Bound exciton model for ``shallow'' impurities in semiconductors

Most impurities can be described by a unified bound exciton model [1] that includes impurity binding energy E$_{\mathrm{I}}$, exciton binding energy E$_{\mathrm{XB}}$, and excitonic transition energy E$_{\mathrm{X}}=$E$_{\mathrm{I}}$-E$_{\mathrm{XB}}$. E$_{\mathrm{XB}}$ corresponds to the commonly known acceptor binding energy E$_{\mathrm{A}}$ or donor binding E$_{\mathrm{D}}$ for the respective case. Analogous to the free exciton problem, E$_{\mathrm{g}}$ or E$_{\mathrm{I}}$ is the single particle transition energy, E$_{\mathrm{XB}}$ is due to many-body effect that can sometimes be simplified as an effective mass (EM) equation with a screened Coulomb interaction between the electron and hole. Although E$_{\mathrm{XB}}$ typically represents a small modification to the inter-band transition energy (e.g., E$_{\mathrm{g}} = $ 1.519 eV and E$_{\mathrm{BX}} = $ 4 meV for GaAs), the excitonic effect is responsible for the strong absorption at E$_{\mathrm{X}}$ and other discrete excitonic transition peaks. The same phenomenon occurs in impurities either known as ``deep'' or ``shallow.'' The standard theory for ``shallow'' impurities overlooks an important aspect of the problem, the E$_{\mathrm{I}}$ part associated with a short-range potential. The attempt to consolidate the discrepancy between the EM model and experimental data by introducing a ``core correction'' into the EM equation is conceptually problematic, equivalent to ``correcting'' E$_{\mathrm{BX}}$ to match the E$_{\mathrm{g}}$ value for a free exciton.\\[4pt] [1] Zhang and Wang, PRB (in press).