Néel Temperature of Antiferromagnets for Phase Transitions Driven by Spin-wave Interactions † Andrés

In a recent article,1 a wide variety of phase transitions, with transition (t) temperature Tt , were shown to be usefully characterized by the form   B t char exp 1/ k T E λ   where λ measured the strength of the quasiparticle interactions driving the phase transition. The present article is concerned primarily with antiferromagnets (AFs) having Néel temperature TN. It is first argued that the characteristic energy Echar can be usefully represented by kBθ, where θ is the Curie-Weiss temperature. This assertion is then confronted with experimental data on four insulating transition metal oxides, these being selected as all having the paramagnetic ions on a facecentered cubic lattice. For four of the five θ / TN is certainly greater than unity, the fifth having the ratio as unity to within experimental error. λ is then argued to be related to physical parameters entering a mean field approximation to AFs. Finally, the above insulating Afs are complemented by a brief discussion of a metallic FeRh alloy where, in addition to having itinerant electrons and antiferromagnetism, elevating the temperature leads to a transition from an AF to a ferromagnetic state. (doi: 10.5562/cca2266)


INTRODUCTION
In a recent work, 1 one of us has been involved in treating a variety of phase transitions driven by quasiparticle interactions, including crystalline melting and the ferromagnetic-paramagnetic Curie point.][4] As such alloys were magnetic, more related studies of the ferromagnetic-paramagnetic transitions were made by looking at the spin spirals using first principles calculations.We found that crucial to a proper description of the transition was the softening of magnons for large values of reciprocal vectors. 5Our common papers 6,7 using an Ising model were concerned with antiferromagnets and their transitions to paramagnetic(ferromagnetic) states.We found that it was also important to look at the staggered magnetization on the two sublattices. 7ere we deal specifically with the case of antiferromagnets around the Néel temperature T N , which we proposed should take the form The parameter λ is taken to measure the strength of the spin-wave interactions.E char is argued to be given by k B θ, where θ is the Curie-Weiss temperature and k B is Boltzmann's constant.Some contact is established with known experimental properties of antiferromagnets, for four diatomic insulators where the paramagnetic ion lattice is fcc.Finally, the metallic alloy FeRh is briefly discussed and in particular, a temperature-induced transition from an antiferromagnetic to a ferromagnetic phase.the prototype oxides materials in Table 1.The common feature of the four selected examples is that all have an fcc paramagnetic ion lattice.Recent works on these oxides 9 have also confirmed these Néel temperatures, while the Curie-Weiss temperature θ showed a larger scattering between various experimental measurements.This is also listed, again from experiment, in Table 1, together with the ratio θ / T N .

FOUR INSULATING ANTIFERROMAGNETIC OXIDES WITH FCC STRUCTURE OF THE PAR-AMAGNETIC ION LATTICE
These oxides have a basic superexchange mechanism between the local magnetic moments of 3d traditional magnets, mediated by oxygen ions, similar to GdO. 10 This mechanism in these oxides is sustained by strong localization, but it is competing with other direct exchanges based on hybridization with band formation, as seen in 11 and references therein.
Theories for the Néel temperatures in such compounds have to take into account the spin fluctuations and the change in the local magnetic moments.Within standard density functional theory, an improvement in the descriptions of correlations is needed in order to describe the temperature dependence of the localized magnetic moments.Sometimes as in NiO their survival is strongly dependent on the treatment of correlations.Such theories based on density functional theory study the fluctuations directly with 12,13 or without 14 quantum fluctuations.It is noteworthy that through these fluctuations at the transitions, local magnetic moments remain non-negligible above the transition temperature.Of course, the remanence of local moments is guaranteed using either the previous [12][13][14] or Hartree-Fock 11 results or the mappings onto Heisenberg models for the temperature description.Furthermore, good experimental and theoretical agreement is found for the Néel temperatures of MnO, FeO, and CoO, except for NiO, for which better correlation schemes or dynamical correlations including temperature dependence seem to be needed.As the controversy about these points has not yet been settled at high temperatures, not even the validity of the Heisenberg-like models used, 15 we shall herein develop another approach that should describe the temperature dependence of antiferromagnets with just two sublattices, following ideas going back essentially to Weiss.

