Magnetic Dipole Interactions in Crystals
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Magnetic Dipole Interactions in Crystals
David C. Johnston ∗ Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011 (Dated: January 18, 2016)The influence of magnetic dipole interactions (MDIs) on the magnetic properties of local-momentHeisenberg spin systems is investigated. A general formulation is presented for calculating theeigenvalues λ and eigenvectors ˆ µ of the MDI tensor of the magnetic dipoles in a line (one dimension1D), within a circle (2D) or a sphere (3D) of radius r surrounding a given moment ~µ i for givenmagnetic propagation vectors k for collinear and coplanar noncollinear magnetic structures on bothBravais and non-Bravais spin lattices. Results are calculated for collinear ordering on 1D chains, 2Dsquare and simple-hexagonal (triangular) Bravais lattices, 2D honeycomb and kagom´e non-Bravaislattices and 3D cubic Bravais lattices. The λ and ˆ µ values are compared with previously reportedresults. Calculations for collinear ordering on 3D simple tetragonal, body-centered tetragonal, andstacked triangular and honeycomb lattices are presented for c/a ratios from 0.5 to 3 in both graphicaland tabular form to facilitate comparison of experimentally determined easy axes of ordering onthese Bravais lattices with the predictions for MDIs. Comparisons with the easy axes measuredfor several illustrative collinear antiferromagnets (AFMs) are given. The calculations are extendedto the cycloidal noncollinear 120 ◦ AFM ordering on the triangular lattice where λ is found to bethe same as for collinear AFM ordering with the same k . The angular orientation of the orderedmoments in the noncollinear coplanar AFM structure of GdB with a distorted stacked 3D Shastry-Sutherland spin-lattice geometry is calculated and found to be in disagreement with experimentalobservations, indicating the presence of another source of anisotropy. Similar calculations for theundistorted 2D and stacked 3D Shastry-Sutherland lattices are reported. The thermodynamics ofdipolar magnets are calculated using the Weiss molecular field theory for quantum spins, includingthe magnetic transition temperature T m and the ordered moment, magnetic heat capacity andanisotropic magnetic susceptibility χ versus temperature T . The anisotropic Weiss temperature θ p in the Curie-Weiss law for T > T m is calculated. A quantitative study of the competition betweenFM and AFM ordering on cubic Bravais lattices versus the demagnetization factor in the absenceof FM domain effects is presented. The contributions of Heisenberg exchange interactions and ofthe MDIs to T m and to θ p are found to be additive, which simplifies analysis of experimental data.Some properties in the magnetically-ordered state versus T are presented, including the orderedmoment and magnetic heat capacity, and for AFMs the dipolar anisotropy of the free energy andthe perpendicular critical field. The anisotropic χ for dipolar AFMs is calculated both above andbelow the N´eel temperature T N and the results are illustrated for a simple tetragonal lattice with c/a > c/a = 1 (cubic) and c/a <
1, where a change in sign of the χ anisotropy is found at c/a = 1.Finally, following the early work of Keffer [Phys. Rev. , 608 (1952)], the dipolar anisotropy of χ above T N = 69 K of the prototype collinear Heisenberg-exchange-coupled tetragonal compoundMnF is calculated and found to be in excellent agreement with experimental single-crystal literaturedata above 130 K, where the smoothly increasing deviation of the experimental data from the theoryon cooling from 130 K to T N is deduced to arise from dynamic short-range collinear c -axis AFMordering in this temperature range driven by the exchange interactions. I. INTRODUCTION
Local magnetic moments generate magnetic dipolefields around them. In local-moment spin systems, thelong-range magnetic dipole interaction between the lo-cal magnetic moments (spins) is always present. How-ever its strength is usually small compared to other in-teractions such as exchange and RKKY interactions be-tween the spins. The thermal-average magnitude of theinteraction energy is of order E ∼ µ /r , where µ isthe thermal-average value of the magnetic moment and r is the distance between nearest-neighbor spins. Tak-ing, e.g., µ = 7 µ B for Gd +3 or Eu +2 ( µ B is the Bohrmagneton) and r = 3 ˚A gives E/k B ∼ .
02 K ( k B isBoltzmann’s constant), which is usually very small com-pared to the other interactions between the spins. How-ever, even when the dipole interactions are weak, these interactions can be decisive in determining the orienta-tions of the ordered moments in magnetic structures oflocal-moment ferromagnets (FMs) or antiferromagnets(AFMs).If the distance between local moments is large enough,the magnetic dipole interaction can dominate the ex-change interactions in local-moment insulators and resultin either FM or AFM dipolar ordering. Examples includeFM ordering between Mn +36 clusters with spin S = 12at the Curie temperature T C = 0 . O Br (Et dbm) , and AFM ordering in theface-centered cubic (fcc) diamond lattice of rare-earth R atoms ( R = Gd, Dy, Er) in R PO (MoO ) · O withN´eel temperatures T N = 0 .
01 K to 0.04 K. The theoretical study of magnetic dipole interactionsand associated magnetic structures in crystals has a longhistory. In 1946 Luttinger and Tisza solved for thepossible magnetic structures of simple cubic (sc), body-centered cubic (bcc) and fcc Bravais spin lattices aris-ing solely from classical magnetic dipole interactions,where the ordered moments all had the same magnitude(equal-moment magnetic structures). They found thatthe ground state for the sc lattice is an AFM state withpropagation vector k = ( , ,
0) r.l.u., whereas a FMstate with k = 0 is the ground state for the bcc and fcclattices if the samples are in the shape of long thin nee-dles, but AFM structures are the most stable structuresotherwise with k = ( , ,
0) r.l.u. and k = ( , , ) r.l.u.,respectively. The abbreviation r.l.u. means reciprocal lat-tice unit, where 1 r.l.u. = 2 π/a for cubic lattices and a is the cubic lattice parameter. Cohen and Keffer con-firmed using spin-wave theory that FM cannot be theground state at T = 0 for pure magnetic dipole inter-actions in thin needles of a sc spin lattice but can bethe ground state for bcc and fcc lattices. The magneticstructures of two-dimensional (2D) Bravais spin latticesinduced by magnetic dipole interactions have also beeninvestigated.
Luttinger and Tisza also showed that in a classical cu-bic dipolar AFM in the magnetically-ordered state attemperature T = 0 with a magnetic field H z appliedperpendicular to the easy axis of ordering, the compo-nent µ z of the ordered moment per spin in the directionof H z is proportional to H z for 0 ≤ H z ≤ H c and isequal to the saturation moment µ sat for H z > H c , where H c is termed the critical field. An expression for themagnetic susceptibility χ z = µ z /H z for 0 ≤ H z ≤ H c was given. The high-field state with H ≥ H c is a field-induced paramagnetic (PM) state in which the magneticmoments are ferromagnetically aligned in the directionof H z with µ z = µ sat . According to the Weiss molecularfield theory (MFT), precisely the same type of µ z ( H z )behavior for the perpendicular magnetization occurs forboth collinear and coplanar noncollinear AFMs with thespins interacting only by Heisenberg exchange. The sus-ceptibility parallel to the easy axis at
T < T N for dipolarAFM ordering in a uniaxial (tetragonal or hexagonal)crystal has not been calculated before to our knowledge.The so-called pyrochlore spin lattice has attractedmuch attention over the past two decades in the con-text of spin-ice compounds. This non-Bravais fcc spinlattice with 16 spins per fcc unit cell consists of a 3D net-work of corner-sharing tetrahedra formed by either the A or B sublattices of a pyrochlore-structure compound A B X or by the B sublattice of a spinel-structure com-pound AB X . An example is the Ho sublattice in thepyrochlore compound Ho Ti O , where due to crystallineelectric field effects the Ho cations behave at low T likeIsing spins that can only point along the [111] and equiv-alent crystal directions (the Ti +4 cations are nonmag-netic). The spin-ice arrangement of the Ho momentsat low T gives rise to a macroscopic degeneracy and anonzero spin entropy at T = 0, as occurs in water ice.Magnetic dipole interactions between the Ho momentshave been determined to be important to this magnetic behavior, and hence these compounds are sometimesreferred to as dipolar spin ices.On another front, dynamic magnetic fluctuations inlong-range ordered 3D AFMs mediated by magneticdipole interactions are stronger than for exchange inter-actions on the same lattice, contrary to what mighthave been anticipated from the classical origin of themagnetic dipole interaction. In particular, in the cubicdiamond lattice dipolar AFMs R PO (MoO ) · O( R = Gd, Dy, Er), White et al. found that the suppressionof the T → /S ( S is the effectivespin quantum number) due to quantum fluctuations wasa factor of two stronger than predicted for the nearest-neighbor Heisenberg model on the diamond lattice. Cor-ruccini and White found that within spin-wave theory,the 3D sc lattice exhibits quantum corrections to theN´eel state that are also a factor of two larger than thoseof the nearest-neighbor Heisenberg AFM on the samelattice, indicating that dipolar magnets are more quan-tum mechanical than generally suspected, whereas the2D dipolar square lattice does not exhibit long-range or-der at finite temperature. On the other hand, severalauthors have found that dipolar interactions in conjunc-tion with Heisenberg interactions can induce long-rangeorder at finite temperatures on 2D spin lattices.
The influence of magnetic dipole interactions on themagnetic properties of 3D Bravais spin lattices with unitcell symmetries lower than cubic has been discussed forparticular cases. Rotter has discussed the predictions ofdipolar interactions for the easy axis of collinear AFMswith AFM propagation vectors k that are determined byisotropic Heisenberg exchange interactions in a variety ofcollinear AFM compounds containing sc, fcc, hexagonaland body-centered tetragonal (bct) Gd sublattices. Hefound that in most cases the easy axis is consistent withthat predicted for magnetic dipole interactions. Severalauthors calculated the local dipolar fields at a lattice sitefor general simple tetragonal and bct Bravais spin latticesversus a parameter not proportional to the c/a ratio.
Maurya et al. calculated the influence of magnetic dipoleinteractions on the magnetization versus field isothermsof three AFMs containing Eu +2 spins-7/2 below theirN´eel temperatures of 12 K to 15 K. Classical Monte Carlo (MC) simulations on Heisenbergspin systems have been carried out on a variety of spinlattices to examine the influence of magnetic dipole inter-actions on the properties with either dipolar interactionsonly or in combination with other spin interactions. Forpurely dipole interactions, Bouchaud and Z´erah studiedFM on the fcc lattice and determined the Curie temper-ature T C . They studied the critical exponents at T C and determined the anisotropy constants K and K ,where they found that the ordered moment direction inthe collinear FM state at the lowest T was along [100],with a crossover from [111] at higher T . Tomita reportedMC simulations on 2D triangular, square, honeycomband kagom´e spin lattices with only dipolar interactionsand studied the ground state magnetic structures andcritical phenomena. One result was that the kagom´elattice has a FM ground state with 1/3 of the spins dis-ordered at T = 0 [an amplitude-modulated (AM) state]with residual entropy (“missing entropy”) at T = 0 re-sulting from macroscopic degeneracy of the ground state.[An AM magnetic structure is one where the magnitudeof the ordered moment is not the same for all (identical)spins in the spin lattice.] Very recent MC simulations onthe kagom´e lattice by Holden et al. and Maksymendo etal. instead found a noncollinear coplanar equal-momentground-state magnetic structure on the kagom´e latice.Thus when an AM magnetic structure is obtained the-oretically for a particular spin lattice, this may indicatethat a lower-energy equal-moment magnetic structure ex-ists in which the moments have their maximum (satura-tion) value. Other MC simulations examined the influ-ence of dipolar interactions on the properties in combi-nation with other interactions.
In this paper previous work on the effects of mag-netic dipolar interactions on the magnetic and thermalproperties of magnetic systems is significantly extended.Usually exchange and/or RKKY interactions are strongerthan dipole interactions and determine the nature (FM orAFM) and k of the magnetic structure. However, whenthe exchange interactions are Heisenberglike (isotropic),some sort of anisotropy is needed to determine the di-rections of the ordered moments in the ordered state asdiscussed above, even if very weak compared to the ex-change interactions. The present work was initially mo-tivated by the lack of systematic studies of this topic foruniaxial tetragonal and hexagonal Bravais spin latticesversus the c/a ratio to compare with experimental re-sults such as for the Eu +2 spins S = 7 / Sb on a bct sublattice that exhibit collinear AFM orderingbelow T N = 5 . We study the influence of dipolar interactions on themagnetic ordering temperature T m , on the collinear or-dered moment directions and the temperature T depen-dence of the ordered moment and other properties at T ≤ T m , and on the Weiss temperature in the Curie-Weiss law at T ≥ T m in a systematic way for a varietyof spin lattices including 1D, 2D and 3D spin lattices us-ing our recent formulation of the Weiss MFT. All spinsin a given system are assumed to be identical and crys-tallographically equivalent. The 3D spin lattices studiedhere include sc, bcc, fcc, simple tetragonal, bct and sim-ple hexagonal (triangular) Bravais lattices. Non-Bravaisspin lattices are also studied which include the honey-comb (chickenwire) lattice, the kagom´e lattice and theShastry-Sutherland lattice. For the uniaxial stacked lat-tices, the eigenvalues and eigenvectors of the magneticdipole interaction (MDI) tensor are calculated for c/a ra-tios from 0.5 to 3. We utilize an appropriately modifiedtheory to calculate the properties of noncollinear AFMstructures and compare the results with calculations as-suming collinear AFM structures for the same k . WithinMFT, the contributions of different sources of molecularfields to the Weiss temperatures and the magnetic order- ing temperatures are additive. Therefore, for example,when dipolar and exchange interactions are simultane-ously present, one can calculate the dipole contributionsto good accuracy and then subtract them from the ob-served values to obtain the contributions from the ex-change interactions. Then with a model for the exchangeinteractions one can estimate their values. In addition tocalculating the magnetic and thermal properties of puredipolar magnets, the anisotropy in the susceptibility ofHeisenberg AFMs in the PM regime with T ≥ T m is alsocomputed. We compare our predictions to the magneticproperties measured for illustrative real materials. In thispaper we do not consider critical phenomena, domainformation and similar effects in FMs or other potentialsources of magnetic anisotropy in a spin system such assingle-ion effects.Our theoretical framework allows easy extensions tocalculate the dipolar contributions to the magnetic prop-erties of spin lattices not discussed here such as collinearor noncollinear ordering on orthorhombic, monoclinicand triclinic Bravais or other non-Bravais spin lattices.In Sec. II we first write down the expressions relat-ing the macroscopic magnetic induction, applied mag-netic field and magnetization including shape (demag-netizing) effects. The expression for the local field seenby a spin is discussed in Sec. II B. The part of that lo-cal field (the near field) due to discrete moments insidea macroscopic Lorentz sphere is discussed in Sec. II C,together with the energy of a spin interacting with thenear field. Applications of the general theory in Sec. II Cto magnetically-ordered states in collinear magnets, non-Bravais spin lattices and coplanar noncollinear helical orcycloidal AFMs are presented in Secs. II D, II E and II F,respectively. The expression for the near field due to mo-ments within a Lorentz line (1D), circle (2D) or sphere(3D) is discussed in Sec. II G. Some details about cal-culations of the MDI tensor are given in Sec. II H. InAppendix A some information useful for implementingthe theory in Sec. II is discussed.The calculations of the eigenvalues and eigenvectorsof the MDI tensor for collinear magnetic structures withspecific magnetic propagation vectors for 1D and 2D spinlattices are given in Secs. III, where the 2D spin latticesinclude the square, triangular, honeycomb and kagom´elattices. Three-dimensional spin lattices are consideredin Sec. IV, where results are given for the three cubicBravais lattices, the two tetragonal Bravais lattices, thesimple hexagonal lattice and the honeyomb lattice. Forthe 3D tetragonal and hexagonal lattices the eigenvaluesand eigenvectors are obtained versus the c/a ratio from c/a = 0 . As and BaFe As , and for the bct spin lattices in GdCu Si ,EuCu Sb and MnF . For these cases we compare theresults of the eigenvalues and eigenvectors versus the c/a ratio in graphical format with the experimental data, andthe graphical results for other cases are placed in Ap-pendix C. The treatment of noncollinear AFMs is pre-sented in Sec. V, with application to the 120 ◦ orderingon the triangular lattice, to the 90 ◦ ordering on the dis-torted Shastry-Sutherland GdB compound and to theundistorted 2D and 3D Shastry-Sutherland lattices.Section VI presents the calculation of the FM order-ing temperature T C and AFM ordering temperature T N arising from dipolar interactions within our recent for-mulation of MFT. A quantitative discussion of thecompetition between FM and AFM ordering on cubicBravais lattices versus the demagnetization factor of asample in the absence of FM domain formation is givenin Sec. VII. The properties of dipolar magnets in themagnetically-ordered state are derived in Sec. VIII. Theordered moment and heat capacity of dipolar magnets inzero magnetic field versus temperature are presented inSec. VIII A, where the results are the same within MFTfor both FMs and AFMs. The dipolar anisotropy param-eter K for uniaxial dipolar AFMs versus temperatureis derived in Sec. VIII B. Calculations of the perpen-dicular susceptibility below T N and the associated criti-cal field for uniaxial AFMs are presented in Secs. VIII Cand VIII D, respectively.The Curie-Weiss law for dipolar magnets in the PMstate is derived in Sec. IX, where the Weiss temperatureis found to be anisotropic in general. In Sec. X we spe-cialize to spherical samples of collinear AFMs, where theanisotropic susceptibilities χ for temperatures above T N as well as both the parallel and perpendicular suscepti-bilities below T N are presented and discussed. Examplesof these anisotropic χ ( T ) behaviors are given in Sec. X Dfor simple tetragonal lattices with c/a < c/a = 1 (sc)and c/a > χ ( T ) of a Heisenberg-exchange AFMat T > T N due to MDIs is derived in Sec. XI and appliedto fit the experimental data for single-crystal MnF . Thepaper concludes with a short summary in Sec. XII.Tables of values of the dipolar eigenvalues and eigen-vectors versus c/a plotted in the text and Appendix Care available in the Supplemental Material. II. THEORY
The magnetization per unit volume of magnetic ma-terials can be significant compared to the applied fieldand results in a demagnetizing field and an internal fieldsmaller than the applied field. In the following the theoryfor this important demagnetizing correction is discussedwithin the Gaussian cgs system of units that is used throughout this paper.
