Nanomagnetic Models

The atomic-scale and mesoscopic physics of magnetic nanostructures is reviewed. Emphasis is on the description of magnetic phenomena and properties by analytical models, as contrasted to numerical approaches. Nanostructuring affects the magnetic properties on different length scales, from a few interatomic distances for intrinsic properties such as magnetization and anisotropy to more than 10 nm for extrinsic properties, such as coercivity. The consideration includes static and dynamic mechanisms, as well as nanoscale fi nite-temperature effects. Some explicitly discussed examples are Curie-temperature changes due to nanostructuring, the effect of narrow and constricted walls, the potential use of magnetic nanodots for fi nite-temperature quantum computing, and exchange-coupled hard-soft nanocomposites. The temperature dependence of extrinsic properties refl ects the atomic-scale static or ‘intrinsic’ temperature dependence of the free-energy barriers and thermally activated dynamic or ‘extrinsic’ jumps over metastable free-energy barriers. 42 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 43 na considered in this book realized on a length scale of a few nanometers, as contrasted to, for example, a few meters? An answer is provided by the relativistic nature of many magnetic phenomena of importance in nanomagnetism [5, 6]. For example, typical domain walls have a thickness of few nanometers. This length refl ects the competition between nonrelativistic exchange and relativistic magnetic anisotropy. A simple but qualitatively correct picture is obtained from the relativistic electron energy mc2 √ 1̄̄ +̄̄ ̄ v̄̄ 2̄/ c̄̄2, where v is the electron velocity. Expanding the energy into powers of v/c yields the rest energy mc2, the electrostatic or ‘nonrelativistic’ energy mv2/2, and the lowest-order relativistic correction (α/2)2mv2/2, where α = 4πεoe 2/ħc is Sommerfeld’s fi ne-structure constant. Here we have exploited that typical electron velocities in solids are of order v = αc. Respective examples of nonrelativistic and relativistic magnetic interactions are exchange, which has the character of an integral over electrostatic interactions, and spin-orbit coupling, which leads to magnetocrystalline anisotropy [7, 8]. On an atomic scale, relativistic interactions are unable to compete against atomic-scale exchange effects. For example, Heisenberg exchange may exceed 1000 K, whereas typical anisotropies are less than 1 K. However, electrostatic and relativistic contribution become comparable on length scales of order ao/α = 7.25 nm [5, 6, 9]. In addition to the range of interactions, there is the question of interference with structural length scales. For example, Bloch wave functions, which form the basis for the band-structure theory of itinerant magnetism, require infi nite crystals with perfect periodicity. How does nanostructuring interfere with this requirement? Similarly, from a thermodynamic point of view, ferromagnetism is limited to infi nite crystals. It fact, the spontaneous magnetization of any fi nite magnet is zero, because thermal fl uctuations cause the magnetization to average. This leads to the next consideration, the dependence of equilibration or averaging times on structural length scales. There is a fundamental distinction between intrinsic and extrinsic properties. Examples of intrinsic properties are the spontaneous magnetization Ms, the Curie temperature Tc, and the anisotropy Kl. Intrinsic properties describe perfect crystals or surfaces, but their physical origin is atomic and involves quantum phenomena such as exchange, crystal-fi eld interaction, interatomic hopping, and spin-orbit coupling [1, 2, 8, 10, 11]. Intrinsic properties tend to approach their bulk values on fairly small length scales. For example, ‘long-range’ thermodynamic fl uctuations, as involved in the realization of the Curie temperature, and deviations from the Bloch character of metallic wave functions yield only small corrections when the size of the magnetic particle exceeds about 1 nm. The dynamics is characterized by fast equilibration times which means that intrinsic properties can be treated by equilibrium statistical mechanics. This makes it possible to treat intrinsic properties as local parameters. For example, Ms(r) and Kl(r) refl ect the local chemistry, and the unit vector n(r) of the easy magnetization direction corresponds to the local c-axis orientation of the crystallites. Extrinsic or hysteretic magnetic properties, such as the coercivity Hc and the remanence Mr, refl ect the magnet’s real-structure [12–16]. For example, the coercivity of technical iron doubles by adding 0.01 wt.