Helimagnon bands as universal excitations of chiral magnets

MnSi is a cubic compound with small magnetic anisotropy, which stabilizes a helimagnetic spin spiral that reduces to a ferromagnetic and antiferromagnetic state in the long- and short-wavelength limits, respectively. We report a comprehensive inelastic neutron-scattering study of the collective magnetic excitations in the helimagnetic state of MnSi. In our study we observe a rich variety of seemingly anomalous excitation spectra, as measured in well over 20 different locations in reciprocal space. Using a model based on only three parameters, namely, the measured pitch of the helix, the measured ferromagnetic spin wave stiffness and the amplitude of the signal, as the only free variable, we can simultaneously account for all of the measured spectra in excellent quantitative agreement with experiment. Our study identifies the formation of intense, strongly coupled bands of helimagnons as a universal characteristic of systems with weak chiral interactions.

Figure 1

(Color online) (a) Depiction of a low-energy spin wave excitation of a ferromagnet with wavelength ${\lambda }_{\text{ex}}$. The spins precess around the uniform magnetization. [(b) and (c)] Sketch of a similar excitation in the case of a helimagnet with a momentum, $\text{q}$, parallel to the propagation vector of the helix, $k_{\text{h}}$. The wavelength ${\lambda }_{\text{ex}}$ (b) is smaller or (c) much larger than the pitch of the helix ${\lambda }_{\text{h}}=\frac{2\pi }{k_{\text{h}}}$. Note that only for this parallel configuration the excitations of the ferromagnet and the helimagnet are very similar for short ${\lambda }_{\text{ex}}$. (d) Reciprocal space map around the nuclear (110) Bragg peak. The locations of the magnetic Bragg satellites ${\mathbf{k}}_{\mathbf{1}}$ to ${\mathbf{k}}_{\mathbf{4}}$ associated with the four domain configurations of the helimagnetic order are shown as color coded throughout the text. Here the actual positions of the satellites for the domains ${\mathbf{k}}_{\mathbf{2}}$ and ${\mathbf{k}}_{\mathbf{3}}$ are out of the scattering plane and only their projection onto the image plane is shown as emphasized by the open symbols. The locations at which the spin excitations were measured are shown by the black open symbols. They may be grouped along three trajectories, where the resolution ellipsoids are shown in gray shading. Note that all of the spectra recorded may be simultaneously accounted for by the model described in the text using just one parameter (the intensity).

Figure 2

(Color online) Typical constant-$\mathit{Q}$ scans at selected locations of the three trajectories shown in Fig. 1. Note that the energy and momentum resolution is at the technical limit currently available. The strong elastic peak at 0 meV is due to incoherent scattering. The inset in each panel shows the precise location in reciprocal space as a black spot where data were recorded. The curves represent the intensity calculated in the model described in the text, where all data are accounted for by the same values of the ferromagnetic spin-wave stiffness, the helical wavelength and the intensity. [(a) and (b)] Data for the trajectory with $\mathbf{q}\parallel {\mathbf{k}}_1$; these scans are dominated by a very broad maximum. [(c) and (d)] Data for $\mathbf{q}\perp {\mathbf{k}}_1$; these scans show very broad, essentially featureless intensity that decreases for increasing energy. [(e) and (f)] Data for an arbitrary trajectory emanating from ${\mathbf{k}}_1$; data for this trajectory are characterized by almost featureless intensity over an extremely wide range of energies and a distinct peak in a small range. The open data points below $-0.4\text{meV}$ in (f) represent a spurious signal that arises from additional incoherent scattering of neutrons from the monochromator crystals of the triple-axis spectrometer.

Figure 3

(Color online) Illustration of characteristic features of helimagnon bands. In the regime investigated, $q⪢k_{\text{h}}$, a crossover to a ferromagnetic dispersion with little weight in side bands was generally expected (see text). Instead, for finite $q_{\perp }$ multiple bands with approximately equal weight are excited due to significant Umklapp scattering. (a) For wave vectors strictly parallel to the helix a special symmetry prohibits higher-order Umklapp processes: a translation of the helix can be compensated by a rotation of all spins. As a consequence only three modes are excited, two modes with minima at $\pm {\mathbf{k}}_{\text{h}}$ and an additional mode centered directly at the position of the nuclear Bragg peak with zero momentum but vanishing intensity for $\mathbf{q}\to 0$. For comparison the dispersion of a ferromagnetic mode $\omega =c{q}^2$ is given (red solid line). (b) The strong Umklapp scattering for finite perpendicular momentum, $q_{\perp }=4k_{\text{h}}$, stops the motion of spin excitations with small $q_{\parallel }$ leading to flat bands [see text and Eq. (6)]. In both panels the spectral weight of the corresponding modes is proportional to the area of the points where we use a maximal size for better visibility. For the calculation of the dispersion we used $c\left(20\text{K}\right)=37\text{meV}{\text{Å}}^2$ and $k_{\text{h}}=0.035{\text{Å}}^{-1}$. For clarity only a single domain is shown, namely, ${\mathbf{k}}_1$.

Figure 4

(Color online) Dispersion curves for various directions as calculated in the model described in the text. Contributions from different domains are color coded as shown in Fig. 1. The area of each point is proportional to the weight of the corresponding peak in neutron scattering (only for the most intense peaks close to the reciprocal lattice vectors a fixed point size was used for better visibility). The complex lineshapes observed experimentally (cf. Fig. 2) result from the large number of excited bands. The gray shaded bars show the direction and parameter range of the constant $Q$ scans recorded in our measurements, where the width in $Q$ of the gray area represents the resolution in $Q$. Data for the gray bars labeled (a)–(d) are shown in Fig. 2. (a) Dispersion for the trajectory with $\mathbf{q}\parallel {\mathbf{k}}_1$. (b) Dispersion for the trajectory with $\mathbf{q}\perp {\mathbf{k}}_1$.