Ordering Due to Quantum Fluctuations in Sr2Cu3O4Cl2

Sr2Cu3O4Cl2 has CuI and CuII subsystems, forming interpenetrating S=1/2 square lattice Heisenberg antiferromagnets. The classical ground state is degenerate, due to frustration of the intersubsystem interactions. Magnetic neutron scattering experiments show that quantum fluctuations cause a two dimensional Ising ordering of the CuII's, lifting the degeneracy, and a dramatic increase of the CuI out-ofplane spin-wave gap, unique for order out of disorder. The spin-wave energies are quantitatively predicted by calculations which include quantum fluctuations. Disciplines Physics | Quantum Physics Author(s) Youngjune Kim, Amnon Aharony, Robert J. Birgeneau, Fangcheng Chou, Ora Entin-Wohlman, Ross W. Erwin, Martin Greven, A. Brooks Harris, Marc A. Kastner, I. Ya Korenblit, Youngsu Lee, and Gen Shirane This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/464 VOLUME 83, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JULY 1999 Ordering due to Quantum Fluctuations in Sr2Cu3O4Cl2 Y. J. Kim,1,2 A. Aharony,3 R. J. Birgeneau,1 F. C. Chou,1 O. Entin-Wohlman,3 R. W. Erwin,4 M. Greven,1,* A. B. Harris,5 M. A. Kastner,1 I. Ya. Korenblit,3 Y. S. Lee,1 and G. Shirane6 1Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 2Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 3School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel 4National Institute of Standards and Technology, Gaithersburg, Maryland 20899 5Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 6Department of Physics, Brookhaven National Laboratory, Upton, New York 11973 (Received 4 January 1999) Sr2Cu3O4Cl2 has CuI and CuII subsystems, forming interpenetrating S 1 2 square lattice Heisenberg antiferromagnets. The classical ground state is degenerate, due to frustration of the intersubsystem interactions. Magnetic neutron scattering experiments show that quantum fluctuations cause a two dimensional Ising ordering of the CuII’s, lifting the degeneracy, and a dramatic increase of the CuI out-of-plane spin-wave gap, unique for order out of disorder. The spin-wave energies are quantitatively predicted by calculations which include quantum fluctuations. PACS numbers: 75.30.Ds, 75.10.Jm, 75.25.+z, 75.45.+ j The classical ground states of many magnetic systems are degenerate due to frustration. Quantum or thermal fluctuations often lift this degeneracy, yielding order due to disorder [1–4]. For example, when a system can be separated into two Heisenberg antiferromagnetic (AFM) sublattices, so that the molecular field of the spins in each sublattice vanishes on the spins of the other, then classically the sublattices order independently of each other, the ground state is degenerate, and the excitation spectrum contains two distinct sets of zero energy (Goldstone) modes, reflecting the fact that these subsystems can be rotated independently without cost in energy. As shown by Shender, quantum spin-wave (SW) interactions prefer collinearity of the spins in the two sublattices [2]. Concomitantly, this also generates a fluctuation driven gap in the SW spectrum. Indeed, such a gap was hypothesized in the garnet Fe2Ca3 GeO4 3 [5]. Since a similar gap could also arise from crystalline magnetic anisotropy, the final identification was rather complex. In parallel to these developments, the discovery of high temperature superconductivity triggered much work on the magnetism in lamellar copper oxides. These materials contain CuO2 planes, whose two-dimensional (2D) spin fluctuations can be modeled well by the S 1 2 square lattice quantum Heisenberg antiferromagnet (SLQHA) [6]. The above two advances combine in the isostructural compounds Sr2Cu3O4Cl2 and Ba2Cu3O4Cl2 (2342). In the present paper, we show that these materials offer a dramatic and clear demonstration of ordering due to fluctuations. As shown in Fig. 1(a), the CuO2 plane is replaced in 2342 by a Cu3O4 one, which contains an additional Cu II ion at the center of every second plaquette of the original CuIO2 square lattice [7]. The configuration in the neighboring plane is obtained by translating the whole plane by a2 ā 2 . In the plane, the CuI and CuII subsystems form interpenetrating S 1 2 SLQHA’s with exchange interactions JI and JII. The isotropic interaction JI-II between these subsystems is frustrated; that is, its molecular field vanishes as described above, and one needs nontrivial theories to explain their ordering and SW gaps. Experimentally, 2342 exhibits AFM order of the CuI’s and CuII’s below the Néel temperatures TN ,I and TN ,II, respectively [8,9]. For temperatures T . TN ,II, the CuII susceptibility behaves as in a SLQHA, with JII 10 meV [10]. JI 130 meV is known from other cuprates. For T , TN ,II, the magnetization data were interpreted assuming Shender