A. Macroscopic Fields
We initially assume that a sample has the shape ofan ellipsoid of revolution and that the applied field isalong one of the three principal axes α . Then the volumemagnetization (net magnetic moment per unit volume) M (units: G) is uniform in the sample and the magneticinduction B (units: G), the magnetic field H (units: Oe= G) and M are collinear with components M α , H α and B α for the external field H α applied along the α axis.For each point in space one has B α = H α + 4 πM α . (1a)Thus internal to the sample one has B int α = H int α + 4 πM α . (1b)The demagnetizing field internal to the sample due to M α is H d α = − πN d α M α , (1c)where here the demagnetizing factor N d α is defined asin the Syst`eme International system of units for which0 ≤ N d α ≤ P α =1 N d α = 1. Thus the internalmagnetic field H int α and the magnetic induction B shapeint α due to sample shape effects and including the appliedfield H α are H int α = H α − πN d α M α , (2a) B shapeint α = H int α + 4 πM α = H α + (1 − N d α )4 πM α . (2b)For a given M α , the internal field is H int α in Eq. (2a).Thus in descriptions of the magnetic behavior of a samplein terms of M α and H α , one can correct for the demag-netizing field by retaining the measured value of M α butreplacing H α by H α − πN d α M α , where N d α is estimatedfrom the sample shape and the field orientation with re-spect to the sample (see below).The magnetic susceptibility of a material is oftendefined as χ = M ( H ) /H , which in general is field-dependent. In the present discussion, M is the volumemagnetization, so χ is the susceptibility per unit vol-ume and is dimensionless. The observed susceptibilityis then χ obs α = M α /H α and the intrisic susceptibility is χ α = M α /H int α . Utilizing Eq. (2a) one obtains χ α from χ obs α according to χ α = M α H int α = M α H α − πN d α M α = M α /H α − πN d α M α /H α = χ obs α − πN d α χ obs α . (3)At each temperature one can correct the observed sus-ceptibility for the demagnetizing field using Eq. (3).Alternatively, using Eq. (3) one can write the observedsusceptibility in terms of the intrinsic one as χ obs α = χ α πN d α χ α . (4)Thus when 4 πN d α χ α ≫
1, one obtains the field-independent susceptibility and linear M α ( H α ) behavior χ obs α = 14 πN d α , M α = 14 πN d α H α . (5)The latter behavior holds until M α reaches it saturation(maximum) value M sat α ; at higher fields M α is of courseequal to its constant saturation value. In practice, thelimiting behaviors in Eqs. (5) are realized only when amaterial is approaching its FM transition temperaturefrom above.An expression for the demagnetizing factors N d α forthe general ellipsoid of revolution was calculated longago. For sample shapes other than ellipsoids, M isnot uniform within the sample except for limiting cases.What is then relevant in the present context is the de-magnetizing field averaged over the sample volume asexpressed in the associated “magnetometric” demagne-tizing factor. Such sample shapes include the cylinderand the rectangular parallelepiped (rectangular prism)for which the magnetometric N d α values have been calcu-lated for arbitrary sample dimensions in Refs. 35 and 36,respectively. B. Local Magnetic Induction from MagneticDipole Interactions
Theoretical predictions of magnetic properties for localmagnetic moments are often cast in terms of the localmagnetic induction B localint i seen by a local moment ~µ i atposition r i . This local magnetic induction along a givenprincipal axis α is traditionally written for a 3D spinlattice in terms of the four contributions B localint α i = B α + B shapeint α + B Lorentzint α + B nearint α i , (6)where B α = H α is the applied magnetic induction aris-ing from currents outside the sample and B shapeint α is thecontribution in Eq. (1b) due to the sample shape. Thecontribution B Lorentzint α = 4 π M α (7)is the Lorentz cavity field inside a spherical cavity of ra-dius R surrounding the point at its center at position r i at which B localint α i is to be calculated. The fourth contribu-tion B nearint α i is the sum of the dipolar fields at position r i arising from the other magnetic dipoles inside the Lorentzcavity at positions r j . This is the only term that dependson the crystal structure of the material. The Lorentzcavity radius R is much larger than the distance betweenmagnetic moments in a sample and is large enough so that the calculated B nearint α i becomes independent of R towithin some specified precision. Substituting Eqs. (2b)and (7) into (6) gives B localint α i = H α + (cid:18) − N d α (cid:19) πM α + B nearint α i . (8a)This is an important fundamental equation for calculat-ing the local field.Two special cases of Eq. (8a) are of use. In the first, onecorrects the applied field for the demagnetizing field inthe measurements as described above which is equivalentto removing N d α from Eq. (8a), yielding B localint α i = H α + 4 π M α + B nearint α i . (8b)This equation is sometimes favored for comparison oftheoretical predictions of the dipolar magnetic proper-ties with experimental data because it is independent ofsample shape. Here M α is the total magnetic momentper unit volume. If all spins are identical and crystallo-graphically equivalent as assumed throughout this paper,one can write M α = µ α /V spin where µ α is the net averageordered and/or induced moment per spin in the α direc-tion and V spin is the volume per spin, so an equivalentform of Eq. (8b) is B localint α i = H α + 4 π V spin µ α + B nearint α i . (8c)Note that M α = µ α = 0 for an AFM in H = 0.Alternatively, one can shape a sample into a sphere,giving N d α = 1 / α , andthen Eq. (8a) becomes B localint α i = H α + B nearint α i , (8d)which eliminates the effect of the Lorentz field but onlyapplies to a spherical sample. This formulation is de-sirable if one wishes to ameliorate the tendency of theLorentz field to enhance dipolar FM ordering with re-spect to AFM ordering, as illustrated in Fig. 13 belowwhere FM is favored for small values of N d α for bcc andfcc Bravais lattices. C. Magnetic Induction Due to Collinear Alignmentof Magnetic Dipoles Inside Lorentz Cavity
The magnetic induction B ij seen by a central moment ~µ i at a position r i due to a point magnetic dipole moment ~µ j at position r j is B ij = 1 r ji [3( ~µ j · r ji ) r ji − r ji ~µ j ] , (9a)where r ji = r j − r i , r ji = | r ji | . (9b)The energy of interaction E i of ~µ i at position r i dueto the magnetic induction B ij is E i = − ~µ i · B ij = − r ji [3( ~µ i · r ji )( ~µ j · r ji ) − r ji ~µ i · ~µ j ] , (10) where the factor of 1/2 in the first equality recognizesthat the interaction energy of the ~µ i with B ij from ~µ j isequally shared between ~µ i and ~µ j . Expanding the firstterm on the right side of Eq. (10) in Cartesian coordi-nates, one can write the term in matrix form as( ~µ i · r ji )( ~µ j · r ji ) = ( µ ix µ jy µ jz ) r jix r jix r jiy r jix r jiz r jix r jiy r jiy r jiy r jiz r jix r jiz r jiy r jiz r jiz µ jx µ jy µ jz = ~µ T i r ji r ji ~µ j , (11)where ~µ T i is the transpose of the column vector ~µ i , ~µ j isa column vector and r ji r ji is a 3 × ~µ i · ~µ j = ~µ T i ~µ j , (12)where is the 3 × ~µ j within a length of chain (1D), a circle of specified radius(2D) or Lorentz sphere (3D), all centered on ~µ i , and thencan be succinctly written in matrix form as E i = − ~µ T i G i ~µ j . (13a)where the 3 × G i is G i = X j = i r ji (3 r ji r ji − r ji ) . (13b)In order to solve Eq. (13a) for the eigenenergies E i andeigenvectors ˆ µ i of the tensor G i , one must first expresseach ~µ j in terms of ~µ i . In the following three sections wediscuss our methods for doing so for collinear magneticstructures on Bravais and non-Bravais spin lattices andcoplanar noncollinear AFM structures, respectively. D. Collinear Magnetic Structures
In this section we consider collinear magnetic struc-tures with magnetic wavevector k where ~µ j = cos( k · r ji ) ~µ i . (14)Since the cosine function is a scalar with a value between ±
1, Eq. (14) expresses that ~µ j can be either parallel orantiparallel to ~µ i . For cos( k · r ji ) = ± ~µ j themagnetic structure is an “equal-moment” (EM) structurewhere the ordered moments all have the same magnitude µ (which depends on T ). For cos( k · r ji ) = ± ~µ j , µ depends on j and the structure is a collinear AMAFM structure. Collinear magnetic structures include both FM ( k = 0) and AFM structures below the magneticordering temperature T m and the FM-aligned magneticstructure induced above T m by an external magnetic fieldapplied along one of the three principal axes of the MDIin Eq. (16c) below. From Eq. (14) one obtains the “ex-tinction condition” ~µ j = 0 if k · r ji = odd multiple of π , (15)as in Eq. (A6) for AM AFM structures associated withspecific k values and spin lattices. A general k corre-sponds to either an EM or AM collinear AFM structure.All simple Bravais lattices have EM magnetic struc-tures. Amplitude-modulated AFM structures occurwhen the simple Bravais lattices have more than onespin in the unit cell such as for bcc, bct and fcc spinlattices. With the cos( k · r ji ) term as given in Eq. (14),EM structures occur for bcc and bct lattices with k = (cid:0) , , (cid:1) , (cid:0) , , (cid:1) and (001) r.l.u., and AM structuresfor k = (cid:0) , , (cid:1) and (cid:0) , , (cid:1) r.l.u. For the fcc lattice,EM structures occur for k = (cid:0) , , (cid:1) and (0,0,1) r.l.u,whereas AM structures occur for k = (cid:0) , , (cid:1) , (cid:0) , , (cid:1) , (cid:0) , , (cid:1) and (cid:0) , , (cid:1) . With the exception of the last one,all AM structures considered can be converted into EMstructures by inserting an additive phase in the cosineterm: cos( k · r ji ) → cos( k · r ji + φ ), where φ = π/ π/
4) = 1 / √
2, which corresponds to a re-duction in the ordered moment by a factor of 1 / / . Alleigenvalues plotted or listed in this paper were obtainedfor φ = 0.In pure magnetic dipole AFMs, the above discussionshows that the AFM ground state can be an AM state,depending on the AFM wavevector. However, even insystems in which the magnetic dipole interaction is notexpected to play an important role, this interaction canstill cause a small modulation of the ordered momentversus position in the magnetic unit cell. Furthermore,large-amplitude AM AFM structures are observed ingeometrically-frustrated systems such as in Gd Ti O . Because AM structures contain at least some fraction ofspins with ordered moments less than the saturation mo-ment and hence show strong quantum fluctuations in theground state, the entropy increase on heating from lowtemperatures would be less than the value R ln(2 S + 1)per mole of spins. This can be checked by calorimetry.The discussion throughout this paper applies to iden-tical crystallographically-equivalent spins with identicalsaturation moments µ sat and with thermal-average (or-dered) magnetic moments ~µ i = µ ˆ µ i , where µ can be dif-ferent for different spins in AM structures. We express r ji in units of the lattice parameter a of the respectivecrystal structure. The crystallographic unit cell oftencontains more than one spin per unit cell in the exam-ples described. Then using Eq. (14), Eqs. (13) become E i = − ǫ ˆ µ T i b G i ( k )ˆ µ i , (16a)where ǫ = µ a (16b)has dimensions of energy and the dimensionless symmet-ric MDI tensor is b G i = X j = i r ji /a ) (cid:18) r ji r ji a − r ji a (cid:19) cos( k · r ji ) . (16c)Labeling the eigenvalues of b G i ( k ) as λ k α , Eq. (16a) givesthe eigenenergies as E iα = − ǫ λ k α , (16d)where the subscript α refers to a Cartesian principal or-dering axis eigenvector of the collinear magnetic struc-ture, where the three principal axes are orthogonal toeach other. Thus the ground state energy and order-ing axis for a given k due to the MDI corresponds tothe largest of the three λ k α eigenvalues. The MDI en-ergy scale is set by the value of ǫ in Eq. (16b) which issystem-dependent. The value of ǫ/k B is typically of order0.01–0.1 K.The magnetic propagation vector k must be specifiedin terms of the reciprocal lattice translation vectors inCartesian coordinates in advance of computing b G ia ( k ).One can calculate the λ k α eigenvalues and correspond-ing eigenvectors (ordered moment axes ˆ µ i ) for various k vectors, including k = 0 for FM-aligned moments whichmay occur due to FM ordering in applied field H = 0 orto H > k vec-tor observed by, e.g., neutron diffraction measurements,is determined by exchange or RKKY interactions ratherthan dipole interactions. In that case one can still testwhether the easy axis predicted by the MDI is consistentwith the observed one. A negative answer would indi-cate that the MDI does not contribute to determiningthe easy axis, and hence some stronger source of magne-tocrystalline anisotropy must be present that overcomesthe preference of MDIs. A positive answer would mean that the MDI at least contributes to ordering along theobserved easy axis; however, this does not rule out othersources of anisotropy that may also contribute.A general feature of the eigenvalues λ k α of the MDItensor b G i for a given k and spin lattice is that their sumover the three eigenvectors α is identically zero when no apriori constraint is placed on the ordering axis of ~µ i . Thissum rule is violated when such a constraint is imposedsuch as for coplanar noncollinear helical or cycloidal AFMorder as discussed in Secs. II F and V A. In those cases,one of the λ k α is the eigenvalue for FM ordering ( k = 0)along the helix or cycloid axis. The other two eigenvaluesand corresponding eigenvectors are the ones associatedwith the actual AFM components of the helix or cycloid. E. Non-Bravais Spin Lattices
A crystal structure consists of a Bravais lattice plusa basis of atoms attached to each Bravais lattice point.Non-Bravais spin lattices are Bravais lattices with morethan one spin in the basis. These include, e.g., the fccdiamond lattice and the 2D hexagonal honeycomb (orchickenwire) lattice, each with two spins in the basis,and the kagom´e lattice with three spins in the basis. Insuch cases one must modify Eq. (16c) to include a sumover the atoms in the basis, in addition to the sum overBravais lattice points already included in Eq. (16c) via,e.g., Eqs. (A1). AFM structures in such non-Bravais spinlattices include those with AFM propagation vector k =0 for N´eel-type ordering on the 2D honeycomb lattice,which is the same propagation vector as for FM order-ing. In such AFM structures where the magnetic andcrystallographic unit cells are the same, in order to cal-culate b G i one must specify the orientations of the orderedmoments within a unit cell with respect to the orienta-tion of a central moment ~µ i . Thus Eq. (16c) is modifiedto read b G i = X j X k r jki /a ) (cid:18) r jki r jki a − r jki a (cid:19) R ki , (17a)where the sum over j again refers to the sum over theBravais lattice positions, the sum over k sums over allatoms in the basis, the position r i of the central moment ~µ i is not necessarily at the origin of of a central unit cell,and the vector from ~µ i to a moment ~µ k is r jki = r j + r k − r i , (17b)where r k is the position of moment ~µ k in the basis withrespect to the position of the associated Bravais latticepoint r j . The term with r jki = 0 is omitted from the sumbecause that term corresponds the difference in positionof moment ~µ i with itself. The Cartesian rotation matrix R ki in Eq. (17a) expresses the moment direction of ~µ k inthe basis with respect to that of the central moment ~µ i via ~µ k = R ki ~µ i , (17c)similar to Eq. (14) for collinear ordering associated witha magnetic propagation vector k . Prior to calculating b G i , the 3 × R ki rotation matrix must be specified foreach spin in the basis via a model for the AFM structure.For example, for the N´eel AFM structure in Fig. 1 below,if ~µ i were at a red position r i /a = ˆ a + ˆ b , then R i fora spin at another red position would be R ki = andthat for a black position would be − , where again is the 3 × b G i for the known coplanar noncollinear AFM structure oftetragonal GdB in Fig. 12 containing four moments inthe basis, each pointing in different directions, and forthe related Shastry-Sutherland spin lattice. F. Coplanar Noncollinear Helical or CycloidalAntiferromagnets
Here we extend the above discussion to coplanar non-collinear helical or cycloidal AFM ordering on tetragonalor hexagonal Bravais lattices. For both types of AFMorder, the ordered moments are defined to lie in the crys-tallographic ab plane. For helical AFM ordering, the or-dered moments are FM-aligned in the ab plane and thehelix wavevector k axis is the c axis. For cycloidal AFMordering, k lies in the ab plane and the moments in planesperpendicular to both k and the ab plane are aligned fer-romagnetically. The Cartesian x -axis is parallel to a ,the y -axis is perpendicular to a in the ab plane and the z axis is perpendicular to the ab plane along the c axis.Pictures of the helical and cycloidal structures are givenin Refs. 40 and 41, respectively. In either structure, theazimuthal angle φ ji = φ j − φ i with respect to the posi-tive a axis between moments ~µ j and ~µ i in the ab planeis given by φ ji = k · r ji . (18a)The relationship between the central moment directionˆ µ i at position r i and that of another moment at position r j in either AFM structure isˆ µ j = cos φ ji φ ji
00 0 1 ˆ µ i , (18b)which can be writtenˆ µ j = (ˆ x ˆ x cos φ ji + ˆ y ˆ y sin φ ji + ˆ z ˆ z )ˆ µ i , (18c)where the Cartesian coordinate system is used through-out. Then b G i in Eq. (16c) becomes b G i = X j = i r ji /a ) (cid:18) r ji r ji a − r ji a (cid:19) (19) × (ˆ x ˆ x cos φ ji + ˆ y ˆ y sin φ ji + ˆ z ˆ z ) . As with collinear AFM ordering, one must specify k in terms of the reciprocal lattice translation vectors inCartesian coordinates in advance of computing b G i ( k ).Note that when b G i ( k ) is diagonalized, one eigenvalue andcorresponding eigenvector are for FM ordering along the z axis and are not relevant to those for the helix, whereasthe other two sets of eigenvalues and eigenvectors are forthe helix. As a result, the sum of the three eigenvalues donot add to zero as they do for all other AFM structuresdiscussed above. G. Near Field
The value of B nearint α i in Eq. (6) that is seen by a givenmoment ~µ i at position r i in a given magnetic structurewith a given ordered moment configuration, due to thesum of the magnetic fields from the magnetic momentsaround it within the Lorentz cavity of radius R , is simplygiven as B nearint α i = − E i µ α = µλ k α a , (20)where the factor of 2 arises because the energy per pairis split evenly between each pair of moments, whereasthe magnetic field arises only from the neighbor of eachpair, the second equality was obtained using Eqs. (16b)and (16d) and B nearint α i can be either positive or negative,depending on the sign of λ k α . If the MDI is the onlysource of anisotropy present, this field must be positivebecause then the ordered moment is parallel to the localmagnetic induction, which minimizes the free energy ofthe moment. The quantity B nearint α i is needed to calcu-late the total local magnetic induction at the site of alocal moment according to Eq. (8a). If E i is expressedin cgs units of erg and those of µ in cgs units of erg/G(= G cm ), then B nearint α i has the correct cgs units of G.Using Eq. (20), the total local field in Eq. (8b) seen bycentral moment ~µ i becomes B localint α i = H α + (cid:18) πµ α /µ V spin /a + λ k α (cid:19) µa , (21)where the first term in parentheses is the Lorentz field,where we distinguish the moment component µ α in the α direction per spin averaged over the sample and themagnitude µ of the average moment per spin. A nonzerovalue of µ α only occurs in a FM or in an AFM in thepresence of an external magnetic field. The second termin parentheses arises from the near field. H. Calculation and Diagonalization of theMagnetic Dipole Interaction Tensor
We chose to carry out the sums in the expressionsfor the dipole interaction tensor b G i in Eqs. (16c), (17a)and (19) directly instead of by using the Ewald-Kornfeldtechnique, because we wanted to study the convergenceproperties of the eigenvalues λ k α versus the radius R ofthe circle or Lorentz sphere for 2D and 3D lattices, re-spectively. The calculations and diagonalizations of b G i were carried out using standard Macintosh laptop anddesktop computers and
Mathematica software. For the1D chain with FM and N´eel AFM states, the eigenvaluesand eigenvectors of b G i are trivially determined exactlyfor the infinite chain as shown in Sec. III A. For 2D lat-tices the sums were carried out within circles of radiusup to R/a = 1000 containing up to 1 × spins (forthe kagom´e lattice containing three spins per unit cell).For the 3D lattices the sums were carried out within aLorentz sphere, usually up to a radius R/a = 50 contain-ing up to 6 × spins. Calculations were also done fortwo AFM structures out to a sphere radius of R/a = 100containing 1 . × spins to check convergence.For the FM spin structures in 2D, the values of b G i versus 1 / ( R/a ) were extrapolated to 1 / ( R/a ) = 0. Asshown in Appendix B, the calculations of b G i for AFMstructures generally converge more rapidly with increas-ing R/a than for FM structures. These procedures deter-mined λ k α to accuracies of < ∼ ± − for 2D lattices and < ∼ ± .
