Phase diagram and magnons in quasi-one-dimensional dipolar antiferromagnetsM. Hummel, F. Schwabl, and C. Pich Citation: Journal of Applied Physics 85, 5088 (1999); doi: 10.1063/1.370099 View online: http://dx.doi.org/10.1063/1.370099 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/85/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The square-kagome quantum Heisenberg antiferromagnet at high magnetic fields: The localized-magnonparadigm and beyond Low Temp. Phys. 40, 513 (2014); 10.1063/1.4881184 Phase diagram of the two-dimensional quantum antiferromagnet in a magnetic field J. Appl. Phys. 99, 08H503 (2006); 10.1063/1.2172209 Ground state phase diagrams of an anisotropic spin- 1 2 Δ -chain with ferro- and antiferromagnetic interactions J. Appl. Phys. 97, 10B306 (2005); 10.1063/1.1851893 Spin waves, phase separation, and interphase boundaries in double exchange magnets J. Appl. Phys. 91, 7508 (2002); 10.1063/1.1448306 A new paradigm for two-dimensional spin liquids J. Appl. Phys. 83, 7387 (1998); 10.1063/1.367682

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Phase diagram and magnons in quasi-one-dimensional dipolarantiferromagnets

M. Hummela) and F. SchwablPhysik-Department, Technische Universita¨t Munchen, D-85747 Garching, Germany

C. PichPhysics Department, University of California, Santa Cruz, California 95064

We investigate antiferromagnetic spin chains, which are coupled by a weak antiferromagneticexchange interaction. The spins are located on a hexagonal lattice, i.e., frustration is present whenthree-dimensional order sets in. Typical realizations of such systems are the halides ABX3. In thiswork we particularly study the role of the long-range dipolar interaction within the framework of aHeisenberg model with nearest-neighbor exchange and additional dipolar interaction. We perform aclassical ground-state analysis and show that the spin configuration is sensitively dependent onk8,the ratio of the dipolar interaction to the interchain interaction, as a consequence of their competingcharacter. Several commensurate and incommensurate phases arise in the different regions of theparameter space. The ground-state investigations are supplemented by a stability analysis by meansof a linear spin-wave calculation. From the magnon spectra we can show that all commensuratephases are stable against fluctuations. In comparison with experiments (CsMnBr3, RbMnBr3) weobtain good agreement for the energy gaps. From this we conclude that the dipolar interaction is themost important source of anisotropy in these Mn compounds. ©1999 American Institute ofPhysics.@S0021-8979~99!75108-X#

I. INTRODUCTION

Unconventional magnetic systems have attracted the in-terest of experimental and theoretical physicists in the lastfew years.1 In these systems competing interactions and/orgeometric frustration due to the underlying lattice can lead tounconventional ground states, magnon spectra, and magneticphase diagrams.2 Furthermore, fluctuations are enhanced insystems with frustrated ground states as well as in low di-mensions.

An interaction, which is often competing with respect tothe exchange interaction is the dipole–dipole interaction.3 Inreal systems the dipole–dipole interaction~DDI! is alwayspresent in addition to the short-range exchange interaction.Although its energy is lower than the exchange energy itplays an important role in low-dimensional systems due to itsanisotropic and long-range character.

The most famous quasi-one-dimensional systems are theternary compounds ABX3 ~A alkaline, B transition metal,and X halogen!, which have been studied intensively theo-retically and experimentally in the context of Haldane’sphase4 and solitonic excitations.5 In these systems the carrierof the magnetic moment, the B ions, are located on a hex-agonal lattice.6 In this work, the Mn compounds are of par-ticular interest. Because the angular momentumL is zero, nocrystal-field splitting occurs in these systems and the dipole–dipole interaction should be the most important anisotropy.

