Pressure-induced formation of rhodium zigzag chains in the honeycomb rhodate LiRhO
We use powder x-ray diffraction to study the effect of pressure on the crystal structure of the honeycomb rhodate LiRhO. We observe low-pressure ( = 6.5 GPa) and high-pressure ( = 14 GPa) regions corresponding to the monoclinic symmetry, while a phase mixture is observed at intermediate pressures. At , the honeycomb structure becomes distorted and features short Rh–Rh bonds forming zigzag chains stretched along the crystallographic direction. This is in contrast to dimerized patterns observed in triclinic high-pressure polymorphs of -LiIrO and -RuCl. Density-functional theory calculations at various pressure conditions reveal that the observed rhodium zigzag-chain pattern is not expected under hydrostatic pressure but can be reproduced by assuming anisotropic pressure conditions.
In recent years, and transition-metal compounds were intensively studied due to their extremely rich physics. In comparison to compounds, where the electronic correlation dominates over the spin-orbit coupling constant and Hund’s coupling , spin-orbit coupling (SOC) becomes more and more important for and transition-metal compounds, whereas the strength of electronic correlations decreases. The actual physics of these compounds thereby depends on a delicate balance between , and , as well as the crystal structure. The class of layered honeycomb-type and transition-metal compounds, such as O (= Li, Na and = Ir, Rh) and -RuCl, is especially interesting in this regard, as this delicate balance of parameters was discussed in terms of Kitaev physics and possible spin-liquid state Kitaev (2006); Jackeli and Khaliullin (2009); Chaloupka et al. (2010); Choi et al. (2012); Plumb et al. (2014); Chun et al. (2015); Winter et al. (2017); Luo et al. (2013); Khuntia et al. (2017). However, in NaIrO, -LiIrO, and -RuCl the quantum spin liquid ground state is not realized, since these materials were found to order magnetically at low temperatures Choi et al. (2012); Sears et al. (2015); Banerjee et al. (2016); Williams et al. (2016).
As for LiRhO, its magnetic ground state is still under debate. No long-range magnetic order could be found down to 0.5 K, but instead at small magnetic fields spin freezing was observed below 6-7 K Luo et al. (2013); Khuntia et al. (2017), although it is suspected that the majority of magnetic moments form a fluctuating liquid-like state Khuntia et al. (2017). Whether this partial spin freezing is due to a proximity to the Kitaev quantum spin liquid ground state or due to unavoidable defects (anti-site disorder and/or stacking faults) is still unclear Khuntia et al. (2017); Katakuri et al. (2015). However, ab initio and effective-model calculations showed that LiRhO bears similar electronic structure to the iridates Mazin et al. (2013) and hosts anisotropic Kitaev interaction terms of the same magnitude as in iridates Katakuri et al. (2015). According to electrical resistivity measurements, LiRhO is insulating at ambient pressure Luo et al. (2013); Mazin et al. (2013).
Another interesting aspect of LiRhO is its behavior under pressure, where honeycomb iridates Hermann et al. (2018); Clancy et al. (2018) and -RuCl Bastien et al. (2018); Biesner et al. (2018) become dimerized and, consequently, non-magnetic. Previously Hermann et al. (2018), we showed that the size of the central ion, the strength of the spin-orbit-coupling, electronic correlations, and Hund’s coupling all act against the dimerization. In comparison to -LiIrO, the in LiRhO is expected to be lower, while the electronic correlations should be enhanced in Rh compared to Ir, as screening by oxygen orbitals is reduced. Thereby, one generally expects a higher transition pressure in LiRhO and a larger pressure range for tuning the putative Kitaev magnetism of this compound. Here, we show that this is the case, but also that the pressure-induced transformations are very different from the dimerization observed in honeycomb iridates.
ii.1 Experimental Details
A powder sample of LiRhO was prepared by a solid-state reaction of LiCO and Rh in oxygen flow at C with several intermediate re-grindings. The sample quality was confirmed by laboratory x-ray diffraction (XRD) using the Rigaku MiniFlex diffractometer (CuK radiation, Bragg-Brentano geometry). This synthesis procedure yields samples with the best structural order achieved so far Manni (2014), although stacking faults are still present. Their concentration is discussed in Sec. III.1 below.