001 for 3D lattices, more than sufficient for ourpurposes. The eigenvectors ˆ µ i usually converged veryquickly with increasing R/a . For the various 3D tetrag-onal and hexagonal lattices, b G i was calculated for c/a ratios from 0.5 to 3 in 0.1 increments.Figures 17 and 18 in Appendix B show the convergenceof λ k α with increasing R/a for FM and N´eel AFM mo-ment alignments along the c axis in the 2D simple squarelattice, respectively. Figures 19(a) and 19(b) show plotsfor a simple tetragonal lattice with FM alignment of themoments along the c axis for c/a = 1 . k = ( , , ) and alignment of the momentsalong the c axis for c/a = 1 . λ k α were typically obtained for R/a = 1 to 50 in incrementsof 1 and the last 10 or 20 values were averaged to obtainthe data in the figures in the text and Appendix C andin the tables in the Supplemental Material. III. EIGENVALUES AND EIGENVECTORSFOR MAGNETIC ORDERING ON ONE- ANDTWO-DIMENSIONAL SPIN LATTICESA. Spin Chain
We assume that the spin chain lattice is oriented alongan axis desigated as the a axis ( x axis) with spacing a between adjacent spins, so r ji a = n a . (22) Ferromagnetic alignment corresponds to k = 0. Thisalignment can occur either in the FM-ordered state orin the PM state in the presence of an applied mag-netic field. The central spin is positioned at n a = 0,so n a of the neighbors runs from −∞ to ∞ , excluding n a = 0. Numerical diagonalization of b G i in Eq. (16c)with k = 0 and | n max x | = 1000 (2000 neighbors of thecentral moment) shows that the principal axes of theinteraction tensor are parallel and perpendicular to the a axis. The lowest energy configuration with a calcu-lated λ (0 , , = 4 . a (chain) axis. This makes sense be-cause the lowest energy configuration of a moment iswhen each moment points along the local field seen bythe moment, which is along the axis of the chain. Theeigenvalues for the two higher-energy orthogonal direc-tions are λ (0 , , = λ (0 , , = − λ (0 , , / λ (0 , , , , exactly. Equation (16c) yields theeigenvalue λ (0 , , , , = 4 ∞ X n a =1 n a . (23)The sum is P ∞ n x =1 = ζ (3), yielding λ (0 , , , , = 4 ζ (3) ≈ .
808 228 (24)as shown in Table I, where ζ ( z ) is the Riemann zetafunction with ζ (3) ≈ . λ (0 , , , , agrees with this exactvalue to six-place accuracy. This shows that the value of | n max a | = 1000 and a spin chain containing 2000 neighborsof the central moment used in the numerical calculationis sufficient to obtain this accuracy.It is of interest to examine the approach to the infinite-chain limit of λ (0 , , , , on increasing | n a | . For large | n a | one can replace the sum in Eq. (23) in the regionwhere n a is large by an integral R n − a dn a ∝ − /n a . Thuswe expect for n a ≫ λ (0 , , , , = 4 ζ (3) − An a = 4 ζ (3) (cid:20) − A ζ (3) n a (cid:21) , (25)where A is a positive constant. An exact series ex-pansion of the sum in Eq. (23) about n a = ∞ indeedgives λ (0 , , , , = 4 ζ (3) − /n a + O ( n − a ), yielding A = 2. Equation (25) then predicts six-place accuracyfor λ (0 , , , , for | n a | = 1000, consistent with the abovecomparison.Here we also examine the N´eel-type AFM wavevector k = (1 / , ,
0) r.l.u., where 1 r . l . u . = 2 π/a is the re-ciprocal lattice unit for this spin lattice. A numericalcalculation using Eq. (16c) shows that the eigenvalues ofEq. (16d) converge to six significant figures even with asmall | n a | max = 70. These calculations also show thatthe most stable ordered moment direction is perpendic-ular to the chain with λ (1 / , , , , = λ (1 / , , , , = 1 . x -axis direction has λ (1 / , , , , = − λ (1 / , , , , = − . . (27)An exact calculation for the a -axis eigenvalue is ob-tained using the AFM version of Eq. (23), yielding λ (1 / , , , , = 4 ∞ X n a =1 ( − n a n a = − ζ (3) ≈ − . , (28a)as listed in Table I. This value is identical to withinsix places with the numerical result for λ (1 / , , , , ob-tained above using only a 141-spin chain (including thecentral spin). Thus the dipole fields seen by a centralmoment converge much faster with increasing n max a forthe AFM structure than for FM one for the spin chain.The two lower-energy eigenvalues are λ (1 / , , , , = λ (1 / , , , , = − λ (1 / , , , , = 32 ζ (3) ≈ . . (28b)Comparing Eqs. (24) and (28b) shows that the eigen-value for the FM-aligned state with the ordered/inducedmoments aligned along the a axis is larger than the max-imum AFM one, and hence the energy per spin is smalleraccording to Eq. (16d) for the FM state than for the AFMstate. The FM state is thus expected to be the magneticground state of the linear spin chain for purely dipolarinteractions provided the ordering is not destroyed byquantum fluctuations. B. Two-Dimensional Spin Lattices
For the simple square lattice, one has a = b and thespin positions given by the first two terms in Eq. (A1a).The normalized wavevectors are the first two terms inEq. (A4). The largest (positive) eigenvector λ k α of b G ia ( k ) in Eq. (16c) with k = 0 for FM alignmentoccurs for the a or b easy axes, with λ (0 , , , , = λ (0 , , , , and λ (0 , , , , = − λ (0 , , , , . Shownin Fig. 17(a) in Appendix B is the dependence of λ (0 , , , , on the inverse radius R − for the c -axiseigenvalue λ (0 , , , , . According to the discussion inSec. III A, in 2D one should have λ (0 , , , , ( R/a ≫
1) = λ (0 , , , , ( a/R = 0) + A/R , in agreement withthe calculations in Fig. 17(a). Fitting the data for0 . ≤ a/R ≤ .
002 gives λ (0 , , , , ( a/R = 0) = − . , (29) A = 6 . . The deviations of the data from the fit are shown inFig. 17(b), where it is seen that the deviations are ofthe order of 1 part in 10 for 0 . ≤ a/R ≤ . a/R . ab FIG. 1: (Color online) Honeycomb lattice. Each honeycombcell (not a unit cell) is bounded by solid blue lines. The hexag-onal unit cells with translation vectors a and b are outlinedby dashed black lines. The 2D space group is p m (No. 17)with two spins in Wyckoff positions 2 b (cid:0) , (cid:1) , (cid:0) , (cid:1) . Bipar-tite N´eel ordering of the two spins per unit cell is shown. Thesolid red circles represent half the magnetic moments pointingin one direction and the open black circles correspond to halfthe moments pointing in the opposite direction. The eigenvalues and eigenvectors for square-latticeAFM propagation vectors k = (cid:0) , , (cid:1) (stripe AFM) and (cid:0) , , (cid:1) (N´eel-type AFM) were also computed as shownin Table I. One sees that of these and the FM propaga-tion vector, the stripe AFM propagation vector with theordered moments aligned along the b axis (perpendicularto k as shown in the last column of Table I) has the low-est energy. Our eigenvalues λ (0 , , and λ ( , , are in agreement with, but are more precise than, thosepreviously reported in Ref. 44, and our λ (0 , , and λ (1 / , / , values are in precise agreement with theresults in Ref. 45.For the 2D simple-hexagonal (triangular) lattice theeigenvalues and eigenvectors were calculated for the FMstate and four AFM states. From Table I, the lowest-energy states are the FM state and the AFM state with k = (1,0,0) (stripe-type), with the moments aligned withinthe ab plane in both cases. Data for the AM AFM statewith k = (cid:0) , , (cid:1) are included in Table I because theclassical ground state of a triangular lattice of spins withHeisenberg interactions is the well-known coplanar non-collinear 120 ◦ state that can be described by k = (cid:0) , , (cid:1) which we consider further in Sec. V A. Our eigenvalues λ (0 , , and λ ( , , − / , −√ / , are in agreement toseven significant figures with those previously reportedin Ref. 46.The 2D hexagonal honeycomb lattice is a non-Bravaisspin lattice containing two spins per unit cell as shownin Fig. 1. The eigenvalues and eigenvectors of b G i forcollinear magnetic ordering were calculated for this lat-tice according to the method of Sec. II E and the resultsare listed in Table I for the FM and N´eel-type (Fig. 1)AFM states, where the magnetic propagation vector forboth states is k = (0,0,0).Rozenbaum found that the ground state of the 2Dhoneycomb lattice is noncollinear with all spins aligned1 TABLE I: One- and two-dimensional spin lattices. Eigenvalues λ k α and eigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinatesof the MDI tensor b G i ( k ) in Eq. (16c) for various values of the magnetic wavevector k in reciprocal lattice units with collinear magnetic moment alignments. The most positive λ k α value(s) corresponds to the lowest energy value according to Eq. (16d).The Cartesian x , y and z axes are along the a , b and c axes of orthogonal-axis lattices, respectively. For the hexagonal lattice,the x axis is parallel to the hexagonal a axis and the y axis is perpendicular the the a axis, rather than along the b axis. Thelinear chain is aligned along the a axis and the square and hexagonal lattices are aligned in the ab plane. k λ k ˆ µ ˆ µ ˆ µ · ˆ k
1D linear chain(0, 0, 0) (FM) 4 ζ (3) ≈ .
808 228 [100] − ζ (3) ≈ − .
404 114 [010], [001] (cid:0) , , (cid:1) (N´eel AFM) ζ (3) ≈ .
803 085 [010], [001] 0 − ζ (3) ≈ − .
606 171 [100] 12D square lattice (cid:0) , , (cid:1) (stripe AFM) 5.098 873 [010] 00.935 462 [001] 0 − .
034 335 [100] 1(0, 0, 0) (FM) 4.516 811 [100], [010] − .
033 622 [001] (cid:0) , , (cid:1) (N´eel AFM) 2.645 887 [001] 0 − .
322 943 [100], [010] 1 / √ ≈ . − .
034 176 [001](1 , ,
0) 5.517 088 [100], [010] 1 − .
034 176 [001] 0( , ,
0) 4.094 909 [ , − √ ,
0] 1/21.839 029 [001] 0 − .
933 939 [ − √ , , −√ / ≈ − . , ,
0) 4.094 909 [ − , − √ , − ( √ / / ≈ − . − .
933 939 [ − √ , ,
0] 1 / √ ≈ . , ,
0) 2.331 796 [001] 0 − .
165 898 [100], [010] 1 / √ ≈ . − .
184 718 [001]( , ,
0) 12.827 051 [ − , √ , − / − .
090 183 [001] 0 − .
736 868 [ √ , , √ / ≈ . , ,
0) (N´eel-type) 12.116 366 [001] − .
058 183 [100], [010]2D hexagonal kagom´e lattice(0, 0, 0) (FM) 51.321 197 [010]11.205 800 [100] − .
526 996 [001](0 , ,
0) (ferrimagnet) 40.458 644 [001] 0 − .
171 624 [100] 0 − .
287 021 [010] 1( , ,
0) 13.213 509 [001] 09.212 253 [100] 1 / √ ≈ . − .
425 762 [010] 1 / √ ≈ . , ,
0) 4.094 910 [100] 01.839 029 [001] 0 − .
933 939 [010] 1 k λ λ = 3a/4(a) ab k λ = 3 a/2(b) ab k λ = 3 a λ (c) ab FIG. 2: (Color online) Two-dimensional hexagonal kagom´elattices. The hexagonal unit cell is shown at the lower leftof each panel oulined in heavy black lines and contains threespins. The unit cell edges a and b are twice the length of thetriangular-lattice unit cell edge. Three magnetic structuresare shown. (a) The red, blue and open circles represent mo-ments that are mutually at an angle of 120 ◦ to each otherwithin the ab plane, so a given moment has no nearest neigh-bors with the same orientation. The cycloid spin configura-tion shown has a wavevector k = (cid:0) , , (cid:1) r.l.u. (b) Collinearmagnetic structure for k = (0 , ,
0) r.l.u. The red circles rep-resent moments in one direction and the blue circles repre-sent moments in the opposite direction. Because there aretwice as many red as blue circles, this magnetic structure isa ferrimagnet (a net FM). (c) Collinear AFM structure for k = (cid:0) , , (cid:1) r.l.u. The red and blue circles have the samemeanings as in (b). There are equal numbers of red and bluecircles, but the open circles represent spins with zero orderedmoment, so the magnetic structure is an AM AFM. in the ab plane, in contrast to the collinear FM andAFM structures assumed in the above calculations. Hegave the ground state energy per spin as E/N = − µ a (4 .
453 809), where a nn = a/ √ N = 2 spins per unit cell according to Eqs. (16) givesthe eigenvalue λ = 3 / . . -depleted triangular lattice containing three spins perhexagonal unit cell. The 2D hexagonal space group ofthe kagom´e lattice is p m (No. 17), with three spins inWyckoff positions 3 c (cid:0) , (cid:1) , (cid:0) , (cid:1) , (cid:0) , (cid:1) . For thekagom´e lattice the cycloid wavevector in Fig. 2(a) is k = (cid:0) , , (cid:1) r.l.u. instead of k = (cid:0) , , (cid:1) r.l.u. for the trian-gular lattice, due to the factor of two increase in the a -and b -axis lattice parameters compared to the triangularlattice. The magnetic structure shown in the figure is thewell-known classical 120 ◦ structure for nearest-neighborAFM Heisenberg interactions. However, the ground statefor collinear magnetic ordering arising from only dipoleinteractions is seen from Table I to be a FM structurewith the moments pointing perpendicular to the plane ofthe lattice. Also shown in the table are results for twoAFM wavevectors directed along the ˆ b ∗ ( y ) direction dis-cussed next.A net FM (ferrimagnetic) collinear structure is shownin Fig. 2(b) with magnetic wavevector k = (0 , ,
0) r.l.u.There are twice as many moments pointing one way com-pared to the other way, so the net ordered FM moment is µ sat /
3, where µ sat is the saturation moment of each spin.An AM collinear AFM structure is shown in Fig. 2(c)with AFM propagation vector k = (cid:0) , , (cid:1) r.l.u. Thered and blue circles have the same meaning as in (b), butthe black open circles represent spins with no ordered mo-ment. Therefore the average AFM ordered moment perspin is 2 µ sat / y direction (vertically upwards in Fig. 2) within the ab plane. The collinear structures shown in Figs. 2(a)–(c) are significantly less stable. Classical MC simula-tions determined that the ground state magnetic struc-ture is an EM noncollinear ferrimagnetic structure withall ordered moments lying in the ab plane. Theground state energy per spin is quoted as E/ spin = − . µ /a , where a nn = a/ a nn is thenearest-neighbor spin-spin distance. In terms of our no-tation, E/ spin = − µ a λ which takes into account the3 TABLE II:
Simple cubic spin lattice.
