University of Pennsylvania ScholarlyCommons

Department of Physics Papers Department of Physics

7-1-1997

Single-Ion Anisotropy, Crystal-Field Effects, Spin Reorientation Transitions, and Spin Waves in R2CuO4 (R=Nd, Pr, and Sm)

Ravi Sachidanandam

Taner Yildirim University of Pennsylvania, [email protected]

A. Brooks Harris University of Pennsylvania, [email protected]

Amnon Aharony

Ora Entin-Wohlman

Follow this and additional works at: https://repository.upenn.edu/physics_papers

Part of the Physics Commons

Recommended Citation Sachidanandam, R., Yildirim, T., Harris, A., Aharony, A., & Entin-Wohlman, O. (1997). Single-Ion Anisotropy, Crystal-Field Effects, Spin Reorientation Transitions, and Spin Waves in R2CuO4 (R=Nd, Pr, and Sm). Physical Review B, 56 (1), 260-286. http://dx.doi.org/10.1103/PhysRevB.56.260

At the time of publication, author Taner Yildirim was affiliated with the University of Maryland, College park, Maryland. Currently, he is a faculty member in the Department of Materials Science and Engineering at the University of Pennsylvania.

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/343 For more information, please contact [email protected]. Single-Ion Anisotropy, Crystal-Field Effects, Spin Reorientation Transitions, and Spin Waves in R2CuO4 (R=Nd, Pr, and Sm)

Abstract We report a detailed study of single-ion anisotropy and crystal-field effects in rare-earth cuprates R2CuO4 (R=Nd, Pr, and Sm). It is found that most of the magnetic properties are mainly due to the coupling between the copper and rare-earth magnetic subsystem which exhibits a large single-ion anisotropy. This anisotropy prefers ordering of rare-earth moments along [100] for R=Pr and Nd and along [001] for R=Sm. Combined with a pseudodipolar interaction arising from the anisotropy of the R-Cu exchange, we can explain the magnetic structures of these materials. The spin reorientation transitions in Nd2CuO4 can be explained in terms of a competition between various interplanar interactions which arises because of the rapid temperature dependence of the Nd moment below about 100 K. Finally we introduce a simple two- dimensional model for the Nd spin-wave spectrum. For zero wave vector, this model gives two optical modes involving Cu spins whose temperature-dependent energies agree with experimental results and an acoustic mode whose energy is predicted to be of order √(2k4Δ)≈5μeV, where k4 is the fourfold in-plane anisotropy constant and Δ is the Nd doublet splitting.

Disciplines Physics

Comments At the time of publication, author Taner Yildirim was affiliated with the University of Maryland, College park, Maryland. Currently, he is a faculty member in the Department of Materials Science and Engineering at the University of Pennsylvania.

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/343 PHYSICAL REVIEW B VOLUME 56, NUMBER 1 1 JULY 1997-I

Single-ion anisotropy, crystal-field effects, spin reorientation transitions, …Nd, Pr, and Sm؍and spin waves in R2CuO 4 „R Ravi Sachidanandam School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

T. Yildirim University of Maryland, College Park, Maryland 20742 and National Institute of Standards and Technology, Gaithersburg, Maryland 20899

A. B. Harris Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396

Amnon Aharony and O. Entin-Wohlman School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel ͑Received 5 February 1997͒

We report a detailed study of single-ion anisotropy and crystal-field effects in rare-earth cuprates R2CuO 4 (RϭNd, Pr, and Sm͒. It is found that most of the magnetic properties are mainly due to the coupling between the copper and rare-earth magnetic subsystem which exhibits a large single-ion anisotropy. This anisotropy prefers ordering of rare-earth moments along ͓100͔ for RϭPr and Nd and along ͓001͔ for RϭSm. Combined with a pseudodipolar interaction arising from the anisotropy of the R-Cu exchange, we can explain the

magnetic structures of these materials. The spin reorientation transitions in Nd 2CuO 4 can be explained in terms of a competition between various interplanar interactions which arises because of the rapid temperature dependence of the Nd moment below about 100 K. Finally we introduce a simple two-dimensional model for the Nd spin-wave spectrum. For zero wave vector, this model gives two optical modes involving Cu spins whose temperature-dependent energies agree with experimental results and an acoustic mode whose energy is

predicted to be of order ͱ2k4⌬Ϸ5␮eV, where k4 is the fourfold in-plane anisotropy constant and ⌬ is the Nd doublet splitting. ͓S0163-1829͑97͒04525-6͔

I. INTRODUCTION ͑SCO͒, coexisting rare-earth magnetism and superconductiv- ity has also been observed.7 Therefore the nature of magnetic

Magnetic interactions in rare-earth (R) cuprate R2CuO 4 interactions which determine the three-dimensional ͑3D͒ ͑RCO͒ systems have been the subject of extensive study1–9 magnetic structure and the correlation between rare-earth for various reasons. First and foremost the R cuprates ͑which magnetism and superconductivity are both of fundamental become superconducting under electron doping͒ have a sim- importance. pler structure than the hole-doped superconducting cuprates. We start by giving a brief overview of some of the mag- 2 netic properties of the RCO systems. An extensive study of In particular, R2CuO 4 crystallizes in the tetragonal structure known as the TЈ phase in which there are no apical O ions. neutron, specific heat, magnetization measurements, Raman, Hence CuO sheets form a planar square lattice. However, it and many more experiments have led to the following con- clusions. First of all, many magnetic properties of the Cu has been observed that for too small or too large rare-earth subsystem in RCO are the same as those in LCO.8 In par- ions, the TЈ phase is not stable, as evident from the distorted ticular, one has ͑1͒ very strong Cu-Cu exchange in the CuO structure of Gd CuO .3,4 Pr CuO PCO is at the limit of 2 4 2 4 ͑ ͒ plane, and ͑2͒ very small Cu-Cu interplane exchange inter- the TЈ phase: the next compound with a lighter R, actions. As a consequence ͑3͒ the antiferromagnetic ͑AFM͒ La 2CuO 4 ͑LCO͒, crystallizes not in the TЈ phase, but in- long-range ordering of Cu spins is characteristic of a 2D stead in the more compressed T-phase, where the out-of- Heisenberg antiferromagnet with weak anisotropies and in- 5 plane oxygens move to apical positions. In this phase there terplanar couplings which lead to a Ne´el temperature, T ,in 10 N is an orthorhombic distortion which allows the existence of the range 250–320 K.9 The fact that these features remain a weak Dzialoshinskii-Moriya interaction, which gives rise the same in the RCO family indicates that the presence of the to weak ferromagnetism.11 Besides the direct structural evi- R subsystem does not significantly modify the properties of dence from x-ray4 and neutron diffraction,6 the absence of the Cu subsystem. For example, the anisotropy of the spin- 4 weak ferromagnetism in the R2CuO 4 system with RϭNd, 1/2 Cu subsystem can obviously not be attributed to a single- Pr, Sm, etc. is clear evidence for the absence of any distor- ion mechanism. Theoretical efforts, culminating in the work tion away from tetragonal symmetry. Second, rare-earth cu- of Refs. 12 and 13 have shown that this anisotropy can be prates exhibit novel magnetic properties involving both the understood as arising from a small anisotropy in the ex- Cu and R subsystems. In the case of Ce-doped Sm 2CuO 4 change tensor due to a mechanism involving the combined

0163-1829/97/56͑1͒/260͑27͒/$10.0056 260 © 1997 The American Physical Society 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 261 effects of Coulomb exchange and spin-orbit coupling. We favors alignment in the plane along25 ͓100͔ for NCO and assume that the CuO planes are not very different in RCO PCO and along ͓110͔ for SCO. than in the tetragonal cuprate Sr 2CuCl 2O 2 ͑SCCO͒ which As mentioned above, we assume that the Cu-Cu interac- has no R ions, so that the anisotropy of the Cu-Cu exchange tions are similar to those in LCO or other cuprates. Next, one is the same in the RCO systems as in SCCO.14,15 may consider the Nd-Cu interactions. It has been suggested22 However, there are several important differences in the that the strongest interaction is a ferromagnetic interaction magnetism and 3D spin structure of rare-earth cuprates and between the Cu ions and the two Nd ions which are its near- other cuprates without a magnetic rare-earth ion, such as est neighbors along the tetragonal axis. In Sec. IV we discuss LCO,8 and SCCO.14,15 Among these differences are the fol- the experimental evidence which implies26 that the dominant lowing. ͑1͒ Although it is now understood that the R ions Cu-Nd interactions are instead those between nearest neigh- exhibit magnetic moments that are mainly induced by the boring Nd and Cu planes. As we shall discuss, these interac- exchange field of the Cu ions,6 the role of the R-R interac- tions cannot be the usual isotropic exchange interactions, be- tions is less well quantified. ͑2͒ Unlike LCO and SCCO, in cause in that case the exchange field on a Nd ion would RCO the Cu spins prefer a noncollinear arrangement16 vanish when summed over the neighboring plaquette of Cu ͑which we will describe in detail below͒. Although it seems ions. Accordingly, it is necessary to consider anisotropic in- clear that this noncollinear structure is due to the presence of teractions, such as dipolar interactions.20 However, the dipo- the R ions, the detailed analysis of the energetics of these lar interaction has the wrong algebraic sign to explain the noncollinear structures on the basis of a microscopic model low-temperature phase of NCO. In any event, the magnitude has not yet been given. ͑3͒ In particular, the sequence of spin of the dipolar interaction is too small to be relevant in this 6 reorientation phase transitions in Nd 2CuO 4 ͑NCO͒ ͑and the context. For NCO it is therefore necessary to introduce a absence of such reorientations in PCO͒ has not been ex- pseudodipolar interaction which results from the anisotropic plained in terms of a microscopic model. ͑4͒ The spin-wave component of the Nd-Cu exchange interaction.21 The domi- spectrum observed in NCO ͑Refs. 17–19͒ has not yet been nance of this interaction implies that a Cu plane together obtained from a microscopic model which is consistent with with the nearest-neighboring Nd planes are tightly coupled. the lowest temperature spin structure and which also cor- We then explain the sequence of spin reorientations in NCO rectly accounts for the temperature dependence of the Cu and the lack of such transitions in PCO in terms of smaller modes at zero wave vector. couplings between adjacent tightly bound units. Within these There have been several theoretical efforts to understand smaller couplings, we infer the existence of competing Nd- these properties. An attempt to explain some of these mag- Nd, Nd-Cu, and Cu-Cu interactions. The rapid temperature netic properties is that of Yablonsky.20 He developed a dependence of the Nd moment has a crucial effect on this theory for the magnetic structure of NCO based on the sym- competition and, with a proper choice of parameters, can metry of the system. He concluded that the noncollinear spin lead to spin reorientation transitions at the observed6 tem- structure was stabilized by biquadratic interactions. Recently peratures. In addition, this result explains the absence of such some of the present authors have developed a theory21 in transitions in PCO at atmospheric pressure. This explanation which the various anisotropic magnetic interactions in the and the inferred dominance of the nearest-neighbor Cu-Nd cuprates can be given a microscopic explanation. From these interactions is the second important result of the present pa- interactions it was possible to have a global understanding of per. the 3D spin structure of various layered magnetic systems The final phenomenon which we address in Sec. V of this but the magnetic structure of NCO remained unexplained. paper is the spin-wave spectrum of NCO. There are two new The spin-wave spectrum of NCO has been the object of sev- ingredients in NCO which are not present in, say, LCO. The eral experimental17–19 and one theoretical investigation.22 As first of these is the existence of low-energy excitations on the a result of these studies one has a reasonable qualitative un- rare-earth sublattices. These excitations will give rise to derstanding of the spectrum. However, as we will discuss, nearly flat optical magnon modes, reminiscent of the analo- there are some inconsistencies in the calculation that should gous rare-earth excitations in the rare-earth iron garnets.27 be removed in order to arrive at a coherent picture of the The second new feature of NCO is the noncollinearity of spin-wave spectrum and its relation to the magnetic struc- both the Cu and Nd moments.16 Another interesting feature tures of NCO. In summary, a detailed consistent microscopic of this system is the existence of a Goldstone mode which explanation of the properties mentioned in the preceding reflects a symmetry of the dipolar interactions with respect to paragraph does not yet exist. It is the purpose of the present a suitable rotation of the moments in the easy plane. When paper to remedy this situation. the fourfold anisotropy which must occur in a tetragonal en- Now we summarize the general features of the micro- vironment is taken into account, this mode will develop a scopic interactions we will invoke in order to explain the small gap. Our model for the calculation of the spin-wave magnetic properties and phases of NCO, PCO, and SCO. In spectrum is somewhat similar to Thalmeier’s22 except that, Sec. III we present detailed calculations of the crystal-field as mentioned, we assume a different Cu-Nd interaction to be states which verify previous work23,24 which showed that dominant than does Thalmeier. Also, because we wish to NCO and PCO have an easy plane perpendicular to the te- reproduce the interesting observed17 temperature dependence tragonal axis. The same approach provides a microscopic of the optical spin-wave modes which involve the Cu spins, explanation for the observation7 that for SCO the tetragonal we introduce a simplified 2D model which includes both Nd axis is an easy axis. This is the first important result of the and Cu spins, rather than assume a static Cu exchange field present paper. We also obtain a systematic treatment of the as Thalmeier does. Our treatment indicates the need for ad- fourfold in-plane anisotropy due to the crystal field which ditional experiments to probe the very low-energy regions of 262 RAVI SACHIDANANDAM et al. 56

FIG. 1. Possible relative orientations of spins in the chemical unit cell of Nd 2CuO 4. Here the open circles are Cu ions and the filled ones R ions. Experiments in a magnetic field ͑Ref. 16͒ show that the actual structures are the noncol- linear ones. We also indicate several of the inter- actions in our models for NCO.

the spin-wave spectrum to locate the Goldstone mode re- and whose existence has been inferred from experiment in a ferred to above. related material.31 As we shall see, such an in-plane anisot- ropy arises naturally in NCO from the much larger single-ion anisotropy of the Nd ion in the crystal electric field of its II. MAGNETIC STRUCTURE OF R CUO 2 4 neighboring ions, as suggested by Yablonsky.20 ͑For SCO ND, PR, AND SM؍R „ … the single-ion mechanism may not be dominant, as we dis- We summarize here various experimental results on the cuss in Sec. III.͒ In the ordered phase the Cu magnetization, structure and properties of the RCO compounds which are i.e., the thermally averaged value of the Cu spin, ͗S͘T , of all relevant to our work. We first discuss the features common the cuprates is the same and can be represented as14,6,32–34 to all these materials. As the temperature is lowered, the 2D AFM correlations between Cu ions, which are well described ␤ by the 2D AFM Heisenberg model,28 grow and, in the pres- ͗S͘TϭB͑1ϪT/TN͒ . ͑1͒ ence of even weak interplanar coupling, lead9 to a phase 32,35,36 transition at a temperature of order TNϷ300 K, below which For many cuprates ␤Ϸ0.25. To reproduce ͗S͘T over there is long-range 3D AFM ordering. ͑The fact that the the entire temperature range for NCO we set ␤ϭ0.3 ͑Ref. magnetic order in some RCO systems is noncollinear, rather 33͒ and take Bϭ0.4. than collinear as in LCO, is not expected to have a signifi- Next we discuss the noncollinear order found in the cant effect on TN .) In contrast to materials like RCO’s. In Fig. 1 we show two forms of noncollinear order 29 K 2MnF 4, the magnetic anisotropy in the cuprates is such and their collinearly ordered counterparts for NCO. In zero that the moments lie in the plane perpendicular to the tetrag- magnetic field the diffraction spectrum of a noncollinear onal axis. In contrast to many materials where such anisot- structure is identical to that from a sample with equal popu- ropy is explained in terms of single-ion anisotropy, here, lations of domains of the two corresponding collinear because the Cu spin is 1/2, this anisotropy has been ex- structures.37 This fact caused some confusion which was re- plained in terms of a Hubbard model in which the combined solved when the application of a symmetry-breaking mag- effect of Coulomb exchange and spin-orbit interactions lead netic field16,33 showed that the noncollinear structures were to a small anisotropy in the exchange interactions between the correct ones for NCO. Apart from field-dependent neighboring Cu ions.30,12,13 In NCO we assume that as far as neutron-diffraction experiments, the strongest evidence for the Cu ions are concerned the picture for LCO remains in- the noncollinear spin structure comes from the single-crystal tact. The fact that the interplanar couplings are stronger in magnetization experiment of Cherny et al.38 They interpreted NCO than in LCO will have only a small effect on the actual their data as showing a first-order phase transition for a field value of TN . In tetragonal SCCO, a small in-plane anisot- H applied along a ͓100͔ direction and a second-order phase ropy in which the ͓100͔ direction for the Cu moments is transition for H applied along a ͓110͔ direction, indicating preferred over the ͓110͔ direction, has been predicted theo- that the easy axis of the magnetization for Cu moments is retically on the basis of zero-point spin-wave fluctuations,21 ͓100͔ in the NCO system. We now summarize the experi- 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 263

FIG. 2. The magnetic unit cell for the magnetically ordered phases I, II, and III of NCO. Note that the magnetic unit cell is twice as large as the chemical unit cell shown in Fig. 1. Also note that the ͓100͔ directions of the chemical unit cell are the diagonals of the square plaquettes shown here. The open circles are Cu ions and the filled ones R ions. Note that in all phases each set of three planes ͑one Cu plane together with its two neighboring Nd planes͒ forms a rigid unit ͑here labeled A and B͒ within which the relative spin orientations remain fixed. In passing from one phase to another the relative orientations of one rigid unit with respect to its neigh- boring rigid unit is reversed. At the far right we indicate the inter- action energies X, Y, and Z, associated with interactions between spins in adjacent sets of planes. In each case, the interactions are those between nearest neighbors of the type in question. FIG. 3. As in Fig. 1. ͑a͒ The magnetic structure of PCO. ͑b͒ The ments which bear on the magnetic structure and single-ion noncollinear structure attributed ͑Refs. 34,47͒ to SCO for TϽ6K. properties of these RCO systems. We also show in ͑c͒ the corresponding collinear structures, since the existing data does not completely exclude them.