% nitrogen [15]. Such small concentrations have little effect on the intrinsic properties but lead to inhomogeneous lattice strains that affect the propagation of magnetic domain walls and explain the observed coercivity increase. The hysteretic character of extrinsic properties means that equilibration times may be very long. At room temperature, the switching of a single atomic moment is a frequent event, but the thermally activated switching of nanoscale cooperative units, such as domain-wall segments, is very rare. This is the thermodynamic origin of hysteresis, enabling us to build permanent magnets and to store information on magnetic disks. This chapter investigates how intrinsic and extrinsic properties are affected by nanostructuring. Emphasis is on model calculations and analytical approximations, as contrasted to Chapter 2 by Kashyap et al. which focuses on numerical calculations, and Ch. 4 by Schrefl et al., where emphasis is on micromagnetic models and simulations. Section 2 is devoted to static properties, whereas section 3 is concerned with magnetization dynamics. Finally, section 4 presents a number of case studies. 2. MESOSCOPIC MAGNETISM 2.1. Nanoscale Spin Structure 2.1.1. Magnetic moment The magnetic moment nearly exclusively originates from the spin and orbital moments of transition-metal electrons. The magnetic moment of iron-series transition-metal atoms in metals (Fe, Co, Ni, YCo5) and nonmetals (Fe3O4, NiO) is largely given by the spin, and the moment, measured in μB, is equal to the number of unpaired spins. The orbital moment is very small, typically of the order of 0.1 μB, because the orbital motion of the electrons is quenched by the crystal fi eld [16–18]. By contrast, rare-earth moments are given by Hund’s rules, which predict the spin and orbital moment as a function of the number of inner-shell electrons [17]. In some cases, atoms are spin-polarized by neighboring atoms. An example of importance in nanomagnetism is L10 magnets such as FePt, where the Pt carries a magnetic moment. 44 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 45 The spin moment is largely determined by intra-atomic exchange. It is an electrostatic many-body effect, caused by the 1/|r r’| Coulomb interaction between electrons located at r and r’. Physically, ↓↑ electron pairs in an atomic orbital are not forbidden by the Pauli principle but are unfavorable from the point of view of Coulomb repulsion. Parallel spin alignment, ↑↑, means that the two electrons are in different orbitals, which is electrostatically favorable. However, the corresponding gain in Coulomb energy competes against an increase in one-electron energies, because one of the two electrons must occupy an excited state. The magnetic moments of insulating transition-metal oxides and rare-earth metals are located on well-defi ned atomic sites. However, in Fe, Co, and Ni, as well as in many alloys, the moment is delocalized or itinerant. Nonmagnetic metals, or Pauli paramagnets, have two equally populated ↑ and ↓ subbands; and an applied magnetic fi eld transfers a few electrons from the ↓ band to the ↑ band. The corresponding spin polarization is very small, of the order of 0.1 %, because the Zeeman interaction is a small relativistic correction [19]. Itinerant ferromagnetism is realized by narrow bands, where the intraatomic exchange is stronger than the band-width related gain in single-electron hybridization (Stoner criterion). The Bloch character of itinerant wave functions means that the wave functions extend to infi nity. This is not realistic for two reasons. First, magnets encountered in reality, in particular nanomagnets, cannot be considered as infi nite. Second, fi nite-temperature excitations create spin disorder and break the Bloch symmetry of the ↑ and ↓ wave functions. The problem of nonequivalent sites can be tackled, for example, by real-space approaches [16, 20–23]. Restricting the consideration to nearest neighbors yields the correct band width, but details of the band structure, such as peaks in the density of states, are ignored. Increasing the number of neighbors improves the resolution of the density of states and makes it possible to distinguish between bulk sites and sites close to surfaces. As a consequence, magnetic moments are determined by the local atomic environment, typically without major nanoscale corrections. 