Spin-orbit coupling effects on spin-dependent inelastic electronic lifetimes in ferromagnets

For the $3d$ ferromagnets iron, cobalt, and nickel we compute the spin-dependent inelastic electronic lifetimes due to carrier-carrier Coulomb interactions including spin-orbit coupling. We find that the spin-dependent density of states at the Fermi energy does not, in general, determine the spin dependence of the lifetimes because of the effective spin-flip transitions allowed by the spin mixing. The majority and minority electron lifetimes computed including spin-orbit coupling for these three $3d$ ferromagnets do not differ by more than a factor of 2, and agree with experimental results.

Figure 1

Energy-resolved majority $({\tau }^{\uparrow }$, blue $+)$ and minority $({\tau }^{\downarrow }$, red $\circ )$ carrier lifetimes for cobalt. There are in general several different lifetime points at the same energy (see text). We used ${17}^3$ $\vec{k}$ points in the full BZ.

Figure 2

Same as Fig. 1 for nickel.

Figure 3

Energetically averaged majority (blue uptriangles) and minority (red downtriangles) lifetimes for (a) Fe, (b) Co, and (c) Ni. The error bars denote the standard deviation obtained from the scatter of the lifetimes as shown in Figs. 1 and 2. The insets show the calculated ratio of majority and minority electrons $(\circ )$, ${\tau }^{\uparrow }/{\tau }^{\downarrow }$, in comparison with experimental data, where the $•$ $(\times )$ correspond to values extracted from Refs. [11, 17] (Ref. [12]).

Figure 4

Calculated energy-resolved electronic lifetimes for cobalt and nickel in analogy to Fig. 3 without the inclusion of spin-orbit coupling effects.