Spin-Lattice Relaxation of Shallow Donor States in Ge and Si through a Direct Phonon Process

Abstract

The many-valley character of the conduction band edge of germanium and silicon causes an anisotropy of the $g$ shift and of the deformation potential for the conduction electrons. It is shown that the combination of these two effects provides a mechanism for spin-lattice relaxations of the donor spins in germanium and silicon that yields $\frac{1}{T_s}$ proportional to the temperature $T$ and to the fourth power of the static magnetic field $H$. Using known data about the deformation potential constant, the $g$ shift, the energy of the intervalley splitting, and the elastic constants, the magnitude of $T_s$ is found to be approximately 2×${10}^{-3\mathrm{}}$ sec for phosphorus donors in germanium, and 1×${10}^4$ sec for phosphorus donors in silicon. These values refer to $T=1.25\mathrm{}°$K, $H=3000\mathrm{}$ gauss, with the field applied along the [111] axis. Our mechanism fails to give a finite $T_s$ for donors in silicon, when the field is applied along the [100] axis.