Dzyaloshinskii-Moriya interaction from unquenched orbital angular momentum

Orbitronics is an emerging and fascinating field that explores the utilization of the orbital degree of freedom of electrons for information processing. An increasing number of orbital phenomena are being currently discovered, with spin-orbit coupling mediating the interplay between orbital and spin effects, thus providing a wealth of control mechanisms and device applications. In this context, the orbital analog of the spin Dzyaloshinskii-Moriya interaction (DMI), i.e., orbital DMI, deserves to be explored in depth since it is believed to be capable of inducing chiral orbital structures. Here, we unveil the main features and microscopic mechanisms of the orbital DMI in a two-dimensional square lattice using a tight-binding model of $t_{2g}$ orbitals in combination with the Berry phase theory. This approach allows us to investigate and transparently disentangle the role of inversion symmetry breaking, strength of orbital-exchange interaction, and spin-orbit coupling in shaping the properties of the orbital DMI. By scrutinizing the band-resolved contributions, we are able to understand the microscopic mechanisms and guiding principles behind the orbital DMI and its anisotropy in two-dimensional magnetic materials, and uncover a fundamental relation between the orbital DMI and its spin counterpart, which is currently being explored very intensively. The insights gained from our work contribute to advancing our knowledge of orbital-related effects and their potential applications in spintronics, providing a path for future research in the field of chiral orbitronics.

Figure 1

Orbital texture of the spinless model. (a) The band structure of the $t_{2g}$ model along the high-symmetry directions for a two-dimensional square lattice calculated without SOC and spin/orbital-exchange interaction. (b) The band structure near $\mathrm{\Gamma }$ corresponding to the region marked with an orange circle in (a). The blue circle marks the region of strong orbital hybridization mediated by inversion symmetry breaking. (c)–(e) Band-resolved orbital textures in the $\mathbf{k}$ space near the $\mathrm{\Gamma }$ point, where $E_{1,\mathbf{k}}, E_{2,\mathbf{k}}$, and $E_{3,\mathbf{k}}$ denote the different bands as indicated in (a). The direction of orbital angular momentum is shown with an arrow, with a length corresponding to its magnitude (in arbitrary units).

Figure 2

Orbital DMI without SOC. (a) The orbital DMI variation with respect to the position of Fermi energy without SOC. (b) The band-projected orbital DMI along the high-symmetry path in the Brillouin zone at $E_F=2.7\mathrm{eV}$. The color marks the value of orbital DMI projected on each band. The projected orbital DMI is pronounced in the vicinity of the band avoided crossing between the $\mathrm{\Gamma }$ and $M$ points. (c) The $\mathbf{k}$-resolved orbital DMI along the high-symmetry path in the Brillouin zone at $E_F=2.7\mathrm{eV}$.

Figure 3

Orbital mixed Berry curvature. Distribution of the orbital mixed Berry curvature ${\mathrm{\Omega }}_{M_y^{\mathrm{L}}{k}_x}$ (left) and the orbital Dzyaloshinskii-Moriya spiralization $D_{yx}^{\mathrm{L}}$ (right) of all occupied bands in the Brillouin zone of the model for two Fermi energy values: (a),(b) $E_F=1.0\mathrm{eV}$ and (c),(d) $E_F=2.7\mathrm{eV}$. All calculations are performed without considering SOC. The data are shown on a logarithmic scale, $\mathrm{sgn}\left(x\right){log}_{10}(1+|x\left|\right)$.

Figure 4

The anatomy of orbital DMI. The orbital DMI $\left(D_{yx}^{\mathrm{L}}\right)$ as a function of the Fermi energy $\left(E_F\right)$ for (a) different orbital-exchange coupling strengths $J_{\mathrm{L}}$ and (b) inversion symmetry-breaking strengths $\gamma$. The orbital DMI $\left(D_{yx}^{\mathrm{L}}\right)$ as a function of the Fermi energy $\left(E_F\right)$ for different angles $\theta$ of the orbital-exchange field with the $z$ axis in the ranges of (c) $0^{\circ }$ to ${45}^{\circ }$ and (d) ${45}^{\circ }$ to ${90}^{\circ }$. All calculations take into account the effect of SOC (${\lambda }_{\text{soc}}=0.04\mathrm{eV}$).

Figure 5

Orbital DMI with varying $J_{\mathrm{L}}$. The band-projected orbital DMI along the high-symmetry path in the Brillouin zone at (a) $E_F=0.909\mathrm{eV}$ for $J_L=0.3\mathrm{eV}$ and (b) $E_F=1.939\mathrm{eV}$ for $J_L=0.7\mathrm{eV}$. The color marks the value of orbital DMI projected on each band. Clearly, the band avoided crossing region contributes primarily to the orbital DMI. (c),(d) The $\mathbf{k}$-resolved orbital DMI along the high-symmetry path in the Brillouin zone corresponding to the case in (a) and (b), respectively.

Figure 6

Orbital DMI with varying $\gamma$. The band-projected orbital DMI along the high-symmetry path in the Brillouin zone at $E_F=2.7\mathrm{eV}$ for (a) $\gamma =0.005\mathrm{eV}$ and (b) $\gamma =0.015\mathrm{eV}$. The color marks the value of orbital DMI projected on each band. Clearly, the band avoided crossing region primarily contributes to the orbital DMI. (c),(d) The $\mathbf{k}$-resolved orbital DMI along the high-symmetry path in the Brillouin zone corresponding to the case in (a) and (b), respectively.

Figure 7

Orbital DMI with varying $\theta$. The band-projected orbital DMI along the high-symmetry path in the Brillouin zone at (a) $E_F=2.7\mathrm{eV}$ for $\theta =0^{\circ }$ and (b) $E_F=0.2\mathrm{eV}$ for $\theta ={90}^{\circ }$. The color marks the value of orbital DMI projected on each band. (c),(d) The $\mathbf{k}$-resolved orbital DMI along the high-symmetry path in the Brillouin zone corresponding to the case in (a) and (b), respectively.

Figure 8

Variation of spin and orbital DMI with SOC. (a) The orbital DMI $\left(D_{yx}^{\mathrm{L}}\right)$ and (b) spin DMI $\left(D_{yx}^{\mathrm{S}}\right)$ as a function of the Fermi energy $\left(E_F\right)$ for various SOC strengths $\lambda$ in small SOC regime. (c) $D_{yx}^{\mathrm{L}}$ and (d) $D_{yx}^{\mathrm{S}}$ in large SOC regime. (e),(f) The band-resolved contributions to the (e) orbital and (f) spin DMI for the spin-orbit strength of ${\lambda }_{\mathrm{soc}}=0.04\mathrm{eV}$ at $E_F=2.2\mathrm{eV}$.