Eigenvalues λ k α andeigenvectors ~µ = [ µ x , µ y , µ z ] in Cartesian coordinates of theMDI tensor b G i ( k ) in Eq. (16c) are listed for various valuesof the magnetic wavevector k in reciprocal lattice units with collinear magnetic moment alignments. The most positive λ k α value(s) corresponds to the lowest energy value accord-ing to Eq. (16d). Also shown are the differences between theeigenvalues for the different eigenvectors for a given k andspin lattice, which are proportional to the respective mag-netic anisotropy energies and fields. The Cartesian x , y and z axes are along the a , b and c axes of the cubic unit cell, re-spectively. The labels A-, B-, C- and G-type for the differentwavevectors are from Ref. 47. The λ k α values agree with the f – f eigenvalues in Table II of Ref. 4.( k x , k y , k z ) [ µ x , µ y , µ z ] λ k α (0,0,0) (FM, B-type) [100], [010], [001] 0 (cid:0) , , (cid:1) (A-type) [100] − . − [100] 14.5311 (cid:0) , , (cid:1) (C-type) [100], [010] − . − [100] 8.0302 (cid:0) , , (cid:1) (N´eel- or G-type) [100], [010], [001] 0 three spins per unit cell. Then also taking into accountthe relation a = 2 a nn , one obtains the ground-state eigen-value λ = 48(2 . . ≈ . λ (0 , , for collinear FM in Table I. Thus the clas-sical MC simulations reveal a noncollinear ground statethat is much more stable than the most stable classicalcollinear FM state.The results for the 2D spin lattices in Table I providevery useful reference points for 3D lattices, where the 2Dresults correspond to the limit c/a → ∞ . Indeed, in plotsof λ k α versus c/a for uniaxial 3D spin lattices below, weinclude horizontal dashed lines in the plots to observe therate at which the 2D limits are approached with increas-ing c/a ratio within the calculated range 0 . ≤ c/a ≤ IV. EIGENVALUES AND EIGENVECTORSFOR THREE-DIMENSIONAL SPIN LATTICESA. Cubic Spin Lattices
The eigenvalues and eigenvectors of the dipolar inter-action tensor for the cubic Bravais lattices are well-knownbut are presented here in modern notation for complete-ness and as a check on our calculation methods. Ourparameters for sc, bcc and fcc lattices are found to agreewith previous results and are listed in Tables II, IIIand IV, respectively, for various values of k along withthe common magnetic structure designations. Belobrov et al. carried out an exact calculation of the ground statespin configuration and energy of the sc lattice and founddegenerate noncollinear and noncoplanar AFM ground
TABLE III: Body-centered cubic spin lattice. Symbol defini-tions are the same as in Table II. The λ k α values agree withthe eigenvalues in Table IV of Ref. 4.( k x , k y , k z ) [ µ x , µ y , µ z ] λ k α (0,0,0) (FM) [100], [010], [001] 0 (cid:0) , , (cid:1) [100] − . − [100] 14.5311 (cid:0) , , (cid:1) [001] 5.3534[1¯10] 7.9437[110] − . − [110] 18.6505[001] − [1¯10] − . (cid:0) , , (cid:1) [100], [010], [001] 0(1,0,0) [100], [010], [001] 0TABLE IV: Face-centered cubic spin lattice. Symbol defini-tions are the same as in Table II. The designations of theAFM type are from Ref. 49. The λ k α values agree with theeigenvalues in Table V of Ref. 4.( k x , k y , k z ) [ µ x , µ y , µ z ] λ k α (0,0,0) (FM) [100], [010], [001] 0 (cid:0) , , (cid:1) (Type-IA AFM) [100] − . − [100] 38.518 (cid:0) , , (cid:1) (Type-IV AFM) [1¯10] 14.383[¯1¯10] − . − [¯1¯10] 34.119[1¯10] − [001] 9.029(0,0,1) (Type-I AFM) [100], [010] 8.6687[001] − . − [001] 26.0061 (cid:0) , , (cid:1) (Type-II AFM) [¯1¯1¯1] − . − [¯1¯1¯1] 43.381 (cid:0) , , (cid:1) [¯1¯1¯1] − . , [0¯11] 15.0293[2¯1¯1] − [¯1¯1¯1] 45.0881 (cid:0) , , (cid:1) (Type-III AFM) [100] 6.3040[010], [001] − . − [010] 9.4560 states with energy per spin corresponding to eigenvalue λ = 5 . which is essentially the same as our value λ (1 / , / , = 5 . k = (cid:0) , , (cid:1) in Table II. The designationsof the AFM type for fcc lattices in Table IV are fromRef. 49. B. Simple Tetragonal Spin Lattices
The eigenvalues and eigenvectors for FM momentalignments [ k = (0,0,0)] for simple tetragonal latticeswith c/a = 0 . For c/a < -6-4-202468100.0 0.5 1.0 1.5 2.0 2.5 3.0c/a [001]moments aligned along [010][100]k = (1/2,0,1/2)simple tetragonal(a)2D limits-8-6-4-2024680.0 0.5 1.0 1.5 2.0 2.5 3.0c/a(b) [001]moments aligned along [100] or [010] k = (1/2,1/2,1/2)simple or body-centered tetragonal2D limits
FIG. 3: (Color online) Eigenvalues (a) λ (1 / , , / for AFMwavevector k = (1/2,0,1/2) r.l.u. and (b) λ (1 / , / , / forAFM wavevector k = (1/2,1/2,1/2) r.l.u. versus the c/a ra-tio for a simple tetragonal lattice with the moments alignedalong [010] ( b axis, solid red circles), [001] ( c axis, solid greendiamonds) and [100] ( a axis, solid blue squares). moment alignment along the c axis is energetically favor-able, whereas for c/a > ab -plane moment alignment ispreferred.The eigenvalues and eigenvectors for a number of 3DAFM structures for simple tetragonal spin lattices weredetermined versus c/a . The 2D limits corresponding to c/a → ∞ are shown as black horizontal dashed linesin the figures. The data for k = (1/2,0,0) are plottedin Fig. 22 in Appendix C. In this case there are threedistinct λ k α values for the three eigenvectors [100], [010]and [001] because this k breaks the fourfold rotationalsymmetry about the c axis, with the easy axis switchingfrom [001] for c/a < c/a >
1. One seesthat the respective 2D limits in Table I are reached for c/a > ∼
2. Similarly, data for k = (1/2,1/2,0) and (0,0,1/2)are plotted in Fig. 23 in Appendix C and the data arelisted in the Supplemental Material. The eigenvalues for AFM wavevectors k = (1/2,0,1/2) ca b BaMnAs ca b
Ba FeAsBaMn As BaFe As FIG. 4: (Color online) Crystallographic structures of bctThCr Si -type BaMn As and BaFe As . The magneticatoms Mn and Fe form simple-tetragonal sublattices. TheAFM structure of BaMn As is N´eel-type (G-type) with AFMpropagation vector k = (cid:0) , , (cid:1) and with the ordered mo-ments aligned along the c axis, whereas the AFM structureof BaFe As is stripe-type with AFM propagation vector k = (cid:0) , , (cid:1) and with the ordered moments aligned along the a axis of the simple-tetragonal Fe sublattice structure (due toan orthorhombic distortion, the a and b axes have slightlydifferent lengths at T ≤ T N in BaFe As ). and (1/2,1/2,1/2) are plotted for the respective eigenvec-tors versus the c/a ratio for a simple tetragonal latticein Figs. 3(a) and 3(b), respectively, with the numericalvalues listed in the Supplemental Material. Here again,the respective 2D limits in Table I are reached ratherquickly with increasing c/a in Fig. 3 at c/a ∼ Si -type crystalstructure (space group I /mmm ) of BaMn As andBaFe As . In both compounds the transition-metalatoms Mn and Fe form a simple tetragonal sublatticewith lattice parameters a Mn / Fe = a bct / √ c Mn / Fe = c bct /
2, yielding c Mn / Fe /a Mn / Fe = ( c bct /a bct ) / √ . As and 2.32 for BaFe As . BaMn As has aG-type (N´eel-type) AFM structure with k = (cid:0) , , (cid:1) inthe tetragonal lattice notation below T N = 625 K withthe Mn ordered local moments aligned along the c axis,whereas BaFe As has a stripe-type itinerant AFM struc-ture with k = (cid:0) , , (cid:1) below T N = 137 K with the Feordered moments aligned along the a axis of the simple-tetragonal sublattice in Fig. 4. The ordered moment axisfor BaMn As agrees with the prediction for the wavevec-tor k = (cid:0) , , (cid:1) in Fig. 3(b). However, as shown inFig. 3(a), for BaFe As MDIs favor the b = [010] easyaxis for k = (cid:0) , , (cid:1) and c/a = 2 .
32, perpendicular tothe in-plane component k ab = (cid:0) , (cid:1) of the AFM prop-agation vector, whereas the easy axis is found to be the a axis, parallel to k ab (see Fig. 40 of Ref. 33). Therefore,5 -4-3-2-1012340.5 1.0 1.5 2.0 2.5 3.0c/abody-centered tetragonalferromagnetic alignmentalong c axis (a)-0.08-0.040.000.040.080.9 1.0 1.1 1.2 1.3 1.4 1.5c/a (b)bcc fccc/a = 2 c/a = 1 FIG. 5: (Color online) (a) Dependence of the eigenvalue λ (0 , , , , on the c/a ratio for a bct lattice with a FM align-ment of the magnetic moments along the c axis. (b) Expandedplot of the data in (a) for 0 . ≤ c/a ≤ .
5. One sees thatFM alignment along the c axis is the most stable for c/a < < c/a < ∼ . a or b axis, for 1 . < ∼ c/a ≤ √ c axis is favored, thenfor c/a > √ a or b axis is again favored. For this magneticstructure, one has λ (0 , , = λ (0 , , = − λ (0 , , / there must be another source of anisotropy in BaFe As that overcomes that due to MDIs to determine the easyaxis. C. Body-Centered Tetragonal Spin Lattices
The behavior of the eigenvalue λ (0 , , of the MDItensor for FM ordering with k = (0,0,0) and the orderedmoment direction along the c axis is shown versus c/a in Fig. 5(a). A list of the numerical data is given in theSupplemental Material. An expanded plot of the datafor 0 . ≤ c/a ≤ . et al. The first zero crossing -15-10-50510150.0 0.5 1.0 1.5 2.0 2.5 3.0c/a(a) moments along [0,1,0] k = (1/2,0,1/2)body-centered tetragonal[(1 − x ) , 0, − x]square lattice limits [x, 0, (1 − x ) ] FIG. 6: (Color online) Eigenvalues for wavevector k = (cid:0) , , (cid:1) r.l.u. versus the c/a ratio for a bct spin lattice withthe moments aligned along (a) [0, ¯1, 0] (solid red circles),[ −√ − x , , x ] (solid green diamonds) or [ x, , √ − x ](solid blue squares), where x versus c/a is shown in (b). occurs at c/a = 1, corresponding to a bcc lattice, andthe third zero crossing occurs at c/a = √
2. This latter c/a value for the bct lattice corresponds to an fcc latticewithin the bct lattice that is rotated by 45 ◦ with respectto the bct lattice as shown in Fig. 15 of Ref. 33. Thelattice parameters are related by a fcc = √ a bct = c bct ,yielding c bct /a bct = √
2. Hence both values c/a = 0and √ λ (0 , , = 0 for all ˆ µ . We verifiedthat our λ (0 , , versus c/a data in Fig. 5(b) calcu-lated by direct summation quantitatively agree with thecorresponding eigenvalue data in Refs. 18–20 that werecalculated using the Ewald-Kornfeld method.The eigenvectors and eigenvalues of b G i were calculatedfor several AFM propagation vectors. The λ (1 / , / , data for k = (cid:0) , , (cid:1) are plotted versus c/a in Fig. 24 inAppendix C and a listing of the numerical data is given inthe Supplemental Material. The eigenvalues for wave k = (cid:0) , , (cid:1) versus the c/a ratio with the moments aligned6 c b a GdCu Si GdCuSi(101) plane
FIG. 7: (Color online) Crystal and magnetic structures ofbct GdCu Si with the ThCr Si -type crystal structure. Onecrystallographic unit cell is shown. The magnetic unit cellhas dimensions 2 a × b × c and contains four crystallographicunit cells. The collinear magnetic structure has an AFMpropagation vector ( , , ) r.l.u. perpendicular to the (101)plane shown, with the magnetic moments oriented along the b axis. Within each such (101) plane, the Gd magnetic mo-ments are FM aligned. in the [0, −
1, 0], [ −√ − x , , x ] or [ x, , √ − x ] direc-tions are plotted in Fig. 6(a), and x versus c/a is plottedin Fig. 6(b). The numerical data in Fig. 6 are listed inthe Supplemental Material. The compound GdCu Si has the bct ThCr Si -typestructure with space group I /mmm as shown in Fig. 7and lattice parameters and z -axis Si positions a =3 .
922 ˚A, c = 9 .
993 ˚A, c/a = 2 .
548 and z Si = 0 . The magnetic structure of GdCu Si iscollinear, with the Gd ordered moments oriented alongthe tetragonal b axis with an AFM propagation vector k = ( , , ) r.l.u., as shown in Fig. 7. The orderedmoment at 2 K is 7.2(4) µ B /Gd, in agreement withthe value of 7 µ B /Gd obtained from the usual relation µ sat = gSµ B , where here S = 7 / g = 2. Thusthe Gd moments in (101) planes are FM aligned and areoriented perpendicular to k . From Fig. 6, dipolar inter-actions for k = ( , , ) and c/a = 2 .
548 predict that themoment alignment should be along the b axis, in agree-ment with the experimental AFM structure in Fig. 7.The eigenvalues for AFM propagation vector k =(0,0,1) in the bct spin lattice versus the c/a ratio withthe moments aligned along the c axis or in the ab planeare plotted in Fig. 8 and listed in the SupplementalMaterial. The compound EuCu Sb has a primitive tetragonalCaBe Ge -type crystal structure (space group P /nmm ) -12-8-4048120.0 0.5 1.0 1.5 2.0 2.5 3.0c/a[0,0,1] moments aligned along [1,0,0] or [0,1,0] k = (0,0,1) (A-type AFM)body-centered tetragonal2D limits FIG. 8: (Color online) Eigenvalues for wavevector k =(0,0,1) r.l.u. versus the c/a ratio for a bct spin lattice with themoments aligned along [001] ( c axis, solid green diamonds) or[100] or [010] ( a or b axis, solid blue squares). The 2D limitsfor the square lattice for c/a → ∞ are shown as horizontalblack dashed lines. EuCu Sb ab c MnF2
Mn F
FIG. 9: (Color online) Crystallographic and AFM A-typestructure with k = (0,0,1) and ˆ µ = [100] of EuCu Sb with c/a = 2 .
401 (Refs. 29, 30) and MnF with c/a = 0 . µ = [001]. Each compound contains a bct sublattice ofmagnetic ions, but with c/a < c/a >
1, respectively,which is the crossover point between [001] and [100]- or [010]-axis ordering, respectively. +2 ions in crystallographically-equivalentsites forming a bct sublattice as shown in one panel ofFig. 9. Like Gd +3 , the Eu +2 ions have spin S = 7 / g = 2, angular momentum L = 0 and a saturation mo-ment µ sat = gSµ B = 7 µ B . The compound orders an-tiferromagnetically below T N = 5 . k = (0,0,1), and with the Eu +2 moments ori-ented in the ab plane as shown in Fig. 9. The powderneutron diffraction measurements can only determinethat the ordered moments lie in the ab plane and nottheir direction within this plane. EuCu Sb has latticeparameters a = 4 .
488 ˚A, c = 10 .
778 ˚A and c/a = 2 . c/a value the ordering direction for k = (0,0,1) is predicted for MDIs to be in the ab plane,in agreement with the experimental data.The compound MnF has the primitive tetragonal ru-tile crystal structure with space group P /mnm andis widely considered to be the prototype for collinearAFM ordering. The crystal and magnetic structures ofMnF are shown in Fig. 9. At T = 298 K, the lat-tice parameters are a = 4 . c = 3 . c/a = 0 . u = 0 . The Mn +2 d ions with expectedhigh-spin S = 5 / +2 spins order in an A-type AFM structure below theN´eel temperature T N = 67 K (Ref. 54) with an or-dered moment at 5 K of 5.12(9) µ B /Mn. The orderedmoment is in good agreement with the expected value µ sat = gSµ B = 5 µ B /Mn for g = 2. A fit to χ ( T ) mea-surements from 200 to 300 K by the Curie-Weiss law gavea molar Curie constant of 4 .
47 cm K / mol and a Weisstemperature θ = − . The Curie constant is closeto the value of 4 .
