Quantum theory of energy dissipation in non-contact atomic force microscopy in Markovian approximation

Using the coarse-grained method of non-equilibrium statistical mechanics, we consider oscillations of a macroscopic tip in the non-contact atomic force microscopy (NC-AFM). Within our approach both the tip and the microscopic system (the surface and the nano-tip) are treated as one combined system. However, comparing to the previous approaches, in which the whole system was treated classically, we suggest a general method, which allows the microscopic subsystem to be treated quantum mechanically. Our method is based on a mixed quantum-classical representation of the statistical operator of the combined system. Using the Markovian approximation, we derive the Fokker–Plank equation and then the equation of motion for the classical tip and obtain the corresponding quantum analog for the friction term well known from the classical theory. The friction is expressed via a special correlation function of the fluctuating tip-surface force which in the classical limit goes over to the previous result. Then, using a simple model, we discuss temperature dependence of the dissipation energy within the “Brownian motion” mechanism.