Furuduy Discuss., 1993, 95, 27-36 Thermal Roughening investigated by Scanning Tunnelling Microscopy Joost W. M. Frenken, Laurens Kuipers and Jaap B. Sanders FOM-Institutefor Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands The scanning tunnelling microscope (STM) can be used to study surfaces of crystals at elevated temperatures. In this paper it is shown how the roughening transition on a metal surface can be observed in STM images. From such observations an accurate value of the transition temperature is obtained. In addition, two technical steps are described that are necessary for the analysis of short-range correlation functions in the images in terms of the kink formation energy and the step-step interaction energy.The first step concerns an algorithm for the pattern recognition of kinks and step edges, which is used to acquire automatically the statistical kink distributions from a large number of STM images. The second step deals with the theory of short-range correlation functions, and is employed to derive the two energy parameters of interest from the experimental statistical distributions. Most surfaces of metals, semiconductors and molecular crystals develop a more or less severe degree of disorder at high temperatures. The mildest form of disordering, usually taking place at the lowest temperatures, is the loss of a surface superstructure, such as the (7 x 7) reconstruction of Si(111)’ and the (2 x 1) reconstruction of Au( 1 The other extreme is the almost complete loss of surface lattice order, known as ‘surface melting’,3 which typically occurs at temperatures close to the bulk melting point.At intermediate temperatures, the roughening transition converts the initially flat surface into an atomically rough configuration, without affecting the lattice periodicity at the ~urface.~ Each of these three types of disorder may influence the behaviour of the material on a mesoscopic or even macroscopic length scale. In particular, the roughening transition has been recognized as a potential cause for changes in crystal growth beha~iour.~ The scanning tunnelling microscope is an ideal tool to study local arrangements of surface defects, such as thermally generated steps and kinks on an otherwise flat surface.Employing a sufficiently symmetrical STM design, crystal surfaces can be imaged at elevated temperatures. In this contribution we demonstrate that the roughening tempera- ture, TR,of a metal surface, in this case Ag( 119, can be determined straightforwardly from a qualitative comparison of temperature-dependent STM images.6 For a quantitative analysis of the densities of, and correlations between, steps or kinks, each measured STM topograph has to be reduced to its underlying pattern of steps and kinks, from which the required densities and correlations can be counted directly. This reduction can be performed automatically by a pattern recognition algorithm. In the interpretation of the resulting statistical information we make use of the theory developed by Villain et al.for the roughening of vicinal surface^,^ adapted to yield Zocal probability distributions.8 From the comparison between the statistical mechanical theory and the STM observations both the stepstep interaction energy and the kink formation energy should be obtained. The required consistency between these energies and the observed roughening temperature should serve as a test of the employed theory.’ 27 Roughening investigated by STM AgW5) The structure of the Ag(l15) surface is illustrated in Fig. 1. It consists of narrow (001) terraces of 7.5 A (2.5 atom rows) in width, separated by monoatomic steps of 1.6 A in height. The roughening transition of such a vicinal surface is the result of the thermal generation of kinks in the pre-existing step edges.Each kink changes the local terrace width by + 1 or -1 atomic row, which is accompanied by a change in the local height of its step edge by -0.8 A or + 0.8 A. Lines or loops of kinks form the boundaries between areas on the surface that differ by 0.8 A in height. In the following we refer to such a boundary as a 'STEP', in order to distinguish it from the ordinary 'step' on the ideal (1 15) surface. It is important to realize that, on a sufficiently large length scale, real surfaces are never perfectly oriented and completely flat. The local deviations from the (1 15) orientation must be accommodated by STEPs. At temperatures well below TRthese are the only STEPs present, and they minimize their length by running as straight as possible.At TRthe free energy for STEP formation vani~hes.