Current-induced switching in transport through anisotropic magnetic molecules

Anisotropic single-molecule magnets may be thought of as molecular switches, with possible applications to molecular spintronics. In this paper, we consider current-induced switching in single-molecule junctions containing an anisotropic magnetic molecule. We assume that the carriers interact with the magnetic molecule through the exchange interaction and focus on the regime of high currents in which the molecular spin dynamics is slow compared to the time which the electrons spend on the molecule. In this limit, the molecular spin obeys a nonequilibrium Langevin equation which takes the form of a generalized Landau-Lifshitz-Gilbert equation and which we derive microscopically by means of a nonequilibrium Born-Oppenheimer approximation. We exploit this Langevin equation to identify the relevant switching mechanisms and to derive the current-induced switching rates. As a by-product, we also derive $S$-matrix expressions for the various torques entering into the Landau-Lifshitz-Gilbert equation which generalize previous expressions in the literature to nonequilibrium situations.

Figure 1

Illustration of the setup. The electronic level splits due to the effective magnetic field, given by $b=b\left(\vartheta \right)$. The number of levels in the current window depends on $b$, the applied bias voltage $eV={\mu }_L-{\mu }_R$, and the gate voltage $e{V}_{\mathrm{g}}=({\mu }_L+{\mu }_R)/2-\epsilon$. It changes from zero ($eV/2>e{V}_{\mathrm{g}}\pm b$) to one ($e{V}_{\mathrm{g}}+b>eV/2>e{V}_{\mathrm{g}}-b$) to two ($e{V}_{\mathrm{g}}\pm b>eV/2$) (assuming $eV,e{V}_{\mathrm{g}}>0$).

Figure 2

Component of the average current-induced torque in a uniaxial situation for unpolarized leads. $J{s}_z\left(\mathbf{M}\right)$ is plotted as a function of the applied bias voltage for different gate voltages $e{V}_{\mathrm{g}}$. As discussed in the text, $J{s}_z$ changes when the number of levels in the current window varies at $V_{\mathrm{g}}\pm eV/2=\pm b$ (see also Fig. 1). The plots are obtained at zero temperature at the potential minimum $\vartheta =0$ for the values $JM/2=0.2$, $\Gamma =0.1$, $g_{e}B/2=0.002$, and $g_e=g_d$. The precession frequency in the absence of coupling to electron spin and magnetic field is ${\omega }_0=2DM=0.01$. All energies are measured in units of the barrier height without magnetic field $D{M}^2$.

Figure 3

Damping and fluctuations as a function of the applied bias voltage. (a) The damping coefficient ${\gamma }_{s,1}$ is plotted for three different gate voltages. It is peaked when the number of levels in the current region changes from zero to one to two at $V_{\mathrm{g}}\pm eV/2=\pm b$ (see also Fig. 1). (b) At these points the fluctuation kernel ${\tilde{D}}_1$ increase steplike. The level broadening results from the interaction with the leads encapsulated in $\Gamma$. (c) The effective temperature $T_{\mathrm{eff}}={\tilde{D}}_1/\left(2{\gamma }_{s,1}\right)$ is shown as a function of the bias voltage. The plots are obtained at the potential minimum $\vartheta =0$ with the same parameters as in Fig. 2. All energies are measured in units of the barrier height without magnetic field $D{M}^2$.

Figure 4

Potential experienced by the molecular spin in the absence of coupling to the electrons. $U\left(\vartheta \right)$ is shown for different magnetic fields. All energies are measured in units of the barrier height without magnetic field $D{M}^2$.

Figure 5

Switching dynamics for unpolarized leads. (a) $\tilde{V}$ is plotted for different bias and gate voltages. (b) The switching rate $1/{\tau }_{12}$ is plotted as a function of the applied bias voltage for different gate voltages. (c) The ratio between the switching rates $1/{\tau }_{21}$ and $1/{\tau }_{12}$ is shown. The plots are obtained with the same parameters as in Fig. 2. All energies are measured in units of the barrier height without magnetic field $D{M}^2$.

Figure 6

Current-voltage characteristics. (a) The differential conductance $dI/dV$ is peaked when the number of levels in the current window changes at $V_{\mathrm{g}}\pm eV/2=\pm b$. (b) Obviously, the current changes when the number of level with energy between the chemical potentials of the two leads changes. (c) The level splitting $2b\left(\vartheta \right)$ depends on the orientation of the molecular spin. The relative difference of the current evaluated at the two potential minima is plotted as a function of the bias voltage. [(a) and (b) are evaluated at $\vartheta =0$. The plots are obtained with the same parameters as in Fig. 2 and all energies are measured in units of the barrier height without magnetic field $D{M}^2$.]

Figure 7

Sketch of the effect of polarized leads inducing spin-torque-transfer (indicated by the magenta arrow) on the molecular moment. Depending on the polarization and the current, the spin-torque tends to align the molecular moment either (a) along or (b) against the magnetic field.

Figure 8

Motion of the molecular moment in the presence of spin-polarized leads. For negative bias voltage $eV=-0.4$, the magnetic moment is driven toward the positive $z$ axis (red curve) while inverting the voltage $eV\to -eV$ causes a flip of the magnetic moment (blue curve). (We consider ${\Gamma }_L^{\uparrow }={\Gamma }_R^{\downarrow }=0.1$, ${\Gamma }_L^{\downarrow }={\Gamma }_R^{\uparrow }=0$, and $e{V}_{\mathrm{g}}=0$; the other parameters are the same as in Fig. 2.)