Magnon-Phonon Effects in Ferromagnetic Manganites

PHYSICAL REVIEW B, VOLUME 65, 014409

Magnon-phonon effects in ferromagnetic manganites

L. M. Woods* Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996 and Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831 ͑Received 1 August 2000; revised manuscript received 27 April 2001; published 30 November 2001͒ A model is presented for the magnon-phonon interaction in three-dimensional cubic ferromagnetic systems. The Heisenberg Hamiltonian for localized spins is used. The calculated magnon and phonon dampings are compared with the experiments and good agreement is found. It is estimated that there is a signiﬁcant broadening in the magnon linewidth at the end of the zone and that the phonon linewidth is not affected by this mechanism. The modiﬁed magnon spectrum is also evaluated and compared to experimental measurements. It is found that the model predicts softening of the magnon mode at the end of the zone consistent with the experiments. The contribution from both Mn and O atoms is included. The developed model can be applied to systems with reduced dimensions and systems with different than cubic symmetry.

DOI: 10.1103/PhysRevB.65.014409 PACS number͑s͒: 75.10.Ϫb, 75.10.Jm, 75.30.Ds

I. INTRODUCTION interaction between magnons and phonons. A generic model was proposed in Ref. 8, but no explicit form of the Hamil- Spin waves have been studied extensively in magnetic tonian was given and no specific calculations were presented. systems. The theory is designed to investigate the ground In principle, the coupling between magnetic moments and state and the low-lying excitation states of a system with lattice can modify the spin waves in two different ways. One localized coupled spins.1 In insulating ferromagnets the spin way is that the anisotropy of the spin waves can be affected waves ͑or magnons͒ are usually well defined through the through mixing with the phonons. The other way is when entire Brillouin zone and studying the magnon dispersion there is a significant magnetoelastic interaction or magnon- relations can provide useful information about the material. phonon coupling. The latter way is the subject of this work. A suitable description is the Heisenberg Hamiltonian with We try to answer the question if the coupling between spin nearest-neighbor exchange coupling constant. and lattice degrees of freedom are responsible for the ob- ͑ 7 Ferromagnets of the type A1ϪxBxMnO3 manganite per- served effects by presenting a simple model and evaluating ovskites͒, where A is a rare earth and B is an alkaline earth, the damping and the energy dispersion of the magnons. can reach different states by varying A and B.2 The theoret- The paper is organized as follows. In Sec. II the Hamil- ical basis for understanding the manganites is the so-called tonian and the explicit matrix elements are written. We also double-exchange ͑DE͒ model. This is a Kondo lattice model discuss different limits. In Sec. III the magnon and phonon with an exchange constant J in the limit of infinite J.3 Some broadenings are estimated and a comparison with the experi- inelastic neutron scattering measurements suggest that the mental results for the ferromagnetic manganites is given. In spin waves can be mapped with a nearest-neighbor Heisen- Sec. IV the renormalized magnon dispersion is estimated and berg Hamiltonian.4 An approximate spin-wave theory found compared with experimental findings. Section V is left for that the DE model in the infinite-J limit and the nearest- conclusions, where we discuss the model and its possible neighbor Heisenberg model are equivalent, and that the latter generalizations. model is independent of the concentration and the spin of the 5 carriers. The dispersions can be reproduced very well for II. FORMALISM FOR THE MAGNON-PHONON those manganites that have relatively high TC and relatively COUPLING low residual resistivity ␳. Recent inelastic neutron scattering measurements show Studying the spin dynamics of materials is very important that new effects are seen in manganites with lower TC and because it gives insight into the underlying physics of the higher resistivity ␳.6,7 The result is that a large magnon compounds. It has been measured that for several manganites broadening and softening are observed at the end of the zone at dopings xϭ͓0.15,0.5͔ the spin dynamics throughout the at low temperatures. The large magnon linewidth cannot be Brillouin zone is equivalent to a nearest-neighbor Heisenberg reproduced by the single DE model, which means that addi- ferromagnet.4 In this doping regime the compounds are fer- tional contributions have to be considered. The experiments romagnetic metals and are described by the double-exchange suggest that the onset of the linewidth broadening and soft- model, which accounts for the strong ferromagnetic coupling ening of the mode appear when the magnon dispersion between the itinerant eg and localized t2g carriers. crosses the longitudinal optical branch of the phonons. Thus, Furukawa5 argued that the result is consistent with the the most probable cause of the anomalous broadening has double-exchange model. He showed that starting from the lattice character and the interaction between the magnetic DE model the material behaves like a nearest-neighbor excitations and the lattice excitations needs to be considered Heisenberg ferromagnet and the stiffness constant reflects the explicitly. itinerant nature of the carriers. According to Ref. 5 in the In this paper we will describe a model that governs the limit of infinite double-exchange constant JH , the stiffness

