Stick-Slip Motion in Spite of a Slippery Contact: Do We Get What We See in Atomic Friction?

Shortly after the invention of the atomic force micro- scope (AFM) this instrument has been applied (1 )a s a friction force microscope (FFM). This seemingly trivial extension of the AFM to the lateral direction has enjoyed great interest (2), since it is believed to provide direct atomic-scale access to the phenomenon of friction. Experiments with atomic resolution typically show peri- odic, sawtoothlike behavior of the lateral force known as stick-slip. The FFM tip is thought to be held periodically in lattice positions of the surface, which is easily modeled using a quasistatic approach, first proposed by Prandtl (3) and often referred to as the Tomlinson model (4 -6). An object (the tip) is considered to move in a periodic potential field formed by the substrate lattice, while being dragged along the surface by an external spring (the cantilever), which is at the same time used to measure the lateral force experienced. Generalizations of the model, from dynami- cal modeling to nonequilibrium statistical mechanics (7), have advanced our understanding of atomic-scale friction, e.g., its velocity dependence, transitions from stick-slip to other regimes, and the role of thermal effects (4,8-10). with thermal effects in the tip-apex-surface contact: under certain realistic conditions they can lead to substantial changes in the observed lateral forces. We interpret the atomically low values of the effective spring constants as the signature of a small group of atoms at the tip apex. Below we start with an estimate of the relevant characteristic frequency of a typical FFM tip apex, concluding it to be very high, of the order of several GHz or even larger. We further show that the low-frequency re- sponse of the cantilever provides a very indirect —aver- aged—view of the complex dynamics of the tip-apex motion. As a result, what the FFM measures can be very different from what the tip apex does. In particular, rapid thermal motion of the tip apex can lead to a very slippery contact with the surface. A strongly counterintuitive sce- nario—''stuck in slipperiness''—is predicted to apply under certain natural conditions. The cantilever can exhibit stick-slip motion, thus revealing a nonzero mean friction force, while the tip-surface contact is completely ''ther- mally lubricated'' by fast activated jumps of the tip apex, back and forth, between the surface potential wells. Importantly, in this case the apparent surface corrugation (derived from experimental data using the traditional one- spring interpretation) can be substantially smaller than the real one. Moreover, the transition from stick-slip motion to continuous, low-dissipative sliding, as predicted by the Tomlinson model and recently observed experimentally (5,6), can take place at a higher value of the relative surface corrugation than expected. We start with a simple description, in which the tip consists of N rigid atomic layers, and introduce spring constants ki characterizing tangential displacement be- tween layers i and i 1. For simplicity, we imagine only shearlike interlayer motion which makes ki proportional to the number of atoms in the layer and to the mass mi: The ratio k i