Possible Quantum Diffusion of Polaronic Muons in Dy 2 Ti 2 O 7 Spin Ice
Résumé
We interpret recent measurements of the zero field muon relaxation rate in the magnetic pyrochlore Dy 2 Ti 2 O 7 as resulting from the quantum diffusion of muons in the material. In this scenario, the plateau observed at low temperature (< 7 K) in the relaxation rate is due to coherent tunneling of muons through a spatially disordered spin state and not to any magnetic fluctuations persisting at low temperature. Two further regimes either side of a maximum relaxation rate at T Ã ¼ 50 K correspond to a crossover between tunneling and incoherent activated hopping motion of the muon. Our fit of the experimental data is compared with the case of muonium diffusion in KCl. The recent measurement of the zero field SR relaxa-tion in Dy 2 Ti 2 O 7 spin ice (DTO) by Dunsiger et al. [1] joins a long series of puzzling experiments on a diverse range of frustrated magnetic materials over roughly the last fifteen years [1-9]. They indicate a relaxation of the spin asymmetry of the muons after they are implanted in the sample in a spin-polarized state. This persists down to the lowest observed temperatures with little temperature de-pendence in the relaxation rate below a temperature vary-ing from 1-10 K depending on the compound. The interpretation of this relaxation has long been a matter of discussion. At its center lies the question whether the origin of the dynamics in each case is due to intrinsic magnetic fluctuations in the material, or whether the im-planted muons are instead more than merely passive probes of the magnetism. The latter is a realistic possibility as the muon couples not only weakly to the magnetic degrees of freedom via its spin [10,11] but also potentially much more strongly to electric degrees of freedom via its positive charge. As we argue here, this can give rise to new physics interesting in its own right, which is in turn elegantly probed via the magnetic degree of freedom. In DTO, SR measurements have been made [12] and an important debate [13-15] is developing over the origin of the muon spin relaxation in spin ice. In spin ices, the mo-ments have an Ising anisotropy and the interactions are frustrated leading to the onset of a highly degenerate spin ice state at low temperatures that is signaled by a heat capacity peak at around 1 K. The dynamics in this material has been explored using several probes besides SR [1,16] including susceptibility [17-20], neutron scattering [21-23], magnetocaloric effect [24], magnetization relaxation experi-ments [20], nuclear forward scattering [25], and nuclear quadrupole resonance [26], and three distinct dynamical regimes have been found. Above around 15 K, the dynamics follows an Arrhenius law controlled by a gap to excited crystal field levels of several hundred kelvin [22]. Between about 1 K and 15 K, the dynamics is dominated by tunneling between magnetic configurations and the temperature depen-dence is correspondingly weaker than at higher temperatures [27]. Below about 1 K, the time scales greatly increase, but there is some evidence for a second Arrhenius regime in ac susceptibility [19] although the moments are static on neu-tron time scales [23]. Between 2 and 16 K, large time scales (! 10 À3 s) have been observed by ac susceptibility mea-surements [17], while for higher temperature, nuclear forward scattering of synchrotron radiation experiments [25] give characteristic fluctuation times between 10 À7 and 10 À10 s between 30 and 70 K, respectively. Below 2 K, the observed time scales considerably increase as the temperature is low-ered, to reach values as large as 10 5 s below 0.3 K [20]. The existence of several dynamical regimes together with the very large time scales measured by the other experiments at low temperature, appears at odds with the featureless SR relaxation 1=T 1 below 7 K observed in Ref. [1] in a zero-field experiment. Moreover, since the moments are, to an excellent approximation, Ising-like, there should be insufficient spectral weight at low tem-peratures to bring about muon spin relaxation. In our scenario we assume that the magnetic spins are to a first approximation frozen at low temperatures T & 70 K relative to characteristic times experienced by the muons and that the relaxation mechanism of the muon is due to its diffusion through the static disordered magnetic back-ground [28]. Such a hypothesis has not been considered in insulating oxides. On the contrary, muons are generally considered localized in such compounds. In some transi-tion metal oxides, muoxyl bonding (a covalent bond be-tween the muon and an oxygen atom) has sometimes been observed (in Fe 2 O 3 and Cr 2 O 3 , but not in V 2 O 3 for ex-ample Refs. [31,32]). However, even this particular muon state diffuses above a certain temperature (of about 100 K [33]). We note that the presence of both muonium (a bound state of an electron and a muon, denoted by Mu) together with a bare (positive) muon has been demonstrated in some