An optical microscope image of a
typical V shaped AFM cantilever, the
part that makes contact is the small
dot on the end

The AFM spring constant, k_z converts the deflection of the cantilever into a force. It is essential for quantitative nano-force experiments to, for example measure the nano-scale elastic modulus values of polymers at surfaces, study the unfolding of multi-domain proteins and DNA unravelling, measure force induced dissociation of specific bonds.

The AFM spring constant is defined as the quotient of the applied normal force at the probe tip by the deflection of the cantilever in that direction at the probe tip position. Manufacturers give nominal values for the spring constant but these can be 100% in error, therefore quick, easy and accurate methods to determine it are necessary. NPL undertakes research into these methods and gives advice on the methods to use. These methods are split into 3 groups:

Dimensional
In dimensional methods, k_z is determined from the cantilever material and the geometrical properties. These either involve approximations to give simple equations or use finite element analysis (FEA) to give a more carefully modelled solution. For rectangular cantilevers, the equations are well-known, for V-shape cantilevers the situation is more complex. With careful comparison with FEA, we have shown that a new NPL equation has excellent agreement with FEA.

  An optical microscope image of a typical rectangular AFM cantilever

Static Experimental

Finite element analysis of a V-shaped cantilever used to determine
the spring constantStatic experimental methods involve a constant force applied to the cantilever and the subsequent measurement of the deflection. These methods generally, but not exclusively, use a pre-calibrated reference beam or device to push on a working cantilever or vice versa. NPL has developed improved methods to do this via a calibrated cantilever on cantilever method and a nanoindenter on cantilever method. NPL has also developed MEMS devices of known spring constant, traceable to the SI system of measurement via electrical MEMS. This method should allow routine and traceable calibration of AFM cantilever spring constants.

Dynamic Experimental
Dynamic experimental methods generally involve finding the cantilever's resonant frequency combined with other measurements. These other measurements, for example, could involve adding masses to the cantilever to measure the change in the resonant frequency.

ISO Standards
NPL is leading the development of an ISO standard in the calibration of AFM normal spring constant and has led an international interlaboratory study in this area. For more information, please email nanoanalysis@npl.co.uk.

References

[1] The determination of atomic force microscope cantilever spring constants via dimensional methods for nanomechanical analysis, C A Clifford and M P Seah, Nanotechnology, 16, 1666-1680 (2005).

[2] Improved methods and uncertainty analysis in the calibration of the spring constant of an atomic force microscope cantilever using static experimental methods, C A Clifford and M P Seah, Measurement Science and Technology, 20, 125501 (2009).