Review Report – Grant GR/L83677/02: Frustration in Pyrochlore Magnets

O.A. Petrenko, University of Warwick, Department of Physics

(I) Introduction

The Grant “Frustration in Pyrochlore Magnets” was transferred to me after the unexpected departure from science of one of the two principal investigators, Mark Harris. At that stage approximately 2/3 of the project had been completed. Naturally, the change over at such a late stage in a project brought with it considerable difficulties. It took a significant effort to get the project back on track, but the efforts required have been generously rewarded. The unusual and, quite often, spectacular results of the project have generated an intense interest in our work by groups around the world. The success of the project is largely due to the excellent neutron scattering facilities available at the UK, along with millikelvin temperature and high magnetic field sample environments.

In this report, I concentrate mostly on one aspect of the project, with which I have been directly involved, namely our studies of several high quality single crystals of the pyrochlore materials. I describe the motivation for, the methodology and the results of our neutron diffraction experiments on these materials. I also list the results of other neutron scattering experiments. The other aspects of this project, such as bulk property measurements and Monte Carlo simulation are omitted, as they have already been described in detail by Prof. Steven Bramwell, UCL, in his Review Report for the other half of this Grant. Prof. Bramwell’s report has been very highly rated by the Referee. As a result no additions or corrections to it are required.

(II) The Spin Ice Concept

Spin ice materials are magnetic substances in which the atomic magnetic moments obey the same ordering rules as the hydrogen atoms in ice, H2O. They provide a bridge between the simple statistical mechanics of ice-type models and the complex behaviour of frustrated magnets, with wider relevance to diverse areas of research such as high temperature superconductivity and neural networks. Furthermore, relatives of spin ice such as R2Ti2O7 (R= rare earth) are a current focus of attention for their interesting electronic properties. It is therefore important and desirable to establish the detailed physics of the simplest spin ice materials, which provide among the best model systems for the study of frustration and its broad consequences.

Experiments on Ho2Ti2O7 provided the original motivation for the concept of spin ice. In Ho2Ti2O7, the magnetic Ho3+ ions occupy a cubic pyrochlore lattice, a corner-linked array of tetrahedra that is identical to the lattice formed by the mid-points of the oxygen-oxygen bonds of cubic ice. The ground state of Ho3+ is an effective Ising doublet of local <111> quantisation axis, with nearest neighbour Ho3+ moments experiencing an overall ferromagnetic coupling. Qualitatively, Ho2Ti2O7 approximates to the nearest neighbour spin ice model, in which ferromagnetically-coupled Ising spins on the pyrochlore lattice lie parallel to the local <111> axes that point towards the centres of the tetrahedra. The ground state of this model requires two spins pointing into and two pointing out of each tetrahedron. Then, if spins represent hydrogen atom displacement vectors, one obtains the ice rules that lead to Pauling's result for the extensive zero temperature entropy of the disordered ground state of ice.

None of the measurements reported so far however, constitute a proof that the spin ice ground state exists in Ho2Ti2O7 or in a related material Dy2Ti2O7 in zero applied field. Moreover, a certain amount of confusion has recently arisen due to some contradictory observations: Siddharthan et al. suggested that the specific heat behaviour of Ho2Ti2O7 possibly indicates a transition to a partially ordered state at 0.8 K, in disagreement with the previous neutron scattering and muon-spin resonance results. It was only the neutron scattering data that unambiguously established the spin ice nature of the zero field spin correlations in Ho2Ti2O7 (Phys. Rev. Lett. 8704 7205 (2001)) and Dy2Ti2O7 (cond-mat/0107414).

1. Spin correlations in Ho2Ti2O7: A dipolar spin ice system

Figure 1a shows the neutron scattering pattern from Ho2Ti2O7 at T~50 mK. One of the main features of the experimental data is the `four-leaf clover' of intense scattering around (000). There is also strong scattering around (003) and a broad region of slightly weaker scattering around (3/2,3/2,3/2). These intense regions are connected by narrow necks of intensity giving the appearance of bow-ties. The width of the intense regions indicates short-range correlations on the order of one lattice spacing. Qualitatively similar scattering has been observed in ice itself.

For the near neighbour spin ice model the calculated pattern is shown in Fig. 1b. It successfully reproduces the main features of the experimental pattern, but there are differences, notably in the extension of the (000) intense region along (hhh) and the relative intensities of the regions around (003) and (3/2,3/2,3/2). Also, the experimental data shows much broader regions of scattering along the diagonal directions. Clearly, the experimental spin correlations do not reflect a completely disordered arrangement of ice-rule states, but some states are favoured over others.

