Spin Coulomb drag in the two-dimensional electron liquid

We calculate the spin-drag transresistivity ρ_↑↓(T) in a two-dimensional electron gas at temperature T in the random-phase approximation. In the low-temperature regime we show that, at variance with the three-dimensional low-temperature result [ρ_↑↓(T)∼T²], the spin transresistivity of a two-dimensional spin unpolarized electron gas has the form ρ_↑↓(T)∼T²lnT. In the spin-polarized case the familiar form ρ_↑↓(T)=AT² is recovered, but the constant of proportionality, A, diverges logarithmically as the spin-polarization tends to zero. In the high-temperature regime we obtain ρ_↑↓(T)=-(ħ/e²)(π²Ry^*/k_BT) (where Ry^* is the effective Rydberg energy) independent of the density. Again, this differs from the three-dimensional result, which has a logarithmic dependence on the density. Two important differences between the spin-drag transresistivity and the ordinary Coulomb-drag transresistivity are pointed out. (i) The lnT singularity at low temperature is smaller, in the Coulomb-drag case, by a factor e^-4k_Fd, where k_F is the Fermi wave vector and d is the separation between the layers. (ii) The collective mode contribution to the spin-drag transresistivity is negligible at all temperatures. Moreover, the spin-drag effect is, for comparable parameters, larger than the ordinary Coulomb-drag effect.