Sign-switching of dimer correlations in SrCu 2 ( BO 3 ) 2 under high pressure

Magnetic and vibrational excitations in SrCu2(BO3)2 are studied using Raman spectroscopy at hydrostatic pressures up to 34 kbar and temperatures down to 2.6 K. The frequency of a particular optical phonon, the so-called pantograph mode, shows a very strong anomalous temperature dependence below about 40 K. We link the magnitude of the effect to the magnetic exchange energy on the dimer bonds in the Sutherland-Shastry spin lattice in this material. The corresponding dimer spin correlations are quantitatively estimated and found to be strongly pressure dependent. At around P2 ∼ 22 kbar they switch from antiferromagnetic to being predominantly ferromagnetic.

The Shastry-Sutherland model (SSM) is arguably one of the most important constructs in the field of quantum magnetism. It demonstrates that a gapped quantum paramagnet can occur in a wellconnected Heisenberg spin Hamiltonian beyond the unique topology of a single dimension. Its key feature is geometric frustration of antiferromagnetic (AF) interactions between AF S = 1/2 dimers arranged on a particular 2-dimensional lattice (Fig. 1a) [1]. For sufficiently strong frustration the exact ground state is a product of AF singlets on each dimer bond J . For weak frustration one recovers the semi-classical Néel-ordered phase. What happens in-between has been hotly debated [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The most intriguing intermediate phase proposed is the so-called plaquette state [5,7,11]. Dimer singlets are destroyed to be replaced by singlets composed of four spins connected via the inter-dimer bonds J. Translational symmetry is broken and some or all of the dimer spin correlations become ferromagnetic (FM).
The only known and much studied experimental realization of the model is SrCu 2 (BO 3 ) 2 , where S = 1/2 Cu 2+ ions form dimers via Cu-O-Cu superexchange pathways, which are connected through (BO 3 ) units (Fig. 1b) [21]. With a frustration ratio J/J ∼ 0.6 [22] it is reliably in the dimer phase, with a spin gap ∆ = 3 meV [21] in the excitation spectrum. We are incredibly lucky that in this material J/J can be continuously tuned by hydrostatic pressure [23]. The frustration ratio increases steadily with pressure, eventually leading to a Néel ordered state above 30 kbar [24]. Moreover, already at P c ∼ 18 kbar the original dimer phase gives way to a new quantum paramagnet, presumed to be the plaquette state [23][24][25][26][27]. Thus, theoretical predictions for exotic phases of the SSM are put to the experimental test.
How can one be sure that the novel phase is indeed plaquette -rather than dimer-based? To date, the only supporting evidence comes from studies of the wave vector dependence of inelastic neutron scattering intensities. [23] Performing such measurements in a bulky cell needed to produce the required pressure for a sufficiently large sample is a formidable task. The resulting data are unavoidably limited and noisy, leaving the interpretation depending on strong assumptions and theoretical modeling [23]. In the present work we use an entirely different approach. We infer the strength of dimer spin correlations in SrCu 2 (BO 3 ) 2 from their effect on certain optical phonons, which can be measured using Raman spectroscopy in a diamond-anvil pressure cell. We show that around P c these correlations switch from AF to dominantly FM, and thereby independently confirm the destruction of the AF-dimer ground state. Moreover, we obtain a quantitative estimate for the dimer bond energy at pressures up to 34 kbar.
The main idea is as follows. As has been established in other dimer systems, the development of pair spin correlations at low temperatures leads to a magnetic contribution to the rigidity of the corresponding bond [28][29][30]. This, in turn, gives rise to anomalous shifts of certain phonon frequencies at unusually low temperatures set by the magnetic energy scale. In general, each phonon may involve distortions of several magnetic superexchange pathways, making the shifts difficult to associate with any particular spin correlators [30]. For the highly symmetric structure of SrCu 2 (BO 3 ) 2 though, the assignment of measured phonons to particular atomic motions is greatly simplified. At ω ∼198 cm −1 (=24.5 meV) there is a specific optical excitation, the so-called pantograph mode, visualized in (Fig. 1b) [31]. It directly modulates the dimer bond length and therefore the intra-dimer coupling con- stant J . At the same time, it has no first order effect on distances between ions coupled via J, where its influence is expected to be negligible [31,32]. Furthermore, its frequency is much larger than the magnetic energy scale, ensuring an adiabatic coupling scenario. To the first approximation, the magnetic shift of the pantograph mode frequency is then sim-ply proportional to the dimer bond energy J S 1 S 2 averaged over the entire crystal.
Optical phonons, pantograph mode included, as well as purely magnetic excitations, have been extensively studied in SrCu 2 (BO 3 ) 2 using Raman spectroscopy at ambient pressure [33][34][35][36]. In the present work we perform similar measurements in a 1 mmculet diamond anvil cell at pressures up to 34 kbar using a Trivista 557 triple-grating spectrometer and a liquid nitrogen cooled CCD detector. The data were collected in a backscattering geometry using a focusing microscope. The incident laser wavelength was 532 nm. The pressure transmitting medium was Argon. The pressure was determined in-situ from ruby fluorescence. The cell environment was a Heflow cryostat with base temperature 2.5 K. All experiments were performed on a ∼0.2×0.1×0.05 mm 3 single crystal sample extracted from cleaving a crystal grown using the floating-zone technique. The incident light was perpendicular to the ab cleavage plane (see insert in Fig. 2). All spectra shown were measured in thec(a b )c scattering geometry. An independently measured background from the pressure transmitting medium was subtracted from all spectra.
Optical measurements on SrCu 2 (BO 3 ) 2 are extremely challenging due to the narrow band gap of the material, which leads to high absorption [36]. The biggest problem is unwanted heating of the sample by the incident laser beam. To suppress this effect, the data were collected at very low power, as low as 0.05 mW at the lowest temperatures. As a result, typical counting times were as much as 72 h. In all cases it was verified that further reducing the power had no effect on the resulting spectra. A typical low-energy Raman spectrum collected in SrCu 2 (BO 3 ) 2 at a pressure of P = 2 kbar and T = 2.6 K is shown in Fig. 2a. The observed frequencies appear fully consistent with previous measurements at ambient pressure [35]. The three visible lowest-energy excitations are magnetic in origin: two triplets and one singlet. All remaining peaks are phonons. The pantograph mode is the peak at around 198 cm −1 . It was identified by a density functional theory (DFT) ab-initio calculation with the Quantum Espresso software package [37], using the SSSP Accuracy (version 1.1) pseudopotential library [38][39][40][41][42][43][44]. The kinetic energy cutoff was 120 Ry and the charge density cutoff 600 Ry. Brillouin zone integration was performed using a 4×4×4 k-point grid. For better convergence a Marzari-Vanderbilt  Fig. 2c. At higher pressures they become undetectably weak or shift outside our measurement window (T 1 ). The observed softening of the triplet modes is fully consistent with previous Electron spin resonance(ESR) [27] and inelastic neutron scattering [23] studies, shown as triangles and squares in the figure. The softening of the singlet mode is a new result, since neither ESR nor neutrons are sensitive to singlet-singlet transitions.

