Dzyaloshinskii-Moriya Interaction as a Consequence of a Doppler Shift due to Spin-Orbit-Induced Intrinsic Spin Current

We present a physical picture for the emergence of the Dzyaloshinskii-Moriya (DM) interaction based on the idea of the Doppler shift by an intrinsic spin current induced by spin-orbit interaction under broken inversion symmetry. The picture is confirmed by a rigorous effective Hamiltonian theory, which reveals that the DM coefficient is given by the magnitude of the intrinsic spin current. Our approach is directly applicable to first principles calculations and clarifies the relation between the interaction and the electronic band structures. Quantitative agreement with experimental results is obtained for the skyrmion compounds ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ and ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Ge}$.

Figure 1

Schematic picture showing the mechanism of the spin current Doppler shift. The polarization $\mathit{s}$ of the spin current intrinsically determined by the spin-orbit interaction is deviated because of a torque due to the localized spin $\mathit{n}$. This deviation corresponds to the change of the electron spin’s laboratory frame and is described by a covariant derivative. The effect leads to a shift in the frequency of the localized spin dynamics, i.e., the Doppler shift.

Figure 2

Spin texture in momentum space for (a) Rashba, (b) Dresselhaus, and (c) Weyl-type Hamiltonians.

Figure 3

The DM coefficients $D$ for ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ and ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Ge}$ calculated using the energies of helical spin structures, $E(q)=Dq+J{q}^2$, and as expectation values of the equilibrium spin current $D=⟨{\widehat{j}}_s⟩$. The error bars for each calculation indicate the fitting errors of $E(q)$ and the variances of $D_x^x$ and $D_y^y$, respectively.

Figure 4

(a) Contribution of each band to the DM interaction, $D_{n\mathit{k}}$, with dominant band anticrossing points circled, and (b) the energy distribution of the DM interaction, $D(E)$ (black line), for FeGe. The Fermi energy dependence of the DM interaction, $D\equiv {\int }^{E_F}D(E^{\prime })f(E^{\prime })d{E}^{\prime }$, within the rigid band approximation is also shown as the red line.