6249
J. Phys. Chem. 1993,97, 6249-6254
Analysis of Fractal Surfaces Using Scanning Probe Microscopy and Multiple-Image Variography. 1. Some General Considerations John M. Williams and Thomas P.Beebe, Jr.’ Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Received: November 12, 1992; In Final Form: February 26, I993
Even when fractal models describe well the topography of a physical surface, the geometry of scanning probe microscopy will generally introduce artifacts into single images that can seriously bias estimates of the fractal dimension D from most techniques of analysis (e.g., power spectrum, correlation function, structure function). Other methods that remain unbiased (e.g., lake/island perimeter-area analysis) cannot extend to crossovers into nonfractal regimes of scale, which are also important to detect. This paper demonstrates that for multiple images fitted to their least-squares planes the topographic variance as a function of image size is a convenient statistic that retains the self-affine scaling form of the original fractal, allows unbiased estimates of D, and also detects crossover scales into nonfractal regimes.
DG = 2, which describes the nonfractal macroscopic geometry of the surface.’* Although the literature makes it clear that SPM In the past several years, researchers have applied fractal topographs often show only the global regime (Le., the horizontal geometric models and techniques, first created by Mandelbrot,’ extent is much greater than the vertical on all scales), the to topographic images from scanning tunneling micro~copy~-~ dimension D = 3 -His reported almost exclusively,as if the local and related methods6 or scanning probe microscopy (SPM) in regime were present.24J8 This might become misleading if general. These techniques provide high-resolution digitized compared to the dimensionas measured by a molecular adsorption topographs of surfaces (or interfaces) on length scales spanning or covering technique; nevertheless, for consistency the authors 4-5 orders of magnitude down to atomic scales. However, the will adhere to the conventional relation D = 3 - H . nature of SPM can introduce artifacts significant to fractal Of course, there are differentiable (nonfractal) functions that analysis into the images; it becomes important to develop robust also have some of the properties of eqs 1, so they are necessary treatments for the images that eliminate these artifacts and but not sufficient c o n d i t i ~ n s ; ~important J~ examples for SPM precisely estimate statistics such as the fractal dimension(s) D are and any fractal crossover length(s)798 em, especially the range cmin to emax within which fractal scaling a p p l i e ~ . ~ , ~ a horizontal corrugated-planar surface has ab2(r) cz C, (2) The topography of a rough physical surface often is stochastic (random) but exhibits statistical self-resemblance over a range but D = 2 and P ( v ) # klvt’ of scales because the random processes that formed the surface (e.g., diffusion,IO wear,” or fracture12) show scale invariance; an inclined corrugated-planar surface has then fractal geometry will describe the topography. The direction ah2(.) C b17I2 and D = 2, but P(v) # klvt3 normal to the surface is, of course, physically distinct from directions parallel to the surface and so in general will scale differently from them:13this behavior is called self-affine. Thus, Figure l a depicts how an SPM tip scans a surface; several the appropriate models for rough surfaces under SPM will be effects are noteworthy. First, the size of the probe tip will continuous, single-valued ( v i ) , stochastic, self-affine fractal determine the spatial resolution of the microscope, and so could functions. The common models, including fractional Brownian bias fractal analysis at small scale~.3*~ Atomic resolution is motion10 and the Weierstrass-Mandelbrot fractal function,14are desirable, since the surface must be nonfractal at the atomic characterized by the Hurst scaling exponent H,as H increases, scale? Also, a topograph is a single-valuedfunction of tip position the fractal becomes smoother, and the followingrelations, treated because the SPM tip does not detect overhangs on the sample fully in the next sections, apply:lJS ~ u r f a c e . ~Finally, J~ unless one employs micrometric techniques that range from the scale of SPM (- 1Q-6m) to the macroscopic O12)>w(x)dx
3 Two examplesrelevant to the SPM experiment produce an explicit form for u2. Example I. Variance over a Finite Interval after Subtracting the Arithmetic Mean (Offset). For convenience, let w(x) be a real function; then fitting to the average value sets g(x,u) = w(u) in eq 10 above, so that u'
= J-IlZ(3)12(l - IW(i)lzJ-Iw(x)d x ) d3 = JmlZ(i)12(1 -m - IW(i)l') d3
The SPM experiment typically treats all values of z(x) on the interval 4 2 Ix Ia/2 with equal weight, where a is the interval width; this corresponds25 to
z(cos a r i - sinc ai)*
The next section will treat the convergence of eqs 12 and 13 for the lZ(i)I2of interest. Convergence of 02 for Scaling Power Spectra Fractal profiles have power spectra of the general 1/fscaling form'
IZ(i)lZ0: 1/14@ Comparison with eq ICimplies f? = 2H + 1. Then, to show that u2 converges for some f? in both examples above, it suffices to show that Jt(l/9)(l- sinc2 a i ) d i converges. As i 0, let sinc air 1 (sin a r i ) / a r i be represented by its Maclaurin series (Taylor series about i o = 0):
-
U22i2
s i n c a i = 1 -so that
sinc' ai = 1 -
sin au3
W(3)= sinc a3 = UTE
(11)
2a2u2i2 + ... 3!