2.1.2. Interatomic exchange The spin structure of a magnetic moment is the relative orientation of the atomic magnetic moments. It includes types of zero-temperature magnetic order, such as ferromagnetism, ferrimagnetism, and antiferromagnetism, and fi nite temperature magnetic order. However, micromagnetic structures, such as domains and domain walls, are usually excluded from the consideration. Figure 1 shows some spin structures of interest in the present context. To a large extent, the spin structure of bulk and nanomagnets is determined by the interatomic Heisenberg exchange, J(Ri Rj) Si·Sj = Jij Si·Sj. For positive and negative values of Jij it favors parallel and antiparallel spin alignment, respectively. In ferromagnets, such as Fe, Co, and Nd2Fe14B, all spins are parallel and the atomic moments add. Ferrimagnets, such as Fe3O4 and BaFe12O19, and antiferromagnets, such as CoO and MnF2, are characterized by two (or more) sublattices with opposite moments. This amounts to a ferrimagnetic reduction or antiferromagnetic absence of a net moment. Sublattice formation may be spontaneous, as in typical antiferromagnets, or imposed by the atomic composition, as in ferrimagnets [24] [25]. In metals, the interatomic exchange may be positive or negative and depends on the atomic environment, on the interatomic distance, and on the band fi lling. Figure 1. Spin structures (schematic): (a) ferromagnetism, (b–c) antiferromagnetism, and (d) noncollinear structure. The shown structure of the L10 type; the small atoms (with the large magnetization arrows) the iron-series transition-metal atoms, as compared to the bigger 4d/4f atoms. Examples of L10 magnets are CoPt and FePt. A simple and asymptotically correct [26–28] model is the Ruderman-Kittel-Kasuya-Yosida or RKKY exchange between two localized moments in a Pauli-paramagnetic matrix. For a free-electron gas of wave-vector kF, The interaction is obtained by second-order perturbation theory, that is, the embedded magnetic moments lead to a mixing of one-electron wave functions. The origin of the oscillations is the sharp Fermi surface, which means 46 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 47 that spatial features smaller than about 1/kF cannot be resolved with available zero-temperature wave functions. RKKY interactions were fi rst considered on an atomic scale, where the oscillation period is on an Å scale. In nanostructures, the fast oscillations do not average to zero but increase with the size of the embedded clusters or nanoparticles. However, the increase is less pronounced than that of magnetostatic interactions, and for particles sizes larger than about 1 nm, the magnetostatic interactions become dominant [27, 29]. In semiconductors and semimetals, such as Sb, the low density of carriers means that kF is small, and the period of the oscillations is nanoscale [16, 28]. This contributes to the complexity of the physics of diluted magnetic semiconductors [30, 31]. Figure 2. Carrier mediated exchange in dilute semiconductors (schematic). The mechanism is similar to RKKY interactions, but due to the essential involvement of donor or acceptor orbitals, J(ri , rj ) can no longer be written as J( |ri rj | ) . In a strict sense, the RKKY interaction is mediated by free electrons, but there exist similar effects in other regimes, for example in the tight-binding scheme [32]. Figure 2 shows the example of a dilute magnetic semiconductor where localized impurity spins are coupled by shallow donor or acceptor carriers. The carrier orbitals have a radius of the order of 1 nm, hybridize, and yield an RKKY -type coupling. A simple case is the exchange mediated by a two weakly overlapping s-orbitals, centered at R1 and R2. The overlap leads to bonding and antibonding orbitals, and the exchange. Jij is obtained from the hybridized electrons by second-order perturbation theory. In terms of local electron densities ρ(r) = ψ*(r)ψ(r), the exchange Jij scales as (ρ(ri – R1) – ρ(ri – R2)) (ρ(rj – R1) – ρ(rj – R2)). This means that the exchange is ferromagnetic if the magnetic ions are located in the same shallow s-orbital and antiferromagnetic if they are in different s-orbitals. Correlations (the Hubbard or ‘Coulomb-blockade’ energy of the orbitals) can be shown to reduce the ferromagnetic exchange while leaving the qualitative picture unchanged. One effect of competing RKKY exchange is noncollinear spin structures, as illustrated in Fig. 1(d). Noncollinearity due to competing exchange is encountered, for example, in some elemental rare earths (helimagnetism), where it refl ects different exchange interactions between nearest and next-nearest rare-earth layers [33]. Note that the corresponding magnetization wave vector is generally incommensurate with lattice spacing, not only in nanostructures but also in perfect crystals. Furthermore, the effect is relatively strong, with angles between neighboring atomic spins from 0 to 180°. This is in contrast to the relativistic effects considered in the next subsection. 