  We investigate the exchange bias and training effects in the FM/AF
heterostructures using a unified Monte Carlo dynamical approach. This real
dynamical method has been proved reliable and effective in simulating dynamical
magnetization of nanoscale magnetic systems. The magnetization of the
uncompensated AF layer is still open after the first field cycling is finished.
Our simulated results show obvious shift of hysteresis loops (exchange bias)
and cycling dependence of exchange bias (training effect) when the temperature
is below 45 K. The exchange bias fields decrease with decreasing the cooling
rate or increasing the temperature and the number of the field cycling. With
the simulations, we show the exchange bias can be manipulated by controlling
the cooling rate, the distributive width of the anisotropy energy, or the
magnetic coupling constants. Essentially, these two effects can be explained on
the basis of the microscopical coexistence of both reversible and irreversible
moment reversals of the AF domains. Our simulated results are useful to really
understand the magnetization dynamics of such magnetic heterostructures. This
unified nonequilibrium dynamical method should be applicable to other exchange
bias systems.
