Spin excitations near the pressure-induced antiferromagnetic transition in SrCu₂(BO₃)₂

This article is part of a collection of articles related to the International Conference on Neutron Scattering, ICNS2025.

SrCu₂(BO₃)₂ is a frustrated quantum magnet which realizes the Shastry–Sutherland model. The system exhibits a sequence of pressure-induced phases, including a long-range antiferromagnetically ordered state. We have performed inelastic neutron scattering experiments on a single crystal at 4.2 (2) GPa, in the antiferromagnetic phase. The observed dispersive spin excitations were modeled using linear-spin-wave theory to extract exchange parameters. Our results are consistent with previously reported parameters at higher pressures and confirm the existence of a Goldstone mode as well as underlining the importance of the interlayer coupling. These results complement previous high-pressure studies and provide further input for theoretical models of SrCu₂(BO₃)₂ under extreme conditions.

1. Introduction

Frustrated quantum magnets are a fertile ground for discovering exotic states of matter and unconventional excitations. In this context, the Shastry–Sutherland model (SSM) describes a two-dimensional network of orthogonal spin-1/2 dimers coupled by intra- and interdimer exchange interactions, respectively, J and J′ (Sriram Shastry & Sutherland, 1981). For small J′/J, the ground state is an exact product of singlet dimers, whereas increasing J′/J introduces frustration and drives the system through a sequence of magnetically correlated phases, including a plaquette state and ultimately long-range antiferromagnetic (AFM) order (Corboz & Mila, 2013; Boos et al., 2019). The model also hosts magnetization-plateau phases in applied magnetic fields (Momoi & Totsuka, 2000; Corboz & Mila, 2014).

The compound SrCu₂(BO₃)₂ provides an almost ideal realization of the SSM (Miyahara & Ueda, 1999; Kageyama et al., 1999b). It is characterized by interactions J and J′, respectively, inside and between the dimers, as illustrated in Figs. 1(a) and 1(b). At ambient pressure, SrCu₂(BO₃)₂ is well characterized by the SSM with J′/J = 0.63 (Miyahara & Ueda, 2000; Shi et al., 2022), placing it near the boundary between the singlet-dimer and plaquette regimes. Experiments have confirmed the singlet-dimer ground state (Miyahara & Ueda, 1999; Kageyama et al., 2000; Gaulin et al., 2004; Kakurai et al., 2005) and shown that applying hydrostatic pressure increases J′/J, sequentially inducing a plaquette phase (Zayed et al., 2017; Jiménez et al., 2021; Waki et al., 2007; Sakurai et al., 2018) and eventually long-range AFM order (Haravifard et al., 2014; Guo et al., 2020). Recently an additional quantum-spin-liquid (QSL) phase has been predicted to reside between the plaquette and AFM phases, but whether this is also realized in SrCu₂(BO₃)₂ remains an open question (Corboz et al., 2026). However the close correspondence between model and material makes SrCu₂(BO₃)₂ an ideal platform for testing theoretical predictions for the highly frustrated SSM.

Figure 1 (a) Unit cell of SrCu2(BO3)2 showing only the magnetic Cu^2+ ions and the accompanying AFM structure. (b) A single layer of Cu^2+ ions corresponding to the Shastry–Sutherland lattice. (c, d) Comparison between predicted and measured excitation spectra for the AFM phase of SrCu2(BO3)2 at T = 4.6 (1) K and P = 4.2 (2) GPa. The blue and gray dashed curves include the interlayer interaction Jc; the gray curve shows all calculated modes, while the blue curve shows the corresponding intensity. This model yields both a gapped and a gapless Goldstone mode at the zone center, with intensity appearing at (1, 0, 0) and (1, 1, 0), respectively. In contrast, the red curves correspond to a model with DM interactions and no interlayer coupling, as detailed by Fogh et al. (2024), resulting in a single gapped mode at both positions.

