- Email: [email protected]
Surface diffusion in temporally and spatially inhomogeneous systems
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Surface Science 307-309 (1994) 465-470
Surface diffusion in temporally and spatially inhomogeneous systems J. Haas, K.R. ROOS,A. Jesina, M.C. Tringides * Ames Laboratory, Department of Physics and Astronomy, Iowa State Uniuersity, Ames, IA 5001 I, USA
(Received 20 August 1993)
Abstract Non-equilibrium processes in an overlayer with the system evolving between two different states (i.e., ordering, disordering, adsorption) can be used to study surface diffusion. Diffraction experiments on O/W(llO) at high coverage measure an activation energy E, = 0.6 k 0.05 eV during ordering and E, = 1 + 0.05 eV during disordering. These results are consistent with other diffusion experiments carried out with the fluctuation and Boltzmann-Matano methods. The difference between the measured activation energies can be accounted for in terms of the interactions which contribute differently in different overlayer configurations. Computer simulations on lattice gas models illustrate the dependence on the configuration, by comparing diffusion coefficients obtained with different methods.
1. Introduction The simple picture [ll of surface diffusion as hopping over a static barrier becomes considerably more complex in the presence of interactions. Spatial or temporal inhomogeneities can exist in the system resulting in a distribution of local environments and diffusion barriers. The measured effective diffusion coefficient by a given technique is an average over the local barriers each atom experiences. Results are sensitive on the technique used, since depending on the technique different configurations of the system are probed. It is useful to search for method-invariant parameters to describe the experiments. The difficulty described above can be expressed
* Corresponding author. Fax: + 1 (515) 294 0689.
differently, but equivalently, in terms of a coverage dependent diffusion coefficient D(0), which implies a non-linear diffusion equation. Surface diffusion studies have been, in some cases, restricted to measuring the single particle diffusion coefficient obtained in the limit 8 + 0. This quantity is “clean”, independent of the experiment and directly comparable to first principle calculations. However, in practice under real growth conditions, many particles are present on the surface and surface diffusion is affected by their mutual correlations. It is important to develop other techniques to deal with these realistic conditions. We will describe non-equilibrium experiments [21 to measure the growth rate activation energy and from it to deduce surface diffusion coefficients, during ordering (i.e., when islands are formed out of a random configuration) and disor-
0039-6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0039-6028(93)E0799-Z
466
J. Huas et al. /Surface
Science 307-309
dering (i.e., elimination of initially ordered islands). These techniques used on O/W(llO) will be compared to other diffusion techniques on the same system to show that results are consistent with each other. These conclusions are supported with simulations on lattice gas models which compare different techniques, and demonstrate that the measured quantities depend on the technique, but they are consistent with each other.
2. Non-equilibrium
diffusion
experiments
A system after a quench of one of the thermodynamic variables (temperature, chemical potential, coverage, etc.) is evolving in time towards a new equilibrium state. Studying the temperature dependence can be used to measure surface diffusion. For inhomogeneous systems, one can write [3] a simple expression for the diffusion coefficient as an average over all the possible local barriers D = CP,D,, where I’,, Di are the probability and diffusion coefficient for local configuration i. The Pi are not determined thermodynamically (as in equilibrium), so one needs to know explicitly their time dependence. For ordering processes, where domains form out of a random configuration, one can distinguish two types of atoms, one inside, A, and the other at the domain boundary, B. They have different local barriers with the boundary atoms easier to move. One can refine this argument further by taking into account the type of walls separating domains, but the energy difference for different wall configurations is less than the energy difference between wall and inside atoms, so we can simply treat all the wall particles equivalently,
One expects (PA> to be a monotonically increasing function of time f(t) (with f(O) = 0, f(m) = 1)
If the is of an in the
D,
measurements are restricted to t = 0, then measured (i.e., when the local environment atom is random) and when t + m diffusion final equilibrium state is obtained; for a
(1994) 465-470
time in between, the corresponding appropriately weighted sum is measured. For self-similar growth, the average domain size follows [4] a very simple power law in time, L =A(T)t”, with the time and temperature dependence separated in the above equation so a growth rate can be easily extracted. A single growth rate A(T) controls the evolution over all times. A a 0; should hold for dimensional consistency in the above equation for L. The diffusion coefficient measured from A(T) corresponds to the initial time, random configuration D,. This is physically justified if the experiment is thought of as one of an initially non-uniform concentration profile, given by the domain morphology. The quantity used to probe the evolution, L(t), is sensitive to the motion of boundary atoms, so only the diffusion coefficient of the atoms at the boundary (i.e., the random configuration) is extracted. For disordering processes the exact time dependence of the average domain size L versus t is not known. Both experimental and theoretical evidence [51 suggests that it is not a simple power law, as for the case of ordering processes. One can restrict the experiment to short times, t = 0, and assume that the decay of the average