Magnetic excitations and amplitude fluctuations in insulating cuprates

TL;DR

This study uses light scattering to analyze magnetic excitations in insulating cuprates, revealing universal spectral features, amplitude fluctuations, and chiral excitations, with results aligning with theoretical predictions.

Contribution

The paper provides the first detailed symmetry-resolved light scattering spectra of multiple cuprates, confirming theoretical models and identifying new amplitude fluctuation and chiral excitation features.

Findings

Spectral shape in B1g symmetry is nearly universal across samples.

Asymmetric line shape relates to amplitude fluctuations of magnetization.

Support for chiral excitations as a continuum below 3J, with an additional excitation in La2CuO4.

Abstract

We present results from light scattering experiments on three insulating antiferromagnetic cuprates, YBa$_2$Cu$_3$O$_{6.05}$, Bi$_2$Sr$_2$YCu$_2$O$_{8+\delta}$, and La$_2$CuO$_4$ as a function of polarization and excitation energy {using samples of the latest generation. From the raw data we derive symmetry-resolved spectra.} The spectral shape in $B_{1g}$ symmetry is found to be nearly universal and independent of the excitation energy. The spectra agree quantitatively with predictions by field theory [\onlinecite{Weidinger:2015}] facilitating the precise extraction of the Heisenberg coupling $J$. {In addition, the asymmetric line shape on the high-energy side is found to be related to amplitude fluctuations of the magnetization. In La$_2$CuO$_4$ alone minor contributions from resonance effects may be identified.} The spectra in the other symmetries are not universal. The variations may be traced back to weak resonance effects and extrinsic contributions. For all three compounds we find support for the existence of chiral excitations appearing as a continuum in $A_{2g}$ symmetry having an onset slightly below $3J$. In La$_2$CuO$_4$ an additional isolated excitation appears on top of the $A_{2g}$ continuum.

Full text

I Introduction

If a continuous symmetry is spontaneously broken the order parameter can fluctuate about either its average direction or magnitude. The changes in direction are usually mass-less (Nambu-Goldstone mode) and decoupled from the amplitude fluctuations (Higgs mode) having a finite energy. The Higgs fluctuations play a crucial role in a variety of interacting systems Podolsky et al. (2011); Pekker and Varma (2015) including the standard model in high-energy physics Higgs (1964), Bose condensates Endres et al. (2012), super-fluidity and -conductivity Sooryakumar and Klein (1980); Littlewood and Varma (1981); Cea and Benfatto (2014) and magnetism Rüegg et al. (2008). The discussion picked up new momentum after the discovery of the cuprates and the iron-based compounds for the putative interrelation of magnetic and other types of fluctuations and superconductivity Perali et al. (1996); Lee et al. (2006); Toyota et al. (2007); Anderson (2007); Mazin et al. (2008); Pfleiderer (2009); Hirschfeld et al. (2011); Scalapino (2012); Fradkin et al. (2015); Pekker and Varma (2015); Lederer et al. (2015) since, at least in the cuprates, magnetic fluctuations can be observed in large regions if not in the entire phase diagram Wakimoto et al. (2007).

In magnetic systems, the order in the ground state, the exchange energies $J_{i}$, the excitation spectra or fluctuations of the phase or the amplitude of the magnetization Zwerger (2004); Podolsky et al. (2011) are still a matter of debate both from an experimental and a theoretical point of view. While neutron and Raman scattering experiments on insulating antiferromagnetic cuprates Lyons et al. (1988a); Sugai et al. (1988); Sulewski et al. (1990) can be described qualitatively in terms of spin-wave theory in a Heisenberg model Fleury and Loudon (1968); Shastry and Shraiman (1990); Knoll et al. (1990); Canali and Girvin (1992); Chubukov and Frenkel (1995a); Rübhausen et al. (1997) with only nearest neighbor exchange coupling $J_{1}$, the role of multi-magnon and cyclic spin excitations or amplitude fluctuations are not well understood yet Shastry and Shraiman (1990); Sulewski et al. (1991); Vernay et al. (2007); Podolsky et al. (2011).

