Phonon Control of Magnetic Relaxation in the Pyrochlore Slab Compounds SCGO and BSZCGO

Phonon Control of Magnetic Relaxation in the Pyrochlore Slab Compounds SCGO and BSZCGO

M. Zbiri zbiri@ill.fr Institut Max von Laue-Paul Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France    H. Mutka Institut Max von Laue-Paul Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France    M. R. Johnson Institut Max von Laue-Paul Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France    H. Schober Institut Max von Laue-Paul Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France    C. Payen Institut des Matériaux Jean Rouxel, Université de Nantes - CNRS, B.P. 32229, 44322 Nantes Cedex 3 - France
Abstract

We are interested in the phonon response in the frustrated magnets SrCrGaO (SCGO) and BaSnZnCrGaO (BSZCGO). The motivation of the study is the recently discovered, phonon-driven, magnetic relaxation in the SCGO compound [Mutka et al. PRL 97 047203 (2006)] pointing out the importance of a low-energy (7 meV) phonon mode. In neutron scattering experiments on these compounds the phonon signal is partly masked by the magnetic signal from the Cr moments and we have therefore examined in detail the non-magnetic isostructural counterparts SrGaO (SGO) and BaSnZnGaO (BSZGO). Our ab-initio lattice dynamics calculations on SGO reveal a peak in the vibrational density of states matching with the neutron observations on SGO and SCGO. A strong contribution in the vibrational density of states comes from the partial contribution of the Ga atoms on the 2b and 12k sites, involving modes at the M–point of the hexagonal system. These modes comprise dynamics of the kagomé planes of the pyrochlore slab magnetic sub-lattice, 12k sites, and therefore can drive magnetic relaxation via spin-phonon coupling. Both BSZCGO and BSZGO show a similar low-energy Raman peak but no corresponding peak in the neutron determined density of states of BSZGO is seen. However, a strong non-Debye enhancement of low-energy phonon response is observed. We attribute this particular feature to the Zn/Ga disorder on the 2 -site, already evoked earlier to affect the magnetic properties of BSZCGO. We propose that this disorder-induced phonon response explains the absence of a characteristic energy scale and the much faster magnetic relaxation observed in BSZCGO.

pacs:
74.25.Kc,78.70.Nx,78.30.-j,71.15.Mb

I Introduction

The interplay between structural, electronic and dynamic degrees of freedom in geometrically frustrated magnetic materials has the consequence of creating highly degenerate ground states, which have generated considerable interest Ramirez (2001); Stewart (2004); Hiroi and Tsunetsugu (2007). Short-range correlations in frustrated magnets lead to the formation of weakly coupled, fluctuating clusters and consequently a macroscopic, collective degeneracy can prevail Moessner and Chalker (1998a, b); Ramirez (2001); Moessner and Ramirez (2006).
At low temperature, frustrated magnets are expected to be sensitive to weak perturbations, which can raise the ground-state degeneracy, produce a hierarchy of closely space energy levels and allow the possibility of slow dynamics at low temperature between the corresponding states. When the associated energy scales of these states fall well below that of magnetic interactions Moessner (2001); Moessner and Ramirez (2006), various perturbations including static Tchernyshyov et al. (2002) or dynamic Sushkov et al. (2005); Fennie and Rabe (2006) lattice effects may play a key role due to coupling between lattice and magnetic degrees of freedom. A recent neutron spin–echo (NSE) examination of the pyrochlore slab antiferromagnet (kagome bilayer) SrCrGaO (SCGO) Mutka et al. (2006) suggested that phonons affect the slow relaxation of this highly frustrated spin system. The SCGO compound has a particular ground-state in which dimer correlations within the pyrochlore slabs show a partial freezing of about 1/3 of the total possible ordered magnetic moment in spite of the high value of the average intraslab superexchange interaction ( 13 meV) Limot et al. (2002). The frozen ground-state does not fit into a spin-glass picture, for example it has been shown that both the freezing transition temperature and the frozen moment decrease with increased level of magnetic dilution due the Ga/Cr substitution Mondelli et al. (2001).
The extremely broad relaxation seen using NSE spectroscopy was fitted with a phenomenological stretched exponential time–dependence to infer an activated temperature dependence with an activation energy of  7 meV Mutka et al. (2006). A vibrational mode was observed close to this same position by Raman spectroscopy and inelastic neutron scattering (INS), leading to the conclusion that phonons drive the relaxation.

