Session T15: Focus Session: Spin Ice and Weakly Disordered Pyrochlores

Challenges in the collective behaviour of spin ice materials

The opportunity to observe magnetic monopoles in spin ice materials has driven a significant theoretical and experimental research effort over the past few years. While a broad class of experimental results have confirmed the monopole picture, some experiments continue to present tantalising puzzles which have not yet been possible to resolve via straightforward application of Coulomb liquid theories a la Debye-H\"{u}ckel. This is illustrated perhaps most strikingly by the departure from the expected asymptotic Arrhenius behaviour of the characteristic relaxation time scales observed at very low temperatures in magnetisation and magnetic susceptibility measurements. Here we investigate some of these phenomena and attempt to identify the necessary extensions of existing theories.   

NMR relaxation in spin ice at low temperature due to diffusing emergent monopoles

At low temperatures, spin dynamics in ideal spin ice is due mainly to dilute, thermally excited magnetic ``monopole'' excitations. I consider how these will affect the longitudinal (T1) and dephasing (T2) relaxation functions of a nuclear spin in the spin-ice pyrochlore Dy2Ti2O4. Up to the time scale for nearby monopoles to be rearranged, a stretched-exponential form of the relaxation functions is expected, due to averaging over nuclei that have different local environments. ror the dephasing (T2) relaxation, the power of time in the stretched exponential is 3/2 in the case of diffusing monopoles, but 1/2 in the case of fixed, fluctuating magnetic impurities. The flip rate and density of fluctuating spins (whatever their nature) can be extracted from the measured relaxation times $T_1$ and $T_2$, and from known parameters. However, the actual experimental relaxation measured by Kitagawa and Takigawa becomes temperature independent in the very low T limit, and the T2 has a power $t^{1/2}$ in the exponential, neither of which can be explained by monopoles. I suggest the very low T behavior could be due to magnetic impurities on the (normally nonmagnetic) Ti sites.   

Impurities in Spin Ice Crystals

Spin ice crystals (and pyrochlore oxides in general) have raised a lot of interest of late thanks to their exotic properties, including emergent gauge symmetries, possible spin liquid behavior, and magnetic monopole excitations. Theoretical and experimental efforts in the study of these materials have benefited from the relative ease of growth of large clean single crystals. Even in such clean systems, however, impurities can play a crucial role in determining the properties at very low temperatures (see e.g., C. Henley, http://arxiv.org/abs/1210.8137). Here we investigate this issue both experimentally and theoretically. We study how controlled non-magnetic Y-dilution in $Dy_2Ti_2O_7$ gradually alters the effective monopole description and the thermodynamic properties of the system at low temperature (extending earlier work by other authors to regimes that have not been investigated so far). We also study how oxygen deficiency affects spin ice samples, and we discuss how the oxygen stoichiometry can be quantified and controlled experimentally.   

Impurity and boundary effects on magnetic monopole dynamics in spin ice

Using a SQUID magnetometer, we measure the time-dependent magnetic relaxation in Dy2Ti2O7 and find that it decays with a stretched exponential followed by a very slow long-time tail. In a Monte Carlo simulation governed by Metropolis dynamics we find that surface effects and a very low level of stuffed spins (0.30\%) - magnetic Dy ions substituted for non-magnetic Ti ions - can explain these signatures in the relaxation. We find that the additional spins trap the magnetic monopole excitations and provide the first example of how defects in a spin-ice material can obstruct the flow of monopoles.   

Low temperature specific heat measurements of the spin ice material Dy$_2$Ti$_2$O$_7$

Recent work on low temperature magnetization [1] and ac-susceptibility [2] of the spin ice Dy$_2$Ti$_2$O$_7$ has revealed a number of inconsistencies with earlier magneto-caloric [3] and thermal relaxation [4] measurements. These unsolved puzzles have motivated us to re-investigate the low temperature specific heat of this material. By measuring the thermal relaxation of Dy$_2$Ti$_2$O$_7$, we extract magnetic spin relaxation times and compare them to previous results in the literature.\\[4pt] [1] K. Matsuhira, et al., J. Phys. Cond. Mat. 13, L737 (2001) \\[0pt] [2] L. R. Yaraskavitch et al., Phys. Rev. B 85, 020410(R) (2012) \\[0pt] [3] M. Orendac, et al., Phys. Rev. B 75, 104425 (2007) \\[0pt] [4] B. Klemke, et al., J. Low Temp. Phys. 163, 345 (2011)   

