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Atomic mechanisms for the diffusion of Mn atoms incorporated in the Cu(100) surface: an STM study
surface science ELSEVIER
Surface Science 371 (1997) 1 13
Atomic mechanisms for the diffusion of Mn atoms incorporated in the Cu(100) surface: an STM study T. F l o r e s , S. J u n g h a n s , M . W u t t i g * lnstitut ffir Grenzfliichenforschung und Vakuumphysik, Forschungszentrum Jfilich, 52428 Jfilich, German), Received 12 March 1996; accepted for publication 18 July 1996
Abstract
The mobility of Mn atoms incorporated into the Cu(100) surface has been studied by scanning tunneling microscopy (STM). For small Mn coverages, a diffusion coefficient of 5.3 x 10 is cm 2 s 1 has been determined from analysis of the mean square displacement. This low mobility can only be explained by a diffusion mechanism that is based on diffusing vacancies. Upper and lower limits of the jump rate of incorporated Mn atoms have been derived from the STM images for several coverages. This analysis shows an enhanced mobility of Mn around 0.3 ML Mn. In this coverage range, fast ripening of alloy islands is observed. This indicates that vacancies also play a decisive role for Ostwald ripening in this system. A more detailed understanding of vacancy mediated motion of incorporated atoms should emerge, if both the diffusion coefficient and jump rate of incorporated atoms could be determined with high precision.
Keywords: Adatoms; Alloys; Compound formation; Copper; Low index single crystal surfaces; Manganese; Scanning tunneling microscopy; Surface diffusion
1. Introduction This decade has witnessed a renaissance of studies of adatom diffusion on surfaces [1 7]. Driven by the need to control and manipulate the growth of ultrathin films on a monolayer scale, interest has focused on characterization of atomic mechanisms governing thin-film growth. In particular, the diffusion of single adatoms on a substrate surface, along island edges and across steps, have been shown to be of paramount importance [-4,5,8]. Hierarchies of diffusion mechanisms have been developed which can account for many of the observed growth morphologies and island shapes [3,6,7]. The focus of this study is a related problem, the * Corresponding author. Fax: +49 2461 6139070.
diffusion of an atom incorporated in (rather than adsorbed on) the substrate surface. So far, this problem has drawn much less attention. For the system studied, Mn on Cu(100), at and above room temperature, alloy formation is observed. In a recent study [-9], we have investigated the atomic mechanism of Mn incorporation. In the present publication, the main goal is the determination of the atomic mechanisms responsible for the motion of incorporated Mn atoms. Our experimental setup, previous studies of Mn on Cu(100), and the mechanisms of alloy formation have been discussed at length in a recent paper [-9]. Therefore we refer the reader to this study for details on the system studied and the experimental set-up. We will present the experimental results in Section 2 and a discussion in Section 3. A summary and the outlook are the subject of Section 4.
0039-6028/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S 0 0 3 9 - 6 0 2 8 ( 9 6 ) 00978-8
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T. Flores et al./Surface Science 371 (1997) 1-13
2. Results
2.1. Mobility of incorporated Mn Fig. 1 shows the surface after deposition of 0.020+0.004 ML Mn. A substrate step and twodimensional islands of monoatomic height are readily discernible. Additionally, small protrusions exist. Neither the islands nor the protrusions are present on the clean Cu(100) surface prior to deposition. We have identified the protrusions as single, incorporated Mn atoms [10]. Their apparent size is larger in many STM images than expected for a single atom. This is probably due to a convolution of the tip shape and the protrusion. The incorporated Mn atoms are mobile, as demonstrated in Fig. 1. The white dots denote the
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Fig. 1. STM images of the Cu(100) substrate after deposition of M n at 300 K. Coverage: 0.020+0.007 M L Mn, scan width: 350 ,& x 350 A. In order to visualize the mobility of the incorporated M n atoms, two subsequently recorded STM images are superimposed. The white dots mark the positions of the incorporated M n atoms 180 s before the image shown was recorded. The white frame indicates the margin of the first with respect to the second recorded image. The images do not match exactly because of thermal drift.
positions of the Mn atoms 180 s before the image shown was recorded. By comparing the position of atoms in Fig. 1, a lower limit for the total jump frequency Ftot of the incorporated Mn atoms was estimated. At 0.1 ML Mn, each Mn atom performs on average at least one jump in the time Atl between the first and second STM picture, taken 180s later. Therefore the total jump frequency satisfies/'tot ~ 1~Aft. An upper limit can be derived from the fact that during the scan of a single Mn atom, the vast majority of atoms does not jump. Therefore, Ftot < 1/Ats, where Ats is the time necessary to scan a single Mn atom. Combining the upper and lower limit, a jump frequency between 1 . 1 x l 0 -2 and 3 . 9 x 1 0 - 1 s 1 is estimated for 0.1 M L Mn. In an analogous way, the jump frequency at 0.5 ML Mn can be obtained. Fig. 2a depicts an image of the c ( 2 x 2 ) surface alloy at 0.45_+ 0.02 M L Mn. Although a pronounced long-range order is observed, a large number of point defects exists. Since only the sublattice formed by the Mn atoms is imaged, the point defects could either be vacancies or Cu atoms located at positions of the Mn sublattice. At room temperature, the equilibrium concentration of vacancies should be much smaller than the concentration of point defects visible in Fig. 2a. Furthermore, vacancy jump rates at room temperature should be of the order of 9 x l0 s s-1. This again is in contrast with our observation, i.e. the fact that we can image a point defect during the time it takes to scan across it. Therefore, we believe that the point defects are Cu atoms located at the positions of the Mn sublattice. Further support for this comes from the fact that the number of incorporated Mn atoms per substrate site equals the fraction of the terrace which is covered by islands. Both "coverages" should be different if the point defects are vacancies. Using the same methodology as described in the case of small coverages, a jump frequency between 1.1 x 10 -2 and 1.3 s -1 is derived at a coverage of approximately 0.45 M L Mn. These values are comparable with the observed jump rate of incorporated Mn atoms at low coverages. However, this similarity might be fortuitous, since at 0.45 ML Mn the mobility of Cu in the substrate surface
T. Flores et al./Surface Science 371 (1997) 1-13
a)
b)
!ii
Fig. 2. STM images of the Cu(100) substrate after deposition of Mn at 300 K. (a) Coverage: 0.45+0.02 ML Mn, scan width: 180 A, x 180 A. (b) Coverage: 0.31 +0.03 ML Mn, scan width:
90A×90A. layer in close proximity to Mn is analyzed. On the contrary, at low coverages, the mobility of single, isolated Mn atoms incorporated into the Cu(100) substrate surface is analyzed. Fig. 2b shows the surface after deposition of 0.31_+0.03 ML Mn. In this coverage regime, one
3
always observes "fuzzy" images as in Fig. 2b, indicating a high mobility. This mobility cannot be caused by Mn or Cu adatoms. Since both Cu and Mn adatoms become mobile on Cu(100) at around 140 and 110 K, respectively, at room temperature the mobility of the adatoms is too large to image them. Therefore, like at small coverages, the mobility has to be caused by diffusing atoms in the surface layer. The STM data show that most of the incorporated Mn atoms do not jump within the time necessary to scan their width. On the other hand, most of the incorporated Mn atoms are not completely imaged. This is in contrast to our findings for coverages below 0.2 ML Mn and above 0.4 ML Mn, where most incorporated atoms are completely imaged (see Figs. 1 and 2a). Hence the mobility is considerably enhanced around 0.3 M L Mn. Since at this coverage most incorporated Mn atoms are immobile during the time it takes to scan their width, but move within the time it takes to image them completely, an upper and lower limit of the mobility can be estimated. This leads to a jump rate between 2.2 and 102 s -1 for 0.3 M L Mn. Also at higher coverages, e.g. around 0.45 ML Mn, where large parts of the substrate already show a c(2 x 2) arrangement, we can see such "fuzzy" regions of varying size. These regions are often in the vicinity of steps or island edges. The estimates discussed so far focus on the jump rate. From the jump frequency, the diffusion coefficient can be derived, once a model for the underlying atomic mechanism exists. We assume that the diffusion takes place via nearest-neighbor jumps of distance a, and additionally neglect correlation effects, which are discussed in Appendix A. Then, the diffusion coefficient D can be derived from the total jump frequency by D=a2Ftot/Z. Here z is the number of nearest neighbors. At very small coverages, one can follow the course of the incorporated Mn atoms. In this case, a direct determination of the diffusion coefficient is possible by evaluating the Einstein relation. To avoid the necessity to correct for thermal drift and distortion effects of the STM, we analyze the change in the distance between two incorporated Mn atoms with time [11]. Then the Einstein relation becomes
4
T. Flores et al./Surface Science 371 (1997) 1 13
projected on a fixed axis, in the time At. Due to the motion of the incorporated Mn atoms, the time interval between the first and the second scan of one particular atom is, in general, not just the time interval between two scans. This effect can be neglected for sufficiently large scan widths, however, since the mobility of the Mn atoms is relative low. Fig. 3a depicts ((AX) 2) as a function of At for a coverage of 0.0033 _+0.0005 ML Mn. The number of considered events decreases from small to large values of At from approximately 1000 to 100. The straight line is the least-square fit to the data points. It describes the experimental observations rather well and justifies an analysis using the Einstein relation. From the slope, the diffusion
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coefficient for incorporated Mn in the limit of small coverages can be determined to 5.3_+0.3 x 10-~8 cm 2 s -1. We have checked carefully whether the diffusion of incorporated Mn atoms could be influenced by the tip of the STM. This should result in different diffusion coefficients in the scan direction and perpendicular to it, which is contrary to our analysis. This implies that the experimentally determined diffusion coefficient is not influenced by the presence of the tunneling tip. If we use the upper and lower limit of the jump frequency at small coverages and assume nearestneighbor jumps and a correlation factor of one, this leads to a diffusion coefficient between 1.8 x 10 -18 and 6.4 x 10 17 cm 2 s-1. The diffusion constant determined directly from the Einstein relation falls into this range. In Fig. 3b, the distribution of AX is shown for At=233 s and At=696 s, respectively, and compared to the theoretical density function which is given by W(AX,At)=(8nDAt) -~ exp(-(AX)2/ 8DAt). Both the experimental and theoretical distributions agree very well, within the statistical error. In Fig. 4, the coverage dependence of the diffusion coefficient is summarized. The single data point represents the diffusion coefficient in the limit of very small Mn coverages, determined by the evaluation of the Einstein relation. The horizontal bars describe the estimated diffusion coefficient derived from the jump frequency. The latter were determined assuming nearest-neighbor jumps and neglecting correlation effects. The jump rates show
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Fig. 3. (a) The mean square of the distance change AX between two incorporated M n atoms is depicted as a function of time At. The number of analyzed distances decreases from approximately 1000 for the smallest to 100 for the largest time At. From the slope of the fit (straight line), the diffusion coefficient of the incorporated M n can be determined to be D = 5.30+_0.32 × 10 is cm 2 s-1. The coverage was 0.0033+ 0.0005 M L Mn. (b) The experimental (solid bars) and theoretical (hollow bars) distributions of the distance change AX in time At are shown for two different times, A t = 2 3 3 s (left side) and A t = 6 9 6 s (right side) at a M n coverage of 0.0033 _+0.0005 ML.
