Colossal spin fluctuations in a molecular quantum dot magnet with ferromagnetic electrodes

We study electronic transport through a magnetic molecule with an intrinsic spin $S$ coupled to two magnetic electrodes in the incoherent regime. The molecule is modeled as a single resonant level with large Coulomb repulsion (no double occupancy). The molecular spin is isotropic and it interacts with the electronic spin through an exchange interaction. Using an alternative method to the usual master-equation approach, we are able to obtain analytical formulas for various physical quantities of interest, such as the mean current and the current fluctuations, and also the mean value of $J_z$—the $z$ component of the total spin on the molecule—and its fluctuations. This allows us to understand how the electronic current between the magnetized electrodes can control the polarization of the molecular spin. We observe in particular that the fluctuations of $J_z$ reach unexpectedly high values.

Figure 1

(Color online) Energy diagram of the system: because of the exchange interaction between the molecular spin $S$ and the electron spin, the occupied dot level is split between the $J=S-\frac{1}{2}$ levels [with $J_z$ ranging from $-\left(S-\frac{1}{2}\right)$ to $\left(S-\frac{1}{2}\right)$] and the $J=S+\frac{1}{2}$ levels (with $J_z$ ranging from $-S-\frac{1}{2}$ to $S+\frac{1}{2}$), with a splitting $\sim J_{\text{ex}}S$. We chose the chemical potential of the electrodes such that the $J=S+\frac{1}{2}$ levels only are in the bias window. The inset shows a schematic view of a possible experimental realization: a magnetic atom with spin $S$ trapped inside a ${\text{C}}_{60}$ molecule, which is placed between two electrodes.

Figure 2

Construction of all the possible sequences—type A and type $B+\text{extensions}$. A sequence starts and ends in the reference state, $Q=0,S_z=S$. Sequence A (solid line) corresponds to the tunneling of a spin-up electron from the left to the right lead. Sequence B (dashed line) to the tunneling of a spin-down electron from left to right lead. Sequence B can be extended by attaching subsequences starting and ending from the state $Q=1,J_z=S-1/2$ (thin dashed lines), forming longer sequences where several electrons are transmitted and where the molecular spin goes through intermediate states with $S_z<S$ and $J_z<S-1/2$.

Figure 3

(Color online) The mean current $\overline{I}$ (top panel) and its fluctuations $S_{II}$ (bottom panel) for the case of a molecular spin $S=1/2$ for the four different cases of electrodes polarizations. The inset in the bottom panel shows the behavior of $S_{II}$ in the parallel case $\left(P\right)$ on a larger scale.

Figure 4

(Color online) The mean value of the $z$ component of the molecular spin, $J_z$ (top panel) and its flucutations $S_{J_z{J}_z}$ (bottom panel) for the case of a molecular spin $S=1/2$, for the four different cases of electrode polarizations.

Figure 5

(Color online) The mean current $\overline{I}$ (top panel) and its fluctuations $S_{II}$ (bottom panel), in the antiparallel configuration of the electrodes, for the values of the spin $S=1/2$, $S=1$, and $S=3/2$ as a function of the electrode polarizations $p$. When increasing the spin $S$, the curves converge toward a “classical” curve (dotted line) corresponding to a fixed spin.

Figure 6

(Color online) The normalized mean value of $J_z$, $J_z/\left(S+1/2\right)$ (top panel), and its normalized fluctuations $S_{J_z{J}_z}/{\left(2S+1\right)}^2$ (bottom panel), in the anti-parallel configuration of the electrodes, for the values of the spin $S=1/2$, $S=1$, and $S=3/2$ as a function of the electrodes polarizations $p$.

Figure 7

(Color online) The third cumulant of the current $C_3\left(I\right)$ (top) and the fourth cumulant of the current $C_4\left(I\right)$ (bottom) as a function of the electrodes polarizations $p$ for the four different cases of electrodes polarizations. The inset on the bottom panel shows the curves on a larger scale.