Anisotropic exchange within decoupled tetrahedra in the quantum breathing pyrochlore Ba3Yb2Zn5O11
The low energy spin excitation spectrum of the breathing pyrochlore Ba3Yb2Zn5O11 has been investigated with inelastic neutron scattering. Several nearly resolution limited modes with no observable dispersion are observed at while, at elevated temperatures, transitions between excited levels become visible. To gain deeper insight, a theoretical model of isolated Yb3+ tetrahedra parametrized by four anisotropic exchange constants is constructed. The model reproduces the inelastic neutron scattering data, specific heat, and magnetic susceptibility with high fidelity. The fitted exchange parameters reveal a Heisenberg antiferromagnet with a very large Dzyaloshinskii-Moriya interaction. Using this model, we predict the appearance of an unusual octupolar paramagnet at low temperatures and speculate on the development of inter-tetrahedron correlations.
Frustrated or competing interactions have been repeatedly found to be at the root of many unusual phenomena in condensed matter physics Lacroix et al. (2011); Tarjus et al. (2005); Watanabe and Maruyama (2012); Balents (2010); Diep (2013). By destabilizing conventional long-range order down to low temperature, frustration in magnetic systems can lead to many exotic phases; from unconventional multipolar Santini et al. (2009); Starykh (2015) and valence bond solid orders Lacroix et al. (2011); Balents (2010) to disordered phases such as classical and quantum spin liquids Lacroix et al. (2011); Balents (2010). Significant attention has been devoted to understanding geometric frustration where it is the connectivity of the lattice that hinders the formation of order. Recently, however, magnets frustrated not by geometry but by competing interactions have become prominent for the novel behaviors that they host. Such competing interactions might be additional isotropic exchange acting beyond nearest neighbors Iqbal et al. (2015); Fåk et al. (2012); Bombardi et al. (2004), biquadratic or other multipolar interactions Läuchli et al. (2006). One possibility attracting ever increasing interest is that competing strongly anisotropic interactions may stabilize a wide range of unusual phenomena.
An exciting research direction in the latter context concerns itself with so-called “quantum spin ice” Gingras and McClarty (2014). This quantum spin liquid can be stabilized by perturbing classical spin ice with additional anisotropic transverse exchange interactions that induce quantum fluctuations. Particularly interesting is the potential realization of such physics in the rare-earth pyrochlores R2M2O7 Molavian et al. (2007); Onoda and Tanaka (2010); Ross et al. (2011), where R is a trivalent rare-earth ion, and M is a non-magnetic tetravalent transition metal ion, such as M=Ti, Sn or Zr. These materials can be described in terms of pseudo spin- degrees of freedom interacting via anisotropic exchanges Gingras and McClarty (2014); Ross et al. (2011), where the effective spin- maps the states of the crystal-electric field ground doublet of the rare-earth ion. These materials display a wealth of interesting phenomena, from the possibility of quantum Savary et al. (2012); Zhitomirsky et al. (2012); Wong et al. (2013) order-by-disorder physics in Er2Ti2O7 Rau et al. (2015), unconventional ordered states Chang et al. (2012); Stewart et al. (2004) as well as several candidates for quantum spin liquids Gardner et al. (1999); Kimura et al. (2013). In many of these compounds, the physics is very delicate, showing strong sample to sample variations Ross et al. (2012) or sensitivity to very small amounts of disorder Taniguchi et al. (2013); Kadowaki et al. (2015). Consequently, an accurate determination of the effective model is crucial in making definite progress in this area. This is particularly true in cases where the idealized disorder-free material may find itself in the vicinity of a transition between competing semi-classical ground states Wong et al. (2013); Robert et al. (2015); Jaubert et al. (2015)
Given the critical importance played by the precise value of the anisotropic exchanges, a number of experiments have been aimed at determining those couplings Ross et al. (2011); Savary et al. (2012). There is, unfortunately, much difficulty in obtaining accurate values for these couplings stemming from two key limitations. First, only approximate methods are available to relate the model to experiment, restricting comparisons to regimes where the theory becomes controlled, such as in high magnetic field Ross et al. (2011); Savary et al. (2012); Hayre et al. (2013) or at high-temperature Thompson et al. (2011); Applegate et al. (2012); Hayre et al. (2013); Oitmaa et al. (2013). Second, to avoid over-fitting the experimental data, one must work with a reasonable number of fitting parameters; for example restricting to a subset of the allowed interactions by ignoring interactions beyond nearest neighbors or possible multi-spin interactions Rau et al. (2015). Even in Yb2Ti2O7, where the latter concern is largely absent, there currently remains no consensus on the values of the anisotropic exchange parameters Ross et al. (2011); Robert et al. (2015). At the present time, a reference rare-earth pyrochlore-like compound with solely bilinear anisotropic interactions and for which essentially exact methods can be employed to compare with experimental data, is badly needed to cement the validity of the effective spin- description of such materials.
