Sign-switching of dimer correlations in SrCu$_2$(BO$_3$)$_2$ under hydrostatic pressure

TL;DR

This study investigates how hydrostatic pressure influences magnetic and vibrational properties in SrCu$_2$(BO$_3$)$_2$, revealing a pressure-induced switch in dimer spin correlations from antiferromagnetic to ferromagnetic.

Contribution

It provides the first quantitative analysis of pressure-dependent dimer spin correlations and identifies a switch in their magnetic nature in SrCu$_2$(BO$_3$)$_2$.

Findings

Dimer spin correlations are strongly pressure dependent.

A switch from antiferromagnetic to ferromagnetic correlations occurs around 22 kbar.

The pantograph phonon mode exhibits anomalous temperature dependence linked to magnetic exchange energy.

Abstract

Magnetic and vibrational excitations in SrCu$_2$(BO$_3$)$_2$ 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 P$_2\sim$22 kbar they switch from antiferromagnetic to being predominantly ferromagnetic.

Equations

$$\delta \omega =\omega -{\omega }_0\propto {\omega }^2-{\omega }_0^2\propto \frac{{\partial }^2{J}^{\prime }}{\partial u^2}⟨{\mathbf{S}}_1{\mathbf{S}}_2⟩,$$

Full text

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 well-connected 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) Shastry and Sutherland (1981). For sufficiently strong frustration the exact ground state is a product of AF singlets on each dimer bond $J^{\prime}$. For weak frustration one recovers the semi-classical Néel-ordered phase. What happens in between has been hotly debated Albrecht and Mila (1996); Miyahara and Ueda (1999); Weihong et al. (1999); Koga and Kawakami (2000); Müller-Hartmann et al. (2000); Takushima et al. (2001); Carpentier and Balents (2001); Chung et al. (2001); Zheng et al. (2001); Läuchli et al. (2002); Al Hajj and Malrieu (2005); Darradi et al. (2005); Isacsson and Syljuåsen (2006); Moukouri (2008); Moliner et al. (2011); Corboz and Mila (2013); Ronquillo and Peterson (2014); Wang and Batista (2018); Boos et al. (2019). The most intriguing intermediate phase proposed is the so-called plaquette state Läuchli et al. (2002); Takushima et al. (2001); Koga and Kawakami (2000). 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 predominantly ferromagnetic (FM).

The only known and much studied experimental realization of the model is SrCu2(BO3)2, where $S=1/2$ Cu2+ ions form dimers via Cu-O-Cu superexchange pathways, which are connected through (BO3) units (Fig. 1b) Kageyama et al. (1999). With a frustration ratio $J/J^{\prime}\sim 0.6$ Knetter and Uhrig (2004) it is reliably in the dimer phase, with a spin gap $\Delta=3$ meV Kageyama et al. (1999) in the excitation spectrum. We are incredibly lucky that in this material $J/J^{\prime}$ can be continuously tuned by hydrostatic pressure Zayed et al. (2017). The frustration ratio increases steadily with pressure, eventually leading to a Néel ordered state above 30 kbar Guo et al. (2019). Moreover, already at $P_{c}\sim 18$ kbar the original dimer phase gives way to a new quantum paramagnet, presumed to be the plaquette state Haravifard et al. (2012); Waki et al. (2007); Zayed et al. (2017); Sakurai et al. (2018); Guo et al. (2019). 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.Zayed et al. (2017) 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 Zayed et al. (2017). In the present Rapid Communication we use an entirely different approach. We infer the strength of dimer spin correlations in SrCu2(BO3)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 spin correlations 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 Raas et al. (2002); Choi et al. (2005); Bettler et al. (2017). This, in turn, gives rise to anomalous shifts of certain phonon frequencies at unusually low temperature. In complex structures it may be difficult to associate these shifts with any particular magnetic bonds Bettler et al. (2017). For the highly symmetric structure of SrCu2(BO3)2 though, the assignment of measured phonons to particular atomic motions is greatly simplified. At $\hbar\omega\sim$198 $\mathrm{cm}^{-1}$ (=24.5 meV) there is a specific optical excitation, the so-called pantograph mode, visualized in Fig. 1b) Radtke et al. (2015).

