Effect of symmetry breaking on short-wavelength acoustic phonons in the chiral magnet MnSi

We have investigated the effects of spatial-inversion and time-reversal symmetry breaking on the acoustic phonon branches in chiral MnSi using high-resolution inelastic x-ray scattering and first-principles calculations. We find a momentum-transfer-dependent ( $q$ -dependent) splitting between transverse phonon bands having angular momentum parallel or antiparallel to $q$ . This is understood by a phenomenological theory using a gyrotropic tensor. We observed no significant impact from the time-reversal symmetry breaking induced by a magnetic field (energy shifts < 0.3 meV). This suggests the effect of time-reversal symmetry breaking is small or is restricted to a very-low-energy regime in this material, possibly due to small spin-orbit interaction.

Figure 1

(a) IXS profiles of MnSi along the high-symmetry line from a reciprocal lattice point (4, −4, 0) to $R$ (4.5, −3.5, 0.5). These plots are offset for clarity. The IXS spectra are fitted to a sum of damped harmonic oscillator functions convoluted with an instrumental resolution function. (b) Comparison of experiment and calculation of IXS data. The color map and white dots represent the calculated dynamical structure factor $S$($Q$, ω) and phonon dispersions. The green, purple, and black circles represent experimental peak positions obtained from the fitting of IXS spectra as shown in (a).

Figure 2

Projections of the phonon angular momentum into (a) the [111], parallel and (b) [11-2], perpendicular direction for dispersion along the [111] direction (Γ-$R$). LA and TA indicate longitudinal and transverse acoustic modes, respectively. (c) Magnitude of phonon angular momentum along the [111] direction for the lowest three branches. In the small-$q$ region, these three modes correspond to two TA modes and one LA mode, respectively. For the third lowest mode (LA mode), the data above $q=0.2$ (r.l.u.) for ($q, q, q$) are omitted because of the mixing with optical modes.

Figure 3

Experimental and calculated energy difference of acoustic phonon modes for MnSi as a function of the square of the wave vector normalized by that at the Brillouin zone edge. The dip in the small-$q$ region of the theoretical data is probably caused by the artifact owing to the calculation based on the finite size of the supercell (3×3×3). The experimental results of some other chiral materials [9, 10] are also shown for comparison.

Figure 4

Magnetic field dependence of transverse acoustic modes. (a) IXS spectra at 0 T and ± 0.7 T, 10 K, and $\mathbf{Q}=\left(4+h,-4+h,h\right)$ at $h=0.33$. A magnetic field is applied along [111], which is parallel to ${\mathit{J}}_{\mathrm{ph}}$. (b) Magnetization curve at 10 K along the [111] direction. (c), (d) Magnetic field dependence of (c) these peak energies and (d) difference between the energies.