PHYSICAL REVIEW B VOLUME 58, NUMBER 19 15 NOVEMBER 1998-I Quantitative assessment of STM images of Fe grown epitaxially on MgO 001... using fractal techniques S. M. Jordan, R. Schad, D. J. L. Herrmann, J.F. Lawler, and H. van Kempen* Research Institute for Materials, University of Nijmegen, Toernooiveld 1, NL-6525 ED Nijmegen, The Netherlands Received 3 June 1998; revised manuscript received 20 July 1998 We have assessed scanning tunneling microscope images of Fe grown on MgO 001 at various temperatures using two different methods. Evaluation of the height-height variance function reported a correlation length very close to the average island radius. The area-perimeter method reported the perimeters above which non-square-law scaling of the islands begins to be somewhat lower than the average perimeters of the discrete islands. A comparison of two common methods for evaluating length-dependent roughness is made. S0163-1829 98 00143-X I. INTRODUCTION of structure sensitivity over several decades, in many cases including the required length scales. For most systems stud- Roughness studies of growing crystals are very attractive ied these measurements have to be done under ultrahigh for several reasons. First, it is the sheer beauty of fractal vacuum UHV conditions to prevent oxidation that can alter systems that has fascinated researchers for decades1 and sec- the surface structure in an unpredictable way. ond, kinetic roughening of the growth front during thin-film As a test system we have chosen the epitaxial growth of deposition2­10 is of eminent technological importance. Fe on MgO(001) at various temperatures for several reasons. The films' physical properties will very much depend on First, the substrate provides an almost uniform template with the smoothness or roughness of the final growth front that up to m wide, atomically flat terraces, which, with respect will form the interface to the adjacent material or the surface to the structure of the Fe films grown on top, can be regarded that interacts with the environment. For instance, the inter- as flat. Second, the epitaxial growth of bcc Fe provides a faces in field-effect transistors or tunnel junctions have to be system with a simple fourfold in-plane symmetry without extremely flat to guarantee homogeneous oxide thicknesses, inhomogeneities like grain boundaries as found in polycrys- whereas the so-called giant magnetoresistance effect in mag- talline samples. Third, the structure parameters estimated here are of importance for the understanding of the magnetic netic multilayers is enhanced by a certain degree of interface properties of such Fe films21 or the transport properties of roughness.11­13 Also, the performance of catalytic materials Fe/Cr superlattices.11­13 relies on a huge surface area. The analysis is done by examining STM micrographs. Proper control of the surface properties requires an under- First, the (2 e)-dimensional surface roughness is analyzed standing of the underlying growth mechanisms. This can be for its length scale dependence. For a self-affine surface the achieved by the detailed structure analysis of surfaces pre- height-height variance function, where L is the lateral dis- pared under various growth conditions. However, the rough- tance between r and r , ness of a surface is a more complicated concept than the widespread use of this simple term might suggest. The size g L z r z r 2 , 1 of the commonly used root-mean-square rms roughness, for example, in most cases depends on the lateral distance should saturate for L at over which it is measured and therefore does not provide a 2 comprehensive description of the surface structure. g L 2 2 Also necessary are quantitative estimates of the surface and vary with L for L as roughness in both vertical and lateral direction. Typically this includes the vertical rms roughness , the lateral corre- g L L2H, 3 lation length and the Hurst parameter H, which describes the fractal dimension D of a self-affine surface via D 3 with and H being as defined above, and being the rms H. The fractal dimension is equally important as since it roughness averaged over an infinitely large image. describes the jaggedness of the surface,14 which, in combi- The function g(L) is related to the height-height correla- nation with , is a measure for the step density which is tion function often the important parameter.15,16 These parameters have to 2 be measured by techniques fulfilling certain requirements. C L exp L/ 2H 4 Their structure sensitivity must range from the smallest pos- via sible length scale the atomic scale up to length scales ex- ceeding and need to be strictly surface or interface sensi- g L 2 2 2C L 5 tive. For surfaces an ideal instrument is the scanning tunneling microscope17­20 STM having a dynamical range yielding 0163-1829/98/58 19 /13132 6 /$15.00 PRB 58 13 132 ©1998 The American Physical Society PRB 58 QUANTITATIVE ASSESSMENT OF STM IMAGES OF Fe . . . 