A. NCO performed24 and the results have been interpreted in terms of

As the temperature is reduced, at TNϷ255 K the Cu mo- a crystalline electric-field model. In the presence of the ex- ments order in the noncollinear structure of Fig. 1͑a͒͑phase change field from the Cu and Nd ions, the lowest doublet of I͒,atT ϭ75 K the Cu spins reorder with the noncollinear the Nd ions has a splitting in energy, ⌬Ϸ0.32 meV in the Ͼ 4 structure of Fig. 1͑b͒͑phase II͒, and at T ϭ30 K the Cu Tϭ0 limit, as determined by specific-heat measurements. Ͻ 44 spins undergo another reorientation back to phase III which Raman experiments give ⌬ϭ0.35 meV at Tϭ20 K and 45 has the same noncollinear order as phase I, the high- inelastic neutron-scattering measurements give ⌬ϭ0.35 temperature phase.32,16,39 Below 2 K, there is evidence for meV. two more transitions.40 The transition at TϷ1.5 K is attrib- uted to ordering of Nd spins due to the Nd-Nd exchange B. PCO interactions and the one at 0.5 K has been attributed to ‘‘hyperfine-induced nuclear polarization’’ by Chattopadhyay Long-range order of the Cu spins develops below 41 and Siemensmeyer. Nd moments have also been observed TNϷ285 K, with an induced Pr ordering observed at lower below TN , but above 1.5 K, which are supposed to be due to temperatures. The Cu spin structure is a simple antiferromag- the exchange interaction with the Cu moments. At 0.4 K the net in the a-b plane and ͑according to the neutron-scattering 6 Nd moment has been measured to be 1.3␮B . experiment with applied field H along the ͓100͔ direction͒ is In phases I and III the Cu and Nd moments along the z noncollinear with the moments alternating along the ͓100͔ axis are parallel, while in phase II they are antiparallel.42,43 and ͓010͔ directions as one moves along the c axis as shown This implies, as shown in Fig. 2, that the relative orientations in Fig. 3͑a͒.33 The ordered moments for the Cu and Pr spins 6 of the Cu spins within one plane and the Nd spins in the at about 10 K are 0.4 and 0.08 ␮B , respectively. As the nearest-neighboring planes above and below this Cu plane temperature is lowered no further transitions have been ob- are fixed and do not change in going from one phase to served in this system. However, under pressure of 0.25 GPa, another. PCO behaves like NCO in having two spin reorientation A systematic investigation of the crystal-field levels of the transitions.46 At atmospheric pressure a nearest-neighbor ex- Nd ion using inelastic neutron scattering has been change constant Jϭ(130Ϯ30) meV and a spin-wave gap of 264 RAVI SACHIDANANDAM et al. 56

ϳ5 meV was observed, which correspond to the reduced ϭ CEFϪJ•hϵ CEFϩVex , ͑2͒ anisotropy constant ␣ϭ(JϪJ )/Jϳ2ϫ10Ϫ4. A systematic H H H xy where is the crystal electric-field ͑CEF͒ potential and investigation of the crystal-field levels of the Pr ion using HCEF inelastic neutron scattering has been performed23,24 and the Vex is the perturbation due to the exchange field. Our aim results have been interpreted in terms of a crystalline here is not to obtain a complete fit of all spectroscopically electric-field model. determined crystal-field energy levels, but rather to explain the anisotropy of the R ions. Accordingly we restrict our treatment to states in the lowest J multiplet. Within this mul- C. SCO tiplet vectors are proportional to J according to the Wigner- 48 SCO differs significantly from NCO and PCO in several Eckart theorem. Accordingly, we arbitrarily define the ex- of its magnetic properties. As the temperature is reduced change field so that it couples to J and has the dimensions of energy. through TNϷ280 K, the Cu moments order in a structure 1 1 with nonzero ͓ 2 20] neutron Bragg intensity, implying exis- tence of either the noncollinear structure of Fig. 1͑b͒ or its A. Crystalline electric-field Hamiltonian CEF H collinear counterparts as shown in Fig. 1͑d͒.34,47 Neutron- 34,47 The crystal electric-field Hamiltonian CEF , constructed scattering experiments with an applied field along a to be the most general one consistent withH the R-site sym- ͓110͔ direction indicate no hysteresis above 20 K, which is metry, D ,is consistent with the noncollinear spin structure shown in Fig. 4 k 3 1͑b͒ and exclude the possibility of collinear ordering. How- 4␲ 1/2 ever, below 20 K, unlike for NCO and PCO, strong hyster- ϭ A͉m͉ ͗rkY m͑⍀ ͒͘, CEF ͚ ͚ k 2kϩ1 ͚ i k i esis effects were observed. Such effects are not expected for H kϭ2,4,6 mϭϪk ͩ ͪ iϭ1 noncollinear spin ordering ͑but are for collinear ordering͒. ͑3͒ The definitive determination of the spin structure in SCO has where the sum over i is over the three electrons in the un- to wait for a magnetization or neutron-scattering experiment filled 4 f shell and the sum over k is restricted to 2, 4, and with an applied field along a ͓100͔ direction. Such experi- 6 because only these values of k have nonzero matrix ele- ments were performed for NCO and unambiguously demon- ments within the manifold of lϭ3 states of the 4 f shell. In 38,16 strated the noncollinear spin structure of NCO. the sum over m only terms for which ͉m͉/4 is an integer or The second major difference between SCO and other R zero are nonzero, as a result of the fourfold axis of rotation cuprates is the magnetic ordering of the R ions. Above about m about the z axis at the R site. The factors Ak are approxi- 10 K, unlike NCO or PCO, no evidence was found for any mately the same for different rare earths in a given environ- magnetic moment associated with Sm ions. In fact, our cal- ment. It has been shown by Stevens49 that when we restrict culations predict this moment to be much smaller than in attention to a manifold of states corresponding to a single NCO or PCO. So, although in principle this induced Sm value of J, then this potential can be rewritten in terms of the moment must exist, it is apparently too small to be observed J operators. In this case, up to now. However, below 6 K Sm ions exhibit long-range ordering with a spin structure totally different than that of ϭB0O0ϩB0O0ϩB4O4ϩB0O0ϩB4O4 , ͑4͒ HCEF 2 2 4 4 4 4 6 6 6 6 NCO and PCO, as shown in Fig. 3͑b͒. The Sm magnetic m structure consists of ferromagnetic sheets within the a-b where the Stevens operators Ok are planes, with the spins in alternate sheets along the c axis 0 2 O ϭ3J ϪJ͑Jϩ1͒, aligned antiparallel to one another.7 In this phase it is not 2 z established whether the structure is the noncollinear one 0 4 2 2 2 2 O ϭ35J Ϫ30J͑Jϩ1͒J ϩ25J Ϫ6J͑Jϩ1͒ϩ3J ͑Jϩ1͒ , shown in Fig. 3͑b͒ or the collinear one shown in Fig. 3͑c͒. 4 z z z The value of Sm moment at about 2 K was measured to be 4 1 4 4 0.37␮B . As the temperature is lowered, another transition of O4ϭ ͑JϩϩJϪ͒, a continuous nature below 1 K was observed. As mentioned, 2 a very similar transition was also observed in NCO and was O0ϭ231J6Ϫ315J Jϩ1 J4ϩ735J4ϩ105J2 Jϩ1 2J2 attributed to nuclear polarization of the R ions.41 6 z ͑ ͒ z z ͑ ͒ z 2 2 Ϫ525J͑Jϩ1͒Jz ϩ294Jz , III. RARE EARTHS IN RCO Ϫ5J3͑Jϩ1͒3ϩ40J2͑Jϩ1͒2Ϫ60J͑Jϩ1͒, We now calculate the magnetic response of Rϩ3-ions sub- ject to tetragonal crystalline fields and a molecular field gen- 1 O4ϭ ͕͓11J2ϪJ͑Jϩ1͒Ϫ38͑J4 ϩJ4 ͒ erated by the copper spins. Except at temperatures below 6 4 z ϩ Ϫ about 10 K, one may neglect the R-R interactions and their 4 4 2 contribution to the molecular field at a R site. Here we cal- ϩ͑JϩϩJϪ͒͑11Jz ϪJ͑Jϩ1͒Ϫ38͔͖, ͑5͒ culate the thermodynamic properties of the R subsystem at and Bm are the CEF parameters temperatures above, say, 10 K. This calculation will explain k the easy axis of the R magnetization in RCO. We will treat BmϭAm͗rk͘␣ /␤m , ͑6͒ the Cu-R interaction within the mean-field approximation. k k k k Therefore the Hamiltonian for the rare-earth ion in the pres- where ͗rk͘ is the average of rk taken over a 4 f radial wave 50 49 ence of an exchange field h is function, the factor ␣k is the Stevens coefficient, and 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 265

m TABLE I. The crystal-field parameters Bk of Eq. ͑4͒͑in ␮eV͒ for R3ϩ in RCO.

m Bk Nd Pr Sm 0 B2 128 170 Ϫ841.7 0 B4 10 23 Ϫ59.1 4 B4 Ϫ62 Ϫ170 Ϫ539.0 0 B6 Ϫ0.064 0.05 4 B6 Ϫ4.877 7.4

m 51 ␤k is given by Kassman. In Table I we list the values of the B coefficients we obtained from various experimental works24,52 and which we have used for our simplified ͑in that we treat only the ground J multiplet͒ calculations. We have m verified that the corresponding values of Ak obtained using Eq. ͑6͒ do not vary greatly from one R ion in RCO to the next. One sees this from Table II where we list the values of m k k Ak ͗r ͘. ͑The values of ͗r ͘ are the same within a factor of 2 from one R ion to the next.50͒

B. Crystal-field levels 3ϩ The ground-state J multiplet of any rare-earth ion, R ,is FIG. 4. Schematic diagrams of the CEF energy levels and states 53 specified by Hund’s rules. The splitting of the 2Jϩ1-fold of PCO ͑left panel͒, NCO ͑middle panel͒, and SCO ͑right panel͒ in degenerate ground state in the tetragonal crystalline environ- the cubic and tetragonal crystalline-electric field ͑CEF͒. Our ap- ment of the various compounds can be qualitatively studied proximate calculations agree qualitatively with more accurate cal- 54 using group theory. For quantitative results the potential of culations including all multiplets ͑Ref. 24͒. Eq. ͑4͒ is diagonalized to get the eigenstates. For our study ؉34 (11/8؍gJ ,2/9؍ground multiplet J ,6؍L ,2/3؍of the rare-earth anisotropy, we confine our attention to the 1. Nd I9/2 (S lowest J multiplet. Accordingly, ͉M͘ will denote the wave Group theory tells us that in view of the Kramers degen- function ͉J,M͘, where the value of J is implicit. Since one eracy the tenfold degenerate ground multiplet will split into expects that the tetragonal symmetry crystalline electric field five doublets. Diagonalizing the CEF Hamiltonian, we find is not extremely different from what one would have under the five doublets, whose wave functions ͑in order of increas- cubic symmetry, we will show how the energy-level scheme ing energy͒ are compares to the cubic symmetry results of Lea et al. ͑LLW͒.55 In the present case, the cubic CEF Hamiltonian, ͉A1͘ϭ0.482͉9/2͘ϩ0.638͉1/2͘ϩ0.601͉Ϫ7/2͘, ͑8a͒ , assumes the form Hcub ͉A2͘ϭ0.971͉5/2͘Ϫ0.237͉Ϫ3/2͘, ͑8b͒

Wx 0 4 W͑1Ϫ͉x͉͒ 0 4 cubϭ ͑O4Ϫ5O4͒ϩ ͑O6ϩ21O6͒, ͑7͒ A ϭ0.543 9/2 ϩ0.321 1/2 Ϫ0.776 Ϫ7/2 , 8c H F͑4͒ F͑6͒ ͉ 3͘ ͉ ͘ ͉ ͘ ͉ ͘ ͑ ͒ A ϭ0.237 5/2 ϩ0.971 Ϫ3/2 , 8d where F(4)ϭ60 and F(6) assumes the values 2520 and ͉ 4͘ ͉ ͘ ͉ ͘ ͑ ͒ 1260 for Nd and Pr, respectively, and is irrelevant for Sm. ͉A5͘ϭ0.688͉9/2͘Ϫ0.700͉1/2͘ϩ0.192͉Ϫ7/2͘. ͑8e͒ The noncubic tetragonal components, tet , are fixed by the 0 H4 0 4 condition Tr cub tetϭ0tobeOϩ7O and O Ϫ3O . Any We have given only one of the partners in each doublet. The H H 4 4 6 6 tetragonal CEF is thus a unique linear combination of a cubic corresponding energies are shown in Fig. 4, where for com- and a noncubic tetragonal CEF. parison we also show the levels scheme when only the cubic component of the CEF is retained, where we have m k TABLE II. The crystal-field parameters Ak ͗r ͘ in meV for xϭϪ0.554 and WϭϪ1.192 meV. R3ϩ in RCO. Since we will be carrying out perturbative and numerical treatments of the effect of the exchange field on these states, Nd Pr Sm we now discuss them briefly. Using the fact that the time-

0 2 reversal operator, ⌰, acting on an angular momentum eigen- A2͗r ͘ Ϫ40 Ϫ16 Ϫ40.8 0 4 state results in A4͗r ͘ Ϫ280 Ϫ251 Ϫ189 4 4 204 221 206 JϪM A4͗r ͘ ⌰͉J,M ϭ͑Ϫ1͒ ͉J,ϪM , ͑9͒ 0 6 ͘ ͘ A6͗r ͘ 27 13 4 6 A6͗r ͘ 183 173 we find the partner of ͉Ai͘ in the ith doublet, denoted ͉Bi͘,to be 266 RAVI SACHIDANANDAM et al. 56

2J 2 C. Effect of an exchange field on the rare-earth ion ͉Bi͘ϭ͑Ϫ1͒ ⌰ ͉Bi͘ϭ⌰͉Ai͘. ͑10͒ 1. Energy levels in the exchange field Naturally, ͗Bi͉Ai͘ϭ0. Note that even though the doublet is degenerate, we may specify the doublet wave functions We start by discussing the calculation of the energy levels uniquely to within a phase factor by requiring that rotation when the exchange interaction, Vex of Eq. ͑2͒, is treated per- about the z axis by ␲/2 gives back the original wave function turbatively. For the present it is not important which Cu ion with at most an added phase. Note that ͉A͘ and ͉B͘ satisfy is responsible for this interaction. As we shall see, we will this requirement but nontrivial linear combinations of them need to obtain the energy levels correctly up to order h4. do not. First we discuss the calculations for NCO. ͑The calcula- tions for SCO, which is also a Kramer’s ion, were done ؉33 analogously.͒ We consider the energies of the 10 levels of (5/4؍gJ ,4؍ground multiplet J ,5؍L ,1؍Pr H4(S .2 Group theory tells us that the ninefold degenerate ground the Jϭ9/2 state in a D4 CEF and a weak exchange field, h, as expansions in powers of h. Thus we write multiplet will split into two doublets (͉Di͘ and ͉di͘) and five singlets. We find the eigenstates ͑in order of increasing en- E ͑h͒ϭE ͑0͒ϩE ϩE ϩE ϩE ϩ ...ϵE͑0͒ϩEЈ, ergy͒ to be i i i1 i2 i3 i4 i i ͑13͒

͉g͘ϵ͉e0͘ϭ0.707͉2͘ϩ0.707͉Ϫ2͘, ͑11a͒ where Ein is the nth order ͑in h) correction to the Ei(hϭ0) energy due to the exchange field. To develop the ͉d1͘ϭ0.876͉3͘ϩ0.483͉Ϫ1͘, ͑11b͒ perturbation series, we note that the states are doubly degen- erate. To implement perturbation theory we must first diag- onalize the perturbation matrix V within the doublet states. ͉e ͘ϭϪ0.427͉4͘ϩ0.797͉0͘Ϫ0.427͉Ϫ4͘, ͑11c͒ ex 1 Then these eigenstates are used to perform the rest of the perturbation calculation. We now evaluate the matrix ele- ͉e2͘ϭ0.707͉4͘Ϫ0.707͉Ϫ4͘, ͑11d͒ ments of the potential between the various states. We start with the relation ͉D1͘ϭϪ0.483͉3͘ϩ0.876͉Ϫ1͘, ͑11e͒ ⌰J⌰ϭϪJ, ͑14͒ ͉e ͘ϭ0.564͉4͘ϩ0.604͉0͘ϩ0.564͉Ϫ4͘, ͑11f͒ 3 where ⌰ is the time-reversal operator. This along with Eq. ͑10͒ gives us ͉e4͘ϭϪ0.707͉2͘ϩ0.707͉Ϫ2͘. ͑11g͒ ϩ ϩ Ϫ ͗Ak͉J ͉Bl͘ϭ͗Ak͉⌰⌰J ⌰⌰͉Bl͘ϭ͗Al͉J ͉Bk͘, ͑15͒ The doublet partners are ͉d2͘ϭ⌰͉d1͘ and ͉D2͘ϭ⌰͉D1͘. Here when the noncubic contributions to the CEF are ne- where JϮ are the usual raising and lowering operators. From glected one has the LLW parameters xϭ0.807 and the form of the wave functions and the relation above we Wϭ2.051 meV. The eigenenergies are shown in Fig. 4 and have they agree with the observed energies to within an error of ͗A ͉ 20%. This error can be reduced if we include admixtures of k ϩ 24 J ͉͑A ͉͘B ͘)ϭx ͑␴ ϩi⑀ ␴ ͒, ͑16a͒ the higher J multiplets. ͩ ͗B ͉ͪ l l kl x kl y k

؉36 (7/6؍gJ ,2/5؍ground multiplet J ,5؍L ,2/5؍Sm H5/2 (S .3 ͗Ak͉ JϪ͉͑A ͉͘B ͘)ϭx ͑␴ Ϫi⑀ ␴ ͒, ͑16b͒ Group theory tells us that in view of the Kramers degen- ͩ ͗B ͉ͪ l l kl x kl y k eracy the sixfold degenerate ground J multiplet will split into three doublets. We find the eigenstates ͑in order of increas- ͗Ak͉ ing energy͒ to be J ͉͑A ͉͘B ͘)ϭz ␴ , ͑16c͒ ͩ ͗B ͉ͪ z l l kl z k ͉A ͘ϭ0.906͉5/2͘Ϫ0.423͉Ϫ3/2͘, ͑12a͒ 1 where ͉Ak͘,͉Bk͘ are the two members of the kth doublet, defined in Eqs. ͑8͒ and ͑10͒, ␴ are the Pauli matrices, and ͉A2͘ϭ͉1/2͘, ͑12b͒ ⑀ij is given by

⑀ ϭ͑Ϫ1͒iϩj. ͑17͒ ͉A3͘ϭ0.423͉5/2͘ϩ0.906͉Ϫ3/2͘, ͑12c͒ ij In Tables III and IV we list the values of the symmetric with the partners in the doublet given by ͉B ͘ϭ⌰͉A ͘. The i i matrices x and z for NCO which we calculated from the corresponding energies are shown in Fig. 4. When the non- kl kl CEF eigenstates of Eqs. 8 . Tables V and VI contain the cubic components of the CEF are neglected, we have the ͑ ͒ analogous results for SCO based on Eqs. ͑12͒. LLW parameters WϭϪ4.763 meV and xϭ1.0. It is also To get the zeroth-order wave function and the first-order important to note that the lowest excited multiplet Jϭ7/2 lies corrections to the energies we diagonalize at about 150 meV above the ground state,52 compared to 270 57 meV for NCO ͑Ref. 56͒ and 300 meV for PCO. Hence, of Ϫi␾ ͗A ͉ z h x h e all our results, those for SCO are the most likely to suffer k kk z kk Ќ Vex͉͑Ak͉͘Bk͘)ϭ i␾ , ͑18͒ ͩ ͗B ͉ͪ ͩ x hЌe Ϫz h ͪ from not including J-mixing effects. k kk kk z 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 267

TABLE III. The xkl matrix for Nd in NCO. TABLE V. The xkl matrix for Sm in SCO.

