While analyzing randomly rough surfaces we often need a statistical approach to determine some set of representative quantities. Within Gwyddion, there are several ways of doing this. In this section we will explain the various statistical tools and modules offered in Gwyddion, and also present the basic equations which were used to develop the algorithms they utilize.
Scanning probe microscopy data are usually represented as a two-dimensional data field of size N×M, where N and M are the number of rows and columns of the data field, respectively. The real area of the field is denoted as Lx×Ly where Lx and Ly are the dimensions along the respective axes. The sampling interval (distance between two adjacent points within the scan) is denoted Δ. We assume that the sampling interval is the same in both the x and y direction. We assume that the surface height at a given point (x, y) can be described by a random function ξ(x, y) that has given statistical properties.
Note that the AFM data are usually collected as line scans along the x axis that are concatenated together to form the two-dimensional image. Therefore, the scanning speed in the x direction is considerably higher than the scanning speed in the y direction. As a result, the statistical properties of AFM data are usually collected along the x profiles as these are less affected by low frequency noise and thermal drift of the sample.
Statistical quantities include basic properties of the height values distribution, including its variance, skewness and kurtosis. The quantities accessible within Gwyddion by means of the Statistical Quantities tool are as follows:
More precisely, RMS (σ), skewness (γ1), and kurtosis (γ2) are computed from central moments of i-th order μi according to the following formulas:
The surface area is estimated by the following method. Let zi for i = 1, 2, 3, 4 denote values in four neighbour points (pixel centres), and hx and hy pixel dimensions along corresponding axes. If an additional point is placed in the centre of the rectangle which corresponds to the common corner of the four pixels (using the mean value of the pixels), four triangles are formed and the surface area can be approximated by summing their areas. This leads to the following formulas for the area of one triangle (top) and the surface area of one pixel (bottom):
The method is now well-defined for inner pixels of the region. Each value participates on eight triangles, two with each of the four neighbour values. Half of each of these triangles lies in one pixel, the other half in the other pixel. By counting in the area that lies inside each pixel, the total area is defined also for grains and masked areas. It remains to define it for boundary pixels of the whole data field. We do this by virtually extending the data field with a copy of the border row of pixels on each side for the purpose of surface area calculation, thus making all pixels of interest inner.
Surface area calculation triangulation scheme (left). Application of the triangulation scheme to a three-pixel masked area (right), e.g. a grain. The small circles represent pixel-center vertices zi, thin dashed lines stand for pixel boundaries while thick lines symbolize the triangulation. The surface area estimate equals to the area covered by the mask (grey) in this scheme.
One-dimensional statistical functions can be accessed by using the Statistical Functions tool. Within the tool window, you can select which function to evaluate using the selection box on the left labeled Output Type. The graph preview will update automatically. You can select in which direction to evaluate (x or y), but as stated above, we recommend using the fast scanning axis direction. You can also select which interpolation method to use. When you are finished, click to close the tool window and output a new graph window containing the statistical data.
The simplest statistical functions are the height and slope distribution functions. These can be computed as non-cumulative (i.e. densities) or cumulative. These functions are computed as normalized histograms of the height or slope (obtained as dreivatives in the selected direction – horizontal or vertical) values. In other words, the quantity on the abscissa in “angle distribution” is the tangent of the angle, not the angle itself.
The normalization of the densities ρ(p) (where x is the corresponding quantity, height or slope) is such that
Evidently, the scale of the values is then independent on the number of data points and the number of histogram buckets. The cumulative distributions are integrals of the densities and they have values from interval [0, 1].
The height and slope distribution quantities belong to the first-order statistical quantities, describing only the statistical properties of the individual points. However, for the complete description of the surface properties it is necessary to study higher order functions. Usually, second-order statistical quantities observing mutual relationship of two points on the surface are employed. These functions are namely the autocorrelation function, the height-height correlation function, and the power spectral density function. A description of each of these follows:
The autocorrelation function is given by
where z1 and z2 are the values of heights at points (x1, y1), (x2, y2); furthermore, τx = x1 − x2 and τy = y1 − y2. The function w(z1, z2, τx, τy) denotes the two-dimensional probability density of the random function ξ(x, y) corresponding to points (x1, y1), (x2, y2) and the distance between these points τ.
From the discrete AFM data one can evaluate this function as
where m = τx/Δx, m = τy/Δy. The function can thus be evaluated in a discrete set of values of τx and τy separated by the sampling intervals Δx and Δy, respectively. The two-dimensional autocorrelation function can be calculated with → → .
For AFM measurements, we usually evaluate the one-dimensional autocorrelation function based only on profiles along the fast scanning axis. It can therefore be evaluated from the discrete AFM data values as
The one-dimensional autocorrelation function is often assumed to have the form of a Gaussian, i.e. it can be given by the following relation
where σ denotes the root mean square deviation of the heights and T denotes the autocorrelation length.
For the exponential autocorrelation function we have the following relation
Autocorrelation function obtained for simulated Gaussian randomly rough surface (i.e. with a Gaussian autocorrelation function) with σ ≈ 20 nm and T ≈ 300 nm.
The difference between the height-height correlation function and the autocorrelation function is very small. As with the autocorrelation function, we sum the multiplication of two different values. For the autocorrelation function, these values represented the different distances between points. For the height-height correlation function, we instead use the power of difference between the points.