13 133 g L 2 2 1 exp L/ 2H . 6 The values of and H can be estimated from the asymptotic behavior for, respectively, large and small values of L. The correlation length is then found by a one- parameter fit to g(L) using Eq. 6 . Equations 4 and 6 were introduced by Sinha et al.,22 and make convenient interpolation formulas. Their use is re- stricted to surfaces that have a Gaussian distribution of heights, which we found to be the case. For surfaces with a non-Gaussian distribution, H can be found from Eq. 3 . The x intercept between the regime where g(L) scales with L and the regime where it saturates then gives . Second, the (1 )-dimensional perimeter of the Fe growth structures is analyzed using the area-perimeter method.1 A collection of similar, nonfractal islands will dis- play a ratio23 perimeter / area 1/2, 7 FIG. 1. STM image of 5 nm of Fe grown at 395 K, with a scan length of 50 nm. A cross section taken along the solid line is both independent of the island's size and the resolution to shown, with a height scale in nm. which the dimensions are measured. This ratio will be 2 in the case of round islands and 4 in the case of squares. 15° to the sample normal, the flux being directed along the It has been found that for islands displaying fractal prop- Fe 110 axis. The sample was maintained at the required erties the value of measured depends on . As , the temperature by electron heating of the sample holder. The ``yardstick length'' decreases, the measured perimeter in- layers were all 5 nm in thickness, which provides a stable creases without limit. The scaling exponent between the pe- and electrically conductive film. The samples were then studied in situ by use of an in- rimeter P and is found to be (1 D ), where D is the house built STM. two-dimensional fractal dimension and varies between 1 and 2. The relation between D and D is not as simple as D D 1,20 since the fractal properties in the vertical z di- III. HEIGHT-HEIGHT VARIANCE FUNCTION rection are important for g(L). The measurements to be de- scribed in Sec. IV are quite distinct from roughness analysis. We have previously reported an empirical study of the It is practically difficult to make repeated STM scans with behavior of layers of bcc Fe deposited on MgO(001).21 This very different resolutions; nonetheless we are able to access STM study showed that the Fe forms round islands approxi- D by studying the relation between the area of an island A mately 10 nm in diameter at growth temperatures at and and P. Here the scaling law is below room temperature. As the deposition temperature in- creases, the islands become square, and increase in diameter P A D /2. 8 to 30 nm. At the highest temperature we used, 595 K, just below the point at which a discontinuous film results, we Another piece of information will be provided by this rela- were able to resolve single atomic steps of approximately a/2 tion in that the point at which D becomes greater than unity in height. provides a scaling length for the onset of fractal-like behav- These topographic properties of the film surfaces are a ior. result of the underlying growth kinetics and thermodynam- In this paper, we report a detailed structure analysis of ics. The typical island diameter is a result of the nucleation epitaxial Fe layers grown on MgO(001) including the fractal density during the start of the growth whereas the develop- analysis in (2 ) and (1 ) dimensions. ment of growth pyramids is a sign of reduced diffusion across step edges due to the so-called Schwoebel barrier.24 II. EXPERIMENT The square-island shape is caused by the preference of step formation along the 100 directions, which can only be Commercial MgO(001) substrates were first cleaned by achieved when diffusion along the step edges is fast enough, washing with organic solvents to remove contaminants. They thus requiring higher deposition temperatures. were then heated to 1070 K in UHV for 1 min and then A 50-nm-wide STM image for a growth temperature of analyzed by Auger electron spectroscopy. A KLL C peak 395 K is shown in Fig. 1. This illustrates the typical form was seen corresponding to 6% of 1 ML. Heating the MgO to and quality of our images. The height is recorded as a 16-bit temperatures as high as 1400 K did not reduce this contami- digital number. A cross section of the surface, which cuts nation. Atomic-force microscopy investigations showed the across several islands is also shown. The vertical z axis is substrates to be of exceptional flatness; single atom