Electrically controllable $g$ tensors in quantum dot molecules

We present a quantitative theoretical analysis of electron, hole, and exciton $g$ tensors of vertically coupled InAs/GaAs quantum dot pairs in external electric and magnetic fields. For magnetic fields lying in the growth plane, we predict a giant electrically tunable anisotropy of hole $g$ factors that is introduced by piezoelectric charges. This effect allows bias controlled $g$ factor switching and single-spin manipulations in a static magnetic field. We use a relativistic eight-band $\mathbf{k}\cdot \mathbf{p}$ envelope function method including strain, which accounts for magnetic fields in a gauge-invariant manner. In a regime where the molecular wave functions form bonding and antibonding orbitals and for vertical magnetic fields, our results reproduce experimentally observed resonant enhancements of exciton $g$ factors without any fitting parameters.

Figure 1

Schematic cross section of vertically stacked (001)-grown InAs/GaAs double dot structure studied in this paper. We assume a height of $h=2.5\text{nm}$, a width of $w=15\text{nm}$, dot distances of $d=1.5$ or 2 nm, a wetting-layer (WL) thickness of 0.5 nm, and a realistic trumpet-shaped alloy profile within the dots (Ref. 31).

Figure 2

(Color online) Cross sections of calculated molecular electron and hole ground-state probability densities for a dot separation of $d=1.5\text{nm}$. The cross sections are taken at the dot center and run along the vertical [001] growth direction. In addition, we have applied a vertical electric field of $+30$ (solid lines) or $-30\text{kV}/\text{cm}$ (dashed lines) relative to the resonance field ${\mathbf{F}}_{\text{res}}$.

Figure 3

(Color online) $g$ factors of coupled quantum dots, for magnetic and electric fields lying in the vertical [001] direction. All electric field values in units of kV/cm are specified relative to the resonance field ${\mathbf{F}}_{\text{res}}$, as discussed in the main text. (a) Calculated hole $g$ factors for ground state (full lines) and excited state (dashed lines). The black lines show results for dot separations of $d=2\text{nm}$, while the red (gray) lines are for $d=1.5\text{nm}$, respectively. (b) Calculated electron $g$ factors for the ground state. Dot separations are taken as in (a). (c) Comparison of calculated neutral exciton $g$ factors with experimental results from Ref. 18 (circles). The excitons are formed by an electron in the ground state and a hole either in the ground (full line) or first-excited state (dashed line). The dot separation is $d=1.5\text{nm}$. The insets indicate schematically the probability density of the hole states for different electric field values.

Figure 4

(Color online) (a) Calculated molecular $g$ factors of hole ground state as a function of vertically applied electric field (relative to the resonance field ${\mathbf{F}}_{\text{res}}$) in kV/cm. The constant lateral magnetic field lies along the [110] (full line) and $\left[1\overline{1}0\right]$ direction (dashed line), respectively. (b) Same for first-excited hole state. For large magnitudes of the electric fields, the molecular hole states are localized predominantly in either the lower or the upper dot, as indicated in the figure.

Figure 5

(Color online) (a) Isosurface of calculated piezoelectric polarization charge densities of magnitude $\pm 1.5\times {10}^{19}{\text{cm}}^{-3}$ near the overgrown quantum dots. Positive and negative charges are indicated in blue (dark gray) and red (light gray), respectively. (b) Isosurface of probability density of molecular hole ground state for an applied bias of $+30$ and $-30\text{kV}/\text{cm}$, indicated in light blue (light gray) and dark blue (dark gray), respectively. The isosurface is chosen at 25% of the maximum density. The dot separation amounts to $d=1.5\text{nm}$. For these large positive and negative electric fields, the hole ground state is localized in the upper or the lower dot, respectively. (c) Cross sections of the calculated electrostatic potential that results from the piezoelectric charges for a dot separation of $d=24\text{nm}$. The cross sections are taken at half of the dot height. (d) Same as (c) for $d=1.5\text{nm}$.

Figure 6

(Color online) Calculated effective $g$ factors and directions of spin precession axes (arrows) of hole ground state (solid line) and first-excited state (dashed line) as a function of the vertical electric field in kV/cm. The constant magnetic field lies in the horizontal [100] direction.