LiRhO powder was loaded into a diamond anvil cell (DAC) for pressure generation, and helium was used as pressure transmitting medium. The powder x-ray diffraction patterns were obtained using synchrotron radiation at the beamline ID15B at the European Synchrotron Radiation Facility (ESRF), Grenoble at room temperature. The wavelength of the radiation was 0.411267 Å, and the patterns were obtained in the 2-range between 2 and 33. The pressure in the DAC was determined in situ by the ruby luminescence method. The resulting patterns were analyzed by Rietveld refinements using the Jana2006 software Petříček et al. (2014). The quality of the fit is gauged by the weighted structure factor , as defined in Ref. Young, 2002 and by the commonly used weighted profile factor , where are intensities corrected for background. The absorption correction for a cylindrical sample was calculated to be lower than the value one (using Ref. von Dreele et al. (2013)), so no absorption correction was applied. The isotropic atomic displacement parameters were fixed to the value of 0.005 Å for all atomic positions, except for Rh(1) and Li(1).
ii.2 Computational details
Structural optimizations were performed under different pressure conditions by using projector-augmented planewave Blöchl (1994) method based on density functional theory (DFT), as implemented in the VASP package Kresse and Hafner (1993). Calculations were done within the generalized gradient approximation (GGA), GGA+U Dudarev et al. (1998), and GGA+SOC+U (including spin-orbit coupling effects for Rh). The value of the on-site Coulomb parameter was chosen based on the reproducibility of the experimental structure, as will be shown below. The cutoff for the wavefunction was set at 650 eV. K-point meshes of size were used for all the structural optimizations.
We performed two types of structural optimizations: (i) allowing relaxation of both lattice parameters and atomic positions under fixed hydrostatic pressure; we refer to this as “full relaxation”, and (ii) keeping the lattice parameters fixed according to given pressure conditions and allowing only the relaxation of the atomic positions. In both cases, the system is allowed to relax until the total force acting on the system was less than 0.005 eV/Å. At each pressure value, several different initial magnetic configurations were considered: (a) ferromagnetic (FM), (b) zigzag antiferromagnetic (AFM), (c) Neél AFM, (d) stripy AFM, and (e) nonmagnetic (NM) (see Fig. 1).
The analysis of the electronic properties was done with the Full Potential Local Orbital (FPLO) basis Koepernik and Eschrig (1999).
Iii Results and Discussion
iii.1 Experimental results
The x-ray powder diffraction patterns at ambient pressure and at the highest studied pressure (25.2 GPa) together with the corresponding fits from the Rietveld refinement are displayed in Fig. 2. Both refinements were performed within the monoclinic unit cell with the symmetry. The same crystal symmetry is found for the closely related honeycomb iridates O’Malley et al. (2008); Freund et al. (2016); Choi et al. (2012). In the refinements, stacking faults associated with shifts between successive LiRh layers were taken into account, as observed in -LiIrO O’Malley et al. (2008); Freund et al. (2016) and other LiO ( = Mn, Pt, Ru) compounds O’Malley et al. (2008); Bréger et al. (2005); Casas-Cabanas et al. (2007). Stacking faults affect the intensity and lineshape of several peaks and lead to an additional intensity between the (020) and (110) peaks as marked by the dashed red arrows in the insets of Fig. 2. The presence of stacking faults was taken into account by introducing the Li/Rh mixing for the Rh(1)/Li(1) and Li(2)/Rh(2) sites while constraining the overall stoichiometry to LiRhO. This reproduces the peak intensity but not their shape O’Malley et al. (2008).
The lattice parameters as a function of pressure, as obtained by the Rietveld fits of the x-ray powder diffraction diagrams, are depicted in Fig. 3. Up to the critical pressure =6.5 GPa, the lattice parameters , , and decrease monotonically with increasing pressure in a very similar manner. The value, shown in the inset of Fig. 3(a), reveals that the strongest pressure-induced effect occurs for the lattice parameter . The monoclinic angle decreases slightly but monotonically within this pressure range.