high ter- not to the same scale as the horizontal axis - the steep sides races of width up to 200 nm were seen. of the ``valleys'' between islands make typical angles of 12° Fe layers were grown using a Knudsen cell at a rate of to the horizontal. Thu¨rmer et al.7 found angles of 30° on 0.13 nm per min. The iron atoms were incident at angle of pyramid faces found in a 300-nm-thick film due to the 13 134 JORDAN, SCHAD, HERRMANN, LAWLER, AND van KEMPEN PRB 58 TABLE I. Fractal dimension (D), correlation length see Sec. III , average island size and rms roughness averaged over a single 200-nm image summarized. The standard deviation in island sizes was approximately 15% in all cases. Growth temp. K D 0.1 nm Mean island nm radius nm 295 2.1 4.7 0.4 3.5 0.52 395 2.1 4.7 0.4 4.5 0.42 495 2.1 7.9 0.3 8 0.58 595 2.4 14.2 1.5 15 0.28 plete scan of the surface. The small spread of the lines indi- cates either that our data is of high quality or that all images FIG. 2. The height-height variance function g(L) for the four give very similar results. Since g(L) changes in several re- temperatures. The horizontal lines show the saturation at 2 2 . The spects when the deposition temperature is varied, we con- difference in H between 495 and 595 K is clearly visible. Curves clude that g(L) measures a useful property of the image. shifted vertically for clarity. As outlined in Sec. I, when the tile-edge length L is much greater than the size of typical features, increasing L still Schwoebel barrier preventing the downward diffusion of at- further does not bring higher features into a tile, and g(L) oms at step edges; however we cannot expect this slope to saturates at 2 2 . Table I gives values of averaged over have reached its final value on a film 5 nm thick. several complete 200 nm images. This number provides little We evaluated Eq. 1 directly. The distance L between information by itself, since many possible differing surfaces points z and z was varied to provide the x axis, the averag- can give the same parameter. Examination of the surfaces ing occurring over typically 106 points in order to provide leads us to conclude that the lower is associated with the good statistics. The parameter L could typically be varied large flat islands that appear at the highest temperature. We over four orders of magnitude; we analyzed scans of widely are confident that g(L) does not increase further when L varying sizes to extend this range. We also removed images 500 nm. from the data set that showed gross and obvious defects such For a temperature of 495 K, the slope of g(L) begins to as large areas where contaminants are present or resolution is fall when L 0.2 nm. This could be due to the finite reso- lost. This was the sole criterion for removing data. lution of the STM, or that an image of side much less than No discussion of roughness measurements can be com- (5 5 nm) could not be plane-fitted correctly. plete without a treatment of the effects of image artifacts. When L is much smaller than , the scaling exponent H There are two common phenomena to take into account: the Eq. 3 can be determined. It is related to the fractal dimen- finite radius of the tip and the slope present in the image. Our sion by D 3 H. Table I gives values of D found by least data were plane-fitted the least-squares fitted plane was re- squares fitting of data to the equation moved in both the x fast-scan and y directions before the division into tiles, providing an image with no overall slope. g L aL2H 9 The effect of plane-fitting is discussed by Kiely and Bonnell,18 who showed that it has a dramatic effect on g(L), with 0.1 L 8 nm. It appears that D is constant at low- reducing g for lengths greater than . ``Flattening,'' that is, deposition temperatures, but increases somewhat at 595 K. adjusting the mean of each scan line to be the same, also The values of were determined from the intercept of Eq. reduced g for all L, but we felt that for our small scan 9 and the saturation of g(L) at large L using the formula lengths this was unnecessary, and likely to remove some of 2 the surface structure. In this study, the images used had a 2 /a 1/H, 10 scan size larger than the correlation length, as opposed to with being fixed at the value presented in Table I. Aver- studies such as that of Krim et al.,25 which used many small age island radii along the major axes are also given; these images that were individually plane-fitted. were made by measuring the distance between the trenches The effect of tip radius has been discussed by several on opposite sides of well-defined islands. The radii and authors.26,27 It is clear that it affects the measured agree closely. roughness;18 however assessing the tip radius without a ref- erence