The influence of the DDI in quasi-one-dimensional, an-tiferromagnetic spin-chain systems has not yet been studiedvery thoroughly. Instead, the DDI is often replaced by a

single-ion anisotropy7 or the coupling between spin chains isneglected.8

II. MODEL

The Hamiltonian of the dipolar antiferromagnet reads

H52 (lÞ l 8

(ab

~Jll 8dab1All 8ab

!SlaSl 8

b , ~1!

with spins Sl at hexagonal lattice sitesxl . The first termdescribes the exchange interactionJll 8 , which includes theintrachain as well as the interchain interaction. In the follow-ing, we consider only nearest-neighbor exchange, i.e.,

Jll 85H 2J l,l 8 along the chains

2J8 l ,l 8 within the basal plane.

For a hexagonal lattice the Fourier transform of the exchangeenergy is given by

Jq522J cosqz22J8Fcosqx12 cosS qx

2 D cosS)2 qyD G .~2!

Here, and in the following, we measure wave vectors in thechain direction in units of 1/c, and wave vectors within thebasal planes in units of 1/a. The second term in Eq.~1! is theclassical dipole–dipole interaction

All 8ab

521

2~gmB!2S dab

uxl2xl 8u3 2

3~xl2xl 8!a~xl2xl 8!b

uxl2xl 8u5 D .

~3!a!Electronic mail: mhummel@physik.tu-muenchen.de

JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 8 15 APRIL 1999

50880021-8979/99/85(8)/5088/3/$15.00 © 1999 American Institute of Physics

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This term is evaluated by means of the Ewald summationtechnique,9,10 which allows the consideration of the long-range nature of the three-dimensional DDI in terms of fastconvergent sums.

III. GROUND STATES

In this section we calculate the classical ground states ofthe system as a function of the ratio of the dipolar energy tothe interchain exchange interaction

k85~gmB!2

VzJ8, ~4!

where Vz5()/2)a2c is the volume of the primitive cell.Due to the quasi-one-dimensionality of the systems~the ratioJ8/J is 1022– 1023! we consider only antiferromagnetic spinconfigurations along the chain axis, i.e., we restrictqz to p.The ground-state energy reads

Eg52 (lÞ l 8

SlMll 8Sl 852(q

SqMqS2q ~5!

with

Mq5S Jq1Aq11

Aq12

Aq13

Aq12

Jq1Aq22

Aq23

Aq13

Aq23

Jq1Aq33D , ~6!

whereAq135Aq

2350 for qz5p. The ground states are speci-fied by those wave vectors which belong to the largest eigen-values of the matrixMq in Eq. ~6!. We obtain the followingphases for decreasingk8 @for the lattice constants we tookthe values of CsMnBr3 ~Ref. 11!#:

~1! Ferromagnetic phase:k8.k185200.50. In the regionwhere the dipolar energy is large compared to the interchainexchange, the minimum of the ground-state energy isreached atq15(0,0,p). This means that spins within basalplanes are ordered ferromagnetically, but still antiferromag-netically along the chains.

~2! Incommensurate phase I: 200.125k28,k8,k18 . Theground state is an incommensurate~IC! phase in this param-eter region. The wave vector moves continuously fromq1 tothe wave-vectorq25(0,2p/),p) ~the thick solid line inFig. 1! or any other path rotated by 60°~dashed lines in Fig.1!.

~3! Collinear phase: 17.255k38,k8,k28 . Spins withinthe basal planes are oriented ferromagnetically in chains that

are aligned antiferromagnetically to one another~see Fig. 2!.Because of the sixfold symmetry, there are six such groundstates resulting from rotation of the ferromagnetic chains.Note that the continuous degeneracy is lifted.

~4! Incommensurate phase II: 0,k8,k38 . In this in-commensurate phase the wave vector moves fromq2 to q3

5(2p/3,2p/),p). The incommensurability appears be-cause of the finite slope of the dipolar tensor atq3 ~Ref. 12!and the parabolic behavior of the exchange energy.

~5! 120° structure:k850. This ground state is character-ized by a three-sublattice spin configuration in each basalplane.The coupling of the spin space to the real space induced bythe DDI forces the spins to align within the lattice basalplanes for all four phases, in which the DDI is nonzero.Thus, the DDI leads to an in-plane anisotropy.

The phase diagram for the whole parameter region ofk8is shown in Fig. 2.

In summary, we find three commensurate and two in-commensurate phases for the arbitrary value ofk8.