In this Letter, we study Ba3Yb2Zn5O11 (BYZO), a so-called breathing pyrochlore (BP) compound Kimura et al. (2014); Haku et al. (2015), which provides an ideal platform for understanding such anisotropic exchange models. As shown in Fig. 1, BYZO consists of small tetrahedra with a short nearest-neighbor bond distance connected by large tetrahedra with size . Because of the large ratio , the inter-tetrahedron couplings are expected to be small compared to the intra-tetrahedron couplings, leading to effectively decoupled small tetrahedra. This can be compared to the Cr-based BP compounds, where the small and large tetrahedra only differ in size by Okamoto et al. (2013); Tanaka et al. (2014); Okamoto et al. (2015). To characterize BYZO spectroscopically, we have investigated its low energy spin excitations using inelastic neutron scattering (INS). We confirm the picture of nearly independent tetrahedra, seeing nearly resolution limited dispersion-less modes at low temperatures. This INS data, combined with the thermodynamic measurements of Ref. [Kimura et al., 2014], allows for a complete and unambiguous determination of the the effective model for BYZO. We find that a single tetrahedron pseudo-spin model can quantitatively account for all of the current experimental data on BYZO, determining the four anisotropic exchanges as well as the -tensor. In addition to the antiferromagnetic Heisenberg exchange postulated in Ref. [Kimura et al., 2014], we find that significant Dzyaloshinskii-Moriya (DM) exchange is needed to obtain the correct level structure determined from INS. The fitted exchange parameters are far from the spin ice limit recently considered in Ref. [Savary et al., 2015] or the purely Heisenberg limits studied in Ref. [Benton and Shannon, 2015]. Instead, we find the ground state of each tetrahedron is doubly degenerate, consistent with the residual entropy observed experimentally at mK Kimura et al. (2014). These -doublets are nearly non-magnetic, carrying both a scalar spin-chirality as well as octupolar, all-in/all-out moments. The state of BYZO at currently studied base temperatures is thus a type of “octupolar paramagnet” without significant inter-tetrahedron correlations. Notwithstanding the broad agenda of accurately determining the anisotropic exchanges in rare-earth pyrochlore materials, the complete characterization of the single-tetrahedron model should provide a useful guide for further experimental studies of BYZO and other BPs. Specifically, we estimate that the inter-tetrahedron correlations could begin to set in below , at the edge of currently explored temperatures, possibly leading to interesting new physics Tsunetsugu (2001a, b); Kotov et al. (2004a, b, 2005) in this material.
Experimental results: Polycrystalline samples of BYZO were synthesized by solid-state reaction in AlO crucibles sup (). The resulting samples were characterized by specific heat and magnetization measurements sup (). The structure was studied via neutron powder diffraction utilizing the POWGEN Huq et al. (2011) diffractometer at the Spallation Neutron Source at Oak Ridge National Laboratory sup (). These measurements confirm the previously reported cubic structure Scheikowski and Müller-Buschbaum (1993); Rabbow and Müller-Buschbaum (1996); Kimura et al. (2014) (space group , no. 216) with lattice parameter =13.47117(3) at 10 K and =13.48997(3) at 300 K.
To explore the low energy spectrum of magnetic excitations in BYZO, INS data was collected using the HYSPEC spectrometer Winn et al. (2015) at the Spallation Neutron Source at Oak Ridge National Laboratory. Measurements were performed at 0.25, 10, and 20 K utilizing a He refrigerator, with fixed incident neutron energies of = 3.8 meV, 7.5 meV and 15 meV.
INS measurements with = 3.8 meV at and are shown in Fig. 2. The data at (Fig. 2(a) and (b)) exhibits several well-defined modes with no observable dispersion. The -dependence of the inelastic scattering intensity exhibits a broad peak centered near (see Fig. 2(b) and the Supplemental Material sup ()). The width in energy of the modes is close to instrumental resolution sup (). At elevated temperatures (Fig. 2(b) and (d)), three new excitations become visible resulting from transitions between excited states.
