated with jumps between adsorption sites—the detail of, for example, long jumps cannot be captured. Similarly, to limit the extent of motion between frames, low temperatures are usually required (i. e., thermal energy, k_BT ≪  activation energy, E_a), which means such data is often from a different regime than to many physical processes of practical interest, such as catalysis or epitaxial growth. Fluctuation correlation methods such as fluorescence correlation spectroscopy (FCS) [2.125] and x-ray photon correlation spectroscopy (XPCS) [2.126] are equilibrium methods that provide more extensive information, as they yield correlation information on specific length scales, using optical and x-ray scattering, respectively. However, as they are sensitive to mesoscopic length scales and relatively long timescales, they still cannot provide a complete atomic-scale description of motion.

The technique of helium spin-echo (HeSE) [2.127, 2.128, 2.129, 2.130] is a relatively recent equilibrium approach to measuring surface diffusion. The method provides perhaps the most comprehensive surface transport data available—giving access to both diffusion and other forms of surface motion. The technique is the surface analogue of quasielastic neutron scattering (QENS) [2.131] and involves scattering a beam of neutral thermal energy helium atoms from mobile species on a surface. The scattered atom distribution is directly related to a pair correlation function that provides a comprehensive statistical description of the motion, with information spanning timescales between picoseconds and nanoseconds and length scales from ångströms up to tens of nanometers. Since HeSE provides such comprehensive data, we will focus on that method here and provide a series of examples.

3.1 Helium Scattering and the Spin–Echo Technique

The broad technique of helium atom scattering has been used routinely in a number of specialist laboratories around the world since the 1970s. The method involves producing a nearly monochromatic, collimated, beam of thermal energy helium atoms (typically 10 to 100 meV), then scattering the atomic beam from a surface in order to study the surface properties [2.132, 2.133, 2.134, 2.135]. Since the incident energy of helium is so low, the atoms scatter from the outermost valence electrons at the surface. Through that interaction, the helium atoms can, in general, exchange momentum, ℏΔk, and energy, ℏω, with the surface, where k and ω relate to wavevector and frequency, respectively. Both elastic and inelastic scattering are possible, enabling surface structure and dynamical processes to be studied, respectively.

In the case of elastic scattering, when the helium atoms experience no energy exchange with the surface, the main processes are diffraction and diffuse scattering. These have been used to study surface structure and growth, often in relation to thin films and delicate adsorbed species [2.133, 2.134]. During inelastic scattering, helium atoms generally interact with surface phonons or adsorbate vibrations. If the interaction is with a single mode, the well-defined energy transfer condition corresponding to the vibrational frequency of the mode means surface dispersion curves can be mapped out [2.135]. Multiphonon interactions are also possible.

If helium atoms interact with an aperiodically moving species (i. e., a diffusing atom), energy changes are still possible, but are now centered around zero energy transfer, i. e., around the elastic peak. These quasielastic energy changes are characteristic of that aperiodic diffusive motion and, hence, enable the motion to be studied—the phenomenon is known as quasielastic helium atom scattering (QHAS) and underpins the HeSE method. Note that these quasielastic energy changes are very small, usually in the μeV range, or even smaller.

In general, quantitative interpretation of helium scattering data can be complex, and various approximations have been developed [2.132]. Fortunately, for helium atoms scattered from mobile adsorbates on a surface, the kinematic approximation can be used to simplify analysis by decomposing the overall scattered intensity, A(ΔK , ω), into the product of an intensity structure factor, S, and an amplitude form factor, F,

$$A(\Updelta\boldsymbol{K},\omega)=S(\Updelta\boldsymbol{K},\omega)\cdot\left|F(\Updelta\boldsymbol{K},\omega)\right|^{2}\;;$$

(2.62)

S(ΔK , ω) is determined by the position of the scatterers on the surface, and F(ΔK , ω) is a result of the shape of each scatterer. For analysis of adsorbate dynamics, the approximation works very well [2.136]; ΔK represents the change in wavevector of the helium atoms during scattering, projected into the surface plane and is given by the scattering geometry according to

$$\Updelta K=k_{\mathrm{f}}\sin(\theta_{\mathrm{SD}}-\theta_{\mathrm{i}})-k_{\mathrm{i}}\sin(\theta_{\mathrm{i}});$$