parameter SGO (calc) SGO (exp) Graetsch and Gebert (1994) SCGO (exp) Obradors et al. (1988)
=473 K =298 K =16 K =4.2 K
a(Å) 5.86 5.80 5.80 5.79 5.80
c(Å) 22.87 22.86 22.82 22.78 22.66
2a: 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0222Cr occupation  90% in least diluted samples Mondelli et al. (2001). The Kagome (12k) and triangular (2a) planes form the pyrochlore slab.
2b(4e):111For refinement of experimental data it is possible to split the 2b occupation over two 4e sites Graetsch and Gebert (1994); Obradors et al. (1988).333Ga occupation 100 %. 0, 0, 1/4 0, 0, 1/4(0.258) 0, 0, 1/4(0.257) 0, 0, 1/4 (0.256) 0, 0, 1/4(0.256)
12k: 0.1678, 0.891 0.1684, 0.891 0.1683, 0.891 0.1682, 0.891 0.1681, 0.892222Cr occupation  90% in least diluted samples Mondelli et al. (2001). The Kagome (12k) and triangular (2a) planes form the pyrochlore slab.
4f:333Ga occupation 100 %. 0.0278 0.0273 0.0273 0.0274 0.0283
4f: 0.190 0.190 0.190 0.190 0.192222Cr occupation  90% in least diluted samples Mondelli et al. (2001). The Kagome (12k) and triangular (2a) planes form the pyrochlore slab.
Table 1: Lattice parameters and the relevant Ga/Cr position parameters from ab-initio calculations on SGO compared with experimental data on SGO and SCGO.

However, why phonons at 7 meV, and not other frequencies, drive magnetic relaxation is not clear and further investigation of phonons and related spin-phonon mechanisms is required. In this context we have studied the isostructural non-magnetic material SrGaO (SGO) using ab-initio lattice dynamics calculations that provide a full picture of the phonon dispersion relations, total and partial density of states, energies of Raman active modes and the neutron scattering cross-section. The choice of the non-magnetic counterpart SGO is motivated by experimental considerations (see below) and the fact that, practically, handling the chemical disorder associated with the Cr/Ga substitution in SCGO efficiently is not feasible in the lattice dynamics calculation.The calculated neutron scattering cross-section is used for the evaluation of the powder averaged –dependent intensity that can be compared with experiment. The calculations are accompanied by new experiments using INS and Raman spectroscopy. Experimental techniques are combined because Raman spectroscopy gives a partial view of the vibrational density of states (–point only) which is complemented by INS that can inform on the -dependence and characteristic energies throughout the Brillouin zone.
In addition to the SCGO/SGO case we have examined the experimental situation in the related pyrochlore slab compound BaSnZnCrGaO (BSZCGO) and its non-magnetic counterpart BaSnZnGaO (BSZGO). Ab-initio lattice dynamics calculations could not be done for BSZCGO and BSZGO due to their complex crystal structures. In BSZCGO the magnetic relaxation rate at low- was observed to be by some two orders of magnitude faster than in SCGO but without a well-defined energy scale Mutka et al. (2006). Nevertheless, both in BSZCGO and BSZGO Raman data reveal a phonon peak at an energy close to the one seen in SCGO and SGO, while no peak is seen in the INS response. As we shall see below the absence of a characteristic energy in the relaxation of BSZCGO makes more sense when we consider the rather strong non-Debye enhancement of the phonon DOS that we attribute to the substitutional disorder, the 50/50 mix of Zn and Ga on the 2 site in this compound Bonnet et al. (2004). Accordingly we conclude that the localized phonon modes associated with the substitutional disorder can also affect the magnetic interactions and induce relaxational dynamics of the magnetic system. These observations also point out the particularity of the frustrated magnets concerning the system dependence of the low–energy properties.
As for SCGO, the aim of the present work is (i) to identify from ab-initio calculations the experimentally observed phonon modes (neutron and Raman spectroscopy), these calculations were used to obtain generalized neutron weighted vibrational density of states (GDOS) and the active Raman modes for SGO, (ii) understand the importance of non--point modes, and (iii) investigate the normal modes having an effect on the dynamics of the magnetic sites and hence the possibility to drive magnetic relaxation.
This paper is organized as follows: The experimental and computational details are provided in Section II and Section III, respectively. Section IV is dedicated to the presentation of the results that are discussed in Section V with conclusions.

