VOLUME 82, NUMBER 14 PHYSICAL REVIEW LETTERS 5APRIL 1999 Zone Boundary Softening of the Spin-Wave Dispersion in Doped Ferromagnetic Manganites I. V. Solovyev 1,2 and K. Terakura 3,4 1 JRCAT-Angstrom Technology Partnership, 1-1-4 Higashi, Tsukuba, Ibaraki 305-0046, Japan 2 Institute of Metal Physics, Russian Academy of Sciences, Ekaterinburg GSP-170, Russia 3 JRCAT-NAIR, 1-1-4 Higashi, Tsukuba, Ibaraki 305-8562, Japan 4 Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8558, Japan (Received 14 December 1998) We argue that the new distinct features observed in doped ferromagnetic manganites R 12x D x MnO 3 (RD  LaSr, PrSr, NdSr, and LaCa), the softening of the spin-wave dispersion at the zone boundary and the increase of the spin-wave stiffness constant with x, have purely magnetic spin origin. They indicate the breakdown of the canonical double-exchange limit, and reflect otherwise natural consequence of the e g -band filling in the half-metallic regime. Details of the realistic electronic structure are important and significantly modify the analysis based on the minimal tight-binding Hamiltonian for e g electrons. [S0031-9007(99)08820-1] PACS numbers: 75.30.Ds, 75.10.Lp, 75.30.Et, 75.50.Cc The understanding of many fascinating properties of colossal magnetoresistive perovskite manganites R 12x D x MnO 3 (R  trivalent rare-earth ion, D  divalent ion) circles around two questions [1]: (i) whether the double-exchange (DE) mechanism alone is enough and (if apparently not) (ii) whether a strong Jahn-Teller based electron-lattice coupling is the trigger for the unusual behavior. Recent studies of the spin-wave dispersion (SWD) of doped ferromagnetic (FM) manganites provide a new piece of information, which is typically argued based on these two standpoints. A brief sketch of the cur- rent situation is given below. Perring et al. [2] measured the SWD of the high Curie temperature (T C  355 K) system La 0.7 Pb 0.3 MnO 3 throughout the Brillouin zone and claimed that a simple Heisenberg Hamiltonian Ee i   2 1 2 X ik J k e i ? e i 1k (1) (here, e i is the direction of magnetic moment at the site i ) with solely the nearest-neighbor (nn) coupling J 1 accounts for the entire dispersion relation vq and also for T C to within 15%. Furukawa [3] argued that the result seems to be consistent with the DE limit W I , 1 (W being the e g bandwidth, I being Hund’s coupling) of the FM Kondo lat- tice model (KLM). Similar classification has been ascribed to higher-T C manganite La 12x Sr x MnO 3 (T C  378 K at x  0.3) in the region 0.175 # x # 0.3, the spin dynam- ics of which was reported to be regular for FM mate- rials [4]. The main controversies came later, when an unexpected softening of the SWD at the zone boundary was observed in several manganites with lower but yet very different T C : Pr 0.63 Sr 0.37 MnO 3 (T C  301 K) [5], La 0.7 Ca 0.3 MnO 3 (T C  242 K) [6], and Nd 0.7 Sr 0.3 MnO 3 (T C  198 K) [7]. Hwang et al. [5] showed that the experimental vq can be reproduced reasonably well by the Heisenberg Hamiltonian with the parameters (for Pr 0.63 Sr 0.37 MnO 3 ) J 1  10.1, J 2  20.6, J 3  0.6, and J 4  2.7 meV, thus clearly demonstrating the importance of the long-range FM coupling J 4 . They also argued that the next important contribution which improves the fit is J 8 [8]. Even more pronounced softening was ob- served in the 50% doped manganites Pr 0.5 Sr 0.5 MnO 3 and Nd 0.5 Sr 0.5 MnO 3 , both in the FM state and within FM lay- ers in the A-type antiferromagnetic (AFM) state [9]. These observations gave rise to several scenarios of the low- temperature behavior of doped manganites, among which are the spin-lattice coupling, the orbital ordering, and the precursor of the AFM spin ordering. The purpose of this work is to show that the observed softening of the SWD at the zone boundary is a natural consequence of the e g band filling in the half-metallic (HM) regime, implying that the canonical DE limit is not appropriate and that neither the lattice deformation nor the orbital ordering are required for the SWD softening. We employ both the tight-binding (TB) analysis and direct adiabatic frozen-spin-spiral calcu- lations of vq [10] performed for the cubic virtual-crystal alloy La 12x Ba x MnO 3 within local-spin-density approxi- mation (LSDA). Our starting point is the fact that the total energy change due to small nonuniform rotations of magnetic moments near an equilibrium, i.e., the processes relevant to the spin-wave excitations, can be exactly mapped onto the Heisenberg model (1) [11,12]. If the magnetic part of an effective one-electron potential is described solely by the on-site exchange splitting b D ex between the majority-spin (") and the minority-spin (#) states [13], the parameters J k can be found as J k  1 2p Im Z ´ F 2` d´ Tr L  b D 0 ex b G " 0k ´ b D k ex b G # k0 ´ , (2) where b G ",# is the one-electron Green function in the real space, and Tr L denotes the trace over the orbital indices L. First, we discuss several basic features of magnetic interactions (2) for HM ferromagnets. Consider the TB Hamiltonian c H s  2 b t 1D ex b 1d s# , where b t  kt LL 0 ij k 0031-9007 99  82(14)  2959(4)$15.00 © 1999 The American Physical Society 2959