Coulomb correlated multi-particle states of weakly confining GaAs quantum dots

TL;DR

We address Coulomb-correlated multi-particle states in weakly confining GaAs/AlGaAs quantum dots. The study uses an eight-band $k\cdot p$ model with continuum elasticity and configuration interaction to predict energies, polarization-resolved oscillator strengths, and radiative rates for $X^0$, $X^+$, $X^-$, and $XX$, including beyond-dipole (BDA) corrections implemented via a Poisson-based formulation equivalent to the dyadic Green’s tensor. They find that BDA reproduces experimental lifetimes (e.g., $\tau^{X}\approx0.279\,\mathrm{ns}$ and $\tau^{XX}\approx0.101\,\mathrm{ns}$) and electric-field tuning, and show sensitivity to CI-basis size and to electron-electron/hole-hole exchange; results emphasize that preparation and detection influence observed spectra. The workflow provides a reproducible link between realistic many-body wavefunctions and nonlocal light-matter coupling, and can be extended to include kinetics (phonons, dephasing) and device-level perturbations.

Abstract

We compute the electronic and emission properties of Coulomb-correlated multi-particle states (X$^0$, X$^\pm$, XX) in weakly confining GaAs/AlGaAs quantum dots using an 8-band $\mathbf{k}\!\cdot\!\mathbf{p}$ model coupled to continuum elasticity and configuration interaction (CI). We evaluate polarization-resolved oscillator strengths and radiative rates both in the dipole approximation (DA) and in a quasi-electrostatic beyond-dipole (BDA) longitudinal formulation implemented via a Poisson reformulation exactly equivalent to the dyadic Green-tensor kernel. For the dots studied, BDA yields lifetimes in quantitative agreement with experiment, e.g., $τ^X=0.279\,\mathrm{ns}$ vs $0.267\,\mathrm{ns}$ (exp.) and $τ^{XX}=0.101\,\mathrm{ns}$ vs $0.115\,\mathrm{ns}$ (exp.). The framework also reproduces electric-field tuning of the multi-particle electronic structure and emission -- including the indistinguishability inferred from $P=τ^X/(τ^X+τ^{XX})$ -- and we assess sensitivity to CI-basis size and to electron-electron and hole-hole exchange.

Figures (8)

