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Incomplete Devil’s Staircase in the Magnetization Curve of SrCu2(BO3)2

Takigawa and Y. Ueda Groups

Magnetization plateaus in frustrated quantum spin systems are manifestation of Wigner crystallization of magnons, resulting from competition between the kinetic energy and repulsive interaction. The layered compound SrCu2(BO3)2 has played a prominent role in this problem since the discovery of a sequence of plateaus at 1/8, 1/4, and 1/3 of the saturation magnetization. Recent experiments have revealed a rich phase diagram below the 1/4 plateau, but with controversial results. We have performed magnetic torque and 11B-NMR measurements in high magnetic field up to 34 T at the Grenoble High Magnetic Field Laboratory and precisely determined the sequence and the spin structures of the plateaus [1].

Fig. 1. Inset: The torque divided by field vs. field obtained at T=60 mK with the sample positioned at 10 mm off the nominal field center. Main panel: The thick black line represents the longitudinal magnetization with the vertical scale appropriately adjusted. The magnetization values at 1/8, 2/15, 1/6, and 1/4 of the saturation are shown by the dashed lines. The horizontal bars indicate the field range of the plateaus determined by NMR. The open circles show the magnetization determined from the Cu NMR shift data (Kodama et al., Science 298, (2002) 395).

Fig. 2 . (left) The distribution of internal magnetic field at the 11B nuclei obtained by deconvoluting the quadrupole splitting structure from the experimental NMR spectra obtained at T=430 mK unless explicitly indicated. (right) The spin superstructure of the plateaus phases. The thin black lines show the lattice of orthogonal Cu dimers in one layer. The thick red lines show the triplet dimers carrying the largest magnetization in the same layer while the blue lines indicate these triplets on the neighboring layers. The unit cell of each superstructure is shown by the shaded area.

The magnetization curve under static magnetic fields was accurately determined from the torque (τ) data (Fig. 1), which consist of two terms: τ = aM × H + b(M•∇)M, the first one proportional to the transverse magnetization and the second one to the longitudinal magnetization. The first term can be eliminated by taking a linear combination of two measurements of τ/H taken at different sample positions with the requirement that the longitudinal magnetization is zero in the dimer singlet phase below 15 T. The longitudinal magnetization thus obtained clearly shows three plateaus at the ratio 1/8:2/15:1/6 below the 1/4 plateau and two intermediate phases below and above the 1/6 plateau.

The plateau phases are associated with symmetry breaking commensurate spin superstructures, which were determined from the 11B-NMR spectrum (Fig. 2 (left)). We have established a systematic way to determine the spin structure from the NMR spectra and the results are displayed in Fig. 2 (right). All plateaus show stripe order of triplets. The structure of 1/6 (1/8) plateau can be obtained by removing every other triplet from the one of the 1/3 (1/4) plateaus. The 2/15 structure exhibits a sequence of domains of 1/8-1/8-1/8-1/6 structure, showing how the proliferation of domain walls leads to a structure of higher order commensurability. This suggests that the plateau sequence can be interpreted as a “devil’s staircase”, which is an infinite sequence of commensurate phases [2]. However, the NMR spectra in the intermediate phase consist of only broad lines, indicating that what is observed here is an example of “incomplete devil’s staircase”, in which the infinite sequence of high order commensurate phases with small steps are replaced by incommensurate phases [3].


References
  • [1] M. Takigawa, M. Horvatić, T. Waki, S. Krämer, C. Berthier, F. Lévy-Bertrand, I. Sheikin, H. Kageyama, Y. Ueda, and F. Mila, Phys. Rev. Lett. 110, 067210 (2013).
  • [2] P. Bak, Rep. Prog. Phys. 45, (1982) 587.
  • [3] S. Arby, Solitons and Condensed Matter Physics, edited by A. R. Bishop and T. Schneider (Springer-Verlag, Berlin, 1979), p. 264.
Authors
  • M. Takigawa, M. Horvatić, T. Waki, S. Krämer, C. Berthier, F. Lévy-Bertrand, I. Sheikin, H. Kageyama, Y. Ueda, and F. Mila