38 cm K / mol expected for S = 5 / g = 2. From the c/a ratio and Fig. 8, the MDI favorsordered moment alignment along the c axis, in agreementwith the easy axis observed in Fig. 9. This ordering axisis perpendicular to the ordering axis for EuCu Sb with c/a > D. Simple Hexagonal (Triangular) and HoneycombSpin Lattices
The eigenvalues and eigenvectors of the MDI tensor b G i for stacked simple hexagonal lattices were calculated ver-sus c/a from 0.5 to 3 for FM alignment ( k = 0) and AFMwavevectors k = (1,0,0), (cid:0) , , (cid:1) , (cid:0) , , (cid:1) , (cid:0) , , (cid:1) and (cid:0) , , (cid:1) , and are plotted in Figs. 25, 26 and 27 in Ap-pendix C and the numerical data are listed in the Sup-plemental Material. In contrast to the AFM cases, forthe FM alignment the approach of the eignevalues to theasymptotic 2D ones with increasing c/a is very slow asseen from comparison of the plots for FM alignments inFig. 25(a) with the AFM ones, which reach their 2D val-ues by c/a ∼ b G i for the hon-eycomb spin lattice in Fig. 1 calculated versus c/a from0.5 to 3 for k = (0,0,0) (FM alignment) and AFM prop- agation vectors k = (cid:0) , , (cid:1) , (cid:0) , , (cid:1) (N´eel-type in alldirections), (0,0,0) r.l.u. (N´eel-type in ab plane and FMalignment along c axis) and (cid:0) , , (cid:1) (FM alignment in-traplane and AFM alignment interplane) are plotted inFigs. 28, 29 and 30 in Appendix C, respectively, and arelisted in the Supplemental Material. Similar to the be-havior of the eigenvalues for the simple hexagonal spinlattice, for FM alignment in the honeycomb lattice theapproach of the eigenvalues to their 2D limits with in-creasing c/a is very slow compared to behaviors for theAFM moment alignments. For the N´eel AFM alignmentsboth just in the ab plane and also along the c axis, theapproach with increasing c/a to the infinite c/a limits isvery fast, being essentially complete by c/a ∼ . V. EIGENVALUES AND EIGENVECTORS FORNONCOLLINEAR ANTIFERROMAGNETS
The relationship between the ordered/induced centralmoment ~µ i and another moment ~µ j at position r ji withrespect to ~µ i in a collinear magnetic structure was givenin Eq. (14). In noncollinear AFMs one must specify thedirections of each of the moments in a crystal in orderto calculate the net dipolar interaction of a given centralmoment ~µ i with its neighbors inside the Lorentz sphere.There are two generic cases. In the first, one can define anonzero AFM propagation vector k such that momentsin a plane perpendicular to k are FM-aligned and allchange their directions from plane to plane along k . Inthe second, the spin lattice is a non-Bravais lattice andthe magnetic and chemical unit cells are the same, wherethe AFM propagation vector is k = (0,0,0) for such cases.We consider the first type of AFM ordering in the 2D tri-angular lattice in the following section and then the sec-ond type of ordering in GdB and the Shastry-Sutherlandlattice. A. 2D Triangular Lattice Antiferromagnets
It is well known that the classical ground state of atriangular lattice AFM interacting by isotropic Heisen-berg exchange is the coplanar noncollinear 120 ◦ struc-ture, where each of the six neigbors of a given momentis at a 120 ◦ angle with the given moment, as in the cy-cloidal AFM structure shown in Fig. 10 where the 2DAFM propagation vector is k = (cid:0) , (cid:1) r.l.u. In theabsence of anisotropy, the energy of the spin lattice inFig. 10 is invariant on rotating each spin by the sameangle, thus retaining the 120 ◦ angles between adjacentmoments. Here we examine whether the MDI can de-termine how the moments are oriented with respect tothe hexagonal unit cell axes for the AFM structure inFig. 10, or indeed whether the MDI alone can stabilizethis magnetic structure.The approach we use is to first calculate the eigen-values of the MDI tensor b G i for noncollinear moments8 ab k^ d kd = 2 π /3 FIG. 10: (Color online) Coplanar noncollinear magnetic unitcell of classical 120 ◦ ordering on the 2D simple hexagonal(triangular) spin lattice for cycloidal AFM ordering with acommensurate wavelength of 3 a /2. The hexagonal latticetranslation vectors a and b ( a = b ) and the direction ˆ k ofthe cycloid wavevector k are indicated. The long-dashed lineis the outline of the hexagonal unit cell containing one spinand the solid line is the outline of the magnetic unit cell con-taining nine spins (nine unit cells). The AFM propagationvector is k = (cid:0) , (cid:1) r.l.u. The quantity d is the distancebetween lines of FM-aligned magnetic moments along the cy-cloid axis (ˆ k ) direction. The rotation angle of the magneticmoments between adjacent lattice lines in the ˆ k direction is φ ji = kd = π rad. and variable k = ( x, x ) r.l.u. and see whether the max-imum eigenvalue is obtained for x = 1 /
3. If so, thenwe are done. If not, we conclude that exchange interac-tions alone determine k = (cid:0) , (cid:1) r.l.u. and then calculatefor this k what the moment orientations should be withrespect to the crystal axes as predicted by the MDI.The MDI tensor b G i was calculated using Eq. (19).Shown in Fig. 11 are plots of the two eigenvalues λ ( x,x ) versus x with k = ( x, x ) r.l.u. for the two eigenvectorsˆ µ i and ˆ µ i shown in the figure for the orientation ofcentral moment ~µ i at the origin of the Cartesian coor-dinate system (the third eigenvalue is for FM orderingalong the c axis as discussed in Sec. II F and is not rele-vant here). From Fig. 11, there is no maximum in λ ( x,x ) at x = 1 / ◦ noncollinear struc-ture. Instead, the MDI favors k = (1/2, 1/2) r.l.u. Set-ting x = 1 /
3, we obtain λ (1 / , / = − . µ i = [100] or [010]. The AFMstructure in Fig. 11 corresponds to ˆ µ i = [010].Interestingly, the eigenvalue λ (1 / , / = − . collinear AM AFM ordering on the triangular lattice with k = (cid:0) , (cid:1) r.l.u. with the same two eigenvectors. This showsthat the net energy of interaction of a moment with themagnetic fields of the other moments inside the Lorentzsphere only depends on the projections of those momentson the eigenvector axis.The fact that λ (1 / , / is negative, whereas the eigen-value for collinear AM ordering along the easy c axis for k = (cid:0) , (cid:1) in Table I is positive, suggests that the MDI -8-40480.0 0.2 0.4 0.6 0.8 1.0x μ i2 = [1/2, 3 /2] μ i1 = [3 /2, − k = (x,x) r.l.u.2D triangular lattice FIG. 11: (Color online) Variation in the eigenvalues λ ( x,x ) versus x in the AFM propagation vector k = ( x, x ) r.l.u. forthe two eigenvectors ~µ i and ~µ i of the MDI tensor for the ori-entation of a representative moment ~µ i . The first eigenvectoris in the hexagonal b direction and the second is in the b ∗ direction, which is rotated clockwise by 90 ◦ from the first (seeFig. 16 in Appendix A). The two curves cross at x = 1 / x = 2 /
3. For x = 1 / a axis) and [010] (perpendicular to the a axis). might tend to cant the moments in the classical 120 ◦ coplanar structure out of the ab plane and also introducean amplitude modulation of the ordered moments. B. GdB and Shastry-Sutherland Antiferromagnets The AFM structure for GdB shown in Fig. 12was deduced from neutron diffraction measurements. The configuration of the exchange interactions J and J shown in the figure is an example of a so-calledShastry-Sutherland Heisenberg exchange model in twodimensions. In GdB , this AFM structure is stackedalong the c axis with FM alignments between nearest-neighbor layers and a corresponding FM interlayer inter-action J c that is not included in the Shastry-Sutherlandmodel.Here we assume that the AFM structure is known,along with the relative orientations of each of the orderedmoments in a unit cell. For noncollinear AFMs, Eq. (14)cannot be used and instead one must express each ~µ k in amagnetic = crystallographic unit cell in terms of the cen-tral moment ~µ i around which the dipolar sum within theLorentz sphere is calculated. Thus we use the methoddescribed in Sec. II E to obtain the orientation (eigen-vector) of ~µ i with respect to the Cartesian coordinatesystem, together with the associated eigenvalue.GdB has a primitive-tetragonal crystal struc-ture with space group P /mbm . The Gd atomsoccupy the Wyckoff 4 g positions (1) (cid:0) − x, x, (cid:1) ,(2) (cid:0) − x, − , (cid:1) , (3) (cid:0) + x, − x, (cid:1) and9 J J GdB Gd ab FIG. 12: (Color online) Four crystallographic and magneticunit cells of the Gd sublattice of the tetragonal GdB com-pound in the ab plane. The Gd ordered moments all lie inthe ab -plane in [110] and equivalent directions. Also shownare the 2D in-plane Shastry-Sutherland exchange interac-tions J and J between nearest- and next-nearest-neighborGd spins, respectively. The four Gd spins in the lower-leftunit cell are numbered counterclockwise as shown. The spininteractions are topologically the same as in the undistortedShastry-Sutherland square lattice model in which the GdB squares are not tilted with respect to the a and b axes. Ad-jacent stacked layers along the c axis are FM-aligned withFM (negative) nearest-neighbor exchange interaction J c (notshown). Since the chemical and magnetic unit cell are thesame, the AFM propagation vector is k = (0,0,0). (4) (cid:0) x, + x, (cid:1) with x = 0 . a -axis lattice parameter are r a = (cid:18) n a + 12 − x, n b + x, n c ca (cid:19) , (30a) r a = (cid:18) n a + 1 − x, n b + 12 − x, n c ca (cid:19) , (30b) r a = (cid:18) n a + 12 + x, n b + 1 − x, n c ca (cid:19) , (30c) r a = (cid:18) n a + x, n b + 12 + x, n c ca (cid:19) , (30d)where n a , n b and n c are positive or negative integers orzero. Taking the central moment ~µ i to be at position r with n a = n b = n c = 0, one obtains the r ki = r k − r i as r i a = (cid:16) n a , n b , n c ca (cid:17) , (31a) r i a = (cid:18) n a + 12 , n b + 12 − x, n c ca (cid:19) , (31b) r i a = (cid:16) n a + 2 x, n b + 1 − x, n c ca (cid:17) , (31c) r i a = (cid:18) n a −
12 + 2 x, n b + 12 , n c ca (cid:19) . (31d)The 3 × R k for the four numbered TABLE V: GdB and Shastry-Sutherland lattice. Eigen-vectors [ µ x , µ y , µ z ] for central spin ~µ i and eigenvalues λ k α for the 3D Gd sublattice in GdB and for the 2D Shastry-Sutherland model. For both cases, the experimental 90 ◦ angles between adjacent spins was assumed with the order φ = φ , φ + 90 ◦ , φ + 100 ◦ and φ + 270 ◦ on going clock-wise around a Gd square as shown in Fig. 12, but with thevalue of φ undetermined for the moment in the lower leftcorner of each square. The experimental x value and c/a ra-tio for GdB are 0.317 46 and 0.567 97, respectively. In the2D Shastry-Sutherland model, x = 1 / c/a = ∞ . Thesymbol ¯1 means − x [ µ x , µ y , µ z ] λ k α
3D GdB − . − [1¯10] 7.883[001] − [110] 76.0001/4 [1¯10] 40.013[001] 26.833[110] − . − [001] 13.180[1¯10] − [110] 106.8582D Shastry- 1/4 [1¯10] 40.790 982Sutherland [001] 7.483 697[¯1¯10] − .
274 678[1¯10] − [001] 33.307 285[1¯10] − [¯1¯10] 89.065 660 moments in the lower-left unit cell in Fig. 12 are R = , (32a) R = yx − xy , (32b) R = − , (32c) R = xy − yx , (32d)where xy and yx are 3 × R/a = 50 for 3D GdB . Then diagonaliz-ing b G i gave the eigenvectors and corresponding eigenval-ues listed in Table V. Recalling that the largest positiveeigenvalue corresponds to the minimum energy accordingto Eq. (16a), the data in Table V show that the MDI fa-vors moment alignment along the c axis, contrary to theexperimental result in Fig. 12 which gives the alignmentof the k = 0 spin as the [1 , ¯1 ,
0] direction, correspond-ing to the second-highest λ k α . The highly unstable [110]direction for central moment ◦ andhence all moments in each Gd square pointing towardsthe center of the square. The RKKY interaction betweenGd spins and/or a high-order crystalline electric field ef-fect evidently give an anisotropic exchange interactionthat is responsible for the observed ordered moment di-rections.Calculations were also carried out for x = 1 /
4, whichcorresponds to untilted Gd squares in Fig. 12, as shownin Table V. One sees significant differences in the0eigenvalues compared to the results for the observed x = 0 . is found to be the same as for x = 1 / c/a = 0 . c/a ratio for GdB . Thusthe ground-state ordering direction predicted by the MDIis sensitive to the tilting angle of the Gd squares. VI. MAGNETIC ORDERING TEMPERATUREDUE TO MAGNETIC DIPOLE INTERACTIONS
The MFT calculations in this and the following sec-tions closely follow the development of the author de-tailed in Ref. 11. Therefore only an outline of the calcu-lations associated with the MDI is given.In this section, an AFM ordering (N´eel) temperaturearising from dipolar interactions only is denoted by T NA and a FM ordering (Curie) temperature by T CA , wherethe subscript A refers to the quantity being the contribu-tion from an anisotropic magnetic interaction. Similarly,a N´eel temperature arising from Heisenberg exchange in-teractions only is denoted by T N J and a Curie tempera-ture by T C J . We use the Weiss MFT to calculate thesetransition temperatures where we assume that the spinsare identical and crystallographically equivalent and weonly treat EM (not AM) magnetic structures on Bravaislattices. Within MFT, the contributions of the dipolarand exchange interactions to the actual ordering temper-atures T N and T C , respectively, are additive: T N = T NA + T N J , T C = T CA + T C J . (33)The magnetic ordering temperature T m J (m = N, C)for both AFMs and FMs due to exchange interactions isgiven by the same expression T m J = − S ( S + 1)3 k B X j J ij cos φ ji , (34)where φ ji is the angle between magnetic moments j and i in the ordered state and φ ji = φ j − φ i = 0 for a FM.We define the reduced ordered and/or applied magneticfield-induced average moment ¯ µ for a spin S as¯ µ ≡ µµ sat = µgSµ B , (35)where µ sat = gSµ B is the saturation moment of the spinand g is the spectroscopic splitting factor. Using Eq. (34),one can write the exchange field seen by a representativemoment i in zero applied field H as H exch i = T m J C µ = 3 k B T m J gµ B ( S + 1) ¯ µ , (36)where the subscript 0 in ¯ µ signifies H = 0, C is thesingle-spin Curie constant (see below) and this expressionapplies to the ordered state. The magnetic ordering temperature is determinedwithin MFT by the criterion that ¯ µ → T → T − m .For magnetic dipole ordering, the near-field contributionto the local magnetic induction is given by Eq. (20). Themagnetic moment µ in that equation is defined in gen-eral as either the ordered moment in a magnetic struc-ture in H = 0 ( µ ) and/or an average moment inducedby H α > µ ). Using Eq. (35), Eq. (20) associated withMDIs becomes B nearint α i = gµ B S ¯ µλ k α a . (37) A. Antiferromagnetic Ordering (N´eel)Temperature
Here we calculate T NA in H = 0 within MFT for aspecified AFM wavevector k and ordered moment axis ˆ µ in the presence of MDIs but in the absence of exchangeinteractions. The standard MFT prediction is obtainedfrom ¯ µ = B S (cid:18) gµ B B localint α k B T (cid:19) , (38)where we have dropped the subscript i because all mo-ments are crystallographically equivalent in H = 0, thesubscript 0 in ¯ µ signifies that H = 0 as above, and B S ( y ) is the Brillouin function for spin S given by ourunconventional expression B S ( y ) = 12 S n (2 S + 1) coth h (2 S + 1) y i − coth (cid:16) y (cid:17)o . (39)There is no demagnetizing field for an AFM in H = 0 be-cause there is no net magnetization, so for AFM orderingin H = 0 the local field is just the near field. Inserting B nearint α i from Eq. (37) into (38) gives¯ µ = B S ( y ) , (40a)where y = g Sµ ¯ µ λ k α a k B T . (40b)Then one obtains for a given k and easy axis α the N´eeltemperature T NA α = g S ( S + 1) µ λ k α a k B . (41)The relevant ordering axis α and hence T NA α is the onewith the largest eigenvalue λ k α for the given AFM struc-ture.The single-spin Curie constant C for spin S is givenby C = g S ( S + 1) µ k B , (42)1so Eq. (41) can be written more succinctly as T NA α = C λ k α a . (43a)Thus one can also write λ k α a = T NA α C = 3 k B T NA α g S ( S + 1) µ . (43b)Then for H = 0 and T ≤ T NA one can write the nearfield in Eq. (37) in the direction of each ordered momentas B nearint α = 3 k B T NA α ¯ µ g ( S + 1) µ B . (44a)The exchange field for H = 0 seen by each moment inits ordering direction due to Heisenberg exchange inter-actions for either FM or AFM ordering can be written inthe same form as H exch = 3 k B T m J ¯ µ g ( S + 1) µ B , (44b)where T m J is the contribution of Heisenberg exchangeinteractions to either a FM Curie temperature T C J oran AFM N´eel temperature T N J . Using Eq. (33), in thecase of AFM ordering the sum of the two local fields inEqs. (44) can be written B local α = 3 k B ( T N J + T NA α )¯ µ g ( S + 1) µ B = 3 k B T N ¯ µ g ( S + 1) µ B , (45)where T N is the N´eel temperature in the presence of bothexchange and MDIs.Because different sources of local fields are additive intheir contributions to the observed T N within MFT, ifboth exchange and dipolar interactions are present T NA is the contribution of dipolar interactions to T N , which isusually but not always a small fraction of T N .Quantum fluctuations generally increase as the dimen-sionality of a spin lattice decreases. These quantum fluc-tuations can prevent long-range magnetic ordering fromoccurring. Corruccini and White found from spin-wavecalculations that AFM order cannot occur at finite tem-perature on the 2D square spin lattice due to dipolarinteractions alone. MFT does not take into accountsuch quantum fluctuations associated with reduced di-mensionality and hence predicts that AFM ordering canoccur in 1D, 2D and 3D spin lattices.
