Coupling of bonding and antibonding electron orbitals in double quantum dots by spin-orbit interaction

We perform a systematic exact diagonalization study of spin-orbit coupling effects for stationary few-electron states confined in quasi-two-dimensional double quantum dots. We describe the spin-orbit-interaction induced coupling between bonding and antibonding orbitals and its consequences for magneto-optical absorption spectrum. The spin-orbit coupling for odd electron numbers (one, three) opens avoided crossings between low energy excited levels of opposite spin orientation and opposite spatial parity. For two electrons the spin-orbit coupling allows for low-energy optical transitions that are otherwise forbidden by spin and parity selection rules. We demonstrate that the energies of optical transitions can be significantly increased by an in-plane electric field but only for odd electron numbers. Occupation of single-electron orbitals and effects of spin-orbit coupling on electron distribution between the dots are also discussed.

Figure 1

(Color online) The shades of gray (blue online) show the in-plane potential of a single dot ($V_b=0$—left column of plots) and of a double dot ($V_b=10\text{meV}$—central and right columns). In the right column of plots an in-plane electric field of $F_x=0.5\text{kV}/\text{cm}$ is included. Inside the light (darker) blue area the potential falls below $-5\text{meV}$ $\left(-45\text{meV}\right)$. The contours indicate the charge density for a single (top row), two electrons (middle row) and three electrons (lowest row of plots) for $B=0$.

Figure 2

(Color online) Lowest single-electron energy levels in function of the magnetic field for a single elongated dot $\left(V_b=0\right)$ (a) and for a double dot $\left(V_b=10\text{meV}\right)$ (b). Solid (dashed) lines correspond to the even (odd) $s$-parity. Black lines show the results without SO coupling. The blue curves show the results obtained with SO coupling for $F_z=0$ and the red curves in (a) for $F_z=188\text{kV}/\text{cm}$. The spin direction for the energy levels without SO coupling are marked with arrows. The thin vertical lines indicate allowed optical transitions from the ground state.

Figure 3

(Color online) Dashed curves show the potential confinement profile for $V_b=10\text{meV}$ calculated for $y=0$. (a) The spin-up components of the even $s$-parity ground state (red lines) and $s$-odd parity excited state (blue lines) in the absence of SO coupling (spin-down components exactly vanish). (b) Same as (a) but with SO coupling present. Spin-down components are presented in (c). The scale for the wave function is the same on all the plots, but in (c) the wave functions were multiplied by 4. At right (left) panels an electric field is $F_x=0.5\text{kV}/\text{cm}$ (zero).

Figure 4

(Color online) Contribution of even-parity orbitals (a) and average value of the $z$ component of the spin (b) (in $\hslash$ units) in the lowest-energy $s$-even and $s$-odd parity eigenstates.

Figure 5

(Color online) The dots show the low-energy absorption from the ground state at $F_x=0$ for the single dot (a) and for the coupled dots (b) (for the energy spectra see Fig. 2). The area of the dots is proportional to the absorption probability. The black dots show the results without SO coupling. The full blue dots correspond to SO coupling with $F_z=0$. The open blue circles in (b) correspond to height of the dot $d$ increased from 5.42 to 7.67 nm which amounts in a twofold reduction of the two-dimensional Dresselhaus constant. The red dots in (a) correspond to a strong linear Rashba coupling present $F_z=188\text{kV}/\text{cm}$. The dashed gray line indicates the Zeeman splitting $g{\mu }_{b}B$.

Figure 6

(Color online) Single-electron energy spectrum (a) and optical absorption spectrum (b) in function of the magnetic field for $F_x=0$ (black color), $F_x=0.2\text{kV}/\text{cm}$ (blue) and $F_x=0.5\text{kV}/\text{cm}$ (green) for coupled quantum dots.

Figure 7

(Color online) Two-electron energy spectrum for a single elongated dot $V_b=0$ (a) and for a couple of dots separated by $V_b=10\text{meV}$ barrier (b). Black (blue) lines show the results without (with) SO coupling. For the results without SO coupling we added labels $S$ for the singlet and $T$ for the triplets (subscript denotes the sign of the $z$ component of the total spin). Dashed (solid) curves correspond to odd (even) $s$-parity. Results were obtained for $F_x=0$.