In the more complete dipolar spin ice model, the spin ice behaviour emerges from the dominant effect of the long-range nature of dipolar interactions. The calculated pattern for this model is shown in Fig 1c. It captures most details of the experimental pattern missed by the near neighbour spin ice model in Fig. 1b, such as the four intense regions around (000), the relative intensities of the regions around (003) and (3/2,3/2,3/2) and the spread of the broad features along the diagonal. Note also the low scattering intensity at (220), consistent with the experimental low intensity around at (220).

Fig.1 (a) Experimental neutron scattering pattern of Ho2Ti2O7 in the (hhl) plane of reciprocal space at T~50 mK. Dark blue shows the lowest intensity level, red-brown the highest. (b) I(Q) for the nearest neighbour spin ice model at T=0.15J. (c) I(Q) for the dipolar spin ice model at T=0.6 K. The areas defined by the solid lines denote the experimental data region of (a).

The difference between Figs. 1b and 1c shows that dipolar interactions, whilst inducing a low energy ice-rules manifold, do cause further correlations to emerge among the spins than those that are due to a nearest neighbour ferromagnetic interaction. This thermal bias towards certain ice-rule configurations in the dipolar model is consistent with mean field calculations. These results settle the question of the nature of the low temperature spin correlations in Ho2Ti2O7 for which contradictory claims have been made.

2. Field Induced Partial Order in the Spin-Ice Dysprosium Titanate

Polycrystalline Dy2Ti2O7 was synthesised by a solid diffusion reaction starting from stoichiometric quantities of TiO2 and isotopically enriched Dy2O7. The single crystal sample of Dy2Ti2O7 was then grown by the floating zone method using infra-red image furnace. The absorption cross section of natural Dysprosium is 994 barn. Through isotopic enrichment, the absorption cross section was reduced by a factor of five to 208 barn. Neutron scattering was carried out at the ISIS facility on the indirect geometry spectrometer PRISMA configured in the diffraction mode. Rotation of the crystal allows a rapid mapping of a large section of reciprocal space, making PRISMA ideal for observing magnetic diffuse scattering.

In the first experiment, the crystal was cooled by a 3He sorption refrigerator for measurements in zero applied magnetic field. It was aligned with [1-10] vertical, such that the (hhl) scattering plane included the three principal symmetry axes [100], [110] and [111]. In the second experiment, an Oxford instruments 7T vertical field cryomagnet, with dilution refrigerator insert, was used. For the applied field measurements two field orientations were studied: [1-10] as above and [100] corresponding to a (0kl) scattering plane. The data were normalised to vanadium and corrected for absorption.

With [100] vertical, a map of reciprocal space was made at the base temperature (~70 mK) in zero field. The diffuse scattering maxima observed agree with those predicted in the (hhl) scattering plane for the dipolar spin ice model. On application of a field the diffuse scattering disappeared and was replaced by magnetic Bragg peaks at the Q=0 positions. A field of ~0.7 T was sufficient to saturate these peaks.

This behaviour is readily explained as the applied field breaks the degeneracy of the six “two in, two out” spin configurations of the elementary tetrahedron. The pyrochlore lattice can be described as a face centred cubic lattice with a tetrahedral basis, and the degeneracy breaking means that every tetrahedron adopts the same “two in, two out” state with a net moment in the direction of the applied field, [100]. This non-collinear ferromagnetic structure allows the observed magnetic Bragg peaks at the Q=0 positions such as (200). Interestingly the experimental magnetisation did not develop smoothly, but in a series of steps. Hysteresis was observed on cycling the field.

In Dy2Ti2O7 the [1-10] direction is a hard direction of magnetisation. In zero field, the magnetic scattering was observed to be diffuse, and characteristic of the disordered low temperature state of dipolar spin ice. Application of a small field (~0.1 T) in this direction again caused the appearance of the Q=0 Bragg peaks; however, unlike the [100] direction, the diffuse scattering features did not disappear. As the field was raised to 1.5 T the diffuse scattering sharpened around the Q=X positions such as (001) without becoming resolution limited. This is seen clearly in Fig. 2. Similar features have been observed in the neutron scattering of Ho2Ti2O7, adding strength to the idea that the true ground state of these systems is a Q=X structure that is dynamically inhibited from being accessed on experimental timescales. Again the magnetisation versus field curve was observed to have several sharp steps and plateaux (Figure 3).

Fig.2. (Top) Diffuse scattering in the (hhl) plane. In zero field at 270 mK no magnetic Bragg peaks are observed. All resolution limited intense features are of nuclear origin. (Bottom) Scattering in the (hhl) plane with a field of 1.5 T applied on [1-10] at ~60 mK. Magnetic Bragg peaks have appeared at positions such as (200) and the diffuse features observed in zero field at positions such as (003) have sharpened into features elongated on [00l].