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D [10-14cm2s1] Fig. 4. The estimate for the diffusion coefficient D at different M n coverages O. For each coverage, the diffusion coefficient falls in the range marked by the error bars. The diffusion coefficient was estimated from the total j u m p frequency, neglecting correlation effects and assuming nearest-neighbor jumps. The single data point is the diffusion coefficient as derived from the Einstein relation.
T. Floreset al./Surface Science 371 (1997) 1 13
a clear m a x i m u m in the intermediate coverage regime around 0.3 M L Mn, where the j u m p frequency is between one and two orders of magnitude higher than at 0.1 M L Mn. As mentioned above, for 0.1 M L Mn, we find a good agreement between the diffusion coefficient as determined directly through the Einstein relation and the estimate based on the j u m p rate. Concerning the higher coverages, around 0.3 M L Mn and above, the diffusion coefficient cannot be determined directly since we are unable to follow the path of individual incorporated Mn atoms with time. At these coverages, however, we can exploit the pronounced mobility of the incorporated Mn atoms. Using a statistical analysis based on time correlation functions, which is explained in detail in Appendix B, we have determined the probability ( 1 - WR) that an incorporated Mn atom makes at least one j u m p in the time interval AtL. This leads to a lower limit of the total j u m p rate as explained in Appendix B. For a coverage of 0.31 M L Mn we obtain/"to t _>3 s 1. 2.2. Late-stage island growth
Section 2.1 focused on the statistical motion of individual atoms. However, the STM images also reveal information about the directional motion required to achieve mass transport. An example is displayed in Fig. 5, which shows a number of islands 1.5 h after deposition (Fig. 5a) and 6 h after deposition (Fig. 5b). A growth of larger islands at the expense of smaller islands is observed. This is characteristic of a ripening process. The STM images reveal that the islands are not mobile and therefore do not grow by dynamic coalescence. Since the coverage does not change during ripening, a net mass transport, like adatom diffusion between small and large islands, has to be effective. Such a late-stage growth is usually called Ostwald ripening [12]. In Fig. 6a, the island density is shown in 1 h intervals after Mn deposition as a function of the coverage. For all coverages, the island density decreases with time, but the magnitude of the change shows a clear coverage dependence. The effect is most pronounced at coverages around 0.3 M L Mn. Here the island density decreases by about a factor of three in the observa-
5
a)
b) f
Fig. 5. STM images of the Cu(100) substrate after deposition of Mn at 300K. Coverage: 0.35_+0.04ML Mn, scan width: 2800 A x 2800 A. The STM images in (a) and (b) were taken 1.5 and 6 h after deposition, respectively. tion time. For this coverage, in Fig. 6b the distribution of island sizes is depicted for different time intervals after Mn deposition. A dramatic transfer from small to large sizes is observed with time. The average island size increases from 3.0 x 104 to 8.7 x 104 /~2.
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T. Flores et aL/Surface Science 371 (1997) 1 13
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Fig. 6. (a) Island density as a function of the Mn coverage. For each coverage, the island density was determined for different times (marked by different symbols) after deposition. For a fixed coverage, the island density decreases due to a ripening process. This effect is most pronounced at intermediate coverages of around 0.3 ML Mn. (b) Island size distribution 1 2, 3 4 and more than 5 h after deposition for a coverage of 0.31_+0.03ML Mn. With increasing time, the island size increases.
3. Discussion: diffusion of incorporated Mn First, we will identify the mechanism responsible for the diffusion of incorporated Mn atoms. To
facilitate this task, we will initially restrict the discussion to small coverages around and below 0.1 M L Mn. Beginning with the simplest mechanisms, an exchange process h o r i z o n t a l to the surface enables the diffusion. A related process is the ring mechanism, where four atoms rotate in the surface. These processes are rather unlikely in a closepacked arrangement of atoms, since the resulting activation barrier for these processes would be extremely high. Therefore, we do not believe that these mechanisms are reasonable candidates for the diffusion of incorporated Mn on Cu(100). Similar arguments have also been put forward for the bulk diffusion of impurities in Si, for example [13]. Another conceivable scenario is the following: in a first step, an incorporated Mn atom is ejected onto the terrace. Possibly this happens via a C u / M n place exchange. Then a subsequent diffusion of the Mn adatom on the surface takes place. This is finally followed by an incorporation of the Mn adatom through a Mn/Cu place exchange. This is illustrated in Fig. 7. For the Mn/Cu exchange process, we can make an Arrhenius "ansatz". The activation barrier for such an exchange can be estimated from the onset temperature for alloy formation, which is around 260 K on Cu(100). This leads to a considerable lifetime of a Mn adatom on the surface. On the other hand, such a Mn adatom can move rapidly on the surface. A lower limit for the diffusion coefficient of Mn adatoms on Cu(100) can be derived from the observation that the c(8 x 2) film structure can be formed at temperatures as low as l l0 K. This gives an upper limit for the onset temperature of diffusion of 110 K. Then, from the Einstein relation the average distance between the ejection and incorporation site can be obtained. This results in a distance of several thousand A, which is at least between two and three orders of magnitude higher than the experimentally observed distance. Hence we have to exclude the scenario described above. The experimental results can be successfully explained, however, if diffusing vacancies cause the mobility of incorporated Mn (Fig. 8). For such a mechanism the total j u m p frequency of incorporated Mn is given by ~r~to t = Z X CV X [ ' V " Here, z is the number of nearest neighbors, F v is the vacancy
T. Flores et al./Surface Science 371 (1997) 1 13
7
a) a)
b) b)
c) c)
d)
Fig. 7. Diffusion of incorporated Mn by an ejection and subsequent surface diffusion process. An incorporated Mn atom is ejected to the surface, e.g. by a Cu/Mn place exchange (a). The ejected Mn atom starts a diffusion process (b) and is finally incorporated by a Mn/Cu place exchange (c), (d). White circles denote Cu atoms, gray circles Mn atoms. j u m p rate, a n d Cv is the c o n c e n t r a t i o n of the vacancies. In the following, we will use for Cv the e q u i l i b r i u m c o n c e n t r a t i o n of vacancies as found on Cu(100). This e q u i l i b r i u m c o n c e n t r a t i o n is given b y Cv = e x p ( - E v / k T ) , where EF is the formation energy of a vacancy. At r o o m t e m p e r a t u r e , the v a c a n c y c o n c e n t r a t i o n is typically very low. If we use a f o r m a t i o n energy of vacancies of E v = 0.47 eV, as c a l c u l a t e d by Stoltze for Cu(100) [ 14], we o b t a i n Cv ,~ 10 - s . T h e j u m p rate of the v a c a n cies is F v = v x e x p ( - E M / k T ) , where v is the a t t e m p t frequency a n d EM the v a c a n c y m i g r a t i o n
d)
Fig. 8. Diffusion of incorporated Mn atoms caused by diffusing vacancies. The vacancies move from the right to the left. As a consequence, the incorporated Mn atom jumps from the left to the right. White circles denote Cu atoms, gray circles Mn atoms.
