Have you ever wondered what is going on electrically at the nanoscale?  Do you have a need to understand the impedance across a nanotube?  The good news - now you can find out.   Agilent engineers and scientists have now married together a network analyzer and an atomic force microscope, making it possible to make impedance and capacitance measurements at the tip of an AFM.  In fact, you can scan across a surface making these measurements. 

I remember visiting a number of research facilities shortly after Agilent announced its line of AFM’s and the big question from the researchers - “When will Agilent be able to make electrical measurements at the nanoscale?”  They were asking for worldclass measurements on objects that weren’t even visible through an optical microscope.  In effect - that day has arrived.   A whole new world of research is now available to us. 

To read more, visit the Agilent nanotechnology website and click on the image in the center of the page. 

Before it is modified and morphed into an SPM image, the underlying data, in a more primitive form, is one continuous, long sequence collected in the time-domain—a single one dimensional waveform that contains the variation of one dependent parameter, e.g. the AFM cantilever deflection, with the one independent parameter, time.

To create the image, this waveform’s independent parameter—its domain—is transformed from time into distance. Then the waveform is folded back and forth several times into a series of equal-length segments that are then successively aligned precisely adjacent to one another to form the image. As a result, the image is a new entity that has transformed the original one dimensional time-domain waveform into a two dimensional, spatial-domain waveform.

Every waveform in the time domain or spatial domain can be constructed from (or decomposed into) a superposition—an algebraic sum—of a series of simpler component waveforms that have different characteristics, such as frequency, amplitude, and relative phase. The component waveforms belong to a larger set, usually called the basis set, within which some members may be entirely absent from or are insignificant contributors to any given waveform. Sine waves are most commonly used as a basis set, but this is not the only choice available.¹

The Frequency Domain
Frequency domain analysis involves transforming the time or spatial domain data—the waveform—into the frequency domain, then measuring and comparing the amplitudes and phases of the component waves at the frequencies that make up the waveform. When sine waves constitute the basis set, this is called Fourier analysis, and involves some implementation of the Fourier transform.

In scanning probe microscopy (SPM) imaging, frequency domain analysis can be performed on time-domain data as raster scanning proceeds, or on recorded data (images or spectroscopy plots). Three classes of instruments are often used to transform time-domain data into the frequency domain: spectrum analyzers, network analyzers, and dynamic signal analyzers. Each instrument class has its own advantages over the others for certain types of measurements. Network analyzers, for example, allow phase measurements, while spectrum analyzers generally do not. On the other hand, spectrum analyzers can measure smaller signals because they usually have lower noise floors than network analyzers. Dynamic signal analyzers allow modal analysis, while the other two classes of instruments do not. Nonetheless, all three classes are frequently interfaced with an SPM to analyze its time-domain data.

In an SPM image, time translates into space via the scan speed: time series of data points are recorded versus spatial sampling points. Usually, the fast scan direction is designated “X”, the slow “Y”. Frequency domain analysis of SPM images is carried out in the SPM software, and concerns spatial wavelengths, rather than frequencies. Hereafter, “frequency” will continued to be used per standard signal processing terminology, but it must be understood that in image processing, including SPM images, (spatial) wavelength, not frequency, is the natural parameter with which to work.

Why the Frequency Domain?
Why would we want to study an SPM image in the frequency (wavelength) domain?

Perhaps the simplest, most intuitive hint to the answer is this: Periodicity is inherent in nature, and influences the way it works. The properties of a sample surface often depend on the periodicities (wavelengths) characteristic to that surface.

In an AFM image, we may consider a line in any direction, as shown in Figure 1, and seek the variations of the topography along this line. We may be interested not only in the rms roughness, but also in the spatial frequency characteristics of the roughness.

Figure 1. Spatial-domain visualization and analysis. AFM image of a reference sample, and the variation of the topography along a line of arbitrarily chosen direction.

We may also want to consider all directions simultaneously, and inquire about the variations of topography across either the entire image or a subsection of the image. But again, we may want to characterize the area of interest beyond measuring parameters such as rms roughness, skewness, and maximum and minimum height, all of which are wavelength-independent and direction-neutral. We may want to find, for example, any preferred direction(s), or quantify any spatial in-plane symmetry that the image holds.

Frequency domain techniques provide numerous ways to extract information from an SPM image. While periodicity at some frequencies may be readily visible in the spatial domain (in the image,) at other frequencies it may be faint and difficult to decipher. The image in Figure 1, in which the periodicity is 10mm along two orthogonal directions, is a case in point. When we transform the spatial domain data along the virtual cross-section line (indicated by the arrow) into the frequency domain via Fast Fourier Transform (FFT), the sine waves that contribute most to the spatial domain data stand out at frequencies that correspond to wavelengths 10μm, 3.33μm, 2μm (Figure 2).

Figure 2. Frequency domain analysis. Fourier spectrum of the cross-section data in Figure 1.

The latter two are called the third and the fifth harmonics of the fundamental wavelength at 10μm. Higher order odd-numbered harmonics (beyond the fifth) also contribute relatively strongly compared to their nearest even-numbered harmonic neighbors, but not as strongly as the third and fifth harmonics. This kind of information is valuable in understanding the structure of the surface, in relating that structure to the properties and functionality of the surface, and also in filtering, as described next.

Filtering
Frequency domain techniques enable the selective admittance of some frequencies to the waveform and the exclusion of others. This is called frequency domain filtering, the results of which can be quite impressive. Figure 3 shows what happens to the AFM image and the cross-section data from Figure 1 when we filter it. We use FFT to transform the image into the frequency domain, filter out the high-frequency components (the higher harmonics) while keeping the low-frequency ones (in this case, only the fundamental frequency), and then use inverse Fast Fourier Transform to reconstruct the image from the low-pass filtered spectrum.

Figure 3. Low-pass filtered image and cross- section, same location as in Figure 1.

On the other hand, if we filter out the low frequencies while keeping the higher ones in the spectrum, and then inverse transform to reconstruct the image, we arrive at the results depicted in Figure 4. The edges of the square pits in the original image are visible, but the height variation across the image has largely disappeared (notice the height scale in the cross-section data, and compare with that in Figures 1 and 3).

Figure 4. High-pass filtered image and cross- section, same location as in Figure 1.

SPMs are extremely susceptible to noise from all types of sources. One of the most frequent uses of analyzing SPM images in the frequency domain is identifying spatial wavelengths (and therefore the temporal frequencies) of the noise source(s), at a single frequency or a collection of frequencies. Once the frequencies are known, the noise source(s) may be easier to identify and suppress, leading to enhanced quality of the subsequent SPM images and accuracy of the measurements made on the same.

More Power in Frequency Domain Techniques
Some of the most powerful methods that arise from understanding and working with the frequency domain involve mapping a relatively difficult operation to be performed on spatial (time) domain data onto a relatively easy operation to be performed on the corresponding frequency domain data. One example is how the relatively complex convolution operation in the spatial (time) domain is substituted for by the relatively simple multiplication operation in the frequency domain, a subject that we shall visit in the near future in this blog.

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

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¹The choice of the working basis set is made in part by considering the symmetries and the boundary conditions that apply to the waveform in the time domain or the spatial domain. See for example Chapter 1, in “Spatial Vision” by R L. De Valois and K. K. De Valois, Oxford Psychology Series, 14, Oxford Science Publications, Oxford University Press, US 1988.