Deflection Measurement

The bending response of the microcantilever due to adsorption of molecules onto the functional layer is caused by stress formation at the interface between functional layer and the forming molecular layer. Because the forces within the functional layer try to keep the distance between molecules constant, the cantilever beam responds by bending because of its extreme flexibility. This property is reflected by the spring constant k of the cantilever. For a rectangular microcantilever of length L, thickness t and width w the spring constant is k is calculated as follows: k = Ewt3 / 4L3, (1) where E is Young’s modulus (ESi = 1.3 × 1011 N/m2 for Si(100)). The microcantilever bends as a response to formation of surface stress caused by adsorption of a molecular layer. In its simplest case the shape of the bent microcantilever can be approximated as part of a circle with radius R. This radius of curvature is given by [1,2] 1/R = 6(1-ν) / Et2, (2) The resulting surface stress change is described using Stoney’s formula [1]: Δσ = Et2 / 6R(1-ν), (3) where E is Young’s modulus, t the thickness of the cantilever, ν the Poisson’s ratio (νSi = 0.24), and R the bending radius of the cantilever. The deflection of microcantilever sensors can be measured in various ways. They differ in sensitivity, effort for alignment and setup, robustness and ease of readout as well as in potential for miniaturization. 1. Stoney GG (1909) Proc R Soc London,Ser.A 82:172. 2. von Preissig FJ (1989) J Appl Phys 66:4262. Fig.: Beam-deflection concept to determine microcantilever bending with an accuracy of one nanometer.

The most frequently used approach to read out microcantilever deflections is optical beam deflection [3], because it is a comparatively simple method with an excellent lateral resolution.

The actual cantilever deflection Δx scales with the cantilever dimensions; therefore deflection responses should be expressed in terms of surface stress Δσ in N/m to be able to compare cantilever responses acquired with different setups. Surface stress takes into account the cantilever material properties, such as Poisson ratio ν, Young’s modulus E and the cantilever thickness t. The radius of curvature R of the cantilever characterizes bending, see Eq. (2). As shown in the drawing in Fig. 2, the actual cantilever displacement  is transformed into a displacement Δd on the position sensitive detector (PSD). The position of a light spot on a PSD is determined by measuring the photocurrents from the two facing electrodes. The movement of the light spot on the linear PSD is calculated from the two currents I1 and I2 and the size L of the PSD by


         Δd = (I1 - I2)L / 2(I1 + I2).        (4)

As all angles are very small, it can be assumed that the bending angle of the cantilever is equal to half of the angle ϴ of the deflected laser beam, i.e. ϴ/2. Therefore, the bending angle of the cantilever can be calculated to be

    ϴ / 2 = Δd / 2s,                                  (5)

where s is the distance between the PSD and the cantilever. The actual cantilever deflection Δx is calculated from the cantilever length L and the bending angle ϴ/2 by

     Δx = L Δd  / 4s.                     (6)

The relation between the radius of curvature and the deflection angle is

    ϴ / 2 = L / R                                          (7)

and after substitution becomes

     R = 2Ls / Δd     or   

   1 / R = 2Δx / L2.                              (8)


3.Meyer G, Amer NM (1988) Appl Phys Lett 53:2400