2.1.3. Exchange stiffness On a continuum level, the Heisenberg exchange energy of a cubic material is where A is the exchange stiffness. More generally, Heisenberg exchange is described by A well-known derivation of Eq. (2) is in terms of magnetization angles. It, assumes φ = 0, so that Eex = ∫A(∇θ) 2dV, and takes into account that Σij Jij cos(θi – θj) ≈ Σij Jij (1 – (θi – θj) 2/2). Using the expansion θj = θi + ∇θ · (rj – ri) and comparing the result with Eq. (2) then yields A ~ Σij Jij (ri – rj) 2. This result is meaningful for nearest-neighbor interactions, but it diverges for long-range interactions Jij. An example is the RKKY interaction, where integration over all neighbors yields A ~ ∫ 1/R3 R2 R2 dR = ∞. This is because θj = θi + ∇θ · (rj – ri) breaks down for large distances R = | ri – rj |. A more general derivation of A is based on Fourier transformation, which diagonalizes J(|r – r’|) and yields a representation in terms of Jk. Since ∫A(∇θ)2dV = ∫ Jk θk 2 dk and ∫ A(∇θ)2dV = – ∫ A k2 θk 2 dk, A is given by the quadratic coeffi cient of the expansion of Jk with respect to k. Putting k = kek, R = R cosθ’ek + R sinθ’e⊥, and dV = 4πR 2 sinθ’ dθ’ dR yields 48 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 49 Here Jk = F(|k|) is the Lindhard screening function [26, 34]. Note that noncollinear or incommensurate spin states then correspond to a minimum of J(k). Figure 3. Exchange energy as a function of the wave vector of the magnetization inhomogenity: Lindhard function (solid line) and exchange-stiffness or continuum approximation (dashed line). For late 3d elements (Cu), k/2kF = 1 corresponds to modulation wavelength of 0.23 nm. Figure 3 compares the Lindhard function (solid line) with the exchangestiffness approximation (dashed line). We see that the exchange-stiffness approximation works well unless k is comparable to kF. In noncubic materials, A must be replaced by the 3 × 3 exchange-stiffness tensor Aμν, and the energy is Σμν ∫ Aμν ∂M/∂xμ · ∂M/∂xν dV. Here the indices μ and ν denote the spatial coordinates x, y, and z of the bonds. The energy is anisotropic with respect to the nabla operator ∇μ = ∂/∂μ (bond anisotropy) but isotropic with respect to the magnetization M. By contrast, the relativistic anisotropic exchange Σαβ ∫ Aαβ ∇Μα ∇Μβ dV is isotropic with respect to ∇ but anisotropic with respect to M. 2.1.4. Curie temperature Thermal disorder competes against interatomic exchange and causes the magnetization of ferromagnets to vanish at a well-defi ned sharp Curie temperature Tc. In a strict sense, ferromagnetism is limited to infi nite magnets, because thermal excitations in fi nite magnets cause the net moment to fl uctuate between opposite directions. The Curie temperature is determined by the site-resolved exchange coeffi cients. Jij, and since Ms and Tc are equilibrium properties, it is suffi cient to know the partition junction Z = Σμ exp(–Eμ/kBT), where the summation includes all microstates or spin-confi gurations μ. However, the number of terms in Z increases exponentially with the size of the magnet, and there exist exact solutions only in a few cases [35]. The simplest approximation is the mean-fi eld approximation, where the interactions are mapped onto a self consistent fi eld. The homogeneous nearest-neighbor Heisenberg ferromagnet has the mean-fi eld Curie-temperature Tc = (S + 1)zJ/3kBS, where S is the spin quantum number and z is the number of nearest neighbors. The mean-fi eld model is easily generalized to two or more sublattices. This site-resolved or lattice mean-fi eld theory includes the case of nanomagnets, which have a very large number N of non-equivalent atomic sites or ‘sublattices’. Since the Jij form an N × N matrix, there are N coupled algebraic equations, and Tc is given by the largest eigenvalue of the matrix. Jij/kB [24, 36]. Using averaged exchange constants < Jij> fails to properly account for the spatial dispersion of the exchange. An extreme example is a mixture of two ferromagnetic phases with equal volume fractions but different Curie temperatures T1 and T2 > T1. In the above approximation, Tc = (T1 + T2)/2, but in reality Tc = T2 [36]. Note that mean-fi eld theory is unable to describe the long-range correlations aspect of the problem, but the involved energy contributions are small, and the long-range features of the thermodynamics are not affected by the nanoscale effects [36]. In practice, it is diffi cult to distinguish the magnetism of particles or nanostructural features larger than about 1 nm from true ferromagnetism, because interatomic exchange ensures well-developed ferromagnetic correlations on a nanoscale. For example, when the radius of a particle is larger than a few interatomic distances, then the Ms(T) curve is diffi cult to distinguish from a ferromagnet. Disordered two-phase nanostructures have a single common Curie temperature close to the Curie temperature of the phase with the strongest exchange coupling [5, 36, 37]. Similar considerations apply to multilayers [38–42] and to systems such as magnetic semiconductors [30]. Figure 4 illustrates that it is not possible to enhance the fi nite-temperature magnetization of a phase having a low Curie temperature by exchange-coupling it to a 50 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 51 phase with a high bulk Curie