Luigi Paolasini
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1 Luigi Paolasini
2 LECTURE 5: MAGNETIC STRUCTURES - Mean field theory and magnetic order - Classification of magnetic structures - Collinear and non-collinear magnetic structures. - Magnetic domains.
3 Hamiltonian describing a periodic ferromagnetic crystal in an applied magnetic field B: Exchange term (Heisenberg) Zeeman term We define an internal effective molecular field B mf acting on a i-atom and due to a net magnetization of the whole neighbours atoms: molecular field at i-site (or Weiss field) The exchange term is replaced by the an effective molecular field: Effective mean field Hamiltonian
4 Phenomenological model valid for ferromagnetic systems, and analogus to the problem of a paramagnetic system placed in an effective magnetic field (B+B mf ). All the magnetic ions experience the same molecular field. At low temperatures the ferro-magnetic order is self-sustained by the effective molecular field B mf and it is proportional to the magnetization M: PIERRE-ERNEST WEISS λ=strength of molecular field The internal magnetic field B mf explains the ferromagnetic order also in absence of an external magnetic field B.
5 The magnetization of a paramagnet can be calculated by considering the simultaneous solution of the two equations: Brillouin function If B=0, the magnetization is linear in y: T>T c => M s (y=0)=0 T<T c => M s (0)=0 (unstable) [ Ms (±y 0 ) 0 (stable) Spontaneous magnetization T~T c => [ M=M s k B Ty/g J µ B JλM s M=M s B J (y) ~ (J+1)/3J
6 In presence of an external field B 0 M 0 for all temperatures and the phase transition is removed. g J µ B JB/k B T The magnetic moment orientations follow the magnetic field direction, but in real ferromagnetic the magnetic anisotropy need to be considered. At T=T C And the magnetization => =>
7 For small applied magnetic field B, and for T>T c, we can use the approximation : And by arranging the terms: The magnetic susceptibility corresponds to the Curie-Weiss law Curie-Weiss law
8 If the exchange interaction is negative J<0, the nearest neighbours magnetic moment lie antiparallel. We suppose to separate the molecular field in two interprenetrating ferromagnetic sublattices associated to the two spin direction up F(+) and down F(-) We assume that the molecular field B + acting on S i at F(+) is proportional of the magnetization M - due to the sublattice F(-), and viceversa:
9 For each sublattice the magnetization is given by the same expression of the ferromagnetic lattice, but with the molecular field constant λ<0: Each sublattice follows the same temperature dependence and the critical temperature correspond to the Néel temperature T N : Néel temperature Notice that the total magnetization M + +M - =0 (moments oppositely directed). We define the order parameter below the Néel temperature as the staggered magnetization: Staggered magnetization
10 The Curie-Weiss susceptibility for an antiferromagnet have the same expression of the susceptibility for a ferromagnet, but the Curie temperature T C is replaced with the Néel temperature -T N We define the Weiss temperature θ, and in general the Curie-Weiss susceptibility in the paramagnetic state is defined as: θ=0 paramagnet θ>0 ferromagnet θ=t C θ<0 anti-ferromagnet θ=-t N
11 In the case of an AF, the magnetic susceptibility depends from the direction of application of the magnetic field. H perpendicular to M + and M :, a canting of the magnetic moments produces a net magnetization. H parallel to M + or M - : the magnetization is zero, because at T=0 the two sublattices are saturated.
12 B parallel to M + or M - : at B=B c the two sublattices snap in spin-flop phase. A further increase of the magnetic field line up the magnetic moments until they reach the saturation value M s. If the magnetic anisotropy is strong, the sublattice magnetization can directly line up parallel to B, in this case the new configuration is called spin-flip phase.
