Non-reciprocal magnons in non-centrosymmetric MnSi

Using two cold-neutron triple-axis spectrometers we have succeeded in fully mapping out the field-dependent evolution of the non-reciprocal magnon dispersion relations in all magnetic phases of MnSi. The non-reciprocal nature of the dispersion manifests itself in a full asymmetry (non-reciprocity) of the dynamical structure factor $S(q, E, \mu_0 H_{int})$ with respect to flipping either the direction of the applied magnetic field $\mu_0 H_{int}$, the reduced momentum transfer $q$, or the energy transfer $E$.


INTRODUCTION
The chiral itinerant-electron magnet MnSi crystallizes in the non-centrosymmetric P2 1 3 space group. The group's lack of inversion symmetry enforces a Dzyaloshinskii-Moriya interaction and gives rise to the rich magnetic phase diagram of MnSi: 1 Below a critical magnetic field of µ 0 H int c1 = 0.1 T and temperatures lower than T c = 29.5 K the Mn electron spins arrange in a helical pattern with a pitch of k h = 0.036 Å −1 . 2 A conical spin phase is found for fields above µ 0 H int c1 and below µ 0 H int c2 = 0.55 T. The small magnitude of k h compared to the chemical unit cell causes a pronounced backfolding of the magnon dispersion branches and an ensuing bandlike dispersion. 2-4 Above µ 0 H int c2 , the spins align along the external field direction, forming the field-polarized phase. Most interestingly, a small region at the border of the conical phase has been found to contain a skyrmion phase where the Mn magnetic momenta align in a vortexlike fashion. 1 In the early 1980s, G. Shirane et al. 5 found first clues to asymmetric scattering phenomena in MnSi caused by its non-centrosymmetric unit cell. They discovered that the helical arrangement of the Mn momenta is single-handed, i.e. either fully left-or fully right-handed. As result they deducted that the dispersion in the field-polarized phase is asymmetric with respect to the spin-flip channels. Recently, the non-reciprocal behavior of the field-polarized state was further investigated by S. Grigoriev et al. 6 as well as T. J. Sato et al.,7 confirming that the dispersion in that region takes the form of asymmetric, non-centered parabolic branches as shown in the right-hand panel of Fig. 1. In the paramagnetic phase above T c , B. Roessli et al. 8 found asymmetric contributions to critical scattering. First investigations into the magnon dynamics of the skyrmion phase were conducted by M. Janoschek. 9 This contribution serves a twofold purpose: We first give a concise summary of our recent work on the non-reciprocal dynamics in the conical and field-polarized phases of MnSi, 10 also presenting new data. Secondly, we show our very recent 11 results which prove that the nonreciprocal phenomena of the conical phase evolve into the elusive skyrmion phase.

II. EXPERIMENTAL SETUPS
For all of our experiments in the helimagnetic, the conical, the field-polarized, and the skyrmion phases of MnSi, we employed the cold-neutron triple-axis spectrometers MIRA 12 and TASP. 13 The results for each phase are discussed in the following sections.
For the experiments at MIRA we used two collimators with a maximally allowed divergence of 30', one before and one after the sample. A beryllium filter was placed in the incoming beam in order to avoid contamination with neutron energies above E i = 5 meV. The monochromator was vertically curved, the analyzer was flat. The experiments at MIRA were performed at fixed incident neutron energies of 3.19 meV ≤ E i ≤ 4.06 meV. At TASP we used collimators of 40' before and after the sample, a beryllium filter, a vertically curved monochromator and a flat analyzer. The TASP experiments were performed at fixed final neutron energies in the range 4.06 meV ≤ E f ≤ 4.66 meV.
All measurements were performed around the G = (110) nuclear Bragg reflection of MnSi. The individual measurements were performed with q µ 0 H int , this is also indicated by the shorthand notation q . Except for the measurements in the skyrmion phase, we utilized T = 20 K. For the skyrmion measurements, both the temperature and the field were scanned to yield optimal intensity on the skyrmion satellites. For all experiments we used a cylindrical MnSi single crystal of 0.5 cm radius and 3 cm height and corrected for demagnetization effects due to the sample geometry. 14 arXiv:1805.08750v1 [cond-mat.str-el] 22 May 2018

III. FIELD-POLARIZED PHASE
Our investigation of the field-polarized magnons 10 comprised const-q scans with q µ 0 H int 110 . Here, the Mn spins fully align along the field and the magnetic structure becomes commensurate in this case. Nevertheless, the dispersion branches ( Fig. 1) are centered around the q = ±k h positions where in the helical and conical phase the incommensurate helimagnetic satellite reflections would be located. It is striking that -depending on whether the magnon is created or annihilated -the dispersion branches are asymmetric in that they center around either q = +k h or q = −k h . This behavior can be seen in the different absolute energies of the (1) and (3') peak in the left panel of Fig. 2. Reversing the field polarity (right panel of Fig. 2) flips the centering of each of the +E and -E parabolic dispersion branches towards the other respective satellite position.

IV. HELICAL AND CONICAL PHASE
Unlike the field-polarized phase, the helimagnetic 2,3 and conical 10 phases are both marked by a seemingly symmetric dispersion relation consisting of three parabolic branches (left panel of Fig. 1), respectively centered around q = ±k h and q = 0. For finite fields, the dispersion nevertheless shows the same asymmetries as the field-polarized phase, only here the effect is purely due to an ever increasing asymmetry in the distribution of the spectral weights with increasing field (the scattering geometry is the same as in the field-polarized measurements). A flipping of the spectral weights between the q = +k h and the q = −k h branches can be observed upon reversing the field polarity, see Fig. 3. Here, the labels (1) and (3) refer to the dispersion branches which are centered around the magnetic satellites and (2) to the branch centered around the nuclear Bragg reflection. For increasing fields, the branch having q = 0 centering loses all spectral weight and is forbidden in the fully field-polarized phase. 10

V. SKYRMION PHASE
We performed const-q scans with q µ 0 H int [110]. For this configuration, which is depicted in the top-left panel of Fig. 4, the skyrmion plane aligns perpendicular to the (hk0) scattering plane with its normal vector parallel to µ 0 H int . For q µ 0 H int the dynamical structure factor S (q, E, µ 0 H int ) shows the same full nonreciprocity in all three variables q, E, and µ 0 H int as in the conical and the field-polarized phases. In Fig. 4

VI. CONCLUSION
We could successfully identify the non-reciprocity of the dispersion in all ordered magnetic phases of MnSi, amending and concluding our work that started in Ref. 10. The non-reciprocity of the magnetic fluctuations in the paramagnetic phases was already demonstrated before. 8 The next goal will be a combination of comprehensive data as obtained recently using triple-axis and time-of-flight techniques with novel theoretical models as obtained within our large collaboration. In difference to the field-polarized phase, all three parabolic branches (see Fig. 1) are theoretically allowed. The asymmetry in the dispersion is solely created by a field µ0H int , momentum q, and energy E dependent asymmetric distribution of the spectral weights. The two panels show the same scan with only the polarity of the field µ0H int reversed. The lines are an instrumental resolution-convolution 15  . Our very recent measurements of the non-reciprocal magnons in the skyrmion phase 11 found the same asymmetries upon reversal of either E, q or µ0H int as in the conical and field-polarized phases. Furthermore, the discernible magnons have a lower stiffness than the magnons in the other phases, consistent with the symmetric dispersion. 9 The lines are Gaussian fits and merely serve as guides to the eye.