Ii Experimental Details

Powder samples of non-magnetic SGO and BSZGO were prepared using a standard solid-state high temperature ceramic synthesis method and characterized by x-ray and neutron diffraction. As for the magnetic SCGO (x=0.95) and BZSCGO (x=0.97) samples, we analyzed those that were already used in our previous studies Mondelli et al. (2001); Bonnet et al. (2004); Mutka et al. (2006, 2007). The INS measurements were performed at the Institut Laue Langevin (ILL) ill () (Grenoble, France), using two instruments; the cold neutron time-of-flight spectrometer IN6 and the thermal neutron time-of-flight spectrometer IN4. IN6 operating with an incident wavelength of  = 4.1 Å provides very good resolution (0.2 meV FWHM) in the lower energy transfer range ( 10 meV) for the anti-Stokes spectrum. On IN4 using incident wavelengths of  = 2.6 or 1.8 Å one has an extended -range and this allows the Stokes spectrum to be measured at low temperature over a broader energy transfer range with a resolution of 0.5 or 1 meV (FWHM), respectively. Experiments were performed on both SCGO/SGO and BSZCGO/BSZGO systems mainly at  = 300K and also at low temperatures 2 K12 K for SCGO/SGO. The data analysis was done using ILL software tools and the GDOS was evaluated using standard procedures Mutka et al. (2008) without applying multiphonon corrections. The experimental GDOS was normalised to 3N modes where N is the number of atoms in the formula unit. Raman measurements at  = 300 K were performed at the Institut des Matériaux Jean Rouxel (IMN) imn ().

Figure 1: (Color online) Measured maps at = 300 K for SCGO (top left) and SGO (top right). Strong paramagnetic scattering dominates the low– response of SCGO. We can note that a peaked phonon reponse comparable to the one seen in SGO at  -5.5 meV with –dependent intensity is observed in SCGO at the highest –range, but at a slightly shifted energy transfer  -7 meV, as shown by the arrows. The bottom panels show the constant energy transfer scans at  - 7 meV for SCGO (bottom left) and at  - 5.5 meV for SGO (bottom right). For SCGO the magnetic form factor –dependence is shown (dashed line) in order to indicate the trend of the magnetic scattering. The phonons in SCGO appear less structured and somewhat blurred due to the weaker coherent scattering power of Cr–atoms as well as due to the overlap with the magnetic signal.

Iii Computational Details

Relaxed geometries and total energies were obtained using the projector-augmented wave (PAW) formalism Blöchl (1994) of the Kohn-Sham formulation of the density functional theory KSDFT Hohenberg and Kohn (1964); Kohn and Sham (1965) at the generalized gradient approximation level (GGA), implemented in the Vienna ab-initio simulation package (VASP) Kresse and Furthmüller (1996); Kresse and Joubert (1999). The GGA was formulated by the Perdew-Burke-Ernzerhof (PBE) Perdew et al. (1996, 1997) density functional. All results are well converged with respect to k-mesh and energy cutoff for the plane wave expansion. The break conditions for the self consistent field (SCF) and for the ionic relaxation loops were set to 10 eV and 10 eV Å, respectively. Hellmann-Feynman forces following geometry optimisation were less than 10 eV Å. Full geometry optimization, including cell parameters, was carried out on the experimentally refined SGO structure Graetsch and Gebert (1994) containing eleven crystallographically inequivalent atoms (5 O, 5 Ga and 1 Sr). A comparison of the ab-initio optimized and experimental structural parameters is given in the Table 1. The space group is P6/mmc with 2 formula-units per unit cell (64 atoms). In order to determine all force constants, the supercell approach was used for lattice dynamics calculations. An orthorhombic supercell was constructed from the relaxed structure containing 8 formula-units (256 atoms). A second partial geometry optimization (fixed lattice parameters) was performed on the supercell in order to further minimize the residual forces. Total energies and Hellmann-Feynman forces were calculated for 90 structures resulting from individual displacements of the symmetry inequivalent atoms in the supercell, along the inequivalent cartesian directions (x, y and z). 192 phonon branches corresponding to the 64 atoms in the primitive cell, were extracted from subsequent calculations using the direct method Parlinski et al. (1997) as implemented in the Phonon software Parlinski (). The coherent dynamic structure factor was calculated by the numerical procedure for evaluation of powder-averaged lattice dynamics, PALD Johnson et al. (2009); Koza et al. (2008). For a list of wave vectors with randomly chosen directions, but lengths corresponding to the Q range of interest, the dynamical matrix is diagonalised for each wave vector and the corresponding eigenfrequencies and eigenvectors are used to calculated for one-phonon creation. The wave vectors, spectral frequencies and intensities are then used to construct the 2-D map, which can be compared with the measured map as will be shown below.