IV. MAGNON SPECTRA

The spin-wave calculation is of interest in its own rightand also serves to scrutinize the stability of the phases foundagainst the fluctuations. To that end, we write the Hamil-tonian ~1! in terms of creation and annihilation operatorsemploying the Holstein–Primakoff transformation.13 For lowtemperatures an expansion up to bilinear terms can be used,leading to linear spin-wave theory. The Hamiltonian then isdiagonalized with a Bogoliubov transformation from whichwe obtain the spin-wave frequencies. This investigation isrestricted to the commensurate phases because the infiniteprimitive cell of the incommensurate phases resists such ananalysis.

A. Ferromagnetic phase

The magnon spectrum of the ferromagnetic phase isstable for the parameter region given in Sec. III. The rota-tional invariance of the spins around the chain axis leads to aGoldstone mode in the spectrum.14

FIG. 1. Paths of the wave vector that minimize the ground-state energy inthe Brillouin zone of the hexagonal lattice (qz5p).

FIG. 2. Spin configurations within the basal plane for different ratios of thedipolar to interchain interactionk8. Only for k850 the 120° structure isestablished. For infinitesimalk8 the phase IC II is favored. The spin con-figurations of incommensurate phases are not sketched.

5089J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Hummel, Schwabl, and Pich

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B. Collinear phase

The collinear phase of Sec. III is also stable against fluc-tuations. There is no Goldstone mode due to the discretedegeneracy of the ground state.

C. 120° structure

We argued in Sec. III that the 120° structure is unstablefor infinitesimal dipolar energy due to a linear slope of thedipolar tensor at the ordering wave vector. However, we per-formed a spin-wave calculation based on a commensurate120° structure, where the spins are located within the basalplanes of the lattice.15 It turns out that the spin-wave spec-trum is stable for weak dipolar energies, from which weconclude that this commensurate ground state is a good ap-proximation.

The spin-wave frequencies resulting for the dipolar 120°structure for CsMnBr3 are shown in Fig. 3, where we usedJ5215 GHz andJ850.41 GHz, respectively.16 This leads tok850.774, i.e., this substance is in the IC II region of thephase diagram in Fig. 2. Including the DDI, from the threeGoldstone modes only one survives, reflecting the un-changed rotational symmetry around the chain axis.

The spin-wave gaps atq50 amount toE0,15198 GHz,E0,25295 GHz, andE0,35410 GHz for CsMnBr3, whichcompares favorably with the experimental values.16,17 Notethat this calculation has no free parameter to fit, since thedipolar energy is determined by the lattice constants. We

also calculated the spin-wave gaps for RbMnBr3 ; neglectingcrystal distortions we also obtain good agreement with theexperiment.18

Thus, we do not need any single-ion anisotropy to ex-plain these results.

V. SUMMARY

We found three commensurate and two incommensuratephases for different values of the ratio of the dipolar to in-terchain interaction due to the competing character of thosetwo interactions. We showed via linear spin-wave theory thatall commensurate phases are stable against fluctuations andthat the incommensurate phase IC II can be approximated bya 120° structure for weak dipolar energies.

The spin-wave gaps of CsMnBr3 and RbMnBr3 are ingood agreement with the experiment, which shows that thedipolar energy is the most important source of anisotropy inthese Mn compounds.

ACKNOWLEDGMENTS

This work has been supported by the BMBF under Con-tract No. 03-SC5-TUM 0 and the DFG under Contract No.PI 337/1-2.

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tallogr. Cryst. Chem.28, 1640~1972!.12H. Shiba and N. Suzuki, J. Phys. Soc. Jpn.51, 3488~1982!.13T. Holstein and H. Primakoff, Phys. Rev.58, 1098~1940!.14M. Hummel, C. Pich, and F. Schwabl~unpublished!.15J. A. Oyedele and M. F. Collins, Can. J. Phys.56, 1482~1978!.16U. Falk, A. Furrer, H. U. Gu¨del, and J. K. Kjems, Phys. Rev. B35, 4888

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FIG. 3. Magnon spectrum of the 120° structure perpendicular to the chaindirection.

5090 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Hummel, Schwabl, and Pich

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