energy. In the following, we will assume t h a t the diffusion of vacancies is n o t affected b y the presence of Mn. Then, the m i g r a t i o n energy of vacancies on Cu(100) can be used, which has been calculated to EM = 0.44 eV [ 14]. A similar value of 0.42 eV has been o b t a i n e d in Ref. [ 15]. A s s u m i n g a typical a t t e m p t frequency of 1013-+1 s -1, a t o t a l j u m p frequency of F t o t = 1 0 -2+1 s -1 is derived. This is in g o o d a g r e e m e n t with the e x p e r i m e n t a l l y o b s e r v e d u p p e r a n d lower limit for the j u m p rate, which lies between a p p r o x i m a t e l y 10 - 2 a n d 5 x 10 -1 s -1 for 0.1 M L Mn. As m e n t i o n e d in Section 2.1, we have also determ i n e d the diffusion coefficient D from an evalua-
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T. Flores et al./Surjace Science 371 (1997) 1 13
tion of the Einstein relation at 0.0033_+ 0.0005 M L Mn. In Appendix A, the relation between the diffusion coefficient D and the total j u m p frequency for a vacancy mediated motion is discussed. There it is shown that for vacancy-mediated diffusion, Frot = zD/fa 2, with f = 0.4669 on an fcc (100) surface in the limit of small vacancy concentrations. The factor f arises since the atoms do not perform a random but a correlated walk [ 16]. This results in a total jump frequency of 6.9_+0.4 x 10 2 s-1. This again is in good agreement with the theoretically estimated jump frequency for vacancy mediated diffusion of Mn. Considering the existence of vacancies, another process seems possible which is very similar to the process illustrated in Fig. 7. Instead of a Mn/Cu place exchange, an alternative way for the last step could be an incorporation by vacancy annihilation. In other words, the ejected Mn adatom diffuses on the terrace until it meets a vacancy and is then incorporated, thereby annihilating the vacancy. However, also in this case, due to the extremely small equilibrium concentration of vacancies at room temperature, the mean diffusion length of incorporated Mn would be between two and three orders of magnitude higher than the experimentally observed value. We can thus conclude that only a vacancydriven motion of incorporated Mn can account for the observed mobility. Vacancy diffusion as a mechanism for diffusion of incorporated atoms has also been discussed in a similar context [17]. These authors studied the adsorption and incorporation of In on Ag(100) and Ag( 111 ) by perturbed angular-correlation (PAC) spectroscopy. They attributed the incorporation to adatom diffusion and subsequent attachment to a step, followed by incorporation through diffusing vacancies into the upper terrace. In the following we want to address the assumptions implicit in our theoretical estimate of Mn mobility in more detail. In applying Ftot= Z XCv x Fv, we have made three assumptions. (i) The equilibrium concentration of vacancies is not affected by incorporated Mn atoms. (ii) The j u m p rate of vacancies is not affected by incorporated Mn atoms. (iii) The concentration of vacancies corresponds to the equilibrium concentration,
i.e. we consider a stage where the rates of vacancy creation and vacancy annihilation are in equilibrium. This last assumption is justified since we observe Mn mobility long after Mn incorporation. Hence, the vacancy concentration should be equilibrated. More difficult to justify are the first two assumptions, which imply that the vacancies are not affected by Mn atoms. The diffusion of Mn depends on both vacancy motion and vacancy concentration, i.e. on E M + E F. The analysis presented above shows good agreement between the mobility of incorporated Mn in the Cu(100) substrate surface and the theoretical model, which strictly only holds for vacancy-mediated motion in the clean Cu(100) surface. Therefore we can only conclude that E M + E F is similar on Cu(100) and after incorporation of small amounts of Mn on Cu(100). A precise determination of both D and Fto t at small coverages would allow us to test assumptions (i) and (ii) individually, at least to some extent. Let us assume, for example, that the j u m p rate of a vacancy in the direct vicinity of a Mn atom is considerably enhanced. Then, the incorporated Mn atom would perform many correlated jumps where the vacancy changes place with the Mn atom. The resulting j u m p rate would be high. The diffusion coefficient, however, would not increase to a similar extent, since this site oscillation would not lead to any significant atomic displacement. Hence the correlation factor f would be much smaller than the ideal value of 0.4669 for an fcc (100) lattice. From a precise determination of D and Ftot, one could determine f with high accuracy. Since for low Mn coverages we can only determine an upper and lower limit for the j u m p rate, we can only estimate in which range f falls. This results in a wide range of values for f, which lies between 0.08 and 2.94. Therefore we cannot determine experimentally, whether assumptions (i) and (ii) are independently fulfilled for Mn atoms incorporated into the Cu(100) substrate surface. To reiterate the statement, a more precise determination of the total j u m p frequency and the diffusion coefficient would allow a more quantitative analysis of the underlying vacancy mechanism. So far, we are not aware of any experimental study of diffusion of incorporated atoms which has
T. Flores et al./Surface Science 371 (1997) 1 13
reached this demanding and ambitious goal. Yet, such an investigation would not only result in a better understanding of the limitations implicit in our theoretical analysis and a precise determination of the correlation factor, but also improve our general understanding of vacancy-mediated diffusion of incorporated atoms. In passing, we note that increasing the temperature from room temperature to 340 K, raises the diffusion coefficient by approximately two orders of magnitude. Therefore, within a few minutes the diffusion length of the incorporated Mn atoms approaches the average island distance. This leads to a homogeneous Mn distribution, in agreement with the experimental observation [9]. For higher coverages, around 0.3 M L Mn and above, the discussion of the diffusion mechanism becomes more difficult. The assumption of a vacancy-driven process is supported by the fact that such a model also offers a plausible path for the ordering of the system at higher coverages. This is illustrated in Fig. 9. A striking point is the strong increase of the mobility of incorporated Mn atoms around 0.3 M L Mn. Using D=faZ/"tot/Z, a lower limit for the diffusion coefficient D_>1.7x10 -16cm 2 s 1 can be derived from the j u m p frequency/"tot (see Fig. 2c). A similar estimate is given by N o h et al. [18] with D > 1.5 x 10 -16 cm 2 s -1 for a comparable coverage [18]. In terms of the vacancy model, the enhancement of the mobility of incorporated Mn is formally explained either by a higher j u m p rate of the vacancies, which corresponds to a decrease of the vacancy migration energy EM, or by a larger equilibrium concentration of vacancies, i.e. a decrease of the vacancy formation energy E v. A change of two orders of magnitude for the total j u m p frequency of incorporated Mn, as experimentally observed from small to intermediate coverages, corresponds to a decrease of the sum E F + E M by approximately 0.12 eV. Such a decrease seems possible within the framework of a coverage dependence of g F and/or EM. Support for a pronounced composition dependence of E v comes from studies of the bulk diffusion in binary alloys. These investigations have shown that in ordered alloys of the DO3-type, a fast diffusion process is often effective. An example is
9
a)
b)
Fig. 9. Atomsin a disordered area (a) are arranged in a c(2 x 2) structure after a vacancy has passed (b). The path of the vacancy is shown by the black line in (b). White circles denote Cu atoms, gray circles Mn atoms and the square the vacancy. Fe3Si. In many of these ordered alloys, the vacancy concentration is up to a few orders of magnitude higher than in pure metals. It is assumed that this high vacancy concentration causes the fast diffusion in several of those alloys [-19,20]. Such a scenario could also account for the high mobility we observe for Mn incorporated in the Cu(100) substrate surface at higher coverage. Interestingly enough, Mn is not the only deposit on Cu(100) for which a coverage dependence of the mobility is found. Qualitatively, the same observation was made for Pb. Pb also forms ordered surface alloys on Cu(100) [-21,22]. The mobility of incorporated Pb atoms shows a pronounced m a x i m u m in an intermediate coverage regime and is strongly reduced at higher and lower coverages [-21,23]. It is remarkable that for both Pb and Mn, the increase in the mobility of the incorporated atoms is clearly correlated with an
10
T. Flores et al./Surface Science 371 (1997) 1 13
ordering process. At small coverages, the surface alloys are disordered. Here, the mobility is relatively low. In the intermediate coverage regime, where the ordering proceeds, the mobility is strongly enhanced. At high coverages, where the c(2 x 2) and c(4 x 4) are observed for Mn and Pb, respectively, the mobility is again relatively low. We attribute this coverage dependence to the interactions between the incorporated atoms. To explain the observed superstructures, the interaction between incorporated atoms should be repulsive on nearest-neighbor sites. At least for Mn, this observation is supported by the fact that, independent of coverage, no nearest-neighbor distances between two incorporated Mn atoms were found in the STM images. At low coverages, the mean distance between the Mn atoms is sufficiently large, allowing an essentially undisturbed diffusion of the atoms. With increasing coverage the mean distance between the Mn atoms drops and the Mn Mn interactions gain increasing influence. This should lead to a rise of the mobility. At coverages where the ordered structures exist, the mobility is again low, since the Mn atoms have the maximum possible distance for the given coverage. These observations are in qualitative agreement with a Monte Carlo study by Uebing and Gomer [24]. These simulations show an enhancement of the diffusion coefficient in the intermediate coverage regime (between 0 and 0.5 ML Mn) for the case of nearest-neighbor repulsive interactions. The enhancement of the mobility of incorporated Mn at 0.3 ML Mn is observed in the same coverage regime where a pronounced ripening process occurs (see Fig. 4a). In the classical approach of Ostwald ripening applied to surfaces, the most important energies for the mass transport are the activation energy for the diffusion and creation/ annihilation of adatoms [25]. We doubt if ripening processes in the binary surface alloy Mn/Cu(100) can be successfully described on the basis of this simple classical approach. However, the same energies should play an important role. Referring again to the energies relevant for Cu(100), we note that the activation energies for vacancy diffusion and creation/annihilation are comparable to that for adatoms, or even smaller [14]. Therefore, it is
thoroughly conceivable that vacancies contribute considerably to the mass transport between islands. This assumption is strongly supported by the observation that the increase of the mobility of incorporated Mn at 0.3 ML Mn is correlated with a pronounced ripening process at this coverage. In Fig. 10, the mass transport between islands is shown schematically for vacancy diffusion.
5. Summary and outlook The atomic mobility of incorporated Mn atoms in the Cu(100) surface can be attributed to diffusing vacancies. Other mechanisms including atomic exchange processes cannot account for the experimental observations. A quantitative analysis
Fig. 10. Mass transport between islands caused by vacancies. At first, a vacancy is created at an island edge (a). This vacancy starts a diffusion process (b) and is finally annihilated at another island edge (c), (d).