~ The resulting strong meandering of the pre-existing STEPs and the additional high concentration of thermally generated STEPs wash out the original STEP structure. It is this feature that appears most prominently in the STM images and that is used to identify the roughening temperature. Experimental Observations The STM results presented here have been obtained with a microscope that had been modified to image warm ( < 300 "C) surfaces.6 The Ag( 1 15) specimen was prepared by standard polishing, etching and in situ sputtering and annealing treatments.6 Fig. 2 shows STM images of the Ag( 1 15)surface, at 20,58,98 and 260 "C.Before each of [oii] Fig.1 Side view (a) and top view (b)of the Ag(ll5) surface. A contour followed by the tunnelling microscope is indicated schematically. A kink in a step edge leads to the narrowing or broadening of the adjacent terraces, thus causing height differences. A line of kinks forms a STEP (see text), separating areas that differ one level in height. J. W.M. Frenken et al. (b) Fig. 2 STM topographs from the Ag( 1 15) surface.6(a)20 "C,585 x 220 &;(b)58 "C,1 180 x 375 A2; (c) 58 "C,enlarged perspective view of a small section of (b);(d)98 "C,450 x 120 A'; (e) 260 "C, enlarged perspective view. these measurements, the surface had been annealed at 600°C and stabilized for several hours at the measurement temperature. At all temperatures, the images clearly show the steps of the (1 15) surface, although the experimental corrugation, of typically 0.25 A, is much smaller than the true step height of 1.6 A.The atomic structure on the (001) terraces has not been resolved. As explained, it is in the meandering of the STEP edges that one most easily recognizes whether the surface is below or above TR.The STEPs at room temperature [Fig. 2(a)]are relatively straight, which indicates that they involve a non-zero free energy. At 58 "C the STEPs meander more strongly than at room temperature, from which we infer that this temperature is much closer to, but still below, the roughening temperature [Fig. 2(b)].Fig. 2(c)shows a perspective representation of a small section of Fig.2(b).A loop of kinks in this section encloses a portion of the surface that is one level (0.8 A) below its surroundings. Also visible in this enlarged view is the effect of surface diffusion. Several scan lines seem to 'protrude' one atom row from individual step edges. This corresponds to temporary displacements of one of the kinks on the edge, displacing the location of the edge by one atom row during one scan line (ca. 1 s). Kink dynamics have been proposed as the Roughening investigated by STM mechanism behind the so-called ‘frizziness’ of individual step edges on Ag( 1 1 1) and cu(001).9~~ At temperatures of 98 “Cand higher [Fig. 2(+(e)] the STEP edges are ‘delocalized’. The surface morphology is dominated by a high density of kinks in the step edges, and the resulting STEPS easily bridge the non-thermal surface height variations.Based on this observation, we conclude that the roughening temperature of Ag( 115) is between 58 and 98 “C, i.e. TR= 78 f20 “C. The density of kinks at the higher temperatures (in particular, above 200°C) seems much higher than one would expect theoretically (see below). Probably, this is caused by the strong surface diffusion, which changes the surface during the scan, thereby introducing apparent kinks. Possibly this could also affect our identification of the rough state, in which case it would render our value for TRa lower estimate of the true roughening temperature. Pattern Recognition We will show below that the kink formation energy and the stepstep interaction energy can be inferred from the statistical distribution of kinks.For the evaluation of the kink distribution in an extensive set of images, it is essential that this statistical information is extracted from the measurements in an automatic manner. Especially at high temperatures, where the kink density is relatively large, an unambiguous identification by hand of the kinks in a large image is almost impossible. Although the signature of the (1 15) steps seems rather strong in Fig. 2, the experimental (1 15) corrugation in most images is below 0.3 A, i.e. much smaller than the geometrical step height of 1.6 A. This easily leads to ambiguities in the interpretation of the images which usually results in inconsistencies in the patterns of kinks obtained from an image by visual inspection. We have developed a simple pattern recognition procedure that avoids inconsistency and finds for each ambiguous situation the best possible ‘guess’ that is consistent with the rest of the information in the image.It is rather insensitive to statistical noise in the image and is extremely robust with respect to bad resolution of the tunnelling tip, and even ‘survives’ tip-resolution changes within one image. The two main difficulties that the method has to overcome are illustrated in Fig. 3. First of all, the data are superimposed on a heavily tilted background. Because of the elevated crystal temperature the thermal drift in the vertical direction has been extremely strong in these measurements.The drift depended on temperature and was not equal for different images measured at the same temperature. The slope of the background is almost entirely due to this drift, so that the precise orientation of the (1 15) plane with respect to the scanning plane has to be determined for each image separately. Secondly, as already mentioned, in most images the corrugation am litude related to the (1 15) steps is much smaller than the geometrical corrugation of 1.6 8:. Additional sources of possible error are noise in the images, temperature-dependent variations in the piezoelectric coefficient of the translation element of the microscope and hysteresis of the piezo