domain size is linear in time, L/L,, = 1 - cDt + . . , with the first term proportional to the diffusion coefficient. By plotting the initial slope of the intensity decay curves versus l/T, the activation energy related to the initial, ordered state is obtained. Other well developed techniques can be used to study surface diffusion in systems with interactions. At equilibrium, the spontaneously induced particle density fluctuations can be measured [6] and the diffusion coefficient can be obtained from the decay of their autocorrelation function. Since the fluctuations are expected to be a small fraction of the average concentration, SC/C -+E1, a small signal is available, so appropriate amplification techniques are necessary. In this case, although the diffusion coefficient is coverage dependent, because the excursions from equilibrium are small, the problem is reduced to one with a constant diffusion coefficient. A different technique [7] to study surface diffusion involves monitoring the time evolution of
.I. Haas et al. /Surface
Science 307-309
the 1D initial step concentration profile. This is a standard method for systems with no interactions. It is still appropriate to extract the full coverage dependence of the diffusion coefficient from the evolving profile lineshape c(x, t), if it can be shown that it obeys the scaling c(.~/t’/~). The so-called Boltzmann-Matano (BM) method gives an expression for the full coverage dependence,
qc’)
=
L
the same activation energy. Table 1 lists the results of five different diffusion techniques [2,6-S] applied on the same system, O/W(llO). Some of the experiments were carried out over a range in coverage and as expected, the activation energies are coverage dependent because of the interactions. We restrict the comparison in Table 1 at high coverage, so any differences will be simply attributed to the configuration of the overlayer and not the coverage. Four of the methods used, including the equilibrium one, give, remarkably, almost the same value for the activation energy, E = 1 k 0.05 eV, while the other one, based on non-equilibrium ordering experiments, measures a [2a] substantially lower value, E, = 0.6 k 0.05 eV. This system has been described [91 very well in terms of a lattice gas model that includes competing attractive nearest neighbor and repulsive nextnearest neighbor interactions, c$, = -&, = 0.09 eV. As explained above, the ordering experiment probes the system in the random configuration where the two types of interactions cancel each other, while the disordering and equilibrium experiments measure diffusion in the well-ordered (2 X 1) + (2 X 2) configuration where there is higher probability to have nearest than nextnearest neighbors occupied. The experiments based on the evolution of initial, abrupt profiles surprisingly give the same value; this is not expected because the local configuration of a diffusing atom at the moving boundary is different from the configuration at equilibrium (even for low resolution probes a sharp boundary can still
1 ndc
‘cI
2t (dc/dn),f
’
where n = n/t ‘I2 is the scaled variable. It is remarkable that a single lineshape c(n) can provide the full coverage dependence when the scaling condition holds. It is still not clear what is the range of validity of the condition. Although, it can always be tested empirically whether it is true, it would be useful to determine its validity in terms of the interaction strength, temperature range, etc. It is most likely that for strong interactions the scaling 4x, t) = c(.x/~‘/~) is violated because of build up of particle correlations. In principle, this step profile method is a spatially non-equilibrium one, with the local environment of the atoms at the boundary (which mainly controls the evolution) different from the equilibrium one, so it should be expected to give different activation energies than the fluctuation method. Only when the step profile is not infinitely steep (which can result from resolution limitations of the probe and imposes an averaging over the resolution distance AX) when compared to the equilibrium density fluctuations, Ax(dc/dx) <
Table 1 The table summarizes disordering, fluctuation,
the activation energies and prefactors Boltzmann-Matano and patch profile
467
(W,., one expects [l] the two techniques to give
rc’
1
(1994) 465-470
obtained on O/W(llO) with different analysis give the same activation energy
experimental
Process
Technique
Ed (eV)
D, km’/s)
Ref.
Ordering
LEED
0.6
10-7
121
Disordering
LEED
1.0
_
151
Field emission
1.0
10-4
161
1D Boltzmann-Matano
Kelvin probe
1.1
0.38
[71
Patch profile
SEM
1.1
0.2
181
Equilibrium
fluctuation
techniques;
16X
J. Hum
rt al. /Surface
Sciencr 307-309
be defined). In addition, the temperature range used in these ID profile experiments is well within the disordered region, where the two types of interactions should cancel each other. It is unclear why there is good agreement between the profile evolution results and the equilibrium results. Table 1 also lists the diffusion prefactors, when available. There is some discrepancy between the listed values. We have not attempted to extract the exact prefactors from ordering/disordering experiments because other parameters, as can be seen from the cases when the growth law is known [LO] exactly, control the relation between the growth rate and activation energy. We can introduce an absolute length scale for the average domain size by using the full-width at half-maximum (FWHM) of the changing diffracted spots. In the ordering experiment [2a] the domains increase from 3 to 120 A, in the disordering experiment [5] the average size stays the same, while the number of domains is changing over the time interval of the experiment, 2000 s. We can estimate a prefactor D,, = 1O.-7 cm2/s in the ordering experiment by assuming that all the other constants are of order unity, but clearly this is an oversimplification. This value is different from the other values listed in Table 1 and is similar to the value obtained at low coverage with the fluctuation method [6].