Figure 8

(Color online) Two-electron ground-state absorption spectrum as a function of the magnetic field for SO coupled single dot (a,b) and double dot (c,d). Panels (a,c) correspond to $F_x=0$ and (b,d) to $F_x=1\text{kV}/\text{cm}$. The area of the dots is proportional to the absorption probability. Transitions are denoted by labels of two-electron spin eigenstates which are found without SO coupling. Without SO coupling all the transitions presented in this figure are forbidden.

Figure 9

(Color online) Electron charge localized in the left dot as a function of the electric field for the double dot at $B=0$. Results with and without SO coupling are not distinguishable.

Figure 10

(Color online) Two-electron energy spectrum of the double dot is shown in (a,c,e) as a function of the electric field. Plots (b,d,f) indicate the charge localized in the left dot. Black (blue) lines show the results without (with) SO coupling. The labels $S$ and $T_{+}$ correspond to eigenstates without SO coupling.

Figure 11

(Color online) Probability that both the electrons occupy the same dot with and without SO coupling in the ground state and first excited state.

Figure 12

(Color online) Occupation of the single-electron orbitals of definite spatial parity and spin for two-electron ground state with (b) and without (a) SO coupling for the electron pair in the double dot.

Figure 13

(Color online) Contributions of the two-electron orbitals to the ground state with (b) and without (a) SO coupling for the electron pair in the double dot.

Figure 14

(Color online) Three-electron energy spectrum for the single elongated dot. Black (blue) lines show the results without (with) SO coupling. Solid (dashed) lines correspond to even (odd) $s$-parity states. The arrows in the plot indicate the $z$ component of the spin without SO coupling.

Figure 15

(Color online) Three-electron energy spectrum (a) for the double dot. Black (color) lines show the results without (with) SO coupling. Solid (dashed) lines correspond to even (odd) $s$-parity. The short arrows in the plot indicate the $z$ component of the spin without SO coupling and the longer ones show the allowed optical transitions from the ground-state. (b) $z$ component of the spin for the lowest even $s$-parity and three lowest odd $s$-parity energy levels. Type and color of curves for these states is adopted of panel (a).

Figure 16

(Color online) Optical absorption spectrum for the three-electron system in the single dot (a) and in the double dot (b). Black (blue) dots correspond to SO coupling absent (present). Area of the dot is proportional to the absorption probability.

Figure 17

(Color online) (a) Three electron energy spectrum for SO-coupled double dot at $F_x=0$ (black dotted lines) and for $F_x=0.5\text{kV}/\text{cm}$ (blue solid curves). The arrow indicates the ground-state avoided crossing which is opened in presence of nonzero $F_x$. (b) Optical absorption spectrum for the SO-coupled double dot. Black (blue) dots correspond to $F_x=0\left(F_x=0.5\text{kV}/\text{cm}\right)$.

Figure 18

(Color online) Occupation of the single-electron orbitals of definite spin orientation and spatial parity without (a,b) and with (c,d) SO coupling for the lowest-energy $s$-even (a,c) and $s$-odd (b,d) state for three electrons in the double dot.

Figure 19

(Color online) Contributions of single-electron orbitals of a given symmetry to the lowest-energy three electron $s$-even (a,c) and $s$-odd (b,d) states, with (c,d) and without (a,b) SO coupling.

Figure 20

Energy spectra (a,c,e,g) and the absorption spectra (b,d,f,h) for a single (a–d) and two (e–h) electrons for a single elongated quantum dot with the lateral sizes increased twice $2R_x=180\text{nm}$, $2R_y=80\text{nm}$ with respect of the results presented above. In (a,b,e,h) we kept $\mu =10$ in Eq. (9) (as above) for which the lateral confinement profile is a square quantum well and in (c,d,g,h) a smoother profile with $\mu =3$ was adopted.