T. Flores et al./Surface Science 371 (1997) 1 13
determines the diffusion coefficient D to be 5.30+_0.32x 10-1Scm 2 s -1 in the limit of small Mn coverage. Furthermore, upper and lower limits for the jump rate have also been determined for low coverages. Reasonable agreement between the experimentally observed diffusion coefficient and the limits of the jump rate with a theoretical estimate based on vacancy-mediated diffusion are obtained. To obtain this estimate, we assumed that the energies of vacancy formation and diffusion are not altered in the close proximity of incorporated Mn atoms. A precise determination of both the diffusion coefficient D and the total jump rate Frot would allow the experimental verification of this assumption. This will remain a challenge for future work. Solving this interesting problem would greatly enhance our understanding of the diffusion of incorporated atoms. Such studies might also help to develop models to understand technologically important problems, such as vacancymediated impurity motion in silicon and on silicon surfaces. In the intermediate coverage regime, around 0.3 M L Mn, the mobility of incorporated Mn is strongly enhanced. This is correlated with the ordering process into the c(2 x 2) structure and a considerable mass transport leading to fast Ostwald ripening. Presumably also here and at higher coverages, vacancies are responsible for the mobility in the alloyed layer. This indicates that vacancies also play a major role for the mass transport required to achieve island ripening in this system.
Acknowledgements Helpful discussions with C.C. Knight, T.L. Einstein and H. Ibach are gratefully acknowledged. We are indebted to the DAAD for travel support within the P R O C O P E program. Financial support by the Deutsche Forschungsgemeinschaft (Wu243/2) is gratefully acknowledged. One of us (M.W.) would like to thank the Alexander von Humboldt foundation for a stipend to work at AT&T Bell Laboratories, Murray Hill, USA, where parts of this manuscript were completed.
11
Appendix A: Relationship between the total jump frequency and the diffusion coefficient for vacancymediated motion of incorporated atoms in a substrate surface In the following, the relationship between the jump frequency and the diffusion coefficient for a random walk is discussed. A detailed description can be found in Ref. [-16]. The prime example for a random walk is given by a single atom which diffuses on a vacant lattice performing jumps between neighboring sites.
j=l
is the diffusion coefficient in an arbitrary direction (called x). Here, the sum is over those neighboring sites which are potential jump targets, e.g. all nearest-neighbor sites, xj is the projection of the vector, pointing to the jth neighboring site, on the x-axis and Fj is the corresponding jump frequency. The formula is valid for diffusion in cubic crystals as well as along principal axes in non-cubic crystals. In the following we consider two-dimensional diffusion, as encountered on an fcc (100) surface with a surface lattice constant a, which corresponds to the nearest-neighbor distance. Assuming only nearest-neighbor jumps and a constant and uniform jump frequency F~=F, the diffusion coefficient is related to the jump frequency via D = Fa z. The diffusion is isotropic in this example. Usually, an Arrhenius ansatz is made for the jump frequency F = v x e x p ( - E o / k T ) , where v is the jump attempt frequency and ED is the activation energy for adatom diffusion. When correlations are involved in the diffusion process, the relationship between the diffusion coefficient and the jump frequency has to be modified. Characteristic for a correlated walk is that the atom jump probabilities depend on the direction of previous jumps. Thereby, successive jumps are no longer independent of each other. Often the correlation effects can be described by formally introducing an effective jump frequency Feff=fF. Here, f is the so-called correlation factor, which depends on the lattice geometry as well as the type of diffusion process. Consequently the jump fie-
12
T. Floreset al./Surface Science371 (1997) 1-13
quency F in the formula for the diffusion coefficient has to be substituted by Feff. A typical example for a correlated process is the diffusion of an atom mediated by diffusing vacancies. This is discussed in the following for the case of an fcc (100) surface, where atoms perform only nearest-neighbor jumps. The jump frequency of an atom is given by FA = Fvcv. Here, Cv is the probability to find a vacancy on a neighboring site, which corresponds to the vacancy concentration. Sometimes instead of the normal jump frequency FA, the total jump frequency Ftot = zFa is used, where z = 4 is the number of the nearest neighbors. Fv = v x e x p ( - EM/kT) is the vacancy jump frequency, and EM is the vacancy migration energy. Then, the relationship between the diffusion coefficient and the jump frequency is given by F A , e f f = f a Z F a • The correlation factor amounts to f = 0.4669 in the limit of a small vacancy concentration [26]. f is smaller than one, since there is a larger probability for a return jump after an atom has changed sites with a vacancy. This reduces D but not FA. The value for flisted above holds only in the case of a clean fcc (100) surface without incorporated foreign atoms. These impurities might alter both the vacancy mobility and vacancy concentration, thereby modifying f. Note that, in contrast to the relationship between the diffusion coefficient and the jump frequency, for the Einstein relation it is irrelevant whether the process is a random or correlated walk.
Appendix B: Determination of jump frequencies from time correlation images To unravel the correlations between the positions of the atoms, we have used a spatial correlation function. Dunphy et al. [27] first applied this to the surface diffusion of sulfur on Re(0001). The correlation function is given by
1 C(r, t) = ~ r~o H(ro, to)H(ro + r, to + t). Here, the sum is over all pairs of pixels in the STM image separated by a spatial and temporal distance r and t, respectively. N is the number of products in the sum and H(ro,to) is the z-position of the tip
at the position r o at the time to. Hence, C(r,t) describes the correlation between two pixels separated by a spatial and temporal distance r and t, respectively. In an STM image, t is a function of r and is determined by the time, in which the tip moves from the start to the end position of r. Since the correlation function has a maximum at each vector r, which is a lattice vector, the atomic arrangement is reflected in the correlation images. Therefore, C(r,t) describes the probability of finding two, possibly identical, atoms after time t at a distance r. In the following, we use the shape of the central peak ( r = 0 , t = 0 ) to estimate a lower limit for the total jump frequency of the incorporated Mn atoms. Due to the high mobility of the incorporated Mn atoms, the central peak is strongly elongated in the scan direction. For a sufficiently small scan width, the maxima in the correlation images contain a large number of pixels. In this case a statistical analysis can be performed. For the two pixels, the pixel PA in the center of the central peak and the pixel PB below PA, separated by a time Atc, the relationship CB/CA=WB/WA is valid. Here, Atc is the time necessary to scan an STM line, CA is the correlation value for r = 0 and t = 0, CB is the correlation value for r = 0 and t = A t e and WA and WB are the corresponding probabilities, respectively. WA represents the probability to find an Mn atom at a given lattice site, therefore WA corresponds to the Mn coverage WA= O. Thus it follows that WB= 0 x CB/CA. WB is the probability that a given site P is occupied at t and t +Atc, either by the same atom or by a different atom. The following events contribute to the probability WB: atoms which remain at P between t and t+AtL (probability: WR) as well as atoms which leave P and afterwards either return or are replaced by other Mn atoms in the time between t and t +Atc (probability: WL). Since WB=Ox(WR+Wc) we can estimate WB > OWR. WR is the probability for an Mn atom to remain at a site P between t and t+Atc, accordingly 1 - WR is the probability that an Mn atom moves between t and t + Ate. This movement corresponds to at least one jump in the time between t and t+Ate. Therefore, the total jump frequency is related to WR via Ftot = ( 1 - WR)/AtL. This allows an estimate of a lower limit for
T. Flores et al./Surface Science 371 (1997) 1-13
the total jump frequency of the incorporated Mn atoms by Ftot=(1-- WR)/AtL_>(1--Wn/O)/AtL= (1 -CB/CA)/AtL. An upper limit for the total jump frequency is easily derived from the STM images since we observe that most of the atoms do not jump in the time Atw. Here Atw is the time necessary to scan the width of a Mn atom. Therefore, it follows that Fro t ~ 1/Atw. The importance of this expression stems from the fact that it should allow a more precise description of the total jump rate. In combination with accurate measurements of the diffusion coefficient D, this offers the possibility to determine the correlation factor f, described in Appendix A.
References [ 1] W.K. Burton, N. Cabrera and F.C. Frank, Trans. R. Soc. A 243 (1951) 299. [2] J.D. Wrigley and G. Ehrlich, Phys. Rev. Lett. 44 (1980) 661. [3] R.Q. Hwang, J. Schr6der, C. G~inther and R.J. Behm, Phys. Rev. Lett. 67 (1991) 3279. [4] H. R6der et al., Nature 366 (1993) 141. [5] R. Kunkel, B. Poelsema, L.K. Verheij and G. Comsa, Phys. Rev. Lett. 65 (1990) 733. [6] T. Michely, M. Hohage, M. Bott and G. Comsa, Phys. Rev. Lett. 70 (1993) 3943. [7] G. Rosenfeld, N.N. Lipkin, W. Wulfhekel, J. Kliewer, K. Morgenstern, B. Poelsema and G. Comsa, Appl. Phys. A 61 (1995) 455. [8] G. Ehrlich and F.G. Hudda, J. Chem. Phys. 44 (1966) 1039.
13
[9] T. Flores, S. Junghans and M. Wuttig, Surf. Sci. 371 (1996) 14. [10] M. Wuttig, S. Junghans, T. Flores and S. Blagel, Phys. Rev. B 53 (1996) 7551. [11] K. Morgenstern, G. Rosenfeld, B. Poelsema and G. Comsa, Phys. Rev. Lett. 74 (1995) 2058. [12] W. Ostwald, Z. Phys. Chem. (Leipzig) 34 (1900) 495. [ 13] S.K. Ghandhi, VLSI Fabrication Principles (Wiley, New York, 1995). [14] P. Stoltze, J. Phys.: Condens. Matter 6 (1994) 9495. [15] G. Tr~glia, B. Legrand, A. Safll, T. Flores and M. Wuttig, Surf. Sci. 352-354 (1996) 552. [16] J.R. Manning, Diffusion Kinetics for Atoms in Crystals (van Nostrand, Princeton, 1968). [17] R. Fink, R. Wesche, T. Klas, G. Krausch, R. Platzer, J. Voigt, U. WOhrmann and G. Schatz, Surf. Sci. 225 (1990) 331. [18] H.P. Noh, T. Hashizume, D. Jeon, Y. Kuk, H.W. Picketing and T. Sakurai, Phys. Rev. B 50 (1994) 2735. [19] G. Vogl and W. Petry, Physikalische Bl~itter 10 (1994) 925. [20] T. Heumann, H. Meiners and H. Sttier, Z. Naturforsch. 25a (1970) 1883. [21] C. Nagl, E. Platzgummer, O. Haller, M. Schmid and P. Varga, Surf. Sci. 331-333 (1995) 831. [22] Y. Gauthier, W. Moritz and W. H6sler, Surf. Sci. 345 (1996) 53. [23] Y. Girard, C. Cohen, J. Moulin and D. Schmaus, Nucl. Instrum. Methods B 64 (1992) 73; C. Cohen, Y. Girard, P. Leroux-Hugon, A. L'Hoir, J. Moulin and D. Schmaus, Europhys. Lett. 24 (1993) 767. [24] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7626. [25] M. Zinke-Allmang, L.C. Feldman and M.H. Grabow, Surf. Sci. Rep. 16 (1992) 377. [26] K. Compaan and Y. Haven, Trans. Faraday Soc. 54 (1958) 1498. [27] J.C. Dunphy, P. Sautet, D.F. Ogletree, O. Dabbousi and M.B. Salmeron, Phys. Rev. B 47 (1993) 2320.
