The aim of this project is to gain a better understanding of the dissipation mechanisms which are responsible for solid friction on the atomic scale. The field of nanoscale friction has experienced an enormous advancement in the last two decades due to the development of the Atomic Force Microscopy (AFM). However, comparing experimental results to results of theoretical models is difficult, as the number of relevant processes involved in the measurement process is rather large. One idea to reduce complications has led to the development of the frequency-modulated dynamic force microscopy technique with which dissipation can be measured directly. This defines our main focus of interest: the modeling and simulation of dissipation effects in dynamic atomic force microscopy, mainly for insulators. However, if different materials are used, new effects might become relevant, especially if ferromagnetic materials are involved. Dissipation effects of purely magnetically interacting systems (magnetic friction) are a second focus (see below). Further interests include mesoscopic friction (see below) as well as electromigration (see below).

The basic principle of frequency-modulated dynamic atomic force microscopy (FM-AFM) is as follows: The height (the topography) of a point on the surface is measured by shifting the probe so that due to surface-tip interaction the resonant frequency of the cantilever oscillation is detuned by a given amount. The amplitude is kept constant which necessitates to drive the oscillation. Energy loss of the oscillation occurs not only due to mechanical damping of the cantilever, but also due to interactions between tip and surface, so that even atomic resolution can be obtained in the dissipation signal. To gain information out of the dissipation signal, a clear understanding of the physical processes during the scan is required.

Fig 1: System setup. The tip is a rotated cube representing the outermost atoms of the tip.Green atoms are potassium, red atoms are bromine.

Our aim is to identify the relevant dissipation processes. To archive this, we use classical molecular dynamics. We consider a system of ionic crystals (KBr), as non-adiabatic processes (electronic friction) can be neglected. We simulate only the outermost atoms of the tip and just a small portion of the substrate (see figure 1). We follow a new strategy for simulating AFM: instead of guiding the tip on a sinuoidal trajectory, we couple the tip to (multiple) harmonic potentials as a replacement for the rest of the cantilever, so that we can a) directly determine the response of the cantilever with respect to frequency shifts and b) determine the damping directly by comparing the energy stored in the cantilever oscillation before and after contact. The total energy of the whole system is conserved in our simulations, which has the advantage that if energy is removed from the cantilever oscillation, we can trace where the energy "goes to".

A common explanation of the experimentally observed damping is adhesion hysteresis. We assume, that the tip oscillates along the z-axis. Further, we call the atom of the substrate, which has the smallest distance to the outermost atom of the tip, projection atom. This atom stays in its energy minimum in the surface layer of the substrate. As the tip approaches the surface, it induces a second energy minimum above the surface. This minimum gets even lower than the energy minimum in the surface layer, but it is separated by a barrier. The closer the distance between tip and substrate, the smaller the barrier gets. At a certain point, the barrier is small enough (or has vanished), so that the atom can jump into the second minimum. The projection atom gains kinetic energy which is then distributed among other atoms in its environment. Finally, the atom has lost its kinetic energy while the local environment has become warmer. When the tip is retracted from the substrate, the atom stays in the second minimum until it can overcome the barrier to jump to its original position. It can happen, that the atom does not have enough energy to overcome this barrier, but in this case the tip would have changed. In the end one finds a hysteresis loop as shown in the right inset of figure 2 and which is visualized in the movie of figure 1. The hysteresis effect sets in at a certain distance between tip and substrate. If the tip does not go below that distance, no damping is observed (see blue line of figure 2). On the other hand, if the tip goes below that distance, the damping rate does not depend on the smallest distance between tip and surface (figure 2 seems to contradict this for distances smaller than 0.35 nm; the increased damping rate is due to another mechanism).

Fig 2: Damping vs. smallest distance. Blue curves: Apex atom and projection atom are oppositely charged. Green curves: Atoms are equally charged. Inset shows the trajectory of the projection atom in the substrate.

