Dzyaloshinskii-Moriya interaction in the presence of Rashba and Dresselhaus spin-orbit coupling

Chiral order in magnetic structures is currently an area of considerable interest and leads to skyrmion structures and domain walls with certain chirality. The chiral structure originates from the Dzyaloshinskii-Moriya interaction caused by broken inversion symmetry and the spin-orbit interaction. In addition to the Rashba or Dresselhaus interactions, there may also exist substantial spin polarization in magnetic thin films. Here, we study the exchange interaction between two localized magnetic moments in the spin-polarized electron gas with both Rashba and Dresselhaus spin-orbit interaction present. Analytical expressions are found in certain limits in addition to what is known in the literature. The stability of the Bloch and N\'eel domain walls in magnetic thin films is discussed in light of our results.