Spin-flip scattering in time-dependent transport through a quantum dot: Enhanced spin-current and inverse tunneling magnetoresistance

We study the effects of spin-flip scatterings on the time-dependent transport properties through a magnetic quantum dot attached to normal and ferromagnetic leads. The transient spin dynamics as well as the steady-state tunneling magnetoresistance (TMR) of the system are investigated. The absence of a definite spin-quantization axis requires the time propagation of two-component spinors. We present numerical results in which the electrodes are treated both as one-dimensional tight-binding wires and in the wide-band limit approximation. In the latter case we derive a transparent analytic formula for the spin-resolved current, and transient oscillations damped over different time scales are identified. We also show that in the presence of spin-flip scatterings the TMR can be inverted even for symmetrically coupled leads. For any given strength of the spin-flip coupling the TMR becomes negative provided the ferromagnetic polarization is larger than some critical value. Finally we show how the full knowledge of the transient response allows for enhancing the spin current by properly tuning the period of a pulsed bias.

Figure 1

(Color online) Schematic illustration of the model Hamiltonian. The lead $L$ with majority-spin $\uparrow$ electrons and the lead $R$ with majority-spin $\uparrow$ $(\downarrow )$ electrons in the P (AP) configurations are separated from the QD by tunnel barriers, which are accounted for with a renormalized hopping $V$. The energy spin splitting is $\Delta \epsilon$ in the leads and $2E_z$ in the QD. Spin-flip scatterings of strength $V_{\text{sf}}$ occur in the QD.

Figure 2

$I_{\text{tot}}\left(t\right)$ and $I_{\text{tot}}\left(\omega \right)$ for $V_{\text{sf}}=0$ [panels (a)-(b)], $V_{\text{sf}}=10$ [panels (c)-(d)], $V_{\text{sf}}=20$ [panels (e)-(f)]. The discrete Fourier transform is calculated using 50 equidistant points of $I_{\text{tot}}\left(t\right)-I_{\text{tot}}\left(\infty \right)$ with $t$ in the range (0.5–1). The rest of the parameters are $U_L=0$, $U_R=200$, ${\epsilon }_{d\sigma }=\theta \left(t\right)U_R/2+\sigma E_z/2$ (with $E_z=10$), and inverse temperature $\beta =100$.

Figure 3

(Color online) $I_{\text{spin}}\left(t\right)$ for $V_{\text{sf}}=0,10,20$ and the same parameters as in Fig. 2.

Figure 4

(Color online) Contour plot of the TMR at the steady state in units of ${10}^{-2}$ as a function of $U$ and $p$ for different values of $V_{\text{sf}}=0.3$ (top left), 0.33 (top right), 0.4 (bottom left), and 0.47 (bottom right). The boundary $\text{TMR}=0$ is displayed with a black line. The remaining parameters are ${\Gamma }_0=0.5$, ${\epsilon }_{d\sigma }=0$, and $\beta =100$.

Figure 5

(Color online) Left panel: in the region ${\Gamma }_0^2<2V_{\text{sf}}^2$ the TMR is always negative while in the region ${\Gamma }_0^2>2V_{\text{sf}}^2$ the TMR changes sign depending on the value of the polarization $p$. Right panel: TMR as a function of $p$ for different values of $V_{\text{sf}}$.

Figure 6

(Color online) $I_{\uparrow }\left(t\right)$ at zero temperature for $N=700$, $U_L=0.5$, $U_R=0$, $E_F=0$ $\left(T_{\text{max}}\approx 700\right)$, and $E_F=-1.7$ $\left(T_{\text{max}}\approx 870\right)$. The rest of the parameters are ${\epsilon }_{d\sigma }=0$, $V=-1$, and $V_{\text{sf}}=0$. The current reaches a well-defined steady state before the occurrence of finite-size effect $\left(t\gtrsim T_{\text{max}}\right)$.