13 Estimation of Weiss field If J=1/2, T C ~1000K => B mf ~ 1500T!!! B mf reflects the strength of exchange interaction!! Weiss constant and exchange interaction Assuming that only the z first neighbours contribute to B mf with the same exchange interaction J, we obtain the expression for the Weiss contant λ and the Curie temperature:
14 Exchange interaction is between the spin degree of freedom! For 3d transition metals, because L=0 is quenched, and the total magnetic moment depends on J=S. For the 4f rare earths ions S is not a good quantum number, but J is. The component of S perpendicular to J must average to 0, whereas its parallel component is conserved! S // projected along J: Heisenberg Hamiltonian: De Gennes factor
15 From Hund s rules Conserved part of S De Gennes factors
16 In the ordered phase, the distribution of the magnetic moments in a crystal lattice is a periodic function in the real space => it can be Fourier expanded Ex.: magnetic Bragg peaks in the reciprocal space! The magnetic symmetries are classified in term of crystallographic symmetries (translations, rotations, mirror planes) with the time-reversal operator T: t!-t which reverses the direction of the magnetic moments. T In general, the symmetry of the magnetic structure is lower than the crystal symmetry. Bi-color space groups introduced by Shubnikov are used to describe the magnetic structures. Alexei Shubnikov
17 - The magnetic propagation vector q m is a reciprocal vector - q m is a eigenfunction of the Translational Group - It is defined inside the 1 st Brillouin zone - Describes the periodicity of a magnetic structures - It is perpendicular to the planes containing atoms with the same orientation of the magnetic moments - The inverse of its modulus is equal to those interplanar distances The propagation vectors give information only on the periodicity of a magnetic structure but NOT on the orientation of the magnetic moments
18 The magnetic moment distribution m ns relative to the s-atom inside the n th -lattice cell is equal to the Fourier transform: m ns - m ns is a REAL vector!!!! - The phase factors are real ONLY if q m correspond to the 1 st Brillouin surface (i.e. for ferro and antiferromagnetics) - In general the for any q m the sum must include -q m such that :
19 Example 1: Ferromagnetic - All the magnetic moment m j are parallel each others - The magnetic and the chemical unit cells coincide - Structural and magnetic reflections coincide chemical unit cell k 2θ k Q First Brilloiun zone q m =(0,0,0) m ns = m 0 = m 01 a 1 +m 02 a 2
20 Example 2: Antiferromagnetic - Adjacent planes contain antiparallel magnetic moment m j - Magnetic cell integral multiple of the chemical unit cell (double) - Magnetic reflections correspond to the symmetry points at the surface of the Brillouin zone chemical unit cell k Magnetic unit cell 2θ k First Brilloiun zone q m =(1/2,1/2,0) m ns = (-) n m 0 = (-) n (m 01 a 1 +m 02 a 2 )
21 Example 3: Non-collinear commensurate - Propagation vector q m have a non-integral periodicity - In general both q m and - q m exists (indicated τ + and τ - ) The propagation vector q m inside the 1 st Brillouin zone chemical unit cell Magnetic unit cell q m =(0,0,1/3) m ns =m 0 [cos(q m. R n +φ)u s +sin(q m. R n +φ)v s ]
22 Example 4: modulated magnetic structures
23 Several equivalent magnetic vectors are present in the same crystal. If only one magnetic atom is present in the unit cell: Examples:
24 Existence of small regions in which the magnetization reaches the saturation and called magnetic domains (Weiss). A weak magnetic field can induce a huge magnetization by the orientation of all the domains. (in contrast to Curie-Weiss law for paramagnets) Existence of domain wall between different saturated regions. Demagnetizing energy Ferromagnetic domain formation Compromise between the energy cost to increase the domain number and to minimize the magneto-static energy (dipolar magnetic field) Magnetization M Demagnetizing field H d
25 Beside the more classical ferromagnetic domains, in the general magnetic structures we distinguish the configurational or the orientation domains: k-domains (configurational domains) For any given propagation vector q m a family of {q m } co-exist in the same crystal s -DOMAINS (orientation domains) For each K-domain different (but energetically equivalent) orientations of the magnetic moments co-exist in the same crystal Ex. Domains in a simple AF structure k-domains q m {0 ½ 0} s-domains q m (0 ½ 0)
26 Boundary between two crystal regions with different K- or S- domain structure Two type of domain walls: cycloidal Néel wall and helical Block wall. (thin films) (bulk) Energy cost to rotate two adiacent spins by θ : Energy cost for rotation across a domain wall: Domain wall energy per unit area for a Block wall: Magnetic moments Domain wall width Na θ=π/n Minimized when N-> Tendency to increase the domain wall width!!
27 Arises from spin-orbit interaction or from partial quenching of the angular momentum. Existence of easy and hard axis of magnetization. Uniaxial anisotropy (Ex. SmCo 5 ): K 1,K 2 anisotropy constants (J/m 3, ~mev) θ angle between the easy axis and H Magnetization M In general for a cubic system: Applied magnetic field H - High symmetry => small anisotropy (Fe, cubic, K 1 ~ 4.8 x 10 4 J/m 3 ) - Low symmetry => high anisotropy (Co, hex., K 1 ~ 5 x 10 5 J/m 3 ) (SmCo 5, hex.,k 1 ~ 1.7 x 10 7 J/m 3 )
28 Anisotropy energy contribution to a Block s domain wall formation Simple uniaxial anisotropy E=K sin 2 θ with K > 0 (favour θ=0 or θ=180 0 ) N =1 Total contribution to the energy per unit area of a Block s wall: Exchange term Anisotropy term The equilibrium is reached when dσ BW /dn=0 The resulting equilibrium thickness of the wall is:
29 - Domain transformations under applied magnetic field H - Increasing the applied magnetic field => domain rotation in favourable direction with respect to the magnetic field direction - Domain wall motion compatible with magnetic anisotropy (easy axis [001]) Induced polarization: Remanence polarization: J r Intrinsic coercitivity: H ic Maximum energy product: (H B) max
30 Soft magnetic materials: small coercitive fields easy to magnetize, broad domain walls (small K) low magnetostriction absence of internal strain small dissipated power Ex. Permalloy Ni/Fe, Bc~ 2 x 10-7 T Hard magnetic materials: large coercitive fields hard to magnetize narrow domain walls (large K) domain wall pinning large energy dissipated Ex. Nd2Fe14B, Bc~ 1.2 T
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