Recently, a field theoretical approach successfully described the line shape of the two-magnon Raman spectra in $B_{1g}$ symmetry considering perturbation theory up to infinite order and amplitude fluctuations of the order parameter. For La2CuO4 (La214) and YBa2Cu3O6.05 (Y123) the line shapes can be well reproduced using slightly different masses for the Higgs modes Weidinger and Zwerger (2015). Similar agreement was found for iridates Gretarsson et al. (2016) suggesting the response in $B_{1g}$ symmetry to be universal. Yet resonance effects and extrinsic contributions such as luminescence were found to have a substantial influence on the line shape Lyons et al. (1988b, 1989); Sugai (1990); Yoshida et al. (1992); Blumberg et al. (1996); Muschler et al. (2010) and need to be considered upon properly interpreting the response.

Resonance effects of the Raman response of the simultaneous excitation of two magnons at $\Omega_{\bf q}$ and $\Omega_{-\bf q}$, where $\Omega_{\bf q}$ is the magnon dispersion, were studied theoretically by several authors. Shastry and Shraiman Shastry and Shraiman (1990) used the one-band Hubbard model as a starting point and described the usual resonances when either the energy of the incident or scattered photon, $\omega_{I,S}$, corresponds to a real transition between the lower and the upper Hubbard band. Chubukov and coworkers Chubukov and Frenkel (1995b, a); Morr and Chubukov (1997) realized that there exists a triple resonance for the parameters of the cuprates when $\omega_{I}$, $\omega_{S}$, and $\omega_{I}-\Omega_{\bf q}$ simultaneously match transitions between the valence and the conduction band. In this case the two-magnon maximum in the Raman spectrum slightly below $3\,J$ has a resonance energy different from $2\max(\Omega_{\bf q})$ at $4J$. As a consequence the line shape depends on $\omega_{I}$. Brenig et al. Brenig et al. (1997) derived the cross section from the more realistic three-band Hubbard model. Zhang and Rice Zhang and Rice (1988) have shown that the two models are equivalent to some extent if the large Hubbard $U$ of 5-10 eV, being relevant for the cuprates, is replaced by an effective energy $\Delta E$ close to the charge transfer energy between copper and oxygen $|\varepsilon_{d}-\varepsilon_{p}|$ in the range of 2 eV which corresponds to the optical gap.

In this publication we present results of light scattering experiments on three different cuprate families, La214, Y123 and Bi2Sr2YCu2O8+δ (Bi2212:Y). In addition to the so far published symmetry-resolved response for Gd214 and La214 Sulewski et al. (1991); Muschler et al. (2010) we derive and compare symmetry-resolved spectra for the three compound classes La214, Y123, and Bi2212:Y and use various laser energies in the range 2.16 and 2.71 eV for excitation so as to separate the intrinsic non-resonant line shape from a putative influence of resonant light scattering and extrinsic contributions. Below three times the magnetic exchange energy $J$ the spectra are essentially universal in that the response $\chi^{\prime\prime}(\Omega)$ depends only on $J$. Above the two-magnon maximum there may be non-universal contributions which are explained in terms of a Higgs mode and minor contributions from resonant light scattering and luminescence.

II Experiment

Y123 was prepared in BaZrO3 crucibles Erb et al. (1995) demonstrated to yield the highest purity Erb et al. (1996). To obtain the best crystal quality and the lowest possible oxygen content of approximately 6.05 the as-grown material was annealed in Ar at 900 C for one day in a tube furnace with the single crystal buried in sintered Y123 and then slowly cooled to room temperature during another day. Bi2212:Y was grown in Y-stabilized ZrO2 crucibles Kendziora et al. (1993) and La214 was prepared using the traveling solvent floating zone (TSFZ) method. The as-grown crystals had a Néel temperature $T_{N}$ of 280 K. After annealing the crystal in Ar, $T_{N}$ increased to 325 K [Muschler et al., 2010] which has not been exceeded so far Kastner et al. (1998); Gozar et al. (2005). Given these facts it is likely that no better samples of Y123 and La214 can be obtained to date. There are only very few reports on the successful preparation of Bi2212:Y, and we do not have data for comparison but the results shown below indicate that these crystals are in the same quality class as Y123 and La214.