Iv Results

iv.1 SGO/SCGO phonon response from experiments

Figure 1 shows the intensity maps measured at  = 300 K using the IN4 spectrometer, for both the magnetic system SCGO and its non-magnetic counterpart SGO. The strong intensity at low centered in the elastic position in SCGO is due to the quasi-elastic magnetic response resulting from the fluctuating moments of the Cr atoms. Note that the –dependence of the magnetic response is modulated with a main maximum at = 1.4 Å and the paramagnetic form factor gives just an overall trend. In spite of the decay of the magnetic form factor with increasing , this signal dominates and swamps the phonon signal. The –map of SGO shows the phonon signal clearly visible in the absence of the magnetic response and it is possible to correlate the 7 meV features at highest –range,   3.5 Å, in SCGO with the 5.5 meV features in non-magnetic sample SGO. As we can see phonons in SCGO are masked due to the presence of the magnetic signal and therefore are difficult to measure.

Figure 2: (Color online) Raman spectra at = 300 K for SGO (red), with calculated Raman active mode energies shown by the ticks at the bottom of the graph, and SCGO (blue, higher intensity at low energy). Both of these compounds have a Raman active low–energy mode in the vicinity of the kink seen in the GDOS, compare with Fig.3. The inset shows in detail the low–energy range highlighting the good match with the position calculated for SGO.
Figure 3: (Color online) Generalized density of states (GDOS) at = 300 K for SCGO (blue) and SGO (red), obtained using the data shown over a limited –range in Fig. 1. For SCGO the weight of the paramagnetic contribution has been minimized using only the high –data ( 3.5Å). Due to the remaining magnetic contribution the GDOS at low energy shows a faster than Debye growth as compared to SGO. There are distinct differences in the phonon bands in the range 20 meV  70 meV, but we can note a similar form at low energy, the peaked phonon response seen in SGO at  5.5 meV and in SCGO at  7 meV (see Fig. 1) produces a characteristic kink at that same position. A rather similar response of the two systems is observed in the high-energy range dominated by the oxygen modes.

In addition, the chemical disorder arising from the Cr substitution on the Ga sites, which can in principle be modelled, leads in practice to phonon instabilities in the calculations. For these reasons, this study is focused on the non-magnetic material SGO. These results are then used with proper care to explain phonon-related phenomena in the magnetic system SCGO. The two systems are isostructural and accordingly the magnetic Cr sub-lattice sites can be identified crystallographically using the corresponding positions of the Ga atoms. Even though it is clear that the phonon response of the two systems cannot be identical both Raman spectroscopy and INS provide justification for our approach since the spectra of SCGO and SGO display similar features in the low-lying frequency range. This is reported in Fig. 2 for the Raman data. Both samples show a low-energy mode at  53 cm ( 6.6 meV) and at  60 cm ( 7.4 meV) for SGO and SCGO, respectively. Also the measured neutron density of states of the two systems follow quite similar trends, see Fig. 3. No attempt was made to eliminate the magnetic contribution that, in spite of being small, influences the GDOS evaluated for SCGO in the low-energy range. Supposing that the low-energy Raman mode are characteristic of Ga/Cr vibrations we can argue that the shift of positions of the main low-frequency features is partly connected to the different masses of the Cr and Ga atoms, 52 and 69.72, respectively. The ratio of the observed Raman frequencies (0.9) is close to the square root of the ratio of the atomic masses (0.86), as would be expected for a vibrational mode in a similar potential involving essentially the Cr or Ga atoms. Comparing the INS response we note a somewhat bigger difference, the peak seen at 5.5 meV in SGO appears at 7 meV in SGCO. Of course for modes with participation from many atomic species and different interatomic potentials the simple comparison based on atomic mass only cannot be quantitatively correct.