Going beyond 4 GPa, the system undergoes a change in crystal symmetry from tetragonal to monoclinic. Below the monoclinic distortion there have been reports of indirect AFM signatures at low temperature (Guo et al., 2020). Above the monoclinic distortion there is strong AFM order, setting in at a high temperature of about 120 K (Zayed, 2010; Haravifard et al., 2014). Recently our own inelastic neutron scattering experiment enabled quantitative determination of the mag­netic interaction parameters in SrCu₂(BO₃)₂ at 5.5 GPa. The spin waves can be described by the same J′ and J in the plane but require a dramatic increase in J′/J and the addition of an interlayer coupling J_c (Fogh et al., 2024). In the model, the orthogonal arrangement of the dimers together with the AFM spin arrangement [Fig. 1(a)] gives rise to a change in mode intensity such that at (1, 0, 0) one observes a gapped mode whereas at (1, 1, 0) the spectral weight is found in the Goldstone mode [Figs. 1(c) and 1(d)]. The splitting between the modes is then given by the interlayer coupling. However, in the previous study, the spectrum at (1, 1, 0) was not directly probed and, although unlikely, a model including a Dzyaloshinskii–Moriya (DM) interaction could also explain the observations. Therefore, to pin down the origin of the observed gap at (1, 0, 0) one needs to study the spectrum at (1, 1, 0).

Here we present complementary inelastic neutron scattering measurements performed at pressures close to the phase boundary between the tetragonal and monoclinic symmetries. We determine the evolution of the spin excitation spectrum and extract exchange interaction parameters from fits within linear-spin-wave theory. We directly confirm the existence of a Goldstone mode and thereby underline the importance of the interlayer interaction in SrCu₂(BO₃)₂, providing further constraints on theoretical descriptions of its behavior under pressure.

2. Experimental details

A single crystal of SrCu₂(BO₃)₂ grown by the floating-zone method (Kageyama et al., 1999a; Jorge et al., 2004) was cut into a disc with a diameter of 4 mm, a thickness of 1.3 mm and a total weight of 36 mg. The crystal was oriented with Q = (q_h, q_k, 0) in the horizontal scattering plane. The sample was mounted in a Paris–Edinburgh press using a TiZr gasket assembly, Zr-toughened alumina anvils and deuterated 4:1 methanol–ethanol as pressure medium following the procedure detailed by Fogh et al. (2024). However, two improvements were implemented: (1) Cooling of the top gasket using liquid nitrogen to freeze the pressure transmitting fluid and thereby avoid the trapping of air in the sample chamber when it is placed on the lower gasket. (2) Inclusion of a 36 mg ring-shaped piece of Pb placed around the sample for pressure determination. The nuclear 111 Bragg peak of Pb was used to calibrate the pressure by using the well characterized equation of state (Strässle et al., 2014). The final pressure at 4.6 K was 4.2 (2) GPa with a pressure drop of less than 10% upon cooling from room temperature. Tracking the 400 reflection during pressure increase revealed a small shift from h = 4.000 to h = 4.025 reciprocal lattice units (r.l.u.), corresponding to an in-plane lattice contraction of approximately 0.6% (Δa ≈ −0.056 Å for a₀ = 8.995 Å), consistent with previous studies (Loa et al., 2005). Note that we stay with the tetragonal notation throughout the experiment, which is a good approximation in this geometry with the a and b lattice parameters changing less than 1% at our maximum pressure and with a monoclinic angle of β = 94° (Loa et al., 2005; Haravifard et al., 2014). Hence, the monoclinic distortion mainly results in a sliding of the Cu²⁺ layers with respect to each other out of plane, but the symmetry within the layers stays approximately tetragonal.

On the thermal triple-axis neutron spectrometer, IN8 (Piovano & Ivanov, 2023), at the Institute Laue–Langevin, we worked with a pyrolytic graphite [PG(200)] analyzer and Si(111) monochromator, both double focused. A PG filter was placed between sample and analyzer to suppress higher-order scattering. Two different final wavenumbers were used during the experiment, k_f = 2.662 Å⁻¹ and k_f = 1.97 Å⁻¹, giving an elastic line resolution full width at half-maximum of, respectively, 1.22 and 0.51 meV. Within each measurement the final wavevector was kept constant but the incident wavevector was changed to realize adequate energy transfer. A series of constant-Q and constant-E scans were performed. In addition, we followed the temperature evolution of the magnetic Bragg peak 100 upon heating (Zayed et al., 2025).

