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

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We calculate the non-linear charge currents in a two-dimensional electron system with Rashba ( α ) and Dresselhaus ( β ) linear spin–orbit interactions (SOIs) in the presence of in-plane electric E and magnetic B fields. Working in a rotated system of coordinates that introduces α ± β as effective couplings on perpendicular directions, we show that the currents, quadratic in the electric field, exist for all values of the chemical potential µ , above or below the band crossing energy E 0 . Our formalism uses a second order correction to the particle distribution function δ f ( 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, where μ > E 0 , 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 with α ± β when E ⊥ B and with ( α ± β ) 2 / ( α ∓ β ) when E ∥ B , a result independent of the presence of any additional SOIs, such as the cubic Dresselhaus or warping.