# Influence of the dipolar interaction on phase diagram, magnons, and magnetization in quasi-one-dimensional antiferromagnets on a hexagonal lattice

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<ul><li><p>PHYSICAL REVIEW B, VOLUME 63, 094425Influence of the dipolar interaction on phase diagram, magnons, and magnetizationin quasi-one-dimensional antiferromagnets on a hexagonal lattice</p><p>M. Hummel and F. SchwablPhysik-Department, Technische Universitat Munchen, D-85747 Garching, Germany</p><p>C. PichPhysics Department, University of California, Santa Cruz, California 95064</p><p>~Received 30 December 1999; published 13 February 2001!</p><p>The role of the dipole-dipole interaction in quasi-one-dimensional antiferromagnets is investigated within aHeisenberg model with nearest-neighbor exchange. We deal with systems in which the magnetic ions arelocated on a hexagonal lattice, i.e., frustration is present when three-dimensional order sets in. We perform aground-state calculation for different ratios of dipolar energy to interchain-exchange energy and find severalcommensurate and incommensurate phases. This is a consequence of the competing character of these twointeractions. For the commensurate phases the influence of fluctuations is studied by means of linear spin-wavetheory. It turns out that all commensurate phases are stable against fluctuations. The theoretical spin-wavespectra are compared with experiments on CsMnBr3 and RbMnBr3. The results suggest that the most importantsource of anisotropy in these Mn compounds is the dipole-dipole interaction. Furthermore the magnetic phase-diagram for small dipolar energies is investigated. A spin-wave calculation allows one to calculate the influ-ence of the fluctuations on physical properties like ground-state energy and magnetization. These resultscompare favorably with measurements on CsMnBr3.</p><p>DOI: 10.1103/PhysRevB.63.094425 PACS number~s!: 75.10.Jm, 75.30.Ds, 75.30.Gw, 75.50.Eeesaam</p><p>sto</p><p>AecnsD</p><p>ti</p><p>eyse</p><p>he</p><p>nteaitw</p><p>eee</p><p>enalal</p><p>m-</p><p>re,lot-arof</p><p>ineen</p><p>ci-larex-re-ge</p><p>e-theIn-</p><p>y,</p><p>I</p><p>o-forto</p><p>eseearteithI. INTRODUCTION</p><p>The interest in unconventional magnetic systems has brenewed in the last few years.1 Competing interactions awell as geometric frustration due to the underlying lattice clead to unconventional ground states, unique phase diagrand novel critical phenomena.2 The fact that the groundstates in these systems are highly degenerate and becauthe low dimensionality the contributions of fluctuationsphysical observables are enhanced.</p><p>In real magnets the dipole-dipole inter-action~DDI! isalways present in addition to the exchange interaction.though the DDI is usually two to three orders of magnitudlower than the exchange interaction, it is nevertheless cruto study its implications in the physics of phase transitioDue to its anisotropic and long-range character the Dchanges the physics in the critical region qualitatively3 andhas also a drastic influence on ground states and excitaspectra4,5 especially in low-dimensional systems.</p><p>In this work we particularly study the influence of thDDI on quasi-one-dimensional antiferromagnetic spin stems on a hexagonal lattice. In these compounds thechange interaction along single chains~intrachain-interaction! is several orders of magnitude larger than tinteraction between different chains~interchain-interaction!.The intrachain as well as the interchain interaction are aferromagnetic. The quasi-one-dimensionality appears duthe fact that one has direct superexchange along the chwhereas the interaction between chains is mediated bynonmagnetic ions.</p><p>The quasi-one-dimensional systems, which have bstudied most intensively in the context of Haldanes phas6</p><p>solitonic excitations7 and the effects of magnetic frustration8</p><p>are the ternary compoundsABX3 (A alkaline, B transition0163-1829/2001/63~9!/094425~18!