Linewidths in excitonic absorption spectra of cuprous oxide

We present a theoretical calculation of the absorption spectrum of cuprous oxide $\left({\mathrm{Cu}}_2\mathrm{O}\right)$ based on the general theory developed by Y. Toyozawa. An inclusion not only of acoustic phonons but also of optical phonons and of specific properties of the excitons in ${\mathrm{Cu}}_2\mathrm{O}$ like the central-cell corrections for the $1S$ exciton allows us to calculate the experimentally observed linewidths in experiments by T. Kazimierczuk et al. [T. Kazimierczuk, D. Fröhlich, S. Scheel, H. Stolz, and M. Bayer, Nature (London) 514, 343 (2014)] within the same order of magnitude, which demonstrates a clear improvement in comparison to earlier work on this topic. We also discuss a variety of further effects, which explain the still observable discrepancy between theory and experiment but can hardly be included in theoretical calculations.

Figure 1

In order to evaluate the quantities ${\mathrm{\Gamma }}_{{\nu }_2{\nu }_1\mathit{0}}\left(\omega \right)$ and ${\mathrm{\Delta }}_{{\nu }_2{\nu }_1\mathit{0}}\left(\omega \right)$, one has to cut the infinite sums over $\nu$ in the formulas at a finite value $n_{\max }$ of the principal quantum number $n$ [cf. Eq. (16)]. Here we show the values obtained for ${\tilde{\mathrm{\Delta }}}_{210\mathit{0}}$ in dependence on $n_{\max }$ for Step 2. The final value ${\tilde{\mathrm{\Delta }}}_{210\mathit{0}}=-2.32\times {10}^{-5}$ eV (dashed line) is then calculated from an extrapolation. We used $f\left(n_{\max }\right)=a/n_{\max }^2+b$ as a fitting function for $n_{\max }\ge 3$ (solid line).

Figure 2

Comparison of (a) the experimental spectrum [10] with (b) the calculated line shapes using Eq. (6) and the values listed in Table 2. Since we do not know the proportionality constant $c$ in Eq. (11), we chose arbitrary values for the amplitudes ${\tilde{F}}_{\nu }\left(\omega \right)$. We shifted the experimental values by an amount of $-6$ meV for a better comparison. The experimental values have been fitted by Lorentzians to obtain the experimental linewidths (red dashed line).