Quantum defects of Rydberg excitons in cuprous oxide: A semiclassical spherical model

TL;DR

This work addresses the problem of explaining the Rydberg-like yellow-exciton spectrum in Cu$_2$O by capturing the nonhydrogenic l-dependent splittings introduced by the crystal band structure. The authors develop a semiclassical, spherically symmetric model based on an adiabatic separation that yields energy surfaces $W_k(p)$, and they determine quantum defects via EBK quantization on the lowest surface, complemented by momentum-space–weighted, angular-momentum–dependent corrections. They obtain analytical expressions for the energy surfaces using Cardano's method, validate the approach against known quantum defects and exact calculations, and show good agreement with experimental data for large $n$ and $l$, while acknowledging limitations in describing l-splittings and central-cell corrections. The results provide a tractable route to describing Rydberg excitons in Cu$_2$O, enable parameter fitting of band-structure constants, and offer a path toward extensions to magnetoexcitons and spin-resolved spectra using first-principles band descriptions.

Abstract

Excitons, i.e. the bound states of an electron and a positively charged hole are the solid state analogue of the hydrogen atom. As such they exhibit a Rydberg series, which in cuprous oxide has been observed up to high principal quantum numbers by T. Kazimierczuk et al. [Nature 514, 343 (2014)]. In this energy regime the quantum mechanical properties of the system can be understood in terms of classical orbits by the application of semiclassical techniques. In fact the first theoretical explanation of the spectrum of the hydrogen atom within Bohr's atomic model was a semiclassical one using classical orbits and a quantization condition for the angular momentum. Contrary to the hydrogen atom, the degeneracy of states with the same principal quantum number $n$ is lifted in exciton spectra. This is similar to the situation in alkali atoms, where these splittings are caused by the interaction of the excited electron with the ionic core. For excitons in cuprous oxide, these splittings occur due to the influence of the complex band structure of the crystal. Using an adiabatic approach and analytically derived energy surfaces, we develop a semiclassical spherical model and determine, via semiclassical torus quantization, the quantum defects of various angular momentum states.

Figures (3)

• Figure 1: Comparison of the spherical average $\overline{W}_{\mathrm{y}}(p)$ in Eq. \ref{['eq:W_avg']} (red line) with the spherical model $\widetilde{W}_\mathrm{y}(p)$ in Eq. \ref{['eq:W_s']} using $\mu'=0.0586$ (green line) and $\mu'=0.47$ (blue line). A fit of the spherical model to the spherical average yields $\mu'=0.497$ (orange line). The spin-orbit splitting is fixed to the experimental value $\Delta=131~\mathrm{meV}$ except for the black dashed line where nearly perfect agreement with the spherical average is obtained when $\tilde{\Delta}=288~\mathrm{meV}$ and $\tilde{\mu}'=0.545$ are used as fit parameters for $\Delta$ and $\mu'$ in Eq. \ref{['eq:W_s']}.
• Figure 2: (a) Quantum defects $\delta_{n,l}$ for $s$, $p$, $d$, and $f$ states obtained with the spherically symmetric energy surfaces $\widetilde{W}_{\mathrm{y}}(p)$ with parameters $\Delta=131~\mathrm{meV}$ and $\mu'=0.47$. (b) Improved quantum defects $\delta_{n,l}$ obtained from a spherical model using values $\mu'_l$ optimized for different $l$ manifolds. Here, $\mu'_l$ acts as an order parameter characterizing the strength of the band structure in the various $l$ manifolds. The solid lines mark the quantum defects in the limit of large principal quantum numbers $n$ given in Ref. heckotter2017scaling.
• Figure 3: Comparison of energies $E_{n,l}$ of odd parity states with $n= 2$ to $5$ obtained by numerically exact quantum calculations (top) with the semiclassical model using order parameters (bottom).