Spin susceptibilities, spin densities, and their connection to spin currents

We calculate the frequency-dependent spin susceptibilities for a two-dimensional electron gas with both Rashba and Dresselhaus spin-orbit interactions. The resonances of the susceptibilities depend on the relative values of the Rashba and Dresselhaus spin-orbit constants, which could be manipulated by gate voltages. We derive exact continuity equations, with source terms, for the spin density and use those to connect the spin current to the spin density. In the free electron model the susceptibilities play a central role in the spin dynamics since both the spin density and the spin current are proportional to them.