Error in dynamic spring constant calibration of atomic force microscope probes due to nonuniform cantilevers

Abstract

Many common atomic force microscope (AFM) spring constant calibration methods regard the AFM probe as a uniform cantilever, neglecting the tip mass and any nonuniformity in the thickness of the probe along its length. This work quantifies the error in the spring constant estimated by the Sader and thermal calibration methods due to nonuniformity in the thickness of the cantilever and the influence of the mass loading effect of the probe tip. Formulae are presented that can be used to compute the uncertainty in cantilever calibration for an arbitrary thickness nonuniformity, or to correct the calibration methods if the thickness nonuniformity is known. The results show that both methods are quite sensitive to nonuniformity. When the first dynamic mode is used in the calibration, the error in the spring constant estimated by either method is between - 4% and 9% for a cantilever whose thickness increases or decreases linearly by 30% along its length. The errors are several times larger if the second or higher dynamic modes are used. To illustrate the proposed methods, a commercial AFM probe that has significant nonuniformity is considered and the error in calibrating this probe is quantified and discussed. For this particular probe, variations in the thickness of the probe over the last 15% of its length are found to significantly reduce the accuracy of the calibration when the thermal method is used, since that method is sensitive to changes in the shape of the eigenmode of the probe near its free end.

Spring constant calibration of atomic force microscope cantilevers of arbitrary shape

“…Since the fundamental mode shape is a simple monotonically increasing function of distance from the clamp, spatial variations in thickness and/or material have a weak effect on this mode shape and hence the general method. This explains recent theoretical findings demonstrating the robustness of the original method, with respect to thickness variations along the cantilever axis, in all but the extreme cases of very strong variations in thickness 32 -in these extreme cases, the mode shape was significantly altered. The same property holds true for the general method.…”

“…25,32 Nonetheless, if the imaging tip is comparable in size to the dominant hydrodynamic length scale of the flow, its presence will enhance the true energy dissipation and thus increase the hydrodynamic function. 24,25 While this can lead to an underestimate of the spring constant obtained using the original method, 25 the general method intrinsically accounts for any such extra energy dissipation.…”

Quantitative Normal Force Measurements by Means of Atomic Force Microscopy Towards the Accurate and Easy Spring Constant Determination

“…Therefore the calibration of one cantilever would provide sufficient data to calculate the spring constant of the other cantilevers with accuracy of the order of 10% or better. However, the work presented by Frentrup and Allen showed that the thickness of the probe may vary along the probe, introducing a significant error while the determination bases on the assumption of perfect shapes of the probe [95]. Moreover, as presented by Webber et al the probes fashioned from that same wafer may differ even by factor of two [148], as the 101 V-shaped probes made of silicon nitride was measured using thermal spectrum method proposed by Hutter and Bechhoefer.…”

“…Also, the models assume shapes uniformity, as in case of the thickness, as critical parameter -it was shown that it may introduce a significant error [94,95]. The model taking into account this particular issue was presented by Fretrup and Allen, where effective thickness t es provides that same outcome that nonuniform probe where thickness profile is h(x) [95]:…”

“…The non-uniformity of the cantilevers was analyzed by Frentrup and Allen [95]. As theoretical calibration approaches assume the certain level of the regularity of the shapes of the cantilever, it can lead to a significant spring constant estimation error.…”