Surface Science 371 (1997) 1 13
Atomic mechanisms for the diffusion of Mn atoms incorporated in the Cu(100) surface: an STM study T. F l o r e s , S. J u n g h a n s , M . W u t t i g * lnstitut ffir Grenzfliichenforschung und Vakuumphysik, Forschungszentrum Jfilich, 52428 Jfilich, German), Received 12 March 1996; accepted for publication 18 July 1996
Abstract
The mobility of Mn atoms incorporated into the Cu(100) surface has been studied by scanning tunneling microscopy (STM). For small Mn coverages, a diffusion coefficient of 5.3 x 10 is cm 2 s 1 has been determined from analysis of the mean square displacement. This low mobility can only be explained by a diffusion mechanism that is based on diffusing vacancies. Upper and lower limits of the jump rate of incorporated Mn atoms have been derived from the STM images for several coverages. This analysis shows an enhanced mobility of Mn around 0.3 ML Mn. In this coverage range, fast ripening of alloy islands is observed. This indicates that vacancies also play a decisive role for Ostwald ripening in this system. A more detailed understanding of vacancy mediated motion of incorporated atoms should emerge, if both the diffusion coefficient and jump rate of incorporated atoms could be determined with high precision.
Keywords: Adatoms; Alloys; Compound formation; Copper; Low index single crystal surfaces; Manganese; Scanning tunneling microscopy; Surface diffusion
1. Introduction This decade has witnessed a renaissance of studies of adatom diffusion on surfaces [1 7]. Driven by the need to control and manipulate the growth of ultrathin films on a monolayer scale, interest has focused on characterization of atomic mechanisms governing thin-film growth. In particular, the diffusion of single adatoms on a substrate surface, along island edges and across steps, have been shown to be of paramount importance [-4,5,8]. Hierarchies of diffusion mechanisms have been developed which can account for many of the observed growth morphologies and island shapes [3,6,7]. The focus of this study is a related problem, the * Corresponding author. Fax: +49 2461 6139070.
diffusion of an atom incorporated in (rather than adsorbed on) the substrate surface. So far, this problem has drawn much less attention. For the system studied, Mn on Cu(100), at and above room temperature, alloy formation is observed. In a recent study [-9], we have investigated the atomic mechanism of Mn incorporation. In the present publication, the main goal is the determination of the atomic mechanisms responsible for the motion of incorporated Mn atoms. Our experimental setup, previous studies of Mn on Cu(100), and the mechanisms of alloy formation have been discussed at length in a recent paper [-9]. Therefore we refer the reader to this study for details on the system studied and the experimental set-up. We will present the experimental results in Section 2 and a discussion in Section 3. A summary and the outlook are the subject of Section 4.
0039-6028/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S 0 0 3 9 - 6 0 2 8 ( 9 6 ) 00978-8
2
T. Flores et al./Surface Science 371 (1997) 1-13
2. Results
2.1. Mobility of incorporated Mn Fig. 1 shows the surface after deposition of 0.020+0.004 ML Mn. A substrate step and twodimensional islands of monoatomic height are readily discernible. Additionally, small protrusions exist. Neither the islands nor the protrusions are present on the clean Cu(100) surface prior to deposition. We have identified the protrusions as single, incorporated Mn atoms [10]. Their apparent size is larger in many STM images than expected for a single atom. This is probably due to a convolution of the tip shape and the protrusion. The incorporated Mn atoms are mobile, as demonstrated in Fig. 1. The white dots denote the
I
looA
I
Fig. 1. STM images of the Cu(100) substrate after deposition of M n at 300 K. Coverage: 0.020+0.007 M L Mn, scan width: 350 ,& x 350 A. In order to visualize the mobility of the incorporated M n atoms, two subsequently recorded STM images are superimposed. The white dots mark the positions of the incorporated M n atoms 180 s before the image shown was recorded. The white frame indicates the margin of the first with respect to the second recorded image. The images do not match exactly because of thermal drift.
positions of the Mn atoms 180 s before the image shown was recorded. By comparing the position of atoms in Fig. 1, a lower limit for the total jump frequency Ftot of the incorporated Mn atoms was estimated. At 0.1 ML Mn, each Mn atom performs on average at least one jump in the time Atl between the first and second STM picture, taken 180s later. Therefore the total jump frequency satisfies/'tot ~ 1~Aft. An upper limit can be derived from the fact that during the scan of a single Mn atom, the vast majority of atoms does not jump. Therefore, Ftot < 1/Ats, where Ats is the time necessary to scan a single Mn atom. Combining the upper and lower limit, a jump frequency between 1 . 1 x l 0 -2 and 3 . 9 x 1 0 - 1 s 1 is estimated for 0.1 M L Mn. In an analogous way, the jump frequency at 0.5 ML Mn can be obtained. Fig. 2a depicts an image of the c ( 2 x 2 ) surface alloy at 0.45_+ 0.02 M L Mn. Although a pronounced long-range order is observed, a large number of point defects exists. Since only the sublattice formed by the Mn atoms is imaged, the point defects could either be vacancies or Cu atoms located at positions of the Mn sublattice. At room temperature, the equilibrium concentration of vacancies should be much smaller than the concentration of point defects visible in Fig. 2a. Furthermore, vacancy jump rates at room temperature should be of the order of 9 x l0 s s-1. This again is in contrast with our observation, i.e. the fact that we can image a point defect during the time it takes to scan across it. Therefore, we believe that the point defects are Cu atoms located at the positions of the Mn sublattice. Further support for this comes from the fact that the number of incorporated Mn atoms per substrate site equals the fraction of the terrace which is covered by islands. Both "coverages" should be different if the point defects are vacancies. Using the same methodology as described in the case of small coverages, a jump frequency between 1.1 x 10 -2 and 1.3 s -1 is derived at a coverage of approximately 0.45 M L Mn. These values are comparable with the observed jump rate of incorporated Mn atoms at low coverages. However, this similarity might be fortuitous, since at 0.45 ML Mn the mobility of Cu in the substrate surface
T. Flores et al./Surface Science 371 (1997) 1-13
a)
b)
!ii
Fig. 2. STM images of the Cu(100) substrate after deposition of Mn at 300 K. (a) Coverage: 0.45+0.02 ML Mn, scan width: 180 A, x 180 A. (b) Coverage: 0.31 +0.03 ML Mn, scan width:
90A×90A. layer in close proximity to Mn is analyzed. On the contrary, at low coverages, the mobility of single, isolated Mn atoms incorporated into the Cu(100) substrate surface is analyzed. Fig. 2b shows the surface after deposition of 0.31_+0.03 ML Mn. In this coverage regime, one
3
always observes "fuzzy" images as in Fig. 2b, indicating a high mobility. This mobility cannot be caused by Mn or Cu adatoms. Since both Cu and Mn adatoms become mobile on Cu(100) at around 140 and 110 K, respectively, at room temperature the mobility of the adatoms is too large to image them. Therefore, like at small coverages, the mobility has to be caused by diffusing atoms in the surface layer. The STM data show that most of the incorporated Mn atoms do not jump within the time necessary to scan their width. On the other hand, most of the incorporated Mn atoms are not completely imaged. This is in contrast to our findings for coverages below 0.2 ML Mn and above 0.4 ML Mn, where most incorporated atoms are completely imaged (see Figs. 1 and 2a). Hence the mobility is considerably enhanced around 0.3 M L Mn. Since at this coverage most incorporated Mn atoms are immobile during the time it takes to scan their width, but move within the time it takes to image them completely, an upper and lower limit of the mobility can be estimated. This leads to a jump rate between 2.2 and 102 s -1 for 0.3 M L Mn. Also at higher coverages, e.g. around 0.45 ML Mn, where large parts of the substrate already show a c(2 x 2) arrangement, we can see such "fuzzy" regions of varying size. These regions are often in the vicinity of steps or island edges. The estimates discussed so far focus on the jump rate. From the jump frequency, the diffusion coefficient can be derived, once a model for the underlying atomic mechanism exists. We assume that the diffusion takes place via nearest-neighbor jumps of distance a, and additionally neglect correlation effects, which are discussed in Appendix A. Then, the diffusion coefficient D can be derived from the total jump frequency by D=a2Ftot/Z. Here z is the number of nearest neighbors. At very small coverages, one can follow the course of the incorporated Mn atoms. In this case, a direct determination of the diffusion coefficient is possible by evaluating the Einstein relation. To avoid the necessity to correct for thermal drift and distortion effects of the STM, we analyze the change in the distance between two incorporated Mn atoms with time [11]. Then the Einstein relation becomes
4
T. Flores et al./Surface Science 371 (1997) 1 13
projected on a fixed axis, in the time At. Due to the motion of the incorporated Mn atoms, the time interval between the first and the second scan of one particular atom is, in general, not just the time interval between two scans. This effect can be neglected for sufficiently large scan widths, however, since the mobility of the Mn atoms is relative low. Fig. 3a depicts ((AX) 2) as a function of At for a coverage of 0.0033 _+0.0005 ML Mn. The number of considered events decreases from small to large values of At from approximately 1000 to 100. The straight line is the least-square fit to the data points. It describes the experimental observations rather well and justifies an analysis using the Einstein relation. From the slope, the diffusion
a)
2°1 t ~<~v._4001001 5:t~~2 ~/~
CoO0 l " 800 " ldOO At [s]
0,06~ 0,051 004 0103
0 , ~ 0,05 0,04 0,03
0,01t 0,00-40-30"-20-t0 0 10 20 3"0 40
~ 0,01 0,00
b)
o,o21