ϭ Ϫ 2 ͚ ϭ † ϩ ͑ ͒ constant is found to be D t(Ri Rj)a /(2S) kf k cos kxa, AQ bϪQ bQ , 5 Ϫ where f k is the Fermi occupation number and t(Ri Rj)is 0 the hopping integral in a single e band. This suggests that in where Ri is the equilibrium positions of the ions on lattice g ␩ˆ order to study the problem of coupling between magnetic sites i, ui is a small displacement, Q is the polarization and lattice excitations for this doping regime, it is correct vector, N is the number of ions, M is the mass of one ion, and † to assume that there are ‘‘effective’’ magnetic moments lo- bϪQ and bQ are the phonon creation and anihilation opera- cated on sites i,j coupled by a ferromagnetic Heisenberg in- tors, respectively. The phonon operator is denoted by AQ . Ϫ teraction. Also the interaction constant J(Ri Rj) can be The next step is to expand the exchange coupling constant 9 Ϫ ϭ␤ 2 ⌬ ␤ taken to be proportional to the hopping integral t. Therefore, J(Ri Rj) V pd/ around the equilibrium positions. de- ͚ 2 the Hamiltonian is notes the numerical factor kf k cos kxa/4S . In this way one obtains up to ﬁrst order in the displacements and for a unit volume ϭϪ ͑ Ϫ ͒ ͑ ͒ H ͚ J Ri Rj Si•Sj . 1 i, j ϭ ϩ ͑ ͒ H H0 HЈ, 6

