Band Structure of Helimagnons in MnSi Resolved by Inelastic Neutron Scattering

A magnetic helix realizes a one-dimensional magnetic crystal with a period given by the pitch length ${\lambda }_h$. Its spin-wave excitations—the helimagnons—experience Bragg scattering off this periodicity, leading to gaps in the spectrum that inhibit their propagation along the pitch direction. Using high-resolution inelastic neutron scattering, the resulting band structure of helimagnons was resolved by preparing a single crystal of MnSi in a single magnetic-helix domain. At least five helimagnon bands could be identified that cover the crossover from flat bands at low energies with helimagnons basically localized along the pitch direction to dispersing bands at higher energies. In the low-energy limit, we find the helimagnon spectrum to be determined by a universal, parameter-free theory. Taking into account corrections to this low-energy theory, quantitative agreement is obtained in the entire energy range studied with the help of a single fitting parameter.

Figure 1

(a)–(e) Universal helimagnon spectrum for momenta $q_{\parallel }$ along the pitch direction ${\mathbf{k}}_h$ in the repeated zone scheme (thin lines) at zero field and ${\chi }_{\text{con}}^{\text{int}}=0.34$ for various values of perpendicular momentum $q_{\perp }$. The thickness of the colored lines indicates the theoretically expected intensity for inelastic neutron scattering on a domain with ${\mathbf{k}}_h\parallel [111]$ with respect to a nuclear $(1\overline{1}0)$ reflection and for ${\mathbf{q}}_{\perp }\parallel [11\overline{2}]$. The colors red and green distinguish between dispersive and practically flat helimagnon bands, respectively. The energy units on the right axes correspond to MnSi at 20 K for which $g{\mu }_B{\mu }_0{H}_{c2}^{\text{int}}=0.062\mathrm{meV}$. (f) Weights of inelastic contributions arising from an equal population of all domains [color coding as in (g)] for $q_{\perp }=3k_h$ as a function of ${\mathbf{q}}_{\parallel }=q_{\parallel }{\widehat{\mathbf{k}}}_h$ with respect to ${\mathbf{k}}_h\parallel [111]$ (red domain). (g) Scattering plane around the nuclear $(1\overline{1}0)$ peak with four domains that are degenerate at zero field; the open circles correspond to out-of-plane pitch wave vectors projected onto the plane.

Figure 2

(a)–(c) show the scattering planes for the setups 1, 2, and 3, respectively. The magnetic field (blue color code) defines the orientation of ${\mathbf{k}}_h$ resulting in magnetic satellite peaks (green squares); for (b) and (c), they are located above and below the scattering plane. Constant-$Q$ scans have been performed at the points in reciprocal space marked by red circles. (d)–(f) show examples of scans obtained with the three setups at 20 K, where the full lines are fits of multi-Gaussian profiles to the data. (d) Three helimagnon branches are measured with setup 1 (incident energies $E_i=3.19$ and 4.06 meV), while (e) setup 2 ($E_i=4.06\mathrm{meV}$) and (f) setup 3 ($E_i=5.04\mathrm{meV}$) clearly resolve the first two and five bands, respectively. The elastic peaks appear due to the $Q$-independent incoherent scattering. Note that the scans in (d) are shifted by 40 counts with respect to each other for clarity.

Figure 3

Comparison between theory and experiment (symbols). (a),(b) Helimagnon spectrum as a function of $q_{\perp }$ and $q_{\parallel }$ with nominally $q_{\parallel }=0$ and $q_{\perp }=0$, respectively. The universal spectrum (dotted lines) agrees with the data at low energies; the correction due to Eq. (6) is necessary (solid lines) for quantitative agreement at higher energies. The instrumental resolution slightly shifts the lower and center branch upwards [black dashed line in (b)]; see the text. (c) $T$ dependence of the two helimagnon modes lowest in energy measured at certain fixed momenta.