It directly modulates the the intra-dimer coupling constant $J^{\prime}$, which is especially sensitive due to an almost 90∘ Cu-O-Cu Radtke et al. (2015). At same time, its effect on $J$ is expected to be negligible in comparisonVecchini et al. (2009); Radtke et al. (2015). The frequency of this phonon is related to the second derivative of the energy of the corresponding lattice distortion with respect to the distortion amplitude $u$: $\omega^{2}\propto\partial^{2}E/\partial u^{2}$. Each dimer’s magnetic contribution is $J^{\prime}\mathbf{S}_{1}\mathbf{S}_{2}$. The frequency of the pantograph mode is much larger than the magnetic energy scale. This ensures an adiabatic coupling scenario where the spin correlations are unaffected by the phonon which is “too fast”. Since the vibration mode spans the entire sample, the magnetic energy needs to be summed (averaged) over all dimers. With this in mind:

[TABLE]

where $\omega_{0}$ is the mode frequency in the absence of any magnetic correlations and it is assumed that $|\omega-\omega_{0}|\ll\omega_{0}$. The anomalous magnetic frequency shift of the pantograph mode is thus proportional to the dimer spin correlator. This also follows from the calculation of the corresponding spin-phonon coupling constant Choi et al. (2003) that assumes a particular power-law $u$-dependence of $J$ Miyahara and Ueda (2003).

Optical phonons and magnetic excitations have been extensively studied in SrCu2(BO3)2 using Raman spectroscopy at ambient pressureLemmens et al. (2000); Choi et al. (2003); Gozar et al. (2005); Homes et al. (2009). In the present work we perform similar measurements in a 1-mm-culet 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 cell environment was a He-flow cryostat with base temperature of 2.5 K. The pressure transmitting medium was argon. The pressure was determined in-situ from the shift of the ruby R1 fluorescence lineFeng et al. (2010); Zha et al. (2000). The temperature dependence of the effect, which becomes particularly important above 50 K, was explicitly corrected for Ragan et al. (1992). The pressure changes below 40 K are marginal (Fig. 2a). Most of the pressure change due to thermal expansion and solidification of argon occurs between room temperature and 40 K (Fig. 2b).

All pressure values quoted below correspond to base temperature readings. All experiments were performed on a $\sim$0.2$\times$0.1$\times$0.05 mm3 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 inset in Fig. 3). Unless otherwise noted, all spectra shown were measured in the $\bar{c}(a^{\prime}b^{\prime})c$ scattering geometry. An independently measured background from the pressure transmitting medium was subtracted from all spectra.

Optical measurements on SrCu2(BO3)2 are extremely challenging due to the narrow band gap of the material, which leads to high absorptionHomes et al. (2009). 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 SrCu2(BO3)2 at a pressure of $P=2$ kbar and $T=2.6$ K is shown in Fig. 3a). The observed frequencies appear fully consistent with previous measurements at ambient pressure Gozar et al. (2005). The three visible lowest-energy excitations are magnetic in origin: two triplets and one singlet. All remaining peaks are phonons. The peak widths are in all cases below 1 cm*-1* and correspond to the expected instrumental resolution. Since data for the entire spectrum are accumulated concurrently, this confirms that temporal drift of spectrometer alignment during the long data collection period is a non issue.

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 Giannozzi et al. (2009), using the SSSP Accuracy (version 1.1) pseudopotential library Prandini et al. (2018a); Lejaeghere et al. (2016); Hamann (2013); Schlipf and Gygi (2015); Prandini et al. (2018b); Garrity et al. (2014); Corso (2014). The kinetic energy cutoff was 120 Ry and the charge density cutoff 600 Ry. Brillouin zone integration was performed using a 4$\times$4$\times$4 k-point grid. For better convergence a Marzari-Vanderbilt cold smearing of 0.01 Ry was applied.

In order to validate the experimental methodology we first checked the behavior of the three magnetic excitations. With increasing pressure all three peaks shift to lower energies and progressively weaken. Their measured frequencies are plotted in colored circles in Fig. 3c). 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) Sakurai et al. (2018) and inelastic neutron scattering Zayed et al. (2017) 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.

The central result of this Rapid Communication is the observation of an anomalous temperature dependence of the pantograph mode, as visualized in Fig. 4. Note the logarithmic temperature scale. At all pressures, the pantograph mode undergoes at most a modest hardening upon cooling down to 40 K. This behavior is due to the usual anharmonicities of lattice vibrations and the temperature dependence of pressure in the loaded cell at higher temperatures.

At temperatures below 40 K the anharmonic thermal frequency shift of the pantograph mode is negligible. In order to establish this, we measured the temperature dependence of the pantograph mode frequency between 35 K and room temperature at ambient pressure. The result is visualized in Fig. 5 (symbols). These data were analyzed using the well-known predictions for anharmonic multi-phonon coupling (Eq. 3.9 in Balkanski et al. (1983)). An excellent fit is obtained using empirical parameters $\omega_{0}=201.2~{}\mathrm{cm}^{-1},\,C=0$ and $D=-2.7(1)\cdot 10^{-3}~{}\mathrm{cm}^{-1}$ (solid line in Fig. 5). Extrapolating this curve to below 40 K yields an estimate of the anharmonic shift in the low-temperature range: $\delta\omega\lesssim 0.1$ cm*-1*. This is considerably smaller than the observed anomalous magnetic frequency shift and can be safely disregarded.