(2.63)

k_i and k_f are the incident and final helium wavevectors, and θ_SD and θ_i are the total scattering angle and incident angles, respectively; S(ΔK , ω) is known as the dynamic structure factor, a quantity used extensively in neutron scattering [2.131], and which was shown by Van Hove [2.137] to be related to a pair correlation function, G(R , t), by a double Fourier transform in both space and time, through the intermediate scattering function, I(ΔK , t),

$$S(\Updelta\boldsymbol{K},\omega)\stackrel{\text{Temporal FT}}{\rightleftharpoons}I(\Updelta\boldsymbol{K},t)\stackrel{\text{Spatial FT}}{\rightleftharpoons}G(\boldsymbol{R},t).$$

(2.64)

The pair correlation function G(R , t) provides a complete statistical description of the motion of the scatterers.

In most energy-resolved helium scattering experiments, the scattering angles are controlled by the experimental setup, and the absolute energy distribution of the scattered beam is measured using time-of-flight methods [2.134]. Hence, providing F can be accounted for or ignored, S(ΔK , ω) is obtained directly from the experiment, which then provides information about diffusion and vibration through the relationship with G(R , t). For QHAS experiments of surface diffusion, the finite velocity spread in the initial helium beam is a serious limit on time-of-flight measurements. The velocity spread masks the tiny quasielastic energy changes that relate to surface diffusion in all but a very few physical systems [2.138, 2.139]. In contrast, HeSE experiments only measure the energy changes during scattering, and so are not limited by the spread of energies in the incident helium beam, and, thus, the HeSE method can be applied much more widely. As we will describe below, HeSE experiments obtain the intermediate scattering function, I(ΔK , t), directly, hence providing a more direct link to the information about diffusion that is encoded in G(R , t). By measuring I(ΔK , t) under a sufficiently wide range of conditions, analysis can reveal the full detail of surface transport, including diffusion, vibration, energy exchange, and collective motion, in a regime that is not accessible to any other experimental technique.

3.2 Measuring the Intermediate Scattering Function with HeSE

Before proceeding to discuss HeSE measurements of diffusion, we give a brief overview of the technique, following the semiclassical description by Gähler et al [2.140]; for more details, the reader is directed to [2.130]. The basic principle of HeSE is shown schematically in Fig. 2.11. An HeSE instrument consists of an extended series of vacuum chambers to convey a beam of helium atoms from the beam source to the sample and then to transfer atoms scattered in a particular direction, to the detector. Along the two beamlines, a series of magnetic components are used to measure the intermediate scattering function, by making use of the nuclear spin of ³He.

Fig. 2.11

The HeSE technique. Atoms produced in the helium source are polarized in a hexapole magnet, spin-aligned in a dipole field, and then passed through a solenoid magnet. The solenoid splits the helium wavepacket into two spin components, which reach the sample separated by a short time delay t_SE. Along the second arm of the instrument, the spin components are recombined and passed through a further dipole and hexapole to spin-analyze the beam. The beam-averaged polarization is proportional to the intermediate scattering function I(ΔK , t).

Working along the beamlines shown in Fig. 2.11, the beam of ³He atoms is first produced in a standard free-jet supersonic expansion, followed by several stages of differential pumping to reduce the background helium pressure diffusing along the beamline from the source chamber. Next, the beam is passed through an intense hexapole magnetic field, where the ³He nuclear spins must align either parallel or antiparallel to the local-field direction. The hexapole field is highly inhomogeneous and has a magnitude B_h, which varies quadratically with the radius. Consequently, the field applies a force to each spin, given by F =  ± μ∇B_h, which focusses one polarization of atoms onto the sample, while the other is defocused and removed for recycling. The focused component of the beam is then passed through a dipole field, which aligns the spins in a particular direction perpendicular to the beamline, which we denote as the state |+⟩_x. The focused atoms then continue into a solenoidal field B_s oriented along the beamline. Inside the solenoid, the helium wavepacket can be represented as a superposition of two spin eigenstates, aligned either parallel or antiparallel to the field,

$$\left|+\right\rangle_{x}=\frac{1}{\sqrt{2}}(\left|+\right\rangle_{z}+\left|-\right\rangle_{z}),$$