• Figure 1: The simulated structure of GaAs "QD1" with 2 nm GaAs wetting layer (WL) in Al$_{0.4}$Ga$_{0.6}$As is shown in panel (a) with marked QD and WL dimensions Yuan2023yuan_xueyong_2023_7748664. Panel (b) gives the single-particle energies of the simulated QD for electrons (blue symbols) and holes (red symbols). For each kind of quasiparticle the energies of 42 states are shown in (b). The doubling of states for each energy level in (b) corresponds to the Kramers doublets of corresponding states. The black broken and green dotted vertical lines in (b) correspond to the largest CI bases used in this work for computations of ${\rm M}\in\{{\rm X}^-, {\rm X}^+, {\rm XX}\}$ and that for X$^0$, respectively. In panel (c) the ground state exciton energy (X$^0$) is shown (by green balls) as a function of symmetric CI basis size. The exciton energy reaches a value of X$^0$=1.5541 eV for a CI basis of 36 $\psi^{(e)}$ and 36 $\psi^{(h)}$ (36x36 CI basis). For comparison, the measured value of X$^0$ was 1.551152 eV Yuan2023. Panel (d) shows the evolution of bright (FSS_B) and dark (FSS_D) FSS of X$^0$ in blue and red balls, respectively, on symmetric CI basis size. That for the bright-dark (B-D) splitting of X$^0$ is given in (d) by violet balls. We see that computed bright FSS value of $7\pm0.5\,\mu$eV almost does not change with size of CI basis while B-D splitting ceases to change appreciably when reaching a value of 68 $\mu$eV. Note that a more detailed analysis of convergence of energies of X$^0$ and B-D splitting in panels (c) and (d) is given in Fig. \ref{['fig:Econv']} (a) and (b) in the Appendix I.
• Figure 2: Panels (a) and (b) show the sketches of the type of the Coulomb exchange considered in CI calculations for X$^-$, X$^+$, and XX. In (a) and (b) the red triangles mark the electron-hole ($K_{eh}$), blue boxes the electron-electron ($K_{ee}$), and balls hole-hole ($K_{hh}$) Coulomb exchange interaction. The empty symbols in (a) and (b) for XX mark $K_{eh}$ of one of the final states of the recombination of XX, i.e. X$^0$. The dimmed colored lines and symbols in (b) mark the exchange interactions omitted in the CI calculations in (d) (see text). In (c) and (d) the variations with respect to the number of single-particle states in CI basis for the binding energies of X$^-$, X$^+$, and XX relative to X$^0$ are shown. In correspondence to (a) and (b), in (c) all Coulomb direct and exchange integrals are considered, while in (d) $K_{ee}$ and $K_{hh}$, and partly $K_{eh}$ are omitted. The meaning of markers in (c) and (d) is the following: (i) full balls represent symmetric CI basis, i.e., same number of $\psi^{(e)}$ and $\psi^{(h)}$; (ii) open squares represent the same but for SDCI approximation; (iii) open upward triangles give SDCI for asymmetric basis composed of twelve $\psi^{(e)}$ and varying number of $\psi^{(h)}$. Note that there is a negligible energy offset $<100\,\mu$eV between the calculations performed using aforementioned methods, seen as steps for the overlapping CI bases {e.g. CI bases of 10 and 18 in (c) and (d)}. The red horizontal broken line denotes experimental binding energy of X$^+$Yuan2023, blue of X$^-$Huber2019, and green of XX DaSilva2021. Notice that calculations reach very close to experimental values of binding energies in (d), i.e., for calculation with $K_{ee}$, $K_{hh}$ and partly $K_{eh}$ omitted [dimmed colored arrows in (b)], corresponding to the situation due to weak confinement effect, see also main text. Note that a more detailed analysis of convergence of binding energies of X$^+$, X$^-$ and XX is given in Fig. \ref{['fig:Econv']} (c) in the Appendix I.
• Figure 3: The evolution of the radiative lifetime of ground states of X, X$^+$, X$^-$, and XX as a function of the CI basis size when the overlap integrals are evaluated considering (a) DA and (b) BDA Stobbe2012, see also main text. The meaning of markers in both panels is the same as that in Fig. \ref{['fig:BindingEXnoEEHH']} (c) and (d). The black (green) dotted horizontal line marks the measured values of exciton (biexciton) lifetime from Ref. Schimpf2019. Note that for both DA and BDA the calculations of lifetime do not change appreciably for CI bases larger than $14$ states. On the other hand, the calculations using BDA reproduce the experiments considerably better than those for DA.
• Figure 4: Panel (a) gives the vertical electric field dependence of X$^0$ energy (open squares, values on the right vertical axis), B-D splitting (full violet balls), bright FSS (full blue balls) and dark FSS (full red balls) of X$^0$. The values of latter three parameters are on the left vertical axis. In panel (b) we show the evolution of the binding energy of X$^+$ (red), X$^-$ (blue), and XX (green) relative to X$^0$ with vertically applied electric field on QD in Fig. \ref{['fig:AFMsp']} (a). The inset in (b) shows an enlarged part of the data corresponding to the band crossings. The meaning of axes in the inset are the same as for the whole panel (b). In (c) we give the radiative lifetime of X$^0$, X$^+$, X$^-$, and XX computed using BDA. In order to facilitate the comparison with Ref. Undeutsch2025, the electric field is given as a voltage applied on 300 nm thick layer, hence the label of horizontal axis of $U_{d300nm}$. The data coloring in (c) is the same as in (b) except for X$^0$ which is given in black. The curves in both panels are guides to the eye. The gray-shaded areas in all panels correspond to voltages not considered in Ref. Undeutsch2025. The calculations of X$^0$ were performed with the CI basis of 36 electron and 36 hole single-particle states while that for X$^+$, X$^-$ and XX using SDCI with basis of 12 electron and 36 hole states and with omitted $K_{ee}$, $K_{hh}$ and partly $K_{eh}$ exchange integrals see Fig. \ref{['fig:BindingEXnoEEHH']} (d).
• Figure 5: The ratio of XX and X$^0$ lifetimes, $\tau^{XX}/\tau^X$ from Fig. \ref{['fig:ELfldBindLife']} (c) is shown by green open squares. The photon indistinguishability $\mathbb{P}$ from Eq. \ref{['eq:GabrielIndisting']} is given by full blue balls. Orange shaded area marks the interval of $\tau^{XX}/\tau^X$ measured in Fig. 2 (d) of Ref. Undeutsch2025. The gray-shaded area correspond to voltages not considered in Ref. Undeutsch2025. The gray horizontal line marks $\tau^{XX}/\tau^X=1$, i.e. the situation when lifetimes of X and XX are the same. In order to facilitate the comparison with Ref. Undeutsch2025, the electric field is given as a voltage applied on 300 nm thick layer, hence the label of horizontal axis of $U_{d300nm}$.