3. Simulations We have performed studies of surface diffusion with Monte Carlo simulations on lattice gas models, where all the factors affecting diffusion are easily controlled. The objective is to compare different experimental techniques on a model with the simplest type of interactions, i.e., nearest neighbor (both repulsive and attractive, which produce drastically different overlayer configurations) over a wide temperature range inctuding T,. The details of the simulation can be found elsewhere [ 111. First, we wanted to address the previous puzzling result, why the 1D BM analysis and fluctuation method give the same activation energy, al-
(1994) 465-470
0.I
A
n
I
. . ‘
.
Fig, 1. Arrhenius plots of D vs l/T for the lattice gas model with nearest neighbor interactions obtained with three different methods. The fluctuation and the Boltzmann-MataIl(~ methods agree with each other while the LID method gives different results.
though the system is interrogated in different configurations. Especially since sharp, atomic scale boundaries are used in the simulations, the issue of experimental resolution is irrelevant. Fig. 1 shows that over a wide temperature range, - 2 < -J/kT < 2 (i.e., both for attractive and repulsive interactions with 1J/kT, ! = 1.86), the diffusion coefficient obtained with the fluctuation method is essentially identical to the one obtained with the 1D BM method. The agreement between the two methods is remarkable but is not clear if it is only true for this type of interaction. The required scaling condition c(x, t) = c(x/ 6) is satisfied at all temperatures used, thus extending the validity of the method at least for this model into the low temperature ordered region. Fig. 1 also shows a comparison with the iaserinduced-desorption (LID) method [ 121, a different technique used to study weakly bound overlayers (a method which cannot be used on O/W(l lo), a strongly bound system). The LID diffusion coefficient is obtained in the simulation from the initial slope of the refilling signal as
J. Haas et al. /Surface
Science 307-309
done experimentally, although other types of analysis have been proposed [13] for the case of coverage dependent diffusion coefficient. The comparison is motivated because of the similar geometry between the LID and fluctuation methods: a probe hole in the middle of the surface having the same coverage as the outside region, method, while 'in 2 'oout~ simulates the fluctuation ein = 0 simulates the LID method. As pointed out before [14], the measured diffusion coefficient for systems with interactions depends on e,,, so the fluctuation and LID methods give different results, shown also in Fig. 1. The reason for the discrepancy between the fluctuation and LID methods can be related to the type of the measured quantity in LID, the refilling signal back to the probe, which is an integral over the moving profile. Information is lost during the integration. Can some type of BM analysis be used [14] with the LID geometry so the correct coverage dependent diffusion coefficient can be extracted? It has been shown before, that even with no interactions, this is not possible and the concentration profile cannot be written in terms of the required scaling for BM analysis, c(r, t) + c(r/ fi). Is there a different type of scaling for the concentration profile so the coverage dependent diffusion coefficient can be extracted? We have looked at the evolving profile, both for repulsive and attractive interactions, and plotted them in a scaling form to test if they follow the same functional form at all times. We have taken the edge of the probe R, area as the origin and define a new variable p = R, - Y.The data were plotted in the form c(p, t)/c(O, t) versus p/w(t), where w(t) is the full-width at half-maximum (FWHM) of the evolving profile. It is interesting that, as seen in Fig. 2, for repulsive interactions the results collapse onto a single universal curve, while for attractive interactions they do not. This conclusion was confirmed for all temperatures, the deviation from scaling for attractive interactions being more pronounced the lower the temperature. One can evaluate in several ways the significance of this result. First, although C(T, t) cannot be written in scaling BM form, at least for repul-
(1994) 465-470
469
1.2 ~~.~
1.0
1
F
Repulsive interactions
1
0.8 -
.
