Spiral spin-liquid state in MnSc 2 S 4

powder neutron diffraction data suggests a cycloidal structure as the ground state, which is at odds with the anisotropy effect of the dipolar interactions [156]. The origin of the second phase transition at 1.9 K is also unclear. This transition is observed in magnetic susceptibility measurements as a sudden drop but has remained elusive in local-probe experiments by NMR [174, 175] andµSR [176].

4.2 Spiral spin-liquid state in MnSc

S

In this section, we present the results of our neutron diffuse scattering study of high-quality single crystals of MnSc2S4. In the short-range correlated regionT>2.2 K, the spiral surface is directly observed, which establishes the spiral spin-liquid state in MnSc₂S₄. The spiral surface is observed to coexist with long-range order down to 1.5 K, which is consistent with previous studies on powder samples [173].

4.2.1 Crystal growth and basic characterizations

MnSc2S4single crystals were grown by Dr. Vladimir Tsurkan at University of Augsburg using the chemical transport technique. Ternary MnSc₂S₄in polycrystalline form was used as the starting material. Iodine, with a concentration of 5∼7 mg/cm³, was used for the transport agent. A quartz ampoule (of length 10∼15 cm, inner diameter 12∼15 mm) was placed in a two-zone furnace in a temperature gradient of 50^◦C with the source temperature between 950 and 1000^◦C. The growth took 1∼2 months depending on the initial mass. The crystals, which have an octahedral form and typical sizes of 1∼3 mm, were characterized by SQUID and X-ray diffraction, and exhibit properties similar to those of the stoichiometric powder samples. Altogether two batches of single crystals (bath number ATR184 and ATR198) whose transition temperatures vary within 0.1 K were used in our measurements.

observed intensity (a. u.)

0 2 4 6 8 10

calculated intensity (a. u.)

0 1 2 3 4 5 6 7 8 9 10

Figure 4.3 – Comparison between the observed and calculated intensities for the MnSc2S4

single crystal synchrotron X-ray diffraction. Space groupF d3mis used in the refinement. The inset shows one typical piece of MnSc₂S₄single crystal with the triangular edge∼1.5 mm.

Figure 4.4 – Magnetic diffuse scattering in the (hk0) plane of MnSc₂S₄measured at tempera-tures 2.9, 3.5, and 7 K. A measurement at 50 K has been subtracted as the background.

In order to check the quality of the grown single crystals, synchrotron X-ray diffraction experi-ments were performed on one crystal from ATR184. At room temperature, altogether 2738 Bragg reflections were collected on the Swiss-Norwegian Beamline at ESRF with the wave-lengthλ=0.6199 Å. By averaging over the symmetry equivalent reflections, 131 reflections were obtained for the refinement with FULLPROF [115]. Fig. 4.3 presents the comparison between the observed and calculated intensities for the MnSc₂S₄structure with theF d3m space group. The refinedxposition for the S site is 0.2574(1) and the agreement factors areR_f

= 1.92 andR_f2= 3.16. No Mn-Sc anti-site disorder is observed, confirming the good quality of our single crystals.

4.2.2 Direct observation of the spiral surface in MnSc₂S₄

To study the short-range spin correlations in MnSc₂S₄, we performed polarized neutron diffuse scattering experiment on the high-flux time-of-flight spectrometer DNS at MLZ (Garching, Germany). Three crystals (ATR184) with a total mass of∼30 mg were co-algined with the (hk0) plane horizontal. The pyrolytic graphite PG(002) monochromator together with a Be-filter were used to select the incoming neutron wavelength of 4.5 Å. To obtain the magnetic diffuse scattering signal, the intensities in thexspin-flip (SF) channel were measured, where thex direction is in the horizontal plane. Thex-SF measurements at 50 K were subtracted as the background.

Fig. 4.4 presents the obtained magnetic diffuse scattering atT =2.9, 3.5, and 7 K. With decreasing temperature, a squared-ring pattern gradually appear. Such a pattern is very different from that observed in CoAl₂O₄(see Fig. 4.2), where intensity concentrates at the Brillouin zone center. Considering that the spiral surface expands with the frustration ratio J₂/J₁, the observed square-ring pattern with intensities lying near the Brillouin zone boundary is an indication of strong frustration in MnSc₂S₄.

4.2. Spiral spin-liquid state in MnSc2S4

Figure 4.5 –(a)Diffuse scattering intensities in the (hk0) plane measured at 2.9 K. The white square outlines the area shown in the inset of panel c. (b)Monte Carlo simulations of the spin correlations in the (hk0) plane using theJ₁-J₂model with the ratio|J₂/J₁| =0.85 and T/|J₁| =0.55.(c)Inset: diffuse scattering intensities around (110) (contour outlined in panel c) measured at 1.9 K, showing the coexistence of Bragg peaks and the diffuse signal. The dashed arc describes the path for the 1D cut presented in the main panel, where the polar angleθ is used as thexaxis. The main panel shows a comparison of the 1D cut atT =2.9,1.9, and 1.5 K, demonstrating the diffuse signal extends down to the base temperature of 1.5 K. The horizontal dashed line indicates the background level averaged from 1D cuts with varying radius.

In order to fix the ratioJ₂/J₁, Monte Carlo simulations were performed using the ALPS package [70]. A 10×10×10 superlattice with periodic boundary conditions was constructed, and the calculated magnetic structure factor was averaged over 25000 sweeps after 10000 sweeps of thermalization. As is compared in Fig. 4.5, the simulation with|J₂/J₁| =0.85 andT/|J₁| =0.55 (J1<0 is ferromagnetic,J2>0 is antiferromagnetic) can reproduce the experimental data very well. This high value of|J₂/J₁|puts MnSc₂S₄deep in the spiral spin-liquid phase and establish it as the firstA-site spinel that realizes the spiral spin-liquid state.

Previous studies of powder sample of MnSc₂S₄revealed a transition into a commensurate long-range ordered state withq=(0.75 0.75 0) atT∼2.3 K [165, 172]. In the inset of Fig. 4.5c, we present the data measured at 1.9 K for the region around (−1 1 0). The same region is also outlined in Fig. 4.5a with the white square. Two strong magnetic Bragg peaks can be observed at (−0.75 0.75 0) and (−1.25 1.25 0), which are compatible with the reported ordering [172].

1D cuts along the arc of the spiral surface at different temperatures are compared in the main panel of Fig. 4.5c, revealing the coexistence of long-range and short-range correlations down to 1.5 K (the base temperature of our diffuse scattering experiment). Such a coexistence demonstrates the presence of strong fluctuations in the ordered phase [173] and calls for a detailed study of the ordering process in MnSc₂S₄.

T (K)

Figure 4.6 –(a)Magnetic susceptibility of MnSc2S4measured in 0.1 T field along the [111]

direction during warming after zero-field-cooling (ZFC) and 0.1 T field-cooling (FC). (b) Magnetic susceptibility along the [111] direction measured during warming in 1, 2, 3, and 6 T field after ZFC.