In the previous section the detection mechanism for the bending of the cantilever under influence of a tip sample force was discussed. The different types of forces that dominate in an AFM/MFM will be discussed in this section. As the tip-sample distance r changes, the forces working on the tip will also change.
For large tip-sample distances long range forces dominate such as magnetic or electrostatic forces. For small tip-sample distances the Van der Waals force dominates.
2.2.1 The Van der Waals force
The Lennard-Jones potential is shown in figure 2.7. The right attractive part of the curve is caused by the Van der Waals force. The left repulsive part is caused by the Pauli exclusion principle. For large tip-sample distances the Van der Waals force decreases to zero. As the distance decreases the attractive Van der Waals force increases. This increase continues until the atoms of the tip and sample come so close together that the electron clouds begin to repel each other due to the Pauli exclusion principle. Around a few angstrom the repulsive force due to the Pauli exclusion principle is just as large as the attractive Van der Waals force and no net force works on the cantilever. When the distance decreases even further the tip and sample are in contact. The slope of the Lennard-Jones potential is steep at low tip-sample distances. This means that the repulsive force grows fast when the tip is pushed further into the sample. If a stiff cantilever is used, or a soft surface, this force will deform the surface instead of bend the cantilever. It is therefore important to use the right cantilever for the right type of surface. This problem can be prevented by keeping the tip-sample distance larger. This can be done by operating the AFM in tapping mode, in the non contact region, which will be discussed in section 2.6.
Force
distance r repulsive
attractive contact region
non contact region
Figure 2.7: The force as a result of the Lennard-Jones potential is given by F = rC13−rD7 Here C and D depend on the tip sample properties.
The Van der Waals force is an intermolecular attractive force based on dipole interactions. There are three types of Van der Waals forces. The first type is
the orientation interaction force between two molecules. Both molecules have an electric dipole moment and tend to align in order to reach energy minimalization.
The second force is between two molecules, one with and one without an electric dipole moment. The electric field induced by the dipole moment polarizes the other molecule. This other molecule gets polarized by the electric field of the first molecule and thus obtains a dipole moment. This dipole moment influences the first molecule again as well as other surrounding molecules. The dispersion interaction is the third type of Van der Waals force and is similar to the induced dipole interaction. The difference is that the initial dipole is not a permanent dipole but a dipole due to spontaneous fluctuations of the electric field. This interaction is the most important for AFM measurements because normally the tip and sample do not have any permanent dipoles. The dipole moment due to spontaneous fluctuations of an atom, p1creates an electric field, E1∝ p1/r3, at a distance r. This electric field induces a polarization in a second atom at r, p2. If the polarizability of the second atom is α, p2∝ αE1. The energy of a dipole with dipole moment p placed in a field E is V = pE. So the potential of the dispersion interaction, and also for the induced dipole interaction, can be given by equation 2.12:
V = p2E1∝ αE12∝ αp21
r6 (2.12)
Despite the different nature of the three Van der Waals forces they all have the same potential dependence on distance, ∝ r16. If the energy has a r16 de-pendence the force, F ∝ ∂V∂r, has a r17 dependence. This is the attractive, right, part of the Van der Waals curve. The left part of the curve was dominated by the Pauli exclusion principle. This part of the potential can be approximated by a r112 dependence with an opposite sign as the Van der Waals potential. The Lenard-Jones potential is thus given by V (r) = rA12−rB6, where A and B depend on the tip sample properties.
2.2.2 The electrostatic force
If the tip and the sample are not grounded, charge can accumulate in one of them generating a voltage difference between the two. A capacitance C is created and the force between tip and sample can be given by equation 2.13 [6]:
F = −V2 2
∂C
∂r, (2.13)
where V is the potential difference between the tip and the sample. By con-necting the sample electrically to the cantilever the electrostatic force can be eliminated.
2.2.3 The magnetic force
In order to measure the magnetic force of a ferromagnetic sample, the tip has to be ferromagnetic. In most cases this is just a normal AFM tip with an extra ferromagnetic coating on the outside. It is complicated to derive the magnetization vector field of a tip and several simplifications are made in order to model this. The ’effective domain model’ [7] approximates the magnetization of the tip by a prolate spheroid with uniform magnetization as is shown in figure
2.8. Outside this spheroid the magnetization is zero. The magnetization is also constant and cannot be changed by external magnetic fields, like the magnetic field produced by the sample. Although these simplifications describe the real tip well it is often difficult to assign the correct magnetic moments to the tip.
A further simplification is made in the ’point probe approximation’ [8]. The representation of such a tip can be seen in figure 2.8 (c).
(a) (b) ( ) c
Figure 2.8: Three schematic representations of magnetic tips. (a)A tip with a ferromagnetic coating. (b)An MFM tip as modeled by the effective domain model. (c) An MFM tip as modeled by the point probe approximation.
The magnetization of the tip in the point probe approximation is considered to be inside an infinitesimal point. A multipole expansion of the magnetic field of a tip results in a magnetic monopole and dipole moment as the first two most important terms. In general the magnetic monopole can be neglected and only a dipole moment remains. However there is a magnetic monopole moment that needs to be taken into account in the point probe approximation since the magnetic stray field produced by the sample decreases with distance according to its decay length. If the decay length is small only a part of the tip senses the sample as is shown in figure 2.9 (a), only the bottom part of the tip senses the magnetic field of the sample and this can be considered as a magnetic monopole. In figure 2.9 (b) the decay length of the sample is larger and the entire tip senses this field. In this case there is no contribution from the magnetic monopole moment.
(a) (b)
Figure 2.9: (a)An MFM tip in close proximity to a magnetic sample producing a magnetic field with a short decay length. (b) The same tip close to a sample with a long decay length.
The infinitesimal point in the point probe approximation contains thus a magnetic monopole and a magnetic dipole and the force working on the tip can be given by equation 2.14 [9]:
F = µ0(q + m · ∇)H (2.14)
Here q and m are the effective magnetic monopole and dipole moment respec-tively and H is the magnetic field produced by the sample. In static mode the force is measured directly. In tapping mode, which will be discussed in paragraph 2.4, the derivative of the force with respect to the sample distance is measured [10].
The magnetic dipole and monopole moment also depend on z, the height of the tip with respect to the sample, since for larger tip sample distances less of the tip senses the magnetic field of the probe. Even with all these simplifications it is still hard to give a good description of the tip. MFM is therefore often used as a qualitative measurement to show the magnetic contrast of the sample and not so much a quantitative method to measure the exact size of the stray field of the sample.
The assumption that the sample magnetization is not affected by the tip magnetic field and the other way around is only valid for hard magnetic tips and samples. A soft magnetic sample is a sample where the stray field is small.
However the field is also sensitive to external magnetic fields and imaging such a sample with a hard magnetic tip will distort the magnetic structure of the sample. A soft magnetic tip also has a small magnetization so the forces working here are much smaller. To still be able to image the magnetic structure the force constant of the cantilever is also smaller. As a general rule the magnetic anisotropy field, Hk the field at which the magnetic structure changes, should be larger than the external magnetization H [11]:
(Hk)sample> (H)tip, (2.16) and
(Hk)tip> (H)sample (2.17)