To obtain a reasonable measure of surface roughness on the nanometer scale, people most often use the atomic force microscope (AFM) or the scanning tunneling microscope (STM), not only because they offer the required resolution, but also, and more importantly, because AFM and STM images are height-encoded. This means we can measure the dimensions of the features in these images both in the plane (in x and y) and out of the plane of the sample surface (in z).
In fact, roughness my be the single most frequently made measurement in industrial applications of the AFM, and certainly an important measurement in academic research applications as well.1
Typically, AFM users rely on root mean square (rms) roughness, Sq, as the measurement of choice. (A quick search on Google for “rms AND AFM” returned about 333,000 hits today.)2
![\displaystyle S_q = \sqrt{\frac{1}{MN} \sum_{k=0}^{M-1} \sum_{l=0}^{N-1} [z(x_k,y_l)-\mu]^2}](http://nano.tm.agilent.com/blog/wp-content/latex/1d3/1d3620acd53c8860bdfff3a119562003-FFFFFF000000.png "\displaystyle S_q = \sqrt{\frac{1}{MN} \sum_{k=0}^{M-1} \sum_{l=0}^{N-1} [z(x_k,y_l)-\mu]^2}")
(Eq. 1)
where μ is the mean value of the height, z, across all in-plane coordinates (x,y):
(Eq. 2)
This measurement, rms roughness, has some inherent limitations that are often neglected. Reporting the rms roughness is almost always useful, but frequently inadequate in accurately describing surface topography in a meaningful quantitative way. In some cases, the consequence of not knowing (or ignoring) the limitations of rms roughness is misperceptions and making poor decisions.
The limitations of rms (roughness) are well-known to those who work often and in some depth with statistics and probability theory, but not to most AFM users. The upshot of these limitations can be summarize by saying that rms roughness measurement can give nearly or identically the same numerical result for two surfaces whose roughness are qualitatively different, or very different; sometimes so different that even a simple visual inspection of the AFM images will reveal.
One important reason rms roughness is sometimes inadequate is that it is computed indiscriminately towards the polarity of the height value at a given pixel, relative to the mean height value across all the pixels in the image. In other words, as the formula (Eq. 1) shows, height values smaller than and larger than the mean value end up contributing to the rms roughness the same way, i.e., as positive numbers. The result is that the rms roughness may measure (very) nearly the same for two different surface, one for example a flat surface with many holes, the other a flat surface with many peaks.
It is clear that to distinguish at least between these two kinds of surfaces, a different kind of parameter from rms roughness is required. Such a parameter exists, and in fact it exists in most if not all commercial AFM image processing software. It is called skewness, and it is not nearly as popular with AFM users as the rms roughness is. (A quick Google search for “skewness AND AFM” returned about 15,000 hits today). The skewness, Ssk, is defined in a way that can quantitatively describe the asymmetry of a height distribution about the mean (and from there, it gets its name):3
![\displaystyle S_{sk} = \frac{1}{MNS_q^3} \sum_{k=0}^{M-1} \sum_{l=0}^{N-1} [z(x_k,y_l)-\mu]^3](http://nano.tm.agilent.com/blog/wp-content/latex/928/9287c3476f4c340c76cb6fdaedb9d4b5-FFFFFF000000.png "\displaystyle S_{sk} = \frac{1}{MNS_q^3} \sum_{k=0}^{M-1} \sum_{l=0}^{N-1} [z(x_k,y_l)-\mu]^3")
(Eq. 3)
The formula is similar to the one for rms, but unlike rms (roughness), the skewness can take on positive and negative values as well as zero (even if the surface is not perfectly smooth), because each term in the double summation is raised to an odd power.
Depending on the way z values are recorded in an AFM image, the mean value, μ, itself can do the same (see Eq.2), that is, take on positive or negative values. The difference is that the appearance of the third power in the double summation in Eq. 3 means that those features whose height is farther above or farther below the mean μ; make relatively heavier contributions to the computed value of the skewness, as compared to features closer to the mean.
For a symmetric surface, that is, a surface the height of whose features are statistically evenly distributed around the mean, the skewness will render a value near or equal to zero–the height distribution is not skewed.
For our example, the skewness will measure positive for the flat surface with peaks, and negative for the flat surface with holes, and yet the rms roughness may measure the same or nearly the same for both. In this example, if the sample is a piece of metal bearing whose friction performance is to be improved, then, it may make a big difference whether the surface is flat with many holes, or flat with many peaks. To settle for the rms roughness then may be to ignore the skewness risk.
I wrote earlier that the limitations of rms roughness measurement are well-known to those who work often and in some depth with probability and statistics, but not to most AFM users. In case I planted any doubt that rms roughness is overly subscribed outside AFM image analysis too, here goes: Millions of people make pretty important decisions about money using statistics and probability often, but not all of them in much depth (that‘s why I italicized the “and“). The celebrated French mathematician Benoit Mandelbrot, the inventor of fractals, has studied the implications of ignoring the skewness risk by analyzing financial markets using models (including perhaps the most famous one, the Black-Scholes model) that assume market “roughness“ has a symmetric distribution (zero skewness). Mandelbrot’s alternative ideas about this subject were published recently.4
Some AFM images are more pleasing to the eye than others. The beauty of every AFM image though is that it is height encoded. The availability, in the form of numerical values, of all three dimensions of every feature in an AFM image, opens up a myriad options for statistical analysis of surface topography on the nanometer scale. These options exist neither in scanning electron and transmission electron microscopy (despite their superior resolution) because the images are slope-encoded, not height encoded. Nor do they exist in other techniques that do offer height encoding, but that lack the required resolution, for example, optical and stylus profiling.
AFM (software) manufacturers are aware of the implications of this advantage beyond what the average AFM user seems to know. They implement numerous formulas in their software (and provide print and electronic support documentation) to give the users options for statistically quantifying surface topography beyond rms roughness (including with fractal dimensions, for example). Surface skewness, or the coefficient of surface skewness, to be more exact, belongs to a family of mathematical functions that work with statistical distributions. There are others like it included in AFM image analysis software. It is up to the users to simply apply them and see what happens beyond rms roughness.
Fadrad Michael Serry
http://www.michaelserry.com/
serry@michaelserry.com
1STM is infrequently used in industrial applications, because it does not work with electrically non-conducting samples. This severely limits the kind of samples that are amenable to STM analysis.
2The choice of symbols in the equations here is similar to those in the software package “Scanning Probe Image Processor“ from Image Metrology A/S, Lyngby, Denmark
http://www.imagemet.com/.
3This is one definition of skewness; others exist. A list of some is available from Wolfram Research Corporation at http://mathworld.wolfram.com/skewness.html.
4Mandelbrot, Benoit B., and Hudson, Richard L., The (mis)behavior of markets: a fractal view of risk, ruin and reward, Profile Press, 2004, London, UK. ISBN 1861977654.
