Micromagnetic Simulation of Thermal Effects in Magnetic Nanostructures
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Micromagnetic Simulation of Thermal Effects in Magnetic Nanostructures Rok Dittrich, Thomas Schrefl, Vassilios Tsiantos, Hermann Forster, Dieter Suess, Werner Scholz, and Josef Fidler Solid State Physics, Vienna University of Technology, Wiedner Haupstr. 8-10/138, A-1040 Vienna, Austria ABSTRACT A path finding method and a stochastic time integration scheme for the simulation of thermally activated magnetization processes are introduced. The minimum energy path and the saddle points for the thermally induced transitions between the ground states of NiFe magnetic nanoelements are calculated. INTRODUCTION With decreasing size of magnetic nanostructures thermal effects become increasingly important. Prominent examples are magnetization noise in magnetic sensor elements [1-3] and the thermal stability of magnetic MRAM (Magnetic Random Access Memory) cells [4] or magnetic storage media [5]. Magnetic sensors require a high sensitivity so that small magnetic fields can be detected. On the other hand thermal fluctuations which will lead to thermal noise should be suppressed. The free layer of a multilayer sensor element is soft magnetic and may have a size well below one micrometer. Thermally induced magnetization processes may cause local or global magnetization rotations which cause the magnetization noise. With decreasing lateral extension of the elements thermal fluctuations become more pronounced. Magnetic storage elements require a low and well defined switching field which in practice is limited by the current through the write line in an array of MRAM cells. On the other hand the shape or the induced anisotropy should guarantee a life time of a stored bit of about 10 years. Again the energy barrier for thermally induced magnetization reversal decreases with increasing size of the storage elements. The corresponding time scales differ by several orders of magnitude: Thermal noise arises on a time scale of a few nanoseconds; thermally induced switching of the magnetization over energy barriers extends over seconds to years. Random thermal fluctuations of the magnetization are the underlying physical process which causes both thermal noise and spontaneous switching. The stochastic fluctuations arise from the interplay between the lattice vibrations and the magnetization. A micromagnetic system will be close to a local minimum of the total magnetic Gibbs’ free energy. Thermal fluctuations of the magnetization cause the magnetization to wander around this minimum. Occasionally the system will reach a region next to a saddle point. The system may cross the energy barrier and move into the basin of attraction of a different energy minimum. This process can be described by the Neel-Brown theory [6,7]. The relaxation time, τ = f 0−1 exp( E b / k BT ) , is the inverse of probability per unit time for crossing the barrier Eb. The attempt frequency, f0, depends on material parameters, like anisotropy, particle shape, and damping [8]. Its value, which ranges from f0 = 109 Hz to f0 = 1012 Hz, sets the time scale for therm
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