/$15.00 63 0944en</p><p>ns,</p><p>e of</p><p>l-sial.I</p><p>on</p><p>-x-</p><p>i-tons,o</p><p>n,</p><p>metal, andX halogen!. In these systems the carrier of thmagnetic moment, the B ions, are located on a hexagolattice,8 leading to frustration effects when three-dimensionorder sets in. We are mainly interested in the Mn copounds; due to the fact that their spin valueS55/2, spin-wave theory should be a good approximation. Furthermoas the orbital angular momentumL is zero apart from smalrelativistic corrections and therefore the crystal-field anisropy is only of the order of at most 20% of the dipolenergy,9 the DDI should be the most important sourceanisotropy.</p><p>The role of the dipolar interaction in ferromagnetic spchains, which are coupled antiferromagnetically has bstudied quite well.10,4,11For CsNiF3, for example, the dipolarorigin of the unexpected collinear ground state was eludated, the spin-wave dispersion calculated in a dipoHeisenberg model was successfully compared with theperiment and furthermore it was possible to determine aliable value of the strength of the interchain exchaninteraction.12</p><p>The influence of the DDI on the properties of quasi-ondimensional antiferromagnetic spin chain systems onother hand, has not yet been studied very thoroughly.stead, the DDI is often replaced by a single-ion anisotrop13</p><p>truncated after the first term14 or the coupling between spinchains is neglected entirely.15 Therefore the effect of the DDis studied in a systematic way in this paper.</p><p>The outline of the paper is as follows. In Sec. II we intrduce the model, in Sec. III we calculate the ground statesthe whole parameter region of the ratio of dipolar energyinterchain exchange energy without magnetic field. Thground-state investigations are supplemented by a linspin-wave calculation in Sec. IV for the commensuraphases; in this section we also compare our findings w2001 The American Physical Society25-1</p></li><li><p>rsiandut</p><p>hafo</p><p>ertrnd</p><p>ut</p><p>w</p><p>io</p><p>ve</p><p>aial</p><p>y</p><p>ee</p><p>st</p><p>-</p><p>n-</p><p>torsthe</p><p>er-iveavedif-</p><p>or-al</p><p>t-</p><p>gyum</p><p>eiven</p><p>M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425neutron scattering measurements of the spin-wave dispein CsMnBr3. In Sec. V we examine the magnetic phase dgram for low dipolar energies for fields both parallel aperpendicular to the chain direction. Based on these grostates we perform a spin-wave calculation and investigateinfluence of fluctuations on the magnetization. We show tquantum fluctuations in our dipolar model are responsiblethe huge anisotropy of the magnetization found in expments. In Appendix A the formulas for the spin-wave specof the dipolar 120 structure with an applied field are givewhereas the dipolar tensor is briefly discussed in AppenB.</p><p>II. MODEL</p><p>The Hamiltonian of the dipolar antiferromagnet withoapplied field reads</p><p>H52 (l l 8</p><p>(ab</p><p>~Jll 8dab1All 8</p><p>ab!Sl</p><p>aSl 8b , ~1!</p><p>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 folloing we consider only nearest-neighbor exchange, i.e.,</p><p>Jll 85H 2J l,l 8 along the chains,2J8 l ,l 8 within the basal plane.For a hexagonal lattice, where the sites of the magneticare indexed byxl5a( l 1m/2,A3/2m,c/an) with l ,m,n in-teger, the Fourier transform of the exchange energy is giby</p><p>Jq522J cosqz22J8Fcosqx12 cosS qx2 D cosSA32 qyD G .~2!</p><p>Here and in the following we measure wave vectors in chdirection in units of 1/c, and wave vectors within the basplanes in units of 1/a.</p><p>The second term in Eq.~1! is the classical dipole-dipoleinteraction</p><p>All 8ab</p><p>52GS dabuxl2xl 8u</p><p>32</p><p>3~xl2xl 8!a~xl2xl 8!b</p><p>uxl2xl 8u5 D , ~3!</p><p>whereG51/2(gmB)2 in cgs units. This term is evaluated b</p><p>means of the Ewald summation technique,16,17 which allowsthe consideration of the long-range nature of the thrdimensional DDI in terms of fast convergent sums.</p><p>III. GROUND STATES</p><p>In this section we calculate the classical ground statethe system as a function of the ratio of the dipolar energythe interchain exchange interaction</p><p>k85~gmB!