Improved calculations of the complex dielectric constant of semiconductors

Abstract

Expressions for the real, static part ${\ensuremath{\epsilon}}_{1}(0)$ and the imaginary part ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ of the dielectric constant of semiconductors in the long-wavelength limit are obtained using the isotropic nearly-free-electron band approximation (Penn model). Earlier calculations of these functions do not satisfy the Kramers-Kronig relations and yield an excessively large result for the $f$-sum rule. The corrected expressions eliminate these inconsistencies. Values of the energy gap between the bonding and antibonding states are obtained for diamond, silicon, and germanium, respectively. ${\ensuremath{\epsilon}}_{1}(\ensuremath{\omega})$ is obtained from ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ through the use of the Kramers-Kronig relation. The theoretical curves for ${\ensuremath{\epsilon}}_{1}(\ensuremath{\omega})$ and ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ are compared with experimental results.