For the measurements on Y123 and Bi2212:Y we used as-grown and freshly cleaved surfaces, respectively. The surface for the experiments on La214 was polished so as to obtain sufficiently large areas of approximately $2\times 2\,{\rm mm}^{2}$. In order to test the surface quality after polishing we cleaved a small piece. The results obtained from the cleaved surface and the polished surface are close to identical. In particular all features appear in the same positions as shown in Fig. 4 in Appendix A.

The samples were mounted on the cold finger of a He-flow cryostat in a cryogenically pumped vacuum. The experiments were performed with a calibrated light scattering setup Opel (2000); Muschler et al. (2010) with the resolution set at 20 cm*-1* at 458 nm. For excitation an Ar ion laser (Coherent, Innova 304C) was used emitting at 528, 514.5, 496.5, 488, 476 or 458 nm. For one experiment on La214 we used a solid state laser (Coherent, Genesis MXSLM) emitting at 575 nm. The laser-induced heating was determined experimentally to be close to 1 K per mW absorbed power. Spectra were measured in the six polarization configurations $xx$, $xy$, $x^{\prime}x^{\prime}$, $x^{\prime}y^{\prime}$, $RR$, and $RL$, where $x^{\prime}=1/\sqrt{2}(x+y)$, $y^{\prime}=1/\sqrt{2}(y-x)$, and $R,L=1/\sqrt{2}(x\pm iy)$, allowing one to extract all in-plane symmetry components, $\mu=A_{1g}$, $A_{2g}$, $B_{1g}$, and $B_{2g}$, of the tetragonal $D_{4h}$ space group by linear combinations of the experimental spectra. For the polarization and symmetry assignment we use the idealized quadratic CuO2 plane. The spectra we show below represent the response $R\chi^{\prime\prime}(\Omega,T)$ as a function of the Raman shift $\Omega$ and the temperature $T$ that is obtained by dividing the cross section by the Bose-Einstein factor $\{1+n(T,\Omega)\}=[1-\exp(-\hbar\Omega/k_{B}T)]^{-1}$. $R$ is an experimental constant.

III Results

The raw data of La214 measured at all six main polarization combinations are displayed in Fig. 5 in Appendix B. In Fig. 1 we show the spectra of Y123, Bi2212:Y and La214 for all four pure symmetries Raman-active for light polarizations in the CuO2 plane.

The spectra in $B_{1g}$ symmetry have the highest intensity in the range 2000 to 4000 cm*-1*. The peak maxima of Y123, Bi2212:Y, and La214 are observed to be at 2735, 2890, and 3300 cm*-1* (339.3, 358.6, and 409.4 meV), respectively. At low energies the continuum is superposed by narrow lines from phonons which are not of interest here and were therefore measured with an energy-dependent resolution in the range $10-20$ cm*-1* and a step width of 50 cm*-1*. Below the maximum the spectra depend strongly on energy. The variation proportional to $\Omega^{3}$ predicted theoretically for low energies Podolsky et al. (2011); Weidinger and Zwerger (2015) cannot directly be observed because of the contributions from phonons and particle-hole excitations which cannot reliably be subtracted. Above the maximum the intensity decays approximately as expected for a Lorentzian oscillator except for the range around 4000 cm*-1* making the peak rather asymmetric. At high energies the intensity becomes very small. Whereas Y123 and Bi2212:Y are similar it is clear at first glance that La214 is different in that a secondary maximum appears above the main peak.