(2.65)

whose energies are split by ±μB_s, respectively. Since the energies are split, the two spin components, |±⟩_z, travel at different velocities and so separate as they travel towards the sample. It can be shown [2.130] that the two components reach the sample with a time delay, t_SE, given by

$$t_{\mathrm{SE}}=\frac{\gamma\hbar}{mv^{3}}\int_{0}^{L}B_{\mathrm{s}}\mathrm{d}l\;,$$

(2.66)

where γ, m and v are the gyromagnetic ratio, mass, and velocity of the ³He atoms, respectively, and the integral is through the solenoid field of length L. In practice, t_SE is controlled by adjusting the current in the solenoid windings, and can range from a fraction of a picosecond to the nanosecond range.

After interacting with the surface, atoms scattered in a particular outgoing direction pass along the second beamline, through an identical but reversed solenoid field, where the separated components, |±⟩_z, are recombined. If the surface remains unchanged over t_SE, both spin components will scatter from the surface in exactly the same way, so the original polarized form, |+⟩_x, will be recovered. However, if the surface has changed, for example, if an atom has jumped between adsorption sites, the recovered spin direction will be changed, and, on average, there will be a reduction in the overall polarization of the beam.

The recovered spins then pass through an analyzer dipole and hexapole, which selects the x component of the polarization, and focuses that proportion of atoms into a mass-spectrometer detector. Quadrature methods, which involve adding an extra π ∕ 4, π ∕ 2, and 3π ∕ 2 of spin rotation near the first dipole field, enable the full polarization to be determined [2.141]. The measured polarization, P(ΔK , t_SE), is usually a good approximation to the normalized intermediate scattering function, I(ΔK , t), given by

$$P(\Updelta\boldsymbol{K},t_{\mathrm{SE}})=\frac{I(\Updelta\boldsymbol{K},t_{\mathrm{SE}})}{I(\Updelta\boldsymbol{K},0)}\;.$$

(2.67)

In other words, by measuring polarization, we make a direct measurement of the surface correlation over time t_SE and on the length scale and in the direction determined by the wavevector ΔK. Note that to measure certain properties, such as phonon linewidths, a more sophisticated wavelength transfer matrix approach is helpful [2.130, 2.142, 2.143], but this is beyond the scope of the present chapter.

The HeSE measurement principle is very general and thus can be applied to a wide variety of experiments, including measuring surface diffusion, surface and adsorbate vibrations, and for probing energy exchange rates. To date, experiments have usually been carried out on single crystal samples under ultra-high vacuum conditions. Samples are mounted on a six axis manipulator to enable ΔK to be aligned along any surface direction. The vacuum systems include standard surface preparation and characterization tools, as well as sample transfer for rapid-exchange into the measurement facility.

3.3 Measuring Diffusion with HeSE

Studying diffusion, or other surface properties, with HeSE involves interpreting the measured intermediate scattering function. Although, in principle, G(R , t) contains a complete description of all the surface dynamics and is related to I(ΔK , t_SE) by a spatial Fourier transform, it is usually not possible to obtain sufficient data to perform an inverse transform directly. Hence, HeSE analysis is generally based on examining changes in the shape of I(ΔK , t_SE) across a range of values of ΔK.

Figure 2.12 shows a cartoon of the typical form of I(ΔK , t_SE) for a measurement of an adsorbed species on a surface. The function takes a decaying form as the surface correlation, as seen by the helium beam over the length scale and direction determined by ΔK, reduces with time. Oscillations in I(ΔK , t_SE) relate to surface vibrations, such as phonons, and generally die away quickly at a rate that is characteristic of energy exchange with the substrate. The overall decay relates to aperiodic processes such as the dephasing of vibrations [2.144], where the characteristic time is usually less than a few picoseconds, or surface diffusion, where the characteristic time can be much longer. Surface diffusion generally results in an exponentially decaying form

$$I(\Updelta\boldsymbol{K},t_{\mathrm{SE}})=A\exp(-\alpha t)\;,$$

(2.68)

as correlations in the surface configuration, measured at a particular value of ΔK, decay with a well-defined rate α(ΔK), known as the dephasing rate. Different analytic models for surface transport, such as simple hopping, long jumping, or continuous Brownian motion, each give rise to a characteristic variation of α with ΔK. Hence, by measuring α(ΔK), the microscopic mechanism of motion can be determined. Once the mechanism is understood, parameters describing that motion can be obtained by fitting a suitable model to the empirical data, and from those model parameters, macroscopic quantities, such as diffusion constants or energy barriers, can be determined [2.130].