One requirement for adhesion hysteresis is that the atom performing the hysteresis loop must interact attractively with the outermost atom of the tip. Therefore, we would not expect dissipation, if the tip is placed such, that the outermost atom is directly above an equally charged atom. The green line of figure 2 shows a surprising result: the damping rate shows a strong dependency on the smallest distance between tip and substrate and its value even exceeds the one of the adhesion hysteresis effect. The origin is completely different from the adhesion hysteresis. The energy is transfered to a lateral oscillation, which can be seen in the left inset of figure 2. However, the adhesion hysteresis is a real dissipation mechanism, while here we have a transfer of energy into a macroscopic degree of freedom. The dissipation takes place as (uncontrolled) mechanical damping of the torsional excitation of the cantilever. This dissipation rate is much smaller than the damping rate shown here. But what happens to the energy? Using simplified models, we can show that after some time energy stored in the torsional mode is transfered back to the bending mode of the cantilever. The real dissipation rate is relatively low, except the torsional and normal frequencies meet certain conditions.

Beside this interesting new aspect, we also study the temperature dependence of the dissipation. Furthermore, we currently develop a theory for AM-AFM (in this technique, the cantilever is driven by a constant driving force and the phase shift between driving force and cantilever oscillation is evaluated) for magnetic tips on non-magnetic substrates as it is used in project C6. We have shown, that current theories on the measurement process do not explain the experimental observations very well.

Considering an atomic force microscope with magnetic tip and/or magnetic substrate, we have shown that current theories cannot explain the experimental observations satisfyingly. We use a two-pronged approach for clarification. First, we develop a new theory for the measurement process by means of classical continuum theory. Second, we are interested in the role of the spin system as a dissipation channel itself, neglecting other forms like phononic or electronic friction (which includes eddy currents). For this, we treat the spin-system by means of classical spin-models: the two-dimensional Ising-model and the three-dimensional Heisenberg-model with Gilbert dynamics.

Fig 3: Setup of Ising model.

Magnetic friction in the classical two-dimensional Ising-model on a square lattice with periodic boundary conditions is investigated by Monte-Carlo simulations. The system box is cut parallel to one axis in two halves. Relative motion of the two subsystems is now introduced by displacing one subsystem by one lattice constant in regular time intervals. If friction is present, this constant motion pumps energy into the system, which is then dissipated by coupling the system to a heat bath (e.g. Metropolis or Glauber dynamics). The effective dissipation rate can be determined by evaluating the net energy transfer to the heat bath per time.

Fig 4: Accumulated energy per spin which is transfered to the heat bath during a time interval t, without motion (blue lines) and with motion (red lines). The system energy E fluctuates around the same value in both cases.

We found, that the system develops a steady non-equilibrium state rather quickly (see figure 3). The dissipation rate appears to be constant from simulation start on. To understand the actual dissipation mechanism, we have to distinguish between the paramagnetic and the ferromagnetic phase. First, we note that the local correlations of spins residing on different subsystems are disturbed. Above the critical temperature the reduced spin correlation length corresponds to an effective temperature increase, which explains the energy flow into the cooler heat bath. Below the critical temperature the correlation length can be associated with the diameter of thermally activated minority clusters of spins pointing into the direction opposite to the spontaneous magnetization. The relative motion distorts minority clusters which extend across the two subsystems. The length of domain walls of the disturbed minority clusters is increased, or if two clusters are cut into two pieces, the total length of both now separated domain walls is bigger. This means, that additional domain wall energy is transfered into the heat bath which has been induced by the relative motion. Obviously, there exists no dissipation for T=0 (as there are no minority clusters) and infinite temperatures (as the spins are already uncorrelated). The implied maximum of dissipation can be found slightly above the critical temperature. The dissipation rate can be connected to a friction force (by dividing by v). Interestingly, the friction force is independent of the velocity (at least for not too high velocities).

Fig 5: System setup of Heisenberg model. Colors indicate the direction of the spins.

The Ising system describes the relative motion of two bodies. To come back to atomic force microscopy, we study a more complex system based on the Heisenberg model: spins are described by a three dimensional spin vector. The dynamics of the spins is given by the Landau-Lifshitz-Gilbert equation, which basically describes the precession of the spins. Additionally, the spins are coupled to a heat bath. Next neighbor spins are coupled by exchange interaction.

