Brownian theory of adsorbate diffusion

'surface

Surface Science 269/270 (1992) 184-188

North-Holland

SCler'lCe

Brownian theory of adsorbate diffusion R. F e r r a a d o , R. Svaci.,,.,,, •-,,,., - "~ ,o.E. ToHlinci Dtpartlmento dt Fisxa dell'Umrerszta dt Genora, Cenbo dt Ftstca delle Superficz e delle Basse Temperature / CNR and Unttk INFM, laa Dodecaneso 33, 16146 Genot a, Italy Received 7 August 1991, accepted for publratlon 9 October 1991

The diffusion of adsorbates on crystal surfaces is usually described by jump models, whxh imply the existence of a large periodic potenlN acting on the diffusing particle. However there are systems where the effective barrier is small, so a transport equation approach must be employed We solve the two-vector variable (position and velocity) Fokker-Planck equation m a periodic potential by the continued fraction method and calculate the dynamx struc',ure factor for a wide range of the surface momentum transfer The results are compared with neutron scattering data from monolayers of CH 4 adsorbed on MgO(001) at different temperatures, it IS shown that the Fokker-Planck theory works noticeably better than the jump theory.

!. Introducti~,n

The way in which atoms migrate on surfaces is a very interesting problem in many fields of physics and chemistry such as for example catalys~s, crystal growth and evaporation. The diffusion of adsorbed or self-adsorbed atoms on cry~tal,~ ~s being extensively studied [1-5] for many years, m pamcular with field 1on microscopy tcchmques which probably are the most appropriate to study some aspects such as the lattices of visited sites. More recently, the quasi-elastic scatte:ing of slow neutrons [61 and atoms [7] has been e,'nployed as a vahd tool to investigate other feat,ires of the surface motion. Besides molecular dynamics calculations [8] and interesting exchange mechamsms proposed for peculiar systems [4,8], usually j u m p dZ~fusion models [2,6,7,9] are employed to describe surface mohhty on CryStals. However, as it is evident in Blenfalt et al data [6], a description by a hopping mechanism between lattice s|tes ,s not alwas,s satisfactory. In th~ incoherent quasi-elastic scattering of neutrons, the relevant measured quantity ~s the self-part of the dynamic structure factor S,(Q, w),

whose full width at half maximum ( F W H M ) gives, at small Q's, the diffusion coefficient and describes, at larger parallel m o m e n t u m trz t i e r , the migration mechanism on the surface. The diffraction peaks, determined by the substrate periodic~ty, are enlarged by small inelastic contributions due t,~ the stochastic positions and velocities assumed by the adsorbed atoms The F W H M behavlour deviates from that predicted by the Chualey and Elliott discrete theory [9] with increasing temperature, although the substrate periodicity is still present. This means, as we will show, that the effective barrier for jumping is of the same order as the thermal energy of the adsorbate; in this case, hopping models are not correct and a transport equation approach is neede,~. In this paper we propose a classical Brownian model which leads to a continuous differential equation (Fokker-Planck equation) for the stochastic process given by the ~ectonal couple pos~tlon-vclocity. We solve thls equation in a 2D periodic potential, obtaining the Van Hove selfcorrelation function from which the incoherent S,(Q, co) and its F W H M are deduced. Finally we compare our theory with the results of Bienfait et

(10~9-6028/~2/$05 00 c 1992 - Elsevier Science Pubhsher~ t] V ,~]1 rights reserved

185

R Ferrando et al. / Browman theory of adsorbate dtffuston

al. [6], ebtaining an estimate of the height of the potential barrier which is the only adjustable parameter contained in the model. Although memory effects [10] are not taken into account, we obtain a very good agreement with the experimental data.

2. The stochastic process

The measured quantity in the purely incoherent tquasi-elastic scattering of neutrons from surfaces, is the self-part of the dynamic structure factor or scattering law [11]:

S,(Q,o,)=f

=f

dt e x p ( - i t o t ) , ~ S ( Q , t) dt e x p ( - i t o t )

x ( e x p ( i Q - ( R - R0))),

(1)

where hQ and h¢o are the surface momentum transfer and the energy loss, respectively; .vs (Q, t) is the intermediate scattering function or selfcorrelation function, and the brackets stand for the statistical average. In our picture an adsorbed at6m is assumed to be a Brownian parhcle moving in a surrounding bath, composed by the particles of the substrate and by the other adsorbates, which furmshes both d~ss~patton and fluctuation. The stochastic problem is treated as an Ornstein-Uhlenbeck process in which the couple position-velocity is assumed to be Markovian, so no restrictive hypothesis is done on the magnitude of the friction and the velocity correlation time is taken into account [12-14]. The Langevin equahon for the process is then written as

dV U I;

r(R) nv+--_

+K(t);

(2)