0.6 1
X=
l
t,=O.I (t$=O.169) t,=0.2 ($,=0.244)
c
0 5 a v
F
0.4 !~
Attractive ink&a
t,=0.01 (eP=o.044) x IJ b 0 0
“so
x
c X
t,=0.05 (eP=o.21) t,=0.4 (C$=O.O65) G=l.O (eP=o.091) t4=l .5 (ep=o. 11) ts=5.0 (eP=o.15)
x x
Fig. 2. The concentration profile in an LID experiment measured with the edge of the probe area as the origin. For
repulsive interactions c(p, t)/c(O, t) vs p/w scale (i.e., the same functional form describes the evolution) while for attractive interactions they do not. ~9~denotes the coverage in the probe at time t,.
sive interactions (which promote the formation of the c(2 x 2) open structures that resemble the non-interactive case), one can write C(T, t> in the scaling form c[(R, - r)/w(t)l and possibly use this to simplify the non-linear diffusion equation. Second, if fits are attempted to the evolving profile at any time (either for the LID geometry or the complementary problem of an initially prepared overlayer patch 8, # 0, 0,,, = 0 diffusing into the vacant surface), then at least for some interactions (i.e., attractive which promote dense, highly correlated (1 X 1) structures) the extracted diffusion coefficient depends on the time chosen, since different functional forms describe the evolution of the profile. Third, if a different generalized scaling combination is searched for, C(T, t) = c(r/tk) with k # l/2, one can show this is not possible. We briefly sketch the argument. The total mass in the probe area increases as m = t2k, if generalized scaling holds. We have also calculated in the simulations the FWHM w of the complementary profile, c(p, t), which grows in time, w = ts (with 8 I l/2), so the generalized scaling of C(T, t) implies scaling of c( IR,,PI/Wk/6)for any R,.If we take R, = 0 it would imply that the generalized scaling should be of the form c(p/w k/6). It can be shown from the conservation of mass that k > 6, so for repulsive
470
J. Ham et al. /Surface
Science 307-309
interactions it is impossible to have c(p, tl to scale with two different powers of w, thus disproving the original assumption of generalized scaling. For attractive interactions it would be interesting to search for the generalized form of scaling in the data of Fig. 2.
4. Conclusions We have shown with a series of experiments on O/W(llO) that useful information about the energetics of surface diffusion can be extracted from non-equilibrium experiments. We have further compared results obtained with different techniques in real experiments and Monte Carlo simulations. The non-linear nature of the problem and the coverage dependence of surface diffusion, when interactions are present, leads to effective surface diffusion coefficients that depend on the technique used.
5. Acknowledgments Ames Laboratory is operated by the US Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This work was
(1994) 465-470
supported by the Director for Energy Office of Basic Energy Sciences.
Research,
6. References [ll R. Gamer. Rep. Prog. Phys. 53. 917. [2] (a) M.C. Tringides. Phys. Rev. Lett. 65 (1990) 1372: (b) M.C. Tringides, K.R. Roo5, A. Jesina. J. Haas and P. Levenberg, Scanning Microsc.. in press. [3] C.P. Flynn. Phys. Rev. 134 (1964) A241. [4] O.G. Mouritsen. in: Kinetic5 of Ordering and Growth at Surfaces, Ed. M.G. Lagally (Plenum. New York, 1990). [51 A. Jesina and M.C. Tringides, Phy5. Rev. B 48 (19Y3) 2694. [6] J.R. Chen and R. Gamer, Surf. Sci. 79 (1979) 413: M.C. Tringides and R. Gomer. Surf. Sci. 155 (1985) 254. [7] R. Butz and H. Wagner, Surf. Sci. 63 (1977) 448. [S] M. Bowker and D.A. King, Surf. Sci. 94 (1980) 564. [9] G.C. Wang, T.M. Lu and M.G. Lagally, J. Chem. Phy5. 69 (1978) 471. [lo] M.C. Tringides, P.K. Wu and M.G. Lagally, Proceedings of LASST-ACSIS Joint Workshop on Interface Phenomena (Compombello. 1987). [ill J. Haas and M.C. Tringides, to be published. [12] R.B. Hall. T.H. Upton and E. Herbolzheimer, J. Vat. Sci. Technol. B 5 (1987) 1470; S.M. George, A.M. DeSantolo and R.B. Hall. Surf. Sci. 159 (1985) L425. 1131 C.H. Mak and SM. George, Surf. Sci. 172 (1986) 509. [I41 M.C. Tringides and R. Gomer, Surf. Sci. 265 (1992) 2X3.
