Ultrafast manipulation of electron spins in a double quantum dot device: A real-time numerical and analytical study

We consider a double single-level quantum dot system with two embedded and nonaligned spin impurities to manipulate the magnitude and polarization of the electron-spin density. The device is attached to semi-infinite one-dimensional leads which are treated exactly. We provide a real-time description of the electron-spin dynamics when a sequence of ultrafast voltage pulses acts on the device. The numerical simulations are carried out using a spin-generalized modified version of a recently proposed algorithm for the time propagation of open systems [Kurth et al., Phys. Rev. B 72, 035308 (2005)]. Time-dependent spin accumulations and spin currents are calculated during the entire operating regime, which includes spin-injection and read-out processes. The full knowledge of the electron dynamics allows us to engineer the transient responses and improve the device performance. An approximate rate equation for the electron spin is also derived and used to discuss the numerical results.

Figure 1

(Color online) Schematic of the double quantum dot device coupled to leads. At $t<T_0=0$ the system is in equilibrium. At $T_0$ a spin bias $U_{L,\uparrow }$ is switched on and, simultaneously, the barrier between QD1 and the left lead is lowered. The injection of spin-up electrons ends at $T_1$ when the bias is turned off and the barrier is raised. Now the spin in QD1 rotates until $T_2$ when the barrier between the dots is lowered and the electron spin is transferred from QD1 to QD2. At $T_3$ the interdot barrier is raised again while the barrier between QD2 and the right lead is lowered. Tuning the electrochemical potential in the right lead $U_R$ to be in between the two levels of QD2, we measure a large (small) spin current if the electron spin is parallel (antiparallel) to the spin impurity.

Figure 2

(Color online) Ratio $S_{2,el}^z\left(t+T_2\right)/S_{1,el}\left(T_2\right)$ versus time $t$ and initial polarization $\theta$ for $J_1=J_2=0.1$ and $V_{\text{QD}}=0.2$.

Figure 3

(Color online) The three components of the electron spin in QD1 versus time for a sudden change in the hopping $V_L$ from $V^{\left(0\right)}=0.01$ to $V^{\left(1\right)}=0.5$ and a simultaneous sudden switching of the spin bias $U_{L,\uparrow }=0.7,0.9,1.1,1.3$. The equilibrium parameters are $J_1=J_2=0.1$ and $V_L=V_R=V_{\text{QD}}=V^{\left(0\right)}$ and the Fermi energy is ${\epsilon }_F=-0.96$.

Figure 4

(Color online) Comparison between the results obtained with one-dimensional (1D) leads and WBL leads for the electron spin in QD1 and $U_{L,\uparrow }=0.7,0.9$. The other parameters are the same as in Fig. 3.

Figure 5

(Color online) The three components of the electron spin in QD1 versus time for a sudden switching of the spin bias $U_{L,\uparrow }=1.3$ and a simultaneous sudden change of the hopping $V_L$ from $V^{\left(0\right)}=0.01$ to $V^{\left(1\right)}=0.1,0.2,0.3,0.4$. The other parameters are the same as in Fig. 3.

Figure 6

(Color online) Electron spin in QD1 and electron density at the left interface during the injection phase $\left(t<T_1\right)$ and the rotation phase $\left(t>T_1\right)$ when $T_1=10$. The equilibrium parameters are the same as in Fig. 5, i.e., $J_1=J_2=0.1$ and $V_L=V_R=V_{\text{QD}}=V^{\left(0\right)}$ and the Fermi energy is ${\epsilon }_F=-0.96$. For $0<t<T_1$ the system is perturbed by a spin bias $U_{L,\uparrow }=1.3$ and a larger hopping $V_L=V^{\left(1\right)}=0.2$. At $t=T_1$ the spin bias is switched off and the hopping $V_L$ is suddenly changed to $V^{\left(2\right)}=0.01$. (a) The three components of ${\vec{S}}_{1,el}$. (b) The trajectory of the projection of ${\vec{S}}_{1,el}$ onto the $yz$ plane. (c) Spin-up and -down densities on the first site of the left lead. The inset is a magnification of both densities $n_{L,\uparrow }$ and $n_{L\downarrow }$ for times $t>25$.

