Analysis of the high-frequency response of atomic force microscope cantilevers

Abstract

In most commercial atomic force microscopes, dynamic modes are now available as standard operation modes. Acoustical vibrations of atomic force microscope cantilevers can be excited either by insonification of the sample or by vibration of the clamped cantilever end. The resulting dynamical system is complex and highly nonlinear. Simplification of this problem is often realized by modeling the cantilever as a one-degree-of-freedom system such that the higher-order flexural modes are neglected. This point-mass model has been successful in advancing material property measurement techniques. The limits and validity of such an approximation have not been fully addressed. In this paper, the flexural beam equation is examined and compared with the point-mass model by using analytical and finite difference numerical techniques. The two systems are shown to have differences in drive-point impedance and are influenced differently by the surface damping and the contact stiffness. The angular deflection at the end of the beam and the influence of lateral sensor tip motion are considered. It is shown that the higher modes must be included for excitations above the first resonance if both the low-and the high-frequency dynamics are to be modeled accurately.The earliest publications on atomic force microscopy (AFM) suggested dynamic modes in which the small microfabricated cantilever vibrates at higher frequencies compared with the tip-sample scanning frequency [1, 2]. Originally these methods were used to increase the force sensitivity in non-contact AFM, as, for example, in magnetic force microscopy [3]. Within the last years, dynamic modes became more widespread and are now available as standard operation modes in most commercial instruments. New dynamic modes (for example the intermittent contact or tapping mode [4] and the force modulation mode [5], where the vibrating cantilever is in repulsive contact with a sample surface) have been * Present address: Department of Engineering Mechanics, 212 Bancroft Hall, University of Nebraska-Lincoln, Lincoln, NE 68588-0347, USA added. In many applications (for example topography imaging of soft surfaces which might be deformed by the forces applied by the sensor tip), dynamic modes have even replaced the contact force and friction mode. But dynamic modes are not only interesting because they supply topography information with less surface damage than the contact mode. They can also provide additional image contrast on sample surface properties, for example stiffness or adhesion. This has been demonstrated for example by the various atomic force acoustic modes [6][7][8]. The range of mechanical vibration frequencies which are used for AFM imaging covers acoustic vibrations (up to 20 kHz) and ultrasonic frequencies from 20 kHz up to the MHz range.

Humidity effects on the determination of elastic properties by atomic force acoustic microscopy

“…Equations that relate the sample-coupled frequencies f n to the contact stiffness k* as a function of the relative tip position L 1 /L have been developed previously for the case of no damping (␥ ϭ0). 9 In other work, 10 similar equations have been derived that include damping (␥ 0) but assume L 1 /Lϭ1. By combining these results, we obtain the following relation between f n and k* if both damping and tip position effects are considered:…”

Contact resonances of U-shaped atomic force microscope probes

“…In order for new material designs to be fully understood, measurements that can quantify behavior at appropriate scales, often down to the nanoscale, are required. The contact resonance atomic force microscope (CR-AFM) technique is a promising materials characterization approach, which can quantify the elastic [1][2][3][4][5][6][7][8][9] as well as viscoelastic [10][11][12][13][14][15][16] properties of materials with spatial resolution on the order of tens of nanometers. CR-AFM uses the vibration spectra of an AFM probe during vibrations for both the noncontact case, and when the tip is in contact with a sample.…”

Analysis of contact stiffness in ultrasound atomic force microscopy: three-dimensional time-dependent ultrasound modeling

“…The motion of the cantilever is measured with the use of the optical beam deflection method [4,5] and gives a high resolution image of the surface [6]. In dynamic mode AFM, the tip-sample interaction influences the vibration mode of the cantilever [7]. The response of the canti lever specifically depends on the tip-sample interaction stiff ness, i.e.…”