125 The atomic force microscope (AFM) was invented in 1986 [1], a close relative of another instrument, the scanning tunneling microscope (STM), invented in 1981 [2]. Both fall under the umbrella of techniques and instruments referred to as scanning probe microscopes (SPMs), with the common thread being that a sharp probe is scanned in a regular pattern to map some sample characteristic. Unlike the STM, the AFM can readily image insulating surfaces. Combined with the ability to study a wide variety of samples and sample environments – ambient, liquid, and vacuum – has made AFM the technique of choice for many high resolution surface imaging applications, including imaging with atomic resolution. Since those early days, AFM techniques have become the mainstay of nanoscience and nanotechnol- ogy by providing the capability for structural imaging and manipulation on the nanometer and atomic scales. Beyond simple topographic imaging, AFMs have found an extremely broad range of applications for probing electrical, magnetic, and mechanical properties – often at the level of several tens of nanometers. One ongoing “holy grail” quest of AFM, since very nearly the beginning [3] has been compositional mapping where materials differences are mapped out with the same nanometer resolution as topographic images. There are many forces acting between an AFM tip and a sample, long-ranged van der Waals, electrostatic and magnetic forces, short-ranged forces stemming from the elasticity of the tip and sample, and dissipative forces associated with adhesion, plasticity, phonon gene- ration, and eddy currents, to name a few. Many if not all of these interactions carry compositional information about the sample. However, the forces also depend on the geometry of both the tip and the sample. The sample topography conspires with the tip geometry to make unraveling of the specific contributions to the net forces very difficult. For example, a situation where there is zero net force on the cantilever tip might mean that there are no forces acting on the tip or that there is a very large adhesive (attractive) force being balanced by a very large elastic (repulsive) force. Much of the theoretical and experimental work being done in AFM originates from R. Proksch (*) Asylum Research, Santa Barbara, CA 93117, USA e-mail: roger@asylumresearch.com Chapter 5 Multi-Frequency Atomic Force Microscopy Roger Proksch S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_5, © Springer Science+Business Media, LLC 2010