Electronic structure and the origin of the Dzyaloshinskii-Moriya interaction in MnSi

The metallic helimagnet MnSi has been found to exhibit skyrmionic spin textures when subjected to magnetic fields at low temperatures. The Dzyaloshinskii-Moriya (DM) interaction plays a key role in stabilizing the skyrmion state. With the help of first-principles calculations, crystal field theory, and a tight-binding model we study the electronic structure and the origin of the DM interaction in the B20 phase of MnSi. The strength of the $\vec{D}$ parameter is determined by the magnitude of the spin-orbit interaction and the degree of orbital mixing, induced by the symmetry-breaking distortions in the B20 phase. Our calculations suggest strong coupling between Mn-$d$ and Si-$p$ states, which is consistent with a mixed valence ground state $|d^{7-x}{p}^{2+x}\rangle$ configuration. Consistent with previous calculations, we find that DFT+U leads to the experimental magnetic moment of $0.4{\mu }_B$, which redistributes electrons between the majority and minority spin channels. We derive the magnetic interaction parameters $J$ and $\vec{D}$ for Mn-Si-Mn superexchange paths using Moriya's theory assuming the interaction to be mediated by $e_g$ electrons near the Fermi level. Using parameters from our calculations, we get reasonable agreement with the observations.

Figure 1

Crystal structure of MnSi in (a) FCC and (b) B20 phases. The Mn and Si occupy equivalent positions. The unit cell is represented by the blue cube. Bonding along the (111) direction (dotted lines) in (c) FCC and (d) B20 structures are also shown. Three different kinds of bonds in B20 phase are marked in Angstrom. In the B20 structure, the ${\mathrm{Mn}}_4{\mathrm{Si}}_4$ cubes are alternately elongated and compressed along the (111) direction as indicated in (d).

Figure 2

(a) Contour plot of the total energy as a function of positional parameters $u_{\mathrm{Mn}}$ and $u_{\mathrm{Si}}$. Endpoints of the line joining the two minima correspond to B20 structure while its center corresponds to FCC. (b) Variation of energy along the line connecting B20 structures.

Figure 3

DFT band structure and density of states for the FCC phase of MnSi. First panel shows nonmagnetic band structure with orbital weights indicated by colored circles: red for Si-$s$, green for Si-$p$, and blue for Mn-$d$. The numbers stand for the symmetry characters of the bands as listed in Table 1. Second and third panel shows ferromagnetic band structures for spin and up and down channels and the corresponding total density of states is shown in the fourth panel. The partial density of states for Mn-$t_{2g}, e_g$, and Si-$p$ states are shown in the last two panels.

Figure 4

The bands of nonmagnetic MnSi in FCC (left) and B20 (right) phases in the four-molecule unit cell. Fermi energy is located at 0 eV. The labels mark dominant band character.

Figure 5

Spin polarized band structure and partial density of states in the B20 phase of MnSi. In the partial DOS, blue and red lines correspond to $e_g$ and $t_{2g}$ states, respectively.

Figure 6

Level splitting for Mn $d$ states in MnSi, as inferred from the DFT results, Figs. 4 and 5. In the trigonal distortion, the structure is expanded (or compressed) uniformly along the cube diagonal direction (111), while in the B20 structure, the cubes are alternately compressed and expanded along the same direction (see Fig. 1). The $“e_g/t_{2g}”$ characters for the B20 structure indicated on right shows the approximate character of the orbitals, referring to the cubic basis.

Figure 7

Effect of Hubbard parameter $U$ on the (a) magnetic moment and (b) density of states of the B20 phase from GGA+$U$ calculations. Spin of Mn ions remains close to 1 ${\mu }_B$ till $U=4$ eV and drops to 0.2 ${\mu }_B$ above 6 eV. The DOS is calculated for $U=6.8$ eV.

Figure 8

A typical superexchange path between two Mn atoms through an intermediate Si atom. Interaction is mediated by the hopping of the two electrons indicated by the black dots occupying the orbitals A0 and B0. ${\mathrm{A}}^{\prime }$ and ${\mathrm{B}}^{\prime }$ are the remaining four $d$ orbitals on each Mn (other than A0 and B0).

Figure 9

One of the superexchange paths from Mn(A) to Mn(B) through an intermediate Si that leads to the DM vector $\vec{D}$. If the spin-flip hopping term ${\vec{C}}_{k^{\prime },A}$ is replaced by standard hopping $V_{k^{\prime },A0}$, then the interaction leads to the Heisenberg interaction $J$.