For AFM measurements, we usually evaluate the one-dimensional height-height correlation function based only on profiles along the fast scanning axis. It can therefore be evaluated from the discrete AFM data values as
where m = τ/Δ. The function thus can be evaluated in a discrete set of values of τ separated by the sampling interval Δ.
The one-dimensional height-height correlation function is often assumed to be Gaussian, i.e. given by the following relation
where σ denotes the root mean square deviation of the heights and T
For the exponential height-height correlation function we have the following relation
In the following figure the height-height correlation function obtained for a simulated Gaussian surface is plotted. It is fitted using the formula shown above. The resulting values of σ and T obtained by fitting the HHCF are practically the same as for the ACF.
The two-dimensional power spectral density function can be written in terms of the Fourier transform of the autocorrelation function
Similarly to the autocorrelation function, we also usually evaluate the one-dimensional power spectral density function which is given by the equation
This function can be evaluated by means of the Fast Fourier Transform as follows:
where Pj(Kx) is the Fourier coefficient of the j-th row, i.e.
If we choose the Gaussian ACF, the corresponding Gaussian relation for the PSDF is
For the surface with exponential ACF we have
In the following figure the resulting PSDF and its fit for the same surface as used in the ACF and HHCF fitting are plotted. We can see that the function can be again fitted by Gaussian PSDF. The resulting values of σ and T were practically same as those from the HHCF and ACF fit.
We can also introduce radial PSDF Wr(K), which of course contains the same information as the one-dimensional PSDF for isotropic rough surfaces:
For a surface with Gaussian ACF this function is expressed as
while for exponential ACF surface as
The Minkowski functionals are used to describe global geometric characteristics of structures. Two-dimensional discrete variants of volume V, surface S, and connectivity (Euler-Poincaré Characteristic) χ are calculated according to following formulas:
Here N denotes the total number of pixels, Nwhite denotes the number of “white” pixels, that is pixels above the threshold. Pixels below the threshold are referred to as “black”. Symbol Nbound denotes the number of white-black pixel boundaries. Finally, Cwhite and Cblack denote the number of continuous sets of white and black pixels respectively.
For an image with continuous set of values the functionals are parametrized by the height threshold value ϑ that divides white pixels from black, that is they can be viewed as functions of this parameter. And these functions V(ϑ), S(ϑ) and χ(ϑ) are plotted.
This tool calculates numeric characteristics of each row or column and plots them as a function of its position. This makes it kind of complementary to Statistical Functions tool. Available quantities include:
In addition to the graph displaying the values for individual rows/columns, the mean value and standard deviation of the selected quantity is calculated from the set of individual row/column values.
Several functions in
→ operate on two-dimensional slope (derviative) statistics.Use local plane fitting is enabled, by fitting a local plane through the neighbourhood of each point and using its gradient. has also another mode operation called Per-angle graph. in which it plots the distribution of r2 over φ where we introduced polar coordinates (r, φ) in the plane of derivatives. The relation between the derivative Cartesian coordinates of the two-dimensional slope distribution and the facet inclination angles are given by the following formula:
calculates a plain two-dimensional distribution of derivatives, that is the horizontal and vertical coordinate on the resulting data field is the horizontal and vertical derivative, respectively. The slopes can be calculated as central derivatives (one-side on the borders of the image) or, if
function is a visualization tool that does not calculate a distribution in the strict sense. For each derivative v the circle of points satisfying
is drawn. The number of points on the circle is given by Number of steps.
→ →
Facet analysis enables to interactively study orientations of facets occuring in the data and mark facets of specific orientations on the image. The left view displays data with preview of marked facets. The right smaller view, called facet view below, displays two-dimensional slope distribution.
The centre of facet view always correspond to zero inclination (horizontal facets), slope in x-direction increases towards left and right border and slope in y-direction increases towards top and bottom borders. The exact coordinate system is a bit complex and it adapts to the range of slopes in the particular data displayed.
Facet plane size controls the size (radius) of plane locally fitted in each point to determine the local inclination. The special value 0 stands for no plane fitting, the local inclination is determined from symmetric x and y derivatives in each point. The choice of neighbourhood size is crucial for meaningful results: it must be smaller than the features one is interested in to avoid their smoothing, on the other hand it has to be large enough to suppress noise present in the image.
Illustration of the influence of fitted plane size on the distribution of a scan of a delaminated DLC surface with considerable fine noise. One can see the distribution is completely obscured by the noise at small plane sizes. The neighbourhood sizes are: (a) 0, (b) 2, (c) 4, (d) 7. The angle and false color mappings are full-scale for each particular image, i.e. they vary among them.
Both facet view and data view allow to select a point with mouse and read corresponding facet normal inclination value ϑ and direction φ under Normal. When you select a point on data view, the facet view selection is updated to show inclination in this point.
Button
sets facet view selection to slope distribution maximum (the initial selection position).Button Tolerance from the selected slope. The facet view then displays the set of slopes corresponding to marked points (note the set of selected slopes may not look circular on facet view, but this is only due to selected projection). Average inclination of all points in selected range of slopes is displayed under Mean Normal.
updates the mask of areas with slope similar to the selected slope. More precisely, of areas with slope within