Cantilever dynamics in static mode

Most cantilevers are rectangular beams with a tip on the free end side. The other side is fixed to a much larger wafer as is shown in figure 2.10 (a).

The cantilever has a length l, a width w and a thickness t. Normally l ≫ w ≫ t. The tip is closest to the sample so it is assumed that all the forces that act on the cantilever, act on the tip. The cantilever can bend in the x-, y- and z-direction. If the bending in the x- and y-directions is small, the laser remains aligned at the back end of the cantilever and there is no change in the interference signal. Since l ≫ w ≫ t, the bending of the cantilever in the z-direction, the direction of the thickness dominates. Therefore we consider

l

Figure 2.10: Schematic view of the cantilever on a wafer. (a) The Si wafer with at the top the cantilever. (b) A zoomed in top view of the cantilever. (c) The side view of the cantilever.

only bending in the z-direction¹. Bending in the z-direction can happen due to three different forces, a force in the x-, y- or z-direction. If the bending of the cantilever is small, the cantilever can be approximated by a perfect spring. The relation between the bending and the force can then be described by Hook’s law [12]:

Fi= −ki∆z (2.18)

Here F is the force acting on the cantilever, k is the spring constant and ∆z is the deflection in the z-direction. The subscript i can be taken as x, y or z.

The cantilever bending resulting from a force in the z-direction is shown in figure 2.11. Two cross-sections are made and figure 2.12 (a) shows a beam element of length L between two cross-sections, (b) shows a cross-section. Here R is the cantilever curvature radius, L is the original length of the cantilever.

∆L is the extension of L at position z, where z is the distance from the neutral plane (the plane where ∆L=0). z can thus be seen as ∆R so ∆L/L = z/R since the relative increase in circumference equals the relative increase of the radius of a circle.

The Young’s modulus, E, of a material is a measure for the stiffness of the material and can be calculated by dividing the stress by the strain [13] and is given by equation 2.19:

E = F (y)L0

A0∆L (2.19)

F (y)² is the force at a distance y acting on a small strip of the cross-section, the area dS in figure 2.12, due to the bending caused by Fz. A0 is the cross-sectional area on which the force is applied. Because the force above the neutral line (z=0) points at the opposite direction as the force under the neutral line the forces cause a bending moment Mz.

1If the detection method used is not an interferometer but for example a quadrant detector, bending in the x- and y-direction can be detected. Nevertheless, with a quadrant detector the bending in the z-direction still dominates.

2Here F (y) is not the force in the z-direction caused by the tip-sample interaction.

y z

x F

Figure 2.11: Schematic view of a bended cantilever due to a force in the z-direction. The force is actually working on the tip but to make the image clearer it has been drawn at the top of the cantilever. The black cantilever is for Fz=0 and the red cantilever is bended under a force in the z-direction.

With the help of figure 2.12 and equation 2.19 an equation for the force working on A0can now be written as dF (y) = ^Ez∆S_R(y) and the bending moment is given by:

Mz(y) = Z

S

zdF (y) = Et³w

12R(y) (2.20)

Mz(y) is the bending moment at a point on the y axis due to Fz on y = l.

This can also be given by Mz(y) = Fz(l − y). If the bending of the cantilever is small the deflection of the cantilever at point y, u(y), can be given by 1/R(y) =

∂²u/∂y². To calculate the deflection at point y = l we have to solve [13]:

∂²u

∂y² = 12Mz

Et³w = 12Fz

Et³w(l − y) (2.21)

Since the cantilever is fixed at the wafer side and cannot move with respect to the dither the boundary conditions become: u|y=0 = 0 and ^∂u_∂y|y=0= 0, this gives:

u(l) = ∆z = kzFz = 4l³

Et³wFz (2.22)

for the force in the z-direction. The force in the y-direction, Fy, also results in a bending moment My as can be seen in figure 2.13. The cantilever deflects in the same way but reacts differently on the size of the force. Here My= Fyltip

with ltip as the length of the tip. The first part of equation 2.21 is still valid and Mz can be replaced by My which gives:

∂²u

∂y² = 12Fyltip

Et³w (2.23)

With the same boundary conditions as for the force in the z-direction this leads to the deflection resulting from a force in the y-direction.

L

Figure 2.12: (a) Beam element of length L between two cross-sections. The dashed line represents the cantilever in equilibrium while the normal line corre-sponds to the bended cantilever. (b) Front view of a cross-section of a bended cantilever. The entire rectangle is the cross-sectional area A0and dS is a small stripe of that area.

u(l) = ∆z =6Fyltipl²

Et³w = 3ltip

2l kzFy (2.24)

The bending in the z-direction due to Fxis much harder to calculate. There are two separate motions of the cantilever when a force in the x-direction is working as is depicted in figure 2.14. First of all a bending in the x-direction will occur. If the bending in the x-direction is small the measured signal does not change. In general this is the case and we will not discuss this motion. The cantilever will also make a torsional motion which is detected by the system.

The torsional motion is much harder to calculate than the previous cantilever motions and only the result will be given [14]:

∆z = 2l³_tip

l⁴ k²_zF_x² (2.25)

The forces in the three directions have been discussed and Hook’s law can be written as:

∆z =2l_tip³

l⁴ k_z²F_x²+3ltip

2l kzFy+ kzFz (2.26) Therefore the largest contribution to ∆z is from the force in the z-direction since the length of the cantilever is much longer than the length of the tip, l ≫ ltip. Typically the length of the cantilever is between 125 and 450 µm and the length of the tip is around 4 µm so from now on we will only consider forces acting on the tip in the z-direction.