Polarized inelastic neutron scattering of nonreciprocal spin waves in MnSi

We report spin-polarized inelastic neutron scattering of the dynamical structure factor of the conical magnetic helix in the cubic chiral magnet MnSi. We find that the spectral weight of spin-flip scattering processes is concentrated on single branches for wave-vector transfer parallel to the helix axis as inferred from well-defined peaks in the neutron spectra. In contrast, for wave-vector transfers perpendicular to the helix the spectral weight is distributed among different branches of the magnon band structure as reflected in broader features of the spectra. Taking into account the effects of instrumental resolution, our experimental results are in excellent quantitative agreement with parameter-free theoretical predictions. Whereas the dispersion of the spin waves in MnSi appears to be approximately reciprocal at low energies and small applied fields, the associated spin-resolved spectral weight displays a pronounced nonreciprocity that implies a distinct nonreciprocal response in the limit of vanishing uniform magnetization at zero magnetic field.

Figure 1

Inelastic scattering processes between a neutron and a left-handed conical magnetic helix in the two spin-flip scattering channels SF($+-$) and SF($-+$). During the spin-flip process angular momentum $\pm \hslash$ is transferred to the helix exerting a torque. If the additional wavevector transfer is along the helix axis and commensurate, ${\mathbf{q}}_{\parallel }\approx \pm {\mathbf{k}}_h$, these processes activate a Goldstone spin wave mode associated with the broken translation symmetry. (Figure 1 was adapted from Ref. [23] under CC BY 3.0 [34].)

Figure 2

Low-energy spin-wave dispersion and spectral weight for the conical helix according to the theory outlined in Ref. [10]. The axes label wave vectors and energies both in dimensionless and dimensionful units where the latter was chosen for MnSi at 20 K with the amplitude of the helix wave vector $k_h=0.036{\AA }^{-1}$, lattice constant $a=4.558\AA$, the critical internal field ${\mu }_0{H}_{c2}^{\mathrm{int}}\approx 530$ mT, and $g\approx 2$ so that $g{\mu }_B{\mu }_0{H}_{c2}^{\mathrm{int}}\approx 0.06$ meV. Panels (a) and (b) show, respectively, the dependence on the wave vector longitudinal, ${\mathbf{q}}_{\parallel }$, and transverse, $q_{\perp }=\left|{\mathbf{q}}_{\perp }\right|$, to the magnetic field with ${\mu }_0{H}_{\mathrm{int}}=170$ mT. The thin gray lines represent the dispersion, whereas the width of the red and blue lines indicates the spectral weight of ${\chi }_{\pm \mp }^{″}(\mathbf{q},E)$ in the two spin-flip scattering channels SF($\pm \mp$), where positive and negative energies correspond to the creation and annihilation of a magnon, respectively. Close to the helimagnetic satellites, $q_{\parallel }=\pm k_h$, we depict the weights in a logarithmic scale due to their strong increase in that region. Note that in the configuration with the nuclear reciprocal lattice vector ${\mathbf{G}}_{\overline{1}\overline{1}0}\parallel {\mathbf{H}}_{\mathrm{int}}$ non-spin-flip scattering (NSF) does not contribute. Experimental energy scans were performed within the gray shaded areas labeled (i)–(iii) (see Figs. 4 and 5).

Figure 3

Elastic scans around the $\left(\overline{1}\overline{1}0\right)$ Bragg peak of MnSi at $T=20\mathrm{K}$ and ${\mu }_0{H}_{\mathrm{int}}=170\mathrm{mT}$. (a) Two magnetic satellites are observed at the positions $(-1.02,-1.02,0)$ and $(-0.98,-0.98,0)$ for spin-flip scattering $\mathrm{SF}(-+)$ and $\mathrm{SF}(+-)$, respectively, proving the single handedness of the crystal. The strong scattering at $\left(\overline{1}\overline{1}0\right)$ is due to nuclear Bragg scattering. (b) The transverse scan indicates the excellent quality of the crystal.

Figure 4

Spin-polarized inelastic neutron scattering (symbols) with a wave-vector transfer ${\mathbf{q}}_{\parallel }$ (i) antiparallel and (ii) parallel to the applied field. The energy scans correspond to the cuts (i) and (ii) indicated by the gray shaded areas in the theoretical spectrum of Fig. 2. The observed peaks can be clearly assigned to the corresponding Goldstone spin-wave modes. The solid lines are convolutions of the theoretical model with the instrumental resolution function. The scans were performed at $T=20\mathrm{K}$ and ${\mu }_0{H}_{\mathrm{int}}=170\mathrm{mT}$. The thick black horizontal lines depict the calculated instrumental energy resolution.

Figure 5

Spin-polarized inelastic neutron scattering (symbols) for a wave-vector transfer ${\mathbf{q}}_{\perp }$ perpendicular to the applied field at ${\mathbf{q}}_{\parallel }=0$. The energy scans correspond to the cut indicated by the gray shaded area (iii) in the calculated spectrum shown in Fig. 2. The contributions of a large number of magnon bands result in broad features. The solid lines are convolutions of the theoretical model with the instrumental resolution. Scans were recorded at $T=20\mathrm{K}$ and ${\mu }_0{H}_{\mathrm{int}}=170\mathrm{mT}$. The data point marked “Sp” is a spurious instrumental artifact. The calculated instrumental energy resolution is shown as a thick black horizontal line.