**2. Nature of the photo-induced surface diffusion process**

The effect of surface photodiffusion is closely related to the well-known process of photodesorption, because both processes begin with the resonance absorption of a light quantum by an atom adsorbed on the surface of a transparent substrate. From the theoretical point of view, the existence of photostimulated diffusion is justified by analysis of the previously established mechanism of photodesorption of resonantly absorbing atoms adsorbed on a transparent substrate (Bonch-Bruevich et al., 1990). Upon the resonant optical excitation of the electronic subsystem of an adsorbed atom, the energy of the photon is converted, rapidly and virtually completely, into vibrational degrees of freedom of the adsorbed atom and its nearest environment on the surface. Only in rare cases is the portion of energy gained by the adsorbed atom sufficient for it to overcome the adsorption potential and be desorbed. It is much more likely that this energy is insufficient for the atom to be desorbed. On the other hand, if we take into account that the activation energy of the surface diffusion is several times smaller than the adsorption energy, it becomes evident that photoexcitation should substantially affect the frequency of hopping of adsorbed atoms between neighboring minima of the adsorption potential even at low intensities of illumination, when the photodesorption rate is still not sufficient to significantly change the surface density of adsorbed atoms. Under nonuniform illumination, this gives rise to a surface flow of particles, which, in the first order in the light intensity, is proportional to the irradiance gradient. Note that photodiffusion does not remove the adsorbed atoms from the surface but rather redistributes them, so that a decrease in the surface density in some places is accompanied by an increase in the density of adsorbed atoms in other locations.

### **3. Case study: Cesium on sapphire**

semiconductor lasers.

As the object of study, a system of cesium atoms adsorbed on the surface of a single-crystal sapphire was used. This system has been thoroughly studied previously (Bonch-Bruevich et al., 1985, 1990, 1997). The advantages of this system are the chemical inertness of sapphire and its transparency in a wide spectral range, covering the entire visible and near-IR regions, in which the adsorbed cesium atoms exhibit strong optical absorption. This allows one, using convenient laser sources, to selectively excite cesium atoms leaving the substrate unexcited. Saturated vapors of cesium atoms were placed into a sealed glass cell with sapphire windows. The volume density of the atoms at room temperature was 1010 cm–3. The experiments were performed under conditions of dynamic equilibrium between the volume phase and the phase adsorbed on the surface. According to (Bonch-Bruevich et al., 1985, 1997) the energy of adsorption of the atoms on the sapphire surface is 0.6 eV, which provides, at room temperature, a surface density of the adsorbed atoms n0= 1013cm–2. Thus, the surface density of the adsorbed atoms appears to be fairly high, though still much lower than a monolayer. Variations of the surface density of the adsorbed atoms induced by high-power radiation were detected by measuring the transmission of a relatively weak probe light of a cw semiconductor laser. Using the value of the absorption cross section of the adsorbed atom (Bonch-Bruevich et al., 1985, 1997) σ=3×10–16cm2, we find that the absorption in the adsorbed layer makes up n0σ =3×10–3 of the total intensity of the light passing through the layer. Thus, absorption of this kind can be reliably measured using low-noise high-intensity

The effect of surface photodiffusion is closely related to the well-known process of photodesorption, because both processes begin with the resonance absorption of a light quantum by an atom adsorbed on the surface of a transparent substrate. From the theoretical point of view, the existence of photostimulated diffusion is justified by analysis of the previously established mechanism of photodesorption of resonantly absorbing atoms adsorbed on a transparent substrate (Bonch-Bruevich et al., 1990). Upon the resonant optical excitation of the electronic subsystem of an adsorbed atom, the energy of the photon is converted, rapidly and virtually completely, into vibrational degrees of freedom of the adsorbed atom and its nearest environment on the surface. Only in rare cases is the portion of energy gained by the adsorbed atom sufficient for it to overcome the adsorption potential and be desorbed. It is much more likely that this energy is insufficient for the atom to be desorbed. On the other hand, if we take into account that the activation energy of the surface diffusion is several times smaller than the adsorption energy, it becomes evident that photoexcitation should substantially affect the frequency of hopping of adsorbed atoms between neighboring minima of the adsorption potential even at low intensities of illumination, when the photodesorption rate is still not sufficient to significantly change the surface density of adsorbed atoms. Under nonuniform illumination, this gives rise to a surface flow of particles, which, in the first order in the light intensity, is proportional to the irradiance gradient. Note that photodiffusion does not remove the adsorbed atoms from the surface but rather redistributes them, so that a decrease in the surface density in some places is accompanied by an increase in the density of adsorbed atoms in other