temperature [36]. This is clear contrast to the nanoscale improvement of extrinsic properties (§2.3.5). Figure 4. Spontaneous magnetization of inhomogeneous magnets: (I) macroscopic mixture, (II) nanostructure, and (III) alloy. In alloys and nanostructures, there is only one Curie temperature, although the Ms(T) curves of nanostructures exhibit a two-phase like infl ection whose curvature may be diffi cult to resolve experimentally [36]. Heisenberg interactions require well-defi ned atomic magnetic moments. where S2 = So 2. In insulators, So 2 = S (S+ 1), whereas in metals, So is an expectation value and S/So has the character of a unit vector that describes the local magnetization direction. The assumption of a constant magnetic moment is often justifi ed, because the total interatomic exchange per atom, of order 100 meV, tends to be much smaller than typical intra-atomic exchange energies of about 1000 meV [43]. Some exceptions are L10 magnets, where the 4d or 5d moments (Pd or Pt) are spin-polarized by the 3d atoms (Fe or Co) [44], and very weak itinerant ferromagnets, such as ZrZn2 [16, 45]. Simplifying somewhat, the overall situation is intermediate between Heisenberg magnetism with stable local moments and a Stoner-like behavior where the moment vanishes at Tc. However, itinerant magnets such as Fe tend to be close to the Heisenberg limit (Ch. 2), which establishes the spin-fl uctuation picture of fi nite-temperature magnetism [16, 34]. 2.1.5. Anisotropic Exchange Heisenberg exchange is magnetically isotropic, that is, coherent rotation of a magnet’s spin system does not change the Heisenberg exchange energy. For example, layered structures, such as YCo5 and L10 magnets, tend to exhibit different intraand interlayer interactions [44, 46], but the exchange does not depend on whether the magnetization is in-plane or normal to the layers. This exchange-bond anisotropy affects the spin structure of a magnet at both zero and nonzero temperatures. For example, it is the main source of spin noncollinearities encountered in elemental rare earths [33] and in magnetoresistive materials, such as NiMnSb [47]. Figure 1 shows some spin structures. The bond anisotropy must not be confused with the relatively weak relativistic exchange anisotropies, which involve spin-orbit coupling and depend on the angle between the magnetization and the crystal axes. Examples are the exchange interactions assumed in the Ising and XY models, the magnetocrystalline anisotropy, and the unidirectional Dzyaloshinskii-Moriya exchange. For example: the exchange anisotropies Jxx Jzz and Jyy Jzz are small corrections to the isotropic exchange J = (Jxx + Jyy + Jzz)/3. Another example is the Dzyaloshinskii-Moriya (or DM) interaction HDM = – 1⁄2 Σij Dij · Si × Sj, where the vector Dij = – Dji refl ects the local environment of the magnetic atom [48]. Net DM interactions require local environments with suffi ciently low symmetry and occur, for example, in some crystalline materials, such as α-Fe2O3 (hematite), amorphous magnets, spin glasses, and magnetic nanostructures [5, 33, 48]. Micromagnetic noncollinearities, such as domain walls, also stem from relativistic effects, because they involve magnetocrystalline anisotropy, but their domain is nanoscale rather than atomic, and they are traditionally treated in the context of micromagnetism (§2.3). Compared to Heisenberg exchange, relativistic contributions are smaller by a factor of order α2, where α = 1/137 [5]. For example, typical DM canting angles are about 0.1°. 2.2. Magnetic Anisotropy The dependence of the magnetic energy on the orientation of the magnetization with respect to the crystal axes is known as magnetic anisotropy. Permanent magnets need a high magnetic anisotropy, in order to keep the magnetization in a desired direction. Soft magnets are characterized by a very low anisotropy, whereas materials with intermediate anisotropies are used as magnetic recording media. In terms of the magnetization angles φ and θ, 52 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 53 the simplest anisotropy-energy expression for a magnet of volume V is Ea = K1V sin 2θ . This anisotropy is known as lowest-order (or second-order) uniaxial anisotropy, and K1 is the fi rst uniaxial anisotropy constant. It is often convenient to express anisotropies in terms of anisotropy fi elds. For example, the expression Ea = K1V sin 2θ yields Ha = 2 K1/μoMs. For magnets of low symmetry (orthorhombic, monoclinic, and triclinic), the lowest-order anisotropy energy is where K1 and K’1 are, in general, of comparable magnitude. This expression must also be used for magnets having a low-symmetry shape, such as ellipsoids with three unequal principal axes, for a variety of surface anisotropies, such as that of bcc (011) surfaces [49], and for nanoparticles with random surfaces. Equation 5 can also be written as Ea = – M·K·M/Ms 2, where K is a 3 × 3 tensor. It obeys TrK = 0, and the two independent eigenvalues of K correspond to K1 and K’1. Higher-order anisotropy expressions contain, in general, both uniaxial and planar terms. For example, Ea/V = K1 sin 2θ + K2 sin 4θ contains secondand fourth-order uniaxial anisotrop terms and describes hexagonal and rhombohedral crystals [16, 50]. 