Dipolar Hyperkagome Spin Ice

Non-magnetic doping of the Pyrochlore spin ices Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$ has been shown\footnote{X. Ke {\it et al}., Phys. Rev. Lett. {\bf 99}, 137203} to exhibit a nonmonotonic residual entropy per spin as a function of doping, with an increase near one quarter doping. Hyperkagome corresponds to a disorder-free one quarter doping of the Pyrochlore lattice with a large residual Pauling entropy $S/N = 1/3\ln(9/2)$. In this talk we discuss the physics of local Ising spins coupled through antiferromagnetic nearest neighbour exchange and a long range dipolar interaction. We generalize the phase diagram\footnote{B.C. den Hertog {\it et al}., Phys. Rev. Lett. {\bf 84}, 3430-3433 (2000)} of dipolar Pyrochlore spin ice to the Hyperkagome case, finding a crossover to a spin ice state followed by a transition to a charge ordered state and finally a transition to an ordered ground state, as first seen on the Kagome lattice\footnote{Gia-Wei Chern {\it et al}., Phys. Rev. Lett. {\bf 106}, 207202 (2011)}. We show that our hybrid single spin flip/loop algorithm Monte Carlo simulations agree with analytical results for small sizes, and present results for systems as large as $12*L^3$ spins with $L=4$.   

Disordered Quantum Spin Ice Ground State of Tb$_2$Sn$_{2-x}$Ti$_x$O$_7$

Inelastic neutron scattering, AC magnetic susceptibility and $\mu$SR measurements have been performed on polycrystalline solid solutions of the pyrochlore magnet, Tb$_2$Sn$_{2-x}$Ti$_x$O$_7$ for seven samples with x between 0 and 2. These measurements probe the crystal field states, low energy spin dynamics and phase behavior to temperatures less than 0.1K. Tb$_2$Ti$_2$O$_7$ is proposed to display a quantum variant of the spin ice ground state, stabilized by virtual excitations between the Tb$^{3+}$ crystal field ground state doublet and its low lying excited state doublet. Isostructural, Tb$_2$Sn$_2$O$_7$, displays ``soft'' spin ice order and its Tb$^{3+}$ ground and excited crystal field states are known to be interchanged relative to those in Tb$_2$Ti$_2$O$_7$. These measurements of the solid solutions of Tb$_2$Sn$_{2-x}$Ti$_x$O$_7$ focus on crystal field excitations between 1meV and 50meV, and show greatly enhanced spin dynamics at low energies for samples with intermediate x. All magnetic order is absent for x$>$0.1, leaving behind a highly fluctuating, disordered spin ice ground state.   

Antiferromagnetic Spin Ice Correlations at (1/2,1/2,1/2) in the Ground State of the Pyrochlore Magnet Tb$_2$Ti$_2$O$_7$

The ground state of the candidate spin liquid pyrochlore magnet Tb$_2$Ti$_2$O$_7$ (TTO) has been long debated. Despite theoretical expectations of magnetic order below 1K based on classical Ising-like Tb spins, muSR and neutron scattering experiments show no long range order down to 50mK. Two theoretical scenarios have been put forward to account for this: the quantum spin ice scenario and a non-magnetic singlet ground state, but no clear consensus has been reached. We present neutron scattering measurements on TTO at 70mK that reveal elastic scattering intensity at (1/2,1/2,1/2) positions in reciprocal space[1]. The corresponding spin configuration can be modeled as a short-range antiferromagnetically ordered spin ice, in which spins obey a variant of the ice rules in each unit cell, and flip directions between adjacent cells. At low temperatures, this elastic scattering is separated from low-lying magnetic inelastic scattering by $\sim$0.05meV. The elastic signal disappears under the application of small magnetic fields and upon elevating temperature. Pinch-point-like elastic diffuse scattering is observed, which together with the elastic spin ice correlations strongly supports the quantum spin ice picture for TTO. [1] K. Fritsch et al., arXiv:1210.1242[cond-mat.str-el]   

Low-Temperature Low-Field Phases of the Pyrochlore Quantum Magnet Tb$_2$Ti$_2$O$_7$

By means of ac magnetic-susceptibility measurements, we have found evidence for a new magnetic phase of Tb$_2$Ti$_2$O$_7$ below about 140~mK in zero magnetic field. In magnetic fields parallel to [111], this phase---exhibiting frequency- and amplitude-dependent susceptibility and an extremely slow spin dynamics---extends to about 70~mT, at which it gives way to another phase. The field dependence of the susceptibility of this second phase, which extends to about 0.6~T, indicates the presence of a weak magnetization plateau below 50~mK, as has been predicted by a single-tetrahedron four-spin model, giving support to the underlying proposal that the disordered low-field ground state of Tb$_2$Ti$_2$O$_7$ is a quantum spin ice.   

