Non-linear charge currents and the planar Hall effect in quantum wells with Rashba and Dresselhaus spin–orbit interactions

We calculate the non-linear charge currents in a two-dimensional electron system with Rashba (<i>α</i>) and Dresselhaus (<i>β</i>) linear spin-orbit interactions (SOIs) in the presence of in-plane electric<b>E</b>and magnetic<b>B</b>fields. Working in a rotated system of coordinates that introducesα±βas effective couplings on perpendicular directions, we show that the currents, quadratic in the electric field, exist for all values of the chemical potential<i>µ</i>, above or below the band crossing energy<i>E</i>₀. Our formalism uses a second order correction to the particle distribution functionδf(2)derived in a semi-classical approximation that takes into account the local change in the single-particle energy under the action of the electric field. In a quantum well, whereμ>E0, for a spin-orbit energy larger than the Zeeman splitting, the non-linear currents that flow perpendicular on the direction of the magnetic field are found to be proportional withα±βwhenE⊥Band with(α±β)2/(α∓β)whenE∥B, a result independent of the presence of any additional SOIs, such as the cubic Dresselhaus or warping.