REVIEW ON MAGNETIC MOMENT CONFIGURATIONS. A. Purwanto Materials Science Research Center, National Atomic Energy Agency, Tangerang 15314, Indonesia

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1 REVIEW ON MAGNETIC MOMENT CONFIGURATIONS A. Purwanto Materials Science Research Center, National Atomic Energy Agency, Tangerang 15314, Indonesia ABSTRACT REVIEW ON MAGNETIC MOMENT CONFIGURATIONs. Magnetic moment configurations in some simplest cases are reviewed to provide a qualitative understanding of magnetic structure in general. The discussion starts from the free energy consideration as a physical basis and proceeds to some 1-D moment configurations. The single-q vs multi-q case is described in a 2-D cube system. The extension to their 3-D counterparts are straightforward. The conclusion is that it is possible to make periodic arrangements for the moment configurations incommensurable with the underlying crytstallographic structure and having either the single- or multi-q magnetic structure. INTRODUCTION Magnetic materials have been very important in advanced technology for a long time. Many interesting applications ranging from bulky materials such as transformer and tiny aparatus such as electronic intruments are based on the magnetic phenomena. One of the most important properties are based on magnetic anisotropy, i.e.; a tendency of magnetic moments to point to a certain direction. The magnetic anisotropy minimizes the free energy of the magnetic system such that it has a certain magnetic moment configuration. The moment configuration in general ranges from a simple collinear and commensurate order to noncollinear and incommensurate order. It is also possible to have a multi- rather than a single-q sample, where q is the magnetic wave propagation vector. However, the full-blown magnetic orders in 3-D often lost people in configuring the possible magnetic structures, especially the multi- and single-q structures. To elucidate possible moment configurations in general, this paper reviews magnetic orders in crystalline material excluding frustrated spin system which is available elsewhere [1]. The review is made to be general irrespective of the data which are readily obtained by either the neutron diffraction [1, 2] or X-ray synchrotron radiation experiments [3]. The former is more widely known than the latter to study magnetic structure and the reactor-based neutron facilities for condensed matter research are available in Indonesia. The latter is a relatively new technique and is based on the interaction between the polarization of high intensity photon arising from resonant X-ray scattering with magnetic moments of atoms [4]. Free energy of simple magnetic systems is considered to provide a physical basis for various possible moment configurations. Unlike most books and papers on the subject, the corresponding moment configurations are then described with 41

2 illustrations in the simplest forms as possible to provide a principal understanding. The extension to the real 3-D case, including helical and umbrella type, is straightforward [1-10]. FREE ENERGY CONSIDERATION The simplest description of moment configuratios for atoms d and d' at positions d and d', respectively, are based on the (localized picture) Heisenberg interaction Hamiltonian: H H dd ' = J ( d d' ) µ d µ (1) d ' where J(d d ) = J (d - d) is the exchange energy or exchange integral between atoms d and d'. For a positive J(d d ), the energy of the magnetic system can only be minimized by arranging the magnetic moments µ d parallel to µ d' and the opposite is true for a negative J(d d ). In other words, ferromagnetic and antiferromagnetic arrangements correspond to positive and negative exchange energies, respectively. These parallel and antiparallel moment configurations are referred to as collinear arrangements. A ferromagnetic system has a net moment while an antiferromagnetic system does not. A more complicated moment configuration occurs if the moments are not parallel but canted with respect to each other. This canted configuration is referred to as noncollinear configuration and, in some cases, can be explained by the Dzialoshinski-Moriya interaction [11-13]: ( DM ) dd ' H = K(d d' ) µ ( µ d d ' where K(d-d ) is the (vectorial) Dzialoshinski-Moriya exchange energy. This type of configuration appears in the tetragonal compounds [5]. Note that, as in collinear order, it is possible to have non-zero (i.e., ferromagnetic) and zero (antiferromagnetic) net moments with noncollinear order. Two further classifications of the magnetic order are based on the value and multiplicity of the (magnetic) wave propagation vector q, which is defined as a vector with the magnetitude of 2π/λ where λ is the wave length of the magnetic moment modulation and with the direction of the propagation. If the value of all three Cartesian components of q in terms of the reciprocal lattice units can be expressed as rational numbers, the magnetic unit cell is commensurate with the underlying crystallographic unit cell. On the other hand, incommensurate order is signified by the apearance of the incommensurate value of at least one component of q. Based on the multiplicity of the q, one can have either single-q or multi-q order. In the next sections, some examples of types of magnetic order are further discussed in both the direct and the reciprocal space for both the crystallographic and magnetic cell. The sections are arranged in a sequence ) (2) 42