element, which is reflected in non-linearities in the distance scale along the scan lines and artificial curving of the steps.Our pattern recognition approach makes use of the fact that the scan lines have been recorded in a direction approximately perpendicular to the steps. It employs the crystallographic information of the step height and stepstep distance on Ag( 119, and combines this with the restriction (see next section) that the distance between two neighbouring steps may differ from the regular distance by no more than plus or minus one atom. Each image is processed line by line, starting from the middle scan line of the image. We follow a prediction-correction procedure, commencing at the step closest to the centre of the line. This step is found as the local maximum nearest to the middle of the line.The next maximum along the line, for example to the right, is expected at or close to one of three locations, either at the regular distance Ax, at one atom too near, Ax -dx (STEPup by one level), or at one atom too far, Ax + dx (STEP down by one level). Corresponding to these three possible step locations are three different heights. Rather than just examining the J. W. M. Frenken et al. +START-* measured STM contour I I I II I I I I identified surface levels -1 -1 -1 -1 0 0 0 -1 -1 Fig. 3 Schematic illustration of the pattern recognition in an individual STM scan line. Starting from the central local maximum, the algorithm searches in both directions for the best match with a distance to the next step edge of Ax (regular distance), Ax -dx (STEP up) or Ax + dx (STEP down).The procedure is iterated to the end points of the line, and repeated for all lines in the image. measured heights at the three lateral distances, the algorithm sums all measured heights between the first maximum and each of the three possible locations for the next maximum, in order to minimize the effect of noise. The three resulting average heights are compared with the average heights expected for the three possible step locations. These expected heights are calculated for a non-corrugated surface, i.e. a surface for which the area between neighbouring steps is interpolated linearly. This makes the method virtually independent of the actual corrugation amplitude in the image. The step location that works best is selected and the procedure is then applied to the three possible stepstep distances measured from this location.This process is iterated up to the image edges. The result is a set of step positions with level numbers, the levels differing by -1,O or 1 between neighbouring steps. Several corrections are necessary to make the procedure perform properly. The tilt angle of the (1 15) plane with respect to the scanning plane has to be estimated in advance. It enters the comparison between measured and predicted height averages. Since a small error in the initial estimate of the tilt angle would lead to the accumulation of errors in the predicted average local heights, we use, at every step in the described iteration, a certain fraction of the difference between measured and predicted heights to correct the subsequently predicted heights.Once a scan line has been processed, the difference between the measured line and the line reconstructed on the basis of the pattern recognition is used to calculate the tilt angle more accurately. This better value for the angle is used in the second and subsequent passes through the same algorithm. The procedure starts from a local maximum near the centre of the scan line. As a result of noise in the image, this local maximum usually is statistically displaced somewhat from its ideal location. In the algorithm the small displacement of the starting location propagates through the line and may lead to errors in the pattern of recognized steps.Therefore, after the first pass of a scan line through the recognition algorithm, the displacement of the starting position is calculated from the comparison between the measured and reconstructed lines, so that the subsequent processing of the line starts from the most suitable location. Finally, after the described processing, the algorithm reinspects each STEP, in order to ensure that a better description of the line cannot be obtained by placing the STEP one step earlier or later along the line. In this way, the recognition is made independent of the order in which the steps are examined. When a line has been analysed by the algorithm, the tilt angle, height offset and starting position are used as first estimates for the analysis of the next line.Once all lines have been converted into step positions and level numbers, the tilt-angle average of the whole image Roughening investigated by STM can be used in subsequent iterations of the entire algorithm, which then again starts near the centre of the image. The final result can be confronted with the original image by reconstructing an image from the pattern of steps and STEPS combined with all non-idealities, such as the finite tip resolution and the piezo creep, recognized by the algorithm. Both a quantitative x2 comparison and a visual comparison show that this type of recognition works extremely well. Each image [e.g.Fig. 