12345 123

1 1.886 0.796 0.444 1.803 Ϫ0.358 1Ϫ0.857 Ϫ0.598 0.718 2 0.796 Ϫ1.056 Ϫ1.694 2.033 0.780 2 Ϫ0.598 1.500 1.281 3 0.444 Ϫ1.694 Ϫ1.008 0.395 Ϫ1.205 3 0.718 1.281 0.857 4 1.803 2.033 0.395 1.056 Ϫ1.575 5 Ϫ0.358 0.780 Ϫ1.205 Ϫ1.575 1.622 easy to see that the energy of the ground state ͑or any state for that matter͒ has a linear dependence on the field, and where zkk and xkk are defined in Eqs. ͑16͒, hЌ is the compo- hence it has a component of J along the exchange field given nent of magnetic field perpendicular to the z axis, and ␾ is by x11 . The higher-order terms give the induced contribution the angle between this component and the x axis. We denote to J due to the exchange field. The analogous results for the the zeroth-order eigenstates of Vex by ͉kϩ,0͘ and ͉kϪ,0͘. ground state of Pr ͑which is not a Kramer’s ion͒ are dis- Under ⌰ we have cussed in Appendix C.

͉kϩ,0͘ϭϪ⌰2͉kϩ,0͘ϭ⌰͉kϪ,0͘. ͑19͒ 2. Susceptibilities and in-plane anisotropy

Using these, with ⌰Vex⌰ϭϪVex , we get Having the perturbation expansion for each energy level we can easily obtain expansions in powers of h for the par- E ϭ kϩ,0 V kϩ,0 ϭϪE , ͑20͒ kϩ1 ͗ ͉ ex͉ ͘ kϪ1 tition function, Zϵ͚iexp(Ϫ␤Ei) and then the free energy, FϵϪkTlnZ. For tetragonal symmetry this expansion takes where E is the first order correction to the energy of the 4 kϮ1 the form ͑up to order h ) eigenstate ͉kϮ,0͘. Similarly, 1 2 1 2 4 4 4 Ek nϭEk n , for even n ͑21a͒ F͑h͒ϭF͑0͒Ϫ 2 ␹ʈhz Ϫ 2 ␹ЌhЌϩ␣4͑hx ϩhy ͒ϩ␤4hz ϩ Ϫ 2gJ 2gJ E ϭϪE , for odd n. ͑21b͒ ϩ␥ h2h2ϩ␦ h2 h2 , ͑23͒ kϩn kϪn 4 x y 4 Ќ z We can calculate the various terms that appear in the expan- the coefficients are given in Appendix B in terms of the sion to fourth order in the perturbation theory to obtain coefficients appearing in Eq. ͑22͒. The Lande´ g value ap- pears here because the external field couples to gJJ rather 2 2 2 2 Ei 1ϭ␭iϭͱx h ϩz h , ͑22a͒ than just to J ͓see Eq. ͑2͔͒. Incorporating the isotropic terms ϩ ii Ќ ii z 2 in F0(h ), we may write E ϭa h2 ϩb h2 , ͑22b͒ iϩ2 2i Ќ 2i z 1 1 F͑h͒ϭF ͑h2͒ϩ ͑␹ Ϫ␹ ͒ h2Ϫ h2 0 2g2 Ќ ʈ ͩ z 3 ͪ E ϭ͓a ͑h4ϩh4͒ϩb h2h2ϩc h2h2 ϩd h4͔/␭ , J iϩ3 3i x y 3i x y 3i z Ќ 3i z i 4 4 2 2 ͑22c͒ ϪK4͑hx ϩhyϪ6hx hy ͒ϩ ..., ͑24͒

4 4 2 2 2 2 4 E ϭa ͑h ϩh ͒ϩb h h ϩc h h ϩd h , which defines the fourth-order anisotropy constant, K4, given iϩ4 4i x y 4i x y 4i z Ќ 4i z ͑22d͒ by K4ϭ(␥4Ϫ2␣4)/8. We have used the standard expres- sions ͑given in Appendix B for NCO and SCO and in Ap- 2 2 2 where hЌϭhx ϩhy . Analytic expressions for the coefficients pendix C for PCO͒ to evaluate the susceptibilities for the are given in Appendix A and their numerical values are systems under consideration, and the results are shown in listed in Table VII for NCO and in Table VIII for SCO. Fig. 5. Keeping in mind that the third-order terms are more impor- Some comments on these results are in order. Note that tant than the fourth-order terms ͑these are relevant only in our results for Nd and Pr are very similar to those of the high-temperature expansion of the free energy͒,wecan Boothroyd et al.24 who took account of all J multiplets. Also see from the fact that 2a3,1Ϫb3,1 is positive that the energy it is interesting that at high temperature the anisotropy be- of the ground state is minimized when the exchange field is tween ␹Ќ and ␹ʈ ͑see Fig. 5͒ for Pr in PCO is much larger along the ͓100͔ or ͓010͔ directions. Let hzϭ0. Then it is than for Nd in NCO. This is an unexpected result: one might have thought that Nd must have larger anisotropy, since it TABLE IV. The zkl matrix for Nd in NCO. has a moment whereas Pr has a nonmagnetic ground state.

123 45 TABLE VI. The zkl matrix for Sm in SCO. 1Ϫ0.015 0.000 2.911 0.000 0.864 12 3 2 0.000 2.275 0.000 0.922 0.000 3 2.911 0.000 Ϫ0.729 0.000 2.090 1 1.784 0.000 1.533 4 0.000 0.922 0.000 Ϫ1.275 0.000 2 0.000 0.500 0.000 4 0.864 0.000 2.090 0.000 2.244 3 1.533 0.000 Ϫ0.784 268 RAVI SACHIDANANDAM et al. 56

TABLE VII. The coefficients in the energy expansion for the ith CEF doublet of Nd 3ϩ in NCO. These coefficients are in units such that when the exchange field is in meV, they give the energy contribution in meV. Listed here are the values of 100a2(i), etc.

iϭ1 iϭ2 iϭ3 iϭ4 iϭ5

2 10 a2(i) Ϫ14.1 Ϫ38.5 15.1 30.4 7.1 2 10 b2(i) Ϫ28.8 Ϫ4.5 21.4 4.5 7.4 3 10 a3(i) Ϫ2.1 Ϫ8.7 Ϫ51.0 Ϫ59.7 Ϫ1.9 2 10 b3(i) Ϫ16.6 Ϫ2.5 17.0 Ϫ9.0 0.2 2 10 c3(i) Ϫ1.8 Ϫ14.9 6.5 8.9 Ϫ1.0 3 10 d3(i) 0.0 Ϫ19.0 Ϫ3.2 Ϫ10.6 Ϫ3.4 2 10 a4(i) 0.0 1.05 Ϫ3.81 2.77 0.00 2 10 b4(i) Ϫ1.27 2.87 1.88 Ϫ3.51 0.02 3 10 c4(i) Ϫ4.0 13.1 Ϫ5.4 Ϫ3.8 0.0 3 10 d4(i) 2.10 Ϫ1.45 Ϫ2.14 1.45 0.04

However as we shall see below, the anisotropy within the plane for Nd is much larger than for Pr at temperatures be- low 150 K. The right panel of Fig. 5 shows the susceptibili- FIG. 5. Temperature dependence of the magnetic susceptibility ties of Sm in SCO. Note that they are completely different parallel (␹ʈ) and perpendicular (␹Ќ) to the tetragonal c axis for three RCO’s. Note that the anisotropy ͑between ␹Ќ and ␹ʈ) for from those of Pr and Nd. First of all, for Sm in SCO, ␹ʈ is larger than ␹ . This indicates that Sm moments prefer to lie PCO is actually larger than for NCO at high temperature, which is Ќ an unexpected result. The right panel shows and for SCO. along the ͓001͔ direction. The second major difference con- ␹ʈ ␹Ќ Note that for SCO ͑unlike for NCO or PCO͒ ␹ is smaller than cerns the magnitudes of ␹ and ␹ , which are both much Ќ ʈ Ќ ␹ . Also the magnitude of the susceptibility of Sm is much smaller smaller than in PCO or NCO. Thus it is not surprising that ʈ 7 than that of Pr and Nd. Our calculations agree with those of Refs. experiments show that in SCO the Sm and Cu sublattices 24 and 23 for PCO and of Ref. 24 for NCO. are nearly decoupled. To analyze the anisotropy within the plane, it is necessary mate evaluation of K4 , by numerically calculating the free to study K4.If␹ʈϽ␹Ќ, then the easy axis is a ͓100͔ direc- energy, F100(T) for h along ͓100͔ and F110(T) for h along tion if K4 is positive and is a ͓110͔ direction if K4 is nega- 4 tive. We should also note that the anisotropy at Tϭ0 is eas- ͓110͔ and associating F110(T)ϪF100(T) with 2K4h . ily deduced from the expansion for the ground-state energy The fact that for NCO K4 is positive at all temperatures given in Eq. ͑22͒. For SCO, we determine the easy direction indicates that the Nd moments prefer the noncollinear struc- of the Sm moments when they are constrained ͑as we might assume by their interactions with the Cu ions͒ to lie in the CuO plane. In Fig. 6 we show our calculations of K4 for NCO and SCO done in two ways. At temperatures large compared to the doublet splitting, we evaluated K4ϭ(␥4Ϫ2␣4)/8 using the analytic expressions for these coefficients in Appendix B. We also carried out an approxi-

TABLE VIII. The coefficients in the energy expansion for the ith CEF doublet of Sm 3ϩ in SCO. These coefficients are in units such that when the field is in meV, they give the energy contribu- tion in meV. The doublets are labeled in order of decreasing energy.

iϭ1 iϭ2 iϭ3

2 10 a2(i) Ϫ3.31 Ϫ6.50 9.82 2 10 b2(i) Ϫ6.22 0.0000 6.22 3 10 a3(i) Ϫ1.32 Ϫ3.33 0.06 2 10 b3(i) 0.83 Ϫ7.75 Ϫ4.36 3 10 c3(i) Ϫ7.63 Ϫ3.89 5.22 103d (i) Ϫ7.55 0.00000 Ϫ3.32 3 FIG. 6. The in-plane anisotropy K4(T) for NCO and SCO ͑full 4 10 a4(i) Ϫ1.05 3.81 Ϫ2.76 line͒ calculated numerically, as described in the text, compared to 3 10 b4(i) Ϫ0.06 Ϫ4.94 4.99 the perturbation result ͑dotted line͒ K4ϭ(␥4Ϫ2␣4)/8. For PCO we 3 10 c4(i) Ϫ0.79 1.63 Ϫ0.84 show only the numerical result. The zero-temperature result implied 4 10 d4(i) Ϫ1.85 0.000000 1.85 by Eq. ͑25͒ agrees perfectly with the numerical result. Note that K4 is at least one order of magnitude larger for NCO than for SCO. 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 269 ture in which they are oriented along ͓100͔ directions. ͑This conclusion assumes that the exchange field acting on the Nd spins is a pseudodipolar one, so that as far as such interac- tions are concerned, the collinear and noncollinear structures would have the same energy, see discussion below.͒ For SCO one sees the reverse result. However, one still has to consider the contribution from the anisotropic Cu-Cu ex- change interactions to the anisotropy within the plane. We study this effect in Appendix E for NCO and show that it is dominated by the intrinsic Nd anisotropy due to the CEF acting on the Nd ions. For SCO, where K4 is at least an order of magnitude smaller, and where the value of h is not known, it is possible that the anisotropy due to the anisotropic Cu-Cu exchange could be dominant. Since this anisotropy favors orientation along ͓100͔, we cannot be certain which direction in the CuO plane is favored. Clearly this topic requires fur- ther theoretical and experimental investigation. For PCO we only carried out the perturbative evaluation of K4 at zero temperature, since for a non-Kramer’s ion, the temperature dependences will be less pronounced. The cal- culation of the ground-state energy is simplified by the fact that only a few of the matrix elements of Vex are nonzero. The details of the calculation are given in Appendix C and the final result is

2 2 4 4 2 2 EgϭϪ0.389hЌϪ0.040hz ϩ0.00696͑hx ϩhy ͒ϩ0.0165hx hy Ϫ5 4 2 2 ϩ1.674ϫ10 hz Ϫ0.00447hЌhz , ͑25͒ where Eg and h are in meV. In the notation of Eq. ͑24͒ this Ϫ4 result implies that K4(Tϭ0)ϭ␥4/8Ϫ␣4/4ϭ3.2ϫ10 meV. From Eq. ͑25͒ we see that the terms of order h2 lead to an easy plane and the fact that K4 is positive indicates that ͓100͔ is an easy direction of magnetization. The numerical evaluation of K4 ͑shown in Fig. 6͒ confirms that the essential results are not very different at nonzero temperature. We would point out an interesting behavior of the two Pr FIG. 7. Fit of experimental magnetization of Nd in NCO ͑bot- doublets in the presence of the exchange field. Normally a tom͒ and Pr in PCO ͑top͒ versus temperature ͑solid circles͒ com- doublet will show an energy splitting linear in h. Here, this pared to theoretical fit ͑solid line͒ based on the parameters dis- happens if the field is oriented along the z axis. However, cussed in the text. under normal conditions the exchange field is in the plane, in which case the splitting is proportional to h2. In general, the NCO and PCO. We obtained these data from a least-squares splittings ⌬d and ⌬D between the two states of the doublets fit to a large number of the neutron magnetic Bragg 58 di and Di , respectively, is given by reflections. At about Tϭ2 K, the Nd and Pr moments are 1.3 and 0.09␮B , respectively, in good agreement with other 2 4 1/2 6,33 ⌬aϭ2͑pahz ϩqahЌ͒ , ͑26͒ studies. In order to understand these observed magnetic moments of Nd and Pr we ought to consider the effect of the with exchange fields on the R subsystem due to both the Cu ions and the other R ions. However, for the purpose of this sec- ͗a ⌰͉J ͉e͗͘e͉J ͉a ͘ 2 2 1 x x 1 tion we will consider only the exchange field due to Cu ions. paϭ͗a1͉Jz͉a1͘ , qaϭ ͚ , ͯ e EaϪEe ͯ This is a good approximation at all T for PCO and at TϾ3K ͑27͒ for NCO. Accordingly we write the magnitude of the ex- where aϭd or aϭD labels the doublet. Numerically we find change field h in Eq. ͑2͒, acting on an R ion as Ϫ2 pdϭ4.28, pDϭ0.0046, and in ͑meV͒ , qdϭ0.031, and qDϭ0.25. hϭ␭R͗S͘T , ͑28͒

3. The rare-earth magnetic moments where ͗S͘T , the thermally averaged value of the Cu spin is In this section we discuss the magnetic moments of Nd given in Eq. ͑1͒. We fix the exchange constants ␭R for and Pr within the framework of the crystal-field approxima- RϭNd and Pr by fitting the experimental temperature depen- tion given in Eq. ͑2͒. In Fig. 7 we show the experimental dence of the magnetization shown in Fig. 7. The magnetiza- results of the rare-earth magnetization versus temperature for tion is calculated from 270 RAVI SACHIDANANDAM et al. 56

Ϫ /kT M R͑T͒ϭ͑1/Z͒Tr͑␮e H ͒

ϪEm /kT ϭ͑1/Z͒ ͗⌿m͉␮͉⌿m͘e , ͑29͒ ͚m where Em and ͉⌿m͘ are the energies and associated eigen- functions of in Eq. ͑2͒, Zϭ͚eϪEm /kT, and ␮ is the mag- H netic moment operator divided by ␮B . The fit to the experimental magnetization is excellent, as shown in Fig. 7. The fitted values of ␭Nd and ␭Pr are 0.1772 and 0.3474 meV, respectively. These values correspond to hϭ0.071 and 0.139 meV for Nd and Pr at Tϭ0 K, respec- tively. For Nd hϭ0.071 meV gives rise to a splitting of 0.27 meV for the ground-state doublet and 1.27␮B zero- temperature magnetization, in good agreement with experiments.4,6,44,45 For NCO, as we shall see in the follow- ing sections, the Nd-Nd interactions are also important, par- ticularly at low temperatures, and one has to include them in order to understand the spin waves, etc. Here it is difficult to separate the contribution to Bragg intensities coming from FIG. 8. Nd ions which are nearest neighboring to a plaquette of the nuclear polarization, so the data below, say, Tϭ3 K are Cu ions, labeled 1, 2, 3, 4. The Nd ions in planes just above ͑below͒ not as decisive as in some other experiments. For Pr, the Cu plane are labeled ‘‘ϩ’’ ͑‘‘Ϫ’’͒. hϭ0.139 meV splits the doublets as we discussed perturba- tively in Eq. ͑27͒. A numerical diagonalization gives values We write which differ at the percent level from those predicted pertur- batively. For hϭ0.139 meV we find the numerical values of ij the splitting to be 6.8 ␮eV for the lower energy doublet and Cuϭ ͚ ͚ ␣␤Si␣Sj␤ , ͑32͒ H ␣␤ J 19.2 ␮eV for the upper energy doublet. These small split- ͗ij͘෈C tings may not be observable via inelastic neutron scattering where ␣ and ␤ label spin components, i and j are Cu site within the current experimental uncertainty, but perhaps they labels, and ͗ij͘෈C ͑and later ͗ij͘෈N) indicates a summa- are accessible via other experimental techniques. tion over pairs of nearest-neighboring Cu ͑Nd͒ sites. Now Finally, we point out that, as an alternative to Eq. ͑29͒,an consider the two pairs of Cu sites (1,4) and (1,2) in Fig. 8. excellent fit to the data can be obtained by treating only the The tetragonal symmetry of the lattice implies that lowest doublet. In this approximation, the magnetic moment 14 12 14 12 14 12 per Nd ion ͑in units of Bohr magnetons͒ can be written as xxϭ yyϵ ʈ , yyϭ xxϵ Ќ , and zzϭ zzϵ z , J J J J J J J J J͑33͒ M Nd͑T͒ϭM 0tanh͑⌬/2kT͒, ͑30͒ and all other elements of the tensor are zero. Since the J where ⌬ϭ2␭͗S͘Tx11 . The best fit to the data using this spins prefer to lie in the x-y plane, we know that equation yields ␭Ndϭ0.17 meV, M 0ϭ1.34, and the doublet ʈϩ ЌϾ2 z . A spin-wave analysis ͑see Ref. 12 or Sec. V, splitting ⌬ϭ0.26 meV at Tϭ0 K, which are in reasonable belowJ J͒ allowsJ us to identify the exchange and anisotropy agreement with other experimental values. fields as