Figure 7

$I_{\text{tot}}\left(t\right)$ and $I_{\text{tot}}\left(\omega \right)$ for $V_{\text{sf}}=0$ [panels (a)-(b)], 0.05 [panels (c)-(d)], and 0.1 [panels (e)-(f)] obtained with a bias $U_L=0.7$ and $U_R=0$ applied to leads of length $N=450$. The discrete Fourier transform is calculated using 280 equidistant points of $I_{\text{tot}}\left(t\right)-I_{\text{tot}}\left(\infty \right)$ with $t$ in the range (70–350). The rest of the parameters are $V=0.03$, ${\epsilon }_{d\sigma }=\theta \left(t\right)U_L/2+\sigma E_z/2$ (with $E_z=0.04$), and zero temperature.

Figure 8

(Color online) $I_{\text{spin}}\left(t\right)$ for the same system parameters of Fig. 7.

Figure 9

$I_{\text{tot}}\left(t\right)$ in units of ${10}^{-6}$ for $U_R=0.4$ and $U_L=0$ applied to leads of length $N=250$. The inset shows $I_{\text{tot}}\left(\omega \right)$ in units of ${10}^{-5}$ calculated using 138 equidistant points of $I_{\text{tot}}\left(t\right)-I_{\text{tot}}\left(\infty \right)$ with $t$ in the range (12–150). The rest of the parameters are $E_z=0$, $E_F=-1.8$, ${\epsilon }_d=-0.5$, $V=0.02$, $V_{\text{sf}}=0$, and zero temperature.

Figure 10

(Color online) Currents in units of ${10}^{-2}$ for $U_L=-U_R=1$ applied to leads of length $N=120$ for different values of $E_z=0.1$ [panels (a)-(b)], 0.6 [panels (c)-(d)], and 2 [panels (e)-(f)]. The rest of the parameters are $V=0.2$, $V_{\text{sf}}=0$, and zero temperature.

Figure 11

(Color online) $I_{\text{tot}}\left(t\right)$ and $I_{\text{spin}}\left(t\right)$ in units of ${10}^{-2}$ for a pulsed bias of period $T=6$ (top left panel) and $T=8$ (bottom left panel) of amplitude $U_L=-U_R=1$ applied to leads of length $N=120$. The polarization ratio $r\left(t\right)$ is displayed in the right panel. The rest of the parameters are the same as in Fig. 10. The dotted lines in the left panel denote the pulsed bias in the left lead in units of ${10}^{-1}$. The numerical calculations have been performed with $\Delta t=0.1$.

Figure 12

(Color online) $I_{\uparrow /\downarrow }\left(t\right)$ in units of ${10}^{-6}$ [panel (a)] and $n_{\uparrow /\downarrow }\left(t\right)-0.5$ in units of ${10}^{-5}$ [panel (b)]. The insets show the discrete Fourier transform $I_{\text{tot}}\left(\omega \right)$ in units of ${10}^{-6}$ [panel (a)] and $n_{\text{tot}}\left(\omega \right)$ in units of ${10}^{-5}$ [panel (b)] calculated using 207 equidistant points of $I_{\text{tot}}\left(t\right)$ and $n_{\text{tot}}\left(t\right)-1$ with $t$ in the range (10–94). The rest of the parameters are $U_L=0.1$, $U_R=0$, $N=350$, ${\epsilon }_{d\sigma }=\theta \left(t\right)U_L/2+\sigma E_z/2$ (with $E_z=0$), $V=0.01$, $V_{\text{sf}}=0.5$, and zero temperature.

Figure 13

(Color online) Contour plot of the TMR at the steady state in units of ${10}^{-2}$ as a function of $U$ and $\Delta \epsilon$ for different values of $V_{\text{sf}}=0$ (a), 0.1 (b) and 0.2 (c). The boundary $\text{TMR}=0$ is displayed with a black line. The remaining parameters are $V=0.25$, ${\epsilon }_{d\sigma }=0$, $\beta =100$, and the length of the leads is $N=100$.