Magnetic ordering, magnetic anisotropy and the mean-field theory
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1 Magnetic ordering, magnetic anisotropy and the mean-field theory Alexandra Kalashnikova Ferromagnets Mean-field approximation Curie temperature and critical exponents Magnetic susceptibility Temperature dependence of magnetization and Bloch law Antiferromagnets Neel temperature usceptibility Magnetic anisotropy Magnetic-dipole contribution Magneto-crystalline anisotropy and its temperature dependence Ferro- and antiferromagnets in an external field 1
2 Magnetic ordering, magnetic anisotropy and the mean-field theory Ferromagnets Mean-field approximation Curie temperature and critical exponents Magnetic susceptibility Temperature dependence of magnetization and Bloch law
3 Mean-field approximation: Weiss molecular field μ J m wm Weiss molecular field - field created by neighbor magnetic moments Alignment of a magnetic moment μ in the total field +wm? 3
4 A brief reminder: independent magnetic moments in external field μ Thermal energy: U T kt At room temperature: J Potential energy of a magnetic moment in the field : U cos In a field of 1MA/m: J U / U T ~
5 Paramagnetism: independent magnetic moments in external field Induced magnetization Probability for a magnetic moment to orient at the angle θ: U exp U M T exp kt Full magnetization along the field: 1 Ncoth cos kt Langevine function L 5
6 Paramagnetism: independent magnetic moments in external field Full magnetization along the field: M 1 Ncoth M kt At RT in the field of 1МА/м: M Paramagnetic susceptibility: NL( ) para N N( / 3...) 3kT M N 3kT Curie law 6
7 Paramagnetism: independent magnetic moments in external field Quantization of the magnetic moment g J z Resulting magnetization along the field: M NgJ B B J z B J z J, J 1... J gj B kt Gd 3+ Fe 3+ Brillouin function Cr 3+ At small : M J NgJ B... 3J1 Curie law para Ng J ( J 1) B 3kT 7 [enry, PR 88, 559 (195)]
8 Ferromagnets: Weiss molecular field J Averaged projected magnetic moment: z g B z B B () () g ( kt B m) Weiss field due to exchange interactions J with N nearest neighbors: m NJ g z B z B NJ g B gb kt z elf-consistent solution 8
9 Back to ferromagnets: Weiss molecular field J =0 z B () < z >() z g ( kt B m) kt NJ gb NJ =0 non-zero solutions: z opposite orientations of the order parameter 9
10 Ferromagnets (=1/): Curie temperature J =0 T 1 T C >T 1 B () T >T C Variations of these curves with T: z B () z kt NJ At T>T C z 0 gb NJ =0 Ordering temperature: z T C nd order phase transition JN k changes continuously with T 10
11 Ferromagnets (=1/): Magnetization in a vicinity of T C J =0 T 1 T C >T 1 B () T >T C z B () z kt NJ gb NJ =0 At T T C z 0 1/ 0 3 T 1 C z 4 T 11
12 Ferromagnets (=1/): Curie temperature J Ordering temperature: nd order phase transition T C JN k Mean-field approximation predicts: 1D case: D case: 3D case: T C J 0 k J k J T C 4 T C 6 k Reality: No magnetic ordering T C. 69 T C J k J k Agreement gets better for the 3D case Reason: fluctuations were neglected 1
13 Ferromagnets (=1/): at low temperatures J =0 T 1 T C >T 1 B () T >T C z B () At T 0 g ( kt B m) z Prediction: z Reality: M 1 T 1 e C / T 3/ ( T ) M 0 1 ( T / TC ) 13
14 Ferromagnets (=1/) in applied field J 0 T C z B () g ( kt B m) < z >() Non-zero magnetization above T C usceptibility Above T C z in small fields k( T TC ) B T T C k( T 1 T C ) Curie-Weiss law 14
15 Ferromagnets Weiss molecular field theory: 1 M M ( T 0) 1 M 0 T C z JN k 0 Temperature Curie T 1 C T 1/ Critical exponent β=1/ T C Curie-Weiss law T T C T 1 k( T T C ) Critical exponent γ=-1 Landau theory of phase transitions gives the same values of critical exponents 15
