Power Spectral Density (PSD)

The Power Spectral Density (PSD) function is useful in analyzing surface roughness. This function provides a representation of the amplitude of a surface’s roughness as a function of the spatial frequency of the roughness. Spatial frequency is the inverse of the wavelength of the roughness features.

The PSD function reveals periodic surface features that might otherwise appear random and provides a graphic representation of how such features are distributed. On turned surfaces, this is helpful in determining speed and feed data, sources of noise, etc. On ground surfaces, this may reveal some inherent characteristic of the material itself such as grain or fibrousness. At higher magnifications, PSD is also useful for determining atomic periodicity or lattice.

PSD and Surface Features

The synthetic surface represented in Figure 1 consists of essentially two dominant wave forms: a long period waveform along the X-axis, and a shorter period waveform along the Y-axis.

Figure 1: Wavelength Depiction

A 2-dimensional power spectral density plot of this surface would consist of two dominant spikes (one for each dominant wavelength), plus some number of extra wavelengths inherent within the image. (These extra wavelengths may appear due to fine surface features and/or side bands of the dominant wave forms.)

Because of the sinusoidal nature of the composite wave form, a relatively small set of spectral frequencies suffices to describe the entire surface. By contrast, an image comprised of angular (saw-toothed or square) waveform contains more spatial frequency components.

PSD and Flatness 

PSD is used increasingly as a metrology tool for evaluating extremely flat surfaces, such as polished or epitaxial silicon. Generally, the desired surface is expected to adhere to certain PSD thresholds, signifying it meets a specified flatness criterion.

The main advantage gained over traditional RMS specifications is that PSD flatness is qualified through the full spectral range of interest. For example, one may specify spectral thresholds at frequencies measured on the atomic scale, thus ensuring surfaces consist largely of uniform lattices. Setting the precise thresholds for various materials remains a matter of debate.

Figure 2: Epitaxial Gallium Arsenide

This surface above is comprised of “terraces” formed from the material’s natural lattice structure; each terrace is one atomic monolayer thick and is spaced at fairly regular intervals. This degree of flatness is handily evaluated with PSD, as the terraces produces a spectral spike corresponding to their spacing wavelength.

Figure 3: PSD Analysis of Epitaxial Gallium Arsenide

This tapered PSD plot is characteristic of flat, isotropic surfaces. Longer wavelengths are present up to the scan width, and are accompanied by uniformly decreasing powers of shorter wavelengths down to 2 pixels. On the plot shown above a spike stands out, corresponding to the wavelength spacing of the terraced features. Depending upon the qualitative standards of the person evaluating such a plot, this spike may exceed a threshold standard of flatness.

PSD Algorithm

Over a given range of spatial frequencies the total power of the surface equals the RMS roughness of the sample squared.

The frequency distribution for a digitized profile of length L, consisting of N points sampled at intervals of d_o is approximated by:

Where frequencies, f, range from 1/L to (N/2)/L. Practically speaking, the algorithm used to obtain the PSD depends upon squaring the FFT of the image to derive the power. Once the power, P, is obtained, it may be used to derive various PSD-like values as follows:

The terms used in the denominators above are derived by progressively sampling data from the image’s two-dimensional FFT center.

Figure 4: Progressive Data Sampling

Each sampling swings a “data bucket” of given frequency f. Since samples are taken from the image center, the sampling frequency, f, is limited to (N/2)/L, where N is the scan size in pixels. This forms the upper bandwidth limit (i.e., the highest frequency or Nyquist frequency) of the PSD plot. The lower bandwidth limit is defined at 1/L.

Power Spectral Density Procedure

Figure 5: Select Power Spectral Density from the workspace

	Or Right-click on a thumbnail in the Multiple Channel window and select Power Spectral Density. Or Select Analysis > Power Spectral Density from the menu bar. Or
	Click the Power Spectral Density icon in the NanoScope toolbar.

Figure 6: The PSD Analysis Window

PSD Interface

Compute PSD Buttons

Once the PSD analysis window is opened, select the type of spectral density analysis you wish to perform by clicking the appropriate button in the Compute PSD window.

• 2D Isotropic—Executes a two-dimensional PSD analysis.
• Horizontal Axis—Executes a one-dimensional PSD analysis along the X-axis.
• Vertical Axis—Executes a one-dimensional PSD analysis along the Y-axis.

The calculation begins as soon as one of these buttons is selected. The PSD and a table of values from the graph are shown in the Results window. Completion of the analysis will also result in a topographical histogram in the spectrum plot window, shown in Figure 7.

Results Display

The Results window displays the Name and Value of the procedures performed during a PSD analysis. The teal shaded area in the display window corresponds to the area designated by the teal cursor on the Power Spectral Density histogram, and the magenta shaded area corresponds to the magenta cursor. You can generate a report by right-clicking in the Results window, selecting Copy Text and pasting the clipboard into another application (e.g. Notepad, Word...).

Figure 7: PSD Results

Table 1: Results Parameters

Select Displayed Parameters

The operator can select which Results will or will not be displayed in the Results window by Right-clicking in the Results window, selecting Show All from the popup menu, and checking or un-checking the appropriate boxes. See Figure 7.

Exporting Text

Selecting Copy Text from the popup menu will copy the text from the Results window, in tab-delimited text format, to the Windows clipboard. You may then paste it into a text or word processing program.

Changing Parameters of the Spectrum Plot

The Spectrum Plot window displays results of the PSD analysis. The window has two cursors whose color corresponds to the shaded areas in the Results window. Move either of the cursors within the Spectrum Plot window by placing the cross hair cursor directly over the cursor, clicking and holding the left mouse button, and dragging the mouse to the left or right. Both cursors can be moved simultaneously by left-clicking the mouse with the cross hair cursor anywhere between the two cursors and dragging to the left or right.

To change the parameters of the Spectrum Plot, right-click in the Spectrum Plot window at the bottom of the display and choose from the popup menu. See Figure 8.

Figure 8: Spectrum Plot Window, Right-click to Change Parameters

Table 2: Spectrum Plot Popup Menu items

Figure 9: Scaling Menu

Figure 10: Line styles

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