Fig. 2.12

The typical form of I(ΔK , t_SE) as measured by HeSE. The general form is a decaying function, as the surface correlation, as seen by the helium beam over the length scale and direction determined by ΔK reduces with spin echo time t_SE. Periodic surface processes, such as phonons, result in oscillations, while aperiodic processes such as diffusion generally result in an exponential decay

3.3.1 Jump Diffusion

Jump diffusion was first analyzed by Chudley and Elliott [2.145] in order to interpret quasielastic neutron scattering data. The same principles were applied to HeSE [2.146] and resulted in I(ΔK , t_SE) taking the general form in (2.68), with α(ΔK) varying sinusoidally,

$$\alpha(\Updelta\boldsymbol{K})=2\sum_{j}\nu_{j}\sin^{2}\left(\frac{\Updelta\boldsymbol{K}\cdot\boldsymbol{j}}{2}\right).$$

(2.69)

The summation is over all the possible adsorbate jumps j within the unit cell, and ν_j are the associated jump frequencies—in other words, each possible jump type contributes a sinusoidal component to the overall dependence. The tracer diffusion constant may then be estimated using [2.147]

$$D_{\Updelta K}=\frac{1}{2}\nu\left\langle j_{\Updelta K}^{2}\right\rangle\;,$$

where the parameters are projected along a particular ΔK direction.

As an example, Fig. 2.13a-c shows measured HeSE data for the jump diffusion of H and D atoms on a Pt(111) surface, from [2.148]. Here, the mechanism of motion almost conforms perfectly to a single jump model, in excellent agreement with the form in (2.69). More widely, measurements have established that show that most systems exhibit jump characteristics, although the precise mechanism often contains a distribution of different jump lengths, elements of continuous motion (e. g., intracell diffusive motion [2.144]), and more complex processes (see later). In the case of different jump lengths, these can be distinguished by fitting a series of Fourier components to the measured α(ΔK) dependence, one for each type of jump, using (2.69).

Fig. 2.13a-c

HeSE measurements (dephasing angle α versus momentum transfer ΔK) of H (brown circles) and D (black triangles) diffusion on a Pt(111) surface at three temperatures: (a) 220 K, (b) 140 K, and (c) 90 K. The measurements are in excellent agreement with a single jump model for hopping between adjacent fcc hollow sites (adapted with permission from [2.148], Copyright 2012 by the American Physical Society)

Other examples where good approximations to jump diffusion have been observed include alkali metals atoms on transition metals [2.149, 2.150, 2.151, 2.152] (although in many situations the motion can be further complicated by lateral interactions, leading to correlated motion) and many small molecules [2.153, 2.154, 2.155, 2.156, 2.157, 2.158, 2.159, 2.160]. Note that (2.69) applies to jumps between adsorption sites that form a Bravais lattice directly. For other networks of adsorption sites, such as the hexagonal lattice that can be formed from the two types of hollow sites on a (111) surface, a more complex model applies [2.161].

3.3.2 Continuous Brownian Motion

If the lateral variation in the potential between adsorbates and substrate is weak, or if the coupling between adsorbate and substrate is very strong, then rather than hopping, continuous Brownian motion can result, even on the atomic scale. Analysis of such motion [2.147] also results in I(ΔK , t_SE) decaying exponentially with t_SE, but this time, α varies quadratically with ΔK,

$$\alpha(\Updelta\boldsymbol{K})=D\Updelta\boldsymbol{K}^{2}\;,$$

(2.70)

scaled by the diffusion constant. Note that in the limit of large length scales (small ΔK), all microscopic mechanisms approach this form, including the limiting case of the previous jump model. An example of perfect Brownian motion at an atomic level is the case of benzene adsorbed on graphite [2.162], as shown in Fig. 2.14. The behavior is a result of the extended nature of the molecule averaging over multiple atomic interactions, combined with a high rate of energy exchange between the molecule and the substrate.