Figure 7

(Color online) (a) The three components of ${\vec{S}}_{1,el}$ and (b) the trajectory of the projection of ${\vec{S}}_{1,el}$ onto the $yz$ plane for the same system as in Fig. 6 except that the hopping parameter $V^{\left(2\right)}=0.06$ is six times larger. (c) and (d) Same as (a) and (b) but with the equilibrium parameter $J_2=0.02$ and hopping parameter $V^{\left(2\right)}=0.01$.

Figure 8

(Color online) The electron spin ${\vec{S}}_{2,el}$ in QD2 before $\left(t<T_2\right)$ and after $\left(t>T_2\right)$ the spin transfer phase when the interdot hopping is set to (a)–(c) $V_{\text{QD}}^{\left(1\right)}=0.2$ and (d)–(f) $V_{\text{QD}}^{\left(1\right)}=0.5$. The transfer phase starts at (a) and (d) $T_2\sim 36.5$, (b) and (e) $T_2\sim 40.4$, and (c) and (f) $T_2\sim 44.3$. All parameters before time $T_2$ are the same as in Fig. 6 except for the exchange coupling $J_2=0.05$.

Figure 9

(Color online) Electron spin in QD1 and QD2 for $T_2=36.5$ and (a)–(c) $V_{\text{QD}}^{\left(1\right)}=0.2$ and (d)–(f) $V_{\text{QD}}^{\left(1\right)}=0.5$. The other parameters are the same as in Fig. 8.

Figure 10

(Color online) (a) Spin-up density and (b) spin-down density in QD1 and QD2 (the inset shows a magnification of the time window $0<t<60$). The spin-polarized current at the right interface [$I_{R,\sigma }\left(t\right)$] is displayed in (c) in units of ${10}^{-3}$ while (d) shows ${\vec{S}}_{2,el}\left(t\right)$. For $t<T_3=39.32$ the system parameters and the perturbations are the same as in Fig. 9 with $V_{\text{QD}}^{\left(1\right)}=0.5$. For $t\ge T_3$ the interdot hopping is lowered to the value $V_{\text{QD}}^{\left(2\right)}=0.001$ and QD2 becomes a well-isolated system. At $t=60>T_3$ the hopping $V_R\left(t\right)$ between QD2 and lead $R$ is raised to $V^{\left(3\right)}=0.05$ and, simultaneously, a bias $U_{R,\uparrow }=U_{R,\downarrow }=0.96$ is switched on in the right lead. At this time the electrochemical potential in lead $R$ is ${\mu }_R={\epsilon }_F+0.96=0$ and lies in between the two spin levels of QD2.

Figure 11

(Color online) (a) Spin-up density and (b) spin-down density on QD1 and QD2. Spin-polarized current at the right interface ($I_{R,\sigma }$) in units of ${10}^{-3}$ (c) and ${\vec{S}}_{2,el}$ (d) . The insets of (a) and (b) show a magnification of the density in the time window $0<t<60$. Same system parameters and perturbations of Fig. 10 but $T_2=52.05$ and $T_3=54.84$.

Figure 12

(Color online) Discrete Fourier transform of the spin current $I_{\text{spin}}=I_{R,\downarrow }-I_{R,\uparrow }$ at the right interface for the parallel and antiparallel configurations of Fig. 10 and Fig. 11, respectively. The inset shows $I_{\text{spin}}$ in time domain with $t$ in the range of (65, 640), which has been used to perform the discrete Fourier transform. Both $I_{\text{spin}}\left(\omega \right)$ and $I_{\text{spin}}\left(t\right)$ are in units of ${10}^{-3}$.

Figure 13

(Color online) Spin current at the left and right interfaces for the parallel and antiparallel configurations of Fig. 10 and Fig. 11, respectively. For the parallel configuration $T_0=0$, $T_1=10.0$, $T_2=36.5$, and $T_3=39.32$, while for the antiparallel configuration $T_0=0$, $T_1=10.0$, $T_2=52.05$, and $T_3=54.84$.