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Sublattice Magnetization of Yttrium Iron Garnet at Low Temperatures

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Sublattice Magnetization of Yttrium Iron Garnet at Low Temperatures University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 3-10-1967 Sublattice Magnetization of Yttrium Iron Garnet at Low Temperatures A. Brooks Harris University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Quantum Physics Commons Recommended Citation Harris, A. (1967). Sublattice Magnetization of Yttrium Iron Garnet at Low Temperatures. Physical Review, 155 (2), 499-510. http://dx.doi.org/10.1103/PhysRev.155.499 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/423 For more information, please contact [email protected]. Sublattice Magnetization of Yttrium Iron Garnet at Low Temperatures Abstract The energies and eigenvectors for long-wavelength acoustic spin waves in yttrium iron garnet are calculated neglecting spin-wave interactions, but including the effects of anisotropy and dipolar z 3/2 5/2 7/2 interactions. The sublattice magnetization is found to be Sα (T)=S−δα−cT−AαT −BαT −CαT ⋯. Here α labels the sublattice a or d, δα expresses the effect of zero-point motion, and cT is the Holstein- Primakoff correction for dipolar interactions. Expressions in terms of the exchange integrals Jaa, Jdd, Jad, and Jad′, where Jad′ describes interactions between next-nearest-neighbor a and d sites, are given for Aα, Bα, and, when Jad′=0, for Cα. The spin-wave spectrum of some substituted garnets and the effect of spin- wave interactions on the zero-point disordering are treated in appendices. Disciplines Physics | Quantum Physics This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/423 PHYSICAL REVIEW VOLUME 1SS, NUMBER 2 10 MARCH 1967 Sublattice Magnetization of Yttrium Iron Garnet at Low Temyeratures* A. BRooKs HARRIs DePartment of Physics, University of Pennsylvania, Philad'e/Phia, Pennsylvania (Received 24 October 1966) The energies and eigenvectors for long-wavelength acoustic spin waves in yttrium iron garnet are calcu- lated neglecting spin-wave interactions, but including the effects of anisotropy and dipolar interactions. The sublattice magnetization is found to be S '(T) =S—8 —cT—A, T'~ —B T5~' —C TI~2 ~ ~ . Here o. labels the sublattice a or d, b„expresses the effect of zero-point motion, and cT is the Holstein-Primakoff correction for dipolar interactions. Expressions in terms of the exchange integrals J „Jzz, J,~, and J,z', where J~z' describes interactions between next-nearest-neighbor u and d sites, are given for A, 8, and, when J,q' —0, for C~. The spin-wave spectrum of some substituted garnets and the effect of spin-wave interactions on the zero-point disordering are treated in appendices. I. INTRODUCTION diets the sublattice magnetization of a ferrimagnet to be of the form HE use of nuclear magnetic resonance' ' as a tool for the investigation of magnetic materials has M(T)/M(0) =1 nT31' —PT'~' —yT'" —. (1.1) enabled one to make very accurate measurements of The purpose of the present is to calculate the the temperature variation of the spontaneous magnet- paper coef5cients and for each sublattice in terms of ization of each magnetically inequivalent sublattice. n, p, y the exchange coefFicients. Previously' we had obtained The possibility of such measurements in turn stimu- expressions for n and in the analogous expansion for lates detailed theoretical calculations of the sublattice P the total magnetization. In order to obtain the sub- magnetizations, especially for low temperatures where lattice magnetization we study in more detail the the spin-wave approximation is valid. Recently such transformation to normal modes. In addition we will accurate measurements have been performed on calculate corrections to Kq. (1.1) when the effects of yttrium iron garnet (YIG)", so that by an accurate anisotropy, externally applied magnetic field, and di- spin-wave analysis one may hope to deduce reliable polar interactions are included. We show that the values of the exchange coefficients. However, for YIG simplest phenomenological extension of the Holstein- an earlier spin-wave analysis gave results in disagree- Primakoff'4 results to the case of an anisotropic ferri- ment with both Wojtowicz's analysis' of the high- magnet is valid. We give these results since it has been temperature susceptibility data, ' and with evidence" " shown' "that dioplar and anisotropy effects cannot be obtained from substituted garnets. This discrepancy neglected when analyzing the Iow-temperature sub- has been largely removed by Gonano e] a/. "who have lattice magnetizations of YIG. analyzed their magnetization data using the calcu- In