PHYSICAL REVIE%' 8 VOLUME 29, NUMBER 7 1 APRIL 1984 Low-energy excitation spectrum for ferromagnetic spin waves on a percolating network: An effective-medium approximation Kin Wah Yu Department of Physics, University of California, Los Angeles, California 90024 (Received 5 December 1983; revised manuscript received 6 March 1984) We develop a theory for the spin waves in a dilute Heisenberg ferromagnet. The d-dimensional bond percolating network of ferromagnetic spins is solved within the effective-medium approxima- tion. It is shown that these excitations obey a classical equation of motion. We have found that for 2&d &4, there is a critical frequency ~„consistent with the mobility edge, such that co, scales as (p — p, ) . When co &co„ the excitations are extended, while when co & co„ they are localized. We also found that there is a rapid increase in the density of states at the mobility edge. The jurnp near the mobility edge is found to scale as (p — p, )' I. INTRODUCTION It is well known that the master equation is applicable to many physical problems, of which the vibrational prob- lem and the diffusion problem are the most interesting. ' The low-energy spin-wave excitations in a Heisenberg model with random coupling constants J-, -, , &0 can also map onto the master equation, ' which we shall show in the next section. Recently, the vibrational properties of a bond percolating network were analyzed by us ' using the effective-medium approximation (EMA) developed by Odagaki and Lax and by Webman. It was found that for 2 & d & 4, the vibrational spectrum exhibits a crossover between the low-energy extended phonon region and the higher-energy localized fracton region. The transition occurs in a narrow region of frequencies marked by the critical frequency to, which scales as p — p„ in the form of a rapid increase in the density of states of the vibrations. Motivated by the above results, one would like to see if such properties would also occur in the case of spin waves of a dilute Heisenberg ferromagnet. There have been many discussions about the localiza- tion of spin waves in dilute Heisenberg magnets in which magnetic ions of concentration p are randomly diluted by nonmagnetic ions of concentration I — p. Only nearest- neighbor exchange interactions between the magnetic spins are considered, while all effects due to the nonmag- netic ions are ignored. This problem is an example of the percolation theory. For concentration p (p„ the percola- tion probability, the magnetic spins are in finite clusters and all excitations are localized. For p p p„ there is an in- finite cluster of magnetic spins, the E =0 Goldstone mode is extended. There exists a mobility edge E, (p) above which all excitations are localized. Near the percolation threshold p =p„however, the infinite cluster is so weakly connected that, by breaking a few bonds, we can separate it into finite clusters. This is the reason why E, (p)~0 as P ~Pe The amplitudes of spin-wave excitations at sites of the infinite cluster obey a classical equation of motion with diagonal and off-diagonal disorder. We shall derive this equation in Sec. II. We want to address two important re- suits here: With only diagonal disorder as in the Anderson model, one finds an exponential localization, E, (p) — exp[ — 2 /(p — p, ) ' i ], while with off-diagonal disorder, one finds a power-law localization, E, (p) -(p — p, )' . This is consistent with the arguments by Economou and Antoniou9 that randomness in site energies (diagonal dis- order) is an opposing factor for extended states while a large transfer rate (off-diagonal disorder) is a favorable factor for extended states. In the case of the Anderson model, one increases the site-energy randomness while keeping the transfer rates constant, thus in favor of locali- zation. However, for off-diagonal disorder, increasing the favorable factor automatically makes larger values of transfer rates more probable; thus complete localization is not achieved. The paper is organized as follows: We derive the equa- tion of motion for the ferromagnetic spin waves in Sec. II. We shall show that this equation maps on the master equation. We then discuss the basic EMA equations developed by Odagaki and Lax" and by Webman. Then we develop some analytic solutions to the above equations. We then calculate the density of states for the excitations, and the dispersion relation is discussed. In Sec. III, we present some numerical solutions to the problem. This includes the use of an approximate Green's function and we solve the self-consistent equation in three dimensions (3D). We will present the results as X(co) vs to and obtain the dispersion relationship as co(q) vs q and co vs Re(q) and Im(q) for differing values of (p — p, )/p, . Fi- nally, in Sec. IV we give a brief discussion of the results and briefly introduce the case of antiferromagnetic spin waves. II. THEORY We consider a lattice of ferromagnetic spins, described by the Hamiltonian in the absence of external magnetic field: 29 4065 Q~ 1984 The American Physical Society