Figure 4: (Color online) Calculated and neutron determined generalized density of states GDOS for SGO show very similar overall features. The results using both IN4 (red dots) and IN6 (blue circles) data are in good agreement with the calculation (green solid line) and display a similar broadened peak at  5.5 meV, see also Fig. 5. The calculated data has been convoluted with a gaussian to account for the experimental resolution.
Figure 5: (Color online) The low–energy part of the SGO-GDOS can be fitted (full line) with a superposition of a Debye term (,  = 46 meV) (dashed line) and single mode peak with a gaussian lineshape. The results using both IN4 and IN6 (not shown) data are in good agreement and display a similar broadened peak at  5.5 meV,  = 2.3 meV (FWHM), indicating that this broadening is not due to instrumental resolution.

iv.2 SGO/SCGO phonon response compared with calculation

Fig. 4 compares the ab-initio determined and measured GDOS for SGO. In order to compare with experimental data, the calculated GDOS was determined as the sum of the partial vibrational densities of states weighted by the atomic scattering cross sections and masses: GDOS()=, where ( = 0.265 (O), 0.096 (Ga), 0.071 (Sr); i={O, Ga, Sr}). A more detailed look at the neutron data on the SGO compound is shown in Fig. 5 where one can see that the low-energy phonon response is well described by a superposition of a Debye term and a broadened peak at =5.5 meV. The observed broadening of the peak ( 2.3 meV (FWHM) does not depend on the the instrumental resolution that is less than the apparent line–width both on IN4 and on IN6. Disregarding the broadening the calculated spectrum is in very good agreement with the observed one, in particular the peak at 5.5 meV is very well reproduced and also the relative weight of the peak is in reasonable agreement with calculation. The width of the peak is not resolution limited and we conclude that the mode seen in Raman spectrum reported in Fig. 2 is included in the peak of the GDOS. Due to the magnetic contribution present in SCGO fitting with the the expected and a gaussian peak is not applicable. However, mimicking the magnetic contribution with an additional linear term, a reasonable fit can be obtained for the gaussian peak at  = 7 meV with a width of 2.4 meV (FWHM) similar to the SGO case (not shown).
Based on the above we can consider that the experimental data validates the computational approach in terms of the generalized density of states and Raman response. Accordingly we can go further in analyzing the calculated partial GDOS. Fig. 6 shows total (GDOS) and the partials of SGO. The major contribution of the Ga atoms to the peaked low-energy response is clear. The Ga partial can be further resolved into the contributions from the different crystallographic sites , see Fig. 7. This is of interest for the case of SCGO since the magnetic Cr atoms occupy almost fully the sites 2a, 12k and 4f that contain only Ga in SGO, see table 1. It appears that the 2b and 12k site partials, and give the strongest contribution to the total density of states. The 2b site is fully occupied by the non-magnetic Ga also in SCGO while the Cr occupation on the 12k and 2a sites, forming the pyrochlore slab, is beyond 90 % in the least diluted samples Mondelli et al. (2001) (see Table 1 and Figure 10). Structural refinements of the isostructural magnetoplumbite compounds, including SGO Graetsch and Gebert (1994), usually indicate large Debye–Waller factor for the cation located in the trigonal bipyramid on the 2b site, and a split occupation on the 4e site can be considered (see Table 1).
The high spectral weight of the partial is consistent with dynamical disorder being the reason for the large and anisotropic thermal displacement of Ga atom. The 12k and 2a sites constitute the pyrochlore slab magnetic sub-lattice, kagome and triangular planes respectively, in the SCGO compound, and we begin to understand how phonons at 7 meV can drive magnetic relaxation in SCGO.

Figure 6: (Color online) Calculated total and partial neutron weighted GDOS for SGO. Arrows indicate the peak in the Ga partial corresponding to the phonon mode of interest at  =  5.5 meV.
Figure 7: (Color online) Ab initio determined partial DOS of Ga-atoms, , in SGO. The strongest contributions to the low-energy peak are from the 2b and 12k sites, respectively, as indicated by arrows. See Table 1 for details of site occupations.
Figure 8: (Color online) Calculated Dispersion relations along the high symmetry directions of the Brillouin zone in SGO, with color coded neutron scattering intensity, blue color corresponds to the maximum intensity. The Bradley-Cracknell notation is used for the high-symmetry points: =(0,0,0), A=(0,0,), and M=(,,0).

Calculated phonon dispersion relationships are shown in Fig. 8 in which the color-coded intensities correspond to the coherent, dynamic structure factor for one-phonon creation. Close to the GDOS low–energy peak ( 6 meV) the dispersion curves consists of flat branches with two modes around the M-point having maximal intensity. These modes make the strongest contribution to the DOS and we claim we see their signature in form of the gaussian peak seen in the experimental GDOS, too. At the -point there are three Raman-active modes close to the GDOS peak but with weaker response in the INS. Due to the fact that the nuclei of the SGO are coherent neutron scatterers the inelastic scattering cross-section has characteristic features even in powder averaged phonon response that can also be resolved in terms of its wave-vector dependence. In order to gain further confidence in the dispersion relations and the overall phonon response, we have calculated the coherent structure factor over the whole Brillouin zone. From this data we have generated the map for a powder sample, using the recently developed, powder-averaged lattice dynamics approach Johnson et al. (2009); Koza et al. (2008). The calculated result can be compared with experimental data from IN4, see Fig. 9.

