Ab initio GW calculations of the time and length of spin relaxation of excited electrons in metals with inclusion of the spin-orbit coupling

An ab initio method has been proposed for calculating the spin relaxation time of excited electrons in metals in the framework of the GW method with inclusion of the spin-orbit coupling. The time and length of spin relaxation in Al, Cu, Au, Nb, and Ta have been calculated. The concept of the spin-flip phase space has been introduced. It has been demonstrated that the ratio between the spin relaxation time and the lifetime of the excited electron is well explained within this concept. The time and length of spin relaxation in Nb appear to be considerably shorter than those in Al, Cu, and Au. These quantities in Ta are especially small in accordance with the strong spin-orbit coupling. A comparison of the results with the previous data on the time and length of spin relaxation due to the interaction with impurities and phonons shows that, at an excited electron energy of the order of 1 eV, the inelastic electron-electron scattering in the presence of spin-orbit coupling is a dominant mechanism of spin relaxation.