Differences between La214 and the other compounds are also present in the other symmetries. In all materials there is a broad maximum between 3000 and 4000 cm*-1* in $A_{1g}$ symmetry, slightly above the position of the $B_{1g}$ peak. In La214 alone there is another broad feature having two substructures between 5000 and 7000 cm*-1* which is stronger than the first one. In Bi2212:Y and Y123 there are just weak shoulders in this energy range. In Y123 this shoulder is barely visible. No qualitative differences between the compounds are found in $B_{2g}$ symmetry. In the range of the $B_{1g}$ peak the $B_{2g}$ spectra differ only quantitatively.

In $A_{2g}$ symmetry there are two remarkable observations: (i) In La214 there is a well resolved peak at 4600 cm*-1*, slightly above that in $B_{1g}$ symmetry, which is completely absent in the other compounds. (ii) In all compounds the intensity vanishes to within the experimental resolution for energies below approximately 3000 cm*-1* close to the $B_{1g}$ maximum. The onset above this energy is well defined and nearly abrupt being reminiscent of gap-like behavior for low energies. As studied in detail for La214 [Fig. 6 (b)], the onset is independent of the excitation energy indicating an intrinsic scattering process.

Since the high-energy parts of the $B_{1g}$ spectra are not universal both close to the maximum and at high energies, the origin of the differences needs to be explored before the line shape will be analyzed. To this end excitation with various laser lines is useful. In Fig. 2 (a)-(c) we show the raw data of the $x^{\prime}y^{\prime}$ spectra ($B_{1g}+A_{2g}$) for all the compounds using green and blue excitation. The spectra taken with green excitation end already slightly above 5000 cm*-1* Raman shift because the range of the spectrometer is limited to 725 nm. In Y123 and Bi2212:Y the scattering cross sections are slightly stronger with blue excitation. In La214 the spectra measured with green light are more intense, and the phonons in the range up to 1500 cm*-1* resonate strongly.

For better comparison constant multiplication factors are applied so as to match the peak intensities [Fig. 2 (d)-(f)]. Obviously the line shape is identical to within the experimental accuracy (statistical error and calibration) for Y123 and Bi2212:Y. For La214 the small extra peak at 4600 cm*-1* is weaker with green excitation. As a consequence it moves closer to the main peak. For better insight we looked at this variation using six different lines as shown in Fig. 7 of Appendix D. The relative intensity of the extra peak at 4600 cm*-1* decreases rather rapidly upon changing the laser line from 488 to 496 nm [Fig. 7 (b)].

IV Discussion

Qualitatively speaking the peak in $B_{1g}$ symmetry comes from a double spin-flip on two neighboring Cu atoms Fleury and Loudon (1968); Sulewski et al. (1990). However, the resulting line shape is a matter of debate as outlined in the introduction. Here we address the question as to whether the line shape is universal or has universal components. There are obvious differences between La214 and the two other materials. To be more quantitative we plot the $B_{1g}$ spectra of Fig. 1 on a normalized energy scale.

Fig. 3 shows a superposition of the spectra of Y123, La214, and Bi2212:Y. Both the intensity and the energy scales are normalized. The spectra of Y123 and La214 are multiplied by factors of 1.11 and 2.27, respectively, to match the peak intensity of Bi2212:Y. The energies are divided by appropriate factors to collapse the peak positions on the canonical value of 2.84 $J$ derived recently (There is a typo in the paper by Weidinger and Zwerger.) Weidinger and Zwerger (2015). From the peak positions alone one obtains 119, 144, and 126 meV, respectively, for $J$. For Y123 and La214 the full theoretical analysis including the Higgs mode yields 126 and 149 meV fairly close to the values of 120 and 143 meV found by neutron scattering Reznik et al. (1996); Headings et al. (2010).

