Emergence of long-range order in sheets of magnetic dimers S. Haravifarda,b, A. Banerjeea,c, J. van Wezeld, D. M. Silevitcha, A. M. dos Santosc, J. C. Langb, E. Kermarrece, G. Srajerb, B. D. Gauline, J. J. Molaisonc, H. A. Dabkowskae, and T. F. Rosenbauma,1,2 aThe James Franck Institute and Department of Physics, The University of Chicago, Chicago, IL 60637; bAdvanced Photon Source, Argonne National Laboratory, Argonne, IL 60439; cNeutron Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge, TN 37831; dSchool of Physics, The University of Bristol, Bristol BS8 1TL, United Kingdom; and eDepartment of Physics and Astronomy and Brockhouse Institute for Material Research, McMaster University, Hamilton, ON, Canada L8S 4M1 Edited by David D. Awschalom, The University of Chicago, Chicago, IL, and approved August 28, 2014 (received for review July 14, 2014) Quantum spins placed on the corners of a square lattice can dimerize whose nearest-neighbor interactions are mediated by in-plane and form singlets, which then can be transformed into a magnetic oxygen bonds; layered Cu–O compounds of this kind are of state as the interactions between dimers increase beyond thresh- broad interest due to their role in driving phenomena such old. This is a strictly 2D transition in theory, but real-world materials as high-temperature superconductivity. At ambient conditions, often need the third dimension to stabilize long-range order. We use SCBO forms a crystal in the tetragonal space group I-42m with + + high pressures to convert sheets of Cu2 spin 1/2 dimers from local a = b = 8.995 Å and c = 6.659 Å. The Cu2 ions form orthogonal singlets to global antiferromagnet in the model system SrCu2(BO3)2. structural and magnetic dimers in the a–b plane (4). As each Single-crystal neutron diffraction measurements at pressures above dimer sits on a mirror plane that passes through the center of its 5 GPa provide a direct signature of the antiferromagnetic ordered orthogonal neighbor, the unit cell contains four dimers on two state, whereas high-resolution neutron powder and X-ray diffrac- layers with alternating orientations (3) (Fig. 1). SCBO has a non- tion at commensurate pressures reveal a tilting of the Cu spins out of magnetic, singlet ground state (5), with 0.6 < x < 0.69 (6). Hy- the plane with a critical exponent characteristic of 3D transitions. drostatic pressure can be used to tune x, with a transition into an The addition of anisotropic, interplane, spin–orbit terms in the ven- intermediate phase observed at P ∼2 GPa (7, 8). The magnetic erable Shastry–Sutherland Hamiltonian accounts for the influence characteristics of the singlet–triplettransitionemergeeveninX-ray of the third dimension. scattering experiments due to a remarkably robust spin–lattice coupling (7, 9, 10). At P ∼4.5 GPa a second phase transition has condensed matter physics | quantum magnetism | phase transition | been observed, marked by a structural transition from tetragonal dimensional cross-over | neutron and X-ray scattering to monoclinic ordering (7, 11). In this article, we use both X-ray and neutron scattering measurements to illuminate the structural wo-dimensional systems lie at the boundary between the and magnetic configurations that permit long-range order to Torder-destroying effects of fluctuations in lower dimension emerge for P > 5 GPa, directly identifying for the first time, to our and the emergence of true long-range order in higher dimension. knowledge, the predicted antiferromagnetic state. How ordered states form and remain stable at finite temperature Results and Discussion is a fundamental question that cuts across the sciences. In physics, surfaces, quantum wells, and layered compounds all We begin with fundamental symmetry considerations. In SCBO’s have provided insights into the peculiar nature of 2D phases, low-pressure tetragonal phase, structural Bragg reflections phase transitions, and dimensional cross-over effects. Of partic- ular interest for quantum phase transitions is the theoretically Significance tractable case of interacting magnetic spins placed on a 2D square lattice, the Shastry–Sutherland model (1). The geometry Magnetic materials are composed of individual spins that interact prevents simultaneous satisfaction of all spin–spin coupling with each other and under suitable conditions can arrange them- terms, leading to a frustrated state that in turn enhances the selves in an ordered array. When spins are confined to two- effects of quantum fluctuations. As the coupling terms are tuned, dimensional sheets, small perturbations can disrupt their order magnetic order can emerge, but its nature and the potential in- and destroy the magnetic state. We show how a set of interact- fluence of other sheets of spins in a real physical system have not