Limit cycles and chaos in the current through a quantum dot

We investigate nonlinear magnetotransport through a single-level quantum dot coupled to ferromagnetic leads, where the electron spin is coupled to a large, external (pseudo)spin via an anisotropic exchange interaction. We find regimes where the average current through the dot displays self-sustained oscillations that reflect the limit cycles and chaos and map the dependence of this behavior on magnetic field strength and the tunnel coupling to the external leads.

Figure 1

Scheme and setup of the investigated system. (a) An electron spin $\widehat{\mathbf{S}}$ (blue arrow) in a QD is coupled via the exchange interaction $\lambda$ with a large spin $\widehat{\mathbf{J}}$ (red arrow). The QD is attached to ferromagnetic electron reservoirs (brown regions), allowing electrons to tunnel through the QD. The large spin is isolated. (b) The spin-dependent rates of the contact barriers are chosen so that a spin-down electron is always trapped in the QD, while spin-up electrons can tunnel through it (see details in the text). The large spin interacts with the spin of the electron trapped in the QD, allowing its spin to flip and, hence, escape form the QD into the right lead.

Figure 2

(a) Parameter space with three regions describing the behavior of solutions of EOMs [see Eq. (6)]. Boundaries between the regions are obtained analytically from Eq. (13). Region I: damped oscillations; region II: both damped and self-sustained oscillations; and region III: self-sustained oscillations only. (b) Numerically obtained small-$\Gamma$ region in the mixed region II. In the dark-colored region, damped oscillations are obtained. In the light-colored region: self-sustained oscillations. Initial conditions: ${\langle {\widehat{S}}_x\rangle }_{t=0}=1/2$ and ${\langle {\widehat{S}}_y\rangle }_{t=0}={\langle {\widehat{S}}_z\rangle }_{t=0}=0$, ${\langle {\widehat{J}}_x\rangle }_{t=0}={\langle {\widehat{J}}_y\rangle }_{t=0}=j/2$ and ${\langle {\widehat{J}}_x\rangle }_{t=0}=j/\sqrt{2}$.

Figure 3

Time evolution in region I of (a) the electron spin components, (b) the large spin components, and (c) the current through the QD obtained by solving numerically the EOMs (6). In this region, the solutions exhibit a slow damped behavior. In the stationary limit, the large spin is completely polarized in the direction parallel to the external magnetic field, and a QD electron trapped in the spin-down state. Current is due only to tunneling through the spin-up level and, in the stationary limit, tends to a constant value of $1/2$. The parameters here are $B_z/\lambda =0.1$ and $\Gamma /\lambda =9$, with initial conditions ${\langle {\widehat{S}}_x\rangle }_{t=0}=1/2$, ${\langle {\widehat{S}}_y\rangle }_{t=0}={\langle {\widehat{S}}_z\rangle }_{t=0}=0$, ${\langle {\widehat{J}}_x\rangle }_{t=0}={\langle {\widehat{J}}_y\rangle }_{t=0}=(5/\sqrt{2})(\sqrt{5}-1)/2$, and ${\langle {\widehat{J}}_z\rangle }_{t=0}=(5/\sqrt{2})\sqrt{5+\sqrt{5}}$.

Figure 4

Results for three different parameter sets are shown, each of which gives rise to very different system behavior. Panels (a)–(c) show fast damping behavior [$B_z/\lambda =0.2$, $\Gamma /\lambda =0.7$, dark region in Fig. 2b]. In the long-time limit, the large spin is almost completely polarized in the direction perpendicular to the external field, but unlike in region I, the spin-down electron can escape from the QD into the right lead due to the interaction with the large spin. Panels (d)–(f) show periodic self-sustained oscillations [$B_z/\lambda =0.1$, $\Gamma /\lambda =0.16$, light region in Fig. 2b], which is a signature of limit cycles in phase space (see Fig. 6). Panels (g)–(i) show chaotic self-sustained oscillations [$B_z/\lambda =0.1$, $\Gamma /\lambda =0.015$, light region in Fig. 2b]. In the oscillatory cases, the oscillations are captured in the current through the QD, and in particular, the chaotic behavior is observed in the current [panel (i)]. The initial conditions are ${\langle {\widehat{S}}_x\rangle }_{t=0}=1/2$, ${\langle {\widehat{S}}_y\rangle }_{t=0}={\langle {\widehat{S}}_z\rangle }_{t=0}=0$, ${\langle {\widehat{J}}_x\rangle }_{t=0}={\langle {\widehat{J}}_y\rangle }_{t=0}=(5/\sqrt{2})(\sqrt{5}-1)/2$, and ${\langle {\widehat{J}}_z\rangle }_{t=0}=(5/\sqrt{2})\sqrt{5+\sqrt{5}}$.

Figure 5

Fourier spectra of the nondamped current time evolutions shown in Fig. 4 in the long-time limit. Panels (a) and (b) show the Fourier transform of Figs. 4f and 4i, respectively, where $\nu$ is the frequency. Panel (a) shows peaks at well defined frequencies, meaning that behavior of the current is periodic. However, panel (b) shows a uniform frequency distribution, which is a signature of chaotic dynamics.

Figure 6

Electron spin (left figures) and large spin (right) trajectories projected on a two-dimensional plane for the nondamped solutions in region II [light region in Fig. 2a]. Panels (a) and (b) show the formation of a limit cycle as seen in the time evolution plots, see Figs. 4d, 4e, 4f ($B_z/\lambda =0.1$ and $\Gamma /\lambda =0.16$). Panels (c) and (d) correspond to the time evolution plots of Figs. 4g, 4h, 4i, which suggests that the trajectories are chaotic ($B_z/\lambda =0.1$ and $\Gamma /\lambda =0.015$).

Figure 7

In this region, the solutions exhibit periodic self-sustained oscillations, which are reflected in the current. The corresponding limit cycles are shown in Fig. 8. The parameters chosen here are $B_z/\lambda =1.0$ and $\Gamma /\lambda =10$. The initial conditions are ${\langle {\widehat{S}}_y\rangle }_{t=0}=1/2$, ${\langle {\widehat{S}}_x\rangle }_{t=0}={\langle {\widehat{S}}_z\rangle }_{t=0}=0$, ${\langle {\widehat{J}}_x\rangle }_{t=0}={\langle {\widehat{J}}_y\rangle }_{t=0}=3\sqrt{11/2}$, and ${\langle {\widehat{J}}_z\rangle }_{t=0}=-1$.

Figure 8

Electron spin trajectories projected in the $\langle {\widehat{S}}_y\rangle$-$\langle {\widehat{S}}_z\rangle$ plane in region III. The solutions of the EOMs given in Eq. (6) are periodic self-sustained oscillations (see Fig. 7). Panels (a)–(f) show the different limit cycles obtained when varying the external magnetic field. $\Gamma /\lambda =1$.