~ o,o2
Ax[lo~cm]
0
6
coefficient for incorporated Mn in the limit of small coverages can be determined to 5.3_+0.3 x 10-~8 cm 2 s -1. We have checked carefully whether the diffusion of incorporated Mn atoms could be influenced by the tip of the STM. This should result in different diffusion coefficients in the scan direction and perpendicular to it, which is contrary to our analysis. This implies that the experimentally determined diffusion coefficient is not influenced by the presence of the tunneling tip. If we use the upper and lower limit of the jump frequency at small coverages and assume nearestneighbor jumps and a correlation factor of one, this leads to a diffusion coefficient between 1.8 x 10 -18 and 6.4 x 10 17 cm 2 s-1. The diffusion constant determined directly from the Einstein relation falls into this range. In Fig. 3b, the distribution of AX is shown for At=233 s and At=696 s, respectively, and compared to the theoretical density function which is given by W(AX,At)=(8nDAt) -~ exp(-(AX)2/ 8DAt). Both the experimental and theoretical distributions agree very well, within the statistical error. In Fig. 4, the coverage dependence of the diffusion coefficient is summarized. The single data point represents the diffusion coefficient in the limit of very small Mn coverages, determined by the evaluation of the Einstein relation. The horizontal bars describe the estimated diffusion coefficient derived from the jump frequency. The latter were determined assuming nearest-neighbor jumps and neglecting correlation effects. The jump rates show
I ~X[lO%m]
Fig. 3. (a) The mean square of the distance change AX between two incorporated M n atoms is depicted as a function of time At. The number of analyzed distances decreases from approximately 1000 for the smallest to 100 for the largest time At. From the slope of the fit (straight line), the diffusion coefficient of the incorporated M n can be determined to be D = 5.30+_0.32 × 10 is cm 2 s-1. The coverage was 0.0033+ 0.0005 M L Mn. (b) The experimental (solid bars) and theoretical (hollow bars) distributions of the distance change AX in time At are shown for two different times, A t = 2 3 3 s (left side) and A t = 6 9 6 s (right side) at a M n coverage of 0.0033 _+0.0005 ML.
~"
0.4
-~
0.2
®
0.0
•
'"'1
0.0001
' ' ''""1
0.001
........ I
0.01
. . . . . . "1
0.1
, ,,,'",1
1
, ' '
D [10-14cm2s1] Fig. 4. The estimate for the diffusion coefficient D at different M n coverages O. For each coverage, the diffusion coefficient falls in the range marked by the error bars. The diffusion coefficient was estimated from the total j u m p frequency, neglecting correlation effects and assuming nearest-neighbor jumps. The single data point is the diffusion coefficient as derived from the Einstein relation.
T. Floreset al./Surface Science 371 (1997) 1 13
a clear m a x i m u m in the intermediate coverage regime around 0.3 M L Mn, where the j u m p frequency is between one and two orders of magnitude higher than at 0.1 M L Mn. As mentioned above, for 0.1 M L Mn, we find a good agreement between the diffusion coefficient as determined directly through the Einstein relation and the estimate based on the j u m p rate. Concerning the higher coverages, around 0.3 M L Mn and above, the diffusion coefficient cannot be determined directly since we are unable to follow the path of individual incorporated Mn atoms with time. At these coverages, however, we can exploit the pronounced mobility of the incorporated Mn atoms. Using a statistical analysis based on time correlation functions, which is explained in detail in Appendix B, we have determined the probability ( 1 - WR) that an incorporated Mn atom makes at least one j u m p in the time interval AtL. This leads to a lower limit of the total j u m p rate as explained in Appendix B. For a coverage of 0.31 M L Mn we obtain/"to t _>3 s 1. 2.2. Late-stage island growth
Section 2.1 focused on the statistical motion of individual atoms. However, the STM images also reveal information about the directional motion required to achieve mass transport. An example is displayed in Fig. 5, which shows a number of islands 1.5 h after deposition (Fig. 5a) and 6 h after deposition (Fig. 5b). A growth of larger islands at the expense of smaller islands is observed. This is characteristic of a ripening process. The STM images reveal that the islands are not mobile and therefore do not grow by dynamic coalescence. Since the coverage does not change during ripening, a net mass transport, like adatom diffusion between small and large islands, has to be effective. Such a late-stage growth is usually called Ostwald ripening [12]. In Fig. 6a, the island density is shown in 1 h intervals after Mn deposition as a function of the coverage. For all coverages, the island density decreases with time, but the magnitude of the change shows a clear coverage dependence. The effect is most pronounced at coverages around 0.3 M L Mn. Here the island density decreases by about a factor of three in the observa-
5
a)
b) f
Fig. 5. STM images of the Cu(100) substrate after deposition of Mn at 300K. Coverage: 0.35_+0.04ML Mn, scan width: 2800 A x 2800 A. The STM images in (a) and (b) were taken 1.5 and 6 h after deposition, respectively. tion time. For this coverage, in Fig. 6b the distribution of island sizes is depicted for different time intervals after Mn deposition. A dramatic transfer from small to large sizes is observed with time. The average island size increases from 3.0 x 104 to 8.7 x 104 /~2.
6
T. Flores et aL/Surface Science 371 (1997) 1 13
a)
2'5t .+ 20
~1,5
~
~
o
li
0,5 ~. ~ n ~ 1,i.0 0~.10. o }~ O, , , ,2 0,3 0,4 cowage[M_Mn] b)
~ 0,175~/
i70
:
°°
!i '
0 5 10 15 20 25 30 island size [104A2]
~_.0 200f , ' ~o 0,175~
°,12 1
>5h
1-120 100 ~.
.80
island size [104A2]
40 .20 .0 0
i 225 -200 :17~
~; o,loot
;loo
~ 0,0'0501/11(h 025 I
; 25 50 ~1"
'
.~ 0,175~
0 5 1'0 1'5 20 ~5 300 islandsize[104A2]
Fig. 6. (a) Island density as a function of the Mn coverage. For each coverage, the island density was determined for different times (marked by different symbols) after deposition. For a fixed coverage, the island density decreases due to a ripening process. This effect is most pronounced at intermediate coverages of around 0.3 ML Mn. (b) Island size distribution 1 2, 3 4 and more than 5 h after deposition for a coverage of 0.31_+0.03ML Mn. With increasing time, the island size increases.
3. Discussion: diffusion of incorporated Mn First, we will identify the mechanism responsible for the diffusion of incorporated Mn atoms. To
facilitate this task, we will initially restrict the discussion to small coverages around and below 0.1 M L Mn. Beginning with the simplest mechanisms, an exchange process h o r i z o n t a l to the surface enables the diffusion. A related process is the ring mechanism, where four atoms rotate in the surface. These processes are rather unlikely in a closepacked arrangement of atoms, since the resulting activation barrier for these processes would be extremely high. Therefore, we do not believe that these mechanisms are reasonable candidates for the diffusion of incorporated Mn on Cu(100). Similar arguments have also been put forward for the bulk diffusion of impurities in Si, for example [13]. Another conceivable scenario is the following: in a first step, an incorporated Mn atom is ejected onto the terrace. Possibly this happens via a C u / M n place exchange. Then a subsequent diffusion of the Mn adatom on the surface takes place. This is finally followed by an incorporation of the Mn adatom through a Mn/Cu place exchange. This is illustrated in Fig. 7. For the Mn/Cu exchange process, we can make an Arrhenius "ansatz". The activation barrier for such an exchange can be estimated from the onset temperature for alloy formation, which is around 260 K on Cu(100). This leads to a considerable lifetime of a Mn adatom on the surface. On the other hand, such a Mn adatom can move rapidly on the surface. A lower limit for the diffusion coefficient of Mn adatoms on Cu(100) can be derived from the observation that the c(8 x 2) film structure can be formed at temperatures as low as l l0 K. This gives an upper limit for the onset temperature of diffusion of 110 K. Then, from the Einstein relation the average distance between the ejection and incorporation site can be obtained. This results in a distance of several thousand A, which is at least between two and three orders of magnitude higher than the experimentally observed distance. Hence we have to exclude the scenario described above. The experimental results can be successfully explained, however, if diffusing vacancies cause the mobility of incorporated Mn (Fig. 8). For such a mechanism the total j u m p frequency of incorporated Mn is given by ~r~to t = Z X CV X [ ' V " Here, z is the number of nearest neighbors, F v is the vacancy
T. Flores et al./Surface Science 371 (1997) 1 13
7
a) a)
b) b)
c) c)
d)
Fig. 7. Diffusion of incorporated Mn by an ejection and subsequent surface diffusion process. An incorporated Mn atom is ejected to the surface, e.g. by a Cu/Mn place exchange (a). The ejected Mn atom starts a diffusion process (b) and is finally incorporated by a Mn/Cu place exchange (c), (d). White circles denote Cu atoms, gray circles Mn atoms. j u m p rate, a n d Cv is the c o n c e n t r a t i o n of the vacancies. In the following, we will use for Cv the e q u i l i b r i u m c o n c e n t r a t i o n of vacancies as found on Cu(100). This e q u i l i b r i u m c o n c e n t r a t i o n is given b y Cv = e x p ( - E v / k T ) , where EF is the formation energy of a vacancy. At r o o m t e m p e r a t u r e , the v a c a n c y c o n c e n t r a t i o n is typically very low. If we use a f o r m a t i o n energy of vacancies of E v = 0.47 eV, as c a l c u l a t e d by Stoltze for Cu(100) [ 14], we o b t a i n Cv ,~ 10 - s . T h e j u m p rate of the v a c a n cies is F v = v x e x p ( - E M / k T ) , where v is the a t t e m p t frequency a n d EM the v a c a n c y m i g r a t i o n
d)
Fig. 8. Diffusion of incorporated Mn atoms caused by diffusing vacancies. The vacancies move from the right to the left. As a consequence, the incorporated Mn atom jumps from the left to the right. White circles denote Cu atoms, gray circles Mn atoms.
