3938 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 10, OCTOBER 2009 Magnetic Friction and the Role of Temperature Martin P. Magiera 1 , Dietrich E. Wolf 1 , Lothar Brendel 1 , and Ulrich Nowak 2 Department of Physics and CeNIDE, University of Duisburg-Essen, D-47048 Duisburg, Germany Department of Physics, University of Konstanz, D-78457 Konstanz, Germany A magnetic dipole moving parallel to a ferromagnetically interacting surface is subject to a friction force du kinetic energy into spin excitations. This phenomenon is studied in the framework of the classical anisotropic the stochastic Landau-Lifshitz-Gilbert equation. The friction force is calculated from dissipation rates, which energy functions. For small velocities, magnetic friction increases linearly (like Stokes’ law for laminar flow). and high-temperature behavior is analyzed and explained by a relaxation time ansatz. Index Terms—Friction, magnetic force microscopy, simulation. I. I NTRODUCTION T HE PHYSICS of friction is one of the oldest scientific problems studied. While on the macroscopic scale the phenomenology of friction is well-known, several new aspects are currently investigated on the micron and nanometer scale. This is made possible by continious advances in experimental techniques like atomic force microscopy. At the same time, magnetic materials, which nowadays can be controlled down to the nanometer scale, become more and more important. This development is stimulated e.g., by the demand of the data storage industry for miniaturized devices. Smaller devices become sensitive to thermal influences, which can lead to data loss in storage devices. Dissipative processes usually generate heat, so that friction in magnetic materials may gain in importance. While in the past friction effects in magnetic ma- terials have been often neglected, recently the contribution of magnetic degrees of freedom to energy dissipation processes has attracted increasing interest [1]–[6]. Our research focusses on dissipative phenomena, where relative motion leads to spin excitations whose energy ends up irreversibly in some heat bath. This is fundamentally different from the well known dissipation due to eddy currents induced in metals by a moving magnet: There energy is transferred via electron scattering into Joule heat (a mechanism used in eddy current microscopy, see e.g., [7]). By contrast, the present study considers the setup ofmagnetic force microscopy (MFM), where a magnetic tip is moved over a magnetic surface, both of them in the simplest cast insulating. Although recent studies have attempted to measure energy dissipation between an oscillating tip and a magnetic sample [8], [9], the occurrence of friction force due to a motion of the tip parallel to the surface has only been considered in a couple of papers starting with C. Fusco et al. [5], which has been extended by [6] to temperatures for weak spin anisotropy. The relative motion of the tip with respect to the surface can lead to the creation of spinwaves which propagate inside the sample and dissipate energy, giving rise to magnetic friction. Beside this tip-substrate interaction Manuscript received March 06, 2009; revised May 07, 2009. Current version published September 18, 2009. Corresponding author: M. P. Magiera (e-mail: m.magiera@uni-duisburg.de). Digital Object Identifier 10.1109/TMAG.2009.2023623 also the planar contact between two magnetic samples mo relative to each other has been discussed in the limit of st uniaxial spin anisotropy [4]. In this case magnetic friction be traced back to modified spin correlations across the sli plane due to the relative motion. While in the point-like co friction force is proportional to the velocity and is nonzero at , the planar contact with strong spin anisotropy lea a constant, velocity independent friction force, which vani at . These recent results make it clear that the physic magnetic friction holds unexpected and versatile phenome store, which have barely started to be explored. II. S IMULATION M ODEL AND F RICTION DEFINITION To simulate a magnetic substrate, we consider a two dim sional rigid lattice of classical normalized dipole moments , where denotes the material dependent magnetic saturation moment. We use open boundary cond The substrate moments, representing magnetic moments of single atoms, can change their orientation but not their ab value, so that there are two degrees of freedom at each la point. Two lattice constants above the substrate, a dipole with a fixed magnetic moment is positioned, which repres the scanning tip. It will become clear, when the friction force will be calculated that it is convenient thatone can simply separate the Hamiltonian of this system into an internal part and an interaction with the external tip without any dynam degrees of freedom of its own. The internal part includes the interaction inside the mag substrate, which is here described by the anisotropic Heis Hamiltonian (1) describes the ferromagentic exchange interaction between two nearest neighbours, expressed by the angular brackets . quantifies the anisotropy which prefers in-plane orientation of the spins. The magnetostatic interactions between the substrate sp are neglected in this work. This is reasonable, as former s have shown that the impact of the magnetostatic interactions on magnetic friction is negligibly small for the used system 0018-9464/$26.00 © 2009 IEEE