Jump processes in surface diffusion - Bilkent University - Faculty of ...

More documents

Recommendations

Info

$SET 7 MATH 543: FUCHSIAN DIFFERENTIAL EQUATIONS ...$

$MATH 225: DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA ...$

40 G. Antczak, G. Ehrlich / Surface Science Reports 62 (2007) 39–61 Fig. 1. Hard-sphere models of bcc (a) (110), (b) (211), and (c) (321) planes. 1.1. Surface diffusivities Diffusion is characterized by the material parameter D, the diffusivity in the (one-dimensional) diffusion equation: J = −D ∂c ∂x connecting the adatom flux J to the concentration gradient ∂c/∂x. The diffusivity is given by the usual Arrhenius relation ( D = D o exp − E ) D , (2) kT where the prefactor D o , usually considered a constant, and the activation energy E D are obtained from the temperature dependence. Langmuir [1], in 1933, already viewed the atoms in surface diffusion as hopping between elementary spaces at a separation a, and was able to show that for an individual adatom the diffusivity was given in terms of the lifetime τ at a particular site by D = a 2 /2τ. (3) In transition-state theory the one-dimensional diffusivity is written more elaborately as: ( ) ( D = υl 2 SD exp exp − E ) D , (4) k kT where υ gives the effective vibrational frequency of the adatom, l its jump length, and S D as well as E D the entropy and activation energy for diffusion. We can, however, introduce the abbreviation: ( ) SD υ o = υ exp , (5) k so that the diffusivity appears as in Eq. (2), but with D o = υ o l 2 . (6) On surfaces it would be difficult to measure atom fluxes and gradients. The mass flux of material over a surface can of course be detected, but it is not clear how to disentangle interactions between the atoms. The diffusivity is therefore obtained alternatively from the Einstein relation, which ties the diffusivity to the mean-square displacement 〈x 2 〉, 〈x 2 〉 = 2Dt, (7) (1) Fig. 2. Models of fcc (a) (111) and (b) (100) planes. where t is the time interval of the measurements. The elementary formalism is now in hand, and using it much data has been gathered about diffusion kinetics [2]. But what are the jump processes participating in surface diffusion? If atoms always jump between nearest-neighbour sites, then since a typical vibrational frequency at the surface is ∼10 12 s −1 , and the spacing is ∼3 Å, the prefactor D o should, if we neglect entropy contributions, be of magnitude ∼10 −3 cm 2 /s. It turns out that this is close to the value of the prefactor for many diffusion systems [2], starting with one of the first studied, W on W(110), for which a value of 2.6×10 −3 cm 2 /s was found [3, 4]. This agreement suggested that this simple view of diffusion processes had considerable merit. However, the story has turned out to be much more interesting than that. 2. Atom exchange 2.1. On fcc(110) planes Early studies of individual metal atoms diffusing on metals were with tungsten atoms on (110), (211), and (321) planes of bcc tungsten, models of which are shown in Fig. 1. Also studied somewhat later were rhodium atoms on rhodium, an fcc metal [4]. Examined were (111) and (100), as well as (110), (311) and (331) planes, shown in Figs. 2 and 3. What is clear is that the (321) and (211) planes on tungsten and (311), (331) and (110) planes on rhodium are channelled, and that an adatom moving along one of these channels might be expected to stay in such a channel and execute strictly one-dimensional diffusion. That was in fact found to be the case as shown in Fig. 4 for W(211), making predictions of the direction in diffusion seemingly straightforward.
G. Antczak, G. Ehrlich / Surface Science Reports 62 (2007) 39–61 41 Fig. 3. Models of fcc (a) (110), (b) (311), and (c) (331) planes. Fig. 5. Mechanism for cross-channel diffusion of adatom on Pt(110) proposed by Bassett and Webber [5]. (a) Adatom in channel. (b) Vacancy forms in channel wall. (c) Adatom has moved into vacancy, lattice atom is now in channel site. Fig. 4. Location of sites on W(211) at which Rh adatom has been detected after diffusion at 197 K. Motion is strictly one-dimensional. Shown at the bottom is a count of the number sighted at each site. This simple picture was suddenly destroyed, however, by the work of Bassett and Webber [5] in 1978, who studied another fcc metal, platinum. On (331) and (311) planes diffusion occurred, as expected, along the channels of the surfaces. On the (110) plane, however, diffusion was two-dimensional! The activation energy for in-channel motion was 0.84 ± 0.1 eV, for cross-channel movement it was smaller, only 0.78 ± 0.1 eV. Similar results were found with the diffusion of iridium atoms on these surfaces. What had been established here was that diffusion on channelled surfaces could occur in two dimensions, depending on the particular system. Predicting the direction in which diffusion occurred was not just a matter of looking at surface geometry. The question still remained how cross-channel motion took place. Bassett and Webber [5] had two ideas. One was that large fluctuations occurred in the rows of atoms constituting the channel walls, creating holes through which adatoms could easily jump. A more likely possibility was that the movement of a channel atom created a hole that could be filled either by the adatom or a surface atom, as shown in Fig. 5. They concluded that “Further experimental and theoretical studies are required to clarify the nature of inter-channel diffusion”. Even without an understanding of the mechanism of cross-channel motion, the phenomenon of cross-channel diffusion had been established. What actually happened in cross-channel atom movement was soon uncovered in experiments by Wrigley and Ehrlich [6]. They took advantage of the atom probe [7], to measure the chemical identity of atoms on Ir(110)-(1 × 1). In this work the system was calibrated by first depositing an iridium and separately a tungsten atom on the surface kept at ∼50 K. The mass of the individual atoms was then obtained by measuring the time of flight to a detector ∼1 m away by field desorbing the atoms; the results are shown in Fig. 6. With this information in hand, a tungsten atom was placed on the clean surface, which was heated until cross-channel diffusion took place. The atom observed by field ion microscopy after diffusing into the adjacent channel was then field evaporated, and as also shown in Fig. 6 proved to be iridium, an atom from the lattice. Thereafter the top layer of the surface was field evaporated and analyzed, as in Fig. 7, and usually a tungsten atom was detected in the surface layer, as expected if atom exchange had occurred. This was not the case when the surface was probed after crosschannel motion had not been seen. These experiments were the first direct proof of an interchange between a substrate and an adsorbed atom during cross-channel diffusion. A number of theoretical investigations were stimulated by the results on Pt(110). Halicioglu [8,9] carried out calculations for diffusion on Pt(110) with Lennard-Jones twobody potentials and found that in cross-channel diffusion an adatom interacts with an atom in the channel wall to form a dumbbell, as in Fig. 8(b). This could decompose by the adatom incorporating into the surface and the surface atom going into the adjacent channel, or else by returning to the original channel, as indicated in Fig. 8(c). Just slightly later, DeLorenzi et al. [10,11] did molecular dynamics calculations, again using Lennard-Jones potentials, on fcc surfaces. Most interesting were the results for (110) planes, shown in Fig. 9, on which both in-channel and
Page 1: Surface Science Reports 62 (2007) 3
Page 5 and 6: G. Antczak, G. Ehrlich / Surface Sc
Page 23: G. Antczak, G. Ehrlich / Surface Sc

Jump processes in surface diffusion - Bilkent University - Faculty of ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?