By designing cantilevers with localised adsorption areas at the terminal end of the cantilever (end loading), we can minimise the contribution of the differential surface stress [the dK/K terms in Eq.]. In that case, the changes in the resonance frequency can be attributed entirely to the mass loading. However, the changes in the spring constant may be due to a number of causes, such as changes in the elastic constant of the surface film and changes in the dimensions of the cantilever or coatings.
A group from the IBM Zurich research division reported an example of measuring mass loading (Berger et al. 1996). They used some zeolite crystals attached to a cantilever to monitor the water adsorption in the pores of the material, which has a high surface-to-mass ratio, as a function of the humidity. They detected an adsorbed water mass in the picogram range. In the case of a multilayer cantilever, such as a piezoelectric unimorph cantilever, the resonant frequency is expressed as follows:
Where EI i is
E is Young's modulus, I is the moment of inertia, h is the thickness, w is the width, L is the length, an2s the dimensionless nth-mode eigenvalue, and the subscripts np and p denote the elastic nonpiezoelectric layer (Lee et al. 2004). Because the nonpiezoelectric layer is composed of multilayers of different materials, we used the rule-of-mixtures approach to combine each individual modulus (Kaw et al. 1997). For example, when a PZT nanomechanical cantilever, which comprises multilayers of SiO2, SiNx, Ta, Pt, PZT, Pt, and SiO2, has a thickness of 2.0 ^m, we can calculate the resonant frequency of the piezoelectric unimorph cantilever structure that is clamped to one end. Theoretically, the first resonant frequency is 16,470 Hz for a structure with the dimensions of 100pm x 300pm and 61,384 Hz for a structure with the dimensions of 50pm x 150pm. Note that because the dimensions of the cantilever decreased from 100pm x 300pm to 50pm x 150pm, the spring constant of the cantilever increased from 1.48 N/m to 5.92 N/m. If molecular adsorption (induced mass) does not affect the spring constant of the cantilever, the beam spring constant, k, remains constant during the mass loading. The adsorbed mass can then be calculated by using the variation in the resonance frequency of a cantilever, as in the following equation:
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