As the object of study, a system of cesium atoms adsorbed on the surface of a single-crystal sapphire was used. This system has been thoroughly studied previously (Bonch-Bruevich et al., 1985, 1990, 1997). The advantages of this system are the chemical inertness of sapphire and its transparency in a wide spectral range, covering the entire visible and near-IR regions, in which the adsorbed cesium atoms exhibit strong optical absorption. This allows one, using convenient laser sources, to selectively excite cesium atoms leaving the substrate unexcited. Saturated vapors of cesium atoms were placed into a sealed glass cell with sapphire windows. The volume density of the atoms at room temperature was 1010 cm–3. The experiments were performed under conditions of dynamic equilibrium between the volume phase and the phase adsorbed on the surface. According to (Bonch-Bruevich et al., 1985, 1997) the energy of adsorption of the atoms on the sapphire surface is 0.6 eV, which provides, at room temperature, a surface density of the adsorbed atoms n0= 1013cm–2. Thus, the surface density of the adsorbed atoms appears to be fairly high, though still much lower than a monolayer. Variations of the surface density of the adsorbed atoms induced by high-power radiation were detected by measuring the transmission of a relatively weak probe light of a cw semiconductor laser. Using the value of the absorption cross section of the adsorbed atom (Bonch-Bruevich et al., 1985, 1997) σ=3×10–16cm2, we find that the absorption in the adsorbed layer makes up n0σ =3×10–3 of the total intensity of the light passing through the layer. Thus, absorption of this kind can be reliably measured using low-noise high-intensity

**2. Nature of the photo-induced surface diffusion process** 

locations.

semiconductor lasers.

**3. Case study: Cesium on sapphire** 

Among the important advantages of the chosen system is a relatively low quantum yield of the photodesorption process. According to (Bonch-Bruevich et al., 1985, 1997), it is as low as 10−5. This allowed us to hope that the main mechanism for light-induced changes in the surface density of the atoms would be the photoinduced diffusion rather than photodesorption. As shown below, these expectations were fulfilled. Since the excitation of an adsorption system by high-power optical radiation induces both the photodesorption of atoms from the surface and their photodiffusion over the surface and, in addition, is accompanied by dark desorption and diffusion, we performed experiments of two types.

In the experiments of the first type, we studied the processes of photodesorption and dark desorption. For this purpose, the cesium atoms adsorbed on the sapphire surface were desorbed by single pulses of a ruby laser generating at a wavelength of 694 nm. The diameter of the irradiated spot varied from 1 to 4 mm. After the end of the strong desorbing pulse, the surface density of the adsorbed atoms gradually recovered. The kinetics of the recovery of the surface density was detected by measuring the intensity of a semiconductor laser beam transmitted through the irradiated area of the sapphire window of the cell. The intensity of the transmitted light at the wavelength 840 nm was measured by an FD-7K photodiode and was monitored by an S8-17 storage oscilloscope. At room temperature, the characteristic time of recovery of the surface density was 25 s. It was verified that this time did not depend on the radius of the irradiated spot within the limits indicated above. This allowed us to assign it exclusively to the process of deposition of atoms from the gas phase, rather than to their surface diffusion from dark regions, because in the latter case the recovery time would be dependent on the radius of the irradiated spot. From these experimental data, we determined the rate of thermal desorption to be τ=0.04 s–1.

The photostimulated diffusion was studied by irradiating the surface with a cw beam of an argon laser at the wavelength 514.5 nm and a power of 1 W. The focused probe beam of a semiconductor laser (20×30 μm2 in size) was scanned in such a way that it could repeatedly pass both regions of the surface subjected and regions not subjected to the optical treatment. The scanning was implemented using a lightweight mirror glued to the diaphragm of a high-power loudspeaker. The loudspeaker was fed by a sine voltage from a low frequency oscillator. The scanning range of the probe beam exceeded by approximately an order of magnitude the diameter of the exciting beam, which varied within the range 30–130 μm. After passing through the cell, the probe beam was detected by a photodiode. The signal thus obtained was amplified by a selective amplifier at a scanning frequency of 30 Hz and was detected by a lock-in amplifier with an XY recorder at the output. The time resolution of the setup was 0.5 s.

Because of different absorption in regions with higher and lower surface densities of adsorbed atoms, the intensity of the transmitted beam was modulated at the beam scanning frequency. Under the conditions described above, when the changes in the transmittance are concentrated within a region that is small compared to the whole scanning length, the amplitude of the first modulation harmonic in the transmitted beam, detected in the experiment, is proportional to the integral of variation of the surface density of the adsorbed atoms along the beam scanning path. In the experiment, we measured the kinetics of changes in the surface density of atoms after the argon laser beam was rapidly switched on and off. It was found that, as in the first experiments, for large spots of the exciting beam (~0.5 mm), this kinetics was exponential, with a rise and decay time of 25 s, and did not depend on the exciting beam power. It was verified that the amplitudes of the observed

Light-Induced Surface Diffusion 419

than the characteristic time for establishment of equilibrium with the volume phase, measured in the previous experiment. Since this time, as before, did not depend on the beam power but decreased quadratically with a decrease in the diameter of the pump beam from 130 to 30 μm (Fig. 2), we concluded that the time of transition to the steady state was governed by the rate of dark diffusion of the atoms over the surface. In addition, it can be concluded that the main contribution to the decrease in the surface density of atoms is made by photodiffusion rather than by photodesorption. Indeed, if the decrease in the surface density were related to the photodesorption, such a decrease of the density would occur until the surface was totally depleted. Moreover, recovery of the steady-state value of the surface density after the exciting radiation has been turned off could occur only from the gas