The origin of the observed low energy excitations appears to be modes originating from decoupled Yb tetrahedra. Several pieces of evidence support this assertion. Low lying crystal field levels can be excluded as the origin of these modes as three higher energy crystal field levels are experimentally observed (the maximum number for Yb3+) with the lowest lying level at Haku et al. (2015); sup (). The magnetic susceptibility and specific heat data do not show any signs of long range magnetic order down to Kimura et al. (2014); sup () that would indicate correlations between the small tetrahedra. Examination of the elastic scattering at is consistent with this conclusion, revealing no indication of long range magnetic order. Finally, the lack of dispersion suggests that these modes arise primarily from isolated tetrahedra and that the interactions connecting the tetrahedra are weak. We note that there is a weak and broad feature at . We have been unable to identify the origin of this feature, but note that it has a -dependence sup () distinct from that of the other nearly resolution limited modes.
Theoretical model: We now use these experimental observations, along with the thermodynamic data from Ref. [Kimura et al., 2014] to construct a model of BYZO. Given the dispersion-less modes seen in the INS, and the large ratio between the large and small tetrahedron sizes, we expect isolated tetrahedra to provide a very good description of the low energy physics. Each of the four Yb3+ ions has a Hund’s rule ground state of , with the manifold strongly split by the () crystalline electric field environment. Since this energy scale is very large, Haku et al. (2015), relative to the expected scale of the intra-tetrahedron interactions, only the ground doublet is relevant at low temperatures. The two states of this doublet define an effective pseudo-spin at each of the four Yb3+ sites. This pseudo-spin is related to the magnetic moment at each site through the -factors, and , present due to the local symmetry. Explicitly,
where are the local axes of tetrahedron site sup (). Regardless of the detailed composition of the ground doublet, since , the interactions between the Yb3+ are expected to be anisotropic and, a priori, not necessarily near the Ising or the Heisenberg limit Rau and Gingras (2015). Symmetry strongly constrains their form; each Yb3+-Yb3+ bond has symmetry () and each small tetrahedron has full tetrahedral symmetry () Rabbow and Müller-Buschbaum (1996); Kimura et al. (2014). Assuming an effective spin-1/2 doublet 111The case of a dipolar-octupolar doublet () is in principle possible as well, with a different anisotropic exchange model Huang et al. (2014). We find such a model does not provide a good description of the specific heat or magnetic susceptibility of BYZO and thus consider only an effective spin-1/2 () doublet. This is consistent with the ground doublet found in Ref. [Haku et al., 2015] by fitting the observed crystal field excitations., there are therefore four allowed anisotropic exchange interactions Ross et al. (2011), taking the form
where the bond dependent phases and are defined in the Supplemental Material sup (). The spectrum of this Hamiltonian is partly determined by tetrahedral symmetry. The four-pseudo-spin states break into the irreducible representations under the action of the tetrahedral group. This gives a level structure of a singlet (), three doublets () and three triplets ( or ). From the observed residual entropy Kimura et al. (2014), it seems plausible that the ground state of the tetrahedron is an doublet, which gives an entropy of / Yb3+.
Best fit parameters: The model of Eq. (Anisotropic exchange within decoupled tetrahedra in the quantum breathing pyrochlore Ba3Yb2Zn5O11), supplemented with the definition of the moment in Eq. (1), is determined by the six parameters , , , , and . To fix these parameters, we perform a fit to the specific heat and susceptibility data of Ref. [Kimura et al., 2014] and a cut of the INS data averaged over the range at . This is a global fit, minimizing squared differences between experimental and theoretical values from each set of experimental data simultaneously. For the specific heat, we fit only the data below to minimize the influence of the subtraction of the lattice contribution, while the susceptibility data up to is used. 222Fitting only the specific heat and susceptibility from Ref. [Kimura et al., 2014] does not produce a unique fit, but many equally good fits. However, these differ significantly when including constraints that arise from fitting the INS data. Three additional fitting parameters were included; a constant shift of the susceptibility, , to account for the Van Vleck and diamagnetic core contributions of the Yb3+ ions, the intensity scale of the INS cut and the overall scale of the Gaussian broadening used in the theoretical INS intensity sup (). Further details of the fitting methodology and comparisons to experimental data can be found in the Supplemental Material sup ().