Above , a second phase with the same symmetry appears and gets more pronounced with increasing pressure. Above the critical pressure =14 GPa, this second phase is dominant and the high-pressure diffractograms can be well described by a single phase with the symmetry. There are only traces of the low-pressure phase found in the diffraction patterns above and up to the highest studied pressure, marked with black arrows in the inset of Fig. 2(b). Most importantly, we can rule out a symmetry lowering above , as such a symmetry lowering would induce peak splittings, for example for the (021) and the (111) diffraction peaks. These peaks are observed at 7.7 and 9.0 in the inset of Fig. 2(b) and are obviously not split. Thus, both the low-pressure () and high-pressure () phases in LiRhO have the symmetry. This result is in contrast to the recent findings for -LiIrO, where a pressure-induced structural phase transition with symmetry lowering from monoclinic to the triclinic symmetry caused by the Ir–Ir dimerization occurs at 3.8 GPa Hermann et al. (2018). Analogously, the monoclinic to triclinic symmetry lowering with the Ru-Ru dimerization is observed in -RuCl at GPa Bastien et al. (2018).
The refinement of the diffraction patterns for the intermediate pressure range, , with a phase mixture of the low-pressure and high-pressure phases did not yield stable fits, as many of the peaks of the two phases are broad and overlapping. Since the refinement with only one phase does not reproduce the actual peak shape, we marked this range with open symbols in Figs. 3, 5, 6 and 7.
Between and , the lattice parameter decreases drastically by about 3 %, while there is only a slight but abrupt increase in the lattice parameter , and follows the pressure-induced monotonic decrease as observed below [see Fig. 3(a)]. The abrupt decrease in the parameter is also revealed by the abrupt increase in the / ratio. Accordingly, the most pronounced pressure-induced change happens along the lattice direction, as will be discussed in more detail later. The monoclinic angle abruptly decreases above , and above it monotonically increases with increasing pressure [see Fig. 3(b)]. The kink in the pressure evolution of in the pressure range 10-12 GPa, i.e., in the intermediate phase, is not discussed here, because the phase mixture affects the refinements in this pressure range.
|low-pressure phase (0 GPa)||high-pressure phase (25.2 GPa)|
The pressure dependencies of the volume and the lattice parameters () were fitted separately for the low- and high-pressure phases, neglecting the intermediate regime, with a second-order Murnaghan equation of state (MOS) Murnaghan (1944), to obtain the bulk moduli and according to:
with fixed to 4. The results are summarized in Table 1. The bulk modulus of the low- and high-pressure phases amounts to 100.4(4) GPa and 118.6(9) GPa, respectively. This means that the material is less compressible in the high-pressure phase. In the low-pressure phase (), the contribution of the direction to the bulk modulus is the lowest with =94.8(7) GPa, as already indicated by the pressure dependence of the / ratio [inset of Fig. 3 (a)]. Thus, the material is most compressible along the direction. The largest contribution to the bulk modulus is attributed to the crystal direction, with =105.7(5) GPa.
In the high-pressure phase (), the contribution of the direction remains low and is even slightly decreased as compared to the low-pressure phase. Most interestingly, the contribution is strongly increased to 155.6(15) GPa in the high-pressure phase, while is much less increased, i.e., to 122.2(12) GPa. Hence, the honeycomb layers along the -plane become less compressible in the high-pressure phase, whereby the pressure-induced hardening has the strongest effect along the direction.
For a more detailed discussion, the atomic parameters of the refinement are shown in Table 2. The partial exchange of Li and Rh accounts for the stacking faults, as described in Ref. O’Malley et al., 2008. Since three Rh atoms are required to change place with one Li atom in order to mimic one stacking fault, each 5.5(1) unit cells one stacking fault occurs at ambient pressure. This value is very similar to previous reports in LiRhO Luo et al. (2013); Todorova and Jansen (2011) and slightly higher than in -LiIrO and LiPtO O’Malley et al. (2008). On the other hand, different studies of NaIrO Ye et al. (2012); Choi et al. (2012) reported the concentrations of stacking faults that are either larger or smaller than in LiRhO. The number of unit cells per stacking fault increases monotonically with increasing pressure and reaches 7.4(5) at 25.2 GPa, as shown in Fig. 4, i.e., the number of stacking faults is slightly reduced by external pressure. The parameters for the oxygen positions are changed in the high-pressure phase as compared to the low-pressure phase, thus affecting the RhO octahedra. The most interesting change, though, is observed for the -parameter of Rh(1) that determines the Rh-Rh distances in the honeycomb network (see Table 2).