From the fact that the times of transition to the steady-state value of the surface density of atoms when the pump beam was turned on and off were approximately the same, while the signal amplitudes varied linearly with the pump beam intensity, we concluded that the photodiffusion process is substantially slower than the dark diffusion and can be described

To derive the equation describing photodiffusion, we will use the standard scheme of the diffusion approximation (Lifshitz & Pitaevskii 1981). Let *n(r)* be the surface density of particles at the point *r= (x,y)* and *ν(r,a)* be the probability of hopping of an atom from the point r to the point *r–a*. Then the change in the surface density of the particles at the point r is determined by the difference between the numbers of particles arriving at this point from

Because the photoinduced inhomogeneities of the surface density and the photoinduced changes in the hopping frequency are characterized by a spatial scale of the order of the light wavelength, while the length of the hops of the adsorbed atoms is much smaller than this value, the integrand can be represented as a series expansion, using the hop length as a small parameter. By retaining the first two terms in the expansion, we may rewrite (1) in the

Since the surface is, by itself, homogeneous and isotropic and its irradiation affects only the hopping rate and does not break this isotropy, among all the quantities presented above, only the *Bxx* and *Byy*, equal to each other, are nonzero. Their common value may be naturally

{ } <sup>2</sup> ∂ ∂= + + *nt t* ( , ) / v( , ) ( , )-v( , ) ( , ) *n t nt d a* ∫ **r r r rr** *aa a a* (1)

/ (A ) (B ) *nt n n jj j <sup>l</sup> jl* ∂ ∂ =∂ +∂ ∂ , (2)

<sup>2</sup> A ( ) v( , )d *j j* <sup>=</sup> *<sup>a</sup>* <sup>∫</sup> **r r** *a a* , <sup>1</sup> <sup>2</sup> B ( ) v( , )d <sup>2</sup> *jl j l* <sup>=</sup> *a a* <sup>∫</sup> **r r** *a a* (3)

**4. Theoretical description of the photo-induced surface transport** 

and the sum over *j* and *l* that attain the values of *x* and *y* is assumed.

phase with a characteristic time of 25 s.

neighboring positions and leaving this point

in the linear approximation..

form

where

denoted by *D(r)*,

Fig. 1. Decrease of the surface concentration under illumination

signals varied linearly with the exciting beam power, while the changes in the surface density of the atoms did not exceed 20% of the equilibrium value. Upon a decrease in the diameter of the pump beam to 110 μm, the kinetics of the variation of the surface density of atoms changed sharply (Fig. 1). It is seen that, for such a small size of the irradiated area, the steady-state surface density of the particles is established during times substantially shorter

Fig. 2. Dependence of the transient time on the focal spot diameter

**time, s** 

signals varied linearly with the exciting beam power, while the changes in the surface density of the atoms did not exceed 20% of the equilibrium value. Upon a decrease in the diameter of the pump beam to 110 μm, the kinetics of the variation of the surface density of atoms changed sharply (Fig. 1). It is seen that, for such a small size of the irradiated area, the steady-state surface density of the particles is established during times substantially shorter

**Focal sport diameter,** μ**m** 

Fig. 2. Dependence of the transient time on the focal spot diameter

Fig. 1. Decrease of the surface concentration under illumination

than the characteristic time for establishment of equilibrium with the volume phase, measured in the previous experiment. Since this time, as before, did not depend on the beam power but decreased quadratically with a decrease in the diameter of the pump beam from 130 to 30 μm (Fig. 2), we concluded that the time of transition to the steady state was governed by the rate of dark diffusion of the atoms over the surface. In addition, it can be concluded that the main contribution to the decrease in the surface density of atoms is made by photodiffusion rather than by photodesorption. Indeed, if the decrease in the surface density were related to the photodesorption, such a decrease of the density would occur until the surface was totally depleted. Moreover, recovery of the steady-state value of the surface density after the exciting radiation has been turned off could occur only from the gas phase with a characteristic time of 25 s.

From the fact that the times of transition to the steady-state value of the surface density of atoms when the pump beam was turned on and off were approximately the same, while the signal amplitudes varied linearly with the pump beam intensity, we concluded that the photodiffusion process is substantially slower than the dark diffusion and can be described in the linear approximation..

