Electric control of the exciton fine structure in nonparabolic quantum dots

We show that the nonparabolic confinement potential is responsible for the nonmonotonic behavior and sign change of the exciton fine-structure splitting (FSS) in optically active self-assembled quantum dots. This insight is important for the theoretical understanding and practical control by electric fields of the quantum state of the emitted light from a biexciton cascade recombination process. We find that a hard-wall (box) confinement potential leads to a FSS that is in better agreement with experimentally measured FSS than a harmonic potential. We then show that a finite applied electric field can be used to remove the FSS entirely, thus allowing for the creation of maximally entangled photons, being vital to the growing field of quantum communication and quantum key distribution.

Figure 1

The biexciton recombination cascade. For degenerate intermediate exciton levels ($\delta \ll \Gamma$, where $\Gamma$ is the linewidth of the emitted light and $\delta$ the FSS) the two decay paths are indistinguishable and the recombination can be described by $\left|\text{XX}\right.〉\left|0\right.〉\to (\left|\text{X}\right.〉\left|\text{H}\right.〉+\left|\text{Y}\right.〉\left|\text{V}\right.〉)/\sqrt{2}\to \left|\text{0}\right.〉(\left|\text{HH}\right.〉+\left|\text{VV}\right.〉)/\sqrt{2}$ where we use the notation $\left|\text{electronic}\right.〉\left|\text{photonic}\right.〉$ and $\left|\mathrm{XX}\right.〉$ is the biexciton state, $\left|\mathrm{X}\right.〉$ ($\left|\mathrm{Y}\right.〉$) denotes the exciton state with horizontal (vertical) polarization, $\left|\mathrm{H}\right.〉$ ($\left|\mathrm{V}\right.〉$) the photonic state with horizontal (vertical) polarization, and $\left|0\right.〉$ is the electronic ground state or photon vacuum.

Figure 2

The FSS $\delta$ calculated for a harmonically confined InGaAs dot with characteristic lengths $30\times 20\times 7$ nm${}^3$ and various directions of the electric field. Regardless of the direction of the field, the FSS is monotonically decreasing and vanishes only asymptotically.

Figure 3

The FSS $\delta$ calculated for In${}_{0.2}$Ga${}_{0.8}$As/GaAs dot of dimensions $55\times 50\times 7$ nm${}^3$ with a hard-wall confinement in the presence of an electric field applied in various directions. The FSS is sensitive to the direction in which the field is applied: In the $\widehat{x}$ direction the FSS decreases and changes sign at a critical field strength where $\delta =0$ (here, at $E_x\simeq 35\mathrm{kV}/\mathrm{m}$). When the field is applied in the $\widehat{y}$ direction, the FSS increases instead.

Figure 4

Hole envelope functions with, $F_{m_j}^{\mathbf{E}}\left(\mathbf{r}\right)$, and without, $F_{m_j}\left(\mathbf{r}\right)$, applied electric field $E_x$ in $x$ direction, for (a) harmonic and (b) hard-wall confinement. For the case of harmonic confinement, the wave functions are translated by ${ł}_{m_j}^{\mathbf{E}}$ and are no longer inside the physical region of the dot when a field $E_x$ is applied but retain their shape. For the hard-wall potential, the main effect of the electric field is to deform the wave functions; this is accompanied by a relatively small shift that leaves the particles inside the dot. The deformation affects the second derivative of $F_{m_j}^{\mathbf{E}}\left(\mathbf{r}\right)$ which determines the FSS; see Eq. (12).