In an earlier posting on this blog about surface roughness measurement with the AFM, I wrote that the rms roughness is too often the only numerical parameter that is computed and reported (February 4, 2008). I wrote about asymmetry in the height distribution, and how the rms roughness does not address it, and used skewness as an example of a parameter that does.

Here, I’d like to continue that thread, and explore another aspect of surface roughness the characterization of which requires computation beyond the rms roughness: the contribution of different spatial wavelengths to roughness.

The texture of a surface is characterized by the height of its peaks and the depths of its troughs. It is also characterized by the distances that separate the peaks and the troughs. This article is concerned primarily with those distances. Are there any dominant values of the spacing between peaks (or, alternatively, between troughs)? If so, what are they, and what is their relative order of dominance? Along which directions in the image are these dominant spacing to be found? So, we speak of dominant wavelengths (or dominant spatial frequencies) contributing to surface roughness, and we seek not only the value of these wavelengths, but also their directions in the surface texture.

Qualitatively, one surface may appear more wavy than another. One surface’s waviness may involve wavelengths shorter than those of another, making it look more rippled, even jagged. Qualitatively, roughness is nothing but waviness at very small wavelengths. One surface may look rougher than another if its dominant in-plane wavelengths are shorter.

All these and many more qualitative characteristics that we can visualize in an SPM image, we can also describe and compare quantitatively by using several features in SPM software. One of these features is called The Power Spectral Density (PSD) Analysis.

PSD analysis belongs to the frequency domain, and as such, it requires transforming the time domain or spatial domain data into the frequency domain, which is almost universally done using Fourier methods.¹

Any waveform in the time domain can be constructed from a linear superposition (an algebraic sum) of a collection of Sine and Cosine waves of different temporal frequencies or, equivalently, of different wavelengths. In an SPM image, the time translates into a distance (distance = time x scanning speed); it is therefore more meaningful to work with spatial wavelengths, rather than (temporal) frequencies.

The AFM image in Figure 1 shows two adjacent regions of a sample’s surface. We can use the AFM analysis software to zoom into each region and “planarize” it (remove slope, bow, …) in order to visualize the roughness better. This can be seen in Figure 2. There emerges a subtle but noticeable difference in the texture and topography between the two regions. This is a result of the way that these regions of the sample surface were processed, each differently from the other. We are interested in a way to possibly quantify the subtle differences between the two regions that we can visualize qualitatively. The region on the right appears to have a weak orderliness to it; there appears to be some faint correlation of the features across this region which is missing in the left region. To put it in a different way, the texture in the left region appears to have a more random character than the one on the right. In this particular software, SPIP Version 4.6.4.0, you can actually see this difference a little easier if you grab one of the small handle bars (arrows) that are attached to the color bars on the right of each image, and move it up and down the along the color bar repeatedly, but we are interested in quantifying the difference that we can visualize..²

Average X and average Y Power Spectral Density (PSD) plots of the two regions are shown in Figure 3 and 4. Although there are some differences in these plots (left region versus right region), there is nothing that immediately and clearly points to the presence of the order, however indistinct and dim, that we can see in the right region and that is absent from the left.

Figures 5 and 6 have what we are after. These figures show maps of the two dimensional PSD for each side of the image, and also a cross section, a profile of each map along a particular direction, the direction that is depicted by the rectangular zoom box in each two dimensional map. For the right side of the image, Figure 5 shows a strong peak at a wavelength of approximately 0.66μm. There are two things that are important about this data. First, this peak is completely absent in the profile along the same direction in Figure 6, which belongs to the left side of the image in Figure 1. It is absent in fact across the entire 180 degrees that you can sweep the profile zoom box around the center of the two dimensional map in Figure 6. This difference between the data in Figures 5 and 6 quantifies the visible, however subtle, difference in the image in Figure 1 between its right and left sides. Along the direction depicted, there is a relatively dominant periodicity, at wavelength of about 0.66μm, in the right side of the image, but not the left. It is visible upon close inspection that there appears to be some weak periodicity in the right side of the image, we see that and we were hoping to quantify it; we have. But the direction along which this periodicity resides is indeed unclear from close inspection. It was in fact surprising when I first saw it! This is the power of PSD analysis, in two dimensions. But there is more.

The second important thing about this data, the one that is not easy to show absent a sequence of profiles for which this blog article does not have the space, is that the peak in the profile in Figure 5 is present not only along the indicated direction, but also across a range of angles centered around that direction, and the wavelength itself varies by some 10% as you sweep across this range of angles. The peak is at its strongest at the angle shown in the profile. This spread of the angles and of the wavelengths is the way that the PSD analysis quantifies the uncertain and indistinct nature of the apparent order that we can visualize on the right side of the image: there is some kind of order, some sort of correlation, it is faint, it is subtle, but it is there, and the PSD analysis—in two dimensions—has quantifies it.

In an upcoming blog article, we will explore how this type of correlation can also be quantified using a different method, a method that belongs to the spatial domain and that is related to PSD in the frequency domain: the Autocorrelation method.

Figure 1. Two regions of a sample surface show subtle differences in their topography.

Figure 2. Each region in Figure 1 separately planarized by removing slope, curvature, and third order departure from planarity.

Figure 3. Top, average X and average Y Power Spectral Density (PSD) plots, and bottom, Isotropic Area PSD plot of the right region in Figure 1.

Figure 4. Top, average X and average Y Power Spectral Density (PSD) plots, and bottom, Isotropic Area PSD plot of the left region in Figure 1.

Figure 5. Two dimensional PSD map of the right region, and the average profile of it along a direction defined by the narrow rectangular white zoom box in the map. There is a dominant peak at wavelength 0.66μm.

Figure 6. . Two dimensional PSD map of the left region, and the average profile of it along the same direction as depicted in Figure 5. There is no peak (other than the one at position zero which carries no significant information.)

Michael Serry
http://www.michaelserry.com/
serry@michaelserry.com

¹For more on Fourier methods applied to SPM images, please see an article that appeared on this blog on July 28, 2008.

²SPIP is Scanning Probe Image Processor, a product of Image Metrology Corporation of Denmark http://www.imagemet.com