Antichiral spin order, its soft modes, and their hybridization with phonons in the topological semimetal ${\mathrm{Mn}}_3\mathrm{Ge}$

We report the magnetic structure and spin excitations of ${\mathrm{Mn}}_3\mathrm{Ge}$, a breathing kagome antiferromagnet with transport anomalies attributed to Weyl nodes. Using polarized neutron diffraction, we show the magnetic order is a $\mathbf{k}=\mathbf{0}$ coplanar state belonging to a ${\mathrm{\Gamma }}_9$ irreducible representation, which can be described as a perfect ${120}^{\circ }$ antichiral structure with a moment of 2.2(1) ${\mu }_B$/Mn, superimposed with weak collinear ferromagnetism. Inelastic neutron scattering shows three collective $\mathbf{Q}=0$ excitations at ${\mathrm{\Delta }}_1=2.9\left(6\right)$ meV, ${\mathrm{\Delta }}_2=14.6\left(3\right)$ meV, and ${\mathrm{\Delta }}_3=17.5\left(3\right)$ meV. A field theory of $Q\approx 0$ spin waves in triangular antiferromagnets with a ${120}^{\circ }$ spin structure was used to classify these modes. The in-plane mode ($\alpha$) is gapless, ${\mathrm{\Delta }}_1$ is the gap to a doublet of out-of-plane spin excitations (${\beta }_x,{\beta }_y$), and ${\mathrm{\Delta }}_2$, ${\mathrm{\Delta }}_3$ result from hybridization of optical phonons with magnetic excitations. While a phenomenological spin Hamiltonian including exchange interactions, Dzyaloshinskii-Moriya interactions, and single-ion crystal field terms can describe aspects of the Mn-based magnetism, spin-wave damping [$\mathrm{\Gamma }=25\left(8\right)$ meV] and the extended range of magnetic interactions indicate itinerant magnetism consistent with the transport anomalies.

Figure 1

(a) The refined antichiral magnetic structure of ${\mathrm{Mn}}_3\mathrm{Ge}$ where the dark orange (blue) dots represent Mn (Ge) atoms. (b) The specific heat and the associated entropy released between 300 K and 400 K in ${\mathrm{Mn}}_3\mathrm{Ge}$. (c) Rocking scans through the $\mathbf{Q}=\left(002\right)$ Bragg peak for four different polarized neutron scattering cross sections measured in an applied field of 2 T. (d) A neutron order parameter measurement collected on the $\mathbf{Q}=\left(001\right)$ Bragg peak of ${\mathrm{Mn}}_3\mathrm{Ge}$ sensitive to magnetic diffraction through a multiple scattering process that involves the magnetic (302) reflection. Calculated and observed polarized beam Bragg diffraction cross sections for ${\mathrm{Mn}}_3\mathrm{Ge}$ are shown in (e) for zero field and in (f) for a 2-T field. For a closer inspection of these data see Table 2. In (c), HF means the neutron polarization vector is parallel to the momentum transfer $\mathbf{Q}$, VF means the polarization vector is perpendicular to the scattering plane and $+/-$ refers to the spin states of the incident or scattered neutrons.

Figure 2

The magnetic structure of ${\mathrm{Mn}}_3\mathrm{Ge}$, which is associated with the ${\mathrm{\Gamma }}_9$ irreducible representation and can be decomposed into (a) an antichiral component and (b) a ferromagnetic component. The in-plane orientation of each component of the spin structure is defined by a global rotation angle for all spin, which we define as ${\theta }_{\chi }$ and ${\theta }_f$ for the antichiral and ferromagnetic components respectively, and as indicated on the figure. The dark orange (blue) dots represent the Mn (Ge) ions. Calculated and observed polarized beam Bragg diffraction cross sections for ${\mathrm{Mn}}_3\mathrm{Ge}$ are shown in (c) for zero field and in (d) for a 2-T field. For a closer inspection of these data see Table 2. HF means the neutron polarization vector is parallel to the momentum transfer $\mathbf{Q}$, while VF means the polarization vector is perpendicular to the scattering plane, while $+/-$ refers to the spin states of the incident or scattered neutrons.

Figure 3

Angular parameters associated with the antichiral (${\theta }_{\chi }$) and the ferromagnetic (${\theta }_f$) components of the magnetic order in ${\mathrm{Mn}}_3\mathrm{Ge}$ in a 2 T field [(a)–(c)]B∥[010] and [(d)–(f)]B∥[120] as determined by polarized and unpolarized neutron diffraction. Panels (a) and (f) show the ${\chi }^2$ goodness of fit for the two field directions as a function of the angular parameters ${\theta }_{\chi }$ and ${\theta }_f$ that define the magnetic structures as depicted in Fig. 2. The white lines in (a) and (f) indicate the experimental constraints on ${\theta }_{\chi }$ and ${\theta }_f$. Panels (b) and (d) represent the antichiral components of the magnetic structure refined for each of the two field directions, while (c) and (e) are the corresponding refined ferromagnetic components.

