In scanning probe microscopy the interaction between a microscopic tip and a surface is measured as a function of the position of the tip relative to the sample. For this purpose, the tip is attached to a cantilever and the deflection of the cantilever is measured. In our microscope, this deflection is measured by means of an interferometric AFM system, which shall be explained in detail in chapter 4. Its principle is shown in Fig. 1.2.
The position of the sample is controlled by a piezoelectric actuator, which is a lead zirconate titanate∗ (PZT) ceramic. Piezoelectric actuators convert electrical energy directly to mechanical energy and they make motion in the sub–nanometer range possible [20]. Scanning the material surface is done by actuating the sample in a raster pattern. For dynamic operation (chapter 2), the cantilever is excited by a bimorph piezoelectric plate called the dither piezo. When tip and sample are brought into close proximity, tip–sample interactions causes the cantilever to bend and as a result the distance d between tip and fiber changes, which results in a change in the interference spectrum. The forces existing between tip and sample can be classified according to the distance dependence of the underlying microscopic interaction potential φst(r, r′). If the variations of the microscopic interaction potential φst(r, r′) are small on the scale of interactomic distances, the integration can be carried out over a continuous tip and sample volume. These forces are short–range forces, for example chemical forces that vary over a distance of fractions of nanometers, contrary to long–range forces, which decay over several nanometers to tens of nanometers. The main long–range forces studied in this thesis result from van der Waals interactions, electrostatic interactions and magnetic interactions. Short–range forces, including chemical bonding forces (adhesion and capillary forces) and short–range electrostatic forces, will be only be discussed shortly in the introduction.
1.2.1 Long–range attractive forces
Long range attractive forces of the van der Waals type arise from instantaneous moments of otherwise nonpolar atoms and molecules, i.e. electromagnetic field fluctuations. Electromagnetic field fluctuations are universal,
∗Lead zirconate titanate (Pb[ZrxTi1-x] O3) is a ceramic perovskite material that shows a marked piezoelectric effect [19].
Feedback
Figure 1.2: Schematic representation of the interferometric AFM system. The deflection of the cantilever is measured by means of a change in the optical path length d of the reflected laser light (chapter 4.3.1). Electronic and/or mechanical connections are indicated in black. Optical connections are indicated in green. The laser source, the fiber–end (ferrule) and the detection system are connected to a 50/50 coupler (chapter 4.3.3). Connection 3 in this case is not used. (a) The unbended cantilever without any tip–sample interaction. (b) The bended cantilever near the sample surface. In this case, the tip–sample interactions cause the cantilever to bend, hereby changing the length of the optical path.
which make van der Waals forces ever–present, independent of the chemical composition of the surfaces or the medium. Beside the spontaneous electric field fluctuations, which are the most important contribution in SPM studies, there are two other types of van der Waals forces. The first type arises from (permanent) electric dipole–
dipole interaction between molecules. In this case, both molecules tend to align to minimize their energy. The second type is the interaction between two molecules, with and without an electric dipole moment. The dipole moment of one molecule induces an electric field, which polarizes the other molecule (i.e. an induced dipole moment). Ever since the classical work of Lifshitz [21], it is commonly known to scientists that the microscopic approach, which is based on assuming simple pairwise additivity of the atomic or molecular contributions, does not rigorously characterize the collective macroscopic nature of the van der Waals interactions. In particular, the effects of multiple reflections of the intermolecular radiation field, and of a dielectric immersion medium surrounding probe and sample can only be characterized by the Lifshitz approach. This leads to a variety of phenomena relevant to long–range scanning force microscopy, see for example Hartmann [22]. In general, the van der Waals force can be described by the Lennard–Jones potential, as can be seen in Fig. 1.3a. At large enough distance the force is dominated by the −D/r7 term is attractive (non contact regime). At close distance, the C/r13 term dominates and the force is repulsive (contact regime). At intermediate distances, the force can both be repulsive and attractive, this is called the ‘semi-contact’ regime. When two spherical particles are brought in close proximity to each other, conservative interaction forces between these particles can be described by the Hamaker approximation [23]. Applying this approximation to SPM studies, where tip and sample are brought in close proximity, we write for the force F(r, t) between tip and surface
