Broadening of Impurity Levels in Silicon

Sharp absorption lines have been observed in $p$- and $n$-type silicon whose absolute and relative positions lead to their interpretation as optical transitions between bound states of trapped holes or electrons that are approximately hydrogenic in character. These lines have a finite breadth of the order of 0.001 electron-volt at liquid helium temperatures. This breadth is determined by the zero-point vibrations of the lattice. At higher temperatures, the squared breadth increases in proportion to the mean squared amplitude of oscillation of those lattice modes that contribute significantly to the broadening. The theory indicates that the modes of importance have wavelengths of the order of the Bohr radius of the trapped carrier state. These are rather long wavelength acoustic modes whose energy $h\omega$ corresponds to 80°K. Thus the squared broadening is expected to increase by a factor of two from helium to nitrogen temperatures—and this increase is confirmed experimentally. This confirms the hypothesis that the broadening is due to the interaction of the trapped electron with the acoustic lattice vibrations. The form of the electron-lattice interaction is taken to be that of the Bardeen-Shockley deformation potential, and its strength is determined from the experimental mobility. Thus the theory contains no adjustable parameters. In absolute magnitude, the theoretical line breadth turns out to be several times too large. Possible reasons for the discrepancy are discussed.