# The Helical Magnet MnSi: Skyrmions and Magnons

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## Abstract

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## 1. What Makes MnSi So Interesting?

## 2. How Everything Began

**Q**” —and showed a disappearance of the magnetic intensity when entering the A-phase in the phase diagram. This was interpreted as a re-orientation of the magnetic helix out of the scattering plane [12], but Lebech had already pointed out that the diffraction pattern in this phase could probably not be explained by pure helical modulation of the spin structure [11].

## 3. From the Neutron Small-Angle Scattering Results to the Skyrmion Interpretation

**Q**, but some of these measurements were also performed along

**Q**. Apart from the A-phase, the results confirmed the picture described above with the hierarchy of the three interactions. However, in the A-phase, when applying

**B**parallel to

**Q**, a six-fold geometry showed up independent of the orientation of the underlying crystal. This result has been explained as the formation of a skyrmion lattice. Theoretically, it can be described by a three-fold Q-structure with Q = 0, which has a local minimum of the free energy inside the A-phase. Due to thermal fluctuations, this structure is stabilized in a global minimum in the A-phase. This leads to the picture of a hexagonal vortex-like skyrmion lattice or crystal in MnSi. The concept of skyrmions originated from Skyrme [14], who used it to describe the low-energy dynamics of mesons and baryons. Bogdanov, Hubert, and Yablonskii [15,16,17] then used the name “skyrmion” to describe vortex-like states in condensed-matter physics, and in 2002, Bogdanov et al. [18] introduced the notation of chiral skyrmions, which is what we simply call “skyrmions” today.

## 4. Magnons in MnSi

#### 4.1. Helimagnons

**B**${}_{c1}$ = 100 mT (see Kugler et al. [27]). This procedure allowed to directly identify the individual bands of the theoretically proposed band structure (see Figure 2) for finite Q. We showed that the theory required further refinements by introducing a higher-order correction term $\Delta F={\rho}_{s}A{\left({\nabla}^{2}n\right)}^{2}/\left(2{k}_{h}2\right)$ in the free energy, with the unit vector

**n**pointing along the magnetization M and ${\rho}_{s}$ being the stiffness density [27]. Fitting the data yields, for the dimensionless constant A, the value $A=-0.0073\pm 0.0004$, providing an excellent description of the data. MnSi can therefore be viewed as a one-dimensional helimagnetic crystal. The low-energy bands are basically equivalent to the physics of particles trapped in a one-dimensional potential. This result is highly relevant because the skyrmions in MnSi are related particles that are trapped by a similar potential and stabilized on the energy scales of the helimagnetic state.

#### 4.2. Conical and in the Field-Polarized Phase

**B**${}_{c1}$, moving up to the second critical field

**B**${}_{c2}$ [30] at much smaller momentum transfer and energies. Observing the transition from the band structure in the helical and conical phases towards a single mode in the field-polarized ferromagnetic phase, we demonstrated an excellent agreement of the theoretically predicted helimagnon energies and their spectral weights.

#### 4.3. Skyrmion Phase

**Q**= 0, the spectral weights condense at three distinct energies, which have been identified as a clockwise, counter-clockwise, and breathing motion of the skyrmion vortex [26]. While these magnons at finite

**Q**possess a still lower spin-wave stiffness than in the conical phase, we recently found that the magnons of both phases share the same asymmetric phenomena as the magnons in the conical phase, where the spectral weight for creating a magnon in the skyrmion phase is markedly different than for annihilation as long as the momentum transfer is along the normal of the skyrmion plane [33].

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DMI | Dzyaloshinsky-Moriya interaction |

SOC | Spin-orbit coupling |

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**Figure 1.**(

**Left**) The (B-T)-phase diagram of MnSi. The A-phase contains the skyrmion lattice (reproduced from [1], © by the American Association for the Advancement of Science (AAAS). (

**Right**) The (T-P)-phase diagram of MnSi. The grey shaded area designates the non-Fermi-liquid phase (NFL). The spheres show directions of strong elastic neutron intensity in reciprocal space (reproduced from [2], © by Springer Nature).

**Figure 2.**The figure (reproduced from [27], © by the American Physical Society, USA) shows an energy scan measured at T = 20 K and for a fixed reduced momentum transfer q perpendicular to the wave vector ${k}_{h}$ of the helix. Five bands are readily resolved and can be parametrized using a multi-Gaussian spectrum. The elastic peak appears due to incoherent scattering.

**Figure 3.**The figure (reproduced from [30], © by the American Physical Society, USA) shows the field-dependent evolution of the helimagnon excitations for the reduced momentum transfer

**q**parallel (

**left**) and

**q**perpendicular (

**right**) to the helix in units of the spiral pitch.

**Figure 4.**The figure (reproduced from [33], © by the American Institute of Physics, USA) shows the non-reciprocal nature of the skyrmion dynamics for momentum transfers along the skyrmion axis. The lines are Gaussian fits.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Georgii, R.; Weber, T.
The Helical Magnet MnSi: Skyrmions and Magnons. *Quantum Beam Sci.* **2019**, *3*, 4.
https://doi.org/10.3390/qubs3010004

**AMA Style**

Georgii R, Weber T.
The Helical Magnet MnSi: Skyrmions and Magnons. *Quantum Beam Science*. 2019; 3(1):4.
https://doi.org/10.3390/qubs3010004

**Chicago/Turabian Style**

Georgii, Robert, and Tobias Weber.
2019. "The Helical Magnet MnSi: Skyrmions and Magnons" *Quantum Beam Science* 3, no. 1: 4.
https://doi.org/10.3390/qubs3010004