Fig. 2.14

HeSE measurements of the diffusion of benzene molecules on a graphite surface, showing continuous Brownian motion on the atomic scale. The solid line is a quadratic fit, as per (2.70), resulting in a diffusion constant of $D=5.39\pm 0.13\times{\mathrm{10^{-9}}}\,{\mathrm{m^{2}{\,}s^{-1}}}$. HeSE experiments (circles) were also shown to be in good agreement with neutron spin-echo measurements (triangles) performed at low coverages and Langevin MD simulations described in [2.162]

3.3.3 Activation Energies

At a particular ΔK, the dephasing rate, α, usually varies with temperature T according to the well-known Arrhenius law

$$\alpha=\alpha_{0}\exp\left(-\frac{E_{\mathrm{a}}}{k_{\mathrm{B}}T}\right),$$

(2.71)

where E_a is an activation energy for motion, and k_B is the Boltzmann constant. Hence, activation energies can be routinely determined from measurements of I(ΔK , t_SE) at different temperatures. However, it is important to approach such analysis with care. Depending on the mechanism of motion, values of I(ΔK , t_SE) at a particular value of ΔK may contain contributions from multiple processes, each with different temperature dependencies. For example, at the sinusoidal minima in Fig. 2.13a-c, at ΔK ≈ 2.5 Å⁻¹, measurements of I(ΔK , t_SE) contain a vanishing amount of information about the jump mechanism. Similarly, it is also possible for the mechanism of motion to change with temperature, further affecting the analysis. Figure 2.15 illustrates Arrhenius plots for ethanethiol adsorbed on Cu(111) [2.163]; at low temperatures the data is dominated by localized rotation of the molecule (see below), whereas as higher temperatures it is overtaken by translation. The mechanism in each regime was determined from the form of α(ΔK), which subsequently enabled the Arrhenius behavior to be understood and accurate activation energies to be extracted.

Diffusion measurements are often used to estimate the adiabatic potential energy barrier to diffusion, and it is possible to use HeSE data to estimate these barriers. The activation energy obtained from Arrhenius analysis alone usually underestimates the adiabatic barrier, and currently, the most accurate way to obtain the true barrier is to use Langevin molecular dynamics simulations to refine a trial potential, including adjustment of the adiabatic barrier heights, to best fit the experimental data (see below).

Fig. 2.15

Arrhenius plots of the dephasing rate, α, extracted from the rotational and translationally dominated measurement regimes for ethanethiol on Cu(111) (adapted with permission from [2.163], Copyright 2012 by the American Physical Society)

3.4 Beyond Diffusion Constants

Most real systems do not conform fully to the simple analytic models of diffusion discussed so far. For example, when alkali metals are adsorbed onto metal surfaces, electron transfer sets up a strong dipole. Consequently, there is strong dipole–dipole repulsion between adsorbed atoms and, thus, strong correlations in the motion result [2.149, 2.150, 2.151, 2.152]. In order to interpret such systems, more sophisticated methods are required and, in general, analytic expressions are insufficient. Simulations are routinely used to predict adsorbate motion given a specific model, from which simulated HeSE data can be produced and subsequently compared with experiment. Free parameters within the model can then be refined to best fit the experimental data. Molecular dynamics simulations that treat all the atoms in the substrate individually can be used but are computationally intensive. For computational speed, the Langevin framework has been used widely, within which only the adsorbed atoms are treated explicitly. The adsorbate–substrate interaction is described by a frozen adiabatic potential, V(R), and a frictional coupling parameter, η, is used to describe energy exchange. The equation of motion for the i-th adsorbate atom then becomes

$$m\boldsymbol{\ddot{R}}_{i}=-\nabla V(\boldsymbol{R}_{i})-\eta m\dot{\boldsymbol{R}}_{i}+\xi_{i}(t)+\sum_{j\neq i}F(|\boldsymbol{R}_{j}-\boldsymbol{R}_{i}|),$$

(2.72)

where the second and third terms are drag and stochastic excitation, respectively, describing interaction with the substrate heat bath in terms of η, which is scaled according to the fluctuation dissipation theorem [2.164]. The final term can be added to include pairwise interaction forces F between adsorbed atoms. The Langevin framework thus provides a much more sophisticated and general description of the dynamical adsorbate–substrate interaction than is possible using diffusion constants alone.