ing, quantum-mechanical spins placed on the corners of a square been probed directly. array evolves from a set of locally bonded entities to a globally The Shastry–Sutherland model can be described by the Ham- ordered structure. The system stabilizes itself against fluctuations iltonian: through subtle local contractions, elongations, and tilts. The com- X X bination of neutron and X-ray scattering at pressures up to 60,000 atmospheres reveals the complex interplay of structural H = J Si · Sj + J′ Si · Sj; [1] nn nnn distortions and spin alignments that permit long-range order to emerge in this model quantum magnet. where a set of S = 1/2 spins sits on a square lattice and interacts via a regular array of diagonal bonds. This creates a network of Author contributions: S.H. and T.F.R. designed research; S.H., A.B., A.M.d.S., J.C.L., E.K., J G.S., B.D.G., and J.J.M. performed research; S.H., B.D.G., and H.A.D. contributed new dimers with antiferromagnetic intradimer coupling and inter- reagents/analytic tools; S.H., J.v.W., D.M.S., and T.F.R. analyzed data; and S.H., J.v.W., dimer coupling J′. The ground state depends on the ratio of D.M.S., and T.F.R. wrote the paper. J′=J = x. For x < 0.7, the ground state consists of S = 0 singlets, The authors declare no conflict of interest. x > whereas for 0.9 a global antiferromagnetic phase is expected. This article is a PNAS Direct Submission. x For intermediate values of , a variety of locally ordered states 1Present address: Division of Physics, Mathematics and Astronomy, California Institute of has been predicted (2). Technology, Pasadena, CA 91125. – The first known experimental realization of the Shastry 2To whom correspondence should be addressed. Email: [email protected]. Sutherland lattice is SrCu2(BO3)2 (SCBO) (3). The magnetism in This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 2+ SCBO consists of S = 1/2 Cu ions in well-separated planes 1073/pnas.1413318111/-/DCSupplemental. 14372–14377 | PNAS | October 7, 2014 | vol. 111 | no. 40 www.pnas.org/cgi/doi/10.1073/pnas.1413318111 Downloaded by guest on September 30, 2021 refinement indicates a space group of P121 (#3). We test this identification using a simulation of the expected neutron diffrac- tion intensity at T = 90 K, including the cross-sections for both SCBO’s structural and antiferromagnetic components, and com- pare experiment and computer simulation in Fig. 3B.Thespace group identifications also permit us to track the lattice parameters as a function of temperature. We see in Fig. 3C that the evolution can be described as an increase in c associated with a simultaneous decrease in a, and a slight decrease in b as temperature is cooled below T ∼ 120 K. The change in lattice constants is accompanied by a decrease in the monoclinic angle β; the overall volume of the unit cell changes by less than 1%. We plot in Fig. 4 a comparison between single-crystal X-ray diffraction in a diamond anvil cell and powder neutron diffraction in a Paris–Edinburgh cell for P = 5.5 GPa at the position and temperature where a forbidden (0 3 0) peak develops. The intensity scaling by 100 is consistent with the relative cross-sections for X-ray and neutron scattering. The simulation of the neutron intensity (dashed line) incorporates the antiferromagnetic structure de- scribed below; different spin configurations contribute from 0 to 30% of the total neutron signal at (0 3 0) (Fig. S1) and provide a quantitative test of the enhancement measured at that Q in Fig. Fig. 1. Crystallographic structure of SrCu2(BO3)2 (SCBO). (Upper) Low-tem- A perature structure of SCBO projected into the a–b plane. (Lower) a–c plane. 3 . The high-resolution single-crystal synchrotron X-ray data per- Solid, dashed, and dotted blue lines show the interactions between the mit us to focus in on the critical behavior of the symmetry change + dimerized Cu2 ions, representing the intradimer (J = 84 K), in-plane inter- from C121 to P121 with decreasing T (Fig. 4, Inset). A power-law dimer (J’ = 54 K), and out-of-plane interdimer (J′′ = 8 K) (6) interactions, fit to the data gives Tc = 122 ± 0.2 K and a critical exponent β = respectively. This hierarchy of direct exchange interactions underlies the 0.36 ± 0.04, consistent with the value of β = 0.37 expected for a 3D effectively 2D nature of the system in the absence of global magnetic order. Heisenberg transition (15) rather than the nominally expected 2D behavior. That this transition is associated with magnetic order is further corroborated by the rapid decrease in the neutron diffrac- satisfying h + k + l = 2n+1 are forbidden due to the symmetry of tion intensity at the forbidden odd-index peaks as expected for the the I-42m space group. Above the monoclinic transition at P ∼ 4.5 Q dependence of Cu’s magnetic form factor.