#### **4. Theoretical description of the photo-induced surface transport**

To derive the equation describing photodiffusion, we will use the standard scheme of the diffusion approximation (Lifshitz & Pitaevskii 1981). Let *n(r)* be the surface density of particles at the point *r= (x,y)* and *ν(r,a)* be the probability of hopping of an atom from the point r to the point *r–a*. Then the change in the surface density of the particles at the point r is determined by the difference between the numbers of particles arriving at this point from neighboring positions and leaving this point

$$\left\{ \left. \partial n(t, \mathbf{r}) \right\} / \left. \partial t \right| = \int \left( \mathbf{v}(\mathbf{r} + \mathbf{a}, \mathbf{a}) n(t, \mathbf{r} + \mathbf{a}) \cdot \mathbf{v}(\mathbf{r}, \mathbf{a}) n(t, \mathbf{r}) \right) d^2 a \tag{1}$$

Because the photoinduced inhomogeneities of the surface density and the photoinduced changes in the hopping frequency are characterized by a spatial scale of the order of the light wavelength, while the length of the hops of the adsorbed atoms is much smaller than this value, the integrand can be represented as a series expansion, using the hop length as a small parameter. By retaining the first two terms in the expansion, we may rewrite (1) in the form

$$
\partial \mathfrak{n} / \partial \mathfrak{t} = \mathfrak{d}\_j(\mathbf{A}\_j \mathfrak{n}) + \mathfrak{d}\_j \mathfrak{d}\_l(\mathbf{B}\_{jl} \mathfrak{n}) \,. \tag{2}
$$

where

$$\mathbf{A}\_{j}(\mathbf{r}) = \int a\_{j} \mathbf{v}(\mathbf{r}, \mathbf{a}) \mathbf{d}^{2} \mathbf{a} \; \prime \; B\_{jl}(\mathbf{r}) = \frac{1}{2} \int a\_{j} a\_{l} \mathbf{v}(\mathbf{r}, \mathbf{a}) \mathbf{d}^{2} \mathbf{a} \tag{3}$$

and the sum over *j* and *l* that attain the values of *x* and *y* is assumed. Since the surface is, by itself, homogeneous and isotropic and its irradiation affects only the hopping rate and does not break this isotropy, among all the quantities presented above, only the *Bxx* and *Byy*, equal to each other, are nonzero. Their common value may be naturally denoted by *D(r)*,

Light-Induced Surface Diffusion 421

inevitably accompanies photodiffusion, as well as the usual thermal desorption and deposition of atoms from the gas phase onto the surface. These processes will be taken into account in the next section. This will allow us to unambiguously separate out the

Under conditions of direct contact between the surface and the gas phase, Eq. (6) should include terms describing the deposition of atoms onto the surface from the gas phase and their thermal desorption from the surface. In addition, the right-hand side of Eq. (6) should be complemented with terms describing the photodesorption, whose rate, along with the photodiffusion coefficient, is proportional to the illuminance. The most complete equation

∂*n t Drnr rn J n* ∂ =Δ −Γ + − [ ()() () ]

*n J* <sup>0</sup> = τ

Let us separate out the relative change in the surface density *p* due to illumination according

Assuming p to be small compared with unity, we may neglect the terms containing the product of *p* and the rates of the photoinduced processes. Then Eq. (7) will acquire the form

> <sup>0</sup> *p t D p p D rn* τ

( ) [ ] ( ) <sup>0</sup> *q tr ptr D r D t* , (, ) exp( )

*f* (*tr D r D r t* , e ) =⎡ −Γ ⎤

Equation (12) may be solved using conventional methods (Vladimirov, 1971) for arbitrary initial conditions and an arbitrary form of the right-hand side, which is determined by the

<sup>∗</sup> ( ) <sup>0</sup> ( ) xp(

we transform Eq. (10) into a standard form of the inhomogeneous diffusion equation

τ

The entire subsequent theoretical analysis is based on Eq. (10).

**4.2 Surface density wells and humps due to photodiffusion** 

By introducing the new sought-for function

distribution of the illumination intensity.

where J is the flow of atoms to the surface from the gas phase, τ is the thermal desorption rate, and Γ(r) is the rate of photodesorption. In the absence of light, the surface density is the

τ

, (7)

(8)

*n n* = <sup>0</sup> (1 + *p*) (9)

( ) <sup>∗</sup> ∂ ∂ − Δ + =Δ −Γ (10)

τ=+ ⎤ <sup>∗</sup> ⎦ (11)

τ) ⎣ ⎦ (13)

∂ ∂− Δ = *q t D*<sup>0</sup> *q f* (*t r*, ) (12)

contribution of photodiffusion to the experimentally observed phenomena.

**4.1 Influence of the surface - Gas phase exchange** 

may be written in the form

same at any point and is equal to

to the formula

where

$$\mathbf{B}\_{\text{xx}}(\mathbf{r}) = \mathbf{B}\_{\text{yy}}(\mathbf{r}) = D(\mathbf{r}) = \frac{1}{2} \int a\_j a\_l \mathbf{v}(\mathbf{r}, a) \mathbf{d}^2 a \tag{4}$$

since, in the absence of irradiation, D(r) is reduced to *D*0, i.e., to the coefficient of the dark surface diffusion, which does not depend on the surface coordinates.