We consider a single spin as a tip model which moves with a constant velocity parallel to the surface of a substrate (see figure 5). Technically, the spin rests in the middle of the system, while the substrate is moved. We use open boundary conditions, which are specialized for the substrate faces in the moving direction: spins are removed and added there blockwise. The tip spin is coupled to all substrate spins via dipole-dipole-interaction. The dissipation rate can be determined by two ways: first by evaluating the energy pumped into the system by the constant motion of the tip spin, second by evaluating the dissipative part of the heat bath (simulations performed at T=0).

Fig 6: Dissipation rate vs. velocity of tip.

In contrast to the Ising model, we find a dissipation effect also for T=0. In figure 6 we show the relation between velocity of the tip and friction force (again given as the dissipated energy rate divided by velocity of the tip). For the Ising model, the friction force does not depend on the tip velocity, where here we find a rather complex dependency. For small velocities, the friction force depends linearly on the velocity. This indicates, that a different mechanism is responsible for the dissipation. For high velocities, the evaluation of the dissipation rate depends on the sketched methods. Evaluating the dissipation done by the heat bath does not lead to the same energy which is pumped into the system, although the total energy of the system remains constant. The remaining energy difference is removed from the system by removing spins due to the specialized boundary conditions.

Fig 7: Experimental setup: rotation and motion of the disc is recorded and evaluated. Independently of the initial rotational or translational velocity, both motions stop simultaneously.

Even though most of the everyday facts about dry macroscopic friction have been known since the works of Leonardo da Vinci, Amontons, Coulomb and Euler, the physics of static friction, especially at the onset to sliding, is not yet fully understood. This applies in particular to bodies undergoing a simultaneous translational and rotational motion which, on account of friction, exhibit nontrivial dynamics. In fact, the sliding friction of a circular disk is reduced if the contact is also spinning with relative angular velocity - a phenomenon which plays an important role in various games such as curling or ice hockey.

In both games, a small disc slides over the ice. Obviously, one would not call this dry friction. The Coulomb friction law must be replaced by a velocity-dependent law with some exponent α. For curling, experiments suggest a negative exponent α for high velocities. This would lead to a divergence of the friction force for v=0. For small velocities, the friction force must still be velocity-dependent, but with an positive exponent. Therefore, we have studied the effect of the exponent α on the motion of a double layer disc: the lower disc controls the area of contact to the surface (with radius R_c), while with the upper disc, the moment of inertia can be controlled independently. We evaluated the ratio of translational velocity to rotational velocity (ε=v/R_cω) by means of a simple analytical model. For Coulomb friction of a disc, this ratio tends towards a fixed value independently of the initial velocities (see figure 7).

Fig 8: Summary of parameter combinations and resulting final motion type.

Figure 8 summarize the results: only in the gray area, spinning and sliding stop simultaneously. For high moments of inertia, the spinning becomes dominant (asymptotically), while for small moments of inertia, sliding becomes dominant. In the fourth area, the dominance of sliding or spinning depends on the initial velocities.

The dynamics of rotating and sliding bodies under the action of dry friction forces is by now fairly well understood. Not so much can be said about the statics of bodies subject to a torque and a force. Here one important question is the minimal force and torque necessary to set the body into motion and how these are coupled. That there must be some coupling between them one knows from daily experience: if a heavy object is to be moved across the floor, it is easier to do so if one applies a torque while pushing it. Some previous analytical studies on disks showed that this is indeed the case but, contrary to the dynamical case, the static situation depends strongly on the model one uses for microscopic displacement: due to the inhomogeneity of the local displacement at each microcontact, some of them are subject to greater lateral stresses than others.

A new aspect of our studies is the effect of electromigration. While there already exist theories for the dynamics of two-dimensional islands, little is known for the three-dimensional case of polycrystalline wires. One reason is, that the electric field as well as the temperature cannot be regarded as constant, neither in space nor in time. As a result, the numerical treatment becomes rather time consuming. Therefore, our first step in this field is the development of a new simulation method, which is much faster than existing methods. We will use our experience made in previous projects dealing with the sinter process of nano-particles, where a similar gap between time scales has been successfully bridged.

This project is in close collaboration with projects C2, C4 and C6.