To evaluate this behavior further, we compare the pressure evolution of the three Rh-Rh bond lengths in the -plane, namely, the bond and the two degenerate bonds as depicted in Fig. 5(b). At ambient pressure, the bond length amounts to 2.985(3) Å, while the bond length is 2.9296(13) Å [see Fig. 5(a)], leading to a slightly distorted honeycomb. The corresponding bond disproportionation , with and being the long and short bonds of the hexagonal Rh network, respectively, amounts to =1.02. For the high-pressure phase, both and bonds are drastically reduced by 0.15 Å, while the bond is increased by the same amount [Fig. 5(a)]. Hence, the bond disproportionation increased to =1.11 at 25 GPa. The bond length of 2.7 Å above is close to but still larger than the interatomic distances in metallic rhodium (=2.69 ÅBale (1958)). We thus conclude that external pressure introduces zigzag chains of rhodium atoms along the direction, as illustrated in Fig. 5(b). A similar structure but with closer bond lengths is found for 5% Na doped crystals (LiNa)RuO at ambient pressure Mehlawat and Singh (2017). On the other hand, pure LiRuO is dimerized at ambient pressure with an armchair pattern of the short Ru–Ru bonds (see Fig. 1(e) for illustration) Mehlawat and Singh (2017); Kimber et al. (2014).
Next, we consider the pressure-induced changes in the RhO octahedra. To this end, we define various Rh–O bond lengths and Rh–O–Rh bond angles that are responsible for the direct metal-to-metal and indirect oxygen-mediated contributions. The octahedra possess a two-fold rotational C-axis which is indicated by the arrow in Fig. 6(c). There are three unique Rh–O bonds labeled , , and two unique Rh–O–Rh angles and , where () involves two Rh atoms connected via the -bond (-bond) [see Fig. 6(c)]. The pressure dependence of the various bonds and bond angles is depicted in Figs. 6(a) and (b), respectively. At ambient pressure, the largest Rh–O bond length is found for the apical oxygen atom, thus the RhO octahedra show a tetragonal distortion with axial elongation. In the low-pressure phase (), the bond length is pressure-independent, whereas and slightly decrease under pressure.
At , the length is increased as compared to the low-pressure phase, whereas and are decreased. Upon further compression, decreases, seems to be unaffected, and shows a small anomaly at 15-20 GPa that may be significant, as the changes exceed the error bars.
The formation of zigzag chains is predominantly due to a change of Rh-O-Rh angles as described in the following. The pressure dependence of the Rh–O–Rh bond angles and is shown in Fig. 6(b). At ambient pressure, the values of and amount to 94.0(3) and 96.4(4), respectively. While is independent of pressure in the low-pressure phase, decreases by increasing pressure. When entering the high-pressure phase above the bond angle is strongly decreased to the value 87. Interestingly, the onset of the intermediate phase at appears at a pressure, when approaches 90, which is a distinct angle for the contributions of the ligand-mediated hopping to the hopping parameters, as discussed in more detail in Refs. Winter et al., 2016 and Winter et al., 2017. The strong pressure-induced decrease in the angle between and confirms the formation of Rh zig-zag chains along the direction. Consistently, the bond angle is strongly increased, as the bond length is increased [see Fig. 5(a)]. Again an anomaly is observed for the Rh–O–Rh bond angles between 15-20 GPa, which is directly related to the anomaly for the Rh–O distances and thereby has the same origin.
The electronic states of LiRhO are affected by the distortion of the RhO octahedra. Therefore, we followed the pressure dependence of the octahedral distortion using the bond-length distortion and the bond-angle distortion Ertl et al. (2002); Kim et al. (2004); Hogan et al. (2017). The bond-length distortion is defined as =, where is an individual Rh–O bond length and the average Rh–O bond length in the RhO octahedron. The bond-angle distortion is calculated according to =, where is an individual O–Rh–O bond angle. At ambient pressure, the distortion parameters are =3.4(11)10 and comparable to the results in Ref. Todorova and Jansen (2011) [=1.2(6)10 and =9.6(5)], although the Rh position seems to be fixed in that report. Comparison of our refinement to previous ones Mazin et al. (2013); Luo et al. (2013) is not straightforward since in those studies some oxygen parameters were fixed or calculated. The distortion parameters for LiRhO reported in our study are comparable to the ones of the related materials -LiIrO and LiPtO O’Malley et al. (2008); Freund et al. (2016). For NaIrO the value is about one magnitude smaller, while the bond-angle distortion is nearly doubled Ye et al. (2012); Choi et al. (2012). A comparison to the octahedral distortions in dimerized LiRuO is difficult, since the reported values determined by various studies are not consistent. For example, the values between 1.410 and 2410 have been reported, and the values for range between 4.7 and 54 Wang et al. (2014); Kobayashi et al. (1995); Mehlawat and Singh (2017).