sample is nontrivial. All images were taken with the IV. AREA-PERIMETER RELATION same mechanically cut Pt-Ir tip; however, the tip's properties can change over time. In situ UHV use makes measurement A classic method for the evaluation of D , which has and maintenance of a particular tip geometry difficult. It is been applied to such diverse systems as STM images20 and difficult to define a radius for our tips; however they are rain cloud formation23 is the area-perimeter method. Experi- sharp enough to resolve atomic steps and have sufficient as- mentally, the perimeters and areas of the objects to be as- pect ratio to follow deep features. sessed are plotted on a log-log graph and the scaling expo- Figure 2 shows the relation g(L) plotted using a log-log nent determined. Mandelbrot, in Ref. 23, gives the relation scale. Each line in the figure is derived from a single com- between the perimeter P and the area A as PRB 58 QUANTITATIVE ASSESSMENT OF STM IMAGES OF Fe . . . 13 135 FIG. 5. STM image of 5 nm Fe grown at 595 K, with a scan length 200 nm. Contours as Fig. 4. FIG. 3. Area-perimeter relation for synthetic data. The top line square islands has been scaled by a factor of 10 over the bottom The sizes of the islands were then measured using com- line round islands . A difference in the value Eq. 11 is visible mercial software. A threshold zt is chosen; points above this between the two lines. The size of the symbols represents the num- level are assigned as ``land,'' the remainder as ``sea.'' An ber of points clustered together. The solid points represent data for automatic algorithm then measures the area and apparent pe- a threshold of 0.5 runtogether allowed ; the hollow 1, which only rimeter, neglecting islands that touch the sides of the image allows islands of the given shape. field. If zt is chosen as 1, then only the islands themselves will be measured, since the ``coastline'' that connects chains P 1 D A D /2. 11 of islands will be ignored. Setting zt 0.5 allows these smaller islands to ``run together,'' forming complex shapes. Figure 3 shows the area-perimeter relations for square and This equation is dependent on the ``yardstick length'' as round islands. The lighter superposed lines follow the law discussed by Mandelbrot.1 In the case of digitized STM im- given in Eq. 7 . For zt 1 open points , no deviation from ages, is the pixel size, which varies with image size. this law is found - the surface is nonfractal. When zt 0 We first demonstrate this relation by applying it to syn- filled points , points falling away from this law above 100 thetic data. Images were generated which consisted of is- units perimeter were seen. These chains of islands give D lands with a randomly generated diameter and lateral posi- values of 1.7 0.05 circles and 1.3 0.3 squares . The res- tion. Each island was the same shape, with height 1. The caling of the image to differing resolutions affects , but in ``coastline'' of the island had height 0.5, forming an inter- this region the behavior is nonfractal, so A/P is constant. In mediate level between ``land'' and ``sea,'' which had a any case, changes to do not affect the power-law relation height of 0. The islands were allowed but not compelled to between A and P. The expected change in between circu- touch at the ``coastline'' level, but not to overlap at the lar and square islands is also found. height 1 level. The images were then digitized to form 512 512 pixel images, which were given an arbitrary length scale of 500 units. A 2.5 magnification of the original im- age was then made, digitized, and assigned a length of 200 units. FIG. 6. Area-perimeter relation for STM images. Top 595 K FIG. 4. STM image of 5 nm Fe grown at 295 K, with a scan growth temperature, middle, 495 K, bottom, 295 K. The sizes of the length 50 nm. The shapes resulting from the highest and lowest circles reflect the number of points clustered together. The variation thresholds are superposed. Islands touching the sides of the image in average island size can clearly be seen. The top two data sets are neglected. have been scaled by factors of 10 to separate the curves. 13 136 JORDAN, SCHAD, HERRMANN, LAWLER, AND van KEMPEN PRB 58 TABLE II. Data from the area-perimeter measurements see Sec. IV . The fractal dimension (D ) is given, as is the point at which the line fitting fractal behavior meets the line D 1. Growth temp. D Perimeter intercept Pc Radius intercept Mean-island area Area s.d. K nm nm (nm)2 (nm)2 295 1.58 0.03 13 2 13 27 495 1.77 0.06 39 6 42 43 595 1.82 0.02 76 12 80 115 Selected STM images used for the roughness study were analyzed by this method. Images of various scan sizes were Phenomenologically, both g(L) and (L) yield the same used, each image providing approximately 100 islands. and approximately the same H: Three different zt values were used, providing islands rang- ing from pinnacles only a few pixels across to complex shapes covering a large area. Typical results of the thresh- olding procedure are shown in Figs. 4 and 5. L aL2H l l2Hdl for l . 