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::‘,;::.‘.y:::
.‘.:r+.‘.:.:
.Y....
kurface
...
science
._;
,.,.,. :.:,::; ::-..>.-_ ,... :‘:::::.‘:i.::+‘. “““‘,‘.‘. ..,.,.,.,::,::.:.:,:,,..:: .....:.:,.,., ....,., ,..,., ““...., “’:>..‘. ..‘......l... ..,....: ..,., .:::::: ;,. i’..‘.‘,‘.:.>. ..)> ::,., .(, ::~:.,.,,, ?.‘.>: ~.=:...3: ::::,,.,.,. “‘.‘.‘I.: .../....:...:.::::‘.::::> ....i,..: .,,.,. ., :: ,,,:,::
‘2
ELSEVIER
.:..,.,
.j:.:. y::..:y .,.,:::::~i:i:,:i:i:,.:~::,~ ::::.,:,,
‘.:.:‘-:-:‘:‘.‘- :.:.:. ... .... .,.,.,.,.. ,,,,, .,,,,, ,,,:, ..:::..:::.:.:.:.:.:.:.x:.:...: :.>:. .. .,....,.,.,.,.,.,:., ,:,:,,,: ~:,,,,
Surface Science 307-309 (1994) 465-470
Surface diffusion in temporally and spatially inhomogeneous systems J. Haas, K.R. ROOS,A. Jesina, M.C. Tringides * Ames Laboratory, Department of Physics and Astronomy, Iowa State Uniuersity, Ames, IA 5001 I, USA
(Received 20 August 1993)
Abstract Non-equilibrium processes in an overlayer with the system evolving between two different states (i.e., ordering, disordering, adsorption) can be used to study surface diffusion. Diffraction experiments on O/W(llO) at high coverage measure an activation energy E, = 0.6 k 0.05 eV during ordering and E, = 1 + 0.05 eV during disordering. These results are consistent with other diffusion experiments carried out with the fluctuation and Boltzmann-Matano methods. The difference between the measured activation energies can be accounted for in terms of the interactions which contribute differently in different overlayer configurations. Computer simulations on lattice gas models illustrate the dependence on the configuration, by comparing diffusion coefficients obtained with different methods.
1. Introduction The simple picture [ll of surface diffusion as hopping over a static barrier becomes considerably more complex in the presence of interactions. Spatial or temporal inhomogeneities can exist in the system resulting in a distribution of local environments and diffusion barriers. The measured effective diffusion coefficient by a given technique is an average over the local barriers each atom experiences. Results are sensitive on the technique used, since depending on the technique different configurations of the system are probed. It is useful to search for method-invariant parameters to describe the experiments. The difficulty described above can be expressed
* Corresponding author. Fax: + 1 (515) 294 0689.
differently, but equivalently, in terms of a coverage dependent diffusion coefficient D(0), which implies a non-linear diffusion equation. Surface diffusion studies have been, in some cases, restricted to measuring the single particle diffusion coefficient obtained in the limit 8 + 0. This quantity is “clean”, independent of the experiment and directly comparable to first principle calculations. However, in practice under real growth conditions, many particles are present on the surface and surface diffusion is affected by their mutual correlations. It is important to develop other techniques to deal with these realistic conditions. We will describe non-equilibrium experiments [21 to measure the growth rate activation energy and from it to deduce surface diffusion coefficients, during ordering (i.e., when islands are formed out of a random configuration) and disor-
0039-6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0039-6028(93)E0799-Z
466
J. Huas et al. /Surface
Science 307-309
dering (i.e., elimination of initially ordered islands). These techniques used on O/W(llO) will be compared to other diffusion techniques on the same system to show that results are consistent with each other. These conclusions are supported with simulations on lattice gas models which compare different techniques, and demonstrate that the measured quantities depend on the technique, but they are consistent with each other.