16 Ferromagnets Weiss molecular field theory: 1 M M ( T 0) 1 0 T C T M T C T ( T ) T C ) ( T C T Landau theory pin-spin correlations 16 and MF
17 Mean-field approximation: Bethe mean-field theory 0 j J From uncorrelated spins to uncorrelated clusters of spins 0 interaction with its 4 neighbors is treated exactly (cluster) j are subject to effective (Weiss) field T J C k ln( N /( N )) Approximation predicts: Reality: 1D case: D case: 3D case: T C 0 T C. 885 T C J k J k No magnetic ordering T C. 69 T C J k J k Agreement is improved 17
18 Temperature dependence of magnetization At T=0 z M(r)=M 0 Minimum of the exchange energy At T>0 Thermal fluctuations M(r) uperposition of harmonic waves thermal spin waves (introduced by Bloch) Nonparallel i Increase of the exchange energy The magnitude of the gradient of M is important! 18
19 Magnetization at T>0 M(r)=M 0 +ΔM(r) Characteristic period of magnetization variation λ λ>> a (inter-atomic distance) λ<< L (sample size) ΔM(r) <<M 0 mall deviation of M from M 0 (M(r)) =M 0 =const Length of M is conserved Plane waves in a continuous medium (some analogy with sound waves) Energy of a ferromagnet as a function of the spatial distribution of magnetization? Increase of the Zeeman energy: -(M(r)-M 0 ) Increase of the exchange energy: Exchange constant 19
20 Magnetization at T>0 Energy of a ferromagnet as a function of the spatial distribution of magnetization: Increase of the Zeeman energy -(M(r)-M 0 ) Increase of the exchange energy 0
21 Magnetization at T>0 z mall deviations of M: M x, M y << M 0 M 0 1
22 Magnetization at T>0 z Introduce complex combinations:
23 Magnetization at T>0: Exchange waves z A sum of the energies of plane waves spin waves 3
24 Magnetization at T>0: magnons Energy of a ferromagnet: number of quasiparticles - magnons in the state with an energy: Quasi-momentum of magnon Effective mass of a magnon Bose-Einstein distribution of thermal magnons 4
25 Magnetization at T>0: magnons Each magnon reduces the total magnetic moment of the sample by μ Quadratic dispersion Bose-Einstein distribution Bloch (3/-) law: correct temperature dependence of magnetization at low T 5
26 Magnetic ordering, magnetic anisotropy and the mean-field theory Antiferromagnets Neel temperature usceptibility 6
27 Antiferromagnets μ 1i μ i Magnetic sublattices: M 1 =Σμ 1i /V M =Σμ i /V M=M 1 +M L=M 1 -M Magnetization (ferromagnetic vector) Antiferromagnetic vector) Equivalent magnetic sublattices same type of magnetic ions in the same crystallographic positions Number of magnetic sublattices is equivalent to the number of magnetic ions in the magnetic unit cell 7
28 Antiferromagnets: Neél temperature A B =0 Equivalent sublattices: Molecular (Weiss) fields for the case of two sublattices: w AA =w BB =w 1 w AB =w BA =w A =w AA M A +w AB M B B =w BB M B +w BA M A M A =-M B Magnetization of a sublattice in the presence of the molecular fields: Neél temperature: B () =0 For each sublattice: T 1 T N >T 1 T >T C 8
29 Antiferromagnets: susceptibility Equivalent sublattices: afm par w AA= w BB =w 1 >0 w AB= w BA =w <0 A =w AA M A +w AB M B + B =w BB M B +w BA M A + M A =-M B Asymptotic Curie temperature: 1 afm 1 k ( T TA) ) A para 1 kt usceptibility above T N : ferro 1 k( T T C ) TA T N T C T 9
30 Antiferromagnets: susceptibility afm par Equivalent sublattices: w AA =w BB =w 1 >0 w AB =w BA =w <0 M A =-M B usceptibility in the Neél point: T N usceptibility at T=0: χ=0 30
31 Antiferromagnets: some examples T N [P. W. Anderson, Phys. Rev. 79, 705 (1950)] 31
32 Magnetic ordering, magnetic anisotropy and the mean-field theory Magnetic anisotropy Magnetic-dipole contribution Magneto-crystalline anisotropy and its temperature dependence 3