3 according to the classification based on the value of the q component; i.e., commensurate and incommensurate order, independent of the classification based on the multiplicity of q. Finally, some remarks on single-q and multiq order are in order. MAGNETIC MOMENT CONFIGURATIONS Commensurate Order The simplest types of commensurate order, which are easy to visualize, are provided by one-dimensional (a) ferromagnetic collinear, (b) antiferromagnetic collinear and (c) ferromagnetic non-collinear configurations in both the direct and the reciprocal lattice as shown in Fig. 1. The magnetic periodicity of the collinear ferromagnet shown in Fig. 1(a) is the same as the crystallographic periodicity. The magnetic contributions then simply add to the nuclear contributions in the reciprocal lattice. In the case of a simple 1-D antiferromagnet as shown in Fig. 1(b), the magnetic and crystallographic periodicities are different in both the direct and reciprocal lattice. In the direct lattice, the magnetic unit cell is twice the size of the nuclear one, often refered as cell doubling. One then observes the magnetic reflections in between the nuclear reflections: the magnetic contributions only appear at half-integer-indexed reflections in the reciprocal space. Note that half-integer indices can be expressed as the ratio between two integers which is then refered to as commensurate order. There are some combinations between the two cases mentioned above; i.e., the appearence of extra half-integer-indexed reflections and the magnetic contributions to the nuclear reflections. The simplest case is the noncollinear ferromagnet illustrated in Fig. 1(c). In the direct lattice, the magnetic periodicity is twice that of the crystallographic periodicity. Consequently, half-integer-indexed magnetic contributions appear in the reciprocal space. However, additional magnetic contributions to the nuclear reflections, also appear indicating non-zero net moments in the planes corresponding to the nuclear reflections. Incommensurate Order To understand incommensurate order, it is instructive to step back to the antiferromagnetic commensurate order and view it as a sine-wave modulation. This adopts the concept of the (magnetic) wave propagation vector q, which is a very useful concept in the analysis of the incommensurate order. Fig. 2 shows sine-wave modulated commensurate order in (a) direct lattice and (b) reciprocal lattice. Basically, Fig. 2 is analogous to Fig. 1(b), only now the sine-wave curve is plotted in (a) and the Brillouin zone boundaries are drawn as vertical lines in (b). The indexing in 43

4 the reciprocal lattice is indicated by τ and ± q. Note that τ is a reciprocallattice vector. Clearly, the moment propagates sinusoidally and commensurate with the nuclear cell. The propagation vector is perpendicular to the moment polarization. The wave propagation vector q is at the first Brillouin zone as q=1/2 for this 1-D case. Note, q=1/2 is equivalent to q=-1/2. For incommensurate order, q is always inside the first Brillouin zone. If it is outside, one can always brings it inside by the translation of reciprocal lattice vectors. Fig. 3 shows sine-wave modulated incommensurate order in (a) direct and (b) reciprocal lattice. The key difference between Fig. 3 and Fig. 2 is that the q of the latter is at the surface of the first Brillouin zone. Note, q for commensurate order is not necessarily at the surface, as one can have commensurate order with q=1/3, q=1/4, q=1/5, etc., which are definitely not at the zone boundary. The incommensurability may be due to the longrange and oscillatory variation of some exchange interaction which is incommensurate with the underlying crystallographic lattice. It is still long range order of the magnetic moments: the magnetization is periodic. One can then (Fourier) expand the moment distribution µ ld on position d of unit cell l such that where q is the wave propagation vector limited to the first Brillouin zone and Σ q denotes a summation of all magnetic contributions belonging to a set of wave propagation vectors belonging to the star of q (see below). Another way of looking at Eq. (3) is that there is a phase difference exp(-iq l) between atom d in the reference unit cell and that in the unit cell l. Single- vs. Multi-q Order µ ld = µ qd exp(-iq l) (3) q The discussion in the previous two sections is valid for both singleand multi-q structures. However, it is possible to further clasify the magnetic structure depending the multiplicity of q. The nature of this classification arises from the fact that the magnetic symmetry should be the same as or at least related to the crystallographic symmetry. Specifically, for multi-q order, the parent crystallographic symmetry would be preserved if the magnetic unit cell by itself contained multiple magnetic periodicities corresponding to the star of q, which is defined as a set of mutually inequivalent q's which are related by the symmetry operations of the crystallographic point group. Two q's are equivalent if they are related by zero or a multiple of one reciprocal lattice. The wave propagation vector q is equivalent to (-q) for q = τ/2. On the other hand, for single-q order, the corresponding magnetic unit cell has a lower symmetry than the crystallographic unit cell. The higher symmetry nature in the crystallographic cell can be matched in the magnetic cell by 44

