Abstract
The gas of magnetic monopoles in spin ice is governed by one key parameter: the monopole chemical potential. A significant variation of this parameter could access hitherto undiscovered magnetic phenomena arising from monopole correlations, as observed in the analogous electrical Coulomb gas, like monopole dimerization, critical phase separation, or charge ordering. However, all known spin ices have values of chemical potential imposed by their structure and chemistry that place them deeply within the weakly correlated regime, where none of these interesting phenomena occur. Here we use highpressure synthesis to create a new monopole host, Dy_{2}Ge_{2}O_{7}, with a radically altered chemical potential that stabilizes a large fraction of monopole dimers. The system is found to be ideally described by the classic Debye–Huckel–Bjerrum theory of charge correlations. We thus show how to tune the monopole chemical potential in spin ice and how to access the diverse collective properties of magnetic monopoles.
Introduction
In the past two decades, remarkable technical advances have been made in pressure cell technology allowing researchers to carry out highpressure investigations in the fields of chemistry, biochemistry, earth and planetary sciences and condensed matter physics. High pressure is used both in the laboratory and on an industrial scale to produce, for example, artificial diamonds, new superconductors and new forms of matter^{1,2}. Pressure is also used to drive materials into new electronic states. Under high pressure, some materials become superconductors, others undergo magnetic phase transitions, and others undergo metal–insulator phase transitions^{3,4}. In magnetic pyrochlore oxides, pressure has been shown to freeze the spinliquid ground state of Tb_{2}Ti_{2}O_{7} (ref. 5). Pressure is therefore an important weapon in a researcher's arsenal for exploring phase space.
The canonical spin ices, Ho_{2}Ti_{2}O_{7} and Dy_{2}Ti_{2}O_{7}, with magnetic Ho or Dy ions^{6,7,8,9}, are part of the pyrochlore family of oxides of general formula A_{2}B_{2}O_{7} (ref. 10). They have facecentred cubic (fcc) lattice constants of a_{fcc}≈10.1 Å. These materials are not very compressible, and studies with high physical pressure have not revealed any significant modification of the spin ice properties^{11}. Significantly reduced lattice constants can be obtained in principle by replacing Ti^{4+} with a smaller B ion, such as Ge^{4+}, but it is found that at ambient pressure the pyrochlore structure is only stable, if the ratio of the ionic radii, ρ=r_{A}/r_{B}, is less than 1.55 (ref. 10). Dy_{2}Ge_{2}O_{7} and Ho_{2}Ge_{2}O_{7}, with ρ≈1.8, adopt a tetragonal structure, when synthesized under ambient pressure^{12}. The range of stability of the cubic pyrochlore form can be increased using a high pressure, high temperature synthesis, which extends the regime of stability beyond ρ≈1.8 (ref. 13). The cubic pyrochlore form of Dy_{2}Ge_{2}O_{7} prepared in this way has a lattice constant of 9.9290 Å, equivalent to a canonical spin ice under enormous physical pressure^{11}.
The microscopic 'dipolar spin ice' Hamiltonian of Ho_{2}Ti_{2}O_{7} and Dy_{2}Ti_{2}O_{7} includes complex dipolar and superexchange interactions^{14,15}. However, to a good approximation, it may be represented by a much simpler spin Hamiltonian that is equivalent to the original spin ice model^{6,7}. In this description^{9,14}, there are three parameters: the lattice constant a_{fcc}, the rare earth magnetic moment μ, and an effective nearneighbour exchange parameter J_{eff}. The equilibrium statistical mechanics then maps onto the statistical mechanics of idealized water ice, such that the lowtemperature magnetic state is equivalent to the proton disordered state of water ice H_{2}O (refs 6,7), and shares with it the Pauling configurational entropy^{8}. The spin ice state is thus equivalent to pure H_{2}O, and its excitations are equivalent to the ionic defects H_{3}O^{+} and OH^{−} (refs 6,7,16,17). The success of this 'nearneighbour spin ice' description may be attributed to the almost perfect selfscreening of the dipole–dipole interaction between rare earth moments in the effective ground state^{14}.
Although dipole–dipole interactions may be ignored in the spin ice ground state, in ref. 17 it was shown that inclusion of the dipole–dipole interaction in the excited states causes the 'ionic defects' of spin ice to behave as magnetic charges that interact via the magnetic Coulomb law. The description of these defects as magnetic monopoles was firmly established in ref. 16 by approximating the microscopic spin Hamiltonian Ho_{2}Ti_{2}O_{7} and Dy_{2}Ti_{2}O_{7} to a 'dumbbell model', where finite dipoles replace spins. The dumbbell model approximately restores dipolar corrections that are integrated out in the nearneighbour description, but still retain three parameters^{16}. These are the monopole 'contact distance', (the lattice constant of the diamond lattice inhabited by the monopoles), the elementary monopole charge Q=2μ/a (refs 16,17) and the monopole 'selfenergy' ν, that replaces J_{eff} (ref. 16). In ref. 18, the self energy ν was equated with a monopole chemical potential in the grand canonical ensemble.