3.4.1 Interaction Potentials

In general, since adsorbate motion explores the entire surface, HeSE data enables the complete shape of the potential energy surface for diffusion to be determined, not just the rate-limiting adiabatic barriers. The Langevin molecular dynamics approach has thus been used to determine many experimental potentials [2.149, 2.150, 2.151, 2.152, 2.154, 2.155, 2.156, 2.157, 2.158, 2.159, 2.160, 2.165]. Usually such a determination can be performed to a level of better than 10 meV, depending on the complexity of the system and the quantity of HeSE data available.

Figure 2.16a,ba shows a typical experimental potential energy surface, in this case determined for CO molecules on Cu(111) [2.156]. The cross section in Fig. 2.16a,bb shows the potential along the high-symmetry directions within the unit cell. Various features are clearly evident, including the adsorption minima (top site), the overall maximum (hollow site), as well as intermediate minima and maxima, the latter which forms the principal rate-limiting barrier for diffusion. In this case, as the coverage of CO is increased, CO interactions were not found to be described by pairwise interactions, but to result in nonpairwise changes in the form of the potential, as shown by the dashed brown line in the figure. Despite being clearly observed, at present, the theoretical origin of this effect is not well understood.

As well as providing a detailed fundamental description of the surface interaction, such interaction potentials have been used as a sensitive test of the validity of first-principles theory for predicting potentials—recent advances in dispersion-corrected density functional theory have been tested [2.160]. Similarly, as more complex molecular systems are beginning to be studied using HeSE, it is becoming possible to explore how higher degrees of freedom can be included in the potential. For example, the coupling between diffusion and molecular orientation has been examined for pentacene molecules [2.166].

Fig. 2.16a,b

Experimental adsorbate–substrate potential energy surface determined for the center of mass motion of CO molecules on a Cu(111) surface (adapted from [2.156]). T, H, B, and TS represent the top site (adsorption site), hollow site (potential maximum), bridge site (local minimum), and the transition state (rate limiting maximum), respectively. (a) shows the lateral variation, while (b) shows cross sections along the directions indicated in (a). Increasing the CO coverage results in changes to the potential energy surface rather than localized interaction effects (adapted from [2.156], © IOP Publishing. Reproduced with permission. All rights reserved)

3.4.2 Energy Transfer and Rate Theory

Transition state theory (TST) can be used to estimate transition rates across a barrier (i. e., hopping, in the present context), by using the energy barrier and partition functions in both the transition and well states [2.5]. More sophisticated rate theory models [2.167] also take account of the rate of energy exchange between the substrate and adsorbate, parameterized by the friction, η. Friction has a significant effect on both the mechanism and rate of diffusion. At a simple level, if the friction is weak, the rate of activation to jumping is low, as it takes a long time for fluctuations to provide sufficient energy to overcome the limiting barrier. However, once activated, it also takes a long time for motion to deactivate, so adsorbates subsequently take long jumps over the surface. Conversely, if the friction is very high, adsorbates are turned around before they actually cross the barrier, and so the rate of jumping is also low, but with well-defined single jumps. In the intermediate regime, the highest rates of transport are achieved.

Through these effects on adsorbate dynamics the different frictional regimes can be identified clearly in HeSE data—using the combined effect on the mechanism and the absolute rate of motion. In general, low-friction regimes can be identified by the presence of long jumps, evident through flattened sinusoidal α(ΔK) curves containing many Fourier components (2.69), such as in the case of Cs∕Cu(111) [2.150]. High-friction regimes result in single jumps on a corrugated surface, as is the case for H on Pt(111) [2.148], or Brownian motion on an uncorrugated surface, which is the situation seen for benzene on graphite [2.162]. Langevin molecular dynamics simulations are, again, used to obtain quantitative values of η, and, thus, most systems where HeSE data has been analyzed in detail have provided precise values of the frictional energy exchange rate.

3.4.3 Adsorbate Interactions and Correlated Motion

Repulsive adsorbate interactions have been studied extensively using HeSE [2.149, 2.152, 2.154, 2.156]. In general, pairwise repulsion between adsorbates leads to the formation of a quasihexagonal overstructure to maximize their separation. These repulsive interactions are superimposed upon the usual adsorbate–substrate interaction potential and can at low coverages lead to a series of larger-scale structures forming. Thermal excitation still leads to motion, which is then determined by both the substrate potential and interactions between adsorbates. The effect on HeSE measurements is the introduction of a peak and dip structure superimposed on the α(ΔK) curve. The peak corresponds to jumping within the preferred hexagonal structure, as in (2.69). The dip corresponds to the preferred length scale of greatest stability, i. e., at the momentum transfer of the diffraction ring associated with the quasihexagonal structure.

