Thermo-mechanical responses of a surface-coupled AFM cantilever

Atomic force microscopy (AFM) has been widely used for measuring mechanical properties of biological specimens such as cells, DNA, and proteins. This is usually done by monitoring deformations in response to controlled applied forces, which have to be at ultralow levels due to the extreme softness of the specimens. Consequently, such experiments may be susceptible to thermal excitations, manifested as force and displacement fluctuations that could reduce the measurement accuracy. To take advantage of, rather than to be limited by, such fluctuations, we have characterized the thermomechanical responses of an arbitrarily shaped AFM cantilever with the tip coupled to an elastic spring. Our analysis shows that the cantilever and the specimen behave as springs in parallel. This provides a method for determining the elasticity of the specimen by measuring the change in the tip fluctuations in the presence and absence of coupling. For rectangular and V-shaped cantilevers, we have derived a relationship between the mean-square deflection and the mean-square inclination and an approximate expression for the specimen spring constant in terms of contributions to the mean-square inclination from the first few vibration modes.