**5. Theory of the photo-induced surface diffusion versus experimental findings**

Let us compare the kinetics of the surface density of adsorbed atoms measured in the experiments (Fig. 2) with the results of theoretical analysis. If the process of adsorption from the gas phase, whose rate is equal to that of the thermal desorption τ, is neglected, then the first integral in Eq. (22), corresponding to the photodesorption process, is reduced to the simple square-root dependence

$$\left(1 + 4t/T\right)^{-1/2} - 1\tag{23}$$

infinitely growing with time, which contradicts the experimental data. Here,

$$T = \left(a^2 + b^2\right) \Big/ D\_0 \tag{24}$$

is the characteristic time of the signal variation, related only to the diffusion processes. Under these conditions, the second integral in Eq. (22), corresponding to the photodiffusion process, is also simplified and yields a time dependence gradually approaching the steadystate value,

$$1 - \left[1 + 4t/T\right] \tag{25}$$

Equation (25) adequately describes the experimental nonexponential kinetics of variation of the surface density of the adsorbed particles, excluding long times, when the effect of adsorption from the gas phase becomes noticeable. In Fig. 1, the results of the measurements for the laser beam diameter of 110 μm are compared with the theoretical curve constructed by using the exact formula (22). By comparing the theoretical and experimental dependences, we can find the characteristic diffusion time T. Fig. 2 shows the dependence of T on the pump beam diameter, which, according to (23), should be quadratic. The best agreement with the experimental data is obtained for *D*0=7×10–6 cm2s–1. If we assume that

2 0

0 0

γ

(1 4 / ) ( )

*Dt a*

∫

**5. Theory of the photo-induced surface diffusion versus experimental** 

infinitely growing with time, which contradicts the experimental data. Here,

2 2

By substituting the solution (15) found above into Eq. (21), we have

π

equilibrium. This condition is well satisfied in the experiment.

0

*S t n ab*

**findings** 

state value,

simple square-root dependence

2 2

2 0

exp( / ) 2( (0) / ) (1 4 / )

exp( / )

τ

*a a u D du D ub a*

<sup>⎧</sup> <sup>⎫</sup> <sup>−</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> + + <sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> = − <sup>⎨</sup> <sup>⎬</sup> <sup>⎪</sup> <sup>−</sup> <sup>⎪</sup> ⎪+ <sup>⎪</sup> ⎪⎩ ⎭ + + <sup>⎪</sup>

Equation (22) describes the process that occurs after the pump beam is turned on. After the pump beam is turned off, the surface density recovers to its equilibrium value. In the approximations adopted here, the kinetics of recovery of the equilibrium concentration does not differ from that of its perturbation, because the rates of the photoinduced processes are assumed to be low compared with those of the processes leading to recovery of the

Let us compare the kinetics of the surface density of adsorbed atoms measured in the experiments (Fig. 2) with the results of theoretical analysis. If the process of adsorption from the gas phase, whose rate is equal to that of the thermal desorption τ, is neglected, then the first integral in Eq. (22), corresponding to the photodesorption process, is reduced to the

is the characteristic time of the signal variation, related only to the diffusion processes. Under these conditions, the second integral in Eq. (22), corresponding to the photodiffusion process, is also simplified and yields a time dependence gradually approaching the steady-

Equation (25) adequately describes the experimental nonexponential kinetics of variation of the surface density of the adsorbed particles, excluding long times, when the effect of adsorption from the gas phase becomes noticeable. In Fig. 1, the results of the measurements for the laser beam diameter of 110 μm are compared with the theoretical curve constructed by using the exact formula (22). By comparing the theoretical and experimental dependences, we can find the characteristic diffusion time T. Fig. 2 shows the dependence of T on the pump beam diameter, which, according to (23), should be quadratic. The best agreement with the experimental data is obtained for *D*0=7×10–6 cm2s–1. If we assume that

0

∫

*Dt a*

\* 0 2 2 3/2

*a u D du D D*

<sup>0</sup> *S t n p t x y e x a dxdy* () (, , )( / ) <sup>=</sup> <sup>−</sup> ∫∫ (21)

2

τ

*ub a*

( ) 1 2 14 1 *t T* <sup>−</sup> <sup>+</sup> <sup>−</sup> (23)

( ) 2 2 *TabD* = + 0 (24)

1 14 − + [ *t T*] (25)

0

(22)