              (a)                                                           (b)                                                         (c)

Figure 9: (Color online) Simulated map (a) for SGO calculated for the Stokes (one phonon annihilation) process at T=0, Fig. 1 shows the measured anti-Stokes (phonon creation) response weighted by the thermal population. Therefore the relative intensity at higher absolute value of energy transfer appears less in Fig. 1. Constant cut (b) of measured and calculated (solid and dotted lines, respectively). Constant energy cut (c) of measured and calculated ( solid and dashed lines, respectively). The measured cuts shown here are the phonon creation response ( 0) at =2 K, note similarity of the constant energy cut (c) with the one shown in the lower right panel of Fig. 1 at = 300 K.

Comparison by eye of the low-energy spectrum is hampered by the elastic intensity in the experimental data, which is not calculated. Measured and calculated maps are, however, in reasonably good agreement as shown by constant energy and constant cuts.
One can also note that the modulations superposed on the overall -dependence of the inelastic signal at energy transfer of 5.5 meV appear periodically at positions close to the successive M-point values at  1.9 Å, 3.1 Å, 4.3 Å. Note that the measured data shown in Fig. 9 is taken at low temperature,  = 2K, and therefore a shorter incident wavelength ( = 1.8 Å) was necessary, compared to the setting used to obtain the data shown in Fig. 1 ( = 2.6  Å) , in order to reach high enough momentum transfer in the Stokes (neutron energy loss, phonon creation) side of the spectrum. The quality of the data at low temperature is worse due to reduced thermal population of phonon states. Also the energy resolution of the instrument is comparably worse. A check of the temperature dependence of the phonon response in SCGO in the vicinity of the spin freezing transition did not indicate any significant change in the limits of experimental precision.

Figure 10: (Color online) Schematic representations of the normal modes in SGO around the M-point at 5.6 meV (left) and at 6.2 meV (right), with arrows indicating the cation displacements. The 2a, 12k and 4f are sites occupied in major part by Cr in SCGO. The 2b and 4f are the sites occupied by Ga only both in SGO and in SCGO.

In view of the importance of the two M-point modes (5.6 meV and 6.2 meV), we show schematically their displacement vectors in Fig. 10 for the case of the SCGO magnetic compound. Therein the displacements of Cr and Ga atoms are highlighted. The sites 12k and 2a belong to the magnetically connected pyrochlore slab network, representing the triangular planes and the kagome planes, respectively, and the vibrational characteristics there should be the most important in the context of the magnetic relaxation. As depicted in Fig. 10 the M-point phonons involve large amplitude displacements of the Ga atoms on the 2b sites, and the associated response of the 2a and 12 k sites will modulate the magnetic interactions between these sites. In this context, the observed phonon-driven magnetic relaxation could be explained as a consequence of the strong sensivity of the magnetic interactions to subtle variations in the magnetic exchange pathways between adjacent Cr magnetic cations. It was shown that the AFM coupling between nearest-neighbour Cr ions within the pyrochlore slab is very sensitive to the Cr-Cr distance Limot et al. (2002). The J value follows a phenomenological law =450K/Å. Because phonon calculations could be done for SGO only, it is not possible however to give an accurate value for the J modulation associated with the M mode in SCGO.

iv.3 The case of BSZGO/BSZGCO

We first examine the Raman response, Fig. 11. Here again we can see that both the magnetic and non-magnetic compounds display a low-energy Raman peak, ressembling the one observed in the SGO/SCGO systems. Also as before, one has a slightly lower energy for the peak in the Cr containing compound, and this situation can be qualitatively ascribed to the difference in atomic mass. The experimental GDOS has a dynamic range comparable to that of SGO/SCGO and especially the high-energy part shows similar form, Fig. 12. However there is a qualitative difference in the experimental GDOS as compared with SGO, the one measured for BSZGO does not show any distinct peak in the range where the Raman peaks are visible. The low-energy phonon response is overall rather different, it does not show the characteristic -dependence typical of acoustic phonon response associated with the Debye-regime. In fact we observe an almost linear dependence GDOS with a progressive upwards curvature that never reaches a quadratic law, see Fig. 13. Such a behavior is quite unusual. For crystalline solids the acoustic phonons with linear dispersion should give a quadratic density of states. To explain the unconventional trend we propose that the substitutional disorder in the form of 50/50 occupation by Zn or Ga on one of the 2d sites creates a situation that affects the lattice dynamics and gives rise to a disorder induced dynamic response. We can argue that the damping of phonon modes due to the lattice disorder leads to the observed non-Debye response of the low–energy modes, not necessarily just the long wavelength acoustic modes but also the ones further out in the Brillouin zone or even at the zone boundary.