2.2.1. Origin of anisotropy Figure 5 illustrates that there are two main sources of anisotropy: shape anisotropy and magnetocrystalline anisotropy. Shape anisotropy is important in magnetic nanostructures made from soft-magnetic materials, for example in Fe, Co, and Ni particles [16, 51] and in nanowires [52–55]. However, the anisotropy of most materials is of magnetocrystalline origin, refl ecting the competition between electrostatic crystal-fi eld interaction and spin-orbit coupling [7]. Note that the same mechanism is responsible for the quenching (or unquenching) of the orbital moment and for phenomena such as magnetic circular dichroism and anisotropic magnetoresistance. For shape anisotropy, K1 = μo(1 – 3D)Ms 2/4, where D is the demagnetizing factor (D = 0 for long cylinders, D = 1/3 for spheres, and D = 1 for plates) [56]. It is important to note that shape anisotropy is limited to very small particles (§2.2). In large particles, shape anisotropy is destroyed by internal fl ux closure, indicated at the bottom of Fig. 5(a). The crystal fi eld [57], which contains both electrostatic and hopping contributions [58], acts on the orbits of the inner d and f electrons. That is, the electron orbits refl ect the anisotropic crystalline environment, and adding spin-orbit coupling translates this anisotropic electron motion into magnetic anisotropy. Figure 5. Physical origin of magnetic anisotropy: (a) compass-needle analogy of shape anisotropy and (b–c) magnetocrystalline anisotropy. In (b) and (c), the anisotropy energy is given by the electrostatic repulsion between the tripositive rare-earth ions and the negative crystal-fi eld charges. The magnitude of the magnetocrystalline anisotropy depends on the ratio of crystal-fi eld energy and spin-orbit coupling. As a relativistic phenomenon, spin-orbit coupling is most pronounced for inner-shell electrons in heavy elements, such as rare-earth 4f electrons. This leads to a rigid or ‘unquenched’ coupling between spin and orbital moment, and the magneto-crystalline anisotropy is given by the relatively small electrostatic crystal-fi eld interaction of the 4f charge clouds [59] with the crystal fi eld [16, 60, 61]. This largely electrostatic mechanism, illustrated in Fig. 5(b) and 5(c), is responsible for the high room-temperature anisotropy of rare-earth permanent magnets, K1 ~ 10 MJ/m3 (see Appendix). In 3d atoms, the spin-orbit coupling is much smaller than the crystal-fi eld energy, and the magnetic anisotropy is a perturbative effect [7, 8, 16]. Typical secondand fourth-order transition-metal anisotropies are of the orders of 1 MJ/m3 and 0.01 MJ/m3, respectively. A manifestation of magnetocrystalline anisotropy is magnetoelastic anisotropy, where the crystal fi eld is changed by mechanical strain [5, 16]. 2.2.2. Surface and interface anisotropy To realize second-order anisotropy, the atomic environment of the transition-metal atoms must have a suffi ciently low symmetry [49, 62 –65]. Figure 6 illustrates that this is often, but not always, the case for surface atoms. Magnetic surface anisotropy, fi rst analyzed by Néel [62], is important in complicated structures and morphologies such as ultrathin transition-metal fi lms [66], multilayers [67], rough surfaces [65], small particles [68], and surface steps 54 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 55 [69]. In a variety of cases it has been possible to calculate surface anisotropies from fi rst principles [64, 67, 70–72]. The same is true for some other low-geometries, such as Fe wires embedded in Cu [73] and free-standing monatomic Co wires [74]. An interesting point is that surface anisotropies easily dominate the bulk anisotropy of cubic materials. From the tables in the appendix we see that bulk anisotropies are about two orders of magnitude smaller than lowest-order anisotropies. Due to the comparatively large number Ns of surface atoms of small particles, the surface contribution dominates the bulk anisotropy in particles smaller than about 3 nm, even if one takes into account that the net surface anisotropy is not necessarily linear in Ns but tends to scale as Ns 1⁄2 due to random-anisotropy effects. Figure 6. Surface anisotropy: (a) atomic origin and (b) realization in a nanoparticle. Lowest-order biaxial anisotropy is realized for bcc (011) but not for bcc (001) and bcc (111). The large surface-to-volume ratio of clusters leads to a comparatively strong diameter dependence of