Here Ri, j stands for the positions of the Mn magnetic ions, Ϫ ϳ ϭϪ ͑ 0Ϫ 0͒ ͑ ͒ Si, j is the ‘‘effective’’ localized spin, and J(Ri Rj) t(Ri H0 ͚ J Ri Rj Si Sj , 7 Ϫ ij • Rj) is the exchange interaction constant. The direct overlap integral t between nearest-neighbor Mn sites in the manganites is zero since due to the perovskite lattice the Mn V pd Mn 0 Јϭ ␤ ͕ ␩ˆ ٌ ͑ Ϫ ͒ iQ Ri Ϫ H 2 ⌬ ͚ XQ i,Q V pd Ri Rl e • atoms are bridged by an O atom. Therefore, t(Ri Rj)is ij,Q • estimated by a second-order perturbation with respect to the ͒ ϪXMn␩ˆ ٌV ͑R ϪR ͒ϪXO␩ˆ ٌ͓V ͑R ϪR 10 electron transfer between Mn 3d andO2p orbitals: V pd . Q j,Q• pd l j Q l,Q• pd i l Thus, tϭV2 /⌬ where V is the overlap integral in Slater- pd pd iQ R0 ϭ͉⑀ Ϫ⑀ ͉ ϩٌV ͑R ϪR ͔͒e • l ͖A S S , ͑8͒⌬ 11 Koster terms and p d is the energy difference be- pd l j Q i• j tween the occupied O 2p and unoccupied 3d levels. It is where the two nearest-neighbor Mn atoms at site i and site j estimated by photoemission experiments that for the manga- are taken to be inequivalent and the O atom is in the middle ϳ 12 nites t 0.72 eV. Furthermore, the charge transfer energy between them at R . The gradient of V is evaluated at equi- ⌬ 13 l is measured to be about 4 eV. Thus, one ﬁnds for the pd librium. The O ion appears because the overlap integral V ϳ pd transfer integral V pd 1.7 eV. depends on the relative distance between the Mn and O po- There are two types of magnon-phonon interactions. One sitions. type is to consider a form of Hamiltonian that is bilinear in 14,15 The unperturbed Hamiltonian can be diagonalized by dif- the magnon and phonon operators. This kind is respon- ferent approaches.1 The Dyson-Maleev transformation ͑ ͒ Ϫ sible for the mixing or hybridization of the magnon and is chosen here: S† ϭͱ2Sa , S ϭͱ2Sa† , Sz ϭS phonon modes and it does not cause the broadening. If hy- i, j i, j i, j i, j i, j Ϫa† a . Considering only nearest neighbors, ␦ϭR0ϪR0 , bridization were signiﬁcant, this would affect both magnon i, j i, j i j the Hamiltonian H and the energy of the magnetic excita- and phonon dispersions through mixing of the excitations. 0 tions after a Fourier transformation are obtained to be The experiments7 show that the phonon linewidth hardly changes and the magnon and phonon excitations exist sepa- ϭ ⑀ † ͑ ͒ rately through the whole zone. The effect of hybridization H0 ͚ kakak , 9 was calculated in the 1970s and 1980s for many rare-earth k materials.14,15 Another way of looking at the problem is when the scat- ⑀ ϭ ␦͒ Ϫ␥ ␦͒ ͑ ͒ k 2S͚ J͑ ͓1 k͑ ͔, 10 tering of a magnon is done with an emission or absorption of ␦ a phonon. In this case the coupling manifests itself through ␦ where a denotes the magnon operator and ␥ (␦)ϭeik• . For the distortion of the lattice. Indeed, studies show the impor- k k a cubic crystal in the nearest-neighbor approximation J(␦)is tance of the fact that conduction electrons strongly couple to replaced by an overall constant J .1 In the limit of small lattice distortions in the manganites.16 To introduce the 0 wave vectors the energy dispersion is quadratic with a stiff- phonons in the picture the magnetic ions are allowed to vi- ness constant Dϭ2SJ a2. Thus, the excitations are isotropic brate around their equilibrium position: 0 and gapless. In further estimates we take the experimental ϭ 2 ϭ 0ϩ ͑ ͒ values cited in Ref. 7: D 0.165 eV Å , stiffness constant, Ri Ri ui , 2 and aϭ3.86 Å lattice parameter. Next, we consider HЈ which describes the interaction be- ϭ ␩ˆ iQ R0 ͑ ͒ tween the localized spins and the phonons. Both type of ions ui ͚ XQ i,Qe • i AQ , 3 Q Mn and O contribute to the magnon-phonon coupling. Spe- cial attention should be given to the gradient of the pd over- ប lap integral. Using the expression for V pd in Ref. 11 the ϭͱ ͑ ͒ symmetry of the p and d functions is such that the derivative XQ ␻ , 4 2NM Q of the exchange constant points along the lattice axis:

014409-2 MAGNON-PHONON EFFECTS IN FERROMAGNETIC MANGANITES PHYSICAL REVIEW B 65 014409

͒ ͑ ˆ␦͒␦ ˜ Ϫ ͒ϭ ٌ V pd͑Ri Rl q0V͑ 11

Here q0 is the Slater coefficient describing the exponential ␦ˆ ϭ 0Ϫ 0 ͉ 0Ϫ 0͉ decrease of the functions and (Ri Rj )/ Ri Rj . q0 is of the order of 1 ÅϪ1. ˜V(␦) is of the order of the original integral V(␦) and for the purposes of these calcuations they will be taken to be equal. FIG. 1. Self-energy diagrams of the first-order magnon-phonon ͑ ͒ ͑ ͒ The interaction Hamiltonian conserves the spin quantum coupling for a magnons, and b phonons. number. The form in Eq. ͑8͒ was obtained for a cubic symmetry and can be written explicitly magnetic excitation are introduced, and the coupling is between two bosonic fields. While in the case of electron- phonon interaction, the coupling is between fermionic and bosonic fields. Јϭ ͓ Mn † ϩ O † ͔ ͑ ͒ H ͚ M k,QakϩQak M k,QakϩQ/2ak AQ , 12 The following Green’s functions for the magnons can be k,Q defined: ͑ ␶͒ϭϪ͗ ͑␶͒ †͑ ͒͘ ͑ ͒ 2 G k, T␶ ak ak 0 , 15 V pd M Mnϭ2i␤ ͚ XMn͑␩ˆ Mn1ϩ␩ˆ Mn2͒␦ˆ ͓sin͑kϩQ͒ a k,Q ⌬ Q Q Q • ˜ ͑ ␶͒ϭϪ͗ †͑␶͒ ͑ ͒͘ ͑ ͒ k,Q G k, T␶ ak ak 0 , 16 Ϫsin k aϪsin Q a͔, ͑13͒ ␶͒ϭϪ ␶͒ ͒ ͑ ͒ • • D͑Q, ͗T␶ AQ͑ AϪQ͑0 ͘. 17 The lowest-order perturbation theory is applied to determine V2 the damping of the magnetic excitations. The unperturbed O ϭϪ ␤ pd O␩ˆ O ␦ˆ ͓ ͑ ϩ ͒ Ϫ M k,Q 4i ͚ XQ Q sin k Q/2 a sin k a Green’s functions are ⌬ k,Q • • • 1 Ϫsin Q/2 a͔. ͑14͒ 0͑ ␶͒ϭ ͑ ͒ • G k, Ϫ⑀ , 18 ik k The lattice constant a and the unit vector ␦ˆ change only in the positive direction of the x, y, and z axes. 1 ˜ 0͑ ␶͒ϭϪ ͑ ͒ G k, ϩ⑀ , 19 ik k III. DAMPING OF THE MAGNONS AND PHONONS ␻ Since the form of the interaction Hamiltonian has been 2 Q D0͑Q,␶͒ϭϪ , ͑20͒ determined, one can analyze it by using Green’s functions ␻2ϩ␻2 technique. It is evident that there is an analogy between this Q type of coupling and the usual electron-phonon coupling— where D0(Q,␶) is the one for the phonons. The self-energy the electron operators which are fermions are substituted diagrams for the magnons and phonons are given in Fig. 1. with boson operators for the magnons. The difference is that The imaginary part of the self-energy of the magnons is a here one takes into account the magnetic ordering of the measure of the damping: ϪIm ⌺(k)ϭប/2␶. The lowest- system. Thus, the bosonic operators ak for the elementary order perturbation expansion reads

Ϫ ⌺͑ ͒ϭ␲ ͉ Mn͉2͕͑ Ϫ ͓͒␦͑⑀ ϩប␻ Ϫ⑀ ͒Ϫ␦͑⑀ Ϫប␻ ϩ⑀ ͔͒ϩ͑ ϩ Ϫ ͓͒␦͑⑀ Ϫប␻ Im k ͚ M k,Q N␻ N⑀ ϩ k Q kϩQ k Q kϩQ 1 N␻ N⑀ ϩ k Q Q Q k Q Q k Q

O 2 Ϫ⑀ ϩ ͒Ϫ␦͑⑀ ϩប␻ ϩ⑀ ϩ ͔͖͒ϩ͉M ͉ ͓͑N␻ ϪN⑀ ͔͓͒␦͑⑀ ϩប␻ Ϫ⑀ ϩ ͒Ϫ␦͑⑀ Ϫប␻ ϩ⑀ ϩ ͔͒ k Q k Q k Q k,Q Q kϩQ/2 k Q k Q/2 k Q k Q/2

ϩ͑1ϩN␻ ϪN⑀ ͓͒␦͑⑀ Ϫប␻ Ϫ⑀ ϩ ͒Ϫ␦͑⑀ ϩប␻ ϩ⑀ ϩ ͔͒, ͑21͒ Q kϩQ/2 k Q k Q/2 k Q k Q/2

1 ϭ ͑ ͒ N␻ ␤ប␻ , 22 Q e QϪ1

1 ϭ ͑ ͒ N⑀ ␤⑀ , 23 k e kϪ1

014409-3 L. M. WOODS PHYSICAL REVIEW B 65 014409

͑ ͒ where M k,Q stands for the matrix elements in Eqs. 13 and ͑14͒. Now let’s look at the different limits. It is instructive to consider the small-wave-vector limit ﬁrst, because one is ⑀ ϭ 2 able to obtain simple expressions. In this case k Dk and ␻ ϭ the longitudinal acoustic phonons with Q sQ are taken. The ␦ functions give the limits of the integration. The matrix elements are expanded with respect to small wave vectors:

V2 MnϭϪ ␤ pd Mn␩ˆ Mn ␦ˆ ͓͑ ͒2͑ ͒ϩ͑ ͒ M k,Q 8Si ⌬ XQ Q • k•a Q•a k•a ϫ͑ ͒2͔ ͑ ͒ Q•a , 24 V2 O ϭ ␤ pd O␩ˆ O ␦ˆ ͓͑ ͒2͑ ͒ϩ͑ ͒ M k,Q 8Si ⌬ XQ Q• k•a Q•a/2 k•a ϫ͑ ͒2͔ ͑ ͒ Q•a/2 . 25 Now the imaginary part of the magnon self-energy is evaluated. Since we are interested in the low-temperature regime, the only signiﬁcant term is the one that has no occupation numbers, since they do not contribute at T→0. Thus,

Ϫ ⌺͑ ͒ϭ␲ ͓͉ Mn͉2␦͑⑀ Ϫប␻ Ϫ⑀ ͒ Im k ͚ M k,Q k Q kϩQ Q ϩ͉ O ͉2␦͑⑀ Ϫប␻ Ϫ⑀ ͔͒ ͑ ͒ M k,Q k Q kϩQ/2 . 26 To the lowest limit of k one calculates

Ϫ ⌺ ͒ϭ 6 ͑ ͒ Im ͑k fJ0͓ka͔ , 27 ϳ ប 6 ϩ ϭ ˆ where f ( J0a /s)(1/M Mn 1/M O) for the case k kz and a similar expression for kϭ(kxˆ ,kyˆ ,0). The following should be noticed here. First, the effect of the magnon-phonon coupling on the damping of the magnons is proportional to k6. This means that the damping is very small since the limit of FIG. 2. Magnon damping as a function of the parameter ka/2␲ ϭ ͑ ͒ ͑ ͒ small wave vectors is considered. If one takes k 0.1kD , a along the z axis and b in the xy plane. where kD is at the end of the zone, then the imaginary part of Ϫ6 the self-energy is proportional to 10 . Second, since the Mn In Fig. 2 the damping is plotted as a function of the param- ion is heavier than the O ion, the contribution from the O to eter ka/2␲ for both cases: k is along the z axis and k is in the the magnon-phonon coupling is actually more significant xy plane. The linewidth of the magnons rises relatively than the contribution to from the Mn. The ratio of the steeply after a certain value of the wave vector, which sug- masses, M /M , shows that the contribution from the O Mn O gests that the effect becomes more important. When com- ion is about 1.6 more. Third, the k6 behavior suggests that pared with the data from Ref. 7 it is evident that the charac- the magnon-phonon damping increases significantly with in- creasing the value of the wave vector. teristic behavior of the damping is relatively well Indeed, experiments indicate that at the end of the zone reproduced. For energies of the magnons smaller than the ␻ for the magnons there is an anomalous increase in the optical phonon frequency 0 the damping of the magnons is damping.7 This happens when the longitudinal optical pho- small. When the crossing in the dispersions occurs, the non dispersion crosses the magnon dispersion. The devel- broadening becomes large, which means that the process of a oped Hamiltonian allows us to calculate the effect—the gen- magnetic excitation scattered into a new one according to the eral expressions from Eqs. ͑12͒–͑14͒ should be used and the conservation energy expressed in the ␦ function becomes energy for the longitudinal optical phonons, ប␻ϭconst, significant. should be taken. The phonon linewidth can also be calculated by evaluat- Again, we calculate the most significant term from Eq. ing the self-energy diagram from Fig. 1͑b͒. Since the cou- ͑21͒, the one that has no occupation numbers. The energy of pling conserves the spin quantum number, the only relevant ប␻ ϭ ϭ the longitudinal optical phonons is Q const 25 meV. term here is