5 introducing multiple domains. Note that domain can actually be defined, in a way analogous to the star of q, as regions of the same magnetic anisotropy mutually related by symmetry operations of crystallographic point group. To provide a clearer insight, one can consider a 2-D square system with single-q and double-q magnetic structures as shown in Fig. 4. Assuming that the magnetic moments are parallel to a <10> direction for the single-q case, one can easily see that there must be two equivalent components µ x [10] and µ x [01]. This can be thought as two separate but equivalent domains, A and B, mutually related through four-fold symmetry; i.e., a double-domain structure. In the reciprocal space, domain A and B would contribute to the (1/2 0) and (0 1/2) reflections, respectively. If one could prepare a single domain sample, only one of the two reflections would be observed. On the other hand, the corresponding magnetic unit cell in double-q structure has the magnetic periodicities corresponding to both q A =(1/2 0) and q B =(0 1/2) simultaneously, as shown in Fig. 4(b). In the reciprocal space, the double-q structure would give magnetic intensity in both the (1/2 0) and (0 1/2) reflections. CONCLUSION Possible magnetic moment configurations have been reviewed and emphasized for some simple cases, ranging from the commensurate to incommensurate including the discussion on the single- and multi-q structure. The extension to the real 3-D cases, including helical and umbrella type, is straightforward [5-10]. In conclusion, it is possible to make periodic arrangements for the moment configurations incommensurable with the underlying crytstallographic structure and having either the single- or multi-q magnetic structure. ACKNOWLEDGEMENT The author is indebted to R. A. Robinson of Los Alamos National Laboratory, USA, for the fruitful and stimulating discussions. 45

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8 Commensurate Order (i) Direct lattice (ii) Reciprocal lattice (a) Ferromagnet a a*= 2 π /a (b) Antiferromagnet an am am * an * am am * (c) Noncollinear- Ferromagnet an * an Fig. 1: Some types of 1-D commensurate order showing (a) the ferromagnetic collinear, (b) the antiferromagnetic collinear and (c) the ferromagnetic noncollinear configurations in (i) the direct lattice and (ii) the reciprocal lattice. In the direct lattice, the circles and arrows represent atoms and magnetic moments, respectively. In the reciprocal lattice, dots and crosses represent the observed intensities (Bragg peaks) of the nuclear and magnetic structures, respectively. an and am denote the nuclear and magnetic a lattice parameters, respectively. 48

9 Commensurate Order as a Modulation am an (a) Direct lattice am * q an * (b) Reciprocal lattice -q τ Fig. 2: A 1-D sine-wave modulated commensurate order in (a) direct lattice and (b) reciprocal lattice. In the direct lattice, the circles and arrows represent atoms and magnetic moments, respectively. In the reciprocal lattice, dots and crosses represent the observed intensities (Bragg peaks) of the nuclear and magnetic structures, respectively. Incommensurate Order λ an (a) Direct lattice q an * -q (b) Reciprocal lattice τ Fig. 3: A 1-D sine-wave modulated incommensurate order with q = 2π / λ in (a) direct lattice and (b) reciprocal lattice, the circles and arrows represent atoms and magnetic moments, respectively. In the reciprocal lattice, dots and crosses represent the observed intensity (interference maxima) of the atom and the magnetic moment, respectively. 49

10 y y domain A domain B x m y x m x (a) single-q y m y x m x (b) double-q Fig. 4: A simple 2-D square system showing (a) single-q structure and (b) double-q structure for q=(1/2 0). In (a), only two domains, A and B, are shown. They correspond to q A =(1/2 0) and q B =(0 1/2), respectively, as indicated by the moment modulation nearby. Clearly, unlike single-q, double-q has q A and q B simultaneously. The moment magnitudes in the moment modulation are not drawn to scale. 50