the intrinsic properties such as anisotropy [68] and magnetization. Magnetocrystalline anisotropy is characterized by a pronounced temperature dependence [16, 61, 75–77]. For example, the leading rare-earth anisotropy contribution of permanent magnet intermetallics such as SmCo5 and Nd2Fe14B scales as 1/T 2 [78]. The main reason is that typical anisotropy energies per atom are quite small, Ea ranging from less than 0.1 K to a few K. The realization of room-temperature anisotropy requires the support of the interatomic exchange fi eld, which suppresses the switching of individual atomic spins into states with reduced anisotropy contributions [16,79,80]. Magnetocrystalline anisotropy is, essentially, a single-ion property, realized by embedding the atom in a metallic or nonmetallic crystalline environment [16, 58]. This must be compared to the popular Néel model [62], which ascribes anisotropy to pair interactions. Figure 7 illustrates the difference. The Néel model requires two interacting magnetic atoms (black), whereas the single-ion or crystal-fi eld model amounts to hopping or crystal-fi eld interactions with atoms that are not necessarily magnetic (white). The principal failure of the Néel model is seen by comparing Sm2Fe17 and Sm2Fe17N3, where the electronegative nitrogen is nonmagnetic but strongly affects the crystal fi eld and changes the room-temperature anisotropy from – 0.8 MJ/m3 to 8.6 MJ/m3 [16]. Figure 7. Models of magnetic anisotropy: (a) Néel model and (b) single-ion crystal-fi eld model. Both models reproduce the correct symmetry, but (b) is physically more adequate for most systems. 2.2.3. Temperature dependence of anisotropy In most materials, including nanostructures, the magnetocrystalline anisotropy is strongly decreases with increasing temperature. This is due to intraatomic excitations. The strong temperature dependence of the leading rareearth anisotropy contribution of hard-magnetic materials such as SmCo5 and Nd2Fe14B refl ects intramultiplet excitations [16, 60, 61]. The excitations may be visualized as changes between Fig. 5(b) and 5(c). The fi gure indicates that these excitations compete against the crystal fi eld. However, the crystal fi eld is only one consideration; the main contribution is from the inter-sublattice exchange, which dominates thermal spin disorder. For one-sublattice magnets, such as Fe and Co, the Akulov or Callen and Callen theory [81] relates the temperature dependence of the anisotropy to the spontaneous magnetization and yields M3 and M10 power laws for uniaxial and cubic magnets, respectively. This theory has become popular far beyond its range of applicability [82] but is unable to describe structures such as rare-earth transition-metal magnets [16, 60], actinide magnets [83], and L10 type compounds [44, 84]. 56 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 57 Figure 8. Nucleation mode in a small L10 particle. Due to reduced anisotropy at the surface, the reversal starts at the surface [89]. An interesting example is the fi nite-temperature magnetization of L10 magnets [44, 84, 85]. Elemental 4d/5d magnets, such as Pd and Pt, are exchanged-enhanced Pauli paramagnets, but in a ferromagnetic environment they are easily spin-polarized by neighboring 3d atoms. [86, 87]. The 4d/5d moment contributes little to the magnetization and Curie temperature, but it plays a key role in the realization of magnetic anisotropy, which is of the order of 5 MJ/m3 at room temperature [88]. The temperature dependence of the anisotropy is refl ects the collapse of the 4d/5d moment. The result of the calculation is an M2 law [44], as compared to the Callen-Callen prediction M3 and to refi ned simulations that yield an M2.08 dependence [84]. By comparison, for uniaxial 3d magnets, such as Co and YCo5, m = 3 [81], cubic and noncubic actinide magnets exhibit m = 1 [83], and for cubic 3d magnets, such as Fe and Ni, m = 10 [81]. Finally, rareearth transition-metal intermetallics exhibit m ≈ 0, that is, the 4f sublattice anisotropy is largely independent of the leading 3d magnetization [16]. The exponents m = 2 and m = 3 are not very dissimilar [85], but the different physics—the crucial involvement of two sublattices—speaks in favor of m = 2. In fact, recent calculations by Mryasov et al. have yielded m = 2.08, amounting to a single-sublattice contribution of the order of 8%. As also pointed out in [84], the reduction of the number of 3d neighbors in magnetic nanoparticles has a very similar surface-anisotropy reduction effect. 