Spin dynamics in MgCr 2 O 4

Inelastic neutron scattering experiments on single crystals were performed on the thermal neutron triple-axis spectrometer EIGER and the cold neutron triple-axis spectrometer TASP at SINQ of PSI. Six crystals with a total mass∼250 mg were co-aligned with (211) plane as the horizontal scattering plane. For the experiments on EIGER, double focusing PG monochroma-tor and horizontally focusing PG analyzer were employed, and the final neutron momentum k_f was fixed at 2.66 Å⁻¹. A PG filter was mounted in between the sample and the analyzer to remove high-order neutrons. For the experiments on TASP, vertically focusing PG monochro-mator and horizontally focusing PG analyzer were employed. And two instrument setups with k_f =1.4 Å⁻¹and 1.17 Å⁻¹were used. On both instruments, an orange cryostat was used and the measurements were performed at the base temperature of 1.5 K.

5.3.1 Resonance-like excitations

Fig. 5.8a and b summarize results from EIGER for the measurements along the (k−1.5k k−3) and (0.5k1−k) directions. Strong resonance-like excitations are observed at around 4.5 meV

5.3. Spin dynamics in MgCr2O4

along both directions, which correspond to the hexagonal modes that have been observed previously [200, 210, 217]. To see this, structure factors of the antiferromagnetic hexagons are calculated with averaging over all possible configurations, and the comparisons with the q-scans measured atE=4.5 meV are presented in Fig. 5.8c and d. The calculations capture the measuredq-dependence of the intensities very well, which confirms the correctness of the effective cluster excitation model.

5.3.2 Low-energy spin wave excitations for theq1components

As is shown in Fig. 5.8a and b, spin wave excitations are observed to emerge from the long-range order Bragg positions and extend up to the resonance modes at 4.5 meV. Using the cold neutron spectrometer TASP, dispersions of these low-energy spin wave excitations were

(k-1.5 k k-3)

Figure 5.9 –(a)2D map for the spin wave excitations around (0 1.5 1.5) measured on TASP along the (k−1.5k k−3) and (0k1−k) directions. The dotted lines indicate the measured q-scans. A logarithmic intensity scale is used.(b)The calculated spin wave excitations for the collinear magentic structure shown in Fig. 5.6a in the same region as that of panel (a). Uniform antiferromagnetic couplings ofJ1=3.0 meV together with an (001) easy-plane anisotropy D=0.005 meV are employed for the calculation. The right-pointing (downwards-pointing) triangles indicate the measured spin wave positions extracted from Gaussian fits of theq-scans (E-scans). A logarithmic intensity scale is used. (c)Representativeq-scan atE=2.25 meV measured along the (k−1.5k k−3) direction. The solid line shows the fitting results using two Gaussian functions plus a constant background.(d)RepresentativeE-scan measured at q=(0 1.5 1.5) showing the existence of a 0.80 meV excitation gap. The solid line shows the fitting results using one Gaussian function for the spin wave excitation and one additional Gaussian function at E = 0 meV for the elastic Bragg peak and incoherent scattering.

mapped out and the results around the (0 1.5 1.5) position are summarized in Fig. 5.9a. Typical q-scan atE=2.25 meV andE-scan at (0 1.5 1.5) are shown in Fig. 5.9c and d, respectively.

By fitting the measurements with Gaussian functions, positions of spin wave excitations can be extracted and the results are summarized in Fig. 5.9b as the triangular marks. Of special importance is the observation of a 0.80 meV excitation gap for the spin wave excitation, which is compatible with the easy-plane anisotropy revealed from ESR measurements [227, 228].

Due to the lack of a unified description of the spin wave and resonance-like excitations, here we try to capture only the low-energy spin wave excitations using linear spin wave theory.

With the SpinW package [72], we implemented a cubic pyrochlore lattice with the collinear magnetic structure shown in Fig. 5.6a. A uniform exchange coupling of J₁=3.0 meV (AF) was assigned for all the nearest-neighbour couplings. And a (001) easy-plane anisotropy of D_z=0.005 meV was assumed for all the spins. As is shown in Fig. 5.9b, the calculated spin wave dispersion fits the measurements very well. For the coplanar magnetic structure shown in Fig. 5.6b, the calculated spin wave excitations will exhibit strong low-energy resonance-like excitations similar to that shown in Fig. 5.10b for theq₁component, although the dispersion of the spin wave branch is still close to that of the collinear structure.