014409-4 MAGNON-PHONON EFFECTS IN FERROMAGNETIC MANGANITES PHYSICAL REVIEW B 65 014409

Ϫ ⌸͑ ͒ϭ␲ ͓͉ Mn͉2͑ Ϫ ͒␦͑⑀ ϩប␻ Im Q ͚ M k,Q N⑀ N⑀ ϩ k Q k k k Q

O 2 Ϫ⑀ ϩ ͒ϩ͉M ͉ ͑N⑀ ϪN⑀ ͒ k Q k,Q k kϩQ/2 ϫ␦ ⑀ ϩប␻ Ϫ⑀ ͒ ͑ ͒ ͑ k Q kϩQÕ2 ͔. 28 For the case of T→0, the occupation numbers are negligible. Thus, the phonon damping is zero. Therefore, even though the magnon-phonon interaction can be signiﬁcant, no broadening in the phonons is shown. This is consistent with the experiments where no signiﬁcant change in the phonon linewidth was measured.6,7

IV. RENORMALIZATION OF THE MAGNON SPECTRUM In the previous section we evaluated the damping of the magnons to the ﬁrst approximation in the perturbation series. We now study the propagation of the magnetic excitations. At low temperatures the dynamics of the spin waves is described by the single-magnon dispersion. The important question which arises is to which extent the magnon spectrum is affected by the magnon-phonon coupling. To answer qualitatively one needs to express the full spectrum in terms of the magnon self-energy

⑀ ϭ⑀ ϩ ⌺ ͒ ͑ ͒ k 0,k Re ͑k , 29 ⑀ where the unperturbed magnon dispersion 0,k was deﬁned in Eq. ͑10͒. Again we look at the lowest-order interaction and one obtains for the real part of the magnon self-energy

1 ⌺͑ ͒ϭ ͉ Mn͉2ͫ͑ Ϫ ͒ͩ Re k ͚ M N␻ N⑀ k,Q Q kϩQ ⑀ Ϫ⑀ ϩប␻ Q k kϩQ Q 1 Ϫ ͪ ϩ͑ ϩ Ϫ ͒ 1 N␻ N⑀ ⑀ ϩ⑀ Ϫប␻ Q kϩQ k kϩQ Q 1 1 FIG. 3. Magnon dispersion as a function of ka/2␲ ͑a͒ along the ϫ Ϫ ͩ ⑀ Ϫ⑀ Ϫប␻ ⑀ ϩ⑀ ϩប␻ ͪͬ z axis and ͑b͒ in the xy plane. The solid line represents the nearest- k kϩQ Q k kϩQ Q neighbor Heisenberg Hamiltonian and the dashed line represents the 1 renormalized dispersion due to the magnon-phonon interaction. ϩ͉ O ͉2ͫ͑ Ϫ ͒ͩ M N␻ N⑀ k,Q Q kϩQ/2 ⑀ Ϫ⑀ ϩប␻ k kϩQ/2 Q ប ͒4 ͑QDa 1 1 1 gϭ ͫ ϩ ͬ. ͑32͒ Ϫ ͪ ϩ͑ ϩ Ϫ ͒ 4 1 N␻ N⑀ ␲ M M ⑀ ϩ⑀ Ϫប␻ Q kϩQ/2 7 sa Mn O k kϩQ/2 Q

1 1 Note that a Debye cutoff QD is introduced in the integration ϫ Ϫ ͑ ͒ ͩ ⑀ Ϫ⑀ Ϫប␻ ⑀ ϩ⑀ ϩប␻ ͪͬ. over the Q variable. For three-dimensional 3D systems k kϩQ/2 Q k kϩQ/2 Q ϭ ␲2 1/3 QD (6 ) /a. If typical constants for the manganites are ͑30͒ used, g is estimated in the range of 0.01. When k is in the xy plane, a similar expression for the Re ⌺(k) is found, al- At low temperatures the terms multiplied by the occupation though the formula for g is more cumbersome, but the nu- factor N␻ or N⑀ are neglected. Consider the case of small Q k merical value is still in the order of 0.01. Thus, in this limit magnon wave vectors and longitudinal acoustic phonons. the magnon-phonon coupling introduces only a small nega- The matrix elements are given in Eqs. ͑24͒ and ͑25͒.To tive correction to the stiffness constant and it is not impor- lowest order of the magnon wave vector when k is along the tant. z axis it is easy to obtain Now we examine the limit of large magnon wave vectors with longitudinally polarized phonons. In this case the inte- Re ⌺͑k͒ϭϪDgk2, ͑31͒ grals are evaluated numerically. In Fig. 3 we plot the magnon

