INSTITUTE OF PHYSICS PUBLISHING NANOTECHNOLOGY Nanotechnology 14 (2003) 1–6 PII: S0957-4484(03)52189-4 Comparison of calibration methods for atomic-force microscopy cantilevers N A Burnham 1 , X Chen 2 , C S Hodges 3 , G A Matei 1 , E J Thoreson 1 , C J Roberts 2 , M C Davies 2 and S J B Tendler 2 1 Department of Physics, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA 2 Laboratory of Biophysics and Surface Analysis, School of Pharmaceutical Sciences, University of Nottingham, Nottingham NG7 2RD, UK 3 Department of Chemical Engineering, University of Leeds, West Yorkshire LS2 9JT, UK Received 9 August 2002, in final form 30 October 2002 Published 3 December 2002 Online at stacks.iop.org/Nano/14/1 Abstract The scientific community needs a rapid and reliable way of accurately determining the stiffness of atomic-force microscopy cantilevers. We have compared the experimentally determined values of stiffness for ten cantilever probes using four different methods. For rectangular silicon cantilever beams of well defined geometry, the approaches all yield values within 17% of the manufacturer’s nominal stiffness. One of the methods is new, based on the acquisition and analysis of thermal distribution functions of the oscillator’s amplitude fluctuations. We evaluate this method in comparison to the three others and recommend it for its ease of use and broad applicability. 1. Introduction Motivated by the need for a fast and accurate method of determining cantilever stiffness, we have conducted a study of calibration methods for atomic-force microscopy (AFM) cantilevers. Although at least a dozen different calibration methods have already been published [1–17], we limited our experimental comparison to four approaches. The first, published by Cleveland et al [1], is based on measuring the cantilever width and length or taking the manufacturer’s values thereof, accepting literature values for elastic modulus and density of the cantilever material and measuring the cantilever’s resonant frequency. The second, for rectangular cantilevers, from Sader et al [3–5], incorporates the viscosity and density of the medium in which the cantilever is immersed along with experimentally determined values of the resonant frequency and quality factor, together with the cantilever dimensions, in order to calculate the stiffness. These first two methods we call ‘geometric’ because the dimensions of the beam occur in the equations. The third and fourth methods we dub ‘thermal’ since they are based on the acquisition of the cantilever’s thermal distribution spectrum (square of the fluctuations in amplitude as a function of frequency). In the third one, published by Hutter and Bechhoefer [6] and later modified by Butt and Jaschke [7], the thermal spectrum is related to thermal energy. In the course of checking their derivation, we discovered the fourth method, an alternative way of measuring cantilever stiffness using thermal spectra that we propose and verify in this paper. Although other existing calibration techniques have been helpful to the development of AFM, many were not included in this study. Finite element analyses [8–12] were thought to be neither general enough nor experimentally verifiable, and too complex for speedy use. Methods using special equipment, for example, pendulums [13] and nanoindentors [15], were also excluded from this work. Approaches involving manipulation of small particles [1] were thought to be too time-consuming as well. The personal experience of the authors is that pushing one spring against the other [16, 17] is subject to stick–slip motion, which interferes with the accuracy of the results. This last method also involves laboriously precise positioning. A comparison of the methods that we surveyed is compiled in the form of table 1. Below, we summarize the salient equations from the two geometric methods and from Hutter and Bechhoefer. We derive the distribution density for the square fluctuations in amplitude of a simple harmonic oscillator (SHO) and show how it leads to Hutter and Bechhoefer’s result and also demonstrate the equivalence of our approach. Then the corrections that must be made to compensate for our instrument’s detection system and configuration are presented, followed by the results of the four methods for ten different cantilevers in air and in water. A discussion of the advantages and disadvantages of the four approaches ensues, just before the conclusions. 0957-4484/03/010001+06$30.00 © 2003 IOP Publishing Ltd Printed in the UK 1