Models in Magnetism: Models in Magnetism: Introduction Introduction - PowerPoint PPT Presentation

Models in Magnetism: Models in Magnetism: Introduction Introduction E. Burzo Faculty of Physics, Babes-Bolyai University Cluj-Napoca, Romania Short review: Short review: • basic models describing the magnetic behaviour • connections between models

General problems General problems Dimensionality Dimensionality of the system, d; Moments coupled: all space directions d=3 in a plane d=2 one direction d=1 polymer chain d=0 Phase transition: Existence of a boundary at d=4, spatial dimensionality can be also continous, ε=4-d Number of magnetization components, n Number of magnetization components, n Heisenberg model n=3 X-Y model n=2 Ising model n=1 Phase transitions: n  spherical model (Stanley, 1968) n=-2 Gaussian model n can be generalized as continous For d  4, for all n values, critical behaviour can be described by a model of molecular field approximation

Comparison with experimental data Comparison with experimental data magnetization versus temperature M=f(T) magnetic susceptibility  =f(T) behaviour in critical region M(T)  t  M(O)  t -  c p  t -  | T T |  - 1 2 t C 10 10     T C

Transition metals: 3d Fe,Co,Ni Transition metals: Fe g= 2.05-2.09 Co g=2.18-2.23 Ni g=2.17-2.22 Moments due mainly to spin contribution For 3d metals and alloys Moments at saturation  =gS 0 , μ g S (S 1)   Effective magnetic moments p p eff generally r=S p /S o >1 Rare-earths: 4f shell presence of spin and orbital contribution Magnetic insulators: localized moments

Localized moments: Localized moments: Heisenberg type Hamiltonian: exchange interactions  J S i S H  J ij exchange integral direct ij j i,j n=3 system Difficulty in exact computation of magnetic properties: many body problem Approximations Ising model (Ising 1925) Ising model Exact results in unidemensional and some bidimensional lattices  S iz S H   2 J jz i,j • Unidimensional Unidimensional neglect the spin components  H strong uniaxial anisotropy

• Linear Linear Ising Ising lattice lattice : not ferrromagnetic 1 exp (2J/k T)   B T Square bidimensional lattice, J 1 ,J 2 • Square bidimensional M=[1-(sh2k 1 sh2k 2 ) -2 ] 1/2 Onsager (1948) J J k 1 k 2   ; 1 2 Yang (1952) k T k T B B • Tridimensional Tridimensional lattice: series development method • Spherical Ising model Spherical Ising model (Berlin-Kac, 1951)  2 S ct  i arbitrary values for spins but i can be solved exactly in the presence of an external field d  4; critical exponents are independent of d and of the geometry of the system

Molecular field models : Molecular field models : Methods which analyse exactly the interactions in a small part of crystal, and the interactions with remaining part are described by an effective field, H m , self consistently determined: small portion  atom (molecular field approach Weiss (1907) •Magnetic domains •Molecular field: aligned magnetic moments in the domains H m =N ii M H  0  M 1 N   Total field H T =H+H m ; M=  0 H  M=  0 (H+N ii M) ii 0

Self consistency: Self consistency: M H N M H N ( H N M)             0 ii 0 0 ii 0 0 ii 0 2 2 H(1 N ) (N )M        0 ii 0 ii 0 2 3 H[1 (N ) (N ) (N )            0 ii 0 ii 0 ii 0 H  0 H    1 N   ii 0 Reverse reaction : corrections are time distributed: n Reverse reaction correction after n-1 one Molecular field: Molecular field: act at the level of each particle

z  2J S S H   2zJ ij i j m ij N  j 1  ii 2 2 Ng μ μ B 0 gμ μ S H H   0 B i m m S  J μ μ gJH 0 B T x  M(T)=M(0)B J (x) k T B M(T) 1 - 3 T   exp C    Low temperatures   1  M(0) J J 1 T    M(T)  3/2 T experimental M(0) T<T C , close to T C M(T)  =1/2   t M(0)  =1/3 exp.