The formation of the Q=X structure is consistent with the spin ice rules. In this field orientation, assuming perfect <111> spins, only two of the spins of the tetrahedral basis have a component along the field direction. These form “in-out” [1-10] chains parallel to the field forcing the remaining two spins per tetrahedra into “in- out” [110] chains perpendicular to the field. The perpendicular chains are not coupled by the spin ice rules which, in the absence of any further neighbour coupling, would lead to two-dimensional [110] Bragg sheets of scattering extended along [00l] in the scattering plane. The diffuse features in the experimental pattern are indeed extended along this direction, but the sharp build up of intensity around the Q=X points indicates a strong tendency to prefer Q=X short range ordering of the perpendicular rods. Neglecting interference between the scattering from the two spin sets (perpendicular and parallel to the field) one arrives at the conclusion that the Q=0 Bragg scattering arises from the parallel rods and the Q=X diffuse scattering arises from the perpendicular rods. However, it may be a crude approximation to separate the scattering in this way.

In conclusion, we have obtained accurate neutron scattering data for Dy2Ti2O7 that is in qualitative agreement with theoretical expectations for a spin ice material. It is of interest that, even in a relatively strong field along [1-10], the system remains only partially ordered. It is also noteworthy that the magnetic hysteresis loop shows several steps and plateaux. An understanding of these effects awaits a detailed study of the static and dynamic properties of the dipolar spin ice model in an applied magnetic field. In zero field no magnetic order was observed down to 50 mK but the magnetic diffuse scattering was in qualitative agreement with that expected for the disordered low temperature state of dipolar spin ice. Application of a field of ~ 0.8 T in the [100] direction led to long range order. With the field applied in the [1-10] direction a coexistence of long range ferromagnetic and short range antiferromagnetic order was observed. This is attributed to the pinning of only half the spins by the field.

Fig.3. Integrated intensity of the (002) magnetic Bragg peak and the (003) diffuse feature as the field is scanned at ~70 mK.

(III) Other Titanate Pyrochlore Magnets

We have characterised all the other members of the titanate pyrochlore series (J. Phys. Cond. Matt. 12, 483 (2000)) and have also produced single crystals of almost all of them (J. Phys. Cond. Matt. 10, L723 (1998)).

1. Order in the Heisenberg Pyrochlore: The Magnetic Structure of Gd2Ti2O7

The rare earth pyrochlore material Gd2Ti2O7 was expected to be the most interesting material, as it should approximate the Heisenberg model antiferromagnet. However it was found to have an ordering transition at 1.1 K. For this system there were several untested theoretical predictions of the ground state ordering pattern. 160 We have established the magnetic structure of isotopically enriched Gd2Ti2O7, using powder neutron diffraction at a temperature of 50 mK. The magnetic structure at this temperature is a partially ordered, non- collinear antiferromagnetic structure, with propagation vector k= 1/2 1/2 1/2. It is quite complex (see Fig.4) and can be described as a set of Q=0 ordered Kagome planes separated by zero interstitial moments.

Fig.4. The proposed magnetic structure of Gd2Ti2O7

2. Neutron Scattering from Yb2Ti2O7

We have grown a crystal of Yb2Ti2O7 and performed neutron diffraction experiments on this material at sub- 1K temperatures using the PRISMA spectrometer. Despite using a sample with a reasonably large volume, no magnetic diffuse scattering has been found in this material down to 0.3 K either in the [hk0] or in the [hhl] scattering planes. At present we do not know for sure the reason for such a sharp contrast in behaviour of Yb2Ti2O7 compared to Dy2Ti2O7 and Ho2Ti2O7. It is possible, that due to a significantly smaller magnetic moment, the diffuse scattering signal in Yb2Ti2O7 is an order of magnitude lower than in other Spin Ice compounds and, therefore, it could not be distinguished from the instrument’s noise.

3. Er2Ti2O7: Order by disorder and giant quantum fluctuations

Fig 5. Inelastic Scattering from Er2Ti2O7 measured on PRISMA at 0.05 K, showing “magnetic excitons” near 6-8 meV and quasielastic scattering near 0 meV.

We have shown that Er2Ti2O7 approximates the XY antiferromagnetic pyrochlore. This model is expected to have a macroscopically degenerate ground state, but with a transition to long range order provoked by quantum fluctuations – a so-called order by disorder transition. The magnetic transition to Q=0 state has been found in Er2Ti2O7 at 1.2 K. Single crystal neutron scattering experiments showed liquid like scattering around the ground state (see Fig 5). We believe this can be interpreted as a giant quantum correction to the classical ground state that drives the material close to a quantum critical point.

