Time-dependent configuration-interaction simulations of spin swap in spin-orbit-coupled double quantum dots

We perform time-dependent simulations of spin exchange for an electron pair in laterally coupled quantum dots. The calculation is based on configuration interaction scheme accounting for spin-orbit (SO) coupling and electron-electron interaction in a numerically exact way. Noninteracting electrons exchange orientations of their spins in a manner that can be understood by interdot tunneling associated with spin precession in an effective SO magnetic field that results in anisotropy of the spin swap. The Coulomb interaction blocks the electron transfer between the dots but the spin transfer and spin precession due to SO coupling is still observed. The electron-electron interaction additionally induces an appearance of spin components in the direction of the effective SO magnetic field which are opposite in both dots. Simulations indicate that the isotropy of the spin swap is restored for equal Dresselhaus and Rashba constants and properly oriented dots.

Figure 1

(Color online) Spin-swap simulation for quantum dots placed on the $x$ axis. In the initial condition one electron is localized in the left dot $\left(x<0\right)$ with spin oriented antiparallel to the $x$ axis and the other electron is localized in the right dot $\left(x>0\right)$ with the spin parallel to the $x$ axis. Results were obtained for pure Dresselhaus coupling. (a) Black lines show the $x$ component of the spin stored in the left (dashed) and right (solid) quantum dots (Ref. 24). The circles indicate the results obtained without SO coupling. Average values of $y$ and $z$ components is zero with grey (red online) lines or without grey (red online) circles SO coupling. (b) Spin-left (left column) and spin-right (right column) densities in selected moments in time.

Figure 2

(Color online) Simulation similar to the one presented in Fig. 1 only for electron spins set initially parallel or antiparallel to the $z$ axis. In the left (right) column of plots the Coulomb interaction is included (neglected). [(a) and (b)] The average position of the spin-up density (black solid line). The electron-electron distance (dashed lines) in the $x$ (black), and $y$ (blue) directions, calculated as $\sqrt{⟨{\left(x_1-x_2\right)}^2⟩}$ and $\sqrt{⟨{\left(y_1-y_2\right)}^2⟩}$, respectively. [(c) and (d)] The spin components stored in the left (dashed curves) and right (solid curves) quantum dots (Ref. 24). Here and in all the other plots the $x$, $y$, and $z$ components of the spin are given by black, blue, and red lines, respectively. Circles in (d)—see text.

Figure 3

(Color online) Simulation for a single electron, pure Dresselhaus coupling and dots placed on the $x$ axis. The electron is initially localized in the left dot with spin-oriented antiparallel to the $x$, $y$, and $z$ axis in the left, central, and right columns of plots, respectively. [(a)–(c)] Average values of spin components are given by curves. The circles indicate the results obtained from the Bloch Eq. (4). [(d)–(f)] The spin stored in the left (dashed lines) and the right (solid lines) quantum dots. Black, blue, and red lines show the results for $x$, $y$, and $z$ components, respectively. [(g)–(i)] The average $x$ position of the electron packet.

Figure 4

(Color online) (a) Same as Fig. 2c. (b) Same as Fig. 2d only for electron spins initially antiparallel in the $y$ direction.

Figure 5

(Color online) Simulation for pure Rashba coupling and centers of dots placed on the $x$ axis. The electrons in the left and right dots initially possess opposite spins in [(a) and (b)] $x$, [(c) and (d)] $y$, [(e) and (f)] and $z$ directions. Black, blue, and red curves show the $x$, $y$, and $z$ spin components stored in the left (dashed curve) and the right (solid curve) dots. Plots (b), (d ), and (f) were obtained for neglected electron-electron interaction which is included in (a), (c), and (e).

Figure 6

(Color online) Simulation for both SO coupling types present $\alpha =\beta /2=5.4\text{meV}\text{nm}$ and centers of dots placed on the $x$ direction. The electrons in the left and right dots initially possess opposite spins along $z$ ([001]) direction in (a) and (b), $x/2+y$ ([1,2,0]) in (c) and (d), and $x-y/2$ $\left(\left[2,\overline{1},0\right]\right)$ in (e) and (f). Purple, green, and red curves show the [1,2,0], $\left[2,\overline{1},0\right]$, and [0,0,1] spin components stored in the left (dashed curve) and the right (solid curve) dots. Plots (b), (d), and (f) were obtained for neglected electron-electron interaction which is included in (a), (c), and (e).

Figure 7

(Color online) [(a) and (b)] Same as Figs. 2c, 2d but for the basis limited to four lowest energy two-electron states. [(c) and (d)] Electron-electron distance $\left({⟨{\left(x_1-x_2\right)}^2⟩}^{1/2}\right)$ as calculated for the wave function given by Eq. (4) (black line) and integrated using Eq. (7) (red crosses) starting from the initial condition. [(e) and (f)] Real parts of right-hand side terms of Eq. (7).

Figure 8

(Color online) Probability densities for the components of the two-electron wave functions corresponding to opposite spin orientations plotted on $x_1,x_2$ plane along the axis of the system $y_1=y_2=0$ for chosen moments in time during the spin swap. Parameters are the same as in Fig. 7. The spins are initially oriented parallel and antiparallel to the $z$ axis for (a) noninteracting and (b) interacting electrons. The spin-swap time is $t_s=1.48\text{ps}$ for (a) noninteracting electrons and $t_s=10.9\text{ps}$ for (b) the interacting ones.

Figure 9

(Color online) Spin stored in separate dots for two interacting electrons with equal linear Dresselhaus and Rashba constants $\alpha =\beta =10.8\text{meV}\text{nm}$. The first and second rows of plots correspond to different alignments of the double dot as shown schematically at the right end of the figure. In (a) and (e), and (b) and (d) the spins are initially set parallel or antiparallel to the $x$ and $z$ axes, respectively. In (c) and (g) the spins are initially set in the $x+y$ (i.e., [110]) direction—within the plane of confinement and perpendicular to ${\mathbf{B}}_{\text{SO}}$ direction $\left(\left[1\overline{1}0\right]\right)$. In (d) and (h) the spins are initially aligned with ${\mathbf{B}}_{\text{SO}}$ vector. The dashed (solid) lines show the results for the left (right) dot. Black, blue, red, purple, and green colors correspond to $x$, $y$, $z$, $x+y$, and $x-y$ components of the spin, respectively.

Figure 10

(Color online) Results of simulations for the pure Dresselhaus coupling and dots placed on the $x$ axis for the interacting electrons with spins initially oriented antiparallel in the $z$ direction. Black, blue, and red curves show the spin components stored in the left (dashed curve) and the right (solid curve) dots. The green curve at the top of the plot is the average electron-electron distance that is referred to the right axis. The results presented in (a), (b), (c), and (d) correspond to the two-electron basis containing $m=4$, 10, 50, and 325 lowest energy eigenstates of Hamiltonian (2).

Figure 11

(Color online) Same as Fig. 10d only for interdot barrier height increased from $V_b=10\text{meV}$ to $V_b=50\text{meV}$.