Atomic Force Microscopy

2.3. Atomic Force Microscopy

(a) (b)

z y x

I

Figure 2.5.: Illustration of the working principle of an AFM: (a) In the most common setup util-ising an optical detection mechanism a laser beam is reflected from the cantilever and the bending motion due to the interaction with the sample is measured with a detector array. (b) In the setup implemented in this system an oscillating quartz tuning fork with a tip attached to it is used as a force sensor that is scanned over the surface. Due to forces acting between tip and sample the frequency and amplitude of the oscillat-ing prong change from its unperturbed values. The oscillation amplitude is directly accessible as a small current, which is induced due to the piezoelectric properties of the quartz.

2.3.1. Working Principle

In AFM a sharp tip is attached to a cantilever and scanned over the sample. Due to the forces that act between tip and surface (see Figure 2.6 (a)) the cantilever bends and thereby the interaction force can be indirectly measured. The cantilever deflection can be measured in different ways: While in the first AFM an STM was used [32], also capacitive, piezoelectric, or optical schemes are deployed. The deflection of a laser beam as sketched in Figure 2.5 (a) is the most common implementation. The most important advantage of AFM over STM is its ability to also investigate insulating samples. This makes AFM also popular in the life sciences.

The different characteristics of AFMs are as broad as the range in which they are used.

To understand the huge advantages of the chosen sensor setup deployed in this work, it is worth quickly summarizing the most popular modes of operation. In ref. [44] an extensive review on the AFM modes is given by Garcia and Perez.

A first reasonable criterion to distinguish operation modes is based on whether or not a feedback is controlling the scan, and if so, which kind. In constant height mode, where the cantilever is scanned at a fixed height above the surface, no feedback is active and only the deflection is measured. While in normal setups this mode is rarely used, as it requires extremely flat samples to avoid crashing, it has to be deployed in a slightly modified way

0.4 0.6 0.8 1.0 -0.1

0.0 0.1

Force (nN)

Distance (nm)

(a) (b)

attractive repulsive

Surface

Cantilever

approx. 10µm: Fluid damping

0.1-1 µm: Electrostatic forces (attractive or repulsive) 10-200 nm: Capillary forces (attractive) few Angstroms: Van der Waals forces (attractive) sub-Angstrom: Pauli repulsion

Figure 2.6.: (a) Forces acting between a tip and a surface, based on [43]. (b) The force-distance curve based on a Lennard-Jones potential (with parameters as discussed in Chapter 4).

The attractive part of the potential is assumed∝r⁻⁶ (based on van der Waals inter-action, green curve) and the repulsive part ∝r⁻¹² (Pauli repulsion, red curve). Note that the forces are plotted and not the potential (F = −∇E).

in order to correctly quantify the tip-sample interaction, as will be discussed below.

On the other hand, several signals like deflection, amplitude, frequency shift, and phase can be used to establish a feedback loop regulating the tip-sample distance. In contrast to STM with its monotonic exponential dependance of the current on the tip-sample distance, no such simple functional dependence between the force and the distance exists. This is due to the interplay of several different types of forces present. Figure 2.6 (a) gives an overview of the forces that might be involved, including typical distances. As the work performed here is carried out in UHV fluid damping and capillary forces can be disregarded. If only van der Waals and repulsive forces due to the Pauli repulsion are considered, a good approximation is given by the force-distance curve shown in Figure 2.6 (b), which is based on a Lennard-Jones potential [45].

Chronologically, the first AFMs operated in contact mode, where the tip, attached to a very soft cantilever, is “scratched” over the surface. This quickly allowed fairly high resolution as reported by Martiet al. in 1987 [46]. A serious advancement was the introduction of tapping mode AFM. This is a dynamic operation mode, where the cantilever is externally excited to oscillate, while the amplitude is used as a feedback signal. Amplitudes in the range of 10 – 100 nm and fast scanning speeds make it popular for application in biology.

The impact from the tip to the sample in contact mode is highly reduced in tapping mode AFM, as is nicely shown by Zhonget al. where the two methods are compared [47].

A further advancement, which is also relevant for this work, is the invention of the true non-contact mode, also a dynamic mode, with amplitudes typically below 10 nm. This mode can be achieved using the oscillation amplitude as feedback (“AM nc-AFM”) [48], or the frequency (“FM nc-AFM”) as introduced by Albrecht et al. in 1991 [49]. The latter requires rather stable oscillation, hence relatively stiff cantilevers are used. This technique provided true atomic resolution to resolve the reactive Si(111) surface and its

2.3. Atomic Force Microscopy

7×7 reconstruction [50].

Similar to STM, where the acquired image is a convolution of topography and electronic structure, the interpretation of dynamic mode AFM images is nontrivial. In the following section the method of choice to calculate the forces quantitatively from the AFM signal will be presented.

2.3.2. Calculation of the Force

Mount / Piezo

Sample

m Tip

Cantilever

Interaction k

k_i

Figure 2.7.: A simple model illustrating the frequency shift of the free cantilever oscillation (with spring constantkand effective massm) due to the interactionk_i between the tip and the sample.

The calculation of the interaction forces between tip and sample from the frequency shift is not straightforward. Because the cantilever oscillation is harmonic it can be characterized by a spring constant k (f ∝ √

k/^m). If it is exposed to a force gradient, which can be described by a spring constant (dF/dz =ki), the oscillation frequency will change [49].

This interplay is depicted in Figure 2.7. The two springs are considered to be in series resulting in the following frequency of the oscillation:

f = 1 2π

√k+ki

m , (2.11)

which is only valid if Hooke’s law holds, i.e. if k_i ≠ k_i(z) (i.e. k_i is constant over the oscillation cycle). For highly nonlinear force-distance laws found at the atomic scale (Figure 2.6 (b) shows an example of a Lennard-Jones type force) this requirement is not met. To circumvent this limitation, the averaged value ki(z) is introduced. If the inter-action is small compared to the stiffness of the used cantilever (k_i(z) ≪k), the frequency shift can be expressed as:

∆f = f−f0 (2.12)

≈ f0

ki(z)

2k . (2.13)

Giessibl calculated k_i(z) with first order perturbation theory using the Hamilton-Jacobi

approach [51]:

with A the amplitude of the oscillation, z the distance of closest approach and u the parametrization of the oscillation. To determine the interaction energy and the forces from the measured frequency shift this equation has to be inverted. Sader and Jarvis found analytical solutions [52]: Those equations enable the calculation of the interaction energy and force at height z from the measured frequency shift ∆f by integration over all farther tip-sample distances (z to∞). In experiment this integration has to be replaced by a summation over a discrete set of a limited number of heights. The implementation will be discussed in Chapter 4.

An extensive comparison of the deconvolution quality of this method (“Sader-Jarvis”) and the “matrix-method” introduced by Giessibl [53] can be found in ref. [54].