Figure 11: (Color online) Raman spectra on BSZGO (red, lower) and BSZSCGO (blue, upper), measured at =300 K. Note the low–energy mode similar to the SGO/SCGO case shown in Fig. 2.

V Discussion and conclusions

In the previous section we have provided evidence of the possibility of particular normal modes being responsible for the magnetic relaxation in SCGO. We suggested that the activated behavior of the relaxation rate is due to the capacity of the phonons at that energy to influence the magnetic system. In the following we attempt to examine more generally the role of phonons in the temperature dependence of the magnetic relaxation. We survey a possible minimal model in which the important issue is the capability of the phonon bath to provide energy for the magnetic system.

v.1 Magnetic relaxation controlled by phonon population

Let us assume that the phonon dominated relaxation rate is determined by the number of occupied phonon states, accordingly the temperature dependence is related to the thermal population .

(1)

In the case of a single mode of energy , we have ; and the phonon population and the relaxation rate have the activated dependence

(2)

where the approximation is valid at low temperatures when . This dependence satisfies the experimental observation concerning the magnetic relaxation in SGCO. However, if any low–energy phonon can induce relaxation, one can expect that the low–energy density of states is the important quantity, and then the expected –dependence of the relaxation rate will depend on the functional form of the density of states . At low temperatures the integration in eq. 1 can be taken from zero to infinity and after a transformation of variable one finds that a power law form leads to a power law in ,

(3)

where the definite integral can be evaluated analytically Blundell and Blundell (2006). Another possible situation might be such that there is a low–energy cut–off due to e.g. – dependence of the effective interaction. In this case, for , the number of phonons affecting the relaxation rate is evaluated to be

(4)

We have seen above that the experimental GDOS is a superposition of the Debye and a single mode energy contribution in the case of SGO , while in BSZGO a linear –dependence appears. In case of independent relaxation processes one can expect that the relaxation rates of each process are additive,

(5)
Figure 12: (Color online) GDOS of BSZGO (blue) compared with that of SGO (red), normalised as explained in the text. For the latter note the Debye like dependence at low temperature (dashed line with  = 46 meV), see also Fig. 5.

It is straightforward to calculate in the eqs. 1 to 4. Note, however, that to determine a quantitative estimate for the relaxation rate it would be necessary to know the prefactor describing the efficiency of the coupling between the spin system and the phonon bath. Experimentally we find for the magnetic relaxation in SCGO that with  s. This means that a Debye phonon controlled relaxation of the type would dominate at low temperatures for values of  s, see Fig. 14, which also shows that both the single mode and the cut-off picture could match equally well the experimental data on SCGO in the activated regime. It is not possible to discern any power law dependence either for SCGO or BSZCGO suggesting that long wavelength acoustic phonons are not effective for the magnetic relaxation process. We can also argue that both in SCGO and in BSZCGO at the lowest temperatures a cross-over to a quantum relaxation regime takes place Mutka et al. (2006, 2007), and we know that at higher temperatures a quasi-elastic response with a width proportional to temperature prevails Mondelli et al. (2000). Meanwhile, it is clear that the limited temperature range of the observations does not allow a full analysis of the relaxational dynamics. We suggest that the most effective modes are the ones in the vicinity of the –point of the reciprocal lattice, indeed these zone boundary modes have a spatial structure on the scale of the in plane lattice parameter, and therefore match the nearest neighbor distance between the localized magnetic moments. With respect to the situation with BSZCGO, we have observed no specific activation energy and it is not possible to see any well-defined power–law dependence in the temperature dependence of the relaxation rate. We evoked earlier (Sect. IV.3) the possibility of the disorder induced modification of the lattice dynamics being the origin of the faster relaxation, due to heavily damped phonon modes. Note that here again the space scale is typical of the pyrochlore slab in–plane unit due to the half/half Zn/Ga substitutional disorder. It is conceivable that this disorder has a strong influence especially on the -point zone boundary modes that, in case of an overdamped response, can achieve a quasi-elastic character, and give rise to the linear GDOS as observed, as well as to the faster relaxation seen in BSZCGO. Anyhow it is clear that the dependence that one might expect with reference to the model calculation is not observed.