The monopole system is a magnetic Coulomb gas of deconfined monopoles and antimonopoles with overall charge neutrality, which closely approximates a magnetic electrolyte ('magnetolyte') in the grand canonical ensemble^{18,19,20,21,22,23,24}. Accordingly, experiments on the canonical spin ices reveal strong evidence of the standard field response of such a system, the Wien effect^{19,23,24}, as well as of the applicability of DebyeHückel theory in zero applied field^{21}. In the magnetolyte description of spin ice, the scale of length may be set by the contact distance a and the scale of energy may be set by the Coulomb energy at contact, μ_{0}Q^{2}/4πa. Different spin ices—that is, different triplets {Q, a, ν}—should have identical monopole interaction potentials if energies and lengths are scaled by the above characteristic quantities—so called 'corresponding states' behaviour. Thus, the zerofield magnetolyte properties should be fully controlled by the dimensionless temperature T*(=4πk_{B}Ta/μ_{0}Q^{2}) and the dimensionless monopole density per lattice site x (∝ca^{3}, where c is the concentration). These two parameters, T* and x, map out a phase space which, as mentioned above, is expected to be surprisingly rich.
The x−T* phase behaviour of spin ice has not been determined in detail, but by analogy with electrolyte models^{25,26,27}, we would expect a gradual transition from a weakly correlated magnetolyte at relatively large T*/x to a strongly correlated magnetolyte at small T*/x (Fig. 1). However, in a given spin ice, T* and x cannot be independently varied as x is a function of T* and the chemical potential ν (Methods). Thus, any one spin ice material maps out a single trajectory in the space of x and T*, and existing spin ices are found to be firmly in the weakly correlated regime (Fig. 1). In this regime, the fraction of bound monopoleantimonopole pairs is sufficiently small that it may be neglected for most purposes (the field response being an exception: see ref. 23). Considering, for example, Dy_{2}Ti_{2}O_{7}, the chemical potential, ν, is found to be −4.35 K (ref. 28), which consists (in magnitude) of half the energy required to create a (+−) contact pair, ɛ_{pair}/k_{B}≈5.7 K, plus half the energy required to unbind the pair, μ_{0}Q^{2}/4πak_{B}≈3 K. As the dipole (+−) pair is much higher in energy than the individual (+ or −) charges, the pairing tendency is weak. In contrast, it can be seen that ν≈3 K is the chemical potential that puts monopole–antimonopole pairs or (hetero)dimers at the same free energy as free monopoles, and thus marks the boundary between the weakly and strongly correlated regimes (Fig. 1).
The only way to experimentally approach the strongly correlated regime in Figure 1 is therefore to change the chemical potential ν by changing the energy of pair creation. Fortunately, the latter depends in large part on exchange constants that vary with distance^{16}. Thus a change of lattice constant from 10.1 to 9.93 Å, as achieved by substituting Ti for Ge in Dy_{2}Ti_{2}O_{7} (Fig. 2), may be sufficient to radically alter the chemical potential. We have discovered that this is indeed the case, and that Dy_{2}Ge_{2}O_{7}, which has a much smaller lattice constant than any spin ice hitherto investigated, lies almost precisely on the boundary of the strongly and weakly correlated regimes in Figure 1. This means that it has significant monopole dimerization at all measured temperatures.
It should be noted that the chemical potential and phase behaviour discussed here is not the same as those discussed in refs 16,22 in connection with a fieldinduced phase transition^{29}. Thus the authors of ref. 16 argue that the magnetic field in that case favours ordering of positive and negative monopoles on different sublattices, leading to the phase transition, and use the terminology 'staggered chemical potential' to describe this. Experimental evidence in support of this scenario is presented in ref. 22. However, the chemical potential we refer to is very different in that it tunes the number density of monopoles without favouring any local ordered arrangement, and is thus equivalent to the chemical potential of ions in an electrolyte (which is not true of the 'staggered chemical potential' of ref. 16).