F(r) = −∇ 1
Here r = (x, y, z) is a coordinate system fixed to the sample and Vsample and Vtip denote the total sample and tip volumes. Considering only small longitudinal probe–sample separations, denoted by z (∼ several
r
(a) Lennard–Jones Potential for van der Waals forces
R a
Dz
z
(b) Micro–tip Figure 1.3: (a) The force exerted between tip and sample as a result of the Lennard–Jones potential U is given by FLJ(r) =rC13
−
rD7. C and D reflect the tip–sample properties. The different regions relevant for SPM studies are indicated in the figure. (b) Tip model used for describing the van der Waals and electrostatic forces. The cluster at the end of the tip is the nanotip. The sample is represented by a molecular layer of undefined species.nanometer), we can neglect retardation effects of the radiation field between probe and sample. In this case, the nonretarded van der Waals force is given by a simple inverse power law of the type
Fn(z) = gnHn/zn, (1.2)
where gn denotes a geometrical constant depending on the actual probe model and Hn is the nonretarded Hamaker constant, which reflects the dielectric contributions involved with the van der Waals force†. As only surfaces covered uniformly with only one chemical species are studied (in the ideal case), the long–range van der Waals force does not vary locally (it does not lead to imaging contrast formation). Therefore, a simple model consisting of a uniform semi–infinite surface is used, in which the tip is described as a cone with a spherical cap (Fig. 1.3b). Then, the van der Waals force is given by [24]
FvdW(z) = −Hn of the cantilever‡, Eq. 1.4 transforms to
FvdW(z) = −HnR
6z2 . (1.5)
The van der Waals force for spherical tips is thus proportional to 1/z2. For large distances, the force is attractive until the atoms of the tip and sample come so close together that the electron cloud begin to repel each other due to the Pauli principle§. However, if the contact area between tip and sample involves tens to
†In SPM studies, the nonretarded Hamaker constant is given by
Hn= 3~
‡The cantilever canting will be neglected throughout this thesis expect where necessary for the discussion of magnetic forces.
§For a modern approach to the calculation of van der Waals forces in SPM please study Hartmann [25] or for a more general description Israelachvili [26].
hundreds of atoms, the description of the effective repulsive force can be obtained without considering the Pauli repulsion and the ionic repulsion of the molecules.
Trapping of electrostatic charge in dielectric surfaces can also give rise to long–range electrostatic interaction forces. This will be the case in situations where an external electric field is applied between tip and sample and/or when an electrostatic potential difference exists between tip and sample. In the absence of an external field, if the separation of the tip and sample z is small in comparison to the tip’s radius R, the electrostatic force existing between tip and sample can be written as [27]
Fel = −πǫR z ΦST2
, (1.6)
with ǫ the electric permittivity and ΦST the contact potential difference caused by the different work functions of tip and sample. The influence of electrostatic force is circumvented by proper grounding of both tip and sample.
1.2.2 Contact and short–range repulsive forces
The surface of two bodies are deformed when they are brought into mechanical contact. The deformation depends on the applied load and the properties of the material. Continuum elasticity theories describes the contact and adhesion between finite bodies under an external load. The problem of two elastic objects in contact, here the tip and the sample, consists in establishing and solving the relationship between the stress (σ) and strain (ε) tensors. This functional relationship is referred to as the constitutive equation and for isotropic materials it is deduced as
σij = ηεllδij + Gεij, (1.7)
where η is the Lamé coefficient and G the shear modulus (G = E/2(1 + υ), where υ is Poisson’s ratio)¶. In ambient conditions a thin film of water is adsorbed on hydrophilic surfaces. At close proximity of tip and surface, a meniscus or liquid bridge may be formed between tip and sample. This meniscus implies an attractive or capillary force that shows a dependence with distance. When cooling down to liquid nitrogen or helium temperatures, the cantilever is placed in a quasivacuum environment (pressure ∼ 20 mbar) after being flushed with 4He gas. Therefore, we assume that capillary forces will be absent. In SPM studies, short–range forces are relevant in the contact regime, as for the magnetic force microscopy studies conducted in this thesis, only the presence of the long–range attractive van der Waals and magnetic forces are considered. The underlying theory relevant to magnetic SPM studies is presented in chapter 3.