Tunnelling transport through a Mn$_{12}$ molecular magnet in an external magnetic field.

Recent single-electron-transistor experiments with Mn$_{12}$ as a central island exhibit a puzzling behaviour in the tunnelling conductance as a function of the external magnetic field, such as the lack of magnetic hysteresis. We describe the system in terms of a phenomenological giant spin model with two charge states: the $N$-electron state (neutral molecule) and the $(N+1)$-electron state (one extra electron added). The parameters of the model, such as the total spin and the magnetic anisotropy barrier, are calculated by state-of-the-art DFT. The addition of the tunnelling electron's spin to the giant spin is represented in terms of Schwinger bosons. We compute transport by means of a quantum rate equation in the sequential tunnelling regime, which is appropriate for experimental conditions. We find that the model cannot display hysteresis in the differential conductance as a function of the magnetic field when the coupling with the leads is the only source of relaxation. Coherence effects and cotunnelling are further analyzed by means of a master equation for the full density matrix. DFT calculations can also shed light on the influence of the tunnelling electron orbital degree of freedom on the tunnelling amplitudes.