AN EXPRESSION FOR NÉEL TEMPERATURES
We begin by comparing the statistical quantities of fluids and magnetic systems, following table 3.5.1 in the book of Chaikin and Lubensky. 16For fluids, we have where n is the number of particles, μ is the chemical potential, κ T is the isothermal compressibility, , and S nn is the long wavelength q limit of the liquid structure factor (see also 1 ).This expression (Eq. 1) has its analogue for the magnetic susceptibility χ where m i and h j denote respectively the magnetization and the field in the i and j directions, while G ij is the order-parameter correlation function.In melts, we have where E char is Ω/κ T with Ω the atomic volume and (0) given by 0.03 exp(−l / λ).In solid-liquid transitions, the parameter λ is related to thermal expansion in the solid, and it is determined in fact by the Grüneisen constant. 1elow, with the assumption that the phase transition at temperature T N is driven by the interaction between anti-ferromagnons (AFM),we shall follow 1 in writing where λ measures the strength of the AFM interac- tions.With respect to Ref. 1 we have that our Eq. 4 is the same as Eq. ( 16) in that reference with γ qp being replaced by 1 / λ.This equation links the transition temperature with the number of magnons.The temperature values are related to a characteristic energy E char .For example, when interaction among modes is negligible, E char will be the average energy of the available excited modes.Similarly to the Debye energy in phonons, E char is given by the Curie-Weiss constant in antiferromagnets.Also there is a dependence on the coupling between modes, given by λ.This coupling expresses both the energy between modes, like anharmonicity, and enters in the exponent of Eq. 4. It is noteworthy that in the limit of vanishing interaction, λ = 0, i.e. the transition temperature goes to zero.However, in the limit of infinitely strong interaction, we have that .λ   The exponential becomes thus unity so that the transition temperature is the characteristic energy E char .
The question of the nature of the characteristic energy E char is then, we believe, the major matter to settle for the four substances listed in Table 1.Early theoretical work summarized in Kittel's book 17 (see Appendix) leads us to the proposal that the characteristic energy should be chosen as where θ is the Curie-Weiss temperature, listed from experiment in Table 1 together with the ratio θ / T N .
Empirically support for the assumption (Eq.5) applied via equation (Eq.4) is then that in all materials in Table 1 the ratio E char = k B T N is greater than unity, as must result from equation (Eq.4).

Trends in Oxides of Late 3d Elements
Then, it clearly follows that there must be a wide spread in the magnitude of the AFM interaction strength measured by λ , in FeO corresponding to 1, λ  while NiO and MnO have AFM strengths increasing from FeO in the above order.So CoO is the only example where one can conclude the strength is less than for FeO.
When comparing all the cases, we found that MnO and NiO behave similarly, while FeO and CoO depart from them.These differences can be explained using the results from previous calculations of the band structures for these oxides. 11On one hand, the oxides MnO and NiO behave similarly because they have half-full or full electronic shells.The basic magnetic mechanism of both compounds is thus superexchange.The difference of NiO with respect to MnO is that, as it has a full electronic shell, the treatment to deal with all the electrons will be largely sensitive to the chosen approach for the correlation.On the other hand, FeO and CoO have open shells, and other magnetic interactions, such as direct exchange, are competing with the superexchange.It seems that the stronger the hybridization with oxygen atoms, the closer is the ratio θ / T N to one.

Meaning of the λ Parameter
At this point, however, we should note that in equation (Eq.4), for the ferromagnetic-paramagnetic Curie temperature T C , it was argued that the spin-wave stiffness D in the dispersion relation in the long wavelength limit, determined k B T C through ~ D / a 2 with a the lattice parameter.We have not followed this, clearly appropriate, choice of E char in ferromagnets for the insulating antiferromagnetic listed in Table 1, since for these materials ω does not go to zero as 0 k  because of the anisotropy field (see Figure 22 on page 344 of Kittel's book, for the spin-wave energy of AFM in tetragonal MnF 2 at 4.2 K observed by neutron inelastic scattering).
In magnetic systems we must also give some physical interpretation for the λ similar to fluids which are driven by phonons as the interacting particles.We now have to rely on magnon-magnon interaction.Well known works 18 have traditionally dealt with spin fluctuations, and they have divided these into transverse and longitudinal susceptibilities.The transverse modes are easily included in oxides, 14 but this is not the case for the longitudinal ones.We feel that the λ constant can be ascribed to how the longitudinal spin fluctuations depart from average values at T = 0; in fact, they could end in spin canted solutions, unless locally and close to the transition.Such assignment compares well with the one given for fluids in terms of the Grüneisen parameter in the solid-liquid transitions.
At elevated temperatures and especially close to the transition temperature we must give up the image of non-interacting magnons.The magnons begin to interact strongly, and we must think more about strong spin fluctuations.Within a mean-field (MF) approach, they could be durable longitudinal oscillations, or beyond MF in the sense that coherent resonances happen in some parts of the sample.We anticipate that the first option is appropriate to describe the FM-AFM transitions while the second one describes probably better the transitions ending in a paramagnetic state.

