Theory of quantum tunneling of the magnetization in magnetic particles

We study the response of the magnetization to a time-dependent applied magnetic field $H\left(t\right)\mathrm{}$ in a model for a uniaxial magnet. It is shown that a staircase structure in the magnetization curve results from Landau-Zener tunneling between different pairs of nearly-degenerate energy levels. This mechanism might be relevant to the analysis of the hysteresis of nanoscale magnets at low temperatures, allowing one to extract the energy splittings from the hysteresis curve. We investigate the dependence of the staircase structure on the sweep rate $\mathrm{dH}\left(t\right)/dt,$ and point out some universal features of the staircase in uniaxial magnets. We also study the effect of a step-wise (instead of continuous) increase in the field, and show that the size of the steps depends sensitively on the procedure used to change the applied field.