2. Non-equilibrium
diffusion
experiments
A system after a quench of one of the thermodynamic variables (temperature, chemical potential, coverage, etc.) is evolving in time towards a new equilibrium state. Studying the temperature dependence can be used to measure surface diffusion. For inhomogeneous systems, one can write [3] a simple expression for the diffusion coefficient as an average over all the possible local barriers D = CP,D,, where I’,, Di are the probability and diffusion coefficient for local configuration i. The Pi are not determined thermodynamically (as in equilibrium), so one needs to know explicitly their time dependence. For ordering processes, where domains form out of a random configuration, one can distinguish two types of atoms, one inside, A, and the other at the domain boundary, B. They have different local barriers with the boundary atoms easier to move. One can refine this argument further by taking into account the type of walls separating domains, but the energy difference for different wall configurations is less than the energy difference between wall and inside atoms, so we can simply treat all the wall particles equivalently,
One expects (PA> to be a monotonically increasing function of time f(t) (with f(O) = 0, f(m) = 1)
If the is of an in the
D,
measurements are restricted to t = 0, then measured (i.e., when the local environment atom is random) and when t + m diffusion final equilibrium state is obtained; for a
(1994) 465-470
time in between, the corresponding appropriately weighted sum is measured. For self-similar growth, the average domain size follows [4] a very simple power law in time, L =A(T)t”, with the time and temperature dependence separated in the above equation so a growth rate can be easily extracted. A single growth rate A(T) controls the evolution over all times. A a 0; should hold for dimensional consistency in the above equation for L. The diffusion coefficient measured from A(T) corresponds to the initial time, random configuration D,. This is physically justified if the experiment is thought of as one of an initially non-uniform concentration profile, given by the domain morphology. The quantity used to probe the evolution, L(t), is sensitive to the motion of boundary atoms, so only the diffusion coefficient of the atoms at the boundary (i.e., the random configuration) is extracted. For disordering processes the exact time dependence of the average domain size L versus t is not known. Both experimental and theoretical evidence [51 suggests that it is not a simple power law, as for the case of ordering processes. One can restrict the experiment to short times, t = 0, and assume that the decay of the average domain size is linear in time, L/L,, = 1 - cDt + . . , with the first term proportional to the diffusion coefficient. By plotting the initial slope of the intensity decay curves versus l/T, the activation energy related to the initial, ordered state is obtained. Other well developed techniques can be used to study surface diffusion in systems with interactions. At equilibrium, the spontaneously induced particle density fluctuations can be measured [6] and the diffusion coefficient can be obtained from the decay of their autocorrelation function. Since the fluctuations are expected to be a small fraction of the average concentration, SC/C -+E1, a small signal is available, so appropriate amplification techniques are necessary. In this case, although the diffusion coefficient is coverage dependent, because the excursions from equilibrium are small, the problem is reduced to one with a constant diffusion coefficient. A different technique [7] to study surface diffusion involves monitoring the time evolution of
.I. Haas et al. /Surface
Science 307-309
the 1D initial step concentration profile. This is a standard method for systems with no interactions. It is still appropriate to extract the full coverage dependence of the diffusion coefficient from the evolving profile lineshape c(x, t), if it can be shown that it obeys the scaling c(.~/t’/~). The so-called Boltzmann-Matano (BM) method gives an expression for the full coverage dependence,
qc’)
=
L
the same activation energy. Table 1 lists the results of five different diffusion techniques [2,6-S] applied on the same system, O/W(llO). Some of the experiments were carried out over a range in coverage and as expected, the activation energies are coverage dependent because of the interactions. We restrict the comparison in Table 1 at high coverage, so any differences will be simply attributed to the configuration of the overlayer and not the coverage. Four of the methods used, including the equilibrium one, give, remarkably, almost the same value for the activation energy, E = 1 k 0.05 eV, while the other one, based on non-equilibrium ordering experiments, measures a [2a] substantially lower value, E, = 0.6 k 0.05 eV. This system has been described [91 very well in terms of a lattice gas model that includes competing attractive nearest neighbor and repulsive nextnearest neighbor interactions, c$, = -&, = 0.09 eV. As explained above, the ordering experiment probes the system in the random configuration where the two types of interactions cancel each other, while the disordering and equilibrium experiments measure diffusion in the well-ordered (2 X 1) + (2 X 2) configuration where there is higher probability to have nearest than nextnearest neighbors occupied. The experiments based on the evolution of initial, abrupt profiles surprisingly give the same value; this is not expected because the local configuration of a diffusing atom at the moving boundary is different from the configuration at equilibrium (even for low resolution probes a sharp boundary can still
1 ndc
‘cI
2t (dc/dn),f
’
where n = n/t ‘I2 is the scaled variable. It is remarkable that a single lineshape c(n) can provide the full coverage dependence when the scaling condition holds. It is still not clear what is the range of validity of the condition. Although, it can always be tested empirically whether it is true, it would be useful to determine its validity in terms of the interaction strength, temperature range, etc. It is most likely that for strong interactions the scaling 4x, t) = c(.x/~‘/~) is violated because of build up of particle correlations. In principle, this step profile method is a spatially non-equilibrium one, with the local environment of the atoms at the boundary (which mainly controls the evolution) different from the equilibrium one, so it should be expected to give different activation energies than the fluctuation method. Only when the step profile is not infinitely steep (which can result from resolution limitations of the probe and imposes an averaging over the resolution distance AX) when compared to the equilibrium density fluctuations, Ax(dc/dx) <
Table 1 The table summarizes disordering, fluctuation,
the activation energies and prefactors Boltzmann-Matano and patch profile
467
(W,., one expects [l] the two techniques to give
rc’
1
(1994) 465-470
obtained on O/W(llO) with different analysis give the same activation energy
experimental
Process
Technique
Ed (eV)
D, km’/s)
Ref.