circles in
The central result of this work is the observation of an anomalous temperature dependence of the pantograph mode, as visualized in Fig. 3. Note the loga- rithmic temperature scale. At all pressures, the pantograph mode undergoes at most a modest hardening upon cooling down to 40 K. This behavior is easily explained by the usual anharmonicities of lattice vibrations. Below this point the mode suddenly becomes strongly T -dependent. At low applied pressures it softens upon cooling to base temperature by as much as 1.5%. At the highest pressures the effect is reversed: in the same low-temperature interval the excitation hardens by as much as 0.5%. Also revealing is the pressure dependence at different temperatures shown in Fig. 4. For T 40 K the pressure dependencies are almost T -independent. They all show a monotonous hardening with a small but reproducible dip at about 20 kbar. At lower temperatures the situation changes drastically, the frequency showing a steep almost step-like increase.
At temperatures below 40 K, the only remaining relevant energy scale is the magnetic one (of the order of the spin gap ∆). We conclude that the anomalous temperature dependence is due to magnetoelastic coupling and the temperature dependence of spin correlations. The corresponding relative frequency shift of the pantograph mode between 40 K and 2.6 K is plotted against pressure in Fig. 5   As discussed above, one can view this quantity as a measure of the dimer bond energy . To within a scale factor, Fig. 5 thus represents the pressure dependence of J S 1 S 2 . Assuming that J is itself only weakly pressure-dependent explains why below P 1 the frequency shift remains roughly constant. Indeed, dimerization in the Shastry-Sutherland model is exact, and S 1 S 2 ≡ − 3 4 regardless of J /J [3]. This provides a calibration for the entire plot (Fig. 5, right axis). At P 2 the relative alignment of nearest neighbor spins switches sign and becomes predominantly ferromagnetic.
The characteristic pressures found in our experiments closely correspond to those recently observed in high-pressure thermodynamic measurements [24]. According to that study, below T c ∼ 2 K the plaquette state emerges in a discontinuous phase transition just about P 1 . Néel magnetic order sets in at approximately P 2 . The lowest attainable temperature in our experiments is slightly above T c . This is consistent with the continuous evolution of J S 1 S 2 observed in our case. The sign-switching is thus to be interpreted as a change in the character of dominant short-range spin correlations, which are just about to order in a new plaquette configuration at slightly lower temperatures. An important question is whether the pressure-induced changes in the magnetic ground state of SrCu 2 (BO 3 ) 2 are driven by magnetic energy, rather than structural changes due to other causes. We do not observe any unambiguous signs of a pressure-induced structural transition, such as splitting of phonon lines or the appearance of new modes. Nevertheless, the observed dip in the pantograph mode frequency at 20 kbar at temperatures much higher than the magnetic energy scale may be indicative of one.
In summary, our measurements of the pantograph mode in SrCu 2 (BO 3 ) 2 provide an indirect but precise quantitative confirmation of a pressure-induced sign switching of nearest-neighbor spin correlations, revealing the demise of dimer-singlets and the emergence of an entirely new ground state where dimer spins are predominantly parallel to one another.
This work was partially supported by the Swiss National Science Foundation, Division 2. We thank D. Blosser (ETHZ) for help with first principles calculations, E. Pomjakushina (PSI) for guidance on the single crystal growth and F. Mila, H. Rønnow and D. Badrtdinov (EPFL) for enlightening discussions.