1
= JmlZ(i)J2(1 -m - sinc'
a i ) d3
(12)
It is important, however, that only the residuals, not the z(x) themselves, are known, so the effect of unknown orientation remains and will still affect this estimate of u2. Example II. Variance over a Finite Interval after Subtracting the Least-Squares Line (Offset and Trend). This technique transforms the data into an orientation-independent frame and so eliminates bias from the unknown tip orientation. For convenience, let w(x) be real and w(-x) = w(x); then fitting to the least-squares line sets
in eq 10, so that
+ ...
Therefore
Then finally u'
3!
-
2a2r2iz+
-(I i+ - s i n 2 a i ) =
-
3!
is integrable as I 0 if and only if 2 - @ As i a,sinc2 a? 0, so that -+
+
1 -(I - sinc' a i )
-
9
-
*"
+ 1 > 0, or @ < 3.
-1 i+
is integrable as I if and only if -6 + 1 C 0 or @ > 1. Therefore, u2 converges if and only if 1 < @ < 3, implying that 0 C H < 1, which agrees with eq l a (the cases H = 0 and H e 1 correspond to nonfractal behavior, where the hypotheses of this treatment do not apply). QJ
Numerical Solution for d(a) The objectiveis to obtain the scaling behavior of u2as a function of sampling interval width for the two examples above, for fractal
SPM and Fractal Variography. 1
l;;> M 0
H
-jl,, ,
,
The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 6253
,; I,!;
1.5
e
~,
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,
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,
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At) = ~(F(v)le'2r"fe'6(u) dv
-6
-3
-2
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2
1
0
(Figures 3) of log u2 versus log a for several 8 indeed have slope = 6 - 1, correlation r2 = 1. The figures also verify that the variance is less for the case of example I1 than for example I; Le., the fit is better, since example I1 accounts for the overall trend as well as the offset. These results naturally apply also to processing time domain signals, by recasting eq 5 in terms of the time domain function f(t) and frequency domain function F(v) familiar in Fourier transforms:25
3
log profile width ( a r b . units)
The extension of these results to functions of several independent variables (e&, SPM topographs) is straightforward: z becomes a function of the position r = ( x , y, ...) and may be treated as a composition of waves of wave vector = ( i l , i 2 , ...) in reciprocal space. Then eq 5 becomes
z(r) = JmIZ(i)le"dre'4(') -m
M 0
- 6 i , , , ' , . , ,,
2.9 :
0 , . . ,
,
,
.
.
.
,
,
.
I
.
.
,
.
,
.