According to Eq. (4), the diffusion coefficient can be interpreted as a product of the hop frequency and the mean square of the displacement per hop. In the absence of irradiation, the frequency of hops can be estimated as a product of the vibration frequency of an adsorbed atom and the Boltzmann factor, which controls the probability of the atom overcoming the diffusion barrier at a given temperature of the surface. In this case, the displacement per hop coincides, in order of magnitude, with the distance between neighboring minima of the adsorption potential. In the case of photostimulated diffusion, the frequency of hops is determined by the probability of a photon being absorbed by an adsorbed atom. The displacement per hop may be, generally speaking, larger than in conventional diffusion, because the energy received by an adsorbed atom upon absorption of a photon substantially exceeds the mean thermal energy and, probably, the height of the diffusion barrier. Note that, in the approximation linear with respect to the pump power, the frequency of the photoinduced hops is directly proportional to the intensity, while their length is intensity-independent.

By separating, in Eq. (4), the contributions of thermal and photostimulated hops, we may represent D(r) in the form

$$D(r) = D\_0 + D\_\star(r) \tag{5}$$

where D\*(r) is the photodiffusion coefficient, proportional to the intensity of irradiation at point r. With these notations, Eq. (2) becomes

$$
\hat{\boldsymbol{\alpha}} \mathbf{u} / \hat{\boldsymbol{\alpha}} \mathbf{t} = \boldsymbol{\Delta} \left[ \boldsymbol{D}(\boldsymbol{r}) \mathbf{u}(\boldsymbol{r}) \right] \tag{6}
$$

where Δ is the two-dimensional Laplace operator. Equation (6) allows us to describe the photodiffusion effect completely. In the next section, we will complement its right-hand side with the terms necessary to describe a number of concomitant processes important under our experimental conditions. The right-hand side of Eq. (6) differs from the conventional form of the diffusion equation with the coordinate-dependent diffusion coefficient div[D(r)gradn(r)] by the additional term div[n(r)gradD(r)]. This important distinction results from the fact that the conventional diffusion equation describes the transition to equilibrium, whereas Eq. (6) describes the transition to an established but not completely equilibrium state, caused by the nonuniform irradiation of the surface. Under these conditions, the diffusion flow is a sum of two terms, namely, the conventional contribution, proportional to the density gradient, and the photoinduced contribution, proportional to the illuminance gradient.

Various forms of the diffusion equation were discussed in (van Kampen, 1992; Zangwill 1988). An expression for the right-hand side of the diffusion equation having a similar structure is obtained for the case of thermodiffusion (Lifshitz & Pitaevskii, 1981), where the incompleteness of the equilibrium results from a specified inhomogeneity in the temperature distribution of the gas mixture.

Equation (6) fully describes the photodiffusion effect. However, to experimentally prove its existence, the theoretical model should be extended to include photodesorption, which

since, in the absence of irradiation, D(r) is reduced to *D*0, i.e., to the coefficient of the dark

According to Eq. (4), the diffusion coefficient can be interpreted as a product of the hop frequency and the mean square of the displacement per hop. In the absence of irradiation, the frequency of hops can be estimated as a product of the vibration frequency of an adsorbed atom and the Boltzmann factor, which controls the probability of the atom overcoming the diffusion barrier at a given temperature of the surface. In this case, the displacement per hop coincides, in order of magnitude, with the distance between neighboring minima of the adsorption potential. In the case of photostimulated diffusion, the frequency of hops is determined by the probability of a photon being absorbed by an adsorbed atom. The displacement per hop may be, generally speaking, larger than in conventional diffusion, because the energy received by an adsorbed atom upon absorption of a photon substantially exceeds the mean thermal energy and, probably, the height of the diffusion barrier. Note that, in the approximation linear with respect to the pump power, the frequency of the photoinduced hops is directly proportional to the intensity, while their

By separating, in Eq. (4), the contributions of thermal and photostimulated hops, we may

where D\*(r) is the photodiffusion coefficient, proportional to the intensity of irradiation at

where Δ is the two-dimensional Laplace operator. Equation (6) allows us to describe the photodiffusion effect completely. In the next section, we will complement its right-hand side with the terms necessary to describe a number of concomitant processes important under our experimental conditions. The right-hand side of Eq. (6) differs from the conventional form of the diffusion equation with the coordinate-dependent diffusion coefficient div[D(r)gradn(r)] by the additional term div[n(r)gradD(r)]. This important distinction results from the fact that the conventional diffusion equation describes the transition to equilibrium, whereas Eq. (6) describes the transition to an established but not completely equilibrium state, caused by the nonuniform irradiation of the surface. Under these conditions, the diffusion flow is a sum of two terms, namely, the conventional contribution, proportional to the density gradient, and the photoinduced contribution, proportional to the

Various forms of the diffusion equation were discussed in (van Kampen, 1992; Zangwill 1988). An expression for the right-hand side of the diffusion equation having a similar structure is obtained for the case of thermodiffusion (Lifshitz & Pitaevskii, 1981), where the incompleteness of the equilibrium results from a specified inhomogeneity in the

Equation (6) fully describes the photodiffusion effect. However, to experimentally prove its existence, the theoretical model should be extended to include photodesorption, which

xx yy

surface diffusion, which does not depend on the surface coordinates.

length is intensity-independent.

point r. With these notations, Eq. (2) becomes

temperature distribution of the gas mixture.

represent D(r) in the form

illuminance gradient.