IV. MAGNETIC REORIENTATION PHASE TRANSITIONS 1 HEϵ ͑ ϩ Ќ͒ϩ z , IN NCO 2 Jʈ J J A. Model of interactions HAϵ ʈϩ ЌϪ2 z . ͑34͒ In this section we will construct a model which can ex- J J J We now set ϭ . In Appendix E we show that including plain the sequence of spin reorientation phase transitions ob- Jʈ JЌ served in NCO and shown in Fig. 2. The model we will the effect of ʈÞ Ќ has only a very small effect on the results for NCO.J TheJ values of the exchange constants are introduce is a minimal model, in that one can add to it some 1,8 other interactions without modifying its main physical char- fixed by many experiments in the cuprates to be acteristics. Some aspects of this model were already pro- HEϭ130 meV and HAϭ0.1 meV. posed in Ref. 26. The model that we treat is described by a We now discuss the remaining interactions between Cu Hamiltonian, , and Nd ions. An important observation concerning the mag- H netic structure of the three phases of NCO is that all three phases can be considered as being constructed from three- ϭ CEFϩ Cuϩ Cu-Ndϩ Nd-NdϩV, ͑31͒ H H H H H plane ͑Nd-Cu-Nd͒ units ͑labeled A and B in Fig. 2͒. At each where the first four terms describe the Hamiltonian of a reorientation transition the orientation of unit A with respect single three plane unit ͑see Fig. 2͒ and V the coupling be- to that of unit B changes, but each unit remains intact. There- tween adjacent units. We now discuss the terms in this fore, it seems clear that the interactions which hold each unit Hamiltonian in turn. was discussed in Eq. ͑3͒. together are dominant over the interactions between different HCEF 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 271 units. This reasoning indicates that the strongest interaction ␪Sϩ␪JϭϪ␲/2). This ordering is maintained in all three between Cu and Nd ions is that between a plaquette of Cu phases of NCO. The angular dependence of this interaction ions in one plane with the Nd ions directly above ͑or below͒ is the same as that of the dipolar interaction,21 but the inter- the center of the plaquette, as shown in Fig. 8. However, it is action required to stabilize the spin structure of NCO is op- also clear that if this interaction were isotropic, the total ef- posite in sign to that for the dipolar interaction. Hence we fective field on a Nd ion due to a plaquette of Cu ions would call this a pseudodipolar interaction. In any event, the mag- sum to zero. Thus we are led to consider an interaction nitude of the pseudodipolar interaction is much larger that which consists of anisotropic exchange interactions HCu-Nd that for dipolar interactions between the magnetic moments between the Cu spin 1 and its neighboring Nd ion ‘‘ϩ’’ in of the spins. Fig. 8. We write this interaction as To summarize: because the orientations of the Nd spins relative to the Cu spins in adjacent planes do not change as ij one passes through the reorientation transitions, it is reason- Cu-Ndϭ ͚ ͚ ͚ K␣␤Si␣Jj␤ , ͑35͒ H i෈C j෈N ␣␤ able to assume that the interactions discussed above are dominant. Considering only these interactions, one sees that the system naturally condenses into structures in which the where i෈C (i෈N) indicates that the sum over i is over all three plane units ͑labeled A and B in Fig. 2͒ remain intact at Cu ͑Nd͒ sites, and J j␤ is the ␤ component of the angular momentum operator for the Nd ion on site j. We will keep all temperatures. For PCO, the sign of Kxy must be opposite only the symmetric part of this exchange tensor. ͑In tetrago- to that for NCO, because in PCO the relative orientations of nal symmetry the effect of the antisymmetric components the Cu and Pr spins are opposite to what they are in NCO. cancels out when summed over a plaquette of Cu ions.͒ The The actual global spin structure now depends on the smaller existence of a mirror plane ͑passing through sites 1, 3, and couplings between adjacent three plane units. ‘‘ϩ’’͒ implies that the exchange tensor between sites iϭ1 We now consider the Nd-Nd interactions within a three- o and jϭϩ and ͑after rotating by 90 ) between sites iϭ2 and plane unit contained in Nd-Nd . Since these interactions H jϭϩ in Fig. 8 are of the form couple collinear spins, we parametrize them in a slightly simplified way, namely, we set

Kxx Kxy Kxz ϩ,1 K ϭ Kxy Kxx Kxz , ϭ ͓N ͑J J ϩJ J ͒ϩN J J ͔ ͩ K K K ͪ Nd-Nd ͚ Ќ ix jx iy jy z iz jz xz xz zz H ͗ij͘෈N

Kxx ϪKxy Kxz ϩ ͚ ͓MЌ͑JixJjxϩJiyJjy͒ϩMzJizJjz͔ ϩ,2 ͗ij͘ K ϭ ϪKxy Kxx ϪKxz . ͑36͒ Ј ͩ K ϪK K ͪ xz xz zz ϩ ͚ ͓OЌ͑JixJjxϩJiyJjy͒ϩOzJizJjz͔, ͑38͒ ͗ij͘Љ We can generate the exchange tensors for other pairs of near- est Nd-Cu neighbors using the symmetry of the lattice. We now use mean-field theory to discuss the effect of this inter- where ͗͘Ј indicates a sum over nearest-neighboring pairs of action when the spins are constrained to lie in the x-y plane Nd spins whose separation vector is parallel to the z axis, and ͗͘Љ a sum over next-nearest-neighbor pairs of Nd spins and are specified by giving the vector S1ϭϪS2ϵS. The Nd angular momentum of site ϩ is taken to be J and that of the in the same plane. These couplings are indicated schemati- oppositely oriented Nd moment is ϪJ. Then the mean-field cally in Fig. 1. interaction free energy per Nd ion is Finally we consider the interaction V between adjacent three-plane units. Referring to Fig. 2 it is natural to imagine that at high temperature ͑when the Nd moments are very FMFϭ4Kxy͓SxJyϩSyJx͔ϭ4KxySJsin͑␪Sϩ␪J͒, ͑37͒ small͒, the interactions ͑labeled ‘‘Z’’͒ between Cu ions in different units are dominant, whereas at very low tempera- where ␪S (␪J) is the angle S (J) makes with the x axis. ture ͑when the Nd moments are comparable in size to the Cu Several aspects of this result are noteworthy. First, be- moments͒, their interaction ͑labeled ‘‘X’’͒ dominates be- cause of the frustration inherent in an antiferromagnetic cause their separation is much less than the Cu-Cu separa- plaquette, only the anisotropic interaction of Kxy contributes tion. To obtain two spin reorientation transitions we also to the total field at an Nd site. Secondly, the resulting inter- invoke an intermediate strength interaction ͑labeled ‘‘Y’’͒ action has the very unique property that the energy is invari- between Cu ions in one unit and Nd ions in an adjacent unit. ant with respect to rotating one sublattice, say, counterclock- In the case of the X and Z interactions, it is necessary to wise and the other clockwise. This unusual symmetry leads invoke an anisotropic pseudodipolar interaction to avoid a to a Goldstone mode in the absence of a fourfold anisotropy. cancellation in the mean field. Since the Y interaction in- Thirdly, we see that when Kxy ͑which by our definition refers volves only pairs of spins, there is no cancellation and we to the coupling tensor for the pair Cu,1 and Nd,ϩ͒ is posi- take this interaction to be isotropic. Accordingly, we write tive, the orientations of the Nd planes relative to their the perturbation V which couples adjacent three-plane units nearest-neighboring Cu planes are as shown in Fig. 8 ͑with in the following form:59 272 RAVI SACHIDANANDAM et al. 56

B. Mechanism for reorientation transitions NN 2xijyij VϭX ͚ Jx͑i͒Jy͑ j͒⌬ij 2 2 i෈N, j෈N xijϩyij We now consider the perturbative contribution, ␦Fuc ,to the mean-field free energy per magnetic unit cell from the CN coupling V between adjacent three-plane units. In analogy to ϩY ͚ S͑i͒•J͑j͒⌬ij i෈C,j෈N Eq. ͑40͒ we have that 2x y CC ij ij 2 2 ϩZ ͚ Sx͑i͒Sy͑j͒⌬ij 2 2 , ͑39͒ ␦FucϭϪ4␴ZmC͑T͒ ͓1ϩy␺͑T͒ϩ4x␺͑T͒ ͔ i෈C,j෈C xijϩyij 2 ϵϪ4␴ZmC͑T͒ ⌽͑T͒, ͑43͒ where rijϵ(xij ,yij ,zij) is the vector connecting sites i and j, and the ⌬ factors are either 1 or 0 so as to limit the sums where yϭϪY/Z, xϭX/Z, ␺(T)ϭx11mN(T)/mC(T), and ␴ to pairs of sites associated with the coupling constant indi- is ϩ1 in phases I and III and is Ϫ1 in phase II. It is clear CC NN cated in Fig. 2: ⌬ij and ⌬ij are nonzero only if sites i and that the free energy is minimized by the structure of phases I j are nearest possible neighbors in nearest neighboring Cu or and III if Z⌽(T) is positive and by that of phase II if CN Nd planes, respectively, and ⌬ij is nonzero only if sites i Z⌽(T) is negative. In order to obtain phase II between the and j are nearest possible neighbors in next nearest neigh- two reorientation transition temperatures, TϽϭ30 K and boring Cu and Nd planes. The geometrical factor TϾϭ75 K, it is necessary that ZϾ0 and 2 2 xijyij /(xijϩyij) has the transformation properties character- istic of a pseudodipolar interaction between moments con- 1 ␺͑TϾ͒ϩ␺͑TϽ͒ strained to be perpendicular to the tetragonal c axis. xϭ ϭ51, yϭϪ ϭϪ31, In Eq. ͑37͒ we have already identified the mean-field en- 4␺͑TϽ͒␺͑TϾ͒ ␺͑TϽ͒␺͑TϾ͒ ergy due to the Nd-Cu interaction we believe to be dominant. ͑44͒ We now give the mean field free energy per Nd spin associ- ated with those terms in Eq. 31 involving the Nd spin. In ͑ ͒ where we used Eqs. ͑1͒ and ͑30͒ to construct mC(T) and ˆ writing this result we set ͗J(i)•n(i)͘Tϭx11mN(T) for Nd mN(T) which we used to obtain the above numerical values. ˆ 1 ˆ ͑These equations give values which are essentially equiva- spins and ͗S(i)•n(i)͘Tϭ 2mC(T) for Cu spins where n(i)is a unit vector along which the ith moment is aligned. ͓In the lent to experimental ones.͒ We reiterate that the magnetic dipole-dipole interaction does not explain the stability of absence of quantum zero-point effects, mC(Tϭ0) these phases.60 Since m (T) is small at both transitions ͑see ϭmN(Tϭ0)ϭ1.͔ We then find N Fig. 7͒, it is clear that xӷϪyӷ1orXӷYӷZ. As we have mentioned, the plausibility of this condition is obvious from 1 F ϭϪ ͓4K Ϫ␴Y͔x m ͑T͒m ͑T͒ the geometry, shown in Fig. 2. MF 2 xy 11 N C We can also now include the effect of these small pertur- bations on the Nd doublet splitting. Referring to Eqs. ͑40͒ 1 2 2 and ͑42͒ we see that the mean field at the Nd site will have a Ϫ ͑4NЌϪMЌϪ4OЌϩ4␴X͒x mN͑T͒ , ͑40͒ 2 11 jump at each of the two reorientation transitions ͑where ␴ changes sign͒, and indeed the Raman data44 shows such a where ␴ϭ1 for phases I and III and ␴ϭϪ1 for phase II. discontinuity. However, the magnitude of the discontinuity is Within this approximation the splitting of the lowest Nd dou- not easy to obtain from the data, because the data gives di- blet is rectly only the sum of the splittings of the doublets of the initial and final Raman states. In principle, a determination of the jumps in the doublet splitting at these transitions would mN͑T͒ϭ͓4KxyϪ␴Y͔x11mC͑T͒ fix the magnitudes of X, Y, and Z, since their ratios areץ/ FMFץT͒ϭϪ2͑⌬ 2 already fixed by Eq. ͑44͒. In any event the sign of the dis- ϩ͑8NЌϪ2MЌϪ8OЌϩ8␴X͒x mN͑T͒ 11 continuity is not consistent with only magnetic dipole-dipole ϵ⌬Cϩ⌬N , ͑41͒ interactions. It remains to consider what this explanation im- plies for PCO, if we assume that the values of x and y for PCO are the same as those for NCO. Note from Figs. 2 and where ⌬Cϭ(4KxyϪ␴Y)x11 is the part of the splitting of the lowest Nd doublet due to the exchange field of the Cu ions 3 that PCO and NCO ͑in phase I͒ differ in the three-plane units because the Cu spins are reversed in PCO from their and ⌬N is the remaining part of the splitting due to the Nd-Nd interactions. Comparing to Eq. ͑2͒, we see that directions in NCO. That means that to treat PCO we should change the signs of the mC(T)’s. This change is equivalent to changing the sign of Y, or equivalently, the sign of y. hϭ2⌬/x11 . ͑42͒ However, then both x and y are positive and ⌽ is positive at all temperatures and the phase analogous to NCO phase I The term in Eq. ͑41͒ proportional to Kxy is the dominant one. ͓i.e., the actual structure of PCO shown in Fig. 3͑a͔͒ is the The next largest terms are those in M, N, and O, which are stable one. This argument is clearly rather speculative, be- intraunit interactions. The effect of the weaker interactions cause then one would have to assert that under high pressure between three-plane units on the mean-field energy will be ͑when PCO does have a sequence of spin reorientations46͒ neglected. the constant y changes sign. 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 273

V. SPIN WAVES IN NCO A. General discussion In this section we use our model to calculate the spin- wave spectrum of NCO. Although Thalmeier22 has given a very thorough treatment of the spin-wave spectrum of NCO, there are some aspects of his model that we find unsatisfac- tory, as we discuss below. In addition, his basic assumption that the dynamics of the Cu spins can be ignored is only appropriate as long as the wave vector is not too small. Thus his approach, although useful in many respects, is not appro- priate for a discussion of the zero wave-vector modes. For that purpose we have had recourse to a simplified 2D model which enables us to easily take account of the motion of the Cu spins, the details of their anisotropic coupling to the Nd FIG. 9. 2D unit cell ͑indicated by the dashed square͒ for our sublattices, and the anisotropy of the Nd spins caused by the simplified model of spin waves in NCO. The open circles are Cu crystalline electric field. In the future, it may be of interest to ions and the filled ones Nd ions. The darker ͑lighter͒ Nd arrows extend our calculation to a full 3D model. represent the Nd spins in planes just above ͑below͒ the Cu planes. The Cu-Cu interactions are taken to be as in other cu- The Nd-Nd interactions scaled by the tensors M, N, and O are prates. The most important Cu-Nd interaction is fixed by the indicated. high-temperature limit of the Nd doublet splitting. We have fixed the Nd-Nd interactions to get reasonable agreement with the experimental results of Henggeler et al.18 for the 3D model, the spin-wave spectrum has 12 branches and each mode energy is a function of the 3D wave vector, spin-wave energies throughout the Brillouin zone. Particu- 18 larly simple results are obtained at zero wave vector. The qϵ(q2 ,qz). The actual spin-wave spectrum consisting of temperature-dependent energies of the optical modes agree 12 branches has essentially no dispersion with respect to qz well with the experimental values of Ivanov et al.17 We also because the coupling between one Nd-Cu-Nd set of planes predict the energy gap in the acoustic mode due to the small and the next such set of planes is relatively weak. Since the fourfold in-plane anisotropy. 3D unit cell contains two nearly noninteracting sets of The present discussion will assume the structure of phase planes, the 12 branch spectrum at some value of I, although as will be seen, most of our results apply to the qϵ(q2 ,qz) is the union of one six branch spectrum evalu- spectrum in all three phases. As discussed in Sec. IV, we ated at q2 and another six branch spectrum evaluated at o may consider the entire system to be built up of weakly Rq2, where R is a rotation by 90 about the z axis. Therefore interacting sets of planes, each set consisting of a Cu plane almost all information is contained in our simplified 2D with one Nd plane above it and another below it. Thus, for model of one set of Nd-Cu-Nd planes. The unit cell for this most purposes it suffices to consider a 2D model consisting model is shown in Fig. 9. of a single set of Nd-Cu-Nd planes. In this 2D model the The exchange interactions for the model that we treat are spin-wave spectrum has six branches, and the energies of the those described previously in Sec. IV, so that the Hamil- modes are functions of the 2D wave vector q2. In the actual tonian is

ϭ CEFϩ ͚ ͓ Ќ͑SixSjxϩSiySjy͒ϩ zSizSjz͔ϩ͚ ͚ ͚ K␣␤Si␣Jj␤ϩ ͚ ͓NЌ͑JixJjxϩJiyJjy͒ϩNzJizJjz͔ H H ͗ij͘෈C J J i෈C j෈N ␣␤ ͗ij͘෈N