In general, the form of such interactions can be tested by adjusting the form of the pairwise interaction, F(R) in (2.72), and comparing simulations with experiment as usual. For alkali metal atoms moving on transition metal substrates, good agreement has been found with a dipole–dipole repulsion model based on the Topping model for the coverage dependence of the dipole moment [2.149, 2.150, 2.151, 2.152]. Conversely, in other systems, it has been possible to rule out the presence of pairwise forces altogether, such as in the case of CO on Pt(111) [2.154]. Here, the signatures for pairwise repulsion between adsorbates were found to be completely absent from the experimental HeSE data. In the existing literature, pairwise forces have been widely invoked to explain changes in the heat of adsorption. The HeSE dynamics data leads to the conclusion that changes in the CO dynamics must be due to more complex longer-scale mean-field changes in the CO∕substrate interaction, which cannot be described in a pairwise manner. Such effects cannot be observed with simpler techniques and require direct observation of the dynamical correlations in order to be distinguished.

4 Perspectives—Towards Complex Surface Motion

The examples above have illustrated the rich variety of diffusion information that can be accessed experimentally, which motivate new theory. We can clearly see that real systems exhibit complex diffusive behavior, which is related to both the substrate and to the mobile species. Looking forwards, we can highlight a few representative examples of areas where HeSE methods are just beginning to be applied and where further new and interesting phenomena can be expected.

One of the most obvious issues is the fact that real surfaces are much more complex than the flat surfaces generally studied so far. Complex surface structure can modulate and confine diffusion, fundamentally altering the nature of transport at the atomic scale. Signatures of confined diffusion in HeSE measurements have been established [2.130] and essentially involve I(ΔK , t_SE) decaying to a finite level at large t_SE—since confinement means the adsorbate always retains some correlation in its position. However, to date, there have been few studies of any form of confined diffusion. Existing work either relates to motion perpendicular to the surface during lateral diffusion [2.149] or diffusion that is confined to narrow stepped terraces of atoms [2.152], where highly anisotropic motion has been observed. In the latter case, anisotropy in the correlations between atoms has also been seen, indicating effective screening of interactions between adsorbates on different terraces.

As molecules become more structurally complex, it becomes crucial to distinguish between many possible dynamical processes. These could include, for example, changes in conformation or motion of a side chain, as well as center of mass motion. To date, HeSE has generally been applied to point particles or rigid molecules with rotation. Specifically, analytic forms of the HeSE signatures for rotational motion have been identified [2.168] and used to distinguish ongoing rotational motion of adsorbed thiols from the onset of translational motion [2.163]. Similarly, jumping, rotating, and flapping motion has been distinguished for thiophene on Cu(111) [2.158], and the contribution of rotational modes in boosting the overall rate of diffusion has been established [2.159]. The next steps are likely to involve decoupling and, thus, an understanding of nonrigid species.

So far, we have only discussed classical mechanisms of diffusion. However, for sufficiently small species and low temperatures, quantum processes can be dominant. The transition towards a quantum diffusion-dominated regime has been observed for H and D atoms on Pt(111) [2.148] and more extensively on Ru(0001) [2.169]. As well as the direct technological relevance of such results to the hydrogen economy, HeSE also provides an ideal opportunity for fundamental testing of quantum-rate theory models, and in particular models for the transition between coherent and incoherent modes of propagation.

All of these areas suggest that a much greater wealth of fundamental information on diffusion is available, which can be accessed experimentally through HeSE, given the method's unique sensitivity to dynamics on the combination of picosecond time and nanometer length scales. Further development of HeSE instrumentation will be important in measuring these increasingly complex forms of motion—for example, to enable better separation of processes by timescales. Fortunately, further increases in resolution and signal level are both realistic prospects [2.170]. We look forward to a next generation of spin-echo measurement facility, and subsequent uptake by the user community, to take forward our scientific understanding in this exciting and unique regime.