ac- and dc-driven noise and $I\text{−}V$ characteristics of magnetic nanostructures

We study a structure consisting of a ferromagnetic (F) layer coupled to two normal metal (N) leads. The system is driven out of equilibrium by the simultaneous application of external dc and ac voltages across the N/F/N structure. Using the Keldysh diagrammatic approach, and modeling the ferromagnet as a classical spin of size $\mathbf{S}⪢1$, we derive the Langevin equation for the magnetization dynamics and calculate the noise correlator. We find that the noise has an explicit frequency dependence in addition to depending on the characteristics of the ac and dc drive. Further, we calculate the current-voltage characteristics of the structure to $\mathcal{O}\left(1/S^2\right)$ and find that the nonequilibrium dynamics of the ferromagnetic layer gives rise to corrections to the current that are both linear and nonlinear in voltage.

Figure 1

(Color online) A sketch of the N/F/N structure. The left and right normal-metal leads are coupled through tunnel barriers to the ferromagnetic layer.

Figure 2

(Color online) Distribution function of electrons in the magnetic layer $f_{neq}\left(\omega \right)$ given by Eq. (11): (a) pure ac voltage case, ${\mu }_{\text{L}}={\mu }_{\text{R}}=0$ and $V_{ac}/{\omega }_0=0.5$; (b) the case of nonzero ac and dc voltages, the parameters are ${\Gamma }_{\text{L}}={\Gamma }_{\text{R}}$, ${\mu }_{\text{L}}=-{\mu }_{\text{R}}=1.5{\omega }_0$, and $V_{ac}/{\omega }_0=0.5$. Frequency $\omega$ is measured in units of ${\omega }_0$.

Figure 3

(Color online) The diagrams for one-loop corrections to the electron Green’s function in the magnetic layer due to coupling to spin fluctuations: (a) exchange (Fock) contribution ${\Sigma }_{\sigma d}^{eR}$ and (b) Hartree contribution ${\Sigma }_{\sigma d}^{hR}$. Wavy lines correspond to spin-spin correlators $⟨S_{-}\left(\omega \right)S_{+}\left(-\omega \right)⟩$.