Hund’s Rule-Driven Dzyaloshinskii-Moriya Interaction at $3d\text{−}5d$ Interfaces

Using relativistic first-principles calculations, we show that the chemical trend of the Dzyaloshinskii-Moriya interaction (DMI) in $3d\text{−}5d$ ultrathin films follows Hund’s first rule with a tendency similar to their magnetic moments in either the unsupported $3d$ monolayers or $3d\text{−}5d$ interfaces. We demonstrate that, besides the spin-orbit coupling (SOC) effect in inversion asymmetric noncollinear magnetic systems, the driving force is the $3d$ orbital occupations and their spin-flip mixing processes with the spin-orbit active $5d$ states control directly the sign and magnitude of the DMI. The magnetic chirality changes are discussed in the light of the interplay between SOC, Hund’s first rule, and the crystal-field splitting of $d$ orbitals.

Figure 1

(a) Calculated magnetic moments of the $3d$ TM monolayers on $5d$ substrates compared to the moments of the $3d$ UML indicated by the dashed black line. (b) The magnetic moments of interface $5d$ atoms [44]. (c) The magnetic order of $3d$ monolayers on $5d$ substrates using different configurations: the FM, row-wise $p(1\times 2)$-AFM, and checkerboard $c(2\times 2)$-AFM states for the square lattice (001); the FM and AFM states for the (111) and (0001) oriented surfaces. A positive $\mathrm{\Delta }E=E_{\mathrm{AFM}}-E_{\mathrm{FM}}$ indicates a FM ground state, while negative values denote an AFM order.

Figure 2

(a) Strength and sign of the Dzyaloshinskii-Moriya interaction $D^{\text{tot}}$ in $3d$ TM monolayers on $5d$ substrates calculated around their magnetic ground state combining the relativistic SOC effect with spin spirals. A positive sign of $D^{\text{tot}}$ indicates a left-rotational sense or “left chirality”. (b) Correlation between $E_{\mathrm{DMI}}\sim D^{\text{tot}}/M_{3d/5d}^2$ averaged over $3d/5d$ interfaces (black line) versus the adlayer, the magnetic moments in the $3d$ TM UML (dashed red line), and the local magnetic moment per atom averaged over $3d/5d$ interfaces (solid red line).

Figure 3

(a) Layer-resolved $E_{\mathrm{DMI}}^{\mu }(\mathbf{q})$ of the long-period lengths for $\mathrm{Mn}/\mathrm{W}(001)$ and (b) the DMI strength $D^{\mu }$ extracted through a linear fit [$E_{\mathrm{DMI}}^{\mu }(\mathbf{q})\approx D^{\mu }q$]. Note that the positive sign of $D^{\text{tot}}$ ($D^{\text{tot}}={\mathrm{\Sigma }}_{\mu }{D}^{\mu }$) indicates left chirality.

Figure 4

Left: filling with electrons of $3d$ TM elements into the five $3d$ orbitals according to Hund’s first rule; spin-up and spin-down are shown by red and blue arrows, respectively. On the right side we show the spin-split band positions of $3d$ states with respect to $5d$ W states. Note, since the $5d$ bandwidth is significantly larger than the crystal-field splitting the $5d$ states are degenerate at the Fermi level. ${\mathrm{\Delta }}_{\mathrm{CF}}$ indicates the crystal-field splitting between the $t_{2\mathit{g}}$ and $e_{\mathit{g}}$ shells.