</p><p>2</p><p>VzJ8, ~4!09442ion-</p><p>ndhetr</p><p>i-a,ix</p><p>-</p><p>ns</p><p>n</p><p>n</p><p>-</p><p>ofo</p><p>whereVz5(A3/2)a2c is the volume of the primitive cell ofthe hexagonal lattice~see Fig. 1!.18 Due to the quasi-onedimensionality of the systems~the ratio J8/J is1022. . .1023) we consider only antiferromagnetic spin cofigurations along the chain axis, i.e., we restrictqz to p. Theground-state energy reads</p><p>Eg52 (l l 8</p><p>SlMll 8Sl 852(q SqMqS2q ~5!</p><p>with</p><p>Mq5S Jq1Aq11 Aq12 Aq13Aq21 Jq1Aq22 Aq23Aq</p><p>31 Aq32 Jq1Aq</p><p>33D , ~6!</p><p>where generallyAqi j 5Aq</p><p>j i for i j and Aq135Aq</p><p>2350 for qz5p. The ground states are specified by those wave vecwhich belong to the largest of the three eigenvalues ofmatrix Mq given by</p><p>l15Jq1Aq33,</p><p>l2,35Jq11</p><p>2~Aq</p><p>111Aq22!6</p><p>1</p><p>2A~Aq112Aq22!214~Aq12!2. ~7!</p><p>The competing quality of the exchange and dipolar intaction becomes clear in Fig. 2. If we change the relatstrength of the two interactions one can see, that the wvector minimizing the ground state energy is situated atferent points of the Brillouin zone.</p><p>Particularly note the linear slope of the dipolar tenscomponents at theH point of the Brillouin zone. For decreasing k8 we obtain the following phases, where the actuvalue of J has no influence on the results, as long asJ@J8,(gmB)</p><p>2/Vz ~the numerical values are valid for the latice of CsMnBr3,</p><p>19 i.e., c53.26 , a57.61 !.</p><p>A. Ferromagnetic phase</p><p>k8.k185200.50. In the region where the dipolar eneris large compared to the interchain exchange, the minimof the ground-state energy is reached at theA point of the</p><p>FIG. 1. Upper half of the crystallographic Brillouin zone of thhexagonal lattice. The wave vectors for the denoted points are gin Appendix C.5-2</p></li><li><p>sti</p><p>athtifve</p><p>nct</p><p>n</p><p>ri-ro-</p><p>tinghatthe</p><p>in</p><p>. 6</p><p>by</p><p>retheby</p><p>or-e</p><p>rg</p><p>II</p><p>tate</p><p>INFLUENCE OF THE DIPOLAR INTERACTION ON . . . PHYSICAL REVIEW B63 094425Brillouin zone, i.e., atq15(0,0,p). This means that spinwithin basal planes are ordered ferromagnetically, but santiferromagnetically along the chains.</p><p>The ground state energy is given by</p><p>Eg52NS2S 2J26J81 12 ~Aq1111Aq122! D . ~8!</p><p>Note that the dipolar componentsAq111 andAq1</p><p>22 have the same</p><p>value. This means that the ground-state energy is the sfor spin configurations where all spins are rotated withinbasal plane and spins in adjacent planes are oriented anromagnetically@see also Goldstone mode in the spin-waspectrum, Eq.~18!#.</p><p>B. Incommensurate phase I</p><p>k18.k8.k285200.12. The ground state is an incommesurate phase in this parameter region. The wave vemoves continuously fromq1 to the wave vectorq25(0,2p/A3,p) ~see Fig. 3, Fig. 4, and thick solid line iFig. 5! or any other path rotated by 60~dashed lines in Fig.</p><p>FIG. 2. Exchange interaction~solid line! and dipolar tensor forsome directions in the Brillouin zone. Long-dashed:Aq</p><p>11, dotted:Aq</p><p>22, and dot-dashed:Aq12. To see the behavior of the dipolar tens</p><p>more clearly, we setc/a5A2/3. The numerical values of the dipolar tensor are given in units ofG/Vz and the values for the exchangconstants are set toJ51 andJ850.2. Aq</p><p>135Aq2350 for qz5p.</p><p>FIG. 3. Wave vector which minimizes the ground state enein the incommensurate phase I as a function ofk8.09442ll</p><p>meeer-</p><p>-or</p><p>5!. This is due to an increase inJq from A to L whereasAqab</p><p>decreases.</p><p>C. Collinear phase</p><p>k28.k8.k38517.25. Spins within basal planes are oented ferromagnetically in chains, that are aligned antifermagnetically to one another~see Fig. 6!. Because of the six-fold symmetry, there are six such ground states resulfrom rotation of the ferromagnetic chains. This means tthe minimum of the ground state energy is reached atpoints L, L8 and the ones rotated by 60 in the Brillouzone. Note that the continuous degeneracy is lifted.</p><p>The ground state energy for the domain drawn in Figreads</p><p>Eg52NS2~2J12J81Aq2</p><p>11!. ~9!