Atomically defined tips in scanning probe microscopy - McGill Physics

3.1 ATOMIC FORCE MICROSCOPY IN ULTRA-HIGH VACUUM 27monic with amplitude A and using canonical perturbation theory, the spring constantof the tip-sample interaction can be averaged over the oscillation cycle in thefollowing way [65]:〈k ts (z)〉 = 1 ∫ Aπk ts (z − q) √ A 2 − q 2 dq, (3.3)2 A2 −A〈kand the induced frequency shift can be calculated from ∆f = f ts(z)〉0 2k N. The followingsubsection discusses different force-distant relations, which can be directlyused to calculate ∆f. In general, for small oscillation amplitudes, the frequencyshift is independent of the amplitude and proportional to the tip-sample forcegradient k ts , as shown above. However, for amplitudes that are large comparedto the range of the tip-sample interaction, the frequency shift is a power functionof the amplitude, ∆f ∝ A − 3 2[66]. This relationship is later used for experimentalcalibration of the oscillation amplitude (see Section 4.2).Conservative forcesForces between the tip and the sample can be divided into various classes.The primary distinction made here is based on energy conservation during theoscillation cycle. Forces that are not delayed in time, i.e. have only an in-phasecomponent and do not depend on the velocity of the cantilever, are conservative.This is in contrast to the e-EFM method (see Section 3.1.3 and Chapter 8), where theelectrostatic force is delayed with respect to the tip oscillation and leads to energydissipation.Van der Waals forces (vdW) arise from the interaction between fluctuatingdipoles in the atoms in the tip and the sample. Considering just two atomicdipoles, the vdW interaction leads to a very weak and usually attractive force.However, if the contributions from all tip and sample atoms are summed up, itcan result in forces of several nN, which under certain experimental conditionscan dominate the total tip-sample interaction. Taking a sphere placed close to aflat surface as an approximation of the tip-sample geometry, the vdW force is describedby:F vdW = − HR6z 2 , (3.4)