33 Magnetic anisotropy Magnetic anisotropy energy energy required to rotate magnetization from an easy to a hard direction Without anisotropy net magnetization of 3D solids would be weak; in D systems it would be absent [Mermin and Wagner, PRL 17, 1133 (1966)] M Magnetization axial vector ources of anisotropy: Deformation Intrinsic electrical fields hape etc. - polar impacts Magnetic anisotropy sets an axis, not a direction 33
34 Magnetic anisotropy: phenomenological consideration Uniaxial anisotropy M E a K u 4 1 sin Ku sin... K u1 0 (001) easy plane Cubic anisotropy K u1 0 Effective anisotropy field a [001] easy axis E a Ku1 cosn... M M 3 w a K K K 1 0 K 1 0 easy axes- {100} easy axes- {111} 34
35 Magnetic anisotropy Compound erg cm -3 erg cm -3 Origins of magnetic anisotropy ingle-ion anisotropy Anisotropic exchange Magnetic-dipolar interactions pin-orbit interaction Interactions between pairs of magnetic ions 35
36 Magnetic-dipolar contribution to the anisotropy Cubic crystal E a Nq x M For a pair of neighboring ions: E( ) w l cos ϕ=α 1 4 q... ex const Dipolar interaction Quadrupolar interaction fcc: K1 Nq N number of atoms per volume unit cubic anisotropy constant vcc: K1 16/ 9Nq K Nq 1 36
37 exagonal crystal θ... sin sin 4 1 u u a K K E Cubic crystal ) ( q l E K E a cos ϕ=α Magnetic-dipolar contribution to the anisotropy θ u1 u a q l K u ~ 1 Contributes to the shape anisotropy This films demonstrate in-plane anisotropy 37
38 ingle-ion anisotropy (crystal-field theory) pinel CoFe O 4 [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] O - [100] plitting of the 3d shell In the ligand field: Co + 3d 7 x -y ; z xy; xz; yz 38
39 ingle-ion anisotropy (crystal-field theory) [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] [100] plitting of the 3d shell In the ligand field + trigonal distortion x -y ; z 3d 7 xy; xz; yz Maxima of electronic density in the plane (111) Maxima of electronic density along the axis [111] 39
40 ingle-ion anisotropy (crystal-field theory) [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] [100] plitting of the 3d shell In the ligand field + trigonal distortion x -y ; z 3d 7 xy; xz; yz Orbital moment L [111] pin-orbit interaction 3 rd unds rule: w O L Co + (3d 7 ) L 0 cos -angle between и [111] 40
41 ingle-ion anisotropy (crystal-field theory) [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] w O L L cos -angle between и [111] [100] Co + (3d 7 ) 0 Averaging over 4 types of Co + positions with distortions along different {111} axes: w a NL( x y y z z x ) K [lonczewski, JAP 3, 53 (1961) ]
42 Temperature dependence of magneto-crystalline anisotropy Classical theory gives: E K a n Cubic anisotropy K K n n n ( T ) M ( ) T ( 0) (0) M Uniaxial anisotropy K K n n ( T ) (0) M M ( T ) (0) 10 3 K K n n Fe ( T ) (0) M M ( T ) (0) n( n1) [Zener, Phys. Rev. 96, 1335 (1954)] Exp: Theory: Competition between different anisotropies gives rise to spin-reorinetation transitions 4
43 Magnetic ordering, magnetic anisotropy and the mean-field theory Ferro- and antiferromagnets in an external field 43
44 Ferromagnets in applied magnetic field: domain walls displacement Cubic anisotropy (K 1 >0) [100] 44
45 Technical Ferromagnets magnetization in applied magnetic processes: field: magnetization rotation of magnetization rotation e.a. M M e.a. Reversible rotation towards the field Irreversible rotation towards the field 45
46 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet Χ max =-1/w M =-w M 46
47 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet If the anisotropy is weak: M Condition for the spin-flop (spin-reorienetation transition) C 47
48 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet If the anisotropy is weak: M If the anisotropy is strong: M pin-flip C C 48
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