Ordering
LEED
0.6
10-7
121
Disordering
LEED
1.0
_
151
Field emission
1.0
10-4
161
1D Boltzmann-Matano
Kelvin probe
1.1
0.38
[71
Patch profile
SEM
1.1
0.2
181
Equilibrium
fluctuation
techniques;
16X
J. Hum
rt al. /Surface
Sciencr 307-309
be defined). In addition, the temperature range used in these ID profile experiments is well within the disordered region, where the two types of interactions should cancel each other. It is unclear why there is good agreement between the profile evolution results and the equilibrium results. Table 1 also lists the diffusion prefactors, when available. There is some discrepancy between the listed values. We have not attempted to extract the exact prefactors from ordering/disordering experiments because other parameters, as can be seen from the cases when the growth law is known [LO] exactly, control the relation between the growth rate and activation energy. We can introduce an absolute length scale for the average domain size by using the full-width at half-maximum (FWHM) of the changing diffracted spots. In the ordering experiment [2a] the domains increase from 3 to 120 A, in the disordering experiment [5] the average size stays the same, while the number of domains is changing over the time interval of the experiment, 2000 s. We can estimate a prefactor D,, = 1O.-7 cm2/s in the ordering experiment by assuming that all the other constants are of order unity, but clearly this is an oversimplification. This value is different from the other values listed in Table 1 and is similar to the value obtained at low coverage with the fluctuation method [6].
3. Simulations We have performed studies of surface diffusion with Monte Carlo simulations on lattice gas models, where all the factors affecting diffusion are easily controlled. The objective is to compare different experimental techniques on a model with the simplest type of interactions, i.e., nearest neighbor (both repulsive and attractive, which produce drastically different overlayer configurations) over a wide temperature range inctuding T,. The details of the simulation can be found elsewhere [ 111. First, we wanted to address the previous puzzling result, why the 1D BM analysis and fluctuation method give the same activation energy, al-
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0.I
A
n
I
. . ‘
.
Fig, 1. Arrhenius plots of D vs l/T for the lattice gas model with nearest neighbor interactions obtained with three different methods. The fluctuation and the Boltzmann-MataIl(~ methods agree with each other while the LID method gives different results.
though the system is interrogated in different configurations. Especially since sharp, atomic scale boundaries are used in the simulations, the issue of experimental resolution is irrelevant. Fig. 1 shows that over a wide temperature range, - 2 < -J/kT < 2 (i.e., both for attractive and repulsive interactions with 1J/kT, ! = 1.86), the diffusion coefficient obtained with the fluctuation method is essentially identical to the one obtained with the 1D BM method. The agreement between the two methods is remarkable but is not clear if it is only true for this type of interaction. The required scaling condition c(x, t) = c(x/ 6) is satisfied at all temperatures used, thus extending the validity of the method at least for this model into the low temperature ordered region. Fig. 1 also shows a comparison with the iaserinduced-desorption (LID) method [ 121, a different technique used to study weakly bound overlayers (a method which cannot be used on O/W(l lo), a strongly bound system). The LID diffusion coefficient is obtained in the simulation from the initial slope of the refilling signal as
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done experimentally, although other types of analysis have been proposed [13] for the case of coverage dependent diffusion coefficient. The comparison is motivated because of the similar geometry between the LID and fluctuation methods: a probe hole in the middle of the surface having the same coverage as the outside region, method, while 'in 2 'oout~ simulates the fluctuation ein = 0 simulates the LID method. As pointed out before [14], the measured diffusion coefficient for systems with interactions depends on e,,, so the fluctuation and LID methods give different results, shown also in Fig. 1. The reason for the discrepancy between the fluctuation and LID methods can be related to the type of the measured quantity in LID, the refilling signal back to the probe, which is an integral over the moving profile. Information is lost during the integration. Can some type of BM analysis be used [14] with the LID geometry so the correct coverage dependent diffusion coefficient can be extracted? It has been shown before, that even with no interactions, this is not possible and the concentration profile cannot be written in terms of the required scaling for BM analysis, c(r, t) + c(r/ fi). Is there a different type of scaling for the concentration profile so the coverage dependent diffusion coefficient can be extracted? We have looked at the evolving profile, both for repulsive and attractive interactions, and plotted them in a scaling form to test if they follow the same functional form at all times. We have taken the edge of the probe R, area as the origin and define a new variable p = R, - Y.The data were plotted in the form c(p, t)/c(O, t) versus p/w(t), where w(t) is the full-width at half-maximum (FWHM) of the evolving profile. It is interesting that, as seen in Fig. 2, for repulsive interactions the results collapse onto a single universal curve, while for attractive interactions they do not. This conclusion was confirmed for all temperatures, the deviation from scaling for attractive interactions being more pronounced the lower the temperature. One can evaluate in several ways the significance of this result. First, although C(T, t) cannot be written in scaling BM form, at least for repul-
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469
1.2 ~~.~
1.0
1
F
Repulsive interactions
1
0.8 -
.