ϩ ͚ ͓MЌ͑JixJjxϩJiyJjy͒ϩMzJizJjz͔ϩ ͚ ͓OЌ͑JixJjxϩJiyJjy͒ϩOzJizJjz͔. ͑45͒ ͗ij͘Ј ͗ij͘Љ

The first line contains the first three terms in Eq. ͑31͒ and the B. Transformation to bosons remaining lines contain the Nd-Nd interactions shown in The transformation to bosons is obtained by the following Figs. 1 and 9. To discuss spin waves we will use the general algorithm for a bilinear interaction involving an op- Holstein-Primakoff ͑HP͒ transformation for the Cu spins and erator R on one site and S on another site. Write a similar transformation to reproduce the dynamics of the Nd spin within the lowest crystal-field doublet. This procedure RSϭ͗R͗͘S͘ϩ͗S͘␦Rϩ␦S͗R͘ϩ␦R␦S, ͑46͒ will lead us to a bosonic Hamiltonian in which terms higher than quadratic in bosonic variables are neglected and in where ͗͘ indicates an average in the mean-field ground state, which quadratic excitations involving higher crystal-field and ␦RϭRϪ͗R͘. By expressing ␦R and ␦S in terms of states are also ignored. bosonic excitations about the mean-field ground state one 274 RAVI SACHIDANANDAM et al. 56 can obtain an expression for the bilinear interaction in terms where ␴␣ is a Pauli matrix and I is the unit 2ϫ2 matrix. In of bosonic variables. ͑In the case of isotropic spins, this pre- Appendix D we develop expressions for the states ͉g͘ and scription is identical to that leading to the HP transforma- ͉e͘ of the lowest doublet in the presence of an exchange field tion.͒ h, as power series in h. We have carried these expansions up There are two sublattices within a copper plane which we to order h3 to obtain results for the constants in Eqs. ͑D7͒– call a and b. In the ground state the a sublattice spins ͗Sa͘ ͑D11͒. The anisotropic response of the Nd ion to a magnetic b point in the ϩx direction, while the b spins ͗S ͘ point in the field is due to the differences in the values of jx , j yϩ , Ϫx direction. The HP transformation may be written as Ϫ j yϪ , and jz . Setting all of them equal to each other ( jxϭ jzϭ j yϩϭϪjyϪ) will make this an isotropic spin-1/2 system. The expansions of the j’s have to be carried to at least second order in h to get anisotropy within the x-y 2 plane. At that order j yϩϭϪjyϪϵjyϭx11ϩO(h ) and Ap- SaϭSϪa†a, SbϭϪSϩb†b, ͑47a͒ x x pendix D gives ( jxϩ j y)/2Ϸ1.886. The anisotropy in the plane is governed by the value of j yϪ jx , which is of order h2 and which is evaluated in Appendix D to be 1.38ϫ10Ϫ4. Thus Eq. ͑48͒ becomes a † b † SyϭͱS/2͑a ϩa͒, SyϭͱS/2͑b ϩb͒, ͑47b͒

J ϭ͗g͉J ͉g͘ϩ͑͗e͉J ͉e͘Ϫ͗g͉J ͉g͒͘n†nϭϪj ϩ2j n†n, a † b † y y y y y y Sz ϭͱS/2͑a Ϫa͒, Sz ϭϪiͱS/2͑b Ϫb͒. ͑47c͒ ͑50a͒

J ϭ e J g n†ϩ g J e nϭ j n†ϩn , ͑50b͒ There are two identically ordered Nd planes, one above, the x ͗ ͉ x͉ ͘ ͗ ͉ x͉ ͘ x͑ ͒ other below the Cu plane. We denote the sublattices above ͑below͒ the Cu plane with moments along the Ϫy direction as nϩ (nϪ) and the other Nd sublattice above ͑below͒ the Cu † † plane as mϩ (mϪ). For the moment we consider a spin in Jzϭ͗e͉Jz͉g͘n ϩ͗g͉Jz͉e͘nϭijz͑n Ϫn͒. ͑50c͒ one of the n sublattices. In the presence of the exchange field due to the other ions its lowest doublet will be split into a ground state ͉g͘ and an excited state ͉e͘. Following the pre- scription given above we write In the m sublattices change the sign of y and z components and replace n with m. The splitting of the doublet ⌬͑when all ions are initially † J␣ϭ͗g͉J␣͉g͘ϩ ͑͗ f͉J␣͉g͘n f ϩ͗g͉J␣͉f ͘n f ϩ͑͗ f͉J␣͉f ͘ in their ground state͒ is ͚f

† † Ϫ͗g͉J␣͉g͒͘n f n f ͒ϩ ͚ ͗ f͉J␣͉f Ј͘n f n f Ј, ͑48͒ f Þ f j 2 2 2 Ј ⌬ϭ 2 j yKxy͗Sa͘ϩ2jy NЌϪ2jy MЌϪ2jy OЌ ͚j ͚j ͚j ͚j † where we define n g ϭ f , and f and f are excited 2 f ͉ ͘ ͉ ͘ ͉ ͘ ͉ Ј͘ ϭ4K j ϩ͑8N Ϫ2M Ϫ8O ͒j states. We henceforth keep only bosonic excitations within xy y Ќ Ќ Ќ y the lowest doublet. Thus in Eq. ͑48͒ the last term is dropped ϵ⌬Cϩ⌬N . ͑51͒ and in the first term the only excited state that enters is ͉e͘ and we let n denote ne . Note that in principle the admixture of higher crystal-field states into ͉g͘ and ͉e͘ is taken into account exactly. However, we did not calculate the moments Here the sums over j encompass the shell of neighbors as- of the Nd ions self-consistently, as this prescription requires. sociated with the exchange interaction in question. This re- The effect of self-consistency is entirely negligible here. sult differs from Eq. ͑41͒ because the interactions between For the n sublattice ͑with Nd spins in the Ϫy direction͒ different three-plane units are not included in the present we have, model. Also here we replace x11 by j y . This replacement has only a small effect numerically, but to treat the R anisotropy correctly we have to include the dependence of the wave functions on h. ͗e͉ 1 1 J͉͑e͉͘g͘)ϭ j ␴ , ͑ j ϩ j ͒Iϩ ͑ j Ϫ j ͒␴ , ͩ ͗g͉ͪ ͫ x x 2 yϩ yϪ 2 yϩ yϪ z C. Spin waves After the above-described transformation to bosons is used, the exchange Hamiltonian becomes Ϫ jz␴yͬ, ͑49͒ 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 275

1 † † 1 † † 1 † † ϭ HEϩ HAϩ2⌬ ͚ apapϩ͚ br br ϩ HA͚ ␥pr͑apbrϩbr ap͒ϩ HE͚ ␥pr͑apbr ϩapbr͒ H ͩ 2 ͪͩ p r ͪ 8 p,r 4 p,r

† † 2 2 † † 2 2 † † ϩ ͕␦ ⌬͑n n ϩm m ͒ϩ␥ ͓͑NЌj ϩN j ͒͑n n ϩn n ͒ϩ͑NЌj ϪN j ͒͑n n ϩn n ͔͒ ͚ ͚ t␣ ,s␣ s␣ s␣ t␣ t␣ t␣ ,s␣ x z z s␣ t␣ t␣ s␣ x z z s␣ t␣ t␣ s␣ ␣ s␣t␣

͑2͒ 2 2 † † 2 2 † † 2 2 † † ϩ␥t ,s ͓͑OЌjxϪOzjz ͒͑ns nt ϩnt ns ͒ϩ͑OЌjxϩOzjz ͒͑ns nt ϩnt ns ͔͖͒ϩ ͓͑MЌjxϪMzjz ͒͑nu nu ϩnu nu ͒ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ͚u ϩ Ϫ ϩ Ϫ

2 2 † † ͑0͒ † ͑1͒ † † ͑0͒ † ͑1͒ † † ϩ͑MЌj ϩM j ͒͑n n ϩn n ͔͒ϩ ͑K a n ϩK a n ϩH. c.͒ϩ ͑K a m ϩK a m x z z uϩ uϪ uϩ uϪ ͚ ͚ p,s␣ p s␣ p,s␣ p s␣ ͚ ͚ p,t␣ p t␣ p,t␣ p t␣ ␣ p,s␣ ␣ p,t␣

ϩH. c.͒ϩ ͑K͑0͒ b†n ϩK͑1͒ b†n† ϩH. c.͒ϩ ͑K͑0͒ b†m ϩK͑1͒ b†m† ϩH. c.͒, ͑52͒ ͚ ͚ r,s␣ r s␣ r,s␣ p s␣ ͚ ͚ r,t␣ r t␣ r,t␣ r t␣ ␣ r,s␣ ␣ r,t␣

where the exchange field, HE , and anisotropy field, HA , 1 1 K ϭ ͑K j ϪK j ͒ϩ iK ͑j ϩj ͒. ͑54b͒ were defined in Eqs. ͑34͒. Also p refers to sites on the a 1 2 xy x zz z 2 xz x z sublattice, r the b sublattice, s␣ the n␣ sublattice, and t␣ the m␣ sublattice, where the subscript ␣ assumes the values ϩ and Ϫ for the Nd sublattices, respectively, above and below the Cu plane. In the third line of the above equation, the sum over u is taken so that uϩ ranges over all sites in the nϩ and The other coupling constants can be obtained using the rela- m sublattices. Also ␦ is unity if uϭv, ␥ is unity if u ϩ u,v u,v tions K(n)(␦ជ )ϭK(n)(Ϫ␦ជ ) and K(n)(␦ជ )ϭK(n)(␦ជ )*. and v are nearest neighbors in the same plane and is zero u u r p To obtain the spin-wave spectrum we introduce spatial otherwise, and ␥(2) is unity if u and v are next-nearest neigh- u,v Fourier transforms via bors in the same plane and is zero otherwise. We now (0) specify the interaction constants Ku,v , where u is a Cu site and v a nearest-neighboring Nd site. For this purpose it is convenient to introduce the notation that vϭuϩa␦ជ , where the x and y components of ␦ជ are each of magnitude 1/2 and † Ϫ1/2 † Ϫiq•r ci ͑q͒ϭNuc ͚ cr e , ͑55͒ (n) (n) r෈i the z component is Ϯ␤. Then we write Ku,vϵKu (␦ជ ). We have

where Nuc is the number of unit cells in the system, r෈i 1 1 1 1 * ͑n͒ ͑n͒ indicates that r is summed over all sites in the ith sublattice, K , ,␤ ϭK Ϫ ,Ϫ ,␤ p ͩ 2 2 ͪ p ͩ 2 2 ͪ and the sublattices are labeled so that 1,2,3,4,5,6 correspond, respectively, to a,b,nϩ ,mϩ ,nϪ ,mϪ . Thus, if r is in the a † † 1 1 1 1 * sublattice, then cr ϭar . With this notation we have ϭϪK͑n͒ Ϫ , ,␤ ϭϪK͑n͒ ,Ϫ ,␤ p ͩ 2 2 ͪ p ͩ2 2 ͪ

ϵKn , ͑53͒ † ϭ͚ ͚ Aij͑q͒ci ͑q͒cj͑q͒ H q ͭ ij where 1 † † ϩ ͚ ͓Bij͑q͒ci ͑q͒cj͑q͒ϩH. c.͔ , ͑56͒ 2 ij ͮ 1 1 K ϭ ͑K j ϩK j ͒ϩ iK ͑j Ϫj ͒, ͑54a͒ 0 2 xy x zz z 2 xz x z where 276 RAVI SACHIDANANDAM et al. 56

1 1 H ϩ H ϩ2⌬ H ͑c ϩc ͒ ͑2K e e ͒Ј Ϫ͑2K e /e ͒Ј ͑2K*e e ͒Ј Ϫ͑2K*e /e ͒Ј E 2 A C 4 A x y 0 x y 0 y x 0 x y 0 y x 1 1 H ͑c ϩc ͒ H ϩ H ϩ2⌬ Ϫ͑2K*e /e ͒Ј ͑2K*e e ͒Ј Ϫ͑2K e /e ͒Ј ͑2K e e ͒Ј 4 A x y E 2 A C 0 y x 0 x y 0 y x 0 x y ϩ Ϫ ϩ Aϭ ͑2K0exey͒Ј Ϫ͑2K0*ey /ex͒Ј ⌬ϩ4O cxcy 2͑cxϩcy͒N M 0 ͑57͒ Ϫ ϩ ϩ Ϫ͑2K0ey /ex͒Ј ͑2K0*exey͒Ј 2͑cxϩcy͒N ⌬ϩ4O cxcy 0 M ϩ ϩ Ϫ 2͑K0*exey͒Ј Ϫ͑2K0ey /ex͒Ј M 0 ⌬ϩ4O cxcy 2͑cxϩcy͒N * ϩ Ϫ ϩ ΄ Ϫ͑2K0 ey /ex͒Ј ͑2K0exey͒Ј 0 M 2͑cxϩcy͒N ⌬ϩ4O cxcy ΅

1 0 H ͑c ϩc ͒ ͑2K e e ͒Ј Ϫ͑2K e /e ͒Ј ͑2K*e e ͒Ј Ϫ͑2K*e /e ͒Ј 2 E x y 1 x y 1 y x 1 x y 1 y x 1 H ͑c ϩc ͒ 0 Ϫ͑2K*e /e ͒Ј ͑2K*e e ͒Ј Ϫ͑2K e /e ͒Ј ͑2K e e ͒Ј 2 E x y 1 y x 1 x y 1 y x 1 x y Ϫ ϩ Ϫ Bϭ ͑2K1exey͒Ј Ϫ͑2K1*ey /ex͒Ј 4O cxcy 2͑cxϩcy͒N M 0 ,͑58͒ ϩ Ϫ Ϫ Ϫ͑2K1ey /ex͒Ј ͑2K1*exey͒Ј 2͑cxϩcy͒N 4O cxcy 0 M Ϫ Ϫ ϩ 2͑K1*exey͒Ј Ϫ͑2K1ey /ex͒Ј M 04Ocxcy2͑cxϩcy͒N * Ϫ ϩ Ϫ ΄ Ϫ͑2K1ey/ex͒Ј͑2K1exey͒Ј0 M2͑cxϩcy͒N4Ocxcy΅

where cxϭcosaqx , cyϭcosaqy , exϭexp(iaqx/2), 90° about the z axis ͑so that the a sublattice spins now point Ϯ 2 2 eyϭexp(iaqy/2), (X)ЈϵReX, and X ϭ(XЌ jx ϮXz jz ), where in the Ϫy direction, and the b sublattice in the ϩy direction, X stands for M, N,orO. The eigenvalues of the matrix etc.͒. From Eqs. ͑57͒ and ͑58͒ one can see that when K0 is (AϩB)(AϪB) give the squares of the energy of the normal real ͑i.e., when Kxzϭ0), the spectrum is invariant under this modes: R4 operation. Even when Kxz is nonzero, this invariance holds for wave vectors in high-symmetry directions. Thus in AϩB AϪB q ϭ q 2 q . 59 ͑ …„ ͒␹␶͑ ͒ ␻͑ ͒ ␹␶͑ ͒ ͑ ͒ the complete model the spectrum consists of six nearly dou- The eigenvalues are invariant with respect to the operation bly degenerate modes which are split by weak couplings q Ϫq, as expected in view of time-reversal symmetry. To between adjacent three-plane units. see→ this explicitly note that changing the sign of q is equiva- lent to interchanging rows and columns 3 and 5 and rows and and on the zone boundary 0؍columns 4 and 6. D. Normal modes at q The complete model for the whole lattice will have two layers of our 2D model per unit cell, with one rotated by For qϭ0, we have the simpler forms

1 1 H ϩ H ϩ2⌬ H 2KЈ Ϫ2KЈ 2KЈ Ϫ2KЈ E 2 A C 2 A 0 0 0 0 1 1 H H ϩ H ϩ2⌬ Ϫ2KЈ 2KЈ Ϫ2KЈ 2KЈ 2 A E 2 A C 0 0 0 0 ϩ Ϫ ϩ Aϭ 2K0Ј Ϫ2K0Ј ⌬ϩ4O 4N M 0 , ͑60͒ Ϫ ϩ ϩ Ϫ2K0Ј 2K0Ј 4N ⌬ϩ4O 0 M ϩ ϩ Ϫ 2K0Ј Ϫ2K0Ј M 0 ⌬ϩ4O 4N Ϫ2KЈ 2KЈ 0 M ϩ 4NϪ ϩ4Oϩ ΄ 0 0 ⌬ ΅ 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 277

0 HE 2K1Ј Ϫ2K1Ј 2K1Ј Ϫ2K1Ј

HE 0 Ϫ2K1Ј 2K1Ј Ϫ2K1Ј 2K1Ј Ϫ ϩ Ϫ 2K1Ј Ϫ2K1Ј 4O 4N M 0 Bϭ ϩ Ϫ Ϫ . ͑61͒ Ϫ2K1Ј 2K1Ј 4N 4O 0 M Ϫ Ϫ ϩ 2K1Ј Ϫ2K1Ј M 04O4N Ϫ2KЈ2KЈ0 MϪ 4Nϩ4OϪ ΄ 1 1 ΅

We can immediately identify several eigenmodes. For in- measure of͒ the Nd-Nd interactions. One of the other modes stance ͉1͘ and ͉2͘ are given, respectively, by the upper and is an out-of-plane Cu mode which would have energy lower choices of sign in (nϩϩmϩϮnϪϮmϪ)/2. The energy Ϸͱ2HAHE in the absence of the Nd ions. The Nd ions con- of these modes is tribute a staggered field of energy 2⌬C , so that HA is here replaced by HAϩ2⌬C . Another mode is an in-plane optical ␻2 ϭ͑⌬ϩ4NϪϩ4OϩϮM ϩ͒2Ϫ͑4Nϩϩ4OϪϮM Ϫ͒2 Ϯ mode in which the staggered field is 2⌬C but this mode does 2 2 not involve the out-of-plane anisotropy, HA . This mode has Ϸ⌬ ϩ2⌬jx͓4OЌϩ4NЌϮMЌ͔. ͑62͒ energy ␻Ͼ given in Eq. ͑66͒. Finally, there is an acoustic The other simple rare-earth mode is ͉3͘ϭ(nϩϪmϩ mode which involves the fourfold anisotropy. We may define ϪnϪϩmϪ)/2, with a phenomenological fourfold anisotropy constant, k4 via