014409-5 L. M. WOODS PHYSICAL REVIEW B 65 014409 dispersion with and without the coupling. It is evident that that with this simple model one is able to describe the ob- for small k the correction to the magnon energy is not sig- served effects of broadening and softening in those mangan- nificant. But at larger k the effect of the interaction is much ites in which the nearest-neighbor Heisenberg model is valid. more pronounced. According to the reported data in Ref. 7 Indeed, the role of the interplay between spin and lattice the magnon mode experiences softening and its dispersion is degrees of freedom needs to be included in the analysis for a very close to the LO phonons after the crossing point of more complete understanding of the physics in these com- about ka/2␲ϭ0.3. Indeed, the estimated values for the cor- pounds. rection to the magnon spectra is always negative and it be- The present model also allows for the Mn and O ions to comes large when k is large. Figure 3 shows that the behav- be treated explicitly. Because the direct overlap between Mn ior of the calculated magnon energy is very similar to the ions is zero, the exchange coupling constant is the second measured one. Thus, the presented model favors the argu- perturbation order of the hopping from Mn to O site. There- ment that such behavior should be attribured to a channel fore, the interaction Hamiltonian contains the positions and related to the coupling between magnons and phonons. masses of both types of sites: Eqs. ͑12͒–͑14͒. Since the O ion is lighter, one concludes that its contribution to the effect is a few times more significant than the contribution from the V. DISCUSSION Mn. This is consistent with experiments reported in Ref. 16 One of the most interesting features of the manganese which show that at temperatures as low as 50 K the displace- perovskites is the existence of colossal magnetoresistance ments of the O ions is about 2 times larger than the displace- ͑CMR͒. Investigating the mechanisms for the magnon damp- ments of the Mn ions. The model also allows for different ing and softening is very important in order to understand types of phonon polarization to be evaluated. their magnetic and electronic properties. It is claimed that to Dimensionality has been shown to be an important con- describe the magnon dispersion properly one needs to in- sideration in the behavior of many materials. The proposed clude orbital fluctuations and phonons via Jahn-Teller model can be generalized to ferromagnets that have reduced distortion.17 The Jahn-Teller-based electron-lattice coupling dimensions. In fact, we calculated the magnon damping in is known to be important at temperatures near and above T . the 2D case using the expression for the magnon damping C ͑ ͒ The experiments, discussed in this paper, were performed at from Eq. 20 with the same values of the exchange constant very low temperatures where the Jahn-Teller effect is very and lattice parameters as used earlier in the paper and we small. They show that the magnons are heavily damped at obtained that the broadening becomes even larger at the end the end of the zone.7 One possibility considered was that a of the zone. This means that the effect of the magnon-phonon strong spin-orbit exchange interaction is the reason.18 An- coupling could be more pronounced in materials that consist other possibility is that this is due to strong magnetoelastic of quasi-2D planes. More experiments for ferromagnetic lay- effects or the interaction between magnons and phonons is ered compounds in this direction are necessary. important. In this paper we focused on the role of the latter In summary, two things are accomplished in this paper. kind of coupling. First, a model for treating the interaction between magnons The interaction between the lattice and the ferromagneti- and phonons in systems with localized spins is established, cally ordered carriers is obtained by allowing a modulation which can be applied to materials with symmetry different of the exchange coupling constant with respect to the lattice than cubic and different types of phonon polarization. The displacements and that the spin quantum number is con- explicit form of the matrix elements is obtained, and thus one served. The performed calculations are for phonons with lon- can look into different limits. Second, the calculations for the gitudinal polarization in the acoustic and optical regimes. It magnon and phonon dampings and magnon softening are in was shown that the long-wavelength magnons would not be agreement with the experimental findings: namely, that there affected, but the damping in the short-wavelength limit be- is an anomalous broadening of the magnon linewidth and comes large. We also find that the phonon linewidth is not softening of the energy dispersion near the zone end and that changed. This is consistent with the experiments. the phonon linewidth is hardly changed. Another feature is the effect on the phonons could be more significant with elevating the temperature. The reason ACKNOWLEDGMENTS is that the damping is proportional to the boson occupation number, which becomes more important at higher tempera- The author would like to thank Professor G. Mahan and tures. This would require more experiments. Finally, we also Dr. J. Cooke for discussions. Research support is acknowl- calculated how the magnon dispersion changes if one takes edged from the University of Tennessee and from the U.S. into account the magnon-phonon coupling. It was found that Department of Energy under Contract No. DE-AC05- the effect is really small at small wave vectors and it is 00OR22725 with the Oak Ridge National Laboratory, man- significant near the zone boundary. Thus, the conclusion is aged by UT-Battelle, LLC.

