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

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 1Tₛ 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 Tₛ is found to be approximately 2×{}10^-3 sec for phosphorus donors in germanium, and 1×{}10⁴ sec for phosphorus donors in silicon. These values refer to T=1.25^∘K, $H=3000$ gauss, with the field applied along the [111] axis. Our mechanism fails to give a finite Tₛ for donors in silicon, when the field is applied along the [100] axis.