</p><p>For a domain where the ferromagnetic chains are rotated60 around thez axis the ground state energy is given by</p><p>Eg52NS2~2J12J8!2</p><p>1</p><p>4NS2~Aq5</p><p>1113Aq52222A3Aq5</p><p>12!,</p><p>~10!</p><p>with q55(p,p/A3,p). These two ground state energies athe same, which can be shown by exact relations fordipolar tensor components. The third domain is describedthe wave vectorq65(p,2p/A3,p).</p><p>y</p><p>FIG. 4. Dependence of the ordering wave vector in the ICphase on the value ofk8.</p><p>FIG. 5. Paths of the wave vector that minimize the ground-senergy in the Brillouin zone of the hexagonal lattice (qz5p).5-3</p></li><li><p>sild</p><p>ha-</p><p>ec</p><p>it</p><p>-</p><p>rrinip</p><p>eeth</p><p>isi</p><p>cep</p><p>ate</p><p>tescu-</p><p>in-</p><p>p-thealra-</p><p>ra-wsed,is</p><p>th-ciested</p><p>itivesis.</p><p>cal-</p><p>m</p><p>ert-</p><p>n</p><p>s a</p><p>M. HUMMEL, F. SCHWABL, AND C. PICH PHYSICAL REVIEW B63 094425Note that if we didnt consider the IC I phase, the trantion from the ferromagnetic to the collinear phase wouappear atk85200.31, i.e., exactly betweenk18 andk28 . How-ever, the ground state energy of the incommensurate pbetweenk18 andk28 is lower than both the one for the ferromagnetic as well as the one for the collinear phase.</p><p>D. Incommensurate phase II</p><p>k38.k8.0. In this incommensurate phase the wave vtor moves fromq2 to q35(2p/3,2p/A3,p). The incommen-surability for low dipolar energy appears because of the finslope of the dipolar tensor atq3 ~Ref. 11! and the parabolicbehavior of the exchange energy.</p><p>The expansion of the largest eigenvalue of Eq.~7! around(2p/3,2p/A3,p) leads to the following ground-state energy:</p><p>EgS 2p3 2qx , 2pA3 2qy ,p D'2NS2S 2J13J82 34 J8~qx21qy2!</p><p>1~gmB!</p><p>2</p><p>Vz~0.0464qx10.0036qy</p><p>2! D . ~11!One can see that without dipolar energy~i.e., k850) thelowest ground state energy is reached forq5q35(2p/3,2p/A3,p). However, for any infinitesimal dipolaenergy the system is driven to an incommensurate ordewave vector. The 120 phase established for vanishing dlar energy is reached asymptotically~see Fig. 4!.</p><p>E. 120 structure</p><p>k850. This ground state is characterized by a thrsublattice spin configuration in each basal plane, i.e., bywave vectorq3 (H8) and the correspondent rotated ones.</p><p>We state that for all four phases, in which the DDInonzero the coupling of the spin space to the real spaceduced by the DDI forces the spins to align within the lattibasal planes. Thus, the DDI leads to an in-plane anisotro</p><p>FIG. 6. Spin configurations within the basal plane for differeratios of dipolar to interchain interactionk8. Only for k850 the120 structure is established. For infinitesimalk8 the phase IC II isfavored. The spin configurations of the incommensurate phasenot sketched.09442-</p><p>se</p><p>-</p><p>e</p><p>go-</p><p>-e</p><p>n-</p><p>y.</p><p>The phase diagram for the whole parameter region ofk8is shown in Fig. 6. In summary, we find three commensurand two incommensurate phases as function ofk8.</p><p>IV. MAGNON SPECTRA</p><p>To see whether fluctuations destroy the ground stafound in the previous section we perform a spin-wave callation. To that end we write the Hamiltonian~1! in terms ofcreation and annihilation operators employing the HolstePrimakoff transformation20</p><p>Sl15A2S flal ,</p><p>Sl25A2Salf l ,</p><p>Slz5S2al</p><p>al ~12!</p><p>with</p><p>f l5A12 alal2S</p><p>~13!</p><p>and S65Sx6 iSy. The tilde denotes the fact that these oerators have to be utilized in the rotated frame, wherelocal z axis is aligned in the spin direction of the classicground state. The Fermi commutation relations for the opetors ~12! are satisfied if the creation and annihilation opetors a and a obey Bose commutation relations. For lotemperatures an expansion up to bilinear terms can be uleading to linear spin-wave theory. The Hamiltonian thendiagonalized with a Bogoliubov transformation or the meods of Greens functions leading to the spin-wave frequenof the system. The investigation described above is restricto t...</p></li></ul>