From this analysis we find a unique best fit which provides excellent agreement with all of the known experimental data on BYZO. The best fit parameters are
Comparison to the specific heat and susceptibility is shown in Fig. 3. Agreement with both is excellent; small differences can be seen in the specific heat at higher temperatures, likely due to some uncertainty in the subtraction of the lattice contribution. Comparison to a cut of the INS data at is shown in Fig. 2(a), along with an illustration of the level structure of the single tetrahedron model with the parameters of Eq. (Anisotropic exchange within decoupled tetrahedra in the quantum breathing pyrochlore Ba3Yb2Zn5O11). The level structure matches very well with the energies of the peaks in the INS cut at . Explicitly one has the spectrum
where the irreducible representation in of each level is indicated. We note that the and levels are very close in energy, but not exactly equal. The model also accurately reproduces the wave vector and temperature dependence of the INS data as can be seen in Fig. 2(c),(d),(f). Additional comparisons to magnetization and INS data can be found in the Supplemental Material sup (). Some of the features of these energy levels can be better understood by adopting global quantization axes and defining global pseudo-spin operators . Using the notation of Ref. [Yan et al., 2013], the model in the global basis is parametrized by four anisotropic exchanges , , and . The best fit parameters of Eq. (Anisotropic exchange within decoupled tetrahedra in the quantum breathing pyrochlore Ba3Yb2Zn5O11) correspond to the values sup ()
Since and to a fair approximation, these fitted parameters describe a Heisenberg antiferromagnet supplemented with large (indirect) DM interaction Canals et al. (2008); sup () and negligible symmetric anisotropies. We can thus understand the doublet ground state as an extension of the pair of singlets that form the ground state in the Heisenberg limit Kimura et al. (2014). Similarly, the approximate quintet can be mapped to the high energy, five-fold degenerate states of the antiferromagnetic Heisenberg model. Indeed, when only Heisenberg and DM interactions are present these remain exact eigenstates and degenerate, leaving only the small symmetric anisotropies to provide any splitting. While this mapping is appealing, there are key differences; for example, the three triplets present in the Heisenberg model are strongly mixed by the DM interactions.
Discussion: The physics at very low temperatures, , should be primarily controlled by the ground doublet. The states of this doublet, , are rather exotic. As in the Heisenberg limit, they are largely non-magnetic, carrying a uniform (scalar) spin-chirality on each triangle of the tetrahedron Kimura et al. (2014). However, due to the large DM interaction, the states additionally acquire all-in/all-out (AIAO) moments. This is generically expected as the AIAO moments and the uniform spin-chirality transform identically under the tetrahedral symmetry Kotov et al. (2004a, b). Explicitly, the projection of a pseudo-spin in the local basis into the doublet takes the form with for the parameters of Eq. (Anisotropic exchange within decoupled tetrahedra in the quantum breathing pyrochlore Ba3Yb2Zn5O11) and . These AIAO moments are octupolar in character, with the net magnetic moment on each tetrahedron vanishing. Due to the smallness of the inter-tetrahedron interactions we thus expect BYZO to be an octupolar paramagnet at temperatures much smaller than . Direct signatures of this unusual paramagnetic state may appear in more indirect magnetic probes, such as the non-linear susceptibilities.
Going to even lower temperatures one can potentially see indications of collective behavior of the small tetrahedra. Depending on the structure of the inter-tetrahedron interactions, a variety of states could be stabilized, such as weak AIAO order or non-magnetic valence bond solid phases Kotov et al. (2004a, b). Tantalizing hints of the onset of such correlations may already be present in the experimental data. We note that the INS data is slightly broader than calculated instrumental resolution (by ) which may be suggestive of weak dispersion, while the specific heat data of Kimura et al. (2014) shows a slight upturn below that is not explained by the single-tetrahedron model. We thus suspect that the current lowest temperatures explored in BYZO are at the threshold of observing such inter-tetrahedron correlations and possibly even ordering of these doublet states. Given the complete characterization of the intra-tetrahedron physics presented in this work, we feel the field is well poised to push the study of BYZO to even lower temperatures and explore such inter-tetrahedra physics.
Acknowledgements.We thank J. Y. Y. Lin for the help with the data reduction. A.D.C., M.D.L. and L.S.W. thank A. Chernyshev, P. Maksimov, G. Ehlers, and I. Zaliznyak for useful discussions. We thank K. Kimura and S. Nakatsuji for kindly providing their data from Ref. [Kimura et al., 2014]. The research at the Spallation Neutron Source (ORNL) is supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy (DOE). AFM was supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. Research supported in part by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U. S. Department of Energy. The work at U. of Waterloo was supported by the NSERC of Canada, the Canada Research Chair program (M.J.P.G., Tier 1), the Canadian Foundation for Advanced Research and the Perimeter Institute (PI) for Theoretical Physics. Research at PI is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. J.G.R. and L.S.W. contributed equally to this work.
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