Calibration of Normal and Lateral Forces in Cantilevers Used in Atomic Force Microscopy

Abstract

Atomic force microscopy (AFM ) is an indispensable technique for nanoscale topographic imaging, as well as quantification of normal and lateral forces exerted on the AFM tip while interacting with the surface of materials. In order to measure these forces, an accurate determination of the normal and lateral forces exerted on the AFM cantilever is necessary. In this chapter, we present a critical review of various techniques for measuring cantilever stiffness in the normal and lateral/torsional directions in order to calibrate the normal and lateral forces exerted on AFM cantilevers. The key concepts of each technique are presented, along with a discussion of their advantages and disadvantages.

Appendix—Nomenclature

1.1 Roman Symbols

A :

Adhesive force

a :

Amplitude

b :

Cantilever width

b _l :

Width of the leg in a triangular cantilever

c :

Photodetector sensitivity

d :

Distance of the tip to the edge of the cantilever

E :

Young’s modulus

F :

Friction force (lateral force)

f :

Frequency of the cantilever

G :

Shear modulus

H :

Piezo tube height in axial sliding method

h :

Cantilever thickness

I :

Area moment of inertia

J :

Mass moment of inertia

k_B :

Boltzmann’s constant

k _x :

Cantilever stiffness in the direction parallel to the longitudinal axis

k _yB :

Cantilever stiffness in the direction perpendicular to the longitudinal axis due to bending

k _yT :

Cantilever stiffness in the direction perpendicular to the longitudinal axis due to applied torque

k _z :

Cantilever stiffness in the normal direction

$k_{\phi }$ :

Cantilever torsional stiffness

L :

Cantilever length

l :

Lever length

$\ell$ :

Tip length

M _T :

Torsion moment

m :

Mass of the cantilever

m* :

Effective mass of the cantilever

m _s :

Mass of added particle

P :

Normal load component in axial sliding method, or applied load in the wedge method

p :

Area of the power spectrum in the thermal noise method

Q :

Quality factor

r :

Radius of added particle

T :

Temperature (in thermal tune method), or friction/horizontal force component (in axial sliding wedge and compliance hysteresis methods)

W :

Normal load

w :

Half width of the friction loop in the wedge method

z :

Cantilever deflection.

1.2 Greek Symbols

α :

One-half the included angle between the legs of a triangular cantilever

α _c, β _c :

Lateral force calibration factors in the wedge method

γ :

Cantilever tilt relative to horizontal axis

Δ:

Friction loop offset in the wedge method

δ_I,II :

Deflection of the cantilever in the parts I and II

ε :

Calibration factor in the lever method

η :

Viscosity of fluid medium

μ :

Coefficient of friction

ν :

Poisson’s ratio

ρ :

Density

θ :

Inclination of calibration standard in the wedge method

θ _II :

Rotation of the legs of a triangular cantilever in the longitudinal direction

$\phi$ :

Cantilever rotation from applied torque

ω :

Angular frequency of the cantilever

Γ :

Hydrodynamic function in the resonance method

ς:

Damping ratio