,~f¢

m, R and V are the mass, position and velocity of the adsorbate and 77 is the friction per unk mass. F(R) = - V U ( R ) is a deterministic periodic force, while K(t) is a purely stochastic force with a white noise, pectrurn. Both these torces originate from the bath and describe the static and rapidly

fluctuating part respectwely of he interaction with the tagged particle. It may be surprising that a Brownian theory is proposed for particles whose mass is comparable with the mass of the bath particles, but in ref. [10] it has been shown that such a treatment is quite accurate at least for a single adatom of the ~zrn,~ mass of the substrate atoms and with a position dependent friction. In this case the fluctuations in the surroundings are on a faster time scale than those of the adparticle and the basic assumptions on g ( t ) are satisfied. In ref. [6] the mass ratio between the substrate (MgO) and the adparticle (CH 4) is 2.5 indeed, but the substrate is very rigid and therefore its typical frequencies are high; this fact should again guarantee the possibility of separating the interaction in a static and m a rapidly fluctuating part. In thts system the adsorbate density is high (coverage 0 = 0.8) and the dynamics should be described by a many-particle Fokker-Planck equation [15]; however the single-particle equation in an effectwe periodic potential should give a good approximation and a significative improvement of the jump theory [9]. Moreover, the frictJon is assumed to bc position independent. From the Langevm equation, as usual, thc Fokker-Planck equation is Uerived [12,16]" /~f ~f F(R) --mV.--+--.--

--n~-~"

Of

V f + - - - - m aV '

(3)

where f(R, V, t) is the probabdity density m the one-particle phase space and k BT ~s the thermal energy of the tagged particle. For a 2D square lattice (e the lattice spacing) we assume a decoupled cosine potential U(R/=const-Acos

-a

X-Acos

~ ,

(4) by which eq. (3) factorizes into two ~dcm~cal 1D equations. Other and more realistic potenuals wdl be treated in an other paper [17], together with the effects of a position dependent friction However, prehmmary calculations show that the

R Ferrando et al / Browman theory of adsorbate dlffi,~ton

186

main conclusions of the present work are not significantly changed introducing these modifications. As described by Risken [16], the FokkerPlanck equation can now be solved with a numerical procedure (matrix continued fraction method). Introducing the following dimensionless quantities: A

a ~-k@

g= 2 k BT , ~ = - ~ a ~/

~7,

m

the result for S~ is:

S,(Q, to)

the value of ~ and tberefore of the friction ")7 (see eq. (5)). In figs. 1 and 2 the Fokker-Planck A E is compared with data from ref. [6] of CH 4 adsorbed on MgO(001) powders at two different temperatures (88 and 97 K), taking the experimental values of D and keeping g as a fitting parameter. In these figures also the single-jump A E is plotted. The comparison between the experimental results and both the theories has been performed convoluting the theoretical results with a triangular shaped experimental resolution having a FWHM of 27/.teV and then averaging over an isotropic distribution of the orientations of the MgO crystallites [18].

=N Re[~.,Gp,,(k,p,r i~)lP-t(g)t'-t(g) 1' 3. Results (6) where N is a normalization factor, I~ ~s the modified Bessel function of order r and Q = 2 w ( k + l ) / •a , w i t h - / <_k -< 51 and l an integer. The Green function G'xs given by: (~(k, ,G) = [ i ~ l + Z,(k, , ~ ) ] - ' ,

(7)

where ,~ can be calculated numerically by the matrix continued fraction 7~(k, ~ ) =B

(i~ + ~ ) I + 2B

n I,

I

B t

(i~ + 2 ~ ) I + ...

(8) 1 is the identity matrix, B is gwen by Bpr(k ) =

(p+k)t~p,r+

2

'

, -

,

(9) and B t is the tianspose of B Knowing g and ~, the scattering iaw is calculated together with ~ts FWHM, A E; at small Q's the diffusion coeffident D is obtained as A E = 2 h D Q 2. Hence D can be numerically calculated as a function of g and ~. At fixed g, D results in a decreasing function of ~; this funchon can be inverted In order to get ~(g, D). If D is taken from the experiment, the value of g dctermmes um~ocal[y

In figs. 1 and 2 the fit is obtained with g = 0.22 and 0.20 at T = 88 and 97 K respectively. These values are consistent with a hypothesis of a constant potential barrier (2A) of 6.7 meV in the range of temperatures considered. This is not surprising because of the substrate stiffness and of the fixed coverage 0 = 0.8 [6]. As can be seen, the Browman theory works considerably better than the single-iump model [9], the mulnple-jump theory would give even worse results because the corresponding F W H M would be even smaller compared to the experimental points. It should be pointed out that the single-jump theory gives A E cc (1 - c o s ( Q a ) ) [9], which has a maximum at the border of the first Brillouin zone. The energy convolution and the orientational average [18] lower the value of the maxim u m and shift its position 'o a higher Q, as can be seen in figs. 1 and 2. In the range of parameters considered, the Fokker-Planck A E has a maximum in the second Bri[Iouin zone [14]; convoluting and averaging, this maximum is shghtly shifted to the right but strongly lowered The net result Is that the difference between the singlej u m p and the Fokker-Planck curves is largely reduced. The fitting parameter g Is .-, 0.2, I e. the height of the potentlal barrier for the migrating parhc!e ~', nearly k bT at both temperatures. U n d e r these