5.3.3 Low-energy spin wave excitations for theq₂components

Fig. 5.10 shows the 2D map for the spin wave excitations around the Bragg peak (0.5 2 1) measured on TASP. Typicalq-scan atE=2.0 meV andE-scan at (0.5 2 1) are presented in Fig.

5.10c and d, respectively. By fitting the measurements with Gaussian functions, the positions of the spin wave excitations can be extracted and the results are summarized in Fig. 5.10 as the triangles. Similar to the gapped excitations around (0 1.5 1.5), a gap of 0.67 meV is observed in theE-scan at (0.5 2 1), which evidences similar easy-plane anisotropy effects for theq₁andq₂ components.

Using the SpinW package [72], the spin wave excitations were calculated for the coplanar structure shown in Fig. 5.7d. We note that a uniform rotation of the spins from theabplane to the (0 0.35 1) plane does not change the spin wave dispersion. Fig. 5.10b shows the calculation results with the parameters J₁=3.0 meV andD_z =0.01 meV. Despite the resonance-like excitations at 0.7 meV, the spin wave branch satisfactorily captures the measured spectra.

5.4 Summary

In this chapter we present our first results on the spin correlations in MgCr₂O₄, where the Heisenberg Cr³⁺spins constitute a pyrochlore lattice and the ground-state degeneracy is relieved through the spin-lattice coupling. Using synchrotron X-ray diffraction, the distorted lattice structure of MgCr₂O₄atT <12 K is found to be eitherF dddorI4₁/amd. Twoq-vectors (0.5 0.5 0) and (1 0 0.5) are observed in the distorted phase. With neutron diffraction and

5.4. Summary

Figure 5.10 –(a)2D map for the spin wave excitations around (0.5 2 1) measured on TASP along the (k−1.5k k−3) and (0.5k1−k) directions. The dotted lines indicate the measured q-scans. A logarithmic intensity scale is used. (b)The calculated spin wave excitations for the collinear magnetic structure shown in Fig. 5.7d in the same region as that of panel (a).

Uniform antiferromagenetic couplings of J1=3.0 meV together with an (001) easy-plane anisotropyD=0.01 meV are employed for the calculation. The right-pointing (downwards-pointing) triangles indicate the measured spin wave positions extracted from the gaussian fits of theq-scans (E-scans). Notice that the spin wave positions at 2.5, 3.0, and 3.5 meV along the (k−1.5k k−3) direction are extracted from the experiment on EIGER. A logarithmic intensity scale is used. The solid line shows the fitting results using two Gaussian functions plus a constant background. (c)Representativeq-scan atE=2.0 meV measured along the (k−1.5k k−3) direction. (d)RepresentativeE-scan measured atq=(0.5 2 1) showing the existence of a 0.67 meV excitation gap. The solid line shows the fitting results using one Gaussian function for the spin wave excitation and one additional Gaussian function at E = 0 meV for the elastic Bragg peak and coherent scattering.

spherical neutron polarimetry, theq₁=(0.5 0.5 0) magnetic structure is found to be either collinear along the (110) direction or coplanar with spins along thea andbaxes, and the q₂=(1 0 0.5) magnetic structure is uniquely determined to be coplanar in the (0 0.36 1) plane.

Using inelastic neutron scattering, we study the spin excitations in MgCr₂O₄. A strong resonance-like mode is observed atE =4.5 meV, whose distribution in reciprocal space can be fitted by anti-ferromagnetic hexagons. Below this resonance-like mode, spin wave exci-tations are observed, and the revealed excitation gap evidences the existence of an easy-plane anisotropy. As the first approximation, the measured spin wave dispersions are fitted using linear spin wave theory. Further work is required to fully understand the spin dynamics in MgCr₂O₄, especially the coexistence of the spin wave and resonance-like excitations, which so far has not been described in a unified theory.