2(a)-(e)] contains no more than just a few locations in individual scan lines where the algorithm prefers to place a STEP in a position one step different from that which we would expect visually.However, at those locations the situation in the original image is usually sufficiently unclear that it indeed allows for different local interpretations. Also at high temperatures, where the STEP density is very high, the reconstructed image is virtually identical to the original. It should be emphasized that in that case the algorithm outperforms us, as we find ourselves unable to analyse the high- temperature images by hand, without running into severe decision problems and substantial internal inconsistencies in the result. Theory of Displacement Statistics The model we adopt to predict the low-temperature displacement statistics on the (1,1,2n + 1) surface of an f.c.c.crystal is based on the terrace-ledge-kink model introduced by Villain et al.7 Two energies determine both the low-temperature statistics and the roughening transition at higher temperature, namely the formation energy, Wo,of a kink and the interaction energy, w,, per step atom for two steps that are one unit, dx, closer to each other than the regular distance Ax. Multiple kinks, displacing a step position by more than one unit, are not allowed in the model. The repulsive interaction between steps is assumed to be sufficiently large to neglect safely, and thus forbid, the possibility of two neighbouring steps reducing their distance by more than one unit from their regular distance: The Hamiltonian constructed according to these rules is7 JF = Wo CIUmOi + 1) -umoi)l+ Cg[um+lcV)-umo/>I (1) m,.v m,?-Here, umo/)= x,o/) -rnAx is the deviation of the rnth step from its regular position mdx, and the function g is defined as X>_O g(x) = { $,; x = -dx 00; x<-dx For the interpretation of the STM images, we consider the vicinal surface at a temperature far below TR.At low temperature, kinks in step edges are very rare, so that the model can be simplified drastically to that of a single step with kinks in either direction between two neighbouring step edges that remain fixed in their regular positions (Fig.4). Then, the two terms of the Hamiltonian reduce to single, instead of double, sums. where now g( fdx) = w, and g(0) = 0. The summation over y is a summation over a one- dimensional row of lattice sites.We start with a row of N sites and impose the periodic boundary condition: site N + 1-site 1. One can rewrite the Hamiltonian (3) as the one for a one-dimensional Ising-system, by assigning a ‘spin’, u= + 1, to every site that is on its regular position, and (Z = -1 to every displaced site: N N JF = --W~~(U,,+,-1) + WnC(a, -(4):[/=1 I= I J. W. M. Frenken et a1 wo L 7-Fig. 4 Schematic representation of a part of the (1 15) surface at a low temperature. One of the step edges contains a short excursion, with two kinks (energy W,) and four displaced step edge sites (energy w,). In our one-dimensional low-temperature model we need to consider only the three step edges marked by the heavy lines.The corresponding partition function is where v and L are reduced energy parameters, depending only on W,, w,and T. In the thermodynamic limit, N +a,the one-dimensional Ising system can be proven to be equivalent to a Markov chain. The conditional probability for the kth spin, ok, to have a particular value, given the values of the preceding spins ol.. .ok -depends only on the value of (Tk -1. In other words, the statistics along the chain can be described on the basis of four transition probabilities P, In ref. 8, these probabilities are derived as: with a, A and B depending only on v and L.* The step edge configurations that appear most useful for the experimental determi- nation of W, and w, are displaced step edge sections between two kinks.An excursion of length r can be described as the spin configuration 1(-1)'l. The probability of finding such an excursion, starting at a specific location along the step edge, is where PI= 1/A is the probability for an arbitrary spin to have the value 1. Using the theory of runs and other recurrent patterns, developed by Feller" and Bizley,12 we have derived the average number of excursions of length r per unit of length along the step edge. Simplifying the problem by counting multiple excursions of length r, separated by just one regularly placed site, e.g. I( -1)'1( -l),l, simply as a single excursion, we find for the excursion frequency* This frequency is lower than E, because of the exclusion of multiple excursions.Explicitly counting the pattern 1( -1)'1( -1),1 as a double excursion and also allowing higher-order multiple excursion patterns, we arrive at the more complicated expression for the frequency* Roughening investigated by STM The latter frequency is slightly higher than E,., because the last '1' in each excursion 1( -1)'l forms the proper starting element of a possible following excursion 1( -1)'1, which makes multiple excursions a little bit more probable than uncorrelated sets of single excursions. In the limit of zero temperature, expressions (7)-(9) all reduce to the simple Boltzmann estimate for the probability of Y displaced sites between two kinks, P,: 2W0+rw, The factor 2 accounts for the fact that we consider two equally probable excursion directions. As an example, we show in Fig.S the excursion length frequency at 300 and 500 K, calculated with expression (9) for the typical values of W, and w,of 1500 and 120 K, obtained by Lapujoulade for Cu( 115).13The higher temperature is above the estimate for the roughening temperature of 380 K given by Lapujoulade,13 but it is below the theoretical value of 635 K obtained for this combination of W, and wn, using the VGL the~ry.