.
profiles having power spectra
(apart from a proportionality factor). The following simple method of numerical integration suffices to calculate u ~ ( u )since , for 6 1 100/a, sinc2 a6 I 10-5, and the integrals take the approximate form dl
where dV = dl, di2 ... is a volume element in reciprocal space. The objective is then to calculate u2 for z . When z is fractal, the power density 12(6)12 is not integrable, so the treatment must introduce the sampling and fitting functions w(r) and f(r) to allow u2 to converge. The SPM experiment collects z over a rectangular region and subtracts either the average height or the least-squares plane from z before computing the variance; these are the two-dimensional extensions of examples I and I1 above (actually, the algorithm processes discrete data over the region instead of weighting and integrating continuously, but since the number of data per image is 16 000 to 65 000, the approximation is adequate). For several independent dimensions or factors, under the conditions of examples I and I1 above, the residual sum of squares partitions so that the n-dimensional variance over the region a* is the sum of the independent one-dimensional variances.26 If z is stochastically isotropic, then u; = nutD,which preserves the linear dependence and slope of log u2 on log a. Subtracting the least-squares plane from every image, the extension of example 11, has the important advantage of eliminating the bias from the orientation effect; the final result then takes the forms
+ Ji3m/a[integrand] d9 +
u2(a) = K&'
(15a)
u2(a) Qc aw
(15b)
u2(a) a
= 2( a2?r2(A9)" + JAm/'[integrand] 3(3 - 8)
dl
+
dV
ubzD
(1 5c)
which have the form of self-affine scaling, eq lb, and allow unbiased estimation of D from the slope of a plot of log u2versus log a, a voriogram.8 The variogram also allows detection of smooth (nonfractal) topographic regimes8 where D = 2: corrugated-planar images, after removal of 'tilt", will have u2(a)that is roughly constant with a, so that the graph of log uz on log a will have zero slope in this region of scale. This case is analogous to eq 2 but is distinct from the extreme roughness of a fractal as H 0 in eq 15b. Since the scaling of the overall topographic variance has a well-defined meaning in both fractal and smooth regimes, it allows the determination of the fractal crossover lengths and cif they lie within the scale range of SPM. In contrast, some other fractal analysis techniques (e&, lake/island perimeterarea analysis5) do not apply to smooth surfaces, which limits their usefulness.
-
This approximationsuggeststhat the variance may take the form $(a) = kapl. This method of numerical integration was applied to eqs 12 and 13 (examples I and 11),with IZ(6)lt as given in eq 14, using double-precision arithmetic and Simpson's rule for the middle integral term, reducing the mesh size A6 until a convergence tolerance of 0.01% was reached for eq 12 (10%for eq 13). The results, in Figures 3, show that for both cases the variance uz is a scaling function of the profile width a, as suggested by the above approximation; this result was obtained previously by another method7.**just for example I. Least-squares h e a r fits
Conclusion
Subtracting the least-squaresplane from an SPM topographic image and then computing the overall topographic variance u2
6254 The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 s .,urface
profile
Williams and Beebe onedimension, between rotation and the residuals from the plane. When the original profile has mean (least-squares) slope m,then Az * m h hdds true on average, so that z‘= (cos e = CJ
AZ‘= c,A( x ’ = x cos 8 X
Figure 4. Diagram of the relationship bctwccn rotation of a profile and its residual from a plane.