APPLICATION TO SOME AVAILABLE RE-SULTS ON METALLIC FERH IN AFM PHASE
We next refer to the pioneering discussion of the metallic alloy FeRh as commented in the book on itinerant electron magnetism by Kübler. 19After Ref. 19, we found that the calculated free energies show a crossover from an antiferromagnetic state at low temperatures to a ferromagnetic phase.The temperature of the phase transition is predicted as 435 K, whereas the experimental counterpart is 328 K. Theory 19 gives the magnetic moment of the Fe atoms as about 3 μ B .Symmetry, in fact, dictates that the magnetic moments of Rh vanish in the antiferromagnetic phase already referred to.We have thus a compound with Fe atoms separated by Rh atoms with zero moment.In principle, this compound can be thought as somewhat similar to the oxides except for the strong hybridization between the Rh and Fe atoms.
It is difficult to know whether such a transition is to be described by a formula such as proposed in equation Eq. 4 above, because this expression was designed to treat a 'simple' Néel temperature, in which the transition occurs to a phase with susceptibility of the Curie-Weiss form: namely χ = constant /(T + θ).We feel this formula is more appropriate to describe the curve labeled AM 2 in Figure 5.13 of Kübler's book; this free energy curve intersects the paramagnetic phase at a Croat.Chem.Acta 86 (2013) 463.temperature ≈ 750 K, even though this transition is 'anticipated' by a crossover to ferromagnetism at 435 K, as already mentioned above.Since the theoretical transition AFM-PM (not the experimental) is of first order, the similarity with the oxides is even reinforced.However, to keep the local magnetic moment through the second order transition as in oxides, the AFM-FM transition could also be considered but is beyond the scope of this letter.
In the review article 9 we find the Néel and Curie-Weiss temperatures recorded for alloys made of FeRh 1−x M x where M is a doping material around Rh in the periodic table.For very small values of x, the Mössbauer experiments show the characteristic signal of Fe in its original lattice.Thus, the M elements mostly exchange sites with the Rh atoms, and we can consider these compounds close to FeRh.The small doping FeRh 0.85 Ir 0.15 expands slightly the lattice, but more important it results in the T AFM−FM being higher than T N so that no ferromagnetic phase experimentally appears.In such measurements the Curie-Weiss law can be applied, and the authors find that the θ / T N ratio is close to 1.This value is close to the one found for CoO which presents the strongest hybridization in previous oxides.It also agrees with the strong hybridization underlying the FeRh results commented on above.
This FeRh compound has been recently revisited by Sandratskii and Mavropoulos in Ref. 20 Already in the AFM phase, before the FM state appears, they found that the local magnetic moment in Rh atoms must be remarkably different from zero.More remarkable, as the temperature increases, they show that within the AFM states, Fe local magnetic moments can adopt a non parallel situation in a canted configuration.The non-alignment of Fe atoms allows for Rh magnetization which indeed suffers longitudinal fluctuations.This finding corroborates, and is in perfect agreement with, the interpretation of the λ parameter in previous sections.

SUMMARY AND FUTURE DIRECTIONS
For the insulating AFs in Table 1, we have argued phenomenologically that in equation (Eq.4), which was used in1 for ferromagnets with E char = D/a 2 , where D measures the spin wave stiffness and a is the lattice parameter, the AF case should take E char ~ k B θ, where θ is the Curie-Weiss temperature.However, λ in equa- tion (Eq.4) continues to measure spin-wave interactions.These, we argue, are the quasiparticle interactions which drive the AF-paramagnetic transition in the insulating transition-metal oxides listed in Table 1.The main reason for the ferro-and AF-systems is that for the AF systems (compare MnF 2 in Refs. 17) there is the anisotropy field which can mean that w does not go to zero as k → 0. But λ is still determined by magnon-magnon or spin-wave interactions.
We have also, but more briefly, selected one metallic alloy, FeRh, to compare and contrast with the insulating transition-metal oxides.One obvious difference is that the transition from the AF low temperature phase is now to a ferromagnetic state at a measured temperature of 328 K.This is to be contrasted, in turn with the theoretically predicted value from itinerant electron calculations 19 of 435 K. Symmetry ensures that in FeRh, the magnetic moments of Rh are zero in the AF phase.Hence, we argue for some similarity of this metallic alloy with the oxides in Table 1, except for the extensive hybridization between the metallic Rh and Fe atoms.
For the future, further calculations on FeRh seem called for, with particular attention to be paid to the transition temperature from the AF to the observed ferromagnetic phase.The itinerant electron magnetic treatment of FeRh recorded in Kübler's book 19 due to the first and second neighbor interactions μ and ε that drive the transition.
We begin with the Heisenberg hamiltonian, as given in,11 and following a mean field treatment.The transition Néel temperature is where z refers to the z-component of the spin z i s , S is the total spin, kB is the Boltzmann's constant, and J l is the exchange coupling that runs over first and second neighbors, used as cutoff for the hamiltonian.The Curie-Weiss temperature that enters into the paramagnetic χ = C/(T + θ) law is The value of θ depends on the total number of neighbors in each neighbor shell and it is independent of the magnetic ordering in a given crystal.As in the expansion we have S(S + 1), we are considering quantum fluctuations important for the exact determination of these temperatures.The ratio θ / T N , however, is independent whether we consider classical or spin fluctuations.It depends via the z values on the local sublattices moments and via J l on their coupling which concerns both first and second neighbors.When com-paring with previous values of ε and μ in Kittel's book, we found that they correspond to the sum of first and second neighbors of the products composed of J l couplings by local magnetic moments in configurations ↑↑ and ↑↓ respectively.
Nagamiya,Yosida and Kubo 8 record the experimental Néel temperature T N for some 15 insulating antiferromagnets, and we summarize results for four of Croat.Chem.Acta 86 (2013) 463.
is, of course, available to provide an excellent starting point for future refinements.