collinearity [10]. This paper reports on theory and neutron scattering experiments in Sr2Cu3O4Cl2. The inelastic neutron data show a dramatic increase of the CuI “out-of-plane” [11] FIG. 1. (a) Magnetic structure of the Cu3O4 plane in 2342 at T , TN ,II. The corresponding 2D reciprocal lattices are shown in (b) for T . TN ,II, and in (c) for T , TN ,II. The shaded area is the 2D Brillouin zone. 852 0031-9007 99 83(4) 852(4)$15.00 © 1999 The American Physical Society VOLUME 83, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JULY 1999 gap below TN ,II (see Fig. 2), which clearly reflects a coupling between the CuI and CuII spins. However, since symmetry indicates that within mean field theory this coupling due to frustrated interactions must vanish, we conclude that the enhanced gap for T , TN ,II is due to fluctuations. This identification is particularly strong, since our data are quantitatively predicted by detailed theoretical calculations, which use parameters determined independently, albeit less accurately, by the static measurements [10]. The SW dispersion along 1 0 L , shown in Fig. 3, depends crucially on the fluctuations. In contrast, most of the in-plane dispersion is described by the linear SW theory. A nontrivial exception arises near the zone boundary of the CuII’s, which we could access with thermal neutrons because of the small JII. Figure 4 exhibits a novel dispersion, which is shown to result from the quantum nature of the SLQHA. Since the SLQHA does not have long-range order at T . 0, such order must arise from spin anisotropy terms or interplane couplings. Our data indicate that TN ,I 385 K is determined by the latter, via j0 TN ,I JI,3D JI 1, with JI,3D 0.1 meV. Here, j0 T is the SLQHA correlation length [6,12]. The spin structure, shown in Fig. 1(a), has been uniquely determined from analyzing 13 different neutron diffraction peaks in the plane [13], confirming the conjecture in Ref. [10]. As in YBa2Cu3O6 [14], the CuI spins order along the CuI-CuII bonds, as expected when the quantum fourfold anisotropy is dominant [15]. Unlike the CuI’s, the interplane CuII-CuII coupling is frustrated, similar to that in Sr2CuO2Cl2 [6], and therefore TN ,II is expected to be determined mainly by in-plane spin anisotropies. Indeed, the 2D nature of the CuII subsystem has been esFIG. 2. (a) Temperature dependence of the CuI out-of-plane gap energy and of the staggered magnetizations Ms of CuI and CuII. Ms is normalized to the extrapolated zero temperature value Ms,0. The solid lines are the results of fits to the form TN 2 T b . (b) The gap energy in a different temperature scale. The solid and dashed lines are interpolations for v4 and v3; see text. tablished from both the existence of rods of 2D scattering for T . TN ,II and the small SW dispersion along L [13]. For T , TN ,I, the ordered CuI spins fluctuate mainly in the directions transverse to their staggered moment Ms,I. JI-II then generates fluctuations in the CuII spins along the same directions, causing an effective reduction in the corresponding transverse exchange components of JII [2]. This yields an effective term 2d̃ Ms,I ? SII 2, where d̃ ~ J I-II JI 1 JII . This implies an Ising-like anisotropy JIIa eff II ~ d̃M 2 s,I, which favors ordering of the CuII spins collinearly with Ms,I, consistent with our measured structure, Fig. 1(a). Indeed, TN ,II 40 K agrees with j0 TN ,II 2a II 1, where a II 0.01 was deduced from our SW gaps. Heuristically, the lowering of the symmetry on the CuII site due to the ordering of the CuI’s is sensed through the quantum fluctuations. Large 2 cm3 single crystals of Sr2Cu3O4Cl2, grown by slow cooling of a melt containing CuO flux, are used in the experiment. The crystal remains tetragonal for 15 , T , 550 K [10]. Our neutron scattering experiments were carried out with the triple-axis spectrometers at the High Flux Beam Reactor, Brookhaven National Laboratory, and at the National Institute of Standards and Technology Research Reactor. Collimation of 400-400-S-400-800 and fixed final neutron energy of 14.7 meV were used for most of the inelastic measurements. When better resolution was required, we used a fixed initial neutron energy of 13.7 meV and tighter collimations. The T dependences of Ms,I and Ms,II [proportional to the square root of the AFM Bragg intensities at the 1 0 1 and the 12 1 2 0 reciprocal positions, respectively] are shown in Fig. 2(a). Since nuclear Bragg scattering is only weakly T dependent, we subtract the hightemperature 1 0 1 nuclear intensity from the observed intensity. The solid lines in Fig. 2(a) represent Ms TN 2 T b . The parameters given there were fitted for FIG. 3. (a) Magnon dispersion along the L direction at the 2D zone center for T 200 K (solid circles). (b) Same for T 10 K. The solid lines show Eq. (2).