In the low-pressure phase, the bond-length distortion only slightly increases with increasing pressure, whereas the bond-angle distortion decreases (Fig. 7). At the critical pressure , both parameters and are drastically enhanced compared to the low-pressure range. Such an enhanced distortion was also reported in Ref. Wang et al. (2014) for dimerized LiRuO compared to the non-dimerized samples. It is therefore likely that the enhancement of and at is caused by the lattice strain due to the formation of the Rh–Rh zigzag chains in LiRhO.
Of further interest is the tetragonal distortion (elongation or compression along the -direction) of the octahedra, as this would cause a splitting of the Rh states. As a measure of the tetragonal distortion we define the parameter as the deviation of the apical Rh–O bond length from the average Rh–O bond length according to Kim et al. (2004). For positive (negative) nonzero values of the octahedra are elongated (compressed) along the apical bond direction. Such a distortion can be explained by a cooperative first-order Jahn-Teller effect neglecting stress on the system Gehring and Gehring (1975); Kaplan and Vekhter (1995). The Jahn-Teller effect is expected to be weak but nonzero in a configuration. The pressure dependence of is depicted in Fig. 7. We note that the tetragonal distortion at ambient pressure amounts to =0.023(4), which is comparable to the value =0.015(3) given in Ref. Todorova and Jansen (2011). In the low-pressure phase, increases slightly but steadily upon compression, i.e., the elongation increases.
Between the low- and high-pressure phases, is nearly doubled, before it decreases upon further compression above . While the tetragonal distortion in the low-pressure phase is comparable to that in LiO (= Ir, Pt), it is much more pronounced than in NaIrO, where is close to zero. We thus conclude that the lattice strain rather than the Jahn-Teller effect is the driving force for the distortion of the octahedra in the honeycomb lattices. The tetragonal distortion in the high-pressure phase of LiRhO is comparable to the tetragonal distortion in the perovskites SrRhO and SrRuO, where the bond angle distortion is zero Huang et al. (1994); Vogt and Buttrey (1996).
iii.2 Computational results
A question that remains open is why LiRhO retains the monoclinic symmetry and shows the zigzag-chain pattern of short Rh–Rh bonds under pressure, whereas -LiIrO Hermann et al. (2018); Clancy et al. (2018) and -RuCl Bastien et al. (2018) become triclinic following the formation of metal-metal dimers.
In previous studies, the experimentally observed dimerization pattern in -LiIrO and -RuCl was identified by DFT calculations within the GGA+SOC+ scheme Hermann et al. (2018), as a consequence of a complex interplay of SOC, magnetism, correlation, and covalent bonding. Following this knowledge, we performed first full relaxations of LiRhO as a function of hydrostatic pressure with and without SOC. As initial guess for the geometrical optimization at each pressure, we considered two structures. The experimental low-pressure ‘undistorted’ structure at 5 GPa and the experimental high-pressure ‘zigzag chain’ structure at 25.2 GPa. In our notation, we assume a structure to be undistorted when the corresponding bond disproportionation . Moreover, for each of these initial geometries, we considered five different spin configurations, as explained in Section II.2.
Test calculations performed at 25 GPa reveal that after relaxation the structure becomes dimerized, regardless of the initial configuration. At a given pressure, the energetics of the various different configurations are obtained by comparing the corresponding enthalpies. Due to the underbinding problem of GGA (relaxed interatomic distances are longer than their experimental counterparts), the volume corresponding to 2 GPa reproduces the experimental volume at ambient pressure. This has been corrected by systematically subtracting GPa from all simulated pressure values.
The value of Hubbard correlation eV was chosen such that at 5 GPa (within GGA+SOC+ scheme): (a) the optimized lowest-enthalpy magnetic configuration corresponds to symmetry and reproduces the experimental value of = 1.02, and (b) the nonmagnetic configuration is dimerized (though parallel type). The latter confirms dimerization at finite pressure for LiRhO as magnetism is known to work against dimerization by pushing the transition pressure to a higher value Hermann et al. (2018).