energy. In the following, we will assume t h a t the diffusion of vacancies is n o t affected b y the presence of Mn. Then, the m i g r a t i o n energy of vacancies on Cu(100) can be used, which has been calculated to EM = 0.44 eV [ 14]. A similar value of 0.42 eV has been o b t a i n e d in Ref. [ 15]. A s s u m i n g a typical a t t e m p t frequency of 1013-+1 s -1, a t o t a l j u m p frequency of F t o t = 1 0 -2+1 s -1 is derived. This is in g o o d a g r e e m e n t with the e x p e r i m e n t a l l y o b s e r v e d u p p e r a n d lower limit for the j u m p rate, which lies between a p p r o x i m a t e l y 10 - 2 a n d 5 x 10 -1 s -1 for 0.1 M L Mn. As m e n t i o n e d in Section 2.1, we have also determ i n e d the diffusion coefficient D from an evalua-
8
T. Flores et al./Surjace Science 371 (1997) 1 13
tion of the Einstein relation at 0.0033_+ 0.0005 M L Mn. In Appendix A, the relation between the diffusion coefficient D and the total j u m p frequency for a vacancy mediated motion is discussed. There it is shown that for vacancy-mediated diffusion, Frot = zD/fa 2, with f = 0.4669 on an fcc (100) surface in the limit of small vacancy concentrations. The factor f arises since the atoms do not perform a random but a correlated walk [ 16]. This results in a total jump frequency of 6.9_+0.4 x 10 2 s-1. This again is in good agreement with the theoretically estimated jump frequency for vacancy mediated diffusion of Mn. Considering the existence of vacancies, another process seems possible which is very similar to the process illustrated in Fig. 7. Instead of a Mn/Cu place exchange, an alternative way for the last step could be an incorporation by vacancy annihilation. In other words, the ejected Mn adatom diffuses on the terrace until it meets a vacancy and is then incorporated, thereby annihilating the vacancy. However, also in this case, due to the extremely small equilibrium concentration of vacancies at room temperature, the mean diffusion length of incorporated Mn would be between two and three orders of magnitude higher than the experimentally observed value. We can thus conclude that only a vacancydriven motion of incorporated Mn can account for the observed mobility. Vacancy diffusion as a mechanism for diffusion of incorporated atoms has also been discussed in a similar context [17]. These authors studied the adsorption and incorporation of In on Ag(100) and Ag( 111 ) by perturbed angular-correlation (PAC) spectroscopy. They attributed the incorporation to adatom diffusion and subsequent attachment to a step, followed by incorporation through diffusing vacancies into the upper terrace. In the following we want to address the assumptions implicit in our theoretical estimate of Mn mobility in more detail. In applying Ftot= Z XCv x Fv, we have made three assumptions. (i) The equilibrium concentration of vacancies is not affected by incorporated Mn atoms. (ii) The j u m p rate of vacancies is not affected by incorporated Mn atoms. (iii) The concentration of vacancies corresponds to the equilibrium concentration,
i.e. we consider a stage where the rates of vacancy creation and vacancy annihilation are in equilibrium. This last assumption is justified since we observe Mn mobility long after Mn incorporation. Hence, the vacancy concentration should be equilibrated. More difficult to justify are the first two assumptions, which imply that the vacancies are not affected by Mn atoms. The diffusion of Mn depends on both vacancy motion and vacancy concentration, i.e. on E M + E F. The analysis presented above shows good agreement between the mobility of incorporated Mn in the Cu(100) substrate surface and the theoretical model, which strictly only holds for vacancy-mediated motion in the clean Cu(100) surface. Therefore we can only conclude that E M + E F is similar on Cu(100) and after incorporation of small amounts of Mn on Cu(100). A precise determination of both D and Fto t at small coverages would allow us to test assumptions (i) and (ii) individually, at least to some extent. Let us assume, for example, that the j u m p rate of a vacancy in the direct vicinity of a Mn atom is considerably enhanced. Then, the incorporated Mn atom would perform many correlated jumps where the vacancy changes place with the Mn atom. The resulting j u m p rate would be high. The diffusion coefficient, however, would not increase to a similar extent, since this site oscillation would not lead to any significant atomic displacement. Hence the correlation factor f would be much smaller than the ideal value of 0.4669 for an fcc (100) lattice. From a precise determination of D and Ftot, one could determine f with high accuracy. Since for low Mn coverages we can only determine an upper and lower limit for the j u m p rate, we can only estimate in which range f falls. This results in a wide range of values for f, which lies between 0.08 and 2.94. Therefore we cannot determine experimentally, whether assumptions (i) and (ii) are independently fulfilled for Mn atoms incorporated into the Cu(100) substrate surface. To reiterate the statement, a more precise determination of the total j u m p frequency and the diffusion coefficient would allow a more quantitative analysis of the underlying vacancy mechanism. So far, we are not aware of any experimental study of diffusion of incorporated atoms which has
T. Flores et al./Surface Science 371 (1997) 1 13
reached this demanding and ambitious goal. Yet, such an investigation would not only result in a better understanding of the limitations implicit in our theoretical analysis and a precise determination of the correlation factor, but also improve our general understanding of vacancy-mediated diffusion of incorporated atoms. In passing, we note that increasing the temperature from room temperature to 340 K, raises the diffusion coefficient by approximately two orders of magnitude. Therefore, within a few minutes the diffusion length of the incorporated Mn atoms approaches the average island distance. This leads to a homogeneous Mn distribution, in agreement with the experimental observation [9]. For higher coverages, around 0.3 M L Mn and above, the discussion of the diffusion mechanism becomes more difficult. The assumption of a vacancy-driven process is supported by the fact that such a model also offers a plausible path for the ordering of the system at higher coverages. This is illustrated in Fig. 9. A striking point is the strong increase of the mobility of incorporated Mn atoms around 0.3 M L Mn. Using D=faZ/"tot/Z, a lower limit for the diffusion coefficient D_>1.7x10 -16cm 2 s 1 can be derived from the j u m p frequency/"tot (see Fig. 2c). A similar estimate is given by N o h et al. [18] with D > 1.5 x 10 -16 cm 2 s -1 for a comparable coverage [18]. In terms of the vacancy model, the enhancement of the mobility of incorporated Mn is formally explained either by a higher j u m p rate of the vacancies, which corresponds to a decrease of the vacancy migration energy EM, or by a larger equilibrium concentration of vacancies, i.e. a decrease of the vacancy formation energy E v. A change of two orders of magnitude for the total j u m p frequency of incorporated Mn, as experimentally observed from small to intermediate coverages, corresponds to a decrease of the sum E F + E M by approximately 0.12 eV. Such a decrease seems possible within the framework of a coverage dependence of g F and/or EM. Support for a pronounced composition dependence of E v comes from studies of the bulk diffusion in binary alloys. These investigations have shown that in ordered alloys of the DO3-type, a fast diffusion process is often effective. An example is
9
a)
b)
Fig. 9. Atomsin a disordered area (a) are arranged in a c(2 x 2) structure after a vacancy has passed (b). The path of the vacancy is shown by the black line in (b). White circles denote Cu atoms, gray circles Mn atoms and the square the vacancy. Fe3Si. In many of these ordered alloys, the vacancy concentration is up to a few orders of magnitude higher than in pure metals. It is assumed that this high vacancy concentration causes the fast diffusion in several of those alloys [-19,20]. Such a scenario could also account for the high mobility we observe for Mn incorporated in the Cu(100) substrate surface at higher coverage. Interestingly enough, Mn is not the only deposit on Cu(100) for which a coverage dependence of the mobility is found. Qualitatively, the same observation was made for Pb. Pb also forms ordered surface alloys on Cu(100) [-21,22]. The mobility of incorporated Pb atoms shows a pronounced m a x i m u m in an intermediate coverage regime and is strongly reduced at higher and lower coverages [-21,23]. It is remarkable that for both Pb and Mn, the increase in the mobility of the incorporated atoms is clearly correlated with an
10
T. Flores et al./Surface Science 371 (1997) 1 13
ordering process. At small coverages, the surface alloys are disordered. Here, the mobility is relatively low. In the intermediate coverage regime, where the ordering proceeds, the mobility is strongly enhanced. At high coverages, where the c(2 x 2) and c(4 x 4) are observed for Mn and Pb, respectively, the mobility is again relatively low. We attribute this coverage dependence to the interactions between the incorporated atoms. To explain the observed superstructures, the interaction between incorporated atoms should be repulsive on nearest-neighbor sites. At least for Mn, this observation is supported by the fact that, independent of coverage, no nearest-neighbor distances between two incorporated Mn atoms were found in the STM images. At low coverages, the mean distance between the Mn atoms is sufficiently large, allowing an essentially undisturbed diffusion of the atoms. With increasing coverage the mean distance between the Mn atoms drops and the Mn Mn interactions gain increasing influence. This should lead to a rise of the mobility. At coverages where the ordered structures exist, the mobility is again low, since the Mn atoms have the maximum possible distance for the given coverage. These observations are in qualitative agreement with a Monte Carlo study by Uebing and Gomer [24]. These simulations show an enhancement of the diffusion coefficient in the intermediate coverage regime (between 0 and 0.5 ML Mn) for the case of nearest-neighbor repulsive interactions. The enhancement of the mobility of incorporated Mn at 0.3 ML Mn is observed in the same coverage regime where a pronounced ripening process occurs (see Fig. 4a). In the classical approach of Ostwald ripening applied to surfaces, the most important energies for the mass transport are the activation energy for the diffusion and creation/ annihilation of adatoms [25]. We doubt if ripening processes in the binary surface alloy Mn/Cu(100) can be successfully described on the basis of this simple classical approach. However, the same energies should play an important role. Referring again to the energies relevant for Cu(100), we note that the activation energies for vacancy diffusion and creation/annihilation are comparable to that for adatoms, or even smaller [14]. Therefore, it is
thoroughly conceivable that vacancies contribute considerably to the mass transport between islands. This assumption is strongly supported by the observation that the increase of the mobility of incorporated Mn at 0.3 ML Mn is correlated with a pronounced ripening process at this coverage. In Fig. 10, the mass transport between islands is shown schematically for vacancy diffusion.