0.6 1
X=
l
t,=O.I (t$=O.169) t,=0.2 ($,=0.244)
c
0 5 a v
F
0.4 !~
Attractive ink&a
t,=0.01 (eP=o.044) x IJ b 0 0
“so
x
c X
t,=0.05 (eP=o.21) t,=0.4 (C$=O.O65) G=l.O (eP=o.091) t4=l .5 (ep=o. 11) ts=5.0 (eP=o.15)
x x
Fig. 2. The concentration profile in an LID experiment measured with the edge of the probe area as the origin. For
repulsive interactions c(p, t)/c(O, t) vs p/w scale (i.e., the same functional form describes the evolution) while for attractive interactions they do not. ~9~denotes the coverage in the probe at time t,.
sive interactions (which promote the formation of the c(2 x 2) open structures that resemble the non-interactive case), one can write C(T, t> in the scaling form c[(R, - r)/w(t)l and possibly use this to simplify the non-linear diffusion equation. Second, if fits are attempted to the evolving profile at any time (either for the LID geometry or the complementary problem of an initially prepared overlayer patch 8, # 0, 0,,, = 0 diffusing into the vacant surface), then at least for some interactions (i.e., attractive which promote dense, highly correlated (1 X 1) structures) the extracted diffusion coefficient depends on the time chosen, since different functional forms describe the evolution of the profile. Third, if a different generalized scaling combination is searched for, C(T, t) = c(r/tk) with k # l/2, one can show this is not possible. We briefly sketch the argument. The total mass in the probe area increases as m = t2k, if generalized scaling holds. We have also calculated in the simulations the FWHM w of the complementary profile, c(p, t), which grows in time, w = ts (with 8 I l/2), so the generalized scaling of C(T, t) implies scaling of c( IR,,PI/Wk/6)for any R,.If we take R, = 0 it would imply that the generalized scaling should be of the form c(p/w k/6). It can be shown from the conservation of mass that k > 6, so for repulsive
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interactions it is impossible to have c(p, tl to scale with two different powers of w, thus disproving the original assumption of generalized scaling. For attractive interactions it would be interesting to search for the generalized form of scaling in the data of Fig. 2.
4. Conclusions We have shown with a series of experiments on O/W(llO) that useful information about the energetics of surface diffusion can be extracted from non-equilibrium experiments. We have further compared results obtained with different techniques in real experiments and Monte Carlo simulations. The non-linear nature of the problem and the coverage dependence of surface diffusion, when interactions are present, leads to effective surface diffusion coefficients that depend on the technique used.
5. Acknowledgments Ames Laboratory is operated by the US Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This work was
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supported by the Director for Energy Office of Basic Energy Sciences.
Research,
6. References [ll R. Gamer. Rep. Prog. Phys. 53. 917. [2] (a) M.C. Tringides. Phys. Rev. Lett. 65 (1990) 1372: (b) M.C. Tringides, K.R. Roo5, A. Jesina. J. Haas and P. Levenberg, Scanning Microsc.. in press. [3] C.P. Flynn. Phys. Rev. 134 (1964) A241. [4] O.G. Mouritsen. in: Kinetic5 of Ordering and Growth at Surfaces, Ed. M.G. Lagally (Plenum. New York, 1990). [51 A. Jesina and M.C. Tringides, Phy5. Rev. B 48 (19Y3) 2694. [6] J.R. Chen and R. Gamer, Surf. Sci. 79 (1979) 413: M.C. Tringides and R. Gomer. Surf. Sci. 155 (1985) 254. [7] R. Butz and H. Wagner, Surf. Sci. 63 (1977) 448. [S] M. Bowker and D.A. King, Surf. Sci. 94 (1980) 564. [9] G.C. Wang, T.M. Lu and M.G. Lagally, J. Chem. Phy5. 69 (1978) 471. [lo] M.C. Tringides, P.K. Wu and M.G. Lagally, Proceedings of LASST-ACSIS Joint Workshop on Interface Phenomena (Compombello. 1987). [ill J. Haas and M.C. Tringides, to be published. [12] R.B. Hall. T.H. Upton and E. Herbolzheimer, J. Vat. Sci. Technol. B 5 (1987) 1470; S.M. George, A.M. DeSantolo and R.B. Hall. Surf. Sci. 159 (1985) L425. 1131 C.H. Mak and SM. George, Surf. Sci. 172 (1986) 509. [I41 M.C. Tringides and R. Gomer, Surf. Sci. 265 (1992) 2X3.