Glassiness in single crystalline Y$_2$Mo$_2$O$_7$

The spin glass transition at T$_g$ = 22 K in the pyrochlore Y$_2$Mo$_2$O$_7$ has remained an enigma in condensed matter physics for over two decades. Despite the results of many experiments which indicate a freezing of the Mo$^{4+}$ spins at low temperatures, a consistent theoretical framework has not been reached to describe how this can occur in the absence of large amounts of chemical disorder. We report on the synthesis of the world's first high quality single crystal of this compound, and its characterization using a variety of thermodynamic and scattering probes. Some of the new results include a non-linear magnetic heat capacity at low temperatures, the presence of liquid-like elastic scattering within the glassy state, and high-Q scattering consistent with orbital or chemical disorder. Possible candidates for the low-T ground state will be discussed.   

Monopole Hopping through Quantum Spin Tunnelling in Spin Ice

The low temperature dynamics in spin ice materials is governed by the density and mobility of elementary excitations that behave as emergent magnetic monopoles. The diffusion of such monopoles proceeds via flipping of large electronic spins with Ising-like anisotropy (due to their crystal field environment). Experimental evidence suggests that, at temperatures relevant for spin ice physics, this flipping occurs as a quantum-mechanical tunnelling through a large anisotropy barrier. Here we investigate this process at the microscopic, single-ion level by computing the quantum dynamics resulting from the interplay between the crystal field Hamiltonian and the Zeeman coupling with magnetic fields (either applied or due to other spins). We interpret our results in terms of monopole hopping rates, and we compare our predictions with existing experiments for both Ho2Ti2O7 and Dy2Ti2O7.   

Numerical Study of Perturbations in Dipolar Spin Ice

Competing interactions in geometrically frustrated magnets can lead to highly degenerate and non-trivially correlated ground states. One topical example, the spin ice compound Dy$_2$Ti$_2$O$_7$, exhibits such a ground state which possesses a Pauling's residual entropy analogous to that of water ice. At temperatures well below the temperature scale set by the frustrated and dominant dipolar interactions, the material displays a classical spin liquid like state. As a result, small perturbations may become significant for the low temperature physics. In this project we consider perturbations from further neighbor interactions and from stuffing impurities in an attempt to account for some of the observed experimental low temperature behaviors. In particular, we determine the third neighbor interactions using Monte Carlo (MC) simulations by fitting to experimental data in a magnetic field near the [112] direction. The effects on the zero-field specific heat due to variation of the exchange parameters are studied using a cumulant method in conjunction with the MC simulations. We also studied the effects of stuffing Dy magnetic ions on the Ti site, which can trigger large variations in the equilibrium value of the specific heat below temperatures of 0.5K.   

Synthesis and Characterization of New Germanate Pyrochlores, A$_2$Ge$_2$O$_7$ (A = Tb, Yb, Er)

The titanate pyrochlores, A$_2$Ti$_2$O$_7$, have yielded some of the most well-studied geometrically frustrated magnetic materials. A new class of pyrochlores with germanium on the B-site is now being investigated. The germanates, which in many cases share ground states with their titanate analogues, are far more highly correlated due to the smaller B-site cation. Two germanate pyrochlores, Ho$_2$Ge$_2$O$_7$ and Dy$_2$Ge$_2$O$_7$, were previously synthesized and characterized as new spin ice compounds [1-3]. We now present the new germanate pyrochlores, A$_2$Ge$_2$O$_7$ with A = Tb, Yb, and Er. Based on the titanates, three distinctly different magnetic ground states can be expected for these materials: Er$_2$Ti$_2$O$_7$ has an ``order-by-disorder'' mechanism, Yb$_2$Ti$_2$O$_7$ is a quantum spin ice and Tb$_2$Ti$_2$O$_7$ is a spin liquid. Preliminary measurements on Tb$_2$Ge$_2$O$_7$ indicate that it too is a spin liquid down to at least 0.35 K. We will present the characterizations of A$_2$Ge$_2$O$_7$ (A = Tb, Yb, Er) and compare them to the titanates.\\[4pt] [1] H. D. Zhou et al., Nature Communications 2, 478 (2011). \\[0pt] [2] H. D. Zhou et al., Phys. Rev. Lett. 108, 207206 (2012).\\[0pt] [3] A. M. Hallas et al., accepted for publication in Phys. Rev. B (2012).