Figure 13: (Color online) The low–energy part of the GDOS of BSZGO, the solid line showing the linear non-Debye dependence for  0 . Data with blue and red symbols from IN6 and IN4, respectively.

v.2 Other aspects of the relaxation mechanism

Our results suggest an important role of lattice vibrations in the low-temperature spin dynamics of the frustrated magnets SCGO and BSZCGO. However, one cannot claim that spin–phonon coupling alone could be the origin of the complex relaxational behavior. The systems studied are disordered due to the non-magnetic dilution that has been widely studied and is a known factor in the pyrochlore slab compounds Mondelli et al. (2001); Limot et al. (2002); Bono et al. (2005). It is even surprising that even though the phonon properties of the SGO/SCGO lattices appear somewhat better defined when compared to the BSZGO, it is in SCGO that very strongly stretched time decay occurs. A complete microscopic picture of the relaxation process is still to be constructed and one can imagine that the fine details of the magnetic states of lowest energy are of major importance in such a pursuit. Pursuing the analysis might be extremely difficult for systems like the pyrochlore slabs but in simpler cases details of spin-lattice coupling have been already examined and could pave the way for further work Sushkov et al. (2005); Fennie and Rabe (2006). In the general context of frustrated magnetism our results point out a trend that has been already evoked, the low–temperature and ground–state properties are highly sensitive to system dependent perturbations. Nevertheless the present work pinpoints the importance of lattice vibrations and their microscopic character in the control of the low–energy spin dynamics.

Figure 14: (Color online) Modeling the relaxation, see details in the text. The single frequency activated model of eq. 2 follows the experimental data on SCGO over the range 3 K 7 K as reported earlier Mutka et al. (2006) and is in that temperature range indiscernable from the model with cut-off at the same frequency, eq. 4. The trend of the power-law models eq. 3 does not match the datapoints of neither SCGO (dots) nor BSZCGO (squares). Note that the two models have been plotted with arbitrarily chosen prefactors.

Vi Summary

We have presented a detailed investigation of the phonon spectra in the pyrochlore slab compounds SrCrGaO (SCGO) and BaSnZnCrGaO (BSZCGO) based on ab-initio lattice dynamics simulations, inelastic neutron scattering and Raman measurements, in order to investigate the origin and mechanisms of the recently observed Mutka et al. (2006) phonon-driven magnetic relaxation in SCGO. Since the magnetic signal dominates the neutron scattering response at low energy transfer, we have performed new experiments on the isostructural non-magnetic material SrGaO (SGO). Moreover, the chemical disorder is difficult to include in the ab-initio lattice dynamics calculations so these have also focussed on SGO. Results of the calculations for the GDOS and for SGO are in good agreement with the experimental data. Neutron and Raman experiments show the similarity of the phonon response in the magnetic system with respect to the non-magnetic counterpart. The calculated partial density of states for SGO indicates that the strongest contribution to the total density of states stems from the vibrations of the Ga–atoms on the 2b and 12k sites of the lattice, associated with flat dispersion branches centered at the zone boundary M–point in the -space. Thus we conclude that the phonons response in SCGO corresponding to that at the M–point in SGO comprises the partial contribution of the atoms residing on the 12k sites of the magnetic sub-lattice of SCGO. The most relevant M-point modes have displacement vectors that modulate the distances between Cr sites in the kagome layers of the pyrochlore slab and therefore can effectively couple with the magnetic moments. The activation energy for magnetic relaxation is equal to the position of the characteristic peak in the experimental GDOS. Calculated energies of two of the Raman active -point modes are included in the energy range of this peak but they appear with less weight in the calculated GDOS.
In comparison BSZCGO, the other pyrochlore slab antiferromagnet shows a qualitatively different low–energy phonon response with an enhanced non-Debye low–energy response. We suggest that the faster relaxation without a particular energy scale is a consequence of this circumstance, which we attribute to the particular Ga/Zn disorder present in this compound. Future computational and theoretical work is called for to examine the interplay of magnetic interactions and the microscopic mechanisms associated with specific phonon displacements for better understanding of the reported phenomena.

Acknowledgements.
C.P. thanks J.Y. Mevellec for the Raman measurements.

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