Close to the maximum between 2.5 and 3.2 $J$ the line shapes of the peaks are rather similar. At low energies the lines of Y123 and La214 coincide down to 2.3 $J$. Below 2.3 $J$ La214 has a higher intensity from resonantly enhanced phonon contributions. Above 3.2 $J$ La214 exhibits a secondary maximum whereas Y123 has only an almost linear tail before the intensity saturates at a value lower than that in all other materials. The two-magnon maximum of Bi2212:Y is generally wider than those of La214 and Y123. The substantially off-stoichiometric composition of Bi2212:Y suggests that the line is inhomogeneously broadened, i.e. the larger width is extrinsic. The most dramatic differences are found above 4.4 $J$ where the saturation values of the intensities differ by more than a factor of three. We do not believe that this part of the spectra results from differences in the spin excitations. Rather, contributions from luminescence should be considered a possible (extrinsic) origin.

Concerning the overall intrinsic line shape a satisfactory description was achieved only recently using field-theoretic methods Weidinger and Zwerger (2015) after various studies on the basis of spin-wave theory Knoll et al. (1990); Canali and Girvin (1992); Rübhausen et al. (1996, 1997). The progress became possible through the inclusion of multi-magnon processes and amplitude (“Higgs”) fluctuations of the staggered magnetization on equal footing up to infinite order as suggested first by Podolsky and coworkers Podolsky et al. (2011). As a result the variation of the low-energy response, the nearly Lorentzian variation above the peak and the shoulder between 3.2 and 4.4 $J$ in Y123 and Bi2212:Y found an almost quantitative explanation. In addition, the differences to La214 can be traced back to a variation of the coupling constant (“Higgs mass”)of the anplitude mode. The detailed comparison between theory and experiment is shown in Ref. [Weidinger and Zwerger, 2015].

The line shape can be explained very well for the La214 data taken with blue excitation by including amplitude (“Higgs”) fluctuations of the magentization (see Ref. Weidinger and Zwerger, 2015). With green excitation the secondary maximum is weaker and completely absent for red photons Gozar et al. (2005). The dependence of the high-energy peak on the excitation line is not easily explained in terms of amplitude fluctuations since the spins and the amplitude fluctuations are excited via the same intermediate electronic states. Alternatively, the secondary maximum could originate in other types of spin excitations, for instance chiral or multi-magnon excitations that could couple to high-energy electronic states different from those coupling to the double spin-flip excitations. The intensity in $A_{2g}$ symmetry which has a well defined peak in La214 at a (nearly) constant Raman shift displaying the same resonance behavior as the two-magnon excitation in $B_{1g}$ symmetry (see Appendix C) argues for the existence of chiral excitations in addition to double spin flip scattering rather than an independent type of scattering. However, the interrelation of the $B_{1g}$ and $A_{2g}$ spectra is not yet clear. First, the similarity of the excitation energies close to 4600 cm${}^{-1}\sim 4\,J$ in both symmetries may be accidental but remains to be explained. Second, since intensity from chiral excitations according to the operator $\hat{C}={\bf S}_{i}\cdot({\bf S}_{j}\times{\bf S}_{k})$ should not appear in first order Ko et al. (2010); Michaud et al. (2011) the cross section should be smaller than in all other symmetries.

No effects similar to those in La214 are observed in Y123 and Bi2212:Y, where the line shapes were found to be independent to within the experimental accuracy for excitation in the blue and green spectral range. For excitation outside and probably also inside this range variations were found earlier Blumberg et al. (1996) but it is difficult to compare those data to our results on a quantitative basis. Hence, a final experimental answer can still not be given and requires more work on samples of the latest generation.

The theoretical studies proposed by Chubukov, Frenkel, and Morr Chubukov and Frenkel (1995b, a); Morr and Chubukov (1997) qualitatively explain changes of the $B_{1g}$ line shape but make also predictions for the other symmetries which we cannot confirm here: As opposed to the prediction the experimental $A_{1g}$ and $B_{2g}$ spectra of LCO resonate towards the green-yellow spectral range as shown in detail in Fig. 6 whereas the two-magnon peak is strongest for 496 nm.

