Spin Superstructures in the 1/8-Magnetization Plateau Phase
of the 2D Orthogonal Dimer Spin System SrCu₂(BO₃

The discovery of a spin-gap and magnetization plateaus [1] in the quasi two-dimensional orthogonal dimer spin system SrCu₂(BO₃ stimulated vast amount of experimental and theoretical activities. The crystal structure realizes a stack of the two-dimensional Shastry-Sutherland spin model, for which the direct product of dimer singlets is the exact ground state. The triplet excitations are known to have an extremely small dispersion width. A striking property of SrCu₂(BO₃ is the plateaus in the magnetization vs. field curve at fractional values (1/8, 1/4 and 1/3) of the fully saturated magnetization. It has been argued that the excited triplets with small kinetic energy localize due to repulsive interactions to form superlattices in the plateau phases, i.e., the Mott insulating state or Wigner crystal of magnetic excitations are realized. Since the nuclei on inequivalent Cu-sites in a superstrtucture should have different magnetic hyperfine fields, which can be observed by NMR, we have performed ⁶⁵Cu and ⁶³Cu NMR measurements at low temperature (35 mK) in the magnetic field up to 28 T using the facilities in The Grenoble High Magnetic Field Laboratory [2]. The field range covers the first 1/8 plateau state, which occurs for 27 - 28.5 T for the field applied perpendicular to the 2D layers.


Figure 1 shows the Cu NMR spectra at 26 T (inset) and at 27.6 T (main panel). Generally, one type of Cu-sites with a specific value of hyperfine field gives six NMR lines; three lines split by electric quadrupolar interaction for each of two isotopes ⁶⁵Cu and ⁶³Cu. The spectrum at 26 T can indeed be represented as superposition of such six lines, indicating that the magnetization is largely uniform. At 27.6 T near the middle of the 1/8 plateau, we observed completely different spectrum with large number of sharp peaks distributed over a wide frequency range, providing clear evidence for a magnetic superstructure breaking the translatioinal symmetry.




Fig. 1. Cu NMR spectra at 35 mK. The main panel (inset) shows the spectrum in the field of 27.6 (26) T. The red line is the fit with 11 distinct hyperfine fields. The arrows in the main panel show the resonance frequencies for zero hyperfine field. The intense peaks for 371 - 379 MHz (at 126 MHz) is due to ¹¹B (¹⁰B) nuclei.


We found that satisfactory fitting of the spectrum requires at least 11 Cu sites with different hyperfine fields. An example is shown by the red line in Fig. 2. In particular, the six sharp lines in the red zone (105 - 165 MHz) are ascribed to the sites with the largest magnetization <S_z> = 0.30. Likewise the spectrum in the yellow zone (165 - 235 MHz) represent another sites with <S_z> = 0.20. The rest of the spectrum comes from the sites with smaller hyperfine fields. The distribution of the hyperfine field obtained by this fitting is shown in Fig. 2 (middle panel in the left part). It should be noted that some sites have positive hyperfine fields, i.e., some spins are polarized opposite to the external field. Thus the magnetization oscillates within the unit cell of the superstructure.




Fig. 2. Right part: Magnetization profile obtained by exact diagonalization of the Shastry-Sutherland model. Red (blue) circles indicate positicve (negative) <S_z>. The circle size represent the magnitude of <S_z>. Left part: Histogram of the hyperfine field distribution. The middle panel (red) shows results of the fitting of the NMR spectrum (Fig. 2, red line). Long (short) lines indicate that the population of the site is 1/8 (1/16). The top panel is obtained from the calculated profile (right part) assuming only the on-site hyperfine coupling. The bottom panel shows the result when the transferred hyperfine couplings, denoted B and C in the right part, are adjusted to best reproduce the experimental results.


By symmetry consideration, we can conclude that the unit cell of the superstructure has the rhomboid shape shown in Fig. 2. The number of inequivalent Cu sites for the rhomboid cell is eight for a single layer, but it may increase up to 16 depending on the pattern of stacking layers along the c-axis. We found that other types of unit cell cannot produce as many as 11 sites.


The magnetization profile calculated by exact diagonalization of the Shastry-Sutherland model on a 16-spin cluster of the rhomboid cell is shown in Fig. 2. One can clearly recognize a building unit consisting of one strongly polarized dimmer surrounded by decaying oscillation of magnetization. This resembles the Friedel oscillation near impurities in metals. The distribution of the hyperfine fields derived from the calculatied magnetization profile captures the essential feature of the experimental results, indicating that such a magnetic superstructure is indeed realized in the 1/8 plateau phase.





References
[1] H. Kageyama et al., Phys. Rev. Lett. 82, 3168 (1999).
[2] K. Kodama et al., Science 298, 395 (2002).