Hole spins in an InAs/GaAs quantum dot molecule subject to lateral electric fields

There has been tremendous progress in manipulating electron and hole-spin states in quantum dots or quantum dot molecules (QDMs) with growth-direction (vertical) electric fields and optical excitations. However, the response of carriers in QDMs to an in-plane (lateral) electric field remains largely unexplored. We computationally explore spin-mixing interactions in the molecular states of single holes confined in vertically stacked InAs/GaAs QDMs using atomistic tight-binding simulations. We systematically investigate QDMs with different geometric structure parameters and local piezoelectric fields. We observe both a relatively large Stark shift and a change in the Zeeman splitting as the magnitude of the lateral electric field increases. Most importantly, we observe that lateral electric fields induce hole-spin mixing with a magnitude that increases with increasing lateral electric field over a moderate range. These results suggest that applied lateral electric fields could be used to fine tune and manipulate, in situ, the energy levels and spin properties of single holes confined in QDMs.

Figure 1

(a) Disk-shaped QDMs without (symmetric) and with (asymmetric) lateral offset. The four lowest-energy hole states as a function of vertical ($z$ direction) electric field in a (b) symmetric QDM for $B_z=12$ T and (c) asymmetric QDM with $\mathrm{\Delta }x=4$ nm and $B_z=6$ T.

Figure 2

Effect of a lateral ($x$ direction) electric field on a symmetric QDM under a constant 12 T magnetic field in the absence of piezoelectric fields. (a) Energy and (b) Zeeman splitting of the four lowest-energy hole states as a function of lateral electric field when $F_z=3.8$ kV/cm. Spatial distribution of a hole wave function in (c) $y=0$ plane and (d) along $z$ axis with different lateral electric fields as derived from the full tight-binding results. (e) Spin-mixing anticrossing splitting $\left({\mathrm{\Delta }}_{\mathrm{sm}}\right)$ and spin-mixing resonance vertical electric field $\left(F_{\mathrm{sm}}\right)$ as a function of lateral electric field.

Figure 3

Effect of lateral electric field on an asymmetric QDM for $B=12$ T, in the absence of piezoelectric field. (a) Energy level of the lowest four hole states for $B=12$ T and $F_z=3$ kV/cm, as a function of lateral electric field $\left(F_y\right)$ perpendicular to the QDM offset direction $\left(x\right)$. (b) Spin-mixing-splitting amplitude ${\mathrm{\Delta }}_{\mathrm{sm}}$ (yellow solid line) and spin-mixing resonance vertical electric field $F_{\mathrm{sm}}$ (blue dashed line) as a function of $F_y$. (c) Energy levels of hole states as a function of lateral electric field along the QDM's offset direction, $F_x$. (d) ${\mathrm{\Delta }}_{\mathrm{sm}}$ and $F_{\mathrm{sm}}$ as a function of lateral electric field $F_x$.

Figure 4

Effect of lateral electric field on a symmetric QDM under 12 T constant magnetic field with the piezoelectric field. Planes of the piezo potential at (a) the edge of the QDs and (b) the middle of the QDs. Spin-mixing-splitting amplitude ${\mathrm{\Delta }}_{\mathrm{sm}}$ (orange) and spin-mixing resonance vertical electric field $F_{\mathrm{sm}}$ (blue) as a function of lateral electric field in different directions, [(c) symmetric QDM; (d), (e) asymmetric QDM]

Figure 5

Effect of lateral electric field with a gradient in the growth direction on asymmetric QDMs with piezoelectric fields included. (a) Electric potential for gradient $G_{yz}=0.002$ (mV/Å)/Å. (b) ${\mathrm{\Delta }}_{\mathrm{sm}}$ (orange) and $F_{\mathrm{sm}}$ (blue) as a function of lateral electric-field gradient $G$.