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 Néel domain walls in magnetic thin films is discussed in light of our results.

Figure 1

Spin eigenfunctions for the 2DEG with Rashba (left) and Dresselhaus (right) spin-orbit coupling.

Figure 2

Numerical results for the ratios $D_y/J$ and ${\gamma }_{yy}/J$ as a function of $\alpha$ (in eV Å) in the large-$E_F$ and large-$R$ limit. The points connected by lines indicate the numerical results, while the smooth lines are analytical results, Eqs. (29, 30, 31). Note from the same equations that the Fermi energy dependence drops out in the ratios, and here, $R=7.5$ Å was used.

Figure 3

Occupied momentum states in the low-$E_F$ limit, valid for cases of both Rashba-only and Dresselhaus-only SOC. The occupied states form a thin circular strip, resembling a 1D electron gas. We have defined ${\mathrm{\Delta }}_k=(k_1-k_2)/2$ and $k_0=(k_1+k_2)/2$, where $k_0=\alpha m/{\hslash }^2$ (Rashba) or $k_0=\beta m/{\hslash }^2$ (Dresselhaus).

Figure 4

The spin eigenfunctions and the Fermi surface for the case $\alpha =\beta$ and $\mathrm{\Delta }=0$, which consists of two circles with their centers shifted by the vector $\vec{Q}=(Q_x,Q_y)=2\sqrt{2}m\alpha (1,1)$. The two bands are indexed by $\nu =1$ for the outer shell and $\nu =2$ for the inner shell.

Figure 5

Computed magnetic interactions for equal Rashba and Dresselhaus SOCs and $\mathrm{\Delta }=0$. Here, $E_F=3.4$ eV, $\alpha =\beta =0.22\mathrm{eV}\AA$, distance $R$ is in angstroms, and the magnetic interactions are in units of ${10}^{-4}{\lambda }^{2}m$.

Figure 6

Bloch and Néel walls, right (RH) and left (LH) handed.