2

<sup>0</sup> *Dr D D r* () () = + <sup>∗</sup> (5)

∂ ∂ =Δ *n t Drnr* [ ()()] (6)

<sup>1</sup> B ( ) B ( ) ( ) v( , )d <sup>2</sup> = = = *D aa* <sup>∫</sup> *j l* **r rr r** *a a* (4)

inevitably accompanies photodiffusion, as well as the usual thermal desorption and deposition of atoms from the gas phase onto the surface. These processes will be taken into account in the next section. This will allow us to unambiguously separate out the contribution of photodiffusion to the experimentally observed phenomena.

#### **4.1 Influence of the surface - Gas phase exchange**

Under conditions of direct contact between the surface and the gas phase, Eq. (6) should include terms describing the deposition of atoms onto the surface from the gas phase and their thermal desorption from the surface. In addition, the right-hand side of Eq. (6) should be complemented with terms describing the photodesorption, whose rate, along with the photodiffusion coefficient, is proportional to the illuminance. The most complete equation may be written in the form

$$
\hat{\alpha}\mathbf{n}\prime\hat{\alpha}t = \Delta\left[D(r)n(r)\right] - \Gamma(r)n + f - n\pi\,,\tag{7}$$

where J is the flow of atoms to the surface from the gas phase, τ is the thermal desorption rate, and Γ(r) is the rate of photodesorption. In the absence of light, the surface density is the same at any point and is equal to

$$m\_0 = \mathbf{J} / \tau \tag{8}$$

Let us separate out the relative change in the surface density *p* due to illumination according to the formula

$$m = n\_0 \left(1 + p\right) \tag{9}$$

Assuming p to be small compared with unity, we may neglect the terms containing the product of *p* and the rates of the photoinduced processes. Then Eq. (7) will acquire the form

$$
\Delta \hat{c} p \{ \hat{c} t - D\_0 \Delta p + p\tau = \Delta D\_\ast - \Gamma(r) n \} \tag{10}
$$

The entire subsequent theoretical analysis is based on Eq. (10).

#### **4.2 Surface density wells and humps due to photodiffusion**

By introducing the new sought-for function

$$q(t,r) = \left[p(t,r)\right] + D\_\*(r) / D\_0\left[\exp(\tau t)\right] \tag{11}$$

we transform Eq. (10) into a standard form of the inhomogeneous diffusion equation

$$\left\| \left\| \mathbf{q} \right\| \left\| \mathbf{t} - \mathbf{D}\_0 \Delta \mathbf{q} = f \left( \mathbf{t}, r \right) \right\| \tag{12}$$

where

$$f(t,r) = \left[\tau D\_\*(r) / D\_0 - \Gamma(r)\right] \exp(\tau t) \tag{13}$$

Equation (12) may be solved using conventional methods (Vladimirov, 1971) for arbitrary initial conditions and an arbitrary form of the right-hand side, which is determined by the distribution of the illumination intensity.

Light-Induced Surface Diffusion 423

and the greatest increase in the density is located at the distance 2 1 *r r* = 2 . At t much smaller than *a*2/4*D*0, the value *r*1 is close to *a* at *t* much larger than *a*2*/*4*D0*, the value of *r1* 

The density minimum, attained at large times at the beam center, corresponds to *p*(*t*=∞, *r*=0)=–*D\**(0)/*D*0, while the maximum increase in the relative density, attained at

When all processes except for the photostimulated diffusion are neglected, the number of adsorbed particles does not change, so that, for any moment of time, the integral of *p*(*t*, *r*)

The results obtained above should be compared with those that would have been obtained in the absence of the photostimulated diffusion, when the effect of light is reduced only to

> <sup>2</sup> 2 2 <sup>0</sup> / <sup>2</sup> 1

*a r <sup>a</sup> t r <sup>u</sup> u du D D a u* −

It is evident from the form of the integrand that the relative role of the processes of thermal desorption and thermal diffusion depends on the radius of the light beam *a*. If *a*2 is much larger than *D*0/*τ*, the main contribution to the integral is made by the region *u*<<1, and we

2 2 p(t, ) [1 exp( )]exp( / ) *r tr a*

In this case, the surface density of the adsorbed atoms decreases in exact correspondence with the local illuminance, and the kinetics of this decrease is controlled only by the rate of thermal desorption. If *a*2>>*D*0/*τ*, the change in the density at the center of the illuminated

