Many novel industrial components, e. g. optoelectronic devices, microchips, diffraction gratings or photonic crystals, are typically formed by highly integrated microstructures arranged on very large areas. The inspection and control of such components are problematic as we need to detect small imperfections on large areas or to evaluate small differences between many semiconductor mask critical dimensions. Scanning probe microscopy techniques (e. g. atomic force microscopy) can be used for this task if sufficiently large areas can be scanned. The bottleneck of using scanning probe microscopes was always the scanning range; however today, specialized equipment can be found in the literature and even on the market 12345 featuring very small uncertainties (less than 1 nm) over very large volumes. However, even if the range of microscopes has changed, the maximum speed of the tip is still nearly the same. Large-scale measurements are therefore very slow.

Even if we measure on very large areas, like those shown on Figure 1 for a microchip surface, only a very small part of the measured data is finally used for the evaluation. From Figure 1, we would probably measure several critical dimensions on observed interconnections only. The rest of the data would be left aside even if it took several hours to acquire them and even if they occupy hundreds of megabytes of disk space. This is a large waste of effort. One of the possibilities how to overcome this is to measure data in a non-equidistant approach (not creating a regular matrix of points during measurements). If we are able to measure the overall sample geometry with low resolution and local features that will be used for surface properties evaluation with high resolution, we can both speed up the measurement and reduce the need for data storage.

In this article, we discuss one possible realization of a non-equidistant measurement method for a long-range scanning probe microscope and discuss its applications for different samples typical for microelectronics and solar cell industry.

The suggested measurement algorithm has three main benefits compared to the matrix-based scanning methods:

1. It can save the amount of necessary measured information while measuring with very high resolution, thus shortening the measurement time and minimizing the probe wear while preserving the necessary resolution on critical surface structures.

2. It allows the user to perform further local refinements easily while the sample is still in the microscope, not necessarily based on the algorithm criteria but also on the user's requirements.

3. It can automatically measure the data allowing the user to perform zooms offline, e. g. in the data processing phase after the measurement. This can be useful namely for automated image processing or inspection purposes.

In order to document the effect of the reduction of the necessary scanned path, we compare the path length saved by the presented algorithm to the standard matrix-based approach for the surface shown in Figure 3. The resulting total tip path reductions for two different final accuracies are summarized in Table 1. It can be noted that the more refinement levels are performed the more data are saved. This is however not valid generally but only on regular structures like the one presented here. As stated above, for completely random data, we cannot save anything using the refinement algorithm.

In order to display the real performance of the algorithm if implemented in the microscope, we compare the results of the matrix algorithm and the presented algorithm on real data of a calibration grating. The same final resolution was requested. In Figure 5, we can see the result of the subtraction of images obtained using the two algorithms. We can see that the local differences are caused only by relatively high noise of the presently used long-range system z scanner and a slight mismatch between the two images (due to a slight shift caused by retraction and approach between the two measurements, the images had to be cropped and aligned manually).

It should be noted that the obtained data have different statistical properties than a typical matrix-based SPM image. There is no difference between the fast and slow axes as both x and y directions are used for the measurement. It has no sense therefore to distinguish between directions while evaluating direct or statistical quantities. All parasitic effects, such as drift or noise, influence data almost isotropically. This could be understood as a drawback in some sense; on the other side, it can simplify the treatment of uncertainties while obtaining profiles of different orientation with respect to the main axes or 2D statistical properties evaluation.

An important issue is the influence of drift on the measurement process. Drift, namely of thermal origin, can be observed in many SPM systems 89. In a typical matrix-based SPM image, drift is mainly seen in the slow scanning axis, leading to image distortion as seen in Figure 6B. Note that the coordinate system origin in this simulation is in the top left corner of the image. Based on the analysis of the known surface structures or based on repetitive scans, we can determine the drift rate, which is usually a time-dependent decaying function, with a maximum right after the instrument start-up or a sample exchange 10. If we use the scanning approach presented in this work, the data in the image are not measured successively, so the straightforward drift determination from AFM image is not possible as the drift influences interleaved values significantly (see Figure 6C). However, drift can still be evaluated. As in each refinement level, we measure on the same area as was already measured in the previous iteration, and we even obtain data at exactly the same points (at row/column crossings); we can determine the drift rate already during refinement process for areas that are being refined. We can do this using the following approach:

• Create an interpolated image from one refinement level (using the standard procedure from the previous section);

• Create an interpolated image from the next level skipping the data measured in the previous level;

• Use cross-correlation to determine the shift between the two data sets (in all three axes), which are the x and y drift values; and

• Shift one data set according to the cross-correlation result and subtract the two data sets, which gives us the z drift value.

As an illustration of the process, we have simulated data measurement with a constant drift vector of (3, 3, 0.3) nanometres per second in Figure 6. We have used the part of the microchip surface seen in Figure 6A, using already measured data (without observable drift) and adding the drift during the simulated measurement (all movements were performed with the same velocity). Drift was evaluated from levels 1 and 2 of the refinement process where, for this sample, we still measure on nearly the whole area (the local refinement criterion still holds everywhere). Using the above-mentioned process, we have evaluated the drift vector as (3.3 ± 0.5, 2.7 ± 0.3, 0.29 ± 0.15) nanometres per second which is a good estimate of the drift rate. In Figure 6D, a simulated measurement with correction based on this estimated drift rate is shown (for drift estimation, the first two iterations were used), obviously leading to significant correction of the image. Of course, if the drift rate is not constant, the above-mentioned approach would not be an optimal one; however, the user could repeat it after several refinement iterations to correct the drift rate estimate. Generally, it can be seen that the drift leads to a much more evident image distortion when our adaptive refinement algorithm is used. On the other side, even the data obtained in a regular matrix approach are influenced by the drift, and if we do not know the measured structure properties, we need to use a similar correlation technique to determine the drift. Data are therefore 'wrong' in both cases, but in the regular matrix case, they look better and allow the user to ignore the systematic errors caused by the drift. The large influence of the drift on the presented algorithm could therefore be seen even as a benefit for a metrologist - if the systematic error is quantitatively the same in both cases anyway, the adaptive approach prevents the fact that it would rest hidden in the data.

We have implemented an adaptive refinement measuring algorithm in a long-range scanning probe microscope. For surfaces with large regular areas, the use of this adaptive scanning instead of a regular matrix scanning can save the time necessary for the measurement and enable high-resolution zooming in the measured data. This can be interesting namely when measuring large areas of different manufactured nano-and micro-structures, like electronic parts, photonic crystals, diffraction gratings, etc. Using the presented approach, we iteratively measure with higher and higher resolution on critical surface areas, determined by a simple and universal criterion. In the data processing phase, the set of xyz data obtained from the measurement is triangulated, and necessary details are regularized in order to reach requested resolution. This enables the user to perform zooming in the data processing phase, with no need of performing further measurements.

The authors declare that they have no competing interests.

PK carried out the development of the adaptive measurement algorithm. MV performed atomic force microscopy measurements on long-range SPM, and PB wrote software routines for general xyz data triangulation. All authors read and approved the final manuscript.