*Present address: Naval Research Laboratory, Washington, D.C. 2 H.Y. Hwang, S.-W. Cheong, P.G. Radaeli, M. Marezio, and B. 20375-5347. Batlogg, Phys. Rev. Lett. 75, 914 ͑1995͒. 1 J. Callaway, Quantum Theory of the Solid State ͑Academic Press, 3 C. Zenner, Phys. Rev. 82, 403 ͑1951͒. San Diego, 1991͒. 4 T.G. Perring, G. Aeppli, S.M. Hayedn, S.A. Carter, J.P. Remeika,

014409-6 MAGNON-PHONON EFFECTS IN FERROMAGNETIC MANGANITES PHYSICAL REVIEW B 65 014409

and S.-W. Cheong, Phys. Rev. Lett. 77,711͑1996͒; Y. Endoh Physics of the Chemical Bond ͑Freeman, San Francisco, 1980͒. and K. Hirota, J. Phys. Soc. Jpn. 66, 2264 ͑1997͒. 12 S. Saitoh, A.E. Bocquet, T. Mizokawa, H. Namatame, A. Fuji- 5 N. Furukawa, J. Phys. Soc. Jpn. 65, 1174 ͑1996͒. mori, M. Abbate, Y. Takeda, and M. Takano, Phys. Rev. B 51,13 6 H.Y. Hwang, P. Dai, S.-W. Cheong, G. Aeppli, D.A. Tennant, and 942 ͑1995͒. H.A. Mook, Phys. Rev. Lett. 80, 1316 ͑1998͒. 13 J.H. Jung, K.H. Kim, T.W. Noh, E.J. Choi, and J.J. Jui, Phys. Rev. 7 P. Dai, H.Y. Hwang, J. Zhang, J.A. Fernandez-Baca, S.-W. B 57, R11 043 ͑1998͒. Cheong, C. Kloc, Y. Tomioka, and Y. Tokura, Phys. Rev. B 61, 14 S. Lovesey, Theory of Neutron Scattering from Condensed Matter ͑ ͒ 9553 2000 . ͑Clarendon Press, Oxford, 1984͒. 8 ͑ ͒ N. Furukawa, J. Phys. Soc. Jpn. 68, 2522 1999 . 15 J. Jensen and A. R. Mackintosh, Rare Earth Magnetism ͑Claren- 9 ͑ ͒ I.V. Solovyev and K. Terakura, Phys. Rev. Lett. 82, 2959 1999 ; don Press, Oxford, 1991͒. P. Mahadevan, I.V. Solovyev, and K. Terakura, Phys. Rev. B 60, 16 P. Dai, J. Zhang, H.A. Mook, S.-H. Liou, P.A. Dowben, and E.W. 11 439 ͑1999͒. Plummer, Phys. Rev. B 54, 3694 ͑1996͒. 10 P.W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 17 G. Khaliullin and R. Kilian, Phys. Rev. B 61, 3494 ͑2000͒. ͑1955͒. 18 S. Ishihara, M. Yamanaka, and N. Nagaosa, Phys. Rev. B 56, 686 11 J.C. Slater and G.F. Koster, Phys. Rev. 94, 1498 ͑1954͒;W.A. ͑1997͒. Harrison, Electronic Structure and the Properties of Solids, The

014409-7