4. Inelastic Neutron Scattering from Tb2Ti2O7

We have performed inelastic neutron scattering experiments on a powder sample of Tb2Ti2O7 using two ISIS spectrometers, MARI and IRIS. The MARI experiment covered the upper energy range of 0.5 to 40 meV (see Fig. 6), while the IRIS experiment covered the lower energy range of 0.05 to 0.8 meV (see Fig. 7). The experiments revealed the complex structure of the energy levels of Tb2Ti2O7 at low temperature. The spectrum of excitations consists of several low lying branches at 0.1-0.35 meV, as well as of another branch around 1.5 meV with a well pronounced dispersion. There is also a triplet of a crystal field excitations at 10, 15 and 17 meV.

Fig 6. Inelastic Scattering from Tb2Ti2O7 Fig 7. Inelastic Scattering from Tb2Ti2O7 measured on MARI at 12K, showing magnetic measured on IRIS at T=9 K (bottom curve) excitations near 1.5 meV. and at T=2 K (top curve).

(IV) Further Research or Activities

We plan further investigations of diluted pyrochlore magnets. For that purposes we have prepared several single crystals with various composition, concentrating our efforts on the preparation of nonmagnetic pyrochlore, Y2Ti2O7, diluted with magnetic ions of the rare-earth materials.

Fig.8. Field dependence of the microwave power absorption in Tb2Ti2O7, H//(111), recorded at the frequency of 26.93 GHz.

From the results of the inelastic neutron scattering experiment on pyrochlore magnets, and in particular on Tb2Ti2O7, it became apparent that the characteristic energy scale in these magnetic systems fits perfectly with the typical frequencies of the Electron Spin Resonance (ESR) technique. The initial ESR measurements conducted in collaboration with Prof. A.I. Smirnov from Kapitza Institute, Moscow, have revealed the potential of this technique for the future studies of the highly frustrated magnets. By measuring the ESR spectrum it is possible to follow precisely the temperature and/or field dependence of magnetic excitations (see Fig. 8). We plan to apply for the EPSRC funding for the ESR studies of the pyrochlores and other geometrically frustrated magnets in a near future.

(V) Research Impact

We are very pleased with the very high impact of the frustrated pyrochlore magnets project (especially of the Spin Ice component). Apart from producing several highly cited publications, such as 1997 paper on Ho2Ti2O7, which has already been cited more than 40 times, this project has stimulated a renewed interest within the scientific community for what used to be a relatively narrow field of highly frustrated magnets. The Spin Ice materials have recently featured twice in Nature and more than a dozen times in Phys. Rev. Letters. The principal investigators of this project have enjoyed invitations to report their results at several major international conferences.

The concept of magnetic frustration could not possibly be understood in any depth while studying only one or two frustrated system. Close connections and numerous links between different systems, such as frustrated square-lattice AFM, AFM on Kagome, pyrochlore, and garnet lattices are very much evident. Therefore, almost all our other research activities in a field of frustrated magnetic materials are heavily influenced by the work on Spin Ice. This is especially so in the case of our computer-based studies. The following four publications have appeared as a result of our Monte Carlo simulations of magnetic properties of various frustrated magnetic systems: A. Honecker, O.A. Petrenko and M.E. Zhitomirsky, “Field-induced order and magnetization plateaux in frustrated antiferromagnets,” to appear in Physica B (2002). O.A. Petrenko and D. McK Paul “Classical Heisenberg antiferromagnet on a garnet lattice: a Monte Carlo simulation” Phys. Rev. B 63, 024409 (2001). M.E. Zhitomirsky, A. Honecker and O.A. Petrenko, “Field induced ordering in highly frustrated antiferromagnets,” Phys. Rev. Lett. 85, 3269 (2000). O.A. Petrenko and D. McK Paul “Geometric frustration in gadolinium gallium garnet: a Monte Carlo study,” AIP Conference Proceedings 479, 90 (1999). Another publication (G. Balakrishnan, O.A. Petrenko, M.R. Lees and D. McK. Paul, “Single crystal growth of rare earth titanate pyrochlores,” J. Phys. Cond. Matt. 10, L723 (1998)) has appeared as a result of the efforts of the Warwick group to produce single crystals of the pyrochlore magnets for the Spin Ice project.

(VI) Explanation of Expenditure

The expenditure followed the original plan, with only one exception. For scientific reasons we decided to transfer 7 days of the beam-time initially allocated to the inelastic spectrometer, HET, to the PRISMA spectrometer operated in the diffraction mode.