5. Summary and outlook The atomic mobility of incorporated Mn atoms in the Cu(100) surface can be attributed to diffusing vacancies. Other mechanisms including atomic exchange processes cannot account for the experimental observations. A quantitative analysis
Fig. 10. Mass transport between islands caused by vacancies. At first, a vacancy is created at an island edge (a). This vacancy starts a diffusion process (b) and is finally annihilated at another island edge (c), (d).
T. Flores et al./Surface Science 371 (1997) 1 13
determines the diffusion coefficient D to be 5.30+_0.32x 10-1Scm 2 s -1 in the limit of small Mn coverage. Furthermore, upper and lower limits for the jump rate have also been determined for low coverages. Reasonable agreement between the experimentally observed diffusion coefficient and the limits of the jump rate with a theoretical estimate based on vacancy-mediated diffusion are obtained. To obtain this estimate, we assumed that the energies of vacancy formation and diffusion are not altered in the close proximity of incorporated Mn atoms. A precise determination of both the diffusion coefficient D and the total jump rate Frot would allow the experimental verification of this assumption. This will remain a challenge for future work. Solving this interesting problem would greatly enhance our understanding of the diffusion of incorporated atoms. Such studies might also help to develop models to understand technologically important problems, such as vacancymediated impurity motion in silicon and on silicon surfaces. In the intermediate coverage regime, around 0.3 M L Mn, the mobility of incorporated Mn is strongly enhanced. This is correlated with the ordering process into the c(2 x 2) structure and a considerable mass transport leading to fast Ostwald ripening. Presumably also here and at higher coverages, vacancies are responsible for the mobility in the alloyed layer. This indicates that vacancies also play a major role for the mass transport required to achieve island ripening in this system.
Acknowledgements Helpful discussions with C.C. Knight, T.L. Einstein and H. Ibach are gratefully acknowledged. We are indebted to the DAAD for travel support within the P R O C O P E program. Financial support by the Deutsche Forschungsgemeinschaft (Wu243/2) is gratefully acknowledged. One of us (M.W.) would like to thank the Alexander von Humboldt foundation for a stipend to work at AT&T Bell Laboratories, Murray Hill, USA, where parts of this manuscript were completed.
11
Appendix A: Relationship between the total jump frequency and the diffusion coefficient for vacancymediated motion of incorporated atoms in a substrate surface In the following, the relationship between the jump frequency and the diffusion coefficient for a random walk is discussed. A detailed description can be found in Ref. [-16]. The prime example for a random walk is given by a single atom which diffuses on a vacant lattice performing jumps between neighboring sites.
j=l
is the diffusion coefficient in an arbitrary direction (called x). Here, the sum is over those neighboring sites which are potential jump targets, e.g. all nearest-neighbor sites, xj is the projection of the vector, pointing to the jth neighboring site, on the x-axis and Fj is the corresponding jump frequency. The formula is valid for diffusion in cubic crystals as well as along principal axes in non-cubic crystals. In the following we consider two-dimensional diffusion, as encountered on an fcc (100) surface with a surface lattice constant a, which corresponds to the nearest-neighbor distance. Assuming only nearest-neighbor jumps and a constant and uniform jump frequency F~=F, the diffusion coefficient is related to the jump frequency via D = Fa z. The diffusion is isotropic in this example. Usually, an Arrhenius ansatz is made for the jump frequency F = v x e x p ( - E o / k T ) , where v is the jump attempt frequency and ED is the activation energy for adatom diffusion. When correlations are involved in the diffusion process, the relationship between the diffusion coefficient and the jump frequency has to be modified. Characteristic for a correlated walk is that the atom jump probabilities depend on the direction of previous jumps. Thereby, successive jumps are no longer independent of each other. Often the correlation effects can be described by formally introducing an effective jump frequency Feff=fF. Here, f is the so-called correlation factor, which depends on the lattice geometry as well as the type of diffusion process. Consequently the jump fie-
12
T. Floreset al./Surface Science371 (1997) 1-13
quency F in the formula for the diffusion coefficient has to be substituted by Feff. A typical example for a correlated process is the diffusion of an atom mediated by diffusing vacancies. This is discussed in the following for the case of an fcc (100) surface, where atoms perform only nearest-neighbor jumps. The jump frequency of an atom is given by FA = Fvcv. Here, Cv is the probability to find a vacancy on a neighboring site, which corresponds to the vacancy concentration. Sometimes instead of the normal jump frequency FA, the total jump frequency Ftot = zFa is used, where z = 4 is the number of the nearest neighbors. Fv = v x e x p ( - EM/kT) is the vacancy jump frequency, and EM is the vacancy migration energy. Then, the relationship between the diffusion coefficient and the jump frequency is given by F A , e f f = f a Z F a • The correlation factor amounts to f = 0.4669 in the limit of a small vacancy concentration [26]. f is smaller than one, since there is a larger probability for a return jump after an atom has changed sites with a vacancy. This reduces D but not FA. The value for flisted above holds only in the case of a clean fcc (100) surface without incorporated foreign atoms. These impurities might alter both the vacancy mobility and vacancy concentration, thereby modifying f. Note that, in contrast to the relationship between the diffusion coefficient and the jump frequency, for the Einstein relation it is irrelevant whether the process is a random or correlated walk.
Appendix B: Determination of jump frequencies from time correlation images To unravel the correlations between the positions of the atoms, we have used a spatial correlation function. Dunphy et al. [27] first applied this to the surface diffusion of sulfur on Re(0001). The correlation function is given by
1 C(r, t) = ~ r~o H(ro, to)H(ro + r, to + t). Here, the sum is over all pairs of pixels in the STM image separated by a spatial and temporal distance r and t, respectively. N is the number of products in the sum and H(ro,to) is the z-position of the tip
at the position r o at the time to. Hence, C(r,t) describes the correlation between two pixels separated by a spatial and temporal distance r and t, respectively. In an STM image, t is a function of r and is determined by the time, in which the tip moves from the start to the end position of r. Since the correlation function has a maximum at each vector r, which is a lattice vector, the atomic arrangement is reflected in the correlation images. Therefore, C(r,t) describes the probability of finding two, possibly identical, atoms after time t at a distance r. In the following, we use the shape of the central peak ( r = 0 , t = 0 ) to estimate a lower limit for the total jump frequency of the incorporated Mn atoms. Due to the high mobility of the incorporated Mn atoms, the central peak is strongly elongated in the scan direction. For a sufficiently small scan width, the maxima in the correlation images contain a large number of pixels. In this case a statistical analysis can be performed. For the two pixels, the pixel PA in the center of the central peak and the pixel PB below PA, separated by a time Atc, the relationship CB/CA=WB/WA is valid. Here, Atc is the time necessary to scan an STM line, CA is the correlation value for r = 0 and t = 0, CB is the correlation value for r = 0 and t = A t e and WA and WB are the corresponding probabilities, respectively. WA represents the probability to find an Mn atom at a given lattice site, therefore WA corresponds to the Mn coverage WA= O. Thus it follows that WB= 0 x CB/CA. WB is the probability that a given site P is occupied at t and t +Atc, either by the same atom or by a different atom. The following events contribute to the probability WB: atoms which remain at P between t and t+AtL (probability: WR) as well as atoms which leave P and afterwards either return or are replaced by other Mn atoms in the time between t and t +Atc (probability: WL). Since WB=Ox(WR+Wc) we can estimate WB > OWR. WR is the probability for an Mn atom to remain at a site P between t and t+Atc, accordingly 1 - WR is the probability that an Mn atom moves between t and t + Ate. This movement corresponds to at least one jump in the time between t and t+Ate. Therefore, the total jump frequency is related to WR via Ftot = ( 1 - WR)/AtL. This allows an estimate of a lower limit for
T. Flores et al./Surface Science 371 (1997) 1-13
the total jump frequency of the incorporated Mn atoms by Ftot=(1-- WR)/AtL_>(1--Wn/O)/AtL= (1 -CB/CA)/AtL. An upper limit for the total jump frequency is easily derived from the STM images since we observe that most of the atoms do not jump in the time Atw. Here Atw is the time necessary to scan the width of a Mn atom. Therefore, it follows that Fro t ~ 1/Atw. The importance of this expression stems from the fact that it should allow a more precise description of the total jump rate. In combination with accurate measurements of the diffusion coefficient D, this offers the possibility to determine the correlation factor f, described in Appendix A.
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