Semiclassical spin-spin dynamics and feedback control in transport through a quantum dot

We present a theory of magnetotransport through an electronic orbital, where the electron spin interacts with a (sufficiently) large external spin via an exchange interaction. Using a semiclassical approximation, we derive a set of equations of motions for the electron density matrix and the mean value of the external spin that turns out to be highly nonlinear. The dissipation via the electronic leads is implemented in terms of a quantum master equation that is combined with the nonlinear terms of the spin-spin interaction. With an anisotropic exchange coupling a variety of dynamics is generated, such as self-sustained oscillations with parametric resonances or even chaotic behavior. Within our theory we can integrate a Maxwell-demon-like closed-loop feedback scheme that is capable of transporting particles against an applied bias voltage and that can be used to implement a spin filter to generate spin-dependent oscillating currents of opposite directions.

Figure 1

Setup of the investigated system. An electronic spin $\vec{\widehat{S}}$ (red) in a single quantum dot (SQD) is coupled to an external spin $\vec{\widehat{J}}$ (blue) via an exchange interaction $\lambda$ (wiggly line). The SQD level is split up by an external magnetic field $\vec{B}$ in the $z$ direction. Due to coupling to the leads $l$ (characterized by inverse temper-atures ${\beta }_l$ and chemical potentials ${\mu }_l)$, electronic transport is taking place on and off the SQD (solid arrows). The feedback mechanism, introduced in Sec. 2c, instantaneously modifies the tunneling barriers conditioned on the system states (dash-dotted arrows).

Figure 2

Plots (a) and (c) show the spin-dependent currents as functions of the applied bias voltage $V$ between the left and right transport leads, plotted for different feedback schemes and compared to the nonfeedback case. $\uparrow$ $(\downarrow )$ currents are depicted as solid (dashed) lines. Scheme B can act as a spin filter, as depicted in (c), where only $\uparrow$ electrons are transported against the bias. For scheme A, shown in (a), all electrons are transported against the bias, but $\uparrow$ currents are preferred. Plots (b) and (d) show a sketch of the SQD with the respective tunneling rates if the system is empty (black) or occupied by an $\uparrow$ (red) or $\downarrow$ (blue) electron. Thickness and directions of the red/blue tunneling arrows show which net currents are reached. The setup is sketched for fixed bias voltages, $V/\lambda =1$, shown by the dotted vertical lines in plots (a) and (c).

Figure 3

Parameter regions I, II, and III in the infinite-bias (IB) setup. Solid lines show the region-I-II-border [as a reference the borders for larger $j$ (dotted) in (a) or the nonfeedback case (dashed) in (b) are shown]. The thick dashed lines show the transition from region II to region III, which is determined by the critical magnetic field $B_c=\lambda /2$. In region I the system always runs into either of the fixed points ${\mathcal{P}}_{\mathrm{IB}}^{\pm }$ performing damped oscillations. If the magnetic field is stronger than $\lambda /2$ (region III) the system performs self-sustained oscillations, either according to the effective model (25) or around the polar fixed points ${\mathcal{P}}_{\mathrm{IB}}^{\pm }$. For region II, we plot exemplarily which dynamics the system exhibits with respect to parameters $B$ and $\Gamma$. We find not only damped oscillations towards the system's fixed points (marked by “$\times$”) and ongoing oscillations (“$+$”) but also quasiperiodic (“$\circ$”) or chaotic oscillations (“$\diamond$”). Plot (a) shows the typical behavior for the nonfeedback case. In plot (b) we show how and in which parameter regions the system's behavior is changed qualitatively if feedback is applied $\left(\delta /\lambda =1\right)$. The red labels show the regions in parameter space, where the dynamics are changed significantly. Labels 1–4 show the parameters for which we plotted solutions in Figs. 4 to 7.

Figure 4

(a)–(c) Electronic currents and (d) the dynamics of $\vec{\widehat{J}}$ for parameters from region III with respect to different feedback strengths $\delta$. The current becomes more sinusoidal while the frequency is decreased; the rotation of $\vec{\widehat{J}}$ remains in the $x\text{−}y$ plane. Parameters: $B/\lambda =0.6,\Gamma /\lambda =5,j/\lambda =10$.

Figure 5

(a)–(c) Electronic currents and (d) the dynamics of $\vec{\widehat{J}}$ for parameters from region III with respect to different feedback strengths $\delta$. The current oscillations become slower and turn to quasiperiodic oscillations. The oscillation of $\vec{\widehat{J}}$ is not planar anymore and the limit cycles become more complicated. Parameters: $B/\lambda =0.6,\Gamma /\lambda =1,j/\lambda =10$.

Figure 6

(a)–(c) Electronic currents and (d) the dynamics of $\vec{\widehat{J}}$ for parameters from region II with respect to different feedback strengths $\delta$. The evolution towards the fixed point ${\mathcal{P}}^y$ [with a corresponding current (28)] is changed into self-sustained oscillations via feedback. Parameters: $B/\lambda =0.05,\Gamma /\lambda =0.1,j/\lambda =10$.

Figure 7

(a)–(c) Electronic currents and (d) the dynamics of $\vec{\widehat{J}}$ for parameters from region II with respect to different feedback strengths $\delta$. Irregular oscillations are regularized for both electronic and large spin. Parameters: $B/\lambda =0.2,\Gamma /\lambda =0.1,j/\lambda =10$.

Figure 8

Comparison between solutions to the full set of Eqs. (8) and (9) (dashed lines) and solutions to the effective model (25) (solid lines) for different setups. Parameters (a) $B/\lambda =0.6,\Gamma /\lambda =5,j/\lambda =10,\delta /\lambda =1$ (red) and $\delta =0$ (black). Parameters (b) $B/\lambda =0.6,\Gamma /\lambda =1,j/\lambda =10,\delta =0$.

Figure 9

(a) The values of the $S_z$ component of the global fixed point ${\mathcal{P}}_{\mathrm{FB}}^{\pm }$ vs the bias voltage. The spin-dependent stationary currents after the system reaches the fixed point ${\mathcal{P}}_{\mathrm{FB}}^{+}$ (b) or ${\mathcal{P}}_{\mathrm{FB}}^{-}$ (c) are shown for feedback schemes A and B and the nonfeedback case.

Figure 10

Average spin-$\sigma$ currents through barrier $l$ for small negative biases and different feedback schemes. Without feedback (a) the currents flow according to the applied bias voltage from the right lead to the left lead. For feedback scheme B (c) an oscillating $\uparrow$ current against the bias is building up, and $\downarrow$ electrons are ejected with the bias, the device is acting as a spin filter. With the given parameters the two transport channels decouple for feedback scheme A (e). A little detuning of the bias voltage leads to the oscillating $\uparrow$ currents against the bias. $\downarrow$ electrons are either ejected to the left reservoir (red squares) or trapped (blue circles). Images (b), (d), and (f) show the SQD with the respective tunneling rates if the system state is empty (black), $\uparrow$ (red), or $\downarrow$ (blue). Thickness and direction of the arrows show which net currents are reached. Level transitions (spin flips) are shown in violet. In (f) the transport channel decouple, no spin flips occur, and the $\downarrow$ current vanishes (dashed lines). Parameters: $B/\lambda =0.1,\Gamma /\lambda =10,\delta /\lambda =10$, and $\left|V\right|/\lambda =20$.