A method to determine the spring constant of a rectangular atomic force microscope cantilever is proposed that relies solely on the measurement of the resonant frequency and quality factor of the cantilever in fluid (typically air), and knowledge of its plan view dimensions. This method gives very good accuracy and improves upon the previous formulation by Sader et al. [Rev. Sci. Instrum. 66, 3789 (1995)] which, unlike the present method, requires knowledge of both the cantilever density and thickness.

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Calibration of rectangular atomic force microscope cantilevers

The vibrational characteristics of a cantilever beam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical analysis of the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Due to its practical importance in application to the atomic force microscope (AFM), we consider in detail the special case of a cantilever beam that is excited by a thermal driving force. This will incorporate the presentation of explicit analytical formulae and numerical results, which will be of value to the users and designers of AFM cantilever beams.

Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope

Using an atomic force microscope, we measured effective spring constants of stacks of graphene sheets (less than 5) suspended over photolithographically defined trenches in silicon dioxide. Measurements were made on layered graphene sheets of thicknesses between 2 and 8nm, with measured spring constants scaling as expected with the dimensions of the suspended section, ranging from 1to5N∕m. When our data are fitted to a model for doubly clamped beams under tension, we extract a Young’s modulus of 0.5TPa, compared to 1TPa for bulk graphite along the basal plane, and tensions on the order of 10−7N.

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Mechanical properties of suspended graphene sheets

This work examines the effect of microstructure upon microcantilever bending stiffness. An existing beam theory model, based upon an isotropic Hooke's law constitutive relationship, is compared to a model based upon a micropolar elasticity constitutive model. The micropolar approach introduces a bending stiffness relation which is a function of any two independent elastic constants of the Hooke's law model (e.g., the elastic modulus and the Poisson's ratio), and an additional material constant (called γ). A consequence of the additional material constant is the prediction of an increased bending stiffness as the cantilever thickness decreases—a stiffening due to the material microstructure which becomes measurable at micron-order thicknesses. Polypropylene microcantilevers, which have a non-homogeneous microstructure due to their semi-crystalline nature, were fabricated via injection molding. A nanoindenter was used to measure their stiffness. The nanoindenter-determined stiffness values, which include the effect of the additional micropolar material constant, are compared to stiffness values obtained from beam theory. The nanoindenter stiffness values are seen to be at least four times larger than the beam theory stiffness predictions. This stiffening effect has relevance in future MEMS applications which employ materials with non-homogeneous microstructures instead of the conventional MEMS materials (e.g., silicon, silicon nitride), which have a very uniform microstructure.

https://iopscience.iop.org/article/10.1088/0960-1317/15/5/024/pdf

Role of material microstructure in plate stiffness with relevance to microcantilever sensors

Two methods commonly used to measure the normal spring constants of atomic force microscope cantilevers are the added mass method of Cleveland et al. [J. P. Cleveland et al., Rev. Sci. Instrum. 64, 403 (1993)], and the unloaded resonance technique of Sader et al. [J. E. Sader, J. W. M. Chon, and P. Mulvaney, Rev. Sci. Instrum. 70, 3967 (1999)]. The added mass method involves measuring the change in resonant frequency of the fundamental mode of vibration upon the addition of known masses to the free end of the cantilever. In contrast, the unloaded resonance technique requires measurement of the unloaded resonant frequency and quality factor of the fundamental mode of vibration, as well as knowledge of the plan view dimensions of the cantilever and properties of the fluid. In many applications, such as frictional force microscopy, the torsional spring constant is often required. Consequently, in this article, we extend both of these techniques to allow simultaneous calibration of both the normal and torsional spring constants. We also investigate the validity and applicability of the unloaded resonance method when a mass is attached to the free end of the cantilever due to its importance in practice.

Normal and torsional spring constants of atomic force microscope cantilevers

A scanning probe microscope having numerous advantages is disclosed. Respective scanning force and scanning tunneling probes are removably mounted in the head using kinematic mounting techniques so that they may be substituted for one another without the need to adjust the cantilever deflection sensor. A linear position-sensitive photodetector in the deflection sensor eliminates further the need for adjustments. A motorized, non-stacked x,y coarse movement stage is kinematically positioned with respect to the base and features a minimized mechanical loop to reduce thermal and vibrational effects on the position of the sample. A z coarse movement stage positions the head kinematically with respect to the base and includes a motorized drive means which allows the height, tilt and pitch of the probe to be adjusted. The scanner includes x,y and z sample position detectors which provide an accurate measurement of the position of the sample with respect to the probe. The z position detector provides an output which is exclusive of sample tilt and which may be used as an output of the scanning probe microscope. The outputs of the x,y and z position detectors may also be connected in feedback loops with the controller to improve the performance of the scanning probe microscope.

Scanning probe microscope

This paper addresses the problem of determining the absolute force constant of Atomic Force Microscope cantilevers. In the method presented, the cantilever under test is deflected against a reference cantilever of known spring constant. The relative deflection of the two cantilevers is related to their spring constants. The novelty of our approach is in the use of a micromachined reference cantilever of a precisely controlled force constant. Preliminary results show that our method is capable of measuring the force constant of cantilevers in the range of 0.1 to 10 N/m with an accuracy of better than 20%. The error is dominated by the non-linear effects in the force versus distance curves used for the measurement.

Characterization of application-specific probes for SPMs

A large scale horizontal translation stage for a microscope or other instrument particularly a scanning probe microscope is disclosed. The translation stage is provided with air bearings which allow it to float over a planar surface. The translation stage is kinematically mounted on a guiding member such that the horizontal position of the translation stage is defined by the guiding member but the translation stage is free to move in a direction perpendicular to the planar surface. To position a sample, the air bearings are actuated and the guiding member moves the translation stage to a desired position. An attractive force, preferably suction in the air bearings, is then applied to hold the translation stage firmly against the supporting surface while the sample is analyzed. The preferred embodiment includes two optical microscopes. The first optical microscope is directed substantially perpendicular to the plane of the sample and has a focal point which coincides with the surface of the sample when it is being positioned by the translation stage. The second optical microscope is directed obliquely and focused on the probe.

Large stage system for scanning probe microscopes and other instruments

A cantilever for a scanning probe microscope is disclosed The cantilever includes a piezoresistor for detecting the deflection of the cantilever, and a tip which is formed integrally with the cantilever A process of fabricating such a cantilever is also disclosed, the process yielding a tip which has a high aspect ratio and a small radius of curvature at its apex A combined atomic force/lateral force microscope including two or more piezoresistors responsive to both the bending and torsion of the cantilever is also disclosed

Piezoresistive cantilever for scanning probe microscope

A scanning force microscope having a sensor head and a base wherein a moveable sample holder is housed in the base and is positioned relative to a probe housed in the sensor head, such sample being monitored by an optical deflection detection system. The detection system is configured to provide direct visual observation of the probe with respect to the sample. The mirror of the detection system is mounted in a cut away portion of a sphere and defines the axis of rotation of a kinematic mount, such providing ease of fine adjustment of the detection system. The sensor head is in communication with the base by a stage kinematic mount, such providing ease of position adjustment of the sensor head with respect to the base.

Michael D. Kirk

Papers

Scanning probe microscope

Characterization of application-specific probes for SPMs

Large stage system for scanning probe microscopes and other instruments

Piezoresistive cantilever for scanning probe microscope

Scanning force microscope having aligning and adjusting means