2 ϩ Ϫ ϩ 2 Ϫ ϩ Ϫ 2 ␻3ϭ͑⌬ϩ4O Ϫ4N ϪM ͒ Ϫ͑4O Ϫ4N ϪM ͒ 1 EϭϪ k4cos4␾, ͑68͒ 2 2 8 Ϸ⌬ ϩ2⌬jx͓4OЌϪ4NЌϪMЌ͔. ͑63͒ These three modes have energy which is split from ⌬ by the where E is the ground-state energy and ␾ is the angle in the Nd-Nd interactions. x-y plane which the exchange field makes with a ͓100͔ di- The other three modes involve the Cu spins. One of these rection. We will identify k4 by finding the ground-state en- ergy for small ␾ when the exchange Hamiltonian is is the out-of-plane Cu mode: ͉4͘ϭ(aϩb)/ͱ2. For it

2 2 2 ␻4ϭ͑HEϩHAϩ2⌬C͒ ϪHEϷ2HE͑HAϩ2⌬C͒. ͑64͒ The remaining two modes are linear combinations of excita- tions on the Cu, ͉5͘ϭ(aϪb)/ͱ2 and on the Nd, ͉6͘ϭ(nϩϪmϩϩnϪϪmϪ)/2. In this subspace we have

HEϩ2⌬C 2ͱ2͑KxyjxϩKzzjz͒ Aϭ 2ͱ2 K j ϩK j ͒ ⌬ϩF ϩF , ͫ ͑ xy x zz z Ќ z ͬ

ϪHE 2ͱ2͑KxyjxϪKzzjz͒ Bϭ 2ͱ2 K j ϪK j ͒ F ϪF , ͑65͒ ͫ ͑ xy x zz z Ќ z ͬ 2 where FЌϭ(MЌϪ4NЌϩ4OЌ) jx and Fzϭ(M zϪ4Nz 2 ϩ4Oz) jz and we used Eqs. ͑54͒. We denote the two eigen- 2 2 values of the matrix (AϩB)(AϪB)as␻Ͼ and ␻Ͻ and find

2 ␻ϾϷ4HE⌬C . ͑66͒ 2 FIG. 10. Full curves are the energies ͑at Tϭ0) of the low- Neglecting terms of order jz and using the values of the parameters given in Sec. V G, below, we have energy modes with respect to the 2D wave vector calculated using the values of the parameters as given in Eqs. ͑84͒–͑88͒. In the full 2 1/2 3D model, each of these modes gives rise to two modes whose jx ␻ ϭ⌬ 1Ϫ Ϸ3.7 ␮eV. ͑67͒ splitting is determined mostly by the small coupling between adja- Ͻ ͩ j2 ͪ y cent sets of three planes. This coupling is neglected in the 2D model. The squares, circles, and triangles are mode energies deter- We can understand these results in the following manner. mined by the inelastic neutron experiments of Ref. 18. ͑The data of Since the Nd’s mix with the Cu’s only via the uniform Nd Ref. 19 is similar to that shown here.͒ Note that the calculations excitation (͉6͘), we have three Nd modes whose energy predict a strong dispersion of the acoustic mode at small wave (␻k for kϭ1,2,3) is approximately ⌬. The energy differ- vector. At the far right we show the density of states ͑DOS͒ ob- ences between these modes is caused by ͑and therefore is a tained from an evaluation over the entire 2D Brillouin zone. 278 RAVI SACHIDANANDAM et al. 56

ˆ ˆ VexϵϪ4Kxyn•J with nϭ(sin␾,cos␾,0). Thus 2 ␾ ͉␾ϭ0. Using the matrix elements of the doubletץ/Eץk4ϭ given in Eq. ͑50a͒, we write the exchange Hamiltonian in terms of Pauli matrices within the lowest doublet as

Vexϭ4Kxy͓Ϫjycos␾␴zϪsin␾ jx␴x͔. ͑69͒ For small ␾ the ground-state energy is

1 E ϭ4K Ϫj 1Ϫ ␾2 Ϫj2␾2/͑2j ͒ . ͑70͒ 0 xyͫ yͩ 2 ͪ x y ͬ This result suggests that we make the identification

2 2 Ϫ5 k4ϭ͑⌬/2͒͑1Ϫ jx / j y ͒Ϸ⌬͑1Ϫjx /jy͒ϭ7.3ϫ10 ⌬ ͑71͒ and we write

2 Ϫ4 2 ␻Ͻϭ2k4⌬Ϸ1.5ϫ10 ⌬ . ͑72͒ In the absence of the fourfold anisotropy the acoustic mode at zero wave vector would have zero energy. This follows from the combined effects of two symmetries. First of all, the diagonal components of the exchange tensor do not con- tribute to the energy of this mode because these Cu-Nd in- teractions are completely frustrated. Secondly, as we saw in Eq. ͑37͒, the pseudodipolar energy is invariant with respect FIG. 11. As in Fig. 10, the full lines are our calculations of the to rotating the Cu and Nd spins through the same angle, but temperature dependence of the spin-wave modes in NCO at zero in an opposite sense. Obviously, introduction of k4 breaks wave vector. In the top panel we show the three modes at zero wave this symmetry and leads to a nonzero acoustic mode energy. vector which do not involve motion of the Cu spins. These energies We can also obtain simple results for wave vectors on the essentially are proportional to the moment of the Cu spins. In the zone boundary, i.e., for a(qxϩqy)ϭ␲. Along that line bottom panel we show the three modes at zero wave vector which cyϭϪcx and the matrices A(q) and B(q) for the Nd sector involve motion of the Cu spins. The experimental points of Ivanov break up into two identical 2ϫ2 matrices. Neglecting the et al. ͑Ref. 17͒ for two of these modes are shown by filled and open very small effect of the coupling to the high-energy Cu circles. The bottom curve is 200␻ac . This mode has not yet been modes, we thereby find the spin-wave energies to be observed.

2 ϩ 2 ϩ 2 Ϫ 2 Ϫ 2 ␻Ϯϭ͑⌬Ϫ4O cx ϮM ͒ Ϫ͑4O cx ϯM ͒ . ͑73͒ ͑47a͒ one replaces S by ͗S͘T , for which we use Eq. ͑1͒. One 1 sees that S ͑which we had previously set equal to 2͒ is now It is a good approximation to set jzϭ0, in which case renormalized by a factor ␰C(T)ϵ2͗S͘T , whereas the trans- 2 2 2 2 2 verse components of the Cu spin, which are proportional to ␻Ϯϭ⌬ Ϫ8⌬OЌ jx cx Ϯ2⌬MЌ jx . ͑74͒ ͱS, are renormalized by a factor ͱ␰C(T). We follow the Approximately, therefore, we have two doubly degenerate same rule for the Nd spin in terms of a factor ␰N(T) where, Nd modes on the zone boundary with energies given by neglecting quantum zero-point motion, one has ͓see Eq. 2 2 2 ͑30͔͒ ␻ϮϷ⌬Ϫ4OЌjxcxϮMЌjx . ͑75͒ ␰N͑T͒ϭtanh͓⌬/͑2kT͔͒. ͑76͒ E. Normal modes at arbitrary wave vectors Thus the temperature dependence of the spin-wave matrices We have evaluated energies of the normal modes for is obtained from Eqs. ͑57͒ and ͑58͒ by the replacements wave vectors in various high-symmetry directions from Eq. ͑59͒. Results for the low-lying ͑Nd͒ modes for selected val- HE HE␰C͑T͒, ⌬C ⌬C␰N͑T͒, ues of the parameters are shown in Fig. 10. We also show the → → density of spin-wave states for energy up to 0.8 meV. K K ͓␰ ͑T͒␰ ͑T͔͒1/2, n→ n C N Ϯ Ϯ F. Temperature dependence of normal modes ⌬ ⌬C␰C͑T͒ϩ⌬N␰N͑T͒ϵ⌬͑T͒, X X ␰N͑T͒. → → ͑77͒ An approximate treatment of the temperature dependence of the mode energies is based on a generalization of the ͓Note that where ⌬C appears it actually represents the ex- random-phase approximation. Within this approximation, as change field acting on the Cu ions due to the Nd moments developed for spin systems, one replaces Sz ͑where z is the and hence it is renormalized by a factor ␰N(T).͔ In this for- direction of long-range magnetic order͒, by its thermally av- mulation we treat HA as a temperature-dependent parameter eraged value, ͗Sz͘T . For instance, in the relations of Eq. and although our prescription indicates that k4(T) is propor- 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 279

TABLE IX. Values for the contributions to the splitting in energy of the lowest Nd doublet from various T interactions according to mean-field theory. In columns labeled hN we list values ͑at Tϭ0) associated with Ќ the interactions In of Thalmeier ͑Ref. 22͒ compared to the corresponding values hJ from the interaction J in our theory. The first four columns refer to the exchange field due to Cu spins and the remaining columns to the exchange field due to other Nd spins. Thalmeier’s theory has no splitting analogous to ours due to Kxy ͑so we leave the first column of the table blank͒. Since we have not made any numerical estimates of X and Y, we leave their entries blank. Entries in the last row are all in meV. The total Nd doublet splitting, as defined by Eq. ͑41͒,is⌬ϭ0.41 meV using our parameters and 0.44 meV using Thalmeier’s.

Cu Nd

T – hK hCu hY h1 h M h2 hX h3 hN h4 hO 1 Ќ 2 Ќ 2 Ќ 2 –4Kxyjy hCu ϪY 2I1 Ϫ2MЌ j y 08XϪ2I3 8NЌjy 2I4 Ϫ8OЌjy – 0.57 0.59 Ϫ0.21 Ϫ0.18 0 0.14 0.11 Ϫ0.08 Ϫ0.09

tional to ⌬(T), it may be more realistic to also regard the values of the Nd crystal-field matrix elements, jx , j y , k4(T) as a temperature-dependent parameter. Following the and jz were calculated in Appendix D. We summarize these: same analysis as for Tϭ0, we now find that the results of Eqs. ͑62͒–͑64͒ become HEϭ140 meV, HAϭ0.1 meV, ͑ jxϩ j y͒/2ϭ1.886,

2 2 j Ϫ j ϭ1.4ϫ10Ϫ4, j ϭϪ0.015. ͑84͒ ␻ϮϷ͓⌬C␰C͑T͒ϩ⌬N␰N͑T͔͒ ϩ2͓⌬C␰C͑T͒ y x z

2 We note the very small value of jz . Considering this, the ϩ⌬N␰N͑T͔͒␰N͑T͒jx͓4OЌϩ4NЌϮMЌ͔, ͑78͒ values of the exchange parameters which involve jz have 2 2 little influence on the results. Therefore, we have set ␻3Ϸ͓⌬C␰C͑T͒ϩ⌬N␰N͑T͔͒ ϩ2͓⌬C␰C͑T͒

2 NzϭM zϭOzϭ0. ͑85͒ ϩ⌬N␰N͑T͔͒␰N͑T͒jx͓4OЌϪ4NЌϪMЌ͔, ͑79͒ and We now fit the other parameters by comparing with the observed spectrum of Nd spin excitations in NCO.18 In mak- 2 ing this comparison we note that the experiment shows more ␻4Ϸ2HE␰C͑T͓͒HAϩ2⌬C␰N͑T͔͒. ͑80͒ than four low-energy spin-wave modes. This observation in- The energy of two mixed modes assumes a simple form in dicates that our assumption that the interaction between ad- the low-temperature limit ͓when HE␰N(T)ӷ⌬C␰C(T)͔: jacent three-plane units is negligible, is not totally correct, at least in this context. However, the dispersion along q is 2 2 z ␻optϷ4⌬CHE␰N␰C , ␻acϷ2k4͑T͓͒⌬͑T͒ϩ2Fz␰N͑T͔͒, small, in conformity with our assumption. As a result, the ͑81͒ comparison ͑shown in Fig. 10͒ of our simple 2D model with where the actual data is somewhat approximate. However, we do seem to capture the main physical effects with our simple 2D 1 model. To estimate the numerical values of the parameters, 2 k4͑T͒ϭ ⌬͑T͓͒1Ϫ͑ jx / j y͒ ͔. ͑82͒ we use Eq. ͑75͒ to identify the observed splitting of 0.3 meV 2 2 at the M point with 2MЌ jx , so that

In the high-temperature limit ͑when HE␰NӶ⌬C␰C): 2 2MЌ jx ϭ0.3 meV. ͑86͒ 2 ␻optϷ⌬C␰C , ␻acϷ8k4͑T͒HE␰N͑T͒. ͑83͒ Also we note that on average the mode energies are about 0.1 meV higher at the X point (cxϭcyϭ0) than at the M point In all these results we assumed that ͑a͒ HE dominates all (cxϭϪcyϭϪ1). Thus we deduce that other coupling constants and ͑b͒⌬␰C(T)ӷk4(T). The temperature dependence of ␰N(T) has a very strong 4O j2ϭ0.1 meV. ͑87͒ effect at temperatures where the thermal energy, kT, passes Ќ x through ⌬. In Fig. 11 we show a graph of the temperature We now adjust the other parameters, Kxy and NЌ to fit the dependence of the modes. In a moment we will discuss the remaining aspects of the observed spectrum. We found that a extent to which these results are consistent with experiments. reasonable fit to the spin-wave spectrum determined by in- elastic neutron scattering18,19 could be obtained by taking ͑all G. Comparison with experiments in meV͒

There are several features of our calculations which can Kxyϭ0.075, Kxzϭ0.01, Kzzϭ0.50, NЌϭ0.004, be compared with the experimental data. To make this com- parison we first discuss how we fix the various parameters MЌϭ0.025,OЌϭ0.003. ͑88͒ which enter the calculation. As mentioned above, the Cu-Cu exchange interactions, which give rise to HE and HA are Note that with this parameter set, we obtain a reasonable fixed from their values in many other cuprates. In addition, fit to various other data. For instance, in Fig. 11, we show the 280 RAVI SACHIDANANDAM et al. 56 temperature dependence of the optical modes involving the the Cu-Nd interaction has a tensor with two components, one Cu sublattices versus temperature. One sees a very good for interactions in the ͓001͔ plane and another for interac- agreement between theory and the observations of Ivanov tions involving z components of spin is not appropriate for et al.17 They interpreted their results in a qualitative way as the local symmetry of the interaction bond. In fact, as showing two zero wave-vector modes with energy gaps pro- pointed out,30 the pseudodipolar Nd-Nd interactions arise portional to ͱHEHA. The detailed theory presented here from the anisotropic exchange interaction only when the cor- gives their argument a firm theoretical basis. Also in Fig. 10 rect symmetry of the bond is taken into account. This pecu- one sees that the density of states indicates a gap to a peak in liar symmetry is particularly important in the case of the the density of states at an energy of about 0.28 meV, in spite nearest-neighbor Cu-Nd interactions of Eq. ͑36͒. As we have of the fact that the mean-field splitting ⌬ given in Eq. ͑41͒,is seen, the crucial part of this interaction is the pseudodipolar somewhat larger ͑0.41 meV, see Table IX͒. This peak value part proportional to Kxy . The other Nd-Nd interactions we in the density of states is in very good agreement with the use are very similar to those of Thalmeier, as can be seen by detailed analysis of the specific heat of NCO ͑Ref. 4͒ which the comparison shown in Table IX, where we give the con- gave a splitting of 0.33 meV ͑3.7 K in temperature units͒. tributions to the splitting from various interactions. Our fit in Fig. 7 to the Nd magnetization in NCO gave 0.30 ͓Thalmeier’s contribution to the doublet splitting from h2 Ќ meV for this splitting. So our theoretical spin-wave spectrum ͑due to his I2 ) vanishes because he does not allow for the is in broad agreement with the various thermodynamic mea- pseudodipolar component of exchange interactions between surements. Finally, based on our calculations we propose Nd nearest neighbors.͔ that a measurement of the lowest gap at qϭ0 and low tem- Finally we mention the earlier calculation of Sobolev peratures would be a useful measure of the fourfold anisot- et al.61 In that calculation the degrees of freedom describing ropy and would confirm the physics of our model. the Nd spins have been removed, so that there are just four Cu spins per unit cell. This actually is not too bad, since the highest mode is exact, and the other mode is reasonably H. Comparison with previous calculations close to one of our modes with Nd-Cu mixed in. Of course, From our results one sees that the approach used by this approach cannot describe either the Nd modes or the Thalmeier22 ͑in which the Cu spins create a fixed exchange low-frequency mode due to Nd-Cu collective excitation. field at the Nd sites͒ is not correct for very small wave vec- tors. In particular, at zero wave vector such an approach, if VI. CONCLUSION used for our model, would give three of the Nd modes cor- rectly, because as we have seen from our exact solution for We may summarize our conclusions as follows. qϭ0, these modes are confined to the Nd sublattices. Of ͑i͒ We show that due to the exchange field acting on the course, treating only the Nd spins cannot possibly give a rare-earth ion and crystalline electric-field interactions, there reasonable estimate of the energy of the lowest ͑acoustic͒ is a strong single ion anisotropy which aligns the Cu and mode for small q, since this mode is a collective mode of the RϭPr, Nd magnetization along the ͓100͔ axis, as observed. Nd and Cu sublattices. Our treatment is only necessary near This same type of calculation also indicates that for Sm in zero wave vector. In fact, from Fig. 10, one sees the ex- SCO the easy axis lies along ͓001͔, again in agreement with tremely strong dependence of the lowest energy mode at observations. Interestingly, our calculation shows that within small wave vector such that aqрͱ⌬/HEϷ0.06. For larger the plane, the Sm anisotropy favors alignment along ͓110͔.If q one has four Nd modes with energies near ⌬. In treating this anisotropy is the dominant in-plane anisotropy, the mag- the acoustic mode it is also important to incorporate the in- netic structure would then be a collinear one. The experi- plane anisotropy of the Nd doublet, as we have done here. ments are not conclusive as to whether or not the magnetic Finally, we give here an approximate treatment of the tem- structure of SCO is noncollinear, especially in the Sm- perature dependence of the spectrum. Because we assume ordered phase for TϽ6K. that the coupling between adjacent three-plane units is small, ͑ii͒ Crystalline electric-field theory with a Cu-R exchange our calculations apply to all three phases of NCO. interaction such that the exchange field, defined to couple to A significant difference between our model and Thalmei- J as in Eq. ͑2͒, of the order 0.080 meV for Nd ͑correspond- er’s is that in his work the values assigned to the various ing to a splitting of the lowest doublet of 0.3 meV͒, and exchange tensors are chosen in a way which seems to be 0.139 meV for Pr successfully explains many properties, inconsistent with the type of order actually found in the vari- such as the induced R magnetization, the splitting of the ous phases. In particular, his choice of the largest Cu-Nd Kramers doublet, etc., at all temperatures. interaction to be a ferromagnetic one ͑presumably between a ͑iii͒ We propose a model in which a Cu plane with its two Cu spin and the Nd ions directly above and below it in the Nd neighboring planes form a tightly bound unit due to in- z direction͒ seems to be contradicted by the fact that these teractions between the Cu plaquette and the Nd ions adjacent spins change their relative orientations during the spin reori- to it. In view of the frustration only pseudodipolar interpla- entation transitions. As discussed in Sec. IV, we would ex- nar interactions21 effectively contribute. We propose a model pect that the spin reorientations would preserve the strongest involving Cu-Cu, Cu-Nd, and Nd-Nd interactions between coupling and break only less dominant couplings. This ob- neighboring tightly bound units. The strengths of the inter- servation motivated our choice of model in which the domi- planar couplings are assumed to decrease rapidly with dis- nant Nd-Cu interaction is that from the anisotropic exchange tance, but in NCO they can compete because the temperature interaction between nearest Cu-Nd neighbors. Also, we may dependence of the Nd is extremely rapid. This is the simplest mention that the form of the exchange anisotropy in which model which explains both the three consecutive phase tran- 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 281 sitions observed6 in NCO as well as the absence of such not yet been observed, but clearly its observation is highly phase transitions in PCO. desirable. ͑iv͒ We have calculated the spin-wave spectrum of ACKNOWLEDGMENTS Nd2CuO4 within a simplified three-plane model which quali- tatively reproduces the spectrum obtained from inelastic neu- 18 We acknowledge stimulating interactions with R. J. Bir- tron scattering, as is shown in Fig. 10. The resulting Cu-Nd geneau, A. Boothroyd, M. A. Kastner, J. W. Lynn, and S. optical modes have a temperature dependence which agrees Skanthakumar. Work at the University of Pennsylvania was 17 quite well ͑see Fig. 11͒ with the experimental results. The partly supported by the National Science Foundation MRL energy of the acoustic spin-wave mode at zero wave vector is Program under Grant No. DMR-95-20175 and that at Tel predicted to be Ϸͱ2k4⌬Ϸ5␮eV, where k4 is the small four- Aviv by the U. S.-Israel Binational Science Foundation, the fold anisotropy in the plane and ⌬ is the splitting of the USIEF, the Israel Science Foundation, and the Sackler Insti- lowest Nd doublet in the Cu exchange field. This mode has tutes for Theoretical and for Solid State Physics. APPENDIX A: COEFFICIENTS IN THE PERTURBATION EXPANSION The coefficients that appear in Eqs. ͑22͒ are