T>T C MF:  -1  T in all temperature range experimental around T C :  t -   =4/3 MF:  =T C θ - T C (2.4 4.8)%   T experimental for Fe,Co,Ni C

Interactions between a finite number of spins +molecular field Oguchi method(1955); Constant coupling approximation (Kastelijn-Kranendonk, 1956); Bethe-Peierls-Weiss method (Weiss 1948) Oguchi: pair of spins H T  molecular field for 2J S S gμ μ (S S )H H     ij i j 0 B iz jz T 0 z-1 neighbours T C ≠   /T C =1.05 (cubic lattice)  -1 nonlinear variation around T C

Spin Waves Spin Waves Slater (1954): exact solution for Heisenberg Hamiltonian: all spins (except one) are paralelly aligned   ' S S S NS, S NS 1    ; t i t t N  number of atoms   gμ B S 2J S S H    B iz j l i neigh. Many spin deviations: additivity law ΔE(n)  nΔE(1) (non rigorous, corrections) repulsion of spin deviations: atoms with S, no more 2S deviations attraction: total exchange energy is lower when two spin deviations are localized on neighbouring atoms

•Semiclassical description of spin wave: Bloch (1930) (Heller-Kramers 1934, Herring-Kittel 1951, Van Kranendonk-Van Vleck, 1958) •Holstein-Primakoff folmalism (1940) M=M(0)(1-AT 3/2 ) T/T C  0.3 •Renormalization of spin waves (M.Bloch, 1962) Keffer-London: effective field proportional with mean magnetization of atoms in the first coordination sphere (1961) replaced by an effective spin at T, proportional with the angle between two neighbouring spins  The system is equivalent, at a given temperature, with a system of independent spin wave, having excitation energy (renormalized energy) equal with the energy of spin wave in harmonical approximation, multiplied by a self consistent term which depends on temperature The model describe the temperature dependence of the magnetization in higher T range

Series development method (Opechowski, 1938, Brown, 1956) Series development method The magnetic properties of the system described by Heisenberg hamiltonian, can be analysed around T C , by series development method in T -1 T>T C  (T-T C ) -   =4/3; For S=1/2 k B T C /J=1.8-1.9 (z=6) =2.70 (z=8) Green function method (Bogolyubov-Tyablikov, 1959) Green function method Bitemporal Green function for a ferromagnet (S=1/2). Temperature dependence of magnetization obtained by decoupling Green function equation. The analysis has been made in lowest decoupling order (random phase approximation) M=M(0)(aT 3/2 +bT 5/2 +cT 7/2 )  =1/2;  =2 Analysis in the second order of Green function decoupling (Callen, 1963) k B T C /J values only little higher than those obtained by series development method.

Antisymmetric exchange interactions: Antisymmetric exchange interactions: (Dzialoshinski 1958) (Dzialoshinski 1958) General form of bilinear spin-spin interaction   J S S α, β x, y, z H  αβ ia jβ α, β S S S J J J  J  αβ αβ βα A A A J J J   αβ αβ βα a d (S xS ) d d H s    ij J S i S H  ij ij i j ij ji ij j Explain weak ferromagnetism in  -Fe 2 O 3 M Fe M Fe

Indirect excahnge interactions through conduction Indirect excahnge interactions through conduction electrons (RKKY) ( Ruderman-Kittel Kasuya, Yoshida (1954-1956)) electrons (RKKY) 4f shell: small spatial extension La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 4f o 4f 7 4f 14 3d dilute alloys in nonmagnetic host H s-d(f) =J sS H = H s-d + H cond.el + H zz First order perturbation theory  Uniform polarization of conduction electrons

Second order J(Rnm)  J 2 F(x) xcosx sinx  F(x) ; x 2k R   F nm 4 x 3 R  Oscillatory polarization: decrease as nm Example: Stearns 1972: Polarization of s and d itinerant 3d electrons: iron T>T C  (g J -1) 2 J(J+1)F(x) Rare earths F(x) are similar

Exchange interactions 4f-5d-3d: Exchange interactions 4f-5d-3d: R-M compounds R=rare-earth M=3d metal M 5d =M 5d (0)+  G G=(g J -1) 2 J(J+1) GdFe 2 0.8 GdFe 2 GdCo 2 M 5d (  B ) 0.6  0.5 M (0) n M  0.4 GdNi 2 5d i i 0.2 0.4 0.0 0 1 2 3 4 M 3d (   /f.u.) i M 5d (0) (  B ) 0.3 n i number of 3d atoms in the first 0.2 GdCo 2-x Si x coordination shell, having M i moment GdCo 2-x Cu x 0.1 GdCo 2-x Ni x YFe 2-x V x 0.0 0 4 8 24 32  n i M i (  B )

Band models Band models •non integer number of  B M Fe =2.21  B M Co =1.73  B M Ni =0.61  B •presence of 3d bands: widths of  1 eV •difference between the number of spins determined from saturation magnetization and Curie constant