Appendix A we show that the ground-state value lation we present here. of the total magnetization of a ferrimagnet one obtains It is well known'3 that, neglecting anisotropy and treating spin-wave interactions perturbatively is the dipolar interactions, linearized spin-wave theory pre- same as the value in the Neel state. Appendix 8 is * Supported in part by the Advanced Research Projects devoted to algebraic details involved in deriving ex- Agency. This paper is a contribution of the Laboratory for pressions for the coeKcients of Kq. (1.1). Finally, in Research on the Structure of Matter, University of Pennsylvania. ' A. C. Gossard and A. M. Portis, Phys. Rev. Letters 3, 164 Appendix C we give formulas for the spin-wave spectra (1959). of substituted garnets in which only the a or d sub- ' A. Narath, Phys. Rev. Letters 7, 410 (1961). lattice is occupied magnetic ions and antiferro- 'A. C. Gossard, V. Jaccarino, and J. P. Remeika, J. Appl. by Phys. 33, 1187 (1962). magnetism results. A. J. Heeger and T. W. Houston, Phys. Rev. 135, A661 One may question the desirability of calculating (1964). 5 analytically the terms in the expansion of R. Gonano, E. Hunt, H. Meyer, and A. B. Harris, J. Appl. Kq. (1.1) Phys. 37, 1322 (1966). since the energies of all the normal modes (optical as ~ J. D.' Litster and G. B. Benedek, J. Appl. Phys. 37, 1320 well as acoustic) and hence the sublattice magneti- (1966). 7 A. B. Harris, Phys. Rev. 132, 2398 (1963). zations could be calculated numerically on an elec- P. J. Wojtowicz, Phys. Letters 11, 18 (1964). tronic computer. Such a calculation is not trivial, ' R. Albonard, J. Phys. Chem. Solids 15, 167 (1960). however. In the first place the exchange integrals are "P.L. Richards and J. P. Remeika, J. Appl. Phys. 37, 1310 (1966). not known precisely, so that the computer program "S. Geller, H. J.Williams, G. P. Espinosa, and R. C. Sherwood, would have to provide for evaluation of the sublattice Bell System Tech. J. 43, 565 (1964). "R.Gonano, E. Hunt, and H. Meyer (to be published). magnetization over a range of parameters with a cri- "See, for instance, L. R. Walker in Magnetism, edited by G. T. terion for selecting the best 6t to the experimental data. Rado and H. Suhl (Academic Press Inc., New York, 1963), Vol. I, Chap. 8. "T.Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940). 499 500 A. B. HARRIS 155 Also, for each set of parameters the eigenvalues of a the Hamiltonian (2.2) is considered. Therefore the 20X20 matrix must be evaluated for a mesh of points magnetization is expected to be the same throughout in reciprocal space. Such a computation would be the entire a (or d) sublattice. When dipolar interactions time-consuming and would involve sophisticated pro- are considered, this argument breaks down because the gramming. In contrast, our results were obtained with Hamiltonian is not invariant under rotation of the only a few weeks labor and in fact appear to be sufficient, direction of magnetization. ""However, this eGect is at least at low temperatures, for the analysis of the probably too small to be observed. experimental data, as is explained by Gonano et al." The simplest spin-wave theory which neglects spin- wave interactions can be obtained by substituting boson II. CALCULATION OF THE SPIN-WAVE operators for the spin operators at each site as follows: SPECTRUM OF YTTRIUM IRON GARNET S'= W [S—ata], (2.3a) The notation we use is as follows. R is a vector of the S+=(25)'/2at (2.3b) bcc lattice gm —(2$)1/2a (2.3c) R= R(22,,n2, 222) =-'2a(n&i+222j+222k), (2.1) where we take the upper choice of sign for the a sites where u is the lattice constant, a, y, and 0 are unit and the lower choice of sign for the d sites. We trans- vectors in the x, y, and s directions, respectively, and form to momentum variables n~, e2, and ms are integers either all even or al1. odd. /2 Let SR+,(„,) denote the spin operator associated with a„,t(k) = V„ P exp(ik [R+e(na)])a„,t(R), (2.4a) the lattice site at R+e(22a), where e(22a) is the position of the 22th a site within (22=1, 8) the unit cell. Let a„dt(k) =E„."'P exp(ik [R+e(/2d)])a„dt(R), (2.4b) SR+,&„d& denote the spin operator associated with the lattice site at R+e(22d), where e(22d) is the position of the 22th d site (22=1, 12) within the unit cell. The where E„is the number of bcc unit cells, so that labeling of the u and d sites in the unit cell is the same as in a previous treatment. 7 1V„,=p 1; V=-2, X„,a2. (2 5) Initially we treat the case of isotropic exchange inter- k actions governed by a Heisenberg Hamiltonian. The Using the substitutions (2. and in the Hamilton- application of linearized spin-wave theory to such a 3) (2.4) ian of (2.
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