B. Ferromagnetic Ordering (Curie) Temperature
As is well-known, whether or not a particular sampleexhibits FM ordering driven by the MDI depends on theshape of the sample via the demagnetizing field as well asthe competition with AFM states. The former is evidentfrom Eq. (8a) which for H α = 0 becomes B localint αi = gµ B Sa (cid:20) λ α + 4 πV spin /a (cid:18) − N d α (cid:19)(cid:21) ¯ µ, (46) where a is the a -axis lattice parameter of the unit cell, V spin is the volume per spin, λ α refers to the FM mo-ment alignment, the magnetic moment per unit volume is µ/V spin = gµ B S ¯ µ/V spin , and we used Eqs. (35) and (37).Then following the same development as in the previoussection gives the Curie temperature T CA α = g S ( S + 1) µ k B a (cid:20) λ α + 4 πV spin /a (cid:18) − N d α (cid:19)(cid:21) = C a (cid:20) λ α + 4 πV spin /a (cid:18) − N d α (cid:19)(cid:21) (FM) , (47)where C was defined in Eq. (42). The system will choosethe easy axis α with the largest value of λ α . For a cubicBravais lattice λ α = 0, so there is no preferred easy axisfor FM ordering according to the present treatment.Using Eq. (47) one can write the local field in Eq. (46)for FM moment alignments as B localint αi = 3 k B T CA α gµ B ( S + 1) ¯ µ. (48)If Heisenberg exchange interactions are present, one addsthe local exchange field in Eq. (44b) to the dipolar con-tribution in Eq. (48) to obtain B localint αi = 3 k B T C α gµ B ( S + 1) ¯ µ. (49)where T C α = T CA α + T C Jα according to Eq. (33).Comparing Eqs. (45) and (49) one sees that the sameform of the local field in the direction of each orderedmoment is obtained for both FM and AFM structuresin the ordered states and one can therefore write thelocal magnetic induction seen by each moment in generalfor either FM or AFM moment alignments and dipolarand/or Heisenberg interactions as B localint αi = 3 k B T m α gµ B ( S + 1) ¯ µ, (50)where T m α is the Curie or N´eel temperature for thecollinear ordering axis α . VII. COMPETITION BETWEENFERROMAGNETIC ANDANTIFERROMAGNETIC ORDERING
One can have a crossover between FM and AFM or-dering depending on the value of the demagnetizing fac-tor N d α and the possible AFM eigenvalues λ k α and FMeigenvalues λ α . The value of N d α depends on the shapeof the sample. For FM ordering, the field direction withthe smallest value of N d α gives the lowest free energy andhence is the FM ordering direction provided that the cal-culated T CA α > T NA α > T NA α a C = λ k α (AFM) , (51a) T CA α a C = λ α + 4 πV spin /a (cid:18) − N d α (cid:19) (FM) . (51b)As an example, we consider the competition betweenFM and AFM ordering due to dipolar interactions on sc,bcc and fcc Bravais lattices, which have λ α = 0 and V spin /a = 1, 1/2 and 1/4, respectively. The reducedCurie temperature in Eq. (51b) is plotted versus N d α forsc, bcc and fcc Bravais spin lattices in Figs. 13(a), 13(b)and 13(c), respectively. Using Eq. (51a) and the data inTables II, III and IV, AFM λ k α values are plotted for themost stable (positive) λ k α value for each k as horizontallines for the sc, bcc and fcc lattices in Figs. 13(a), 13(b)and 13(c), respectively. One sees from Fig. 13 that forthe magnetic structures considered, the ground state ofthe sc lattice is AFM-ordered with k = (cid:0) , , (cid:1) r.l.u.and ordering axis ˆ µ = [001] for all values of N d α , thebcc lattice is unstable to FM ordering only for N d α ≈ N d α < ∼ .
03. These inferences areconsistent with early results. A sample with the shapeof a long thin needle with the magnetization directedalong the axis of the needle has a demagnetizing factor N d α ≈ VIII. PROPERTIES OF THEMAGNETICALLY-ORDERED STATEA. Ordered Moment and Magnetic Heat Capacity
For either an AFM or FM with Heisenberg and/orMDIs, Eq. (50) gives the same form of the local magneticinduction seen by each spin in its ordering direction for T ≤ T m . Using Eq. (50), the behavior of ¯ µ versus t is thesame as for pure Heisenberg interactions and is shown forseveral values of the spin S in Fig. 10 of Ref. 59.The magnetic energy per spin is given by E mag i = − µ i B local i , (52a)where the factor of 1/2 derives from the fact that B local i is attributed to the neighbors of µ i whereas the energyis equally shared by pairs of interacting spins. Inserting B local i from Eq. (50) into (52a) for a mole of spins with N = N A where N A is Avogadro’s number, one obtains E mag = − RS S + 1) T m ¯ µ , (52b)where R = N A k B is the molar gas constant. Then themagnetic heat capacity C mag per mole of spins is obtained -10-505100.0 0.2 0.4 0.6 0.8 1.0N d α (1/2,1/2,0)[001] simple cubic (0,0,0) (FM) (a) (1/2,0,0)[010],[001](1/2,1/2,1/2)AFM -10-50510150.0 0.2 0.4 0.6 0.8 1.0N d α (1/2,1/2,0)[1, − body-centered cubic (0,0,0) (FM) (b) (1/2,0,0)[010],[001](1/2,1/2,1/2), (1,0,0)AFM d α (1/2,1/2,0)[1, − face-centered cubic (0,0,0) (FM) (c) (1/2,0,0)[010],[001](1/2,1/2,1/2)[2, − − − − − − FIG. 13: (Color online) Reduced magnetic ordering temper-ature T mag a /C versus the demagnetizing factor N d α with0 ≤ N d α ≤ m , m , m ) r.l.u. and the ordered moment axisas [ µ x , µ y , µ z ] in Cartesian coordinates. Values of T mag < k = (1/3,1/3,1/3) r.l.u. C mag R = − S ¯ µ ( t ) S + 1 d ¯ µ ( t ) dt , (52c)where t = T /T mag and the reduced ordered moment ver-sus temperature ¯ µ ( t ) in H = 0 is obtained as describedin Ref. 11. This equation is identical to that obtained forpure Heisenberg interactions, where plots of C mag /R ver-sus t for several values of S are shown in Fig. 11 of Ref. 59.For quantum spins, C mag decreases exponentially to zerofor t →
0, whereas for classical spins C mag /R → t → B. Dipolar Anisotropy of UniaxialAntiferromagnets in the Ordered State
Here we calculate the dipolar anisotropy of the freeenergy between EM orthogonal principal collinear mag-netic ordering axes denoted as the α and β axes. We con-sider collinear AFMs with noncubic spin lattices contain-ing identical crystallographically-equivalent spins. Thelowest-order expression for the anisotropy free energy perspin F i is given by the usual expression F i = K sin θ, (53)where θ is defined as the angle between the ordered mo-ment axis and the α axis. We derive an expression for K associated with the anisotropic MDI in terms of theeigenvalues and eigenvectors of the MDI tensor.The orientation of a representative T -dependent or-dered moment ~µ i in the α - β plane in H = 0 with µ = | ~µ i | is ~µ i = µ (cos θ ˆ α + sin θ ˆ β ) , (54)where µ is the T -dependent ordered moment in H = 0and θ = 0 corresponds to ~µ i parallel to the α axis. Thecorresponding T -dependent internal local field is B localint i = B localint α i cos θ ˆ α + B localint β i sin θ ˆ β. (55)where the expression for B localint α i is given in Eq. (8a) with H α = 0. The differential dF i of the magnetic free energyof the moment is dF i = − ~µ i · d B localint i , (56)where the factor of 1/2 is present because B localint i arisesfrom the neighboring moments of ~µ i whereas the freeenergy per moment is equally shared between each pairof moments. Inserting Eqs. (54) and (55) into (56) gives dF i = µ B localint α i − B localint β i ) sin θ cos θ dθ. (57)Integrating dF i from θ = 0 to θ yields F i = µ B localint α i − B localint β i ) sin θ. (58) This expression for F i applies to moments along thecollinear ordering axis with angles of either ± θ to the α axis because the sine function is squared. ComparingEq. (58) with (53) gives the anisotropy parameter K as K = µ B localint α i − B localint β i ) . (59)For an AFM in the ordered state, one has B localint α i = B nearint α i . Inserting B nearint α i in Eq. (20) into (59) gives K = µ a ( λ k α − λ k β ) . (60)From Eqs. (53) and (60) one obtains F i = ( θ = 0) µ ( T )4 a ( λ k α − λ k β ) ( θ = π/ . (61)Therefore if λ k α − λ k β >
0, the minimum free energyoccurs if the moments are aligned along the α axis ( θ = 0)and hence the easy axis is the α axis, whereas if λ k α − λ k β <
0, the β axis ( θ = π/
2) is favored for the orderingaxis over the α axis. These results are consistent withexpectation because one expects a moment ~µ i to line upalong the axis with the largest value of B nearint i in Eq. (20),i.e., with largest value of λ k . C. Perpendicular Magnetic Susceptibility ofCollinear Antiferromagnets in the Ordered State
The Heisenberg exchange Hamiltonian has no intrinsicmagnetic anisotropy to determine the directions of theordered moments in the ordered state with respect tothe spin-lattice axes. In this paper the only source ofmagnetic anisotropy is the MDI, and in this section weonly consider collinear magnetic ordering. The easy axisis the eigenvector of the interaction tensor b G i ( k ) thatcorresponds to the largest eigenvalue for the given AFMpropagation vector.The single-spin magnetic susceptibility χ is rigorouslydefined as χ = lim H → µ ( H ) /H where µ is the thermal-average moment of a spin in the direction of H that isinduced by H . Here we take the easy axis to be the x axisand the applied infinitesimal field to be along a z axis,perpendicular to the x axis. The magnitude of each or-dered moment in zero field is µ , which is T -dependentas shown in Ref. 59. In the presence of the perpendicu-lar field, the magnitude of the moment does not changein the AFM phase and the induced moment acquiresa component along the z axis. Including the applied in-finitesimal perpendicular field and both the exchange anddipolar fields and setting the net torque on a represen-tative moment equal to zero following the procedure ofRef. 11 yields the perpendicular susceptibility χ ⊥ = C ( T N J + T NA x − T CA z ) − θ p J . (62)4The T -dependent ordered moment µ canceled out, so χ ⊥ is independent of T for T ≤ T N , as also obtained forpure Heisenberg spin interactions. Several special cases occur for Eq. (62). If exchangeinteractions are negligible, the pure magnetic dipole pre-diction is obtained by setting T N J = θ p J = 0, yielding χ ⊥ = C T NA x − T CA z (63a)= a λ k x − λ z − π V spin /a . (63b)For cubic Bravais spin lattices for which λ α = 0 for all α ,Eq. (63b) gives χ ⊥ = a λ k x − π V spin /a . (64)This result agrees, e.g., with χ ⊥ obtained from the equa-tion between Eqs. (29) and (30) in Ref. 4 which includesin the denominator of Eq. (64) the ground state eigen-value λ k x = λ (1 / , / , = 5 .
351 ( f in their notation)for the sc dipolar AFM, in good agreement with our valueof 5.3535 in Table II.When dipolar interactions are negligible, Eq. (62) givesfor the pure Heisenberg exchange model χ ⊥ = C T N J − θ p J ( T ≤ T N J ) , (65a)in agreement with Ref. 11. In the PM state at T ≥ T N J ,the isotropic susceptibility per spin is given by the Curie-Weiss law χ = C T − θ p J ( T ≥ T N J ) . (65b)Comparing Eqs. (65a) and (65b) gives χ ⊥ = χ ( T N J ) ( T ≤ T N J ) . (65c) D. Perpendicular Critical Field
As the perpendicular field is increased from zero at
T < T N , the induced perpendicular moment µ ⊥ increasesas µ ⊥ = χ ⊥ H, (66)where χ ⊥ is given by Eq. (62). When µ ⊥ reaches the or-dered moment µ ( T ), the induced moments become par-allel to H and the system enters the PM state in a second-order transition. Setting µ ⊥ = µ with increasing H ,the critical field H c at which this happens is defined by µ = χ ⊥ H c , yielding H c ( T ) = µ ( T ) χ ⊥ . (67) Thus one obtains H c ( T ) H c (0) = µ ( T ) µ (0) = µ ( T ) µ sat = ¯ µ ( T ) , (68)where ¯ µ is plotted versus t ≡ T /T N in Ref. 59. Sincewithin MFT µ ( T ) depends on the spin S of the moment,so does H c ( T ) H c (0) . Near t = 1, one obtains H c ( T ) H c (0) ∝ √ − t ( t → − ) . (69)Previous classical calculations (not utilizing the WeissMFT and hence not the Brillouin function for quantumspins) yielded the behavior in Eq. (69) for the wholetemperature range 0 ≤ t ≤
1, with the proportional-ity replaced by an equality. In that case, expanding theright-hand side of Eq. (69) in a Taylor series about t = 0gives the linear dependence H c ( T ) H c (0) = 1 − t ( t ≪
1) insteadof the exponential approach to unity for t → IX. CURIE-WEISS LAW IN PARAMAGNETICSTATE
In the PM state above the N´eel or Curie temperature,all moments are aligned in the direction α of the magneticfield H α applied along a principal axis of the spin lattice[the magnetic propagation vector is k = (0 , , ≡ ].For Heisenberg exchange interactions, the exchange fieldin the PM state is isotropic and given by H exch = 3 k B θ p J gµ B ( S + 1) ¯ µ, (70)where ¯ µ is the normalized moment induced by H α and θ p J = − S ( S + 1)3 k B X j J ij (71)is the contribution to the Weiss temperature in the Curie-Weiss law due to Heisenberg exchange interactions. Thenadding H exch and H α to the local dipolar field for H α = 0in Eq. (46) gives the total local field seen by each momentas B localint αi = H α + 3 k B θ p J gµ B ( S + 1) ¯ µ (72)+ gµ B Sa (cid:18) λ α + 4 π V spin /a (cid:19) ¯ µ, where we assume that the demagnetizing field has beencorrected for in experimental data and hence the demag-netizing factor N d α does not appear in this expression.To include it, replace the multiplicative factor in thelast term by − N d α .5Analogous to Eq. (38) for H α = 0, in the present caseone has ¯ µ = B S (cid:18) gµ B B localint α i k B T (cid:19) . (73)Inserting B localint α i from Eq. (72) into (73), Taylor expand-ing the Brillouin function B S ( y ) to first order in y , solvingfor ¯ µ and using Eq. (35) gives the Curie-Weiss law χ α = C T − θ p α , (74a) θ p α = θ p J + θ pA α , (74b)where the single-spin Curie constant C is given inEq. (42), θ p J is given in Eq. (71) and the magnetic dipolecontribution θ pA α to the Weiss temperature is θ pA α = C a (cid:18) λ α + 4 π V spin /a (cid:19) . (74c)A comparison of Eq. (74c) with (47) shows that thecontributions of dipolar interactions to the Weiss temper-ature and the Curie temperature of a FM are the same,i.e., θ pA α = T CA α , (74d)which is the same result as obtained from MFT for asystem of local moments exhibiting a FM transition andinteracting by Heisenberg exchange only. On the other hand, a comparison of Eqs. (43a) and(74c) shows that in general the contribution of dipolarinteractions to the Weiss temperature for AFMs is notequal to the negative of the dipolar N´eel temperature inEq. (43a), as is also found in general for local-momentHeisenberg AFMs. Thus the ratio f = θ p /T C for a FMwithin MFT is f = 1 (FM) , (75a)whereas in general for an AFM it is f α = θ p α T N α = θ pA α + θ p J T NA α + T N J < . (75b) X. ANISOTROPIC MAGNETICSUSCEPTIBILITY OF A SPHERICAL SAMPLEOF A PURE DIPOLAR ANTIFERROMAGNET
In the following, we assume that the sample is in theshape of a sphere, which cancels the Lorentz field withinthe Lorentz cavity according to Eq. (8a) and hence ame-liorates the competition of FM with AFM ordering.
A. Paramagnetic State
For a dipolar collinear AFM at
T > T NA x where theeasy axis is defined as the x axis, the Curie-Weiss law in Eq. (74a) becomes χ α = C T − θ pA α ( T > T NA x ) , (76)where θ pA α is given by setting the second term inEq. (74c) to zero for a spherical sample, yielding θ pA α = C λ α a . (77)This would be zero for a cubic Bravais spin lattice be-cause in that case λ α = 0 for all α . The N´eel tempera-ture in Eq. (43a) for the easy x axis is T NA x = C λ k x a (78)and we define the ratio f A α as f A α = θ pA α T NA x = λ α λ k x , (79)where the subscript A in f A α signifies that the value of f arises only from the anisotropic MDI and α can be anyof the principal axes x , y or z .Using Eqs. (78) and (79), the Curie-Weiss law (76) fora single spin can be written in dimensionless form as χ α T NA x C = 1 t A − f A α ( t A > , (80a)where the reduced temperature t A is defined as t A = TT NA x . (80b)Note that Eq. (80a) is a law of corresponding states forall quantum spins S , since S only appears in C .The reduced PM susceptibility at T NA x from the Curie-Weiss law (80a) is then χ α ( t A = 1 + ) T NA x C = 11 − f A α ( t A = 1 + ) . (81)From Eqs. (80a) and (81) one obtains χ α ( t A ) χ α ( t A = 1 + ) = 1 − f A α t A − f A α ( t A > , (82)which yields the identity χ α ( t A = 1 + ) χ α ( t A = 1 + ) = 1 , (83)as required. B. Perpendicular Susceptibility in theAFM-Ordered State
In the AFM state at
T < T NA of a strictly dipolarAFM, one sets T N J = θ p J = 0 and for spherical samples6Eqs. (63) yield χ ⊥ ( T ≤ T NA x ) = a λ k x − λ z (84a)= C T NA x − T CA z , (84b)where, as above, the x axis is the easy axis for thecollinear AFM ordering, T NA x is the associated N´eel tem-perature and the z axis is perpendicular to the x axis, i.e., χ ⊥ = χ z . One can write Eq. (84b) in dimensionless formas χ ⊥ T NA x C = 11 − r z ( t A < , z ⊥ x ) , (85a)where according to Eq. (43a) and Eq. (47) modified fora spherical sample one has r z = T CA z T NA x = λ z λ k x . (85b)Using Eqs. (81) and (85a) one obtains χ ⊥ ( t A < χ α ( t A = 1 + ) = 1 − f A α − r z . (86)Comparing Eqs. (83) and (86), one sees that in generalthe hard-axis χ z is continuous on cooling below T NA x ,where χ ⊥ = χ z below T NA x . If λ α = 0 for all α as incubic Bravais lattices, χ ⊥ is obtained for all axes below T NA x . C. Parallel Susceptibility in the AFM-OrderedState
When an infinitesimal field H = H ˆ i is applied in thepositive x direction along the collinear AFM orderingeasy axis at a temperature 0 < T < T NA x , an orderedmoment initially pointing parallel (antiparallel) to H in-creases (decreases) slightly in magnitude, where the vec-torial change d~µ = dµ ˆ i is the same for both moments.Therefore in this section we only consider the changein the x -axis component of a representative moment ~µ i pointing towards the positive x axis due to the appliedfield.Following Ref. 11 we obtain the dimensionless equation χ k T NA x C = 1 τ ∗ ( t A ) − f A x , (87a)where t A is defined in Eq. (80b) and τ ∗ ( t A ) = ( S + 1) t A B ′ S ( y ) , f A x = θ pA x T NA x = λ x λ k x . (87b) µ ( T ) is obtained by numerically solving µ = gµ B SB S ( y ) , (88a) where y = gµ B k B T µ λ k x a . (88b)Here B S ( y ) is the Brillouin function in Eq. (39) and B ′ S ( y ) ≡ [ dB S ( y ) /dy ] | y = y . Note that the parallel sus-ceptibility in the dimensionless form in Eq. (87a) stilldepends on S since the Brillouin function on the right-hand side does. This contrasts with the dimensionlessforms of the Curie-Weiss and perpendicular susceptibil-ities above for dipolar interactions that do not dependon S .Useful limits are τ ∗ ( t A →
0) = ∞ , τ ∗ ( t A →
1) = 1 , (89)yielding χ k T NA x C = 0 ( t A → , (90a) χ k T NA x C = 11 − f A x ( t A → − ) . (90b)The latter χ k expression is identical with χ x T NA x C = 11 − f A x ( t A = 1 + ) (91)obtained from Eq. (81) for the Curie-Weiss law at t A =1 + for the field applied along the x axis. Thus χ k = χ x for t A < χ x for t A > D. Example
As an example, we consider the simple tetragonal Bra-vais spin lattice with c/a = 0 .