~ The open circles in Fig. 5 illustrate the result of a two-dimensional Monte Carlo simulation of the VGL model described in eqn. (1) and (2). The analytical expression for a one-dimensional king chain gives an adequate description of the low-temperature statistics of step edge excursions on the two-dimensional surface.At 300 K the Boltzmann factor deviates from the exact result by no more than a few per cent. At higher temperatures, e.g. 500 K, collective displacements of neighbouring step edges increase the excursion frequencies above our one-dimensional estimate. The recipe for determining the energies W,and w,from the semilogarithmic plot of the measured low-temperature excursion length frequency is evident from Fig. 5 and expression (10). The (initial) slope of the curve is equal to w,, while the extrapolated value for Y = 0 corresponds to 2W0. Summary In this paper, all ingredients have been supplied for an experimental test of theories of the roughening transition on crystal surfaces vicinal to a low-index orientation.From STM images measured as a function of the crystal temperature, we directly obtain the transition temperature, TR.The images can be further analysed by use of a pattern recognition algorithm. From the experimental frequency distribution of the length of excursions in individual step edges, at temperatures well below TR,the values for the kink formation energy and the stepstep interaction energy can be determined within the framework of the employed theoretical model for the roughening transition.' The STM images presented here have not been subjected to the described type of statistical analysis, since these measurements have been performed too slowly (ca. 1 s per line) with respect to the typical time between subsequent diffusion events, even at room temperature.Consequently, the surface diffusion would have strongly corrupted the experimental excursion length distributions, which would have led to unacceptably large errors in the derived energy parameters. There are two possible experimental solutions to the diffusion problem, namely faster scanning and measurements on a cooled surface. At present, we are repeating our measurements on Ag( 1 15) with a high-speed high-temperature STM. With this new apparatus, the observation speed has been increased by more than two orders of magnitude (< 10 ms per line), which should be sufficient to measure the true room-temperature kink statistics. One of the authors (J.W.M.F.) would like to express his gratitude to J. E. Demuth and R. J.J. W. M. Frenken et al. 10-2c, I I I I,, , , , $,I,, , I,, , , (b) length Fig. 5 (a)Frequency distribution for excursions of length Y along the step edge at a temperature of 300 K. W,and MI,are set to the experimental values of 1500and 120K for Cu( 115).j3 (-), excursion length frequency F,, eqn. (9),for a one-dimensional Ising chain. (--), simple Boltzmann expression (10). The dashed line almost coincides with the solid line. (0),frequency of excursions of length Y in a two- dimensional Monte Carlo simulation according to the model by Villain et al.’ (b)Same as (a) for a temperature of 500 K. Hamers for their support of the STM experiments at the T. J. Watson Research Centre (IBM). This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM) and was made possible by financial support from the Netherlands Organisation for the Advancement of Research (NWO).References 1 W. Telieps and E. Bauer, Surf. Sci., 1985, 162, 163. 2 J. C. Campuzano, M. S. Foster, G. Jennings, R. F. Willis and W. Unertl, Phys. Rev.Lett., 1985,54,2684. 3 J. F. van der Veen, B. Pluis, A. W. Denier van der Gon, in Chemistry and Physics of Solid Surfaces, ed. R. Vanselow and R. F. Howe, Springer, Berlin, 1988, vol. VII, p. 455. Roughening investigated by STM 4 H. van Beijeren and I. Nolden, in Structure and Dynamics of Surfaces, ed. W. Schommer and P. von Blanckenhagen, Springer, Berlin, 1987, vol. 2, ch. 7. 5 W. K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. R. SOC. London, Ser. A., 1951,243, 299. 6 J. W. M. Frenken, R. J. Hamers and J. E. Demuth, J. Vac. Sci. Technol. A, 1990,8, 293. 7 J. Villain, D. R. Grempel and J. Lapujoulade, J. Phys. F, 1985, 15, 809. 8 J. B. Sanders and J. W. M. Frenken, Surf. Sci., 1992, 275, 142. 9 M. Poensgen, J. F. Wolf, J. Frohn, M. Giesen and H. Ibach, Surf. Sci., 1992, 274,430. 10 S. Rousset, personal communication. 11 W. Feller, An Introduction to Probability Theory and its Applications I, Wiley, Chichester, 3rd edn., 1968. 12 M. T. L. Bizley, J. Inst. Actuaries, 1962, 88, 360. 13 J. Lapujoulade, Surf. Sci., 1986, 178,406. Paper 31002651; Received 14th January, 1993