Below this point the mode suddenly becomes strongly $T$- dependent. At low applied pressures itsoftens upon cooling to the 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. 6 a). For $T\gtrsim 40$ 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.

As established above, at temperatures below 40 K anharmonic effects extrapolated from measurements above $35$ K are entirely negligible. The only remaining relevant energy scale is the magnetic one. We conclude that the anomalous temperature dependence of the pantograph mode is due to magneto-elastic coupling. The corresponding relative frequency shift between 40 K and 2.6 K is plotted against pressure in Fig. 6 b) (left axis). As mentioned above, the cell pressure is practically constant in this temperature range. The shift remains flat up to about $P_{1}\sim 15$ kbar and then steadily increases up to the highest attainable pressures, switching sign at about $P_{2}\sim 22$ kbar.

As shown by ESR experiments Sakurai et al. (2018), $J^{\prime}$ decreases by only about 12% from 0 to 15 kBar. This allows us to assume that the 2nd derivative in Eq. 1 is also only weakly pressure-dependent to some approximation. The lack of any pressure dependence for $P\lesssim P_{1}$ in Fig. 6 b) confirms this. Here the system is clearly in the dimer phase Zayed et al. (2017). In this regime the exact result is $\langle\mathbf{S}_{1}\mathbf{S}_{2}\rangle\equiv-\frac{3}{4}$ regardless of $J^{\prime}/J$ Miyahara and Ueda (1999), and the measured constant frequency shift implies a constant ${\partial^{2}J^{\prime}}/{\partial u^{2}}$. Assuming the trend continues at higher pressures, to within a scale factor Fig. 6 b) represents the pressure dependence of $\langle\mathbf{S}_{1}\mathbf{S}_{2}\rangle$. The exact value for the dimer phase provides a calibration for the entire plot (Fig. 6b), right axis). At $P_{2}$ the relative alignment of nearest neighbor spins switches sign and becomes predominantly ferromagnetic.

That AF dimers are replaced by a completely new spin correlation pattern already at rather low pressures is significant. The only other indication of that is the rather tenuous 22 kbar strucure factor data from the neutron study of Ref. Zayed et al. (2017). The transformation that we observe starts already at about 15 kbar. It can only be driven by a subtle change in the $J^{\prime}/J$ ratio, as assumed in most theoretical studies Albrecht and Mila (1996); Miyahara and Ueda (1999); Weihong et al. (1999); Koga and Kawakami (2000); Müller-Hartmann et al. (2000); Takushima et al. (2001); Carpentier and Balents (2001); Chung et al. (2001); Zheng et al. (2001); Läuchli et al. (2002); Al Hajj and Malrieu (2005); Darradi et al. (2005); Isacsson and Syljuåsen (2006); Moukouri (2008); Moliner et al. (2011); Corboz and Mila (2013); Ronquillo and Peterson (2014); Wang and Batista (2018); Boos et al. (2019). It takes place long before the exchange constants themselves can be expected to switch sign Sakurai et al. (2018). It also seems not to be associated with any structural transition, of which we do not observe any obvious signs such as splitting of phonon lines or the appearance of new modes.

According to a recent calorimetric study Guo et al. (2019), there is a distinct thermodynamic phase that emerges below $T_{c}\sim 2$ K just about $P_{1}$. The authors suggest that it may be a plaquette state. The sign-switching reported here is consistent with that interpretation. While in the dimer phase the spin correlations are AF, in the plaquette state they at least partially switch to FM and may evolve continuouslyCorboz and Mila (2013); Boos et al. (2019). Unfortunately, the lowest attainable temperature in our experiments is slightly above $T_{c}$. The observed behavior can be interpreted as a change in the character of dominant short-range spin correlations, which are just about to order in another plaquette configuration upon further cooling. According to Guo et al. (2019), Néel magnetic order sets in above approximately $P_{2}$. Here even stronger FM correlations are predicted Corboz and Mila (2013), and indeed observed in our measurements.

In summary, our measurements of the pantograph mode in SrCu2(BO3)2 provide an indirect but precise measurement of a pressure-induced sign switching of nearest-neighbor dimer spin correlations. We hope that a quantitative comparison with theoretical studies will become possible in the future.

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.