Results
Pauling entropy
Phase pure cubic Dy_{2}Ge_{2}O_{7} and Ho_{2}Ge_{2}O_{7} were prepared and characterized as described in the Methods. Here we describe our results in detail for the Dy compound only, and simply note that we have performed a similar characterization of the Ho compound, which proved to be less interesting in the present context, as it has a more typical chemical potential (Fig. 1). The magnetic entropy determined by integrating the specific heat divided by temperature (c_{m}/T) is shown in Figure 2, where the Pauling residual entropy expected for spin ice^{8} is extremely well reproduced. Magnetometry measurements on Dy_{2}Ge_{2}O_{7} showed this material to have a very similar magnetic moment to Dy_{2}Ti_{2}O_{7}. Incurring negligible error, we henceforth assume that the magnetic moment per Dy is equal in the two materials (9.87 μ_{B})^{15}.
DebyeHückel theory
In Figure 3, we show the measured c_{m}/T plotted against temperature and fitted to DebyeHückel theory with monopole chemical potential ν=(3.35±0.05) K. The method we use has been developed to extend DebyeHückel theory to a good approximation into the hightemperature regime; when applied to Dy_{2}Ti_{2}O_{7} this method gives a similarly good fit to c_{m}(T)/T with a chemical potential of the expected magnitude. It is based on mapping the system to a lattice gas with site exclusion. The lattice gas is considered to have a temperaturevarying chemical potential equal to the sum of the true chemical potential as defined in ref. 28 and the standard DebyeHückel Coulombic correction to the chemical potential^{30}. Without the latter correction, the predicted specific heat (dotted line in Fig. 3) describes the experimental data well in the limit of high and low temperature, highlighting that this approach is a robust method of deriving an experimental estimate of the monopole chemical potential that is not significantly biased by the limitations of DebyeHückel theory. In passing, we note that the origin of the approximate collapse of the experimental data and DebyeHückel calculation onto the ideal lattice gas model at high temperature has a different origin to that at low temperature. In the latter case, the monopole gas is sufficiently dilute that interactions can be neglected, whereas in the former case, a dense and interacting monopole gas reproduces apparent ideal gas behaviour, as the result strong Coulombic screening.
As a further test for consistency, we may use our fitted value of ν to derive a value of the effective nearneighbour coupling constant, J_{eff}, according to the relationship discussed in ref. 16, and then compare this with a J_{eff} estimated from the temperature of the specific heat maximum, as discussed in ref. 31. The result is J_{eff}=(0.62±0.1) K, (0.60±0.1) K, respectively, estimates that are equal, within experimental error. For Dy_{2}Ti_{2}O_{7}, the corresponding value is J_{eff}≈1.1 K, which is roughly twice as large. The large difference is accounted for by a more negative (antiferromagnetic) exchange contribution to the spin–spin interaction in Dy_{2}Ge_{2}O_{7} that opposes the positive (ferromagnetic) dipolar coupling, that is almost the same in the two compounds.
Bjerrum pairing
The measured chemical potential, −3.35 K, puts Dy_{2}Ge_{2}O_{7} in a regime where monopole dimerization should be very significant. To confirm this, we have used the classic theory of Bjerrum^{32}, who separated the contribution of closely spaced charges out of DebyeHückel theory and regarded these as distinct chemical entities to be considered alongside the free charges. In Figure 4, we present the energy u(T) of the system found by integrating the specific heat. The result of DebyeHückel+Bjerrum theory, as described in the Methods, is shown to give an excellent description of the lowtemperature data, without the addition of any new fitting parameters. This result is not in contradiction with the fit of Figure 3, as the modified DebyeHückel theory described in the Methods, including the shortdistance contribution of charge interactions, naturally incorporates the effect of Bjerrum pairs, to an excellent approximation. The monopoles of Dy_{2}Ge_{2}O_{7} are found to be about 50% dimerized at all measured temperatures.
Discussion
We may put these results in the context of the corresponding states diagram (essentially the phase diagram of T* versus x) for the restricted primitive model electrolyte, a basic model of electrolyte behaviour. In the case of a continuum electrolyte, there are three significant boundaries on this diagram marking, respectively, the onset of significant dimerization, the conductance minimum, and phase separation (see, for example, Fig. 1 in ref. 25). In Dy_{2}Ge_{2}O_{7,} we have reached the first of these boundaries for spin ice. To reach the other boundaries would require us to find a spin ice material with ν≪3.3 K. However, a lattice Coulomb gaslike spin ice may show yet more complex phase behaviour in this limit, including chargeordered phases^{26,27}. In fact, the ultimate limit of tuning the monopole chemical potential to ν≪3 K has already been identified through numerical studies on the dipolar spin ice model^{14}. In monopole language, this structure consists of the ordering of 'double charges' ±2Q to give a magnetic structure with '4 spins in/4 spins out' on alternate tetrahedra. This structure (also known as the FeF_{3} structure) becomes stable at ν=2.4 K, J_{eff}=0.2 K. In the unchartered region between ν≈3.3 K and ν≈2.4 K, we would expect to find a great deal of interesting physics associated with increasing monopole correlations and the gradual appearance of double charges. Our results illustrate that this region should be accessible to experiment, as we have shown how highpressure methods afford the opportunity of dramatically altering the chemical potential of magnetic monopoles in spin ice, to the degree where new aspects of monopole physics can be revealed.