An additional IN8 measurement was performed at 3.6 GPa using a different load of the Paris–Edinburgh cell under otherwise identical instrumental conditions. A 60 mg sample cut from the same crystal growth was used, corresponding to a pressure just below the transition to the monoclinic phase (Fogh et al., 2021).

3. Results and discussion

Fig. 2(c) shows the temperature dependence of the 100 magnetic Bragg peak. Five sample rotation points were collected: two off-peak for background (orange) and three near the peak to approximate the integrated intensity (blue). Dashed lines are guides to the eye. Below 150 K, additional intensity appears at the (1, 0, 0) position, demonstrating the onset of AFM order. The apparent ordering temperature is slightly higher than previously reported (Haravifard et al., 2014), probably due to thermal lag between the thermometers and the sample during heating.

Figure 2 (a) Constant-E scans at (1, qk, 0) and (b) constant-Q scans performed with two different fixed kf. For both panels (a) and (b), the fitted mode positions are shown with the colored lines and filled-in areas as described in the text. The predicted mode positions from linear-spin-wave calculations are plotted as vertical black bars and colored using the our fitted model parameters. (c) The 100 magnetic Bragg peak intensity as a function of temperature at 4.2 GPa. The dashed lines are guides to the eye. (d) Neutron intensity as a function of energy transfer at Q = (1, 1, 0) and (1, 0, 0) measured at 4.6 K and approximately 110 K with kf = 1.97 Å^−1. The inset shows the effect on the fit quality when including an additional Gaussian peak with the same width and area as the mode observed at (1, 0, 0). The goodness of fit remains unchanged until about 0.5 meV, indicating that if a spin gap exists it must be smaller than 0.5 meV. (e) Constant-Q scans at 3.6 GPa. Expected AFM spin-wave mode positions (from the 4.2 GPa exchange parameters) are indicated by colored vertical bars for each Q. For Q = (0, 2, 0), the expected mode is at 6.1 meV. Gray points: reference scan at Q = (0.7, 1.3, 0).

A selection of constant-E scans is shown in Fig. 2(a). Mode positions were obtained by fitting the observed neutron intensities with Gaussian functions. These were constrained to be symmetrically displaced, in q_k, around either (1, ±q_k, 0) or (1, 1 ± q_k, 0), while the widths and intensities of the peaks were left as free parameters.

A selection of constant-Q scans is shown in Fig. 2(b). The elastic line was fitted with a Voigt function, representing the instrumental resolution as a convolution of a Gaussian with a Lorentzian broadening. A sloping background was included, and the magnetic excitations were modeled with Gaussian peaks. Mode positions were then extracted from the fitted peak centers.

In Fig. 2(d) we compare constant-Q scans at the key positions (1, 0, 0) and (1, 1, 0). At base temperature a clear excitation is observed at 1.89 (2) meV at (1, 0, 0). It disappears at high temperatures, confirming its magnetic origin. This magnetic signal is not seen at (1, 1, 0). We investigate the possibility of a low-energy mode at this position by including an additional Gaussian peak with a fixed position close to the elastic line and the same width and integrated intensity as the mode observed at (1, 0, 0), since a strong mode is in general expected at the Γ point. In the inset of Fig. 2(d) we plot the resulting reduced chi-squared as a function of peak position with . Here N is the number of data points, p the number of fitted parameters, y_i the measured intensities, f_i the model values and σ_i their respective uncertainties. The additional mode does not improve the fit quality and for mode placement above 0.5 meV the quality significantly decreases, indicating that our data do not support the presence of a gapped mode at (1, 1, 0).

Therefore, our results are consistent with the existence of a Goldstone mode at (1, 1, 0) and favor the model previously put forward including an interlayer coupling rather than the Dzyaloshinskii–Moriya interaction (Fogh et al., 2024). A comparison between predicted mode positions for the two models and measured spectra shown in Figs. 1(c) and 1(d) underlines this point.

