Spin relaxation of two-dimensional electrons with a hierarchy of spin-orbit couplings

The density matrix formalism is applied to calculate the spin-relaxation time for two-dimensional systems with a hierarchy of spin-orbit couplings, such as Rashba-type, Dresselhaus-type and so on. It is found that the spin-relaxation time can be infinite if those coupling strengths $\alpha$, $\beta$, $\gamma_1$ and $\gamma_2$ satisfy either condition (i) $\alpha=\beta, \gamma_1=0$ or (ii) $\alpha=-\beta, \gamma_2=0$, which correspond to the vanishing Yang-Mills "magnetic" field. The effect caused by the application of an external magnetic field is also discussed. It is found that the longitudinal and in-plane spin components can possess infinite life time when the spin components, the Larmor precession frequency and the external magnetic field satisfy certain relations.