0 2 2 1/2 the diffusion is of activation nature, with the length of an elementary hop being equal to a few tenths of a nanometer and with attempts to overcome the barrier occurring at the frequency of vibrational motion of the adsorbed atom, i.e., every several tenths of a picosecond, then the activation barrier for the diffusion can be estimated to be about 0.1 eV. This value agrees with the empirical "one sixth" rule (George et al., 1985) for the ratio of diffusion barrier height to adsorption energy. Since the photodiffusion coefficient *D\**(0) is proportional to the pump intensity *I*(0), it may be naturally written in the form *D\**(0) = κ*I*(0), where the value of κ equals, according to the results of the measurements, 2×10–10 cm4J–1. If we compare this value with the absorption cross section for the adsorbed atoms, which controls the frequency of the photoinduced hops, we come to the conclusion that the mean square of displacement of an adsorbed atom per absorbed photon is of the order of 100 nm2. The above findings mean, first, that the photoinduced diffusion does not require overcoming any energy barrier and, second, that the rate of relaxation of the excess atomic energy is rather low (of the order of 1010–1011 s–1). The first assertion agrees fairly well with the mechanism of the photoinduced diffusion discussed above, because the effective temperature of an adsorbed atom that has absorbed a photon is comparable with the value of the diffusion barrier. The low rate of the energy relaxation of an adsorbed atom upon its interaction with the surface of sapphire was already noted by us previously, both in measuring the accommodation coefficients of atoms upon their impact on the surface (Bonch-Bruevich et al., 1990) and in studying the quasi-thermal regime of photodesorption of sodium atoms from the sapphire surface (Bonch-Bruevich et al., 1999), where the energy relaxation rate was also estimated to be of the order of 1010 s–1. The role of long free paths in surface diffusion, for other mechanisms of transfer of excess energy to an adsorbed atom, has been discussed in the literature (Brune et al., 1992; Tully et al., 1979).

### **6. Novel aspects of the photoinduced surface transport**

As a result of these studies, we have found a new light-induced effect in the processes of transfer on a solid surface—the effect of surface photodiffusion. The physical basis of the effect is the conversion of the energy of radiation into the energy of motion of an adsorbed atom over a surface. At the first stage, the system of the atom passes into an excited state. Then, the electronic excitation is quenched and its energy is distributed over the vibrational degrees of freedom of the substrate and the adsorbed atom. A fraction of adsorbed atoms acquire an energy sufficient to overcome the diffusion barrier and gain the probability to shift to another adsorption site. The frequency of the light-induced hops is proportional to the surface illuminance, while the length of the hops measures tens of the substrate lattice constants. Photoinduced desorption insignificantly affects the photoinduced diffusion, because the desorption is associated with overcoming of a much higher energy barrier, equal to the surface adsorption energy of the atom, and thus its probability is much lower.

It is shown that, under nonuniform illumination of a surface, the adsorbed atoms move away from the illuminated area and, on the boundary between the light and the shadow, the density of the adsorbed atoms exceeds the equilibrium density. Thus, we have discovered and studied a new phenomenon, which makes it possible to efficiently control, by light, the diffusion processes on a surface. Unlike the known process of photoinduced desorption, which removes adsorbed particles from the surface, in the process of photoinduced diffusion, the total number of adsorbed particles on the surface does not change. An important feature of photodiffusion is the possibility to control, by means of light, the

Light-Induced Surface Diffusion 427

time at T= 553 K. A certain decrease in the area under the curves of the extinction spectra (this quantity characterizes the total amount of substance in the film) is connected, as shown by numerical estimates, with the type of the frequency dispersion (differing from that of

a)

b)

Fig. 5 shows the transmission electron microscopy images of (a) an as-grown film and (b) of the film annealed during 30 minutes at T= 553 K. One can see that agglomerates formed on the as-grown film (except for the smallest ones) are diverse in shape. As a result, the

Fig. 4. Annealing kinetics of the granular silver films (a) 1 – as prepared, 2 to 4 - after annealing at 473 K for 8, 16 and 56 minutes. (b) 1 – as prepared, 2 to 5 - after annealing at

553 K for 2, 4, 8 and 20 minutes

Drude–Lorenz) of the silver complex permittivity in the region of 2–4 eV.

density of adsorbed particles on a surface (i.e., both to increase and to decrease it). Since the processes of origination of a new phase on a surface essentially depend on diffusion processes the use of photodiffusion is promising for controlling processes of nucleation and growth of surface nanostructures.