8, 1.0 and 1.2 and AFMpropagation vector k = (cid:0) , , (cid:1) r.l.u. for temperaturesboth above and below the N´eel temperature. Recall thatfor f A α , the x and α axes are the easy principal axisfor AFM ordering and any of the three principal axes,respectively, whereas for r z , the z axis must be an axisperpendicular to the x axis. In a real material, one mustidentify x , z and α with the appropriate crystal axes.The eigenvalues and eigenvectors of the dipolar in-teraction tensor taken from tables in the Supplemen-tal Material are shown in Table VI along with therespective values of f A α and r z defined in Eqs. (79)and (85b). One sees that the AFM state is stable againstthe FM state below T NA x for both c/a values, but thatthe anisotropy in the PM state at T > T NA changes signbetween the two c/a values.Using the data in Table VI, Eq. (86) yields χ ⊥ ( t A ) χ a ( t A =1 + ) =1 for the easy a axis for both c/a = 0 . c/a = 1, one has λ α = f A = r α = θ pa α = 0 for all α . Therefore χ ( T ) follows a Curie lawfor t A ≥
1. Also, there is no restoring force for keepingthe easy axis parallel to the field, so the magnetization7
TABLE VI: Eigenvalues λ and eigenvectors ˆ µ = [ µ a , µ b , µ c ] of the dipolar interaction tensor for simple-tetragonal spin latticeswith c/a = 0 . m , m , m ) r.l.u. The data were taken from tables in the Supplemental Material. The largest eigenvalue for k = 0 is labeled as λ α . For k = (cid:0) , , (cid:1) r.l.u. the maximum eigenvector is denoted as λ k x and thevalue for the perpendicular direction as λ k z . For each k , the values of f A and r z are listed as defined in Eqs. (79) and (85b),respectively. According to Eqs. (80a) and (85a), the parameter f A α is relevant for the PM T range and r z is relevant for theAFM-ordered T range. In the table, the assignments of the x and z Cartesian axes to the c and a crystal axes, respectively,are shown in the subscripts to the parameters. c/a k (r.l.u.) [100] [001] f A r z λ z,a = − . λ x,c = 3 . f A x,c = 0 . f A z,a = − . (cid:0) , , (cid:1) λ k z,a = − . λ k x,c = 9 . r z,a = − . λ z,a = 0 . λ x,c = − . f A x,c = − . f A z,a = 0 . (cid:0) , , (cid:1) λ k z,a = − . λ k x,c = 3 . r z,a = 0 . flops to the perpendicular orientation whenever this isattempted. Thus only χ ⊥ ( T ) = χ ( T NA x ) is measured for t A ≤ χ ( t A ) T NA x C versus t A for c/a = 0 .
8, 1.0 and 1.2 illustrating the progression of theanisotropy in χ as c/a traverses the sc of unity. To ourknowledge no theoretically-predicted behaviors such asin Figs. 14(a) and 14(c) have appeared before in the lit-erature. XI. ANISOTROPY OF MAGNETICSUSCEPTIBILITY OF A HEISENBERGPARAMAGNET DUE TO MAGNETIC DIPOLEINTERACTIONS
In this section we assume that demagnetizing fieldshave been corrected for in experimental data and hencethe demagnetizing factor N d α does not appear.In the PM state above T N , according to Eq. (74c) theanisotropy in χ can only arise from a difference in thedipolar Weiss temperatures along different principal axisdirections α and β , given by Eq. (74c) as θ pA α − θ pA β = C a ( λ α − λ β ) . (92)For cubic Bravais lattices, one has no dipolar anisotropyin the PM state because λ α = 0 for all α . Here we followthe approach of Keffer. For two susceptibilities χ α and χ β measured along the α and β principal axes, one has the identity1 χ β − χ α = χ α − χ β χ α χ β , (93a)or χ α − χ β = χ α χ β (cid:18) χ β − χ α (cid:19) . (93b)Using Eqs. (74), Eq. (93b) yields χ α − χ β = χ α χ β C ( θ p α − θ p β ) . (94) If the dipolar anisotropy in θ is small compared to themeasured average Weiss temperature θ p , one can definethe geometric-mean susceptibility χ = √ χ α χ β and useEq. (92) to obtain χ α − χ β = χ a ( λ α − λ β ) . (95)Here the Curie-Weiss χ ’s are per spin and a is the a -axis lattice parameter for the particular Bravais spin lat-tice considered. The susceptibility difference per mole ofspins is obtained by multiplying each χ on the left sideof Eq. (95) and one χ on the right by Avogadro’s number N A and Eq. (95) yields the molar susceptibility difference χ M α ( T ) − χ M β ( T ) = χ ( T ) N A a ( λ α − λ β ) . (96)Here we apply Eq. (96) to the primitive tetragonalrutile-structure collinear AFM MnF with T N = 69 K,which is often considered a prototype for collinear AFMordering. This compound contains a bct sublattice ofMn +2 cations with spin S = 5 / g = 2and orders into a A-type AFM structure with AFMwavevector k = (0,0,1) as shown in Fig. 9. The latticeparameters are a = 4 . , c = 3 . , ca = 0 . . (97)For the given c/a ratio and FM k = we find λ [001] =4 . λ [100] , [010] = − . λ − λ = 6 . , (98)whereas for the ordering wavevector k = (0,0,1) r.l.u.we obtain λ (001)[001] = 13 . λ (001)[100] , [010] = − . λ (001)[001] − λ (001)[100] , [010] = 20 . T N , in agreement with experiment as follows.The anisotropic χ ( T ) of MnF crystals is shown inFig. 15(a). Above T N , χ is found to be nearlyisotropic. Below T N , the data are a textbook exampleof the anisotropy expected for collinear AFM ordering,where in this case the easy axis is the c axis. According8 NAx k = (1/2,1/2,0)simple tetragonalc/a = 0.8(a) χ a χ c χ ⊥ = χ a χ || = χ c (S = 7/2)0.00.40.81.21.60.0 0.5 1.0 1.5 2.0 2.5 3.0T / T NAx c/a = 1.0 (b) χ (isotropic) χ ⊥ NAx c/a = 1.2 (c) χ a χ c χ ⊥ = χ a χ || = χ c (S = 7/2) FIG. 14: (Color online) Anisotropy of the magnetic suscep-tibilities χ a and χ c due to MDIs versus reduced tempera-ture t A = T /T NA x for a simple tetragonal spin lattice with(a) c/a = 0 .
8, (b) 1.0 (sc lattice) and (c) c/a = 1 .
2. TheAFM propagation vector in the ordered AFM state at t A < k = (cid:0) , , (cid:1) r.l.u. and the easy axis is the c axis [001] forboth c/a = 0 . t A ≥
1, and (85) ( χ ⊥ ) and (87) ( χ k ) for t A ≤ χ c χ ab MnF T N = 69 K (a) χ c -202460 50 100 150 200 250 300T (K) T N = 69 K (b) Data Theory
FIG. 15: (Color online) (a) Magnetic susceptibility χ ver-sus temperature T of tetragonal MnF crystals for appliedfields along the c axis ( χ c ) and in the ab plane ( χ ab ). (b) Anisotropy χ c − χ ab versus T (solid blue squares). Notethe factor of 100 difference between the two ordinate scalesin (a) and (b). The red solid curve is the MFT prediction formagnetic anisotropy arising from magnetic dipole interactionsobtained using Eq. (96). to MFT, χ ⊥ = χ ab for T ≤ T N should be independent of T , which is well satisfied. On the other hand, χ k = χ c should go to zero as T →
0, as also observed. We ob-tained a fairly good fit to χ k ( T ≤ T N ) using our MFTwith no adjustable parameters. The fit function usedwas similar to the equation we obtained for χ k ( T ) forthe pure dipole AFM in Eqs. (87) and Fig. 14.The anisotropy ∆ χ ( T ) ≡ χ c ( T ) − χ ab ( T ) was measuredwith a torque magnetometer and the results are shownin Fig. 15(b). The ∆ χ data measured with the torquemagnetometer for T < T N agree with the anisotropycalculated from the direct measurements in Fig. 15.For T > ∼ T N , a comparison of the data in Figs. 15(a)and 15(b) shows that | ∆ χ | /χ ∼ .
1% for
T > T N . From9Eq. (96), the anisotropy of χ is predicted to be∆ χ M ( T ) = χ ( T ) N A a ( λ − λ ) . (99)Using the values of a and λ − λ in Eqs. (97)and (98), respectively, and the χ M ( T ) data in Fig. 15(a),∆ χ M ( T ) was calculated from Eq. (99) and the resultis shown as the solid red curve in Fig. 15(b) (see alsoRef. 60). The calculation is in excellent agreement withthe data for T > ∼
150 K, suggesting that the MDI is re-sponsible for the χ anisotropy in this T range, or at leastreinforces this anisotropy. However, the data are increas-ingly suppressed to lower values below 130 K, which likelyresult from the onset of dynamic short-range collinearAFM correlations along the c axis with a correlationlength that eventually diverges at T N = 69 K, wherefrom Fig. 15(a), ∆ χ M grows to become large and evenmore negative below that temperature. XII. SUMMARY
A detailed summary of the paper is given in the Ab-stract to the paper. Here we provide a few additionalcomments.The eigenvalues and eigenvectors of the MDI tensorwere determined for specified magnetic wavevectors andspin lattices. The eigenvalues give the energy of a spin inthe magnetic fields of the local moments inside a Lorentzsphere of radius R in units of the a -axis lattice param-eter a . For 3D lattices, R/a = 50 was usually used,for a 2D circle
R/a ≤ R/a = ∞ the eigenvalues were determined exactly. Theeigenvectors are the three orthogonal principal axis di-rections for collinear magnetic ordering. For uniaxial 3Dspin lattices, these were calculated for c/a = 0 . We also calculated the eigenvalues and eigenvectors fornoncollinear AFM structures including the 2D 120 ◦ tri-angular lattice and for the 2D and 3D coplanar non-collinear Shastry-Sutherland lattice and GdB magneticstructure. We compared the ordering-direction predic-tions with data for some Mn +2 , ( S = 5 / +3 andEu +2 , ( S = 7 /
2) compounds and found good agreement.Disagreement occurred for the itinerant AFM BaFe As and for the coplanar noncollinear AFM GdB , which in-dicates that a stronger anisotropy source must be presentin these compounds that defeats the preferences of theMDI.A significant contribution of this paper was to ap-ply our formulation of the Weiss MFT to predictmany properties of the ordered and PM states arisingfrom MDIs. These include the magnetic ordering tem-perature T m , the ordered moment, the magnetic heatcapacity, and for AFMs the perpendicular critical field,the anisotropic magnetic susceptibility versus tempera-ture for T ≤ T N , and the parameters of the Curie-Weiss law for the anisotropic susceptibility for both FMs andAFMs at T ≥ T m . Within MFT, the contributions ofdifferent molecular field sources to these properties areadditive. This means that the same theory can be usedto treat purely magnetic dipole magnets or spin systemscontaining both exchange and dipole interactions. Werecently used the theory to separate the magnetic dipoleand exchange contributions to the properties of the bctcompound EuCu Sb with c/a = 2 . T N = 5 . Acknowledgments
The author is grateful to Andreas Kreyssig for helpfuldiscussions, to Vivek Anand for experimental collabora-tions relating to this work and to Jiping Huang for send-ing a list of published eigenvalue data. This research waspartially supported by the U.S. Department of Energy,Office of Basic Energy Sciences, Division of Materials Sci-ences and Engineering. Ames Laboratory is operated forthe U.S. Department of Energy by Iowa State Universityunder Contract No. DE-AC02-07CH11358.
Appendix A: Direct and Reciprocal Lattices1. Orthogonal Bravais Lattices
In a Bravais spin lattice each spin position is a pointof inversion symmetry with respect to the other spins.For orthogonal lattices which include as special cases thelinear chain, the simple square lattice, sc, bcc, fcc, simpletetragonal and bct lattices, the unit cell origins are at r ji a = n a ˆ a + ba n b ˆ b + ca n c ˆ c , (A1a)where n a , n b and n c are positive or negative integersor 0. For all spin lattices, we normalize all spin positionsand interspin distances by the a -axis lattice parameter a .For body-centered spin lattices one also has atoms at thebody centers r a = (cid:18) n a + 12 (cid:19) ˆ a + (cid:18) n b + 12 (cid:19) ˆ b + ca (cid:18) n c + 12 (cid:19) ˆ c , (A1b)where c/a = 1 for the bcc lattice. The central magneticmoment ~µ i is placed at r i = 0 and hence the sum overneighbors ~µ j at positions r j = r ji in Eq. (16c) excludesthe set ( n a , n b , n c ) = (0 , ,
0) in Eq. (A1a). With ourformulation, b G i ( k ) does not explicitly contain the latticeparameters a or c , and for tetragonal Bravais lattices justthe dimensionless c/a ratio appears as in Eqs. (A1).The reciprocal-lattice vectors in reciprocal-lattice unitsare k = m a ∗ + m b ∗ + m c ∗ , (A2)0where the m i satisfy 0 ≤ m i ≤ a ∗ = 2 πa ˆ a , b ∗ = 2 πa ˆ b , c ∗ = 2 πc ˆ c , (A3)and a , b and c are the corresponding direct-lattice trans-lation vectors. We normalize k by 1 /a , yielding k a = 2 π (cid:16) m ˆ a + m ˆ b + 1 c/a m ˆ c (cid:17) . (A4)Using Eqs. (A4) and (A1a), for the unit cell origins onehas k · r ji = 2 π ( m n a + m n b + m n c ) (A5a)and for the body-center positions k · r ji = 2 π (cid:20) m (cid:18) n a + 12 (cid:19) + m (cid:18) n b + 12 (cid:19) + m (cid:18) n c + 12 (cid:19) (cid:21) , (A5b)where the c/a ratio has canceled out of both expressions.The sum in Eq. (16c) gives an “extinction condition”for the contribution to the sum in Eq. (16c) of the body-centered spins in the bcc lattice in Eq. (A5b), where thecontribution is zero if k · r ji is an odd multiple of π/ k = (cid:18) , , (cid:19) , (cid:18) , , (cid:19) , (cid:18) , , (cid:19) . (A6)For such cases, according to Eq. (15) which assumes acollinear magnetic structure, the spins at the body cen-ters of the unit cells have zero ordered moment and theymake no contribution to the dipolar interaction tensor inEq. (16c). The interaction tensor is then the same as fora simple tetragonal lattice of moments with the same c/a ratio and k value.For the fcc lattice the lattice points are at the positionsin Eq. (A1a) and at r a = (cid:18) n a + 12 (cid:19) ˆ a + (cid:18) n b + 12 (cid:19) ˆ b + 0 , (A7) r a = (cid:18) n a + 12 (cid:19) ˆ a + 0 + (cid:18) n c + 12 (cid:19) ˆ c , (A8) r a = 0 + (cid:18) n b + 12 (cid:19) ˆ b + (cid:18) n c + 12 (cid:19) ˆ c , (A9)with corresponding changes to the expressions for k · r ji .