13 Figure 6 shows the area-perimeter relation for the STM 0 images. The lines represent Eq. 7 for circular islands. As with the synthetic data, the smaller islands fall on the ex- pected square law, fractal behavior being seen in the larger However, is shifted in the positive x direction with respect islands. Deviations from Eq. 7 can also be seen at smaller to g(L) in the case of a log-log plot . perimeters; this is an artifact due to the software measuring We were able to show that our images were correctly islands that consist of few pixels. Table II gives the fitted plane-fitted, since g(L) had truly saturated. However, plane- values of D , which are seen to increase with increasing fitting must be used with discretion, since its use on images deposition temperature. The perimeter at which the line rep- of side smaller than will result in corruption of the image resenting Eq. 11 meets the squarelaw P and give incorrect values for g(L). The height-height vari- c is presented, as is its equivalent radius, P ance function allows us not only to measure , but to as- c/2 . The high standard deviation from the mean-island area is accounted for by the fact that sess its validity since artifacts such as tip collisions and dirt islands with an area 2 orders higher than the mean are particles prevent g(L) from saturating. present. For this reason, these figures are not directly com- We conclude that Eq. 1 gives more meaningful results parable with those in Table I. than the MIV since (L) yields only approximate values of H and . However, MIV requires slightly less computational effort. The correct values for D allowed us to resolve a clear V. DISCUSSION increase in the fractal dimension at the highest temperature, The results of Sec. III should be compared with previous which could not be resolved using variography.28 This reports14,18,28 in which multiple image variography MIV is change supported the report of Thu¨rmer et al.7 that different used to assess length-dependent roughness. MIV is the mea- film morphology results at temperatures above 500 K due to surement of roughness over various sized sections of images, atoms being able to diffuse downwards at step edges. The and yields curves similar in appearance to Fig. 2. The param- values for are numerically close to the average island radii. eter ``rms roughness'' (L) is in fact a measure of the de- We attribute the difference at 295 K to the fact that there is a viation of the surface from the mean height. As smaller and wider variation in island heights at this temperature, increas- smaller sections of the surface are examined, the mean ing . heights for each section will begin to differ from the mean The thresholding procedure used prior to the area- height of the overall image. Thus, the mean becomes closer perimeter analysis reduces the three-dimensional STM image to the average height of the section, and (L) becomes to a two-dimensional set of points. This results in a dimen- smaller. sion D between 1 and 2, which is not directly comparable MIV is also generally used to average over square sec- to D. There is evidence for an increase in D with tempera- tions of images, so to compare with direct evaluation of Eq. ture. The parameter Pc/2 is consistently smaller than the 1 we must consider the fact that MIV averages over all the mean-island radius. lengths present in the square. We can derive the exact rela- tion between g(L) and the more commonly reported (L) ACKNOWLEDGMENTS using the distribution of lengths within a discrete unit square, This work has been financially supported by the Dutch d(l).29 We can now write the relation using Foundation for the Fundamental Research of Matter, which d(l) as a weighting function: is, in turn, financially supported by the Dutch Organization for Scientific Research. R.S. would like to acknowledge the assistance of the Brite-Euram program of the European Community Contract No. BRE 2-CT93-0569 . D.J.L.H. L would like to acknowledge the support of the Studienstiftung d l/L g l dl. 12 0 des Deutschen Volkes. PRB 58 QUANTITATIVE ASSESSMENT OF STM IMAGES OF Fe . . . 13 137 *Author to whom correspondence should be addressed. Fax: 18 J. D. Kiely and D. A. Bonnell, J. Vac. Sci. Technol. B 15, 1483 31243652190. Electronic address: hvk@sci.kun.nl 1997 . 1 B. B. Mandelbrot, The Fractal Geometry of Nature Freeman, 19 J. M. 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