2 2 p( , ) ( / 4 )ln(1 4 / ) 0 0 *tr a D Dt a* = −

This equation shows that thermal diffusion cannot stabilize a surface density perturbed by the photodesorption process. The surface density, in this case, decreases infinitely until the process of deposition from the gas phase becomes noticeable or until the conditions for applicability of the linear approximation in the pump beam intensity are violated. When Eq. (17) is valid, the radial dependence of the surface density is monotonic. No increase in the

In the experimental study of the photostimulated diffusion, a Gaussian beam with the effective radius b was scanned through the center of a region with the modified surface density. As shown in the next section, at *b*<< *a*, the experimentally measured quantity can be represented in the form of an integral *S*(*t*) of the surface density variation over the straight

> <sup>0</sup> *St nb* ( ) p(x 0, , ) π

intensity along the scanning path. In this case, Eq. (20) should be replaced by

∞

−∞

When *b* is comparable with *a*, then Eq. (20) should be replaced by an integral of variation of the surface density over the whole surface with a weight proportional to the probe beam

*y t dy*

γ

τ

= +−−

0 0 0 p( , ) (1 4 ) exp[ ] (1 4 )

2

τ

<sup>+</sup> ∫ (17)

=− − − (18)

+ (19)

<sup>=</sup> <sup>=</sup> ∫ (20)

*t*=2.513*a*2/*D*0 at the distance *r*=1.87*a*, equals 7.5% of *D\**(0)/*D*0.

photodesorption. By setting *D\**(0)=0 in Eq. (15), we have

region, for times *t* <1/*τ*, is described by the function

surface density is possible in the absence of photodiffusion.

line passing through the center of the illuminated region,

*Dt a*

γ

τ

slowly increases.

over the whole surface is zero.

may approximately write

Turning back to the function *p*(*t*, *r*), let us write out the solution of Eq. (10) corresponding to instantaneous switching on, at *t*=0, of a normally incident axisymmetric Gaussian beam. By denoting the effective radius of the beam by *a*, the intensity-dependent photodesorption rate and photodiffusion coefficient may be represented in the form

$$\Gamma\left(r\right) = \mathcal{Y} \exp\left(-r^2 \left/a^2\right), \quad D\_\*\left(r\right) = D\_\*\left(0\right) \exp\left(-r^2 \left/a^2\right). \tag{14}$$

Assume that the surface density of the adsorbed atoms at *t*=0 was equilibrium *p*(*t*=0, *r*)=0. Then, for *t* > 0, we have

$$\begin{split} \mathbf{p}(t,r) &= \frac{D\_{\ast}(0)}{D\_{0}} [\exp(-\frac{r^{2}}{a^{2}}) - \frac{a^{2}}{a^{2} + 4D\_{0}t} \exp(-\frac{r^{2}}{a^{2} + 4D\_{0}t} - t\tau)] - \\ &\frac{a^{2}}{D\_{0}} [\mathbf{y} - \tau \frac{D\_{\ast}(0)}{D\_{0}}] \int\_{0}^{D\_{0}t/a^{2}} (1 + 4u)^{-1} \exp[-\frac{r^{2}}{a^{2}(1 + 4u)} - \frac{a^{2}\tau}{D\_{0}}u] du \end{split} \tag{15}$$

Consider, first, the photostimulated diffusion effect in its pure form, when one may neglect not only the photodesorption, but also the processes of adsorption of atoms from the gas phase and their thermal desorption. For this purpose, we suppose that, in Eq. (15), *γ*= *τ*=0.

Fig. 3. A well and a hump in the surface number density due to the photodiffusion out of the illuminated region

The dependence of the surface density of the adsorbed atoms on the distance from the center of the light beam at different time moments is shown in Fig. 3. It is seen that, as a result of the photodiffusion, the surface density at the beam center decreases while increasing at the periphery. The boundary between the regions of increasing and decreasing density, *r*1, shifts in time according to the equation

$$r\_1 = a[(1 + a^2 \;/\ 4D\_0 t) \ln(1 + 4D\_0 t \;/\ a^2)] \tag{16}$$

Turning back to the function *p*(*t*, *r*), let us write out the solution of Eq. (10) corresponding to instantaneous switching on, at *t*=0, of a normally incident axisymmetric Gaussian beam. By denoting the effective radius of the beam by *a*, the intensity-dependent photodesorption rate

( ) ( ) () () ( ) 2 2 2 2 *r r a Dr D r a*

Assume that the surface density of the adsorbed atoms at *t*=0 was equilibrium *p*(*t*=0, *r*)=0.

(0) p( , ) [exp( ) exp( )] 4 4

*D ra r t r <sup>t</sup> D a a Dt a Dt*

(0) [ ] (1 4 ) exp[ ] (1 4 )

Consider, first, the photostimulated diffusion effect in its pure form, when one may neglect not only the photodesorption, but also the processes of adsorption of atoms from the gas phase and their thermal desorption. For this purpose, we suppose that, in Eq. (15), *γ*= *τ*=0.