Line shapes of the two-magnon spectra similar to those found for the cuprates in $B_{1g}$ symmetry were also observed in equivalent symmetry projections in some iridates Gretarsson et al. (2016) suggesting a universal behavior. However, the spectra in the other symmetries and in metallic materials such as doped cuprates or Fe-based compounds remain widely unexplained and cannot be treated in the same fashion as the $B_{1g}$ spectra of insulating CuO2 using field theory where convergence is granted by the $B_{1g}$ form factor $\propto k_{x}^{2}-k_{y}^{2}$. For addressing subjects such as resonances and the symmetry dependence, only numerical methods are currently available Moritz et al. (2011); Chen et al. (2011); Jia et al. (2014).

V Conclusions

We have measured high-energy spectra of three different cuprates at all main polarization configurations using at least two different excitation lines and, in addition to earlier work, derived all pure symmetries, $A_{1g}$, $A_{2g}$, $B_{1g}$, and $B_{2g}$. In $B_{1g}$ symmetry, the spectral shapes close to the two-magnon maximum are universal. If Higgs fluctuations of the magnetization are included all three compounds can be described consistently Weidinger and Zwerger (2015). However, since the spectral shape of La214 changes slightly with the photon energy additional contributions to the cross section may also exist including the triple resonance Chubukov and Frenkel (1995b, a); Morr and Chubukov (1997) or multi-magnon or chiral spin excitations Knoll et al. (1990); Shastry and Shraiman (1990); Vernay et al. (2007). In spite of these open issues the $B_{1g}$ spectra of the cuprates can be considered understood by now including the numerical value of $J$ which agrees with the neutron results to within a few percent. Yet, the other symmetries which display high intensities as well are still far from being understood, in particular the role of chiral excitation remains elusive and cannot be treated with field-theoretical methods.

Acknowledgements.

We acknowledge very fruitful discussions with P. Knoll, S. Weidinger, and W. Zwerger. Financial support for the work came from the DFG via the Transregional Collaborative Research Center TRR 80 and the Bavarian Californian Technology Center BaCaTeC (project no. A5 [2012-2]).

Appendix A Surface treatment

The surface preparation is crucial only for La214 since natural surfaces can be used in the cases of Y123 and Bi2212:Y. For La214 polishing and cleaving are possible treatments Muschler et al. (2010). However, polishing may result in damage and a severe influence on the spectra. On the other hand, a sufficiently large area is necessary for quantitative measurements and can only be obtained by polishing in the case of La214. For making sure that only intrinsic properties are observed we compared the results from the polished surface with those from a cleaved one which are usually reliable. However, La214 has no cleavage plane, and the flat areas necessary for maintaining the proper angle of incidence across the laser spot and avoiding wavelength dependent diffraction effects of the scattered photons.

When properly polished the differences between the two types of surfaces are small as shown in Fig. 4. The intensity in the maximum is slightly higher for the cleaved surface. The positions of the main peak and the secondary maximum at 4600 cm*-1* are nearly identical and the relative spectral weights as well. There is a strong increase of the intensity towards high energies for the cleaved surface. It is clear, however, that this increase is not intrinsic. Rather, the irregular shape of the surface and the relatively small flat part lead to various extrinsic contributions. For instance, both the incoming and the scattered photons are diffracted on edges close to the spot. In addition, since the surface close to the spot is not exactly flat, the angle of incidence varies across the spot. Therefore the light inside the bulk does not have a well defined polarization in cases other than $x$ or $y$. This complication completely hinders a symmetry analysis - apart from the other problems.

