Influence of electron-hole plasma on Rydberg excitons in cuprous oxide

We develop a many-body approach to the behavior of exciton bound states and the conduction electron band edge in a surrounding electron-hole plasma with a focus on the absorption spectrum of Rydberg excitons in cuprous oxide. The interplay of band edge and exciton levels is analyzed numerically, whereby the self-consistent solution is compared to the semiclassical Debye approximation. Our results provide criteria which allow us to verify or rule out the different band edge models against future experimental data.

Figure 1

Contributions to the excitonic line shift for the $1S$ (blue) and $2P$ (red) excitons at $T=10\mathrm{K}$ vs density ${\rho }_{\mathrm{eh}}$. Sum of Fock and Pauli blocking contributions $\mathrm{\Delta }{E}_{nl}^{\mathrm{PBF}}$ [Eq. (14)] (dotted) and sum of the dynamic contributions $\mathrm{\Delta }{E}_{nl}^{\mathrm{DSP}}$ [Eq. (15)] (dashed). The solid lines denote the total shift $\mathrm{\Delta }{E}_{nl}^{\mathrm{PBF}}+\mathrm{\Delta }{E}_{nl}^{\mathrm{DSP}}$, cf. Eq. (13).

Figure 2

Same representation as in Fig. 1, but for the $n=5$ (blue) and $n=10$ (red) $P$ excitons at $T=2\mathrm{K}$ (thick lines) and $T=10\mathrm{K}$ (thin lines) vs density ${\rho }_{\mathrm{eh}}$. For obvious reasons, the total shift $\mathrm{\Delta }{E}_{nl}^{\mathrm{PBF}}+\mathrm{\Delta }{E}_{nl}^{\mathrm{DSP}}$ is not shown.

Figure 3

Total excitonic line shift $\mathrm{\Delta }{E}_{nl}^{\left(1\right)}$, Eq. (13), for the $n=10$ (solid blue line), $n=12$ (green dashed), $n=14$ (orange dotted), and $n=16$ (red dash-dotted) $P$ excitons at $T=2\mathrm{K}$ (thick lines) and $T=18\mathrm{K}$ (thin lines) vs density ${\rho }_{\mathrm{eh}}$.

Figure 4

Band edge (black lines) and $n=1S$-exciton level (blue lines; upper panel) and $n=2P$-exciton level (red lines; lower panel) vs density for a carrier temperature of $T=10\mathrm{K}$. Approximations for the band edge: $E_{\mathrm{be}}^{\mathrm{mod}}$ [Eq. (2) with Eq. (8); solid line], $E_{\mathrm{be}}^{\mathrm{iter}}$ [Eq. (2) with Eq. (7); dashed], $E_{\mathrm{be}}^{\left(0\right)}$ [Eq. (2) with Eq. (6); dotted], and $E_{\mathrm{be}}^{\mathrm{Debye}}$ [Eq. (9); dash-dotted]. Thick (thin) colored lines denote the shifted (unperturbed) exciton levels, whereupon the shift is given by $\mathrm{\Delta }{E}_{nl}^{\left(1\right)}$ [Eq. (13); solid] and $\mathrm{\Delta }{E}_{nl}^{\mathrm{Debye}}$ [Eq. (19); dash-dotted].

Figure 5

Same presentation as Fig. 4, but for several $P$-exciton levels $[n=12$ (green lines), 14 (orange), and 16 (red)] for a carrier temperature of $T=10\mathrm{K}$.

Figure 6

Logarithmic plot of the maximum excitonic line shifts at the respective Mott density for the $n=10$ (blue lines), $n=12$ (green), $n=14$ (orange), and $n=16$ (red) $P$ excitons vs temperature $T$ using two different approximations: iterative solution of Eq. (2) with Eq. (8) for the band edge and fully dynamic line shift (solid) and Debye approximation for both quantities (dashed).

Figure 7

Mott density vs temperature for several $P$-exciton levels. Line styles, see Fig. 6.

Figure 8

Band edge vs temperature $T$ for a carrier density of ${\rho }_{\mathrm{eh}}={10}^{10}{\mathrm{cm}}^{-3}$ (upper panel) and ${\rho }_{\mathrm{eh}}={10}^{13}{\mathrm{cm}}^{-3}$ (lower panel): $E_{\mathrm{be}}^{\mathrm{mod}}$ [Eq. (2) with Eq. (8); solid line], $E_{\mathrm{be}}^{\mathrm{iter}}$ [Eq. (2) with Eq. (7); dashed], $E_{\mathrm{be}}^{\left(0\right)}$ [Eq. (2) with Eq. (6); dotted], and $E_{\mathrm{be}}^{\mathrm{Debye}}$ [Eq. (9); dash-dotted].

Figure 9

Band edge (black lines) and exciton levels (red lines) vs principal quantum number n for a carrier density of ${\rho }_{\mathrm{eh}}={10}^{13}{\mathrm{cm}}^{-3}$ and two temperatures: $T=3\mathrm{K}$ (upper panel) and $T=20\mathrm{K}$ (lower panel). Line styles as in Fig. 8, i.e., solid lines: self-consistent model; dash-dotted lines: Debye model. Thin red lines denote the unperturbed exciton levels. Excitons with n lying in the shaded areas cannot exist in both models (double shaded) or exist only in the Debye model (single shaded). Note that $n_{\max }=8$ for both models (upper panel) by chance.