produces a convenient statistic whose scaling with image size, u2(u), when plotted double logarithmically as a variogram, indicates fractal scaling regimes where slope = 6 - 20. It also indicates smooth regimes where slope = 0, and thus it can indicate crossover lengths between regimes. The ability of d ( a ) to analyze smooth (nonfractal) regimes is an advantage over some other techniques (e.g., lake/island perimeter-area analysis) which apply only to fractals. Moreover, this statistic is unbiased by the unknown orientation of the SPM probe tip relative to the true surface normal, which biases most single-image techniques for computing D. Acknowledgment. This material is based upon work supported under a National Science Foundation Graduate Fellowship. Additional support was provided by the National Science Foundation under Grant CHE-9206802. Appendix More strictly, it is therotutedropogruphy (neglecting overhang effects) that is recorded by SPM, rather than the topographic residual from a plane; this record still loses self-affine scaling (v.s), however. In one dimension, for example, an SPM profile z’(x’) rotated by 8 from the original profile z(x) obeys
x = x’cos tJ- z‘sin 8 z = x’sin 0
((Ax’sin
e + AZ’COS e)z)
+ Z’COS 8
Ik2@x’cos 81
;.(lAZl2) #
+ IAz’sin
k”Ax1W
The rotated profile z‘(x’), however, has approximately the same scaling form as the residuals t(x) of the original profile z(x) from the SPM planef(x). Figure 4 shows the relationship, in
+ z sin 0
+ AZ sin 0 = Ax cos 0 + mAx sin 0 + m sin e)
Axf = Ax cos f3 = Ax (cos 8 C,Ax
Thus, the scaling form of the residuals remains in the rotated profile. It is this rotation inuariance that motivates the development in terms of residuals from a plane, throughout this paper. References and Notes (1) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco, 1977 and references therein. (2) Denley, D. R. J. Vac. Sci. Technol. A 1990,8, 603-607. (3) Mitchell, M. W.; Bonnell, D. A. J. Mater. Res. 1990,5,2244-2254. (4) Carrejo, J. P.; Thundat, T.; Nagahara, L. A,; Lindsay, S. M.; Majumdar, A. J . Vac. Sci. Technol. B 1991,9, 955-959. (5) G6mez-Roddguez, J. M.; Bar6, A. M.; Vgzquez, L.; Sslvarezza, R. C.; Vara, J. M.; Arvia, A. J. J . Phys. Chem. 1992, 96, 347-350. (6) George, T.; Anderson, M. S.;Pike, W. T.;Lin, T. L.; Fathauer, R. W.; Jung, K. H.; Kwong, D. L. Appl. Phys. Lett. 1992,60,2359-2361. (7) Feder, J. Fractals; Plenum: New York, 1988. (8) Burrough,P. A. FractalsandGeochemistry. In The FractalApprwch to Heterogeneous Chemistry; Avnir, D., Ed.; Wiley: New York, 1989. (9) Pfeifer, P.; Obert, M. Fractals: Basic Concepts and Terminology. In The Fractal Approach to Heterogeneous Chemistry;Avnir, D., Ed.; Wiley: New York, 1989. (10) Mandelbrot, B. B.; Van Ness, J. W. SIAM Rev. 1968,10,422437. (1 1) Stupak, P. R.; Kang, J. H.; Donovan, J. A. Wear 1990,141,73-84. (12) Mandelbrot, B. B.; Passoja, D. E.; Paullay, A. J. Nature 1984,308, 721-722. (13) Bursill, L. A.; XuDong, F.; JuLin, P. Philos. Mag. A 1991,64,443464. (14) Berry, M. V.; Lewis, Z. V. Proc. R. Soc. London A 1980,370,459484. (15) Pfeifer, P. Appl. Surf. Sci. 1984, 18, 146-164. (16) Vas, R. F. Physica D 1989, 38, 362-371. (17) Dubuc, B.; Quiniou, J. F.; Roqua-Carmes, C.; Tricot, C.; Zucker, S.W. Phys. Reu. A 1989,39, 15W1512. (18) Chiarello, R.; Panella, V.; Krim, J.; Thompson, C. Phys. Reo. Lett. 1991,67, 3408-3411. (19) Brown, S.R.;Scholz, C. H. J. Geophys.Res. 1985,90,12575-12582. (20) (a) Presentations and discussions of 12 March 1992 at Pittcon, New Orleans. (b) Discussions at Royaumont Abbey (France) Meeting, April 1991, (21) Clemmer, C. R.; Beebe, T. P., Jr. Scanning Microsc. 1992,6, 319333. (22) Vamier, F.; Llebaria, A.; Rasigni, G. J . Vac. Sci. Technol.A 1991, 9, 563-569. (23) Osborne, A. R.; Provenzale, A. Physica D 1989,35, 357-381. (24) Robinson, Enders A. Least Squares Regression Analysis in Terms of Linear Algebra; Goose Pond: Houston, 198 1. (25) Bracewell, R. N. The Fourier Transform and Its Applications, 2nd ed., Rev.; McGraw-Hill: New York, 1986. (26) Scheff6, H. The Analysis of Variance;Wiley: New York, 1959.
