The functionality of many micro-optical structures such as Bragg gratings, phase masks, and photonic crystals depends critically on the dimensions. In particular, the exact height is often a key parameter. For vertical dimensions below 10 µm, atomic force microscopes (AFM) are often the only nondestructive possibility. Unfortunately, no written standards are available. AFM measurement accuracy is frequently compromised by severe hysteresis caused by the scanning piezos, and by measurement nonlinearities. Proper calibration can address some of these concerns.
Figure 1. A data fit (black) to an AFM profile (blue) of a step height artifact, yields step height (hm) and width (w).
The first step in calibrating an AFM is to obtain a commercial reference step height with a certificate that states the height hr and the associated uncertainty.1 The step heights h, and step widths w, should be as close as possible to the dimensions of the samples to be measured, as this will decrease the errors (see figure 1). A calibration certificate can only be issued for samples of high quality, which is fulfilled for most commercially available samples. Only a measurement or test report can be issued for many dummy products, and this may not be acceptable for a quality control system like ISO 9000.
The calibration factor Cz, is expressed by Cz = hr /hm, where hm is the measured step height of the reference. The calibration and measurement procedure should be as near identical as possible; in particular, users should always level the sample and keep the offset in the x, y, and z directions close to zero.
Figure 2. AFM error sources include hysteresis between the trace (red) and retrace (blue) directions, and image bow (green), introduced by coupling between the horizontal and vertical motion of the AFM probe.
In systems without distance sensors, the hysteresis of the scanning piezo probably introduces the most significant error in the measured height. We estimate hysteresis by measuring the difference in observed height for the trace and retrace of a typical profile (see figure 2). The deviation due to hysteresis ranges from below 1% up to several percent of the height and it cannot be corrected easily. The effect can be decreased by estimating the step height as the average of a trace and retrace profile and as the average of both a left and a right step edge. Scanning piezos suffer from aging and must be recalibrated as often as every three months.
Some commercial microscopes are enhanced with distance sensors such as strain gauge or capacitive sensors, which remove most of the hysteresis. For these systems, the primary error source consists of nonlinearities in the sensor response; that is, variation of Cz as function of the height z. We can estimate this deviation by measuring the same step height with different offsets, or different average positions, of the scanner relative to the surface. Variations due to sensor nonlinearities typically range from 0.1% to 1% over the total height range of the scanners, and can be corrected offline. Estimating the step height with a zero offset minimizes the variation. Reasonable calibration intervals for systems with sensors are three to 12 months.
A particular error source is coupling between the horizontal and vertical motion of the AFM probe, which causes a flat surface to appear bowed (see figure 2). It can be estimated as the primary bow in the profile of a flat sample. Over a length of 100 µm, the peak-to-peak values for image bow range from a few nanometers for high-quality flexure stages to more than 100 nm for the commonly used tube scanners. The bow caused by the system's geometry can be corrected for, and for a narrow step width of, for example, 1 mm it is usually small ( 1 nm or less).
Using commercial software, profiles can be averaged, corrected, and the step height can be fitted as the vertical distance between the solid lines indicated in figure 1. We have calibrated a commercial microscope with distance sensors and achieved a combined standard uncertainty (95% significance) of Uc ~ 1 nm + 0.01 x hm.2 Microscopes without distance sensors have uncertainties from two to five times higher.
These facts make calibrated AFMs very good instruments for accurate and nondestructive vertical measurements. oe
1. Manufactured by Nanosensors (www.nanosensors.com).
2. J Garnaes, et al., Precision Engineering 27, p. 91 (2003).
Joergen Garnaes and Anders Kühle are staff scientists at Danish Fundamental Metrology, Lyngby, Denmark.