Using the implementation of linear-spin-wave theory in the Sunny package (Dahlbom et al., 2025) for evaluating the excitation spectrum, we extracted the model parameters shown in Table 1, and these are compared with those given by Fogh et al. (2024). There is a good agreement between the results, keeping in mind that the two experiments were performed at different pressures. In particular, the decrease in J is consistent with the expected behavior of this interaction at higher pressure. This interaction is highly pressure sensitive because it goes via the Cu–O–Cu superexchange path which has a bond angle close to the Goodenough–Kanamori critical angle of 95° (Goodenough, 1955; Kanamori, 1959). On the other hand, J′ changes less because the superexchange path is via the stiff BO₃ unit. The interlayer coupling also increases with pressure, as would be naively expected when increasing orbital overlaps without changing the bond angles.

Table 1
Comparison of model parameters determined in this work and given by Fogh et al. (2024)

	P (GPa)	Δ (meV)	J (meV)	J′ (meV)	Jc (meV)	J′/J	Jc/J
This work	4.2 (2)	1.83 (6)	2.5 (2)	4.3 (1)	0.049 (6)	1.7 (2)	0.019 (2)
Fogh et al. (2024)	5.5 (5)	1.90 (8)	2.26 (26)	4.28 (14)	0.053 (3)	1.8 (2)	0.023 (3)

Since the splitting between gapped and non-gapped modes at (1, 0, 0) is given by Δ = 8S(J′J_c)^1/2, where S = 1/2 is the spin quantum number of the Cu²⁺ ions (Fogh et al., 2024), and the bandwidth given by J′ is not well determined in our experiment, the quantity that we can determine to a high precision is the product J′J_c. Nevertheless, the results highlight the importance of the three dimensionality in the system, which may also be important at lower pressures. The interlayer coupling has little consequence in the dimer phase, but theoretical studies show that the plaquette phase in the SSM destabilizes above a critical value of J_c/J ∼ 0.0–0.05 (Koga, 2000; Vlaar & Corboz, 2023). Although the values in Table 1 fall well below this point, the three dimensionality in SrCu₂(BO₃)₂ may still play a role in the formation of plaquettes (Zayed et al., 2017; Corboz & Mila, 2013; Boos et al., 2019) as well as the predicted deconfined quantum critical point around the plaquette–AFM transition (Lee et al., 2019; Yang et al., 2022; Cui et al., 2023; Guo et al., 2025).

Inspecting the data collected at 3.6 GPa and shown in Fig. 2(e) we observe broad magnetic excitations at (2, 2, 0), (0, 1, 0) and (0, 1.5, 0), while no clear signal is observed at (0, 2, 0). When contrasted with the scattering expected for an AFM phase, using the exchange interaction obtained from the 4.2 GPa measurements, it becomes evident that the observed spectrum is not compatible with the same AFM phase. Similarly, the data are not consistent with the sharp, well defined excitations expected for the plaquette state (Zayed et al., 2017). The broad spectral weight is, on the other hand, not inconsistent with the continuum-like response predicted by Corboz et al. (2025) and as such could be a first glimpse of the anticipated QSL phase. While the limited counting statistics and the relatively coarse energy resolution prevent any conclusive characterization of the excitation spectrum at this pressure, these preliminary results highlight the need for further experiments to elucidate this pressure region and the possible existence of a QSL in the phase diagram of SrCu₂(BO₃)₂.

4. Conclusion

We have investigated the spin excitation spectrum of SrCu₂(BO₃)₂ at 4.2 GPa using inelastic neutron scattering. Our measurements reveal dispersing magnetic excitations consistent with long-range AFM order. We use linear-spin-wave calculations to fit model parameters, obtaining values close to those previously reported at higher pressure (Fogh et al., 2024). Most importantly, our measurements confirm the existence of a Goldstone mode at Q = (1, 1, 0) which further supports the model. In particular, we confirm that an interlayer coupling is needed to explain the measured spectrum, and hence three dimensionality may be important in SrCu₂(BO₃)₂ also at lower pressures. Moreover, our findings underscore the feasibility and importance of precise inelastic neutron scattering studies under extreme conditions and provide valuable input for ongoing theoretical efforts to map the complete pressure–temperature phase diagram of SrCu₂(BO₃)₂.