x2 Ј il a2iϭ͚ , ͑A1a͒ l Ei,l

z2 Ј il b2iϭ͚ , ͑A1b͒ l Ei,l

x x x x2 Ј il lm mi Ј il a3iϭxii ͚ Ϫxii͚ 2 , ͑A1c͒ ͩ l,m Ei,lEi,m l Ei,lͪ

x x x ͑⑀ Ϫ⑀ ϩ⑀ ͒ x2 Ј il lm mi il lm mi Ј il b3iϭ2xii ͚ Ϫxii͚ 2 , ͑A1d͒ ͩ l,m Ei,lEi,m l Ei,lͪ

͚ ͑Ϫ1͒PP͑x x z ͒ x2 ͚ ͑Ϫ1͒PP͑z z x ͒ z2 Ј P il lm mi Ј il Ј P il lm mi Ј il c3iϭzii ͚ Ϫzii͚ 2 ϩxii ͚ Ϫxii͚ 2 , ͑A1e͒ ͩ l,m Ei,lEi,m l Ei,lͪ ͩ l,m Ei,lEi,m l Ei,lͪ

z z z z2 Ј il lm mi Ј il d3iϭzii ͚ Ϫzii͚ 2 , ͑A1f͒ ͩ l,m Ei,lEi,m l Ei,lͪ

x x x x 2x x x x ϩx2 x2 x2 Ј il lm mn ni Ј ii il lm mi il im 2 Ј il a4iϭ ͚ Ϫ͚ 2 ϩxii͚ 3 , ͑A1g͒ l,m,n Ei,lEi,mEi,n l,m Ei,lEi,m l Ei,l

x x x x ͑⑀ Ϫ⑀ ⑀ ϩ⑀ ͒ x2 4x x x x ͑⑀ Ϫ⑀ ϩ⑀ ͒ϩ2x2 x2 Ј il lm mn ni im il mn ln 2 Ј il Ј ii il lm mi il lm mi il im b4iϭ2 ͚ ϩ2xii͚ 3 Ϫ͚ 2 , ͑A1h͒ l,m,n Ei,lEi,mEi,n l Ei,l l,m Ei,lEi,m

x2 z2 ϩz2 x2 ͚ ͑Ϫ1͒PP͑x x z z ͒ z2 x2 ϩz2 x2 2z ͚ ͑Ϫ1͒PP͑x x z ͒ Ј ii il ii il Ј P il lm mn ni Ј il im im il Ј ii P il lm mi c4iϭ͚ 3 ϩ ͚ Ϫ͚ 2 Ϫ͚ 2 l E l,m,n Ei,lEi,mEi,n l,m E E l,m ͩ E E i,l i,l i,m i,l i,m

P 2xii ͑Ϫ1͒ P͑zilzlmxmi͒ ͚P ϩ , ͑A1i͒ E2 E ͪ i,l i,m

z z z z 2z z z z ϩz2 z2 z2 Ј il lm mn ni Ј ii il lm mi il im 2 Ј il d4iϭ ͚ Ϫ͚ 2 ϩzii͚ 3 , ͑A1j͒ l,m,n Ei,lEi,mEi,n l,m Ei,lEi,m l Ei,l where Ei,lϭEiϪEl , ͚Ј indicates that the summed index ͑or indices͒ may not assume the value i, and

P ͑Ϫ1͒ P͑zilzlmxmi͒ϭ͑zilzlmxmiϪzilxlmzmiϩxilzlmzmi͒. ͑A2͒ ͚P Numerical values of the coefficients are listed in Table VII. 282 RAVI SACHIDANANDAM et al. 56

APPENDIX B: COEFFICIENTS OF THE FREE-ENERGY 5 1 EXPANSION Ϫ1 Ϫ␤Ei0 2 FZ͑T͒ϭ2 ͑0͒ ͚ e Ϫd4iϩ ␤b2iϩ␤d3i Z iϭ1 ͩ 2 Here we list the coefficients that appear in the expansions 3 for the free energy and the partition function. 1 ␤ Ϫ ␤2z2 b ϩ z4 . ͑B8͒ 2 ii 2i 4! iiͪ 1 2 ␹ЌϭAZ͑T͒, ͑B1a͒ APPENDIX C: GROUND-STATE ENERGY AND 2gJ SUSCEPTIBILITY OF PR

1 We first give the matrix elements involving hЌ . Here ␾ 2 ␹zϭEZ͑T͒, ͑B1b͒ refers to the angle which hЌ makes with the x axis: 2gJ i␾ ͉d1͘ e ͗g͉J h ϭ1.883h ; 1 • ͩ ͉d ͪ͘ Ќͩ ϪeϪi␾ ͪ 2 2 Ϫ␣4ϭBZ͑T͒Ϫ ␤AZ͑T͒, ͑B1c͒ 2 i␾ ͉D1͘ e ͗g͉J h ϭ0.675h , ͑C1a͒ • ͩ ͉D ͪ͘ Ќͩ ϪeϪi␾ ͪ 1 2 Ϫ␤ ϭF ͑T͒Ϫ ␤E2͑T͒, ͑B1d͒ 4 Z 2 Z Ϫi␾ ͉d1͘ e ͗e ͉J h ϭ0.332h ; 1 • ͩ ͉d ͪ͘ Ќͩ Ϫei␾ ͪ 2 2 Ϫ␥4ϭCZ͑T͒Ϫ␤AZ͑T͒, ͑B1e͒ Ϫi␾ ͉D1͘ e ͗e ͉J h ϭ1.853h , ͑C1b͒ 1 • ͩ ͉D ͪ͘ Ќͩ Ϫei␾ ͪ 2 Ϫ␦4ϭDZ͑T͒Ϫ␤AZ͑T͒EZ͑T͒, ͑B1f͒ Ϫi␾ ͉d1͘ e where ͗e ͉J h ϭ0.876h ; 2 • ͩ ͉d ͪ͘ Ќͩ ei␾ ͪ 2 5 Ϫi␾ Ϫ␤E ͉D1͘ Ϫe ͑0͒ϭ2͚ e i0, ͑B2͒ ͗e ͉J h ϭ0.483h , ͑C1c͒ Z iϭ1 2 • ͩ ͉D ͪ͘ Ќͩ Ϫei␾ ͪ 2

Ϫi␾ 5 ͉d1͘ e 1 e J h ϭ1.351h ; Ϫ1 Ϫ␤Ei0 2 ͗ 3͉ • Ќ i␾ AZ͑T͒ϭ2 ͑0͒ ͚ e Ϫa2iϩ ␤xii , ͑B3͒ ͩ ͉d2ͪ͘ ͩ Ϫe ͪ Z iϭ1 ͩ 2 ͪ Ϫi␾ ͉D1͘ e 5 ͗e ͉J h ϭ0.797h , ͑C1d͒ 1 3 • ͩ ͉D ͪ͘ Ќͩ Ϫei␾ ͪ 2 Ϫ1 Ϫ␤Ei0 2 EZ͑T͒ϭ2 ͑0͒ ͚ e Ϫb2iϩ ␤zii , ͑B4͒ Z iϭ1 ͩ 2 ͪ i␾ ͉d1͘ Ϫe ͗e ͉J h ϭ0.434h ; 4 • ͩ ͉d ͪ͘ Ќͩ ϪeϪi␾ ͪ 2 5 1 Ϫ1 Ϫ␤Ei0 2 BZ͑T͒ϭ2 ͑0͒ ͚ e Ϫa4iϩ ␤a2iϩ␤a3i i␾ Z iϭ1 ͩ 2 ͉D1͘ e ͗e ͉J h ϭ1.953h . ͑C1e͒ 4 • D Ќ eϪi␾ 1 3 ͩ ͉ 2ͪ͘ ͩ ͪ 2 2 ␤ 4 Ϫ ␤ xiia2iϩ xii , ͑B5͒ 2 4! ͪ The only nonzero matrix elements involving hz are

͉d1͘ 10 5 (͗d1͉͗d2͉͒Jzhz ϭ2.069hz ; Ϫ1 Ϫ␤Ei0 2 2 2 ͩ ͉d ͪ͘ ͩ 0Ϫ1ͪ 2 CZ͑T͒ϭ2 ͑0͒ ͚ e Ϫb4iϩ␤a2iϩ␤b3iϪ␤ xiia2i Z iϭ1 ͩ ͗e͉Jh͉g͘ϭϪ2.000h , ͑C2a͒ 2␤3 4zz z ϩ x4 , ͑B6͒ 4! iiͪ ͉D1͘ Ϫ10 (͗D ͉͗D ͉͒J h ϭ0.068h ; 1 2 z zͩ ͉D ͪ͘ zͩ 01ͪ 2 5 Ϫ1 Ϫ␤Ei0 DZ͑T͒ϭ2 ͑0͒ ͚ e Ϫc4iϩ␤a2ib2iϩ␤c3i ͗e3͉Jzhz͉e2͘ϭ3.190hz , ͑C2b͒ Z iϭ1 ͩ

3 1 2␤ ͉d1͘ Ϫ10 Ϫ ␤2͑x2 b ϩz2 a ͒ϩ x2 z2 , ͑B7͒ (͗D ͉͗D ͉͒J h ϭ1.692h ; 2 ii 2i ii 2i 4! ii ii 1 2 z zͩ ͉d ͪ͘ zͩ 01ͪ ͪ 2 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 283

͗e2͉Jzhz͉e1͘ϭϪ2.415hz . ͑C2c͒ where ␮ is the magnetic moment operator, ͉⌫͘ is a crystal- field state for hϭ0, and E⌫ is the corresponding eigenvalue. Now we use these results to implement perturbation Here ͚Ј indicates a sum over states ͉⌫Ј͘ that are identical to theory. In our numerical results, all energies and h are ex- or degenerate with ͉⌫͘, and ͚Љ indicates a sum over states pressed in meV. First-order perturbation of the ground state ͉⌫Ј͘ that are nondegenerate in energy with ͉⌫͘. The first is zero. For the second-order perturbation of the ground state term gives the temperature-dependence of the magnetic sus- we evaluate terms of the form ceptibility and the second term gives a relatively temperature independent paramagnetism. ͗g͉Vex͉␣͗͘␣͉Vex͉g͘ ͚ . ͑C3͒ ␣ ͑EgϪE␣͒ APPENDIX D: THE STATES ͦg‹ AND ͦe‹ There are two kinds of quantities from this term Here we develop expressions for the states ͉g͘ and ͉e͘ of the lowest doublet in the presence of an exchange field. For ͗g͉Vex͉d͗͘d͉Vex͉g͘ ͗g͉Vex͉D͗͘D͉Vex͉g͘ ͚ ϩ ͚ that purpose it is convenient to label the ten zero-field states d1 ,d2 EgϪEd D1 ,D2 EgϪED as doublets from 1 to 5 in order of increasing energy. We use perturbation expansions identical to those of Sec. III.C. to 2͑1.883͒2h2 2͑0.675͒2h2 Ќ Ќ 2 develop expansions for ͉g͘ and ͉e͘ in terms of the ten doubly ϭ ϩ ϭϪ0.389hЌ , ͑C4͒ EgϪEd EgϪED degenerate states of the Jϭ9/2 multiplet. The exchange field at a Nd ion in an n sublattice due to all its Cu and Nd and neighbors lies along the y axis. The magnitude of the ex- 2 change field, h, will be fixed to give the observed doublet ͗g͉Vex͉e4͗͘e4͉Vex͉g͘ Ϫ4hz ϭ ϭϪ0.0406h2 . ͑C5͒ splitting. E ϪE 98.5 z g e4 We first diagonalize the potential due to interactions with this magnetic field within the ground-state doublet to give The third-order terms vanish. The fourth-order terms are pro- the states ͉1Ϯ͘. We will use the fact, shown in Sec. III C, portional to (h4ϩh4), h2h2 , h4 and h2 h2 . There are two x y x y z Ќ z that the matrix elements of the J operators between any two types of matrix elements that we have to evaluate: sets of doublets are like Pauli spin matrices. For the n sub- lattice ͑with Nd moment along the Ϫy direction͒ we have ͗g͉Vex͉␣͗͘␣͉Vex͉␤͗͘␤͉Vex͉␥͗͘␥͉Vex͉g͘ ͑1͒ . ␣͚,␤,␥ ͑E ϪE ͒͑E ϪE ͒͑E ϪE ͒ g ␣ g ␤ g ␥ ͗iϩ͉ 4 Ϫ4 2 J͉͑jϩ͉͘jϪ͘)ϭ͑xij␴x ,⑀ijxij␴z ,Ϫzij␴y͒, From this term we get ϪhЌ͓9.68ϫ10 cos (2␾) ͩ ͗iϪ͉ͪ Ϫ4 2 2 2 ϩ3.23ϫ10 sin (2␾)͔Ϫ0.00546hz hЌ which can be written ͑D1͒ as where ␴ are the Pauli spin matrices, i, j are labels of the Ϫ4 4 4 Ϫ4 2 2 2 2 Ϫ9.68ϫ10 ͑hx ϩhy ͒ϩ6.44ϫ10 hx hyϪ0.00546hz hЌ . doublets, and ⑀ij , xij , and zij are defined in Eq. ͑16͒. ͑C6͒ From the diagonalization of the potential matrix ͑for a field in the Ϫy direction, carried out in Sec. IIIC͒,wecan ͗g͉Vex͉␣͗͘␣͉Vex͉g͗͘g͉Vex͉␤͗͘␤͉Vex͉g͘ see that the zeroth-order ground state ͉g͘0 is the state labeled ͑2͒ Ϫ͚ 2 . ␣,␤ ͑EgϪE␣͒͑EgϪE␤͒ ͉1Ϫ͘, while the zeroth-order first excited state ͉e͘0 is labeled ͉1ϩ͘. Explicitly we have 4 4 This will only give quantities proportional to hЌ , hz and h2 h2 which are Ќ z 1 1Ϫ ϵ g ϭ ͓ A Ϫi⌰ A ], 4 Ϫ5 4 2 2 ͉ ͘ ͉ ͘0 ͉ 1͘ ͉ 1͘ 0.00793hЌϩ1.674ϫ10 hz ϩ0.000988hz hЌ . ͑C7͒ ͱ2 Hence the ground-state energy in a field is given by 1 ͉1ϩ͘ϵ͉e͘0ϭ ͓⌰͉A1͘Ϫi͉A1͘], ͑D2͒ 2 2 4 4 2 2 ͱ2 EgϭϪ0.389hЌϪ0.0406hz ϩ0.00696͑hx ϩhy ͒ϩ0.0165hx hy Ϫ5 4 2 2 ϩ1.674ϫ10 hz Ϫ0.00447hЌhz . ͑C8͒ where ͉A1͘ and ⌰ are given in Eqs. ͑8e͒ and ͑9͒, respec- tively. At this order there is no difference between the x and We also quote here the general formula for the suscepti- y directions, but there is one between the z direction and the bility for a non-Kramer’s ion, x or y directions, reflecting the tetragonal symmetry of the lattice. To see an anisotropy in the xϪy plane we need to 1 1 carry the expansion to higher order. We can find the correc- ␹ ͑T͒ϭ eϪE⌫ /kT Ј ͉͗⌫͉␮ ͉⌫Ј͉͘2 ␣␣ Z kT͚ ͚ ␣ tions to the zeroth-order wave function to first (͉1Ϯ,1 ) sec- ͭ ⌫ ⌫Ј ͘ ond (͉1Ϯ,2͘) and third order (͉1Ϯ,3͘) in the fields. These eϪE⌫ /kTϪeϪE⌫Ј /kT will be orthogonal to the original state, that is, the correc- ϩ Љ ͉͗⌫͉␮ ͉⌫Ј͉͘2 , ͚ ͚ ␣ E ϪE tions to any state will only involve the states belonging to the ⌫ ⌫Ј ⌫Ј ⌫ ͮ four other doublets. In this formulation the eigenstate is not ͑C9͒ normalized to unity. 284 RAVI SACHIDANANDAM et al. 56