2. Simple Hexagonal (Triangular) Bravais Lattice
The normalized vectors r ji for the simple hexagonallattice with a = b are given by r ji a = n a ˆ a + n b ˆ b + ca n c ˆ c , (A10) ab a*b* ^ ^^ ^ xy ˚ FIG. 16: (Color online) In-plane hexagonal lattice translationunit vectors ˆ a and ˆ b of the direct lattice and ˆ a ∗ and ˆ b ∗ of thereciprocal lattice, respectively. where here the b axis is at an angle of 120 ◦ with respectto the positive x axis as shown in Fig. 16 and the n i are again positive or negative integers or zero. In twodimensions one sets n c = 0. In Cartesian coordinates thetranslation unit vectors areˆ a = ˆ i , ˆ b = − i + √
32 ˆ j , ˆ c = ˆ k . (A11)A magnetic ordering wavevector k is written in termsof the respective simple hexagonal reciprocal lattice vec-tors as k = m a ∗ + m b ∗ + m c ∗ , (A12a)where the m i are chosen to satisfy 0 ≤ m i ≤ a ∗ = 2 πa (cid:18) ˆ i + 1 √ j (cid:19) , b ∗ = 4 πa √ j , c ∗ = 2 πc ˆ k , (A12b) | a ∗ | = | b ∗ | = 4 π √ a ≡ a, b -axis r . l . u ., (A12c) | c ∗ | = 2 πc ≡ c -axis r.l.u. (A12d)In terms of ˆ a ∗ and ˆ b ∗ , the direct lattice unit vectors areˆ a = 1 √ a ∗ − ˆ b ∗ ) , ˆ b = 1 √ b ∗ − ˆ a ∗ ) , ˆ c = ˆ c ∗ . (A13)The expression for k · r ji is the same as in Eq. (A5a).1 Appendix B: Figures Showing the Approach to theLarge-Radius Asymptotic Eigenvalues for MagneticOrdering on 2D and 3D Spin Lattices -9.1-9.0-8.9-8.8-8.70.00 0.01 0.02 0.03 0.04 0.051/(R/a)primitive square latticeferromagnetic alignmentalong c-axis (a) (R/a) − = 0.001 − − -101230.000 0.001 0.002 0.003 0.004 0.0051/(R/a) (b) FIG. 17: (a) Eigenvalue λ (0 , , , , for FM spin alignmentalong the c axis versus the inverse of the circle radius R aroundthe central moment in units of the square lattice parameter a for the 2D simple square lattice. The a and b -axis eigenvaluesare each equal to − λ (0 , , , , /
2. (b) Deviation of the datafrom the fit. The “noise” is due to the discrete nature ofthe lattice, not to numerical inaccuracy. The lines in (b) areguides to the eye.
R/a = 900 − λ (1/2,1/2,0)[0,0,1] = 2.6458865(1) -3-2-10123 R/a (b)200 500 1000 FIG. 18: (a) Eigenvalue λ (1 / , / , , , for the N´eel-typeAFM moment alignment along the c axis versus the circleradius R around the central moment in units of the squarelattice parameter a for the 2D simple square lattice. The a -and b -axis eigenvalues are each equal to − λ (1 / , / , , , / -3.425-3.423-3.421-3.419-3.4170.00 0.01 0.02 0.03 0.04 0.051/(R/a)simple tetragonal latticeferromagnetic alignmentalong c axis (a)c/a = 1.5-6.252-6.248-6.244-6.240-6.236-6.2320.00 0.01 0.02 0.03 0.04 0.051/(R/a) (b)c/a = 3 FIG. 19: Dependences of the eigenvalue λ (0 , , , , on theinverse radius ( R/a ) − of the Lorentz sphere for FM momentalignments [ k = (0,0,0)] along the c axis in 3D simple tetrag-onal spin lattices with (a) c/a = 1 . c/a = 3. Thelines are guides to the eye. λ FIG. 20: Dependences of the eigenvalue λ (1 / , / , / , , forN´eel-type ordering with k = (1/2,1/2,1/2) on the inverse ra-dius ( R/a ) − of the Lorentz sphere for AFM moment align-ments along the c axis in 3D simple tetragonal spin latticeswith (a) c/a = 1 . c/a = 3. The lines are guides tothe eye. The averages for R/a = 51–100 are shown. With in-creasing c/a , the averages of λ (1 / , / , / , , for R/a = 51–100 approach the 2D square-lattice limit λ (1 / , / , , , =2 .
645 887 in Table I, as shown in Fig. 3(b). Appendix C: Figures Showing Dipolar Eigenvectorsand Eigenvalues versus the c/a
Ratio for Tetragonaland Hexagonal Bravais Spin Lattices and for theHoneycomb Lattice -6-4-202460.5 1.0 1.5 2.0 2.5 3.0c/asimple tetragonal k = (0,0,0) rluferromagnetic alignment along c-axis FIG. 21: (Color online) Dependence of the eigenvalue λ (0 , , , , on the c/a ratio for a simple tetragonal latticewith a FM alignment of the magnetic moments along the c axis. From the figure, one sees that FM alignment alongthe c axis is the most stable for c/a <
1, but for c/a > a or b axis is energetically favorable. -20-10010200.0 0.5 1.0 1.5 2.0 2.5 3.0c/amoments along [010] k = (1/2,0,0) simple tetragonal orbody-centered tetragonal[001]2D limits [100] FIG. 22: (Color online) Eigenvalues for wavevector k =(1/2,0,0) r.l.u. versus the c/a ratio for a simple tetragonalor bct lattice with the moments aligned along [010] ( b axis,solid red circles), [001] ( c axis, solid green diamonds) or [100]( a axis, solid blue squares). The 2D limits for c/a → ∞ areshown as horizontal dashed lines. -10-50510150.0 0.5 1.0 1.5 2.0 2.5 3.0c/amoments alignedalong [0,0,1][1,0,0] or [0,1,0] k = (1/2,1/2,0)simple tetragonal (a)-20-100100.0 0.5 1.0 1.5 2.0 2.5 3.0c/amoments alignedalong [1,0,0] or [0,1,0] k = (0, 0,1/2)simple tetragonal[0,0,1] (b) FIG. 23: (Color online) Eigenvalues (a) λ (1 / , / , for AFMwavevector k = (1/2,1/2,0) r.l.u. and (b) λ (0 , , / for AFMwavevector k = (0,0,1/2) r.l.u. versus the c/a ratio for a simpletetragonal lattice with the moments aligned along [1, 0, 0] or[0, 1, 0] ( a or b axis, solid blue squares) or [0, 0, 1] ( c axis,solid green diamonds). -20-10010200.0 0.5 1.0 1.5 2.0 2.5 3.0c/a[1, − FIG. 24: (Color online) Eigenvalues for wavevector k =(1/2,1/2,0) r.l.u. versus the c/a ratio for a bct spin lattice withthe moments aligned along [1, −
1, 0] (solid red circles), [001]( c axis, solid green diamonds) or [110] (solid blue squares). -12-8-40480.0 0.5 1.0 1.5 2.0 2.5 3.0c/a(a) [0,0,1]moments aligned along [1,0,0] or [0,1,0] k = 0 (FM) or (1,0,0)simple hexagonal2D limits-8-40480.0 0.5 1.0 1.5 2.0 2.5 3.0c/a[1/2,(3/2) ,0]moments aligned along [0,0,1][(3/2) , − FIG. 25: (Color online) Eigenvalues for wavevectors (a) k =0 (FM) or (1,0,0) and (b) k = (1/2,1/2,0) r.l.u. versus the c/a ratio for a simple hexagonal (stacked triangular) spin latticewith the moments aligned along the indicated principal axes.The 2D limits of the respective eigenvalues for c/a → ∞ areshown by horizontal dashed lines. -4-20240.0 0.5 1.0 1.5 2.0 2.5 3.0c/a(a) [1,0,0], [0,1,0]moments aligned along [0,0,1] k = (1/3,1/3,1/3)simple hexagonal2D limits-8-4048120.0 0.5 1.0 1.5 2.0 2.5 3.0c/amoments aligned along[(3/2) , − ,0] FIG. 26: (Color online) Eigenvalues for wavevectors (a) k =(1/3,1/3,1/3) and (b) k = (1/2,1/2,1/2) r.l.u. versus the c/a ratio for a simple hexagonal (stacked triangular) spin latticewith the moments aligned along the indicated principal axes.The 2D limits of the respective eigenvalues for c/a → ∞ areshown by horizontal dashed lines. -12-8-4048120.0 0.5 1.0 1.5 2.0 2.5 3.0c/a(a) [1,0,0], [0,1,0]moments aligned along [0,0,1] k = (1/3,1/3,0)simple hexagonal2D limits-12-8-4048120.0 0.5 1.0 1.5 2.0 2.5 3.0c/a(b) [1,0,0], [0,1,0]moments aligned along [0,0,1] k = (1/3,1/3,1/2)2D limits FIG. 27: (Color online) Eigenvalues for wavevectors (a) k =(1/3,1/3,0) and (b) k = (1/3,1/3,1/2) r.l.u. versus the c/a ratio for a simple hexagonal (stacked triangular) spin latticewith the moments aligned along the indicated principal axes.The 2D limits of the respective eigenvalues for c/a → ∞ areshown by horizontal dashed lines. -40-30-20-1001020300.0 0.5 1.0 1.5 2.0 2.5 3.0c/a(a) [0,0,1] moments alignedalong [1,0,0] or [0,1,0] k = (0,0,0) (FM)honeycomb2D limits-20-10010200.0 0.5 1.0 1.5 2.0 2.5 3.0c/amoments aligned along[(3/2) ,1/2,0]k = (1/2,0,0)(b)2D limits [0,0,1][1/2, − (3/2) ,0] FIG. 28: (Color online) Eigenvalues for propagation vectors(a) k = (0,0,0) (FM) and (b) k = (1/2,0,0) r.l.u. versus the c/a ratio for a honeycomb spin lattice with the moments alignedalong the indicated principal axes. The 2D limits of the re-spective eigenvalues for c/a → ∞ are shown by horizontaldashed lines. -15-10-505101520250.0 0.5 1.0 1.5 2.0 2.5 3.0c/a[100], [010]moments aligned along [001]k = (0,0,1/2) (Néel-type AFM)honeycomb lattice2D limits (a)-15-10-505101520250.0 0.5 1.0 1.5 2.0 2.5 3.0c/a[100], [010]moments aligned along [001] k = (0,0,0)(Néel-type in ab plane)2D limits (b) FIG. 29: (Color online) Eigenvalues for AFM propagationvectors (a) k = (0,0,1/2) (N´eel-type in all directions) and (b) k = (0,0,0) r.l.u. 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Ratiofor Tetragaonal and Hexagonal Bravais Spin Latticesand for the Non-Bravais Honeycomb Lattice
TABLE VII:
Simple Tetragonal Spin Lattices with Fer-romagnetic Alignment and k = (0,0,0).
Eigenvalues λ k α and eigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinatesof the magnetic dipole interaction tensor b G i ( k ) in Eq. (16c)for various values of the magnetic wavevector k in reciprocallattice units (r.l.u.) for simple tetragonal spin lattices with collinear magnetic moment alignments. The most positive λ k α value(s) corresponds to the lowest energy value accord-ing to Eq. (16d). The Cartesian x , y and z axes are alongthe a , b and c axes of the tetragonal lattices, respectively.The accuracy of the values is estimated to be < ∼ ± . k , which determine the anisotropyenergies via Eq. (16d). c/a λ (0 , , α [001] [100], [010] [100] − [001]0.5 30.0834 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE VIII:
Simple Tetragonal Antiferromagnetic Spin Lattices with k = (1/2,0,1/2) and (1/2,1/2,0) r.l.u.
Thesymbol descriptions are the same as in the caption to Table VII. λ k α (cid:0) , , (cid:1) (cid:0) , , (cid:1) c/a [100] [010] [001] [010] − [001] [010] − [100] [100], [010] [001] [001] − [100]0.5 13.6119 15.1664 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE IX:
Simple Tetragonal and Body-CenteredTetragonal Spin Lattice with k = (0,0,1/2) r.l.u.
Thesymbol descriptions are the same as in the caption to Ta-ble VII. λ k α c/a [100], [010] [001] [100] − [001]0.5 15.0417 − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE X:
Simple Tetragonal and Body-Centered Tetragonal Spin Lattices with k = (1/2,0,0) and(1/2,1/2,1/2) r.l.u.
The symbol descriptions are the same as in the caption to Table VII. λ k α (cid:0) , , (cid:1) (cid:0) , , (cid:1) c/a [100] [010] [001] [010] − [001] [010] − [100] [100], [010] [001] [001] − [100]0.5 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE XI:
Body-Centered Tetragonal Spin Latticewith ferromagnetic moment alignment and k =(0,0,0) r.l.u.
The symbol descriptions are the same as inthe caption to Table VII. λ k α c/a [001] [100], [010] [100] − [001]0.50 21.9948 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − √ − − − − − − − − − − − − − − − − − − − − TABLE XII:
Body-Centered Tetragonal Spin Lattices with k = (1/2,0,1/2) r.l.u.
Eigenvalues λ k α and eigenvectorsˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinates of the magnetic dipole interaction tensor b G i ( k ) in Eq. (16c) with collinear magneticmoment alignments. The accuracy of the eigenvalues is estimated to be < ∼ ± . µ [010] [ √ − x , , − x ] [ x, , √ − x ] [010] − [ √ − x , , − x ] [010] − [ x, , √ − x ]c/a x − − − − − − − − − − − − − − − − − − − √ − . − − − − − − − − − − − − − − − − TABLE XIII:
Body-Centered Tetragonal Spin Lattices with k = (1/2,1/2,0) and (0,0,1) r.l.u. Eigenvalues λ k α andeigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinates of the magnetic dipole interaction tensor b G i ( k ) in Eq. (16c) for two valuesof the magnetic wavevector k in reciprocal lattice units (r.l.u.) and collinear magnetic moment alignments. The accuracy ofthe eigenvalues is estimated to be < ∼ ± . k . λ k α (cid:0) , , (cid:1) (0,0,1)c/a [001] [1¯10] [110] [001] − [1¯10] [001] − [110] [100],[010] [001] [100] − [001]0.5 38.4705 8.0868 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − √ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE XIV:
Simple 3D Hexagonal (Stacked Triangular) Spin Lattices with k = (1/2,1/2,0) and (0,0,0) or(1,0,0) r.l.u.
Eigenvalues λ k α and eigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinates of the magnetic dipole interactiontensor b G i ( k ) in Eq. (16c) for two values of the magnetic wavevector k in reciprocal lattice units (r.l.u.) and collinear magneticmoment alignments. The x axis is in the direction of the hexagonal a axis. The accuracy of the eigenvalues is estimated to be < ∼ ± . k . λ k α (cid:0) , , (cid:1) h , √ , i h , √ , i (0,0,0), (1,0,0)c/a [001] h √ , − , i h , √ , i − [001] − h √ , − , i [001] [100], [010] [100] − [001]0.5 38.4698 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE XV:
Simple 3D Hexagonal (Stacked Triangular) Spin Lattices with k = (1/2,1/2,1/2) and (1/3,1/3,1/3).
Eigenvalues λ k α and eigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinates of the magnetic dipole interaction tensor b G i ( k ) inEq. (16c) for two values of the magnetic wavevector k in reciprocal lattice units (r.l.u.) and collinear magnetic momentalignments. The x axis is in the direction of the hexagonal a axis. The accuracy of the eigenvalues is estimated to be < ∼ ± . k . λ k α (cid:0) , , (cid:1) h , √ , i h , √ , i (cid:0) , , (cid:1) c/a [001] h , √ , i h √ , − , i − [001] − h √ , − , i [001] [100], [010] [001] − [100]0.5 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE XVI:
Simple 3D Hexagonal (Stacked Triangular) Spin Lattices with k = (1/3,1/3,0) and (1/3,1/3,1/2).
Eigenvalues λ k α and eigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinates of the magnetic dipole interaction tensor b G i ( k )in Eq. (16c) for two values of the magnetic wavevector k in reciprocal lattice units (r.l.u.) and collinear magnetic momentalignments. The x axis is in the direction of the hexagonal a axis. The accuracy of the eigenvalues is estimated to be < ∼ ± . k . λ k α (cid:0) , , (cid:1) (cid:0) , , (cid:1) c/a [001] [100], [010] [001] − [100] [001] [100], [010] [001] − [100]0.5 38.4703 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE XVII:
3D Honeycomb Spin Lattices with k = (0,0,0) (ferromagnetic) and (1/2,0,0).
Eigenvalues λ k α andeigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinates of the magnetic dipole interaction tensor b G i ( k ) in Eq. (16c) for twovalues of the magnetic wavevector k in reciprocal lattice units (r.l.u.) and collinear magnetic moment alignments. The x axisis in the direction of the hexagonal a axis. The accuracy of the eigenvalues is estimated to be < ∼ ± . k . λ k α (0 , ,
0) [100] (cid:0) , , (cid:1) h , − √ , i h , − √ , i c/a [001] [100], [010] − [001] h √ , , i [001] h , − √ , i − [001] − h √ , , i − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE XVIII:
3D Honeycomb Spin Lattices with k = (0,0,0) and (0,0,1/2) (N´eel-type in the ab plane andthen also along the c axis, respectively. Eigenvalues λ k α and eigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian coordinates of themagnetic dipole interaction tensor b G i ( k ) in Eq. (16c) for two values of the magnetic wavevector k in reciprocal lattice units(r.l.u.) and collinear magnetic moment alignments. The x axis is in the direction of the hexagonal a axis. The accuracy of theeigenvalues is estimated to be < ∼ ± . k . λ k α (0 , , (cid:0) , , (cid:1) c/a [001] [100], [010] [001] − [100] [001] [100], [010] [001] − [100]0.5 39.6849 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − TABLE XIX:
3D Honeycomb Spin Lattices with k =(0,0,1/2) (FM intraplane and AFM interplane).
Eigen-values λ k α and eigenvectors ˆ µ = [ µ x , µ y , µ z ] in Cartesian co-ordinates of the magnetic dipole interaction tensor b G i ( k ) inEq. (16c) for the magnetic wavevector k in reciprocal lat-tice units (r.l.u.) and collinear magnetic moment alignments.The x axis is in the direction of the hexagonal a axis. The ac-curacy of the eigenvalues is estimated to be ≈ ± . k . λ k α c/a [001] [100], [010] [100] − [001]0.5 − − − − − − − − − − − − − − − − − − − − − − − − − −−