Mechanism of exchange bias for isolated nanoparticles embedded in an antiferromagnetic matrix

Abstract

Single magnetic nanodots, exchange coupled to an antiferromagnetic (AF) matrix, can produce large exchange bias, while superparamagnetic behavior of the nanodots is suppressed. The exchange bias originates from the formation of a (quasi)spherical domain wall inside the AF matrix when the particle moment rotates under the influence of an external magnetic field. Micromagnetic calculations show that for isolated nanodots the energy of this domain wall increases nearly quadratically with the deflection angle of the nanodot moment. By introducing the corresponding quadratic energy term in a modified Stoner-Wohlfarth model, a two-parameter family of hysteresis loops is obtained, depending on scaled anisotropy energy and field direction. The loops are represented in a phase diagram with three main regions, containing (1) reversible loops, (2) irreversible loops with a metastable $180\ifmmode^\circ\else\textdegree\fi{}$ AF domain wall, and (3) loops with metastable AF domain walls with $360\ifmmode^\circ\else\textdegree\fi{}$ or higher rotation angles. According to this model, isolated nanodots display reversible negatively biased loops for all field directions, if their anisotropy energy is small in comparison to the AF domain-wall energy. For higher anisotropy, irreversible, mostly negatively biased, loops result from switching between the ground state and an higher-energy inverse state with a $180\ifmmode^\circ\else\textdegree\fi{}$ AF domain wall. At even higher anisotropy energy, the loops can show positive exchange bias after an initial ``training branch.'' Switching after ``training'' takes place between states having a $180\ifmmode^\circ\else\textdegree\fi{}$ and a $360\ifmmode^\circ\else\textdegree\fi{}$ AF domain wall, respectively. While for thin films, the bias field increases in inverse proportion to thickness, for nanodots it increases in inverse proportion to the square of particle diameter. Therefore, nanodots can show significantly larger exchange bias than thin films of similar dimension. Hysteresis loops, obtained from averaging over directions and sizes using the modified Stoner-Wohlfarth model, were compared to measurements from a natural sample with nanometer-scale ilmenite-exsolution lamellae in a hematite matrix. The shapes of the hysteresis difference, the difference between upper and lower branches, are similar for model and experiment, whereby increasing temperature in the measurement corresponds qualitatively to decreasing the relative anisotropy energy in the model.