R Ferrando et al / B,owman theory of adsorb,ate dtffu~ton

r / = 2 8 x 10 t~ s - t at lower t e m p e r a t u r e and 77 = 1.3 x 10 '~ s - J at higher temperature These values differ considerably and this is somewhat surprising in such a narrow range of temperatures; however the same large difference is found in the measured diffusion coefficients (erratum to ref. [6D. From the values of the friction, the velocity correlation time % ~ l / r / can be evaluated, obtainip.g % = 4 × 1 0 - ' 4 and 8 × 10-l~ s at 88 and 97 K respectively. These times are much shorter than both zo, c and ¢6 therefore the adsorbed atom diffuses in a regime of relatively high friction and low potential barriers and in that case also the Smoluchowski equation would be a good approximation [13]; however slightly different estimates of the pot,-ntial b~rrier and of the friction would be obtained.

AE (ueV) 80

70

60

5O

40

30

~'~'~'-'~'"',,.

187

,. \ \°

:I

20

10

AE (#eV) 0

02

04

06

08

1

12

14

o

O (A-'/

175

Ftg 1. Comparison wtth experimental data of B~enfa~t ct al [6] FWHM, AE, of S,(Q, to), for Q up to the centre of the second Bnllou,n zone, al T = ~ g K, I ) = 1 5 × 1 ( ) ~ cm -~ ~ (erratum to ref [61) and g = 0 22 Fokkcr-Planck lhe~)rs' {lull hne) and stogie-lump model h-dotted hne) 125

conditions j u m p models are not appropriate because the particle does not s p e n d most of its time oscillating near the bottom of the potential well [9,13]. In this inhomogeneous system, the j u m p diffusion condition can be summarized as:

100

)

¢

75 /*/ \ \

%,, << "rf, where ~-,,,, ~ non period

(10)

50 I

#,'I

a~m-/A

(a = 4.21 ,~,) is the oscillain the potsnt~al well and ~-~ ~ avm/( kBT ) is the ume of flight from a site to its nearest neighbours. The results are T,,,~ ~ 3 X 10 -~z s and ~'f~ 2 x 10 -~2 s; therefore condition (10) ~s clearly not satisfied at both temperatures. As explained at the end of section 2, g and D fix the value of the friction, whlch results m

2f j ~

J

I.~,~11,,¢~', I . . . .

0 0

02

I ....

04

I ....

06

I ....

OB

I ....

'~

I , I ,

"~ 2

/°-I Q,,A )

Ftg 2 As m fig I but T = 9 7 K, D = 3 6 x 1 0 -~ era2 ~-1 ferratum to re~ [6]1and ~' = 0 20

188

R Ferrando et al / Browman theory of adsorbate dtffuston

Acknowledgement The authors are grztefui to Prof. M. Bienfait for rtelpful discussions. References [1] G. Ayrault and G. Ehtach, J t"hem Phy~ 60 {I074, 781 [2] G Ehrhch and K. Stolt, Ann Rev. Phys Chem 31 (1980) 603; G. Ehrhch, J Vac. ScL Technol. 17 (1980) 9. [3] M Trmgldes and R Gomer, Surf Scl 155 (1985) 254, J Chem Phys 84 (1986) 4049 [4] G L. Kellogg and P.J Feibelman, Phys Rev. Lett 64 (1990) 3143 [5] C Chen and T T Tsong, Phys Rev Lett 64 (1990) 3147 [6] M Blenfalt, J.P. Coulomb and I P Palman, Surf So 182 (1987) 557, and Erratum, Surf Scl 241 (1991) 454. [7] B.J Hmch, J.W.M Frenken, G, Zhang and J P Toenrues, Surf Sct 259 (1991) 288

[8] G De Lorenzl, G lacuccl and V Pontlkls, Surf Scl. 116 (1982) 391, R M Lynden-Bell, Surf Scl 259 (1991) 129 [9] C T Chudley and R J EIhott, Proc Phys Soc. (London) 77 (1961) 353 [10] G. Wahnstrom, Surf. SCL 159 (1985) 311. [ll] L Van Hove, Phys. Rev. 95 (1954) 249. [12] S Chandrasekhar, Rev. Mod, Phys. 15 (1943) 1, II.A. Kramers, Physlca 7 (1940) 284. [13] R Ferrando, R. Spadacml. G.E. Tommel and A C Levi, Physlca A 173 (1991) 141. [14] R Ferrando, R. Spadacml and G.E Tommel, Phys. Rev B, m press; Surf. So 251/252 (1991) 773 [15] T Munakata and A. Tsurui, Z. Phys. B 34 (1979) 203. [16] H Risken, The Fokker-Planck Equation (Springer, Berlin, 1989). [17] R Ferrando, R Spadacini and G E Tomme~, Surf SOl 265 (1992) 273. [18] M Blenfalt, J.M Gay and H Blank, Surf Sc~ 203 (1988) 331