5 5 5 ⑀1kxk1 ͑1͒ ͉1Ϯ,1͘ϭϮ͚ h ͉kϮ͘ϵϮh͚ ⑀1kfk ͉kϮ͘, CeCg jzϭϪi͗g͉Jz͉e͘ϭ z11ϩ ͚ zk1͓ f k͑h͒ϩ f k͑Ϫh͔͒ kϭ2 E1k kϭ2 ͩ kϭ2 ͑D3a͒ 5 ϩ ͚ zkpfk͑h͒fp͑Ϫh͒ 5 x x x x k,pϭ2 ͪ 2 kp p1 k1 11 ͉1Ϯ,2͘ϭ ͚ h ⑀1k Ϫ 2 ͉kϮ͘ kϭ2 ͩ E1kE1p E ͪ ϭi e J g , ͑D8͒ 1k ͗ ͉ z͉ ͘ 5 2 ͑2͒ 5 ϵh ͚ ⑀1k f k ͉kϮ͘, ͑D3b͒ 2 kϭ2 Ce j yϩϭ͗e͉Jy͉e͘ϭx11ϩ xk1͓ f k͑h͒ϩ f k͑Ϫh͔͒ k͚ϭ2

5 5 x x x x x x x x x ϩ ͚ xkpfk͑h͒fp͑h͒, ͑D9͒ 3 kp pq q1 kp p1 11 kp p1 11 k,pϭ2 ͉1Ϯ,3͘ϭϮ͚ h ⑀1k Ϫ 2 Ϫ 2 kϭ2 ͩE1kE1pE1q E1kE E E1p 1p 1k 2 xk1x11 xk1x1px p1 5 ϩ 3 Ϫ 2 ͉kϮ͘ 2 E E1pE ͪ C j ϭC g J g ϭϪx Ϫ x ͓ f ͑h͒ϩ f ͑Ϫh͔͒ 1k 1k g yϪ g͗ ͉ y͉ ͘ 11 k1 k k k͚ϭ2 5 3 ͑3͒ 5 ϵϮh ͚ ⑀1k f k ͉kϮ͘, ͑D3c͒ kϭ2 Ϫ xkpfk͑Ϫh͒fp͑Ϫh͒, ͑D10͒ k,͚pϭ2 where the repeated indices (p,q) are summed from 2 to 5. 2 2 Let where Ce ϭ͗e͉e͘ and Cgϭ͗g͉g͘. One sees that 3 ␦ j yϵ j yϩϩ j yϪ is of order h , and ␦ jЌϵ j yϪ jx is of order 2 h , where j yϭ( j yϩϪ j yϪ)/2, whereas jz and the average in 0 ͑1͒ ͑2͒ 2 ͑3͒ 3 plane javϵ( j yϩ jx)/2 both are of order h . For ⌬ϭ0.3 meV, f k͑h͒ϭ f k hϩ f k h ϩ f k h . ͑D4͒ i.e., for hϭ⌬/(2jy)ϭ0.07954, we find that ␦ j ϭϪ1.0ϫ10Ϫ6, ␦ j ϭ1.38ϫ10Ϫ4, j ϭϪ0.0151, and Then the two lowest states are given by y Ќ z javϭ1.886.

5 APPENDIX E: ANISOTROPY DUE TO ZERO-POINT ͉e͘ϭ͉1,ϩ͘ϩ ͚ f k͑h͒⑀1k͉kϩ͘, ͑D5a͒ kϭ2 FLUCTUATIONS

5 In this appendix we consider the effect of the in-plane anisotropy of the Cu-Cu exchange interactions when ͉g͘ϭ͉1,Ϫ͘ϩ ͚ f k͑Ϫh͒⑀1k͉kϪ͘. ͑D5b͒ kϭ2 ␦Jϵ ʈϪ Ќ is nonzero. We consider only the calculation for Tϭ0.J ThenJ it is convenient to follow the analysis of Sec. V For the n sublattice ͑with Nd spin in the Ϫy direction͒ we of Ref. 12. There one sees in Eq. ͑67͒ that within noninter- have acting spin-wave theory there is no gap in the spin-wave spectrum even when ␦J is nonzero. Although one can calcu- late the gap due to ␦J using nonlinear spin-wave theory, it is much easier to estimate this gap by constructing an effective ͗e͉ 1 Hamiltonian, , from the dependence of the quantum J(͉e͉͘g͘)ϭ jx␴x , ͑ j yϩϩ j yϪ͒I ZP ͩ ͗g͉ͪ ͫ 2 zero-point energyH on the orientation of the staggered mo- ment. This anisotropy is not a long-wavelength 1 phenomenon—in simple cases it can be estimated from the ϩ ͑ j yϩϪ j yϪ͒␴z ,Ϫ jz␴y , ͑D6͒ 2 ͬ short-wavelength fluctuations.62,63 Therefore, it is justified to use the effective Hamiltonian given in Eq. ͑76͒ of Ref. 12 for where I is the unit 2ϫ2 matrix and the expressions for the Cu system without any rare-earth spins. In the present jx ...jz are notation ͓and with J ϭ( ϩ )/2͔ we have av JЌ Jʈ

Ϫ3 2 2 5 ZPϭ4Jav␦inS ͚ SixSiy H i෈Cu CeCg jxϭ͗g͉Jx͉e͘ϭx11ϩ ⑀k1xk1͓ f k͑h͒ϩ f k͑Ϫh͔͒ k͚ϭ2 2 † 2 † 5 ϭ4Jav␦in ͚ ͓a pϩ͑a p͒ ϩ2a pa p͔ ͩ p ϩ ⑀kpxkpfk͑h͒fp͑Ϫh͒ k,͚pϭ2 2 † 2 † ϩ͚ ͓ar ϩ͑ar ͒ ϩ2ar ar͔ , ͑E1͒ ϭ͗e͉J ͉g͘, ͑D7͒ r ͪ x 56 SINGLE-ION ANISOTROPY, CRYSTAL-FIELD . . . 285

2 Ϫ9 where ␦inϭ2C2(␦J/Jav) Ϸ10 involves a sum over the now have the altered matrix elements A11ϭHE zero-point energy of modes in the entire Brillouin zone. This ϩ2⌬Cϩ4Jav␦in and B11ϭϪHEϩ4Jav␦in . Then, when ␦in effect is clearly negligible except possibly for zero wave vec- can be treated perturbatively, we find tor. This interaction can be included in the dynamical matri- ces, in which case in Eq. ͑60͒ we should replace HE by 2Jav␦in ␻2 Ϸ␻2 ͑␦ ϭ0͒ 1ϩ . ͑E2͒ HEϩ4Jav␦in and the zero entries in the Cu sector of Eq. ͑61͒ Ͻ Ͻ in ͩ ⌬ ͑1Ϫ j / j ͒ͪ c x y should be replaced by 4Jav␦in . One sees that this modifica- tion has no effect at all on the energy of the modes ␻Ϯ and Using the known values of the parameters we see that includ- ␻3, and a completely negligible effect on ␻4.InEq.͑65͒ we ing the effect of ␦in has an effect of about 1% on ␻Ͻ .

1 For a review, see J. W. Lynn, in Physical and Material Properties 18 W. Henggeler, T. Chattopadhyay, P. Thalmeier, P. Vorderwisch, of High Temperature Superconductors, edited by S. K. Malik and A. Furrer, Europhys. Lett. 34, 537 ͑1996͒. and S. S. Shah ͑Nova Science, New York, 1994͒, p. 243. 19 H. Casalta, P. Bourges, D. Petitgrand, and A. Ivanov, Solid State 2 Y. Tokura, H. Takagi, and S. Uchida, Nature ͑London͒ 337, 345 Commun. 100, 683 ͑1996͒. ͑1989͒. 20 D. A. Yablonsky, Physica C 182, 105 ͑1991͒. 3 M. Braden, W. Paulus, A. Cousson, P. Vigoureux, G. Heger, A. 21 T. Yildirim, A. B. Harris, O. Entin-Wohlman, and A. Aharony, Goukassov, P. Bourges, and D. Petitgrand, Europhys. Lett. 25, Phys. Rev. Lett. 72, 3710 ͑1994͒. 625 ͑1994͒. 22 P. Thalmeier ͑unpublished͒. 4 P. Adelmann, R. Ahrens, G. Czjzek, G. Roth, H. Schmidt, and C. 23 P. Allenspach, A. Furrer, R. Osborn, and A. D. Taylor, Z. Phys. B Steinleitner, Phys. Rev. B 46, 3619 ͑1992͒.InEq.͑4͒of this 85, 301 ͑1991͒. 24 reference kBT should be replaced by 2kBT. However, the au- A. T. Boothroyd, S. M. Doyle, D. Mc. K. Paul , and R. Osborn, thors apparently used the correct equation in calculating the spe- Phys. Rev. B 45,10075͑1992͒. cific heat, because we did reproduce their Fig. 10͑b͒. 25 Throughout this paper crystallographic directions are referred to 5 D. Vaknin, S. K. Sinha, D. E. Moncton, D. C. Johnston, J. M. the chemical unit cell shown in Fig. 1. Newsam, C. R. Safinya, and H. E. King , Jr., Phys. Rev. Lett. 58, 26 Ph. Bourges, L. Boudare`ne, and D. Petitgrand, Physica B 2802 ͑1987͒; J. D. Jorgensen, H.-B. Schuttler, D. G. Hinks, D. 180&181, 128 ͑1992͒. W. Capone II, K. Zhang, M. B. Brodsky, and D. J. Scalapino, 27 A. B. Harris, Phys. Rev. 132, 2398 ͑1963͒. ibid. 58, 1024 ͑1987͒. 28 S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. B 6 M. Matsuda, K. Yamada, K. Kakurai, H. Kadowaki, T. R. Thur- 39, 2344 ͑1989͒. ston, Y. Endoh, Y. Hidaka, R. J. Birgeneau, M. A. Kastner, P. 29 H. W. de Wijn, R. E. Walstedt, L. R. Walker, and H. J. Guggen- M. Gehring, A. H. Moudden, and G. Shirane, Phys. Rev. B 42, heim, Phys. Rev. Lett. 24, 832 ͑1970͒. 10 098 ͑1990͒. 30 T. Yildirim, A. B. Harris, O. Entin-Wohlman, and A. Aharony, 7 I. W. Sumarlin, S. Skanthakumar, J. W. Lynn, J. L. Peng, Z. Y. Phys. Rev. Lett. 73, 2919 ͑1994͒. Li, W. Jiang, and R. L. Greene, Phys. Rev. Lett. 68, 2228 31 F. C. Chou, A. Aharony, R. J. Birgeneau, O. Entin-Wohlman, M. ͑1992͒. Greven, A. B. Harris, M. A. Kastner, Y. J. Kim, D. S. Kleinberg, 8 R. J. Birgeneau and G. Shirane, in Physical Properties of High Y. S. Lee, and Q. Zhu, Phys. Rev. Lett. 78, 535 ͑1997͒. Temperature Superconductors I, edited by D. M. Ginsberg 32 Y. Endoh, M. Matsuda, K. Yamada, K. Kakurai, Y. Hikada, G. ͑World-Scientific, Singapore, 1989͒, and references therein. Shirane, and R. J. Birgeneau, Phys. Rev. B 40, 7023 ͑1989͒. 9 B. Keimer, A. Aharony, A. Auerbach, R. J. Birgeneau, A. Cas- 33 I. W. Sumarlin, J. W. Lynn, T. Chattopadhyay, S. N. Barilo, D. I. sanho, Y. Endoh, R. W. Erwin, M. A. Kastner, and G. Shirane, Zhigunov, and J. L. Peng, Phys. Rev. B 51, 5824 ͑1995͒. Phys. Rev. B 45, 7430 ͑1992͒. 34 S. Skanthakumar, J. W. Lynn, J. L. Peng, and Z. Y. Li, J. Appl. 10 V. B. Grande, Hk. Mu¨ller-Buschbaum, and M. Schweizer, Z. An- Phys. 73, 6326 ͑1993͒. org. Allg. Chem. 428, 120 ͑1977͒. 35 T. Thio and A. Aharony, Phys. Rev. Lett. 73, 894 ͑1994͒. 11 T. Thio, T. R. Thurston, N. W. Preyer, P. J. Picone, M. A. Kast- 36 P. Allenspach, S. W. Cheong, A. Dommann, P. Fischer, Z. Fisk, ner, H. P. Jenssen, D. R. Gabbe, C. Y. Chen, R. J. Birgeneau, A. Furrer, H. R. Ott, and B. Rupp, Z. Phys. B 77, 185 ͑1989͒. and A. Aharony, Phys. Rev. B 38, 905 ͑1988͒. 37 G. Shirane, Acta Crystallogr. 12, 282 ͑1959͒. 12 T. Yildirim, A. B. Harris, A. Aharony, and O. Entin-Wohlman, 38 A. S. Cherny, E. N. Khats’ko, G. Chouteau, J. M. Louis, A. A. Phys. Rev. B 52,10239͑1995͒. Stepanov, P. Wyder, S. N. Barilo, and D. I. Zhigunov, Phys. 13 O. Entin-Wohlman, A. B. Harris, and A. Aharony, Phys. Rev. B Rev. B 45,12600͑1992͒. 53,11661͑1996͒. 39 J. Akimitsu, H. Sawa, T. Kobayashi, H. Fujiki, and Y. Yamada, J. 14 D. Vaknin, S. K. Sinha, C. Stassis, L. L. Miller, and D. C. Phys. Soc. Jpn. 58, 2646 ͑1989͒. Johnston, Phys. Rev. B 41, 1926 ͑1990͒. 40 J. W. Lynn, I. W. Sumarlin, S. Skanthakumar, W-H. Li, R. N. 15 M. Greven, R. J. Birgeneau, Y. Endoh, M. A. Kastner, B. Keimer, Shelton, J. L. Peng, Z. Fisk, and S-W. Cheong, Phys. Rev. B 41, M. Matsuda, and G. Shirane, Phys. Rev. Lett. 72, 1096 ͑1994͒. R2569 ͑1990͒. 16 S. Skanthakumar, J. W. Lynn, J. L. Peng, and Z. Y. Li, Phys. Rev. 41 T. Chattopadhyay and K. Siemensmeyer ͑unpublished͒. B 47, 6173 ͑1993͒. 42 D. Petitgrand, L. Boudare`ne, P. Bourges, and P. Galez, J. Magn. 17 A. S. Ivanov, P. Bourges, D. Petitgrand, and J. Rossat-Mignod, Magn. Mater. 104-107, 585 ͑1992͒. Physica B 213-214,60͑1995͒. 43 S. Skanthakumar, Ph. D. thesis, University of Maryland. 286 RAVI SACHIDANANDAM et al. 56

44 P. Dufour, S. Jandl, C. Thomsen, M. Cardona, B. M. Wanklyn, 55 K. R. Lea, M. J. M. Leask, and W. P. Wolf, J. Phys. Chem. Solids and C. Changkang, Phys. Rev. B 51, 1053 ͑1995͒. 23, 1381 ͑1994͒. 45 U. Staub, P. Allenspach, A. Furrer, H. R. Ott, S.-W. Cheung, and 56 S. Jandl, P. Dufour, T. Strach, T. Ruf, M. Cardona, V. Nekvasil, Z. Fisk, Solid State Commun. 75, 431 ͑1990͒. C. Chen, B. M. Wanklyn, and S. Pin˜o, Phys. Rev. B 53, 8632 46 S. Katano, R. M. Nicklow, S. Funahashi, N. Mori, T. Kobayashi, ͑1996͒. and J. Akimitsu, Physica C 215,92͑1993͒. 57 M. L. Sanjuan and M. A. Laguna, Phys. Rev. B 52,13000 47 S. Skanthakumar, J. W. Lynn, J. L. Peng, and Z. Y. Li, J. Magn. ͑1996͒. Magn. Mater. 519, 104 ͑1992͒. 58 T. Yildirim ͑unpublished͒. 48 M. E. Rose, Elementary Theory of Angular Momentum ͑Wiley, 59 This form assumes that the x and y axes are parallel to the ͓100͔ New York, 1957 . ͒ and ͓010͔ axes of the chemical unit cell shown in Fig. 1. 49 K. W. H. Stevens, Proc. Phys. Soc. London, Sec. A 65, 209 60 From purely magnetic dipole-dipole interactions one would have ͑1952͒. XϭϪ0.74, Yϭ1.00, and ZϭϪ0.30, all in ␮eV. The algebraic 50 A. Abragam and B. Bleaney, Electron Paramagnetic Resonance signs of these interactions are such that they tend to destabilize of Transition Ions ͑Clarendon, Oxford, 1970͒. the structure of phase I. They have the wrong sign to explain the 51 A. J. Kassman, J. Chem. Phys. 53, 4118 ͑1970͒. 52 jump in the doublet splitting seen by Dufour et al. ͑Ref. 44͒. T. Strach, T. Ruf, M. Cardona, C. T. Lin, S. Jandl, V. Nekvasil, 61 D. I. Zhigunov, S. N. Barilo, and S. V. Shiryaev, Phys. Rev. B V. L. Sobolev, H. L. Huang, Yu. G. Pashkevich, M. M. Larionev, 54, 4276 ͑1996͒. I. M. Vitebsky, and V. A. Blinkin, Phys. Rev. B 49, 1170 53 N. W. Ashcroft and N. D. Mermin, Solid State Physics ͑Saunders, ͑1994͒. 62 Philadelphia, 1976͒, p. 650. M. W. Long, J. Phys. Condens. Matter 1, 2857 ͑1989͒. 63 54 M. Tinkham, Group Theory and Quantum Mechanics ͑McGraw- A. B. Harris, in Disorder in Condensed Matter Physics, edited by Hill, New York, 1964͒. J. Blackman and J. Taguen˜a ͑Clarendon, Oxford, 1991͒.