Appendix B Raw data

In Fig. 5 (a) we show the raw data of La2CuO4 as measured at the six main polarization configurations $xx$, $xy$, $x^{\prime},x^{\prime}$, $x^{\prime},y^{\prime}$, $RR$, and $RL$ (in Porto notation). Each configuration contains two symmetries such as $A_{1g}+B_{1g}$ and $A_{2g}+B_{2g}$ for $xx$ and $xy$, respectively, hence the sum of two having the same incident and two orthogonal scattering polarizations comprises always the full set of in-plane symmetries. Therefore, one can check the consistency of the experiments as shown in Fig. 5 (b). With small corrections, which result from insufficient adjustment of the absorbed laser power, the agreement in the energy range up to 5000 cm*-1* is within the statistical error. Above 5000 cm*-1* there are discrepancies between spectra with the incident polarization along $x^{\prime}$ and the other two sets. The agreement is better for Y123 and Bi2212, and the origin of these discrepancies is hard to pin down. Polarization dependent contributions from luminescence but also accumulating surface layers are possible explanations. In this range the derived spectra cannot be considered quantitative. In addition, polarization-independent contributions from luminescence are very like to occur in systems with defects in the crystal structure where charges can be trapped Bahrs et al. (2005).

Appendix C Symmetry dependence of the resonances

For La214 we measured all pure symmetries for three different excitations lines at $\hbar\omega_{I}=2.41$, 2.50, and 2.71 eV (514.5, 496.5, and 457.9 nm) as shown in Fig. 6. The first symmetry analysis was performed for Gd214 using the line at 488 nm alone. Sulewski et al. (1991) Except for details, partially coming from the lower temperatures here, but also from variations across the samples the results in Ref. Sulewski et al., 1991 are close to our results measured with 496 nm excitation.

The line shapes in the range of the two-magnon scattering (2000-5000 cm*-1*) in $A_{2g}$ and $B_{1g}$ symmetry are only weakly dependent on $\hbar\omega_{I}$ [Fig. 6 (b) and (c)] In either case the maximal intensity is found for the laser line at 496 nm. The onset of the intensity in $A_{2g}$ symmetry is independent of the laser energy indicating an intrinsic inelastic process to set in above a gap of approximately 3000 cm*-1* [Fig. 6 (b)]. Since the variation with excitation energy is similar to that of the two-magnon peak in $B_{1g}$ symmetry the $A_{2g}$ response is likely to be excited via the same intermediate electronic states. The most probable explanation of the $A_{2g}$ response is a chiral spin excitation. The variation with excitation energy makes it unlikely that the small peak on the high-energy side of the $B_{1g}$ peak [Fig. 6 (c)] is related to the $A_{2g}$ response.

The shapes of the $A_{1g}$ and $B_{2g}$ spectra depend substantially on the incoming energy, and the overall intensities become monotonously stronger as opposed to those of the $A_{2g}$ and $B_{1g}$ spectra. The phonons are strongly resonant towards the green as observed earlier Sugai (1990). Part of the variation in the range between 1500 and 3000 cm*-1* originates most probably from resonant multi-phonon processes. Above 3000 cm*-1* luminescence seems also to have an influence and is obviously not flat and symmetry independent.

Appendix D Resonance behavior of the $x^{\prime}y^{\prime}$ spectra

In Fig. 7 we show the detailed resonance behavior of the spectra measured in $x^{\prime}y^{\prime}$ configuration corresponding to $A_{2g}+B_{1g}$ symmetry. In the main text in Fig. 2 we show only spectra for two laser lines. Similar as in $A_{1g}$ symmetry the phonons resonate towards the green and become very strong for yellow excitation. Note that the individual lines cannot be resolved for the large steps between the measurement points. The two main observations being particularly clear in the normalized spectra [Fig. 7 (b)] are (i) that the shape in the range around the two-magnon peak is only weakly varying and (ii) that the secondary maximum close to 4600 cm*-1* fades away for green excitation. We could not measure the entire spectra with yellow light since the range of the spectrometer is limited.