Methods
Sample preparation and characterization
Batches of up to 50 mg of the pyrochlore dysprosium germanate, Dy_{2}Ge_{2}O_{7}, were made in a Walkertype, multianvil press. Stoichiometric amounts of Dy_{2}O_{3} and GeO_{2} were ground thoroughly, wrapped in gold foil, compressed to 7 GPa and heated to 1,000 °C. Rietveld refinement of the Xray powder diffraction pattern confirmed the facecentred cubic space group, (Fd3m, No. 227) and the absence of any tetragonal pyrogermanate. The room temperature lattice parameter was determined to be 9.9290(5) Å, (Fig. 2 (inset)). Temperature and fielddependent magnetization measurements confirmed a rare earth magnetic moment of ∼10 μ_{B} and a Curie–Weiss constant of 0.0 K. The heat capacity was measured using a thermal relaxation method between 0.34 and 25 K. The lattice contribution was subtracted from the measured specific heat to reveal the magnetic contribution. The energy u(T) was found by numerical integration of the measured specific heat.
Heat capacity analysis
To calculate the specific heat, the energy per diamond lattice site, u, was written u=νx, where x is the number of monopoles per lattice site: . Here ν_{DH} is the standard DebyeHückel correction to the chemical potential (related to the electrochemical activity coefficient): ν_{DH}/k_{B}T=l_{T}/(l_{D}+a), where l_{T}=μ_{0}Q^{2}/(8πk_{B}T), is the Bjerrum length, is the Debye length, and V_{d} is the volume per diamond lattice site (for a detailed discussion of these quantities, see ref. 23). Using these equations x and l_{D} were determined self consistently, and then the specific heat was found by differentiating u.
To incorporate charge dimers, we considered them as nearneighbour pairs, which is appropriate on a lattice^{26}. Their chemical potential is ν_{d}=2ν−μ_{0}Q^{2}/(4πa), giving the number of pairs per diamond lattice site: x_{B}≈2exp−ν_{d}/(k_{B}T). The DebyeHückel correction was modified to avoid double counting these pairs as follows: ν_{DH}/(k_{B}T)→l_{T}/(l_{D}+2a). The energy was then calculated as: u=ν_{d}x_{B}+νx, which was compared with the measured u(T).
These methods have been comprehensively tested and shown to provide a robust analysis of specific heat data on spin ice materials. They were used to estimate the curves in Figures 1,3 and 4. In Figure 1, the chemical potentials used are: 3.35 K (DyGe), 4.35 K (DyTi), 5.5 K (HoGe) and 5.8 K (HoTi) in an obvious notation; for the Ho materials, these are only rough estimates, owing to the difficulty of accurately isolating the electronic specific heat from the nuclear component^{9}.
Additional information
How to cite this article: Zhou H. D. et al. High pressure route to generate magnetic monopole dimers in spin ice. Nat. Commun. 2:478 doi: 10.1038/ncomms1483 (2011).
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Acknowledgements
This work utilized facilities supported in part by the NSF under Agreement No DMR0944772 and DMR0654118, and the State of Florida. S.T.B. and J.A.B. are grateful to the EPSRC (UK) for financial support. C.R.W. is grateful for support through the Discovery grant program at NSERC and the ACS Petroleum Fund. J.S.Z. and J.B.G. are grateful for financial support from NSF DMR0904282 and Robert A Welch Foundation (No. F1066). L.B. is supported by DOEBES through award DESC0002613. We thank P.C.W. Holdsworth and B.A. Pettitt for numerous useful discussions.
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H.D.Z., C.R.W., J.S.G. and S.T.B. conceived the method; J.S.G. and S.T.B. wrote the paper. S.T.B. derived the theory, which was tested by J.B.; H.D.Z. and J.G.C. prepared the samples. H.D.Z., G.L. measured the bulk properties. L.B., C.R.W., J.B.G. and J.S.J. contributed to the experimental methodology. H.D.Z. and H.J.S. analysed the crystal structure. H.D.Z., G.L., J.S.G. and S.T.B. analysed the heat capacity. All authors contributed to the manuscript.
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Zhou, H., Bramwell, S., Cheng, J. et al. High pressure route to generate magnetic monopole dimers in spin ice. Nat Commun 2, 478 (2011). https://doi.org/10.1038/ncomms1483
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