Figure 4

Time-of-flight neutron-scattering data for ${\mathrm{Mn}}_3\mathrm{Ge}$ collected for $E_i=40$ meV [(a), (b), (c), (d), (f), and (h)] and for $E_i=300$ meV [(e) and (f)]. (a) inelastic neutron scattering from ${\mathrm{Mn}}_3\mathrm{Ge}$ acquired for $E_i=40$ meV for wave vector transfer along high symmetry trajectories of the hexagonal Brillouin zone. (b) The energy transfer weighted neutron-scattering cross section at low energies and for momentum transfer near the zone center. Both (a) and (b) show data averaged over $\pm 0.05{\text{Å}}^{-1}$ in the perpendicular $\mathbf{Q}$ directions. (c) Shows a constant energy slice within the $\left(h0\ell \right)$ plane centered at $\mathbf{Q}=\left(101\right)$, and (d) within the $\left(hk0\right)$ plane. Data from (c) and (d) were averaged over $\pm 0.07{\text{Å}}^{-1}$ in the perpendicular momentum directions. High-energy inelastic neutron scattering acquired for $E_i=300$ meV and wave vector transfer along high symmetry trajectories of the hexagonal Brillouin zone. (f) Energy cuts at 4 different symmetry points are plotted in (f) where the horizontal line represents the FWHM energy resolution. Panel (g) shows the $\left(00\ell \right)$ dependence of inelastic neutron scattering between two magnetic zone centers, which reveals acoustic phonon dispersion. The momentum dependence of the neutron-scattering cross section from phonons is shown in (h), where the black dashed line is proportional to ${\left|\mathbf{Q}\right|}^2$. The error bars in (h) and in all other figures of the paper correspond to 1 standard deviation. The data in panels (e)–(h) were averaged over $\pm 0.1{\text{Å}}^{-1}$ in perpendicular $\mathbf{Q}$ directions.

Figure 5

Interlayer exchange interactions (dashed and dotted lines) between blue spins in the blue bottom layer and red spins in the red top layer. Spins are oriented as in the long-range ordered structure of ${\mathrm{Mn}}_3\mathrm{Ge}$. (a) $J_1$, which is weak and antiferromagnetic for ${\mathrm{Mn}}_3\mathrm{Ge}$ is marked with dotted lines, and $J_3$, which is ferromagnetic is marked with dashed lines. We drop $J_3$ in our model as it produces a similar effect on the spin-wave dispersion as $J_1$. (b) $J_4$, which is strong and ferromagnetic for ${\mathrm{Mn}}_3\mathrm{Ge}$ is indicated by dashed lines.

Figure 6

(a) A single plaquette of the Mn sites in ${\mathrm{Mn}}_3\mathrm{Ge}$ showing the ground-state spin structure with blue arrows. The spins carry the same labels as the site, i.e., spin $S_i$ is at site $r_i$. (b) The normal modes for the spin structure with the in-plane $\alpha$ modes, the blue arrows indicate the ground state, while the orange arrows indicate the distorted state on the top. The out-of-plane $\mathit{\beta }$ modes are shown at the bottom. The three hard modes carry net spin indicated by the thick orange arrow beside $({\alpha }_x,{\alpha }_y)$ and the out-of-plane arrow beside the ${\beta }_0$ mode. The three soft modes are surrounded by a dashed line.

Figure 7

(a) The excitation spectrum of ${\mathrm{Mn}}_3\mathrm{Ge}$ at $\mathbf{Q}=\left(110\right)$ compared to the calculated spin-wave spectrum based on two spin Hamiltonian modes including the effects of the instrumental resolution. Model 1 has the $\alpha$, ${\beta }_x$, and ${\beta }_y$ modes located near ${\mathrm{\Delta }}_1$, ${\mathrm{\Delta }}_2$, and ${\mathrm{\Delta }}_3$ respectively. Model 2 has the $\alpha$ mode gapless, while ${\beta }_x$ and ${\beta }_y$ are degenerate and gapped by ${\mathrm{\Delta }}_1$. (b) The excitation spectrum at $\mathbf{Q}=\left(120\right)$ where phonon contributions to the scattering intensity dominate. The scattering data were averaged over perpendicular directions of momentum transfer covering $\pm 0.15{\text{Å}}^{-1}$ in (a) and over $\pm 0.4{\text{Å}}^{-1}$ in (b). The error bars within both panels correspond to 1 standard deviation.

Figure 8

Results of refinement for diffraction at $B=0$ T. (a) ${\chi }^2$ against the antichiral component and (b) the ferromagnetic moment.(c) Rocking scans of (002) in and out of the magnetic phase with the HF configuration, the non-spin-flip scattering is also displayed for reference. (d) The $\chi$ scan for (002) performed at room temperature indicates the spin-flip scattering is not produced by multiple scattering.

Figure 9

Bulk magnetization of the ${\mathrm{Mn}}_3\mathrm{Ge}$ single crystal that was used for the diffraction experiments reported in this paper. The data were collected at $T=10$ K and the field was applied along the [100] direction. The inset details the low field regime showing a zero-field magnetization of about $0.007{\mu }_B$/Mn.

Figure 10

Representation of the atomic layers at the boundary of a (112) $D{0}_{22}$ tetragonal phase of ${\mathrm{Mn}}_3\mathrm{Ge}$ and a (001) $D{0}_{19}$ hexagonal phase of ${\mathrm{Mn}}_3\mathrm{Ge}$. The bottom shaded layer is the hexagonal phase, and the dark orange and blue dots represent the Mn and Ge ions, respectively.