Impact of the valence band on Rydberg excitons in cuprous oxide quantum wells

TL;DR

This work develops a complete Luttinger-Kohn Hamiltonian for excitons in Cu$_2$O quantum wells, explicitly including the complex valence-band structure and quantum confinement. By expanding the exciton envelope in a B-spline basis and leveraging $D_{4h}$ symmetry, the authors diagonalize a large sparse matrix to obtain energy spectra and relative oscillator strengths for circularly polarized light. They demonstrate that valence-band nonparabolicity induces significant energy shifts and degeneracy lifting beyond hydrogenlike models, and they map how non-diagonal couplings reshape the yellow and green exciton spectra in QWs. The approach provides quantitative predictions for exciton energies and optical strengths under circular polarization, enabling targeted comparisons with future experiments and guiding refinements of realistic QW models.

Abstract

The complex valence band structure of bulk cuprous oxide necessitates going beyond the parabolic approximation to precisely estimate exciton binding energies. The same is true for excitons in cuprous oxide quantum wells, for which many effects have been obtained so far only qualitatively within a hydrogenlike two-band model. Here, we derive the complete Hamiltonian for excitons in cuprous oxide quantum wells based on the Luttinger-Kohn model, taking into account the full complex valence band structure. Symmetry properties of the system are discussed. Numerical results based on the diagonalization of the Hamiltonian using B-spline functions reveal the energy shifts and the lifting of degeneracies due to the nondiagonal coupling terms of the complex valence band. The relative oscillator strengths of the excitonic transitions induced by circularly polarized light are also calculated.

Figures (5)

• Figure 1: Sketch of the general setup for the creation of excitons in a cuprous oxide QW. The $[001]$ axis of the crystal is aligned perpendicular to the QW plane. Excitons are excited by circularly polarized light along this axis.
• Figure 2: Sketch of the band structure of the $\Gamma_6^+$ conduction bands, as well as the $\Gamma_7^+$ and $\Gamma_8^+$ valence bands near the $\Gamma$ point.
• Figure 3: Illustration of symmetry operations of the group $D_{4\mathrm{h}}$. $C_2$ and $C_4$ denote two- and fourfold rotations, respectively, $\sigma_h$, $\sigma_v$, and $\sigma_d$ reflections with respect to different kinds of symmetry planes. In addition to the depicted operations, there is also the inversion $I$ and the improper rotation $S_4 = \sigma_h C_4$.
• Figure 4: Exciton spectra in a cuprous oxide QW. In (a) the QW width is increased from $L=0$ to $10\,$nm for excitons described by the hydrogenlike part of the Hamiltonian given by the first two terms in Eq. \ref{['eq:H_lambda']}. In (b)-(e) the non-diagonal coupling terms of the are successively switched on using the parameters $\lambda_{0,1,2,3}\in [0,1]$ in the Hamiltonian \ref{['eq:H_lambda']} (see text). The photoabsorption spectrum for transitions with circularly polarized light and considering the complete structure of the valence band is given in (f). Some states are labeled by approximate quantum numbers $(n,m,N_{\mathrm{e}},N_{\mathrm{h}})$. The colors of symbols or lines indicate the expectation value $\langle m\rangle$ of the angular momentum of the exciton state.
• Figure 5: Photoabsorption spectrum of a cuprous oxide quantum well of width $L=10\,$nm for transitions with left-handed circularly polarized light ($\hat{\bm e}_x+i\hat{\bm e}_y$). The relative oscillator strengths $f_+^{\mathrm{rel}}$ of parity allowed transitions to states in the subspaces $\Gamma_6^-$ and $\Gamma_7^-$ are represented by vertical light blue and green bars, respectively. Parity forbidden even states in the subspaces $\Gamma_6^+$ and $\Gamma_7^+$ are marked by dark blue and dark green arrows. The insets show the transition amplitude density $\mathcal{M}_\rho^+(Z)$ (orange lines) and $\mathcal{M}_z^+(Z)$ (red lines) given in Eqs. \ref{['eq:transition_amplitudes1']} and \ref{['eq:transition_amplitudes2']}, respectively, together with their sums (black lines). Some states are marked by approximate quantum numbers $(n,m,N_{\mathrm{e}},N_{\mathrm{h}})$.