Theory of phonon-modulated electron spin relaxation time based on the projection—reduction method

This paper introduces a new method for a formula for electron spin relaxation time of a system of electrons interacting with phonons through phonon-modulated spin—orbit coupling using the projection-reduction method. The phonon absorption and emission processes as well as the photon absorption and emission processes in all electron transition processes can be explained in an organized manner, and the result can be represented in a diagram that can provide intuition for the quantum dynamics of electrons in a solid. The temperature ( T ) dependence of electron spin relaxation times ( T 1 ) in silicon is T 1 ∝ T −1.07 at low temperatures and T 1 ∝ T −3.3 at high temperatures for acoustic deformation constant P ad = 1.4 × 10 7 eV and optical deformation constant P od = 4.0 × 10 17 eV/m. This means that electrons are scattered by the acoustic deformation phonons at low temperatures and optical deformation phonons at high temperatures, respectively. The magnetic field ( B ) dependence of the relaxation times is T 1 ∝ B −2.7 at 100 K and T 1 ∝ B −2.3 at 150 K, which nearly agree with the result of Yafet, T 1 ∝ B −3.0 ∼ B −2.5 .