2.3. Hysteresis of Magnetic Nanostructures Magnetic anisotropy yields easy magnetization directions corresponding to local energy minima and energy barriers that separate the easy directions. On an atomic scale, the barriers are easily overcome by thermal fl uctuations, but on nanoscale or macroscopic length scales the excitations are usually too weak to overcome the barriers. This is observed as magnetic hysteresis. Zeeman and selfi nteraction (demagnetization) magnetic fi elds, interatomic exchange, and magnetic anisotropy all contribute to the rotation, which occurs on a mesoscopic scale and has been known historically as micromagnetism [90], although nanomagnetism would be a better name to characterize the involved length scales. Magnetic nanostructures exhibit a particularly rich extrinsic behavior, but even traditional ‘microstructured’ magnets exploit nanometer-scale features for performance optimization [91]. For example, the best room-temperature permanent magnets are now made from Nd-Fe-B [92], but as-cast samples with the correct stoichiometry exhibit a disappointingly low coercivity unless the grain-boundary structure is optimized by a specifi c heat treatment. Figure 9 shows a typical hysteresis loop and illustrates how magnetic hysteresis is realized in real space. In the example of Fig. 9, the hysteresis refl ects domain-wall pinning in a small particle. This means that magnetic domains are separated by domain walls (dotted lines) whose motion is impeded by real-structure defects or ‘pinning centers’ in the bulk or at the surface. Aside from a few basic hysteresis mechanisms, such as pinning, coherent rotation, curling, and localized nucleation, there exist many variations and combinations. The reason is the real-structure dependence of magnetic hysteresis, which makes it necessary to consider each material or each class of materials separately. Figure 9. Magnetic hysteresis: origin and phenomenology of hysteresis. The coercivity of the particles shown in this fi gure is caused by domain-wall pinning at the grainboundary phase. 58 SKOMSKI & ZHOU IN ADVANCED MAGNETIC NANOSTRUCTURES (2006) NANOMAGNETIC MODELS 59 2.3.1. Micromagnetic free energy A key theoretical problem is to derive magnetization curves by simulating or modeling the magnet’s nanostructure. This requires the determination of the local magnetization M(r), from which the hysteresis loop is obtained by averaging. The large strength of the intra-atomic exchange means that typical magnetization changes in magnetic solids are caused by moment rotations rather than by changes in the moments’ magnitude. The result is the complicated nonlinear, nonequilibrium, and nonlocal problem of hysteresis. In addition, there is a strong real-structure infl uence. Properties related to hysteresis (extrinsic properties) are also known as micromagnetic properties [90], but this term is somewhat unfortunate, because most micromagnetic phenomena are nanoscale. Hysteresis problems are usually treated on a continuum level [16, 90, 93]. Narrow-wall phenomena, which have been studied for example in rare-earth cobalt permanent magnets [94] and at grain boundaries [95, 96], involve individual atoms and atomic planes and lead to comparatively small corrections to the extrinsic behavior (§4.2). Furthermore, in contrast to the intrinsic phenomena considered in Section 2, which affect the spontaneous magnetization Ms = |M|, micromagnetic phenomena are realized by local rotations of the magnetization vector. This is because typical micromagnetic energies are much smaller than the quantum-mechanical energy contributions that establish Ms. To explain the hysteresis loop of magnetic materials one needs to trace the local magnetization M(r) = Mss(r) as a function of the applied fi eld H. This is achieved by considering the free-energy functional Here Ms(r) is the spontaneous magnetization, K1(r) is the fi rst uniaxial anisotropy constant, A(r) denotes the exchange stiffness, and n(r) is the unit vector of the local anisotropy direction. H is the external magnetic fi eld, and Hd is the magneto static self-interaction fi eld. The latter can be written as The free-energy character of F refl ects the intrinsic or equilibrium temperature dependence of the parameters A, K1, and Ms. Furthermore, these parameters are local parameters, because they depend on chemistry, crystal structure, and crystallite orientation. Depending on the considered system, additional terms must be added to the micromagnetic equation. In lowest order, DM interactions amount to a random fi eld Σj(Dij,yex – Dij,xey)/2 where the summation (or integration) over j includes all atomic neighbors; the resulting structure may be called a ‘spin colloid.’ 2.3.2. Micromagnetic length scales Equation (6) yields not only the hysteresis loop but also the underlying micromagnetic spin structure. This includes features such as domains and domain walls. An aspect of great importance in nanomagnetism is the length scales on which these features are realized. Dimensional analysis of Eq. (6) yields two fundamental quantities, namely the wall-width parameter ∂o = √ ― A ― / ― K1 and the exchange length lex = √ ― A ― /μ ―