At a pressure 11 GPa, we find that LiRhO undergoes a phase transition from a homogeneous to a dimerized phase with bond disproportionation =1.146 within the GGA+ scheme (not shown here). Upon dimerization, LiRhO becomes nonmagnetic. However, there are a few discrepancies with the experimental structures: (i) below , in the homogeneous structure, the shorter bond corresponds to the bond, rather than to the and bonds as observed in experiments, and (ii) the dimerized phase does not have the symmetry, rather it has the triclinic () symmetry, similar to -RuCl Bastien et al. (2018) and -LiIrO Hermann et al. (2018); Clancy et al. (2018). The inclusion of SOC reproduces the shorter bond to be in the homogeneous phase (below ) and shifts the transition pressure to 27 GPa. However, the high-pressure structure still becomes triclinic with =1.15 (see Fig. 8).
The above results show that the experimental zigzag pattern of the short Rh–Rh bonds cannot be obtained from hydrostatic pressure simulations. We therefore proceed by simulating uniaxial pressure with the and parameters fixed to their experimental values at 25.5 GPa and the ratio varied systematically (Fig. 9). This approach yields the zigzag-chain structure observed experimentally. However, we had to increase the ratio to 1.95, in order to reproduce the ratio between the long and short bonds.
For obtaining the pressure evolution of Rh-Rh bonds under uniaxial condition, we next repeated the above calculations but this time with 1.95, while fixing the and parameters to their experimental values at the corresponding pressure. By comparing the results of the hydrostatic and uniaxial pressure simulations, we conclude (see Fig. 10) that the evolution of LiRhO up to is compatible with hydrostatic pressure conditions, whereas at higher pressures the system progressively moves toward the behavior expected under uniaxial pressure. The uniaxial pressure accounts for the formation of zigzag chains instead of dimers, although it does not fully account for the evolution of the longer Rh–Rh bonds that evolve smoothly in the simulation but show a step-like anomaly experimentally (Fig. 5).
At ambient condition, LiRhO is an insulator as shown in Ref. Mazin et al., 2013. Our calculated density of states (DOS) for the experimentally obtained structures at 25.2 GPa (Fig. 11) show that unlike other dimerized phases in -LiIrO and -RuCl, in LiRhO the degeneracy between and orbitals of Rh states does not get lifted as the symmetry remains the same. Moreover, the system probably becomes metallic under pressure due to the formation of zigzag chains, which provide new hopping pathways.
The origin of the uniaxial-like pressure conditions requires further investigation. Experimental pressure conditions in a DAC with helium as pressure-transmitting medium are expected to be hydrostatic. Therefore, we consider the nature of the LiRhO sample as a more plausible reason. In particular, stacking faults that occur, on average, at every 6-7 layers could act as a local strain and affect the evolution of the structure under pressure. Our data show that the concentration of stacking faults in LiRhO is higher than in the polycrystalline samples of -LiIrO and in single crystals of -LiIrO that were used in our previous study Hermann et al. (2018). Interestingly, NaIrO shows different pressure evolution of the crystal structure in powders Xi et al. (2018) and single crystals Hermann et al. (2017). Given the proclivity of NaIrO to the formation of stacking faults, a similar mechanism may be operative there and deserves further systematic investigation.
In contrast to -LiIrO and -RuCl, where a dimerized triclinic phase is stabilized under pressure, LiRhO retains its ambient-pressure monoclinic symmetry and develops zigzag chains of short Rh–Rh bonds. This structural phase transition is not abrupt, since traces of the low-pressure phase can still be found even at the highest pressure of 25.2 GPa, but above 14 GPa the high-pressure phase is dominant. Our density-functional calculations suggest that such a behavior is not anticipated in LiRhO under hydrostatic pressure, where conventional dimerization should occur. On the other hand, uniaxial pressure may explain the experimental observations and promote the formation of zigzag chains instead of dimers.
Acknowledgements.We thank the ESRF, Grenoble, for the provision of beamtime at ID15B. DKh, RV, AAT, and PG acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG), Germany, through TRR 80, SPP 1666, TRR 49, and SFB 1238. AJ acknowledges support from the DFG through Grant No. JE 748/1. SB thanks Stephen Winter for helpful discussions. AAT acknowledges financial support from the Federal Ministry for Education and Research via the Sofja-Kovalevskaya Award of Alexander von Humboldt Foundation, Germany.
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