Fig. 3. A well and a hump in the surface number density due to the photodiffusion out of the

The dependence of the surface density of the adsorbed atoms on the distance from the center of the light beam at different time moments is shown in Fig. 3. It is seen that, as a result of the photodiffusion, the surface density at the beam center decreases while increasing at the periphery. The boundary between the regions of increasing and decreasing

2 2

1 00 *r a a Dt Dt a* =+ + [(1 / 4 )ln(1 4 / )] (16)

22 2

2

*u u du*

τ

(15)

τ

2 2 2 0 0 0

<sup>=</sup> −− − − − + +

exp , 0 exp . Γ= − ∗ ∗ = − (14)

and photodiffusion coefficient may be represented in the form

2 0

*Dt a*

<sup>−</sup>

\* 1

2 2 / 2

*a D r a*

*D D a u D*

0 0 0 0

− +−− <sup>+</sup> ∫

γ

\*

γ τ

Then, for *t* > 0, we have

illuminated region

density, *r*1, shifts in time according to the equation

and the greatest increase in the density is located at the distance 2 1 *r r* = 2 . At t much smaller than *a*2/4*D*0, the value *r*1 is close to *a* at *t* much larger than *a*2*/*4*D0*, the value of *r1*  slowly increases.

The density minimum, attained at large times at the beam center, corresponds to *p*(*t*=∞, *r*=0)=–*D\**(0)/*D*0, while the maximum increase in the relative density, attained at *t*=2.513*a*2/*D*0 at the distance *r*=1.87*a*, equals 7.5% of *D\**(0)/*D*0.

When all processes except for the photostimulated diffusion are neglected, the number of adsorbed particles does not change, so that, for any moment of time, the integral of *p*(*t*, *r*) over the whole surface is zero.

The results obtained above should be compared with those that would have been obtained in the absence of the photostimulated diffusion, when the effect of light is reduced only to photodesorption. By setting *D\**(0)=0 in Eq. (15), we have

$$\mathbf{p}(t,r) = \frac{a^2}{D\_0} \int\_0^{D\_0 t/d^2} (1+4u)^{-1} \exp[-\frac{r^2}{a^2(1+4u)} - \frac{a^2 \tau}{D\_0} u] du \tag{17}$$

It is evident from the form of the integrand that the relative role of the processes of thermal desorption and thermal diffusion depends on the radius of the light beam *a*. If *a*2 is much larger than *D*0/*τ*, the main contribution to the integral is made by the region *u*<<1, and we may approximately write

$$\exp(\mathbf{t},r) = \frac{\mathcal{Y}}{\tau} [1 - \exp(-t\tau)] \exp(-r^2 \;/\; a^2) \tag{18}$$

In this case, the surface density of the adsorbed atoms decreases in exact correspondence with the local illuminance, and the kinetics of this decrease is controlled only by the rate of thermal desorption. If *a*2>>*D*0/*τ*, the change in the density at the center of the illuminated region, for times *t* <1/*τ*, is described by the function

$$\text{tr}(t, r) = -(a^2 \gamma \;/\; 4D\_0) \ln(1 + 4D\_0 t \;/\; a^2) \tag{19}$$

This equation shows that thermal diffusion cannot stabilize a surface density perturbed by the photodesorption process. The surface density, in this case, decreases infinitely until the process of deposition from the gas phase becomes noticeable or until the conditions for applicability of the linear approximation in the pump beam intensity are violated. When Eq. (17) is valid, the radial dependence of the surface density is monotonic. No increase in the surface density is possible in the absence of photodiffusion.

In the experimental study of the photostimulated diffusion, a Gaussian beam with the effective radius b was scanned through the center of a region with the modified surface density. As shown in the next section, at *b*<< *a*, the experimentally measured quantity can be represented in the form of an integral *S*(*t*) of the surface density variation over the straight line passing through the center of the illuminated region,

$$S(t) = n\_0 b \sqrt{\pi} \int\_{-\phi}^{\phi} \mathbf{p}(\mathbf{x} = 0, y, t) dy \tag{20}$$

When *b* is comparable with *a*, then Eq. (20) should be replaced by an integral of variation of the surface density over the whole surface with a weight proportional to the probe beam intensity along the scanning path. In this case, Eq. (20) should be replaced by

Light-Induced Surface Diffusion 425

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).

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

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

$$S(t) = n\_0 \iint p(t, x, y)e(-x^2 \ne a^2) dx dy\tag{21}$$

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

$$S(t) = -n\_0 ab \sqrt{\pi} \begin{bmatrix} \frac{a^2 \gamma}{D\_0} \int\_0^{D\_0 t} \frac{\exp(-a^2 \tau \mu \, / \, D\_0) du}{(1 + 4u + b^2 \, / \, a^2)^{1/2}} \\\\ + 2(D\_\*(0) \, / \, D\_0) \int\_0^{D\_0 t} \frac{\exp(-a^2 \tau \mu \, / \, D\_0) du}{(1 + 4u + b^2 \, / \, a^2)^{3/2}} \end{bmatrix} \tag{22}$$

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 equilibrium. This condition is well satisfied in the experiment.
