Chapter 6 – Forces  235

However, approximating the tip as a sphere of radius Rtip and assuming interactions with a

planar sample surface, we can integrate the incremental contributions from the sphere using

Equation 6.26 to give

(6.27)

U

zA R

z

LJ

H

rip

≈−

6

where AH is the Hamaker constant depending on the tip and sample electrical polarizability

and density. As Figure 6.6d illustrates, the combination of multiple independent poten­

tial energy functions experienced by an AFM tip, which operate over different length scale

regimes, results in a highly nonlinear force–​distance response curve.

6.5.3  AFM IMAGING MODES

During AFM imaging, the tip–​cantilever force actuator can be used either in contact mode,

noncontact mode, tapping mode, or a relatively newly developed torsional mode. During con­

tact mode imaging, the cantilever deflection is kept constant throughout as the tip is scanned

across the sample surface using fast feedback electronics from the photodiode detector to a

piezo actuator controlling the cantilever z position, to maintain a constant force on the tip

(and hence constant height above the surface, assuming the material properties of the sample

remain the same). Here, although the tip itself does not make direct contact as such with the

sample, it is placed in relatively close contact to it (typically less than the equilibrium atom

separation of ~0.2 nm) such that the overall force detected by the tip from the sample is in

the short-​range repulsive force regime.

As Figure 6.6d suggests, the force, as the gradient of the potential energy curves, varies

dramatically with vertical displacement, with typical forces being in the range 10⁻⁶ to 10⁻⁹

N. This high sensitivity to vertical displacement allows potentially atomic-​level resolution to

be obtained in contact mode. However, shearing forces at short distances from the sample are

high potentially resulting in sample distortion, in addition to sample damage from scraping of

soft sample features by the AFM tip during lateral scanning.

Although atomic-​level resolution in z can in principle be obtained in contact mode, the

finite AFM tip radius of curvature results in a limit on the absolute maximum measurable

z displacement (i.e., height) between neighboring surface features. If, for example, similar

sharp surface features are separated by a characteristic displacement d in the lateral surface

plane, then the maximum height Δz, which an AFM tip of radius of curvature Rtip could

measure, is given from simple space constraints as

(6.28)

∆z

tip

d

R

≈

2

8

In contact mode imaging, the AFM tip can penetrate beyond water layers bound to the

surface to image the sample molecules directly, manifested in a greater spatial resolution.

However, the finite sharpness of the AFM tip itself means that some sample surface features

will be inaccessible with a resultant tip broadening convolution artifact (see Worked Case

Example 6.2). The AFM tip experiences a lateral force from a stiff object on the surface when

the AFM tip is pushed down vertically during imaging. If the half angle of the tip’s triangular

cross-​section is θ, then simple geometrical considerations indicate that the tip broadening

coefficient κ, defined as the ratio of the apparent measured width r′ of the stiff object (mod­

eled as a sphere with a circular cross-​section of radius r), satisfies

(6.29)

κ

θ

θ

=

=

+

′r

r

tan

sec