Magnetic Ordering, Magnetic Anisotropy and the Mean-Field Theory

Magnetic ordering, magnetic anisotropy and the mean-field theory

Alexandra Kalashnikova [email protected] Ferromagnets Mean-field approximation Curie temperature and critical exponents Magnetic susceptibility Temperature dependence of magnetization and Bloch law Antiferromagnets Neel temperature Susceptibility Magnetic anisotropy Magnetic-dipole contribution Magneto-crystalline anisotropy and its temperature dependence

Ferro- and antiferromagnets in an external field 1 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

2 Mean-field approximation: Weiss molecular field

H m  wM Weiss molecular field μ J - field created by neighbor magnetic moments

Alignment of a magnetic moment μ in the total field H+wM ?

3 A brief reminder: independent magnetic moments in external field

μ Thermal energy: UT  kT

At room temperature: 4.11021J

H Potential energy of a magnetic moment U  H cos in the field H: H

In a field of 1MA/m: 1.2 1023 J

U H /UT ~ 0.003

4 Paramagnetism: independent magnetic moments in external field Induced magnetization Probability for a magnetic moment to orient at the angle θ:

 U H   H  exp   exp cos   UT   kT  Full magnetization along the field: H  1  H M  Ncoth        kT Langevine function L 

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5 Paramagnetism: independent magnetic moments in external field

Full magnetization along the field:  1  MH M  Ncoth        kT

H At RT in the field of 1МА/м:   0.003



N 2 M  NL()  N( / 3....)  H 3kT Paramagnetic susceptibility: M N 2  para   Curie law H 3kT 6 Paramagnetism: independent magnetic moments in external field  Quantization of the magnetic moment gJ H z  gB J z   B kT J z  J, J 1....  J H Gd3+ Resulting magnetization along the field: Fe3+ M  NgJB BJ  

Brillouin function Cr3+  J 1  At small  : M  NgJB   ...  3J 

Curie law 2 2 Ng J(J 1)B  para  3kT 7 [Henry, PR 88, 559 (1952)] Ferromagnets: Weiss molecular field H Averaged projected magnetic moment:

z  gB SBS () g S(H  H ) S J   B m kT Sz  SBS ()

Weiss field due to exchange interactions J NJ Sz with N nearest neighbors: H m  gB   NJ S    g S H  z    B  g    B  Self-consistent solution Sz  SBS    kT      8  Back Back to  S S  z S z   g ferromagnets  J NJS kT B SB  S ( S kT H (    ) H H g NJ m : Weiss: field molecular B ) =0 H

< Sz>() H =0 of the orderparameterof the 2 oppositeorientations 2 non - zerosolutions:   S z 9 Ferromagnets VariationsT: with curves these of At T S S z S z  >  T   C J NJS SB kT S S  (S=1/2): Curie temperatureCurie (S=1/2): ( z  )  H g 0 NJ B =0 H

BS() 2 Ordering temperature: S nd H z order phase transitionorderphase =0 changes continuouslyT with  T C  JN k T T 1 T C 10 2 >T >T 1 C Ferromagnets (S=1/2): Magnetization in a vicinity of TC H

T1 J H=0 S TC>T1

T2>TC )  ( S B

Sz  SBS ()

kT gB Sz    H  NJS NJ =0

1/ 2 At T→TC S  0 z 3  TC  S z  1    0 4  T  11 Ferromagnets (S=1/2): Curie temperature H JN Ordering temperature: TC  S J k 2nd order phase transition

Mean-field approximation Reality: predicts: J 1D case: TC  2  0 No magnetic ordering Agreement k gets better J J for the 3D case 2D case: TC  4 TC  2.269 k k Reason: J J fluctuations 3D case: T  6 TC  4.511 C k k were neglected 12 Ferromagnets (S=1/2): at low temperatures H

T1 J H=0 S TC>T1

T2>TC )  ( S B

Sz  SBS () g S(H  H )   B m  kT Prediction:

1 2TC /T At T→0 S z  1 2e  S z  S 2    Reality: M (T )  M 1 (T /T )3/ 2  0 C 13  Ferromagnets in fields small Above T Susceptibility  S  S z  g  C J SB B S S ( kT (S=1/2) in applied (S=1/2)in applied field ( H  S ) H z H  m  ) k ( T B 

H < S >()

T z C ) H Non   - 0 zeromagnetization above T  T C  k (  T 1  T C T C ) law Curie T C - Weiss 14 Ferromagnets Weiss molecular field theory:

M S 1 1 M S (T 0) 

M H 0

0 TC T Temperature Curie Curie-Weiss law 1 Critical JN   exponent TC  T TC k k(T TC ) γ=-1 1/ 2  T  C Landau theory of phase transitions gives S z  1  Critical  T  exponent the same values of critical exponents β=1/2 15 Ferromagnets Weiss molecular field theory:

M S 1 1 M S (T 0) 

0 TC T  M S  TC T 

   (TC T )

  (T T) C Landau theory 0.5 1 0.5 Spin-spin correlations 16 and MF Mean-field approximation: Bethe mean-field theory From uncorrelated spins to uncorrelated clusters of spins  S J S0 interaction with its 4 neighbors is treated exactly 0 (cluster)  S are subject to effective (Weiss) field Sj j 2J T  C k ln(N /(N  2))

Approximation predicts: Reality:

1D case: TC  0 No magnetic ordering J J Agreement 2D case: T  2.885 T  2.269 is improved C k C k J J 3D case: T  4.993 TC  4.511 C k k 17 Temperature dependence of magnetization

At T=0 M(r)=M0 S z  S Minimum of the exchange energy

At T>0 Thermal fluctuations

M(r)

Superposition of harmonic waves – thermal spin waves (introduced by Bloch)

Nonparallel Si Increase of the exchange energy

The magnitude of the gradient of M is important! 18 Magnetization at T>0 H M(r)=M0+ΔM(r) … … ……

|ΔM(r)|<> a (inter-atomic distance) Plane waves in a continuous medium λ<< L (sample size) (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)-M0)H

Increase of the exchange energy:

Exchange constant 19 Magnetization at T>0 H … … ……

Energy of a ferromagnet as a function of the spatial distribution of magnetization:

Increase of the Zeeman energy -(M(r)-M0)H

Increase of the exchange energy

20 Magnetization at T>0 H z … … ……

Mx, My<< M0 Small deviations of M:

21 Magnetization at T>0 H z … … ……

Introduce complex combinations:

22 Magnetization at T>0: Exchange waves H z … … ……

A sum of the energies of plane waves – spin waves 23 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 24 Magnetization at T>0: magnons

Each magnon reduces the total magnetic moment of the sample by μ

Quadratic dispersion Bose-Einstein distribution

Bloch (3/2-) law: correct temperature dependence of magnetization at low T

25 Magnetic ordering, magnetic anisotropy and the mean-field theory

Antiferromagnets Neel temperature Susceptibility

26 Antiferromagnets

μ1i μ2i Magnetic sublattices:

M1=Σμ1i/V M2=Σμ2i/V

M=M +M Magnetization 1 2 (ferromagnetic vector)

L=M1-M2 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 27 Antiferromagnets: Neél temperature A B Molecular (Weiss) fields HA=wAAMA+wABMB for the case of two sublattices: HB=wBBMB+wBAMA H=0 wAA=wBB=w1 Equivalent sublattices: MA=-MB wAB=wBA=w2

Magnetization of a sublattice T in the presence of the molecular fields: H=0 1 T >T For each sublattice: N 1 T >T ) 2 C  ( S B Neél temperature:

 28 Antiferromagnets: susceptibility H H =w M +w M +H afm   par A AA A AB B HB=wBBMB+wBAMA+H

wAA=wBB=w1>0 Equivalent sublattices: MA=-MB wAB=wBA=w2<0 1   1 afm  para  k(T TA ) kT Asymptotic Curie temperature: 1 

1  ferro  k(T TC ) Susceptibility above TN:

TA TN TC T

29 Antiferromagnets: susceptibility H afm   par

wAA=wBB=w1>0 Equivalent sublattices: MA=-MB wAB=wBA=w2<0

Susceptibility in the Neél point: TN

Susceptibility at T=0: χ=0

30 Antiferromagnets: some examples

[P. W. Anderson, Phys. Rev. 79, 705 (1950)]

31 Magnetic ordering, magnetic anisotropy and the mean-field theory

Magnetic anisotropy Magnetic-dipole contribution Magneto-crystalline anisotropy and its temperature dependence

32 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 2D systems it would be absent [Mermin and Wagner, PRL 17, 1133 (1966)]

M Magnetization – axial vector

Sources of anisotropy: •Deformation •Intrinsic electrical fields •Shape etc…. - polar impacts

Magnetic anisotropy sets an axis, not a direction 33 Magnetic anisotropy: phenomenological consideration

Uniaxial anisotropy 2 4  E  K sin   K sin  ... M a u1 u2

Ku1  0 (001) – easy plane

Ku1  0 [001] – easy axis Effective anisotropy field E 2K H   a   u1 cosn ... a M M Cubic anisotropy

w  K  2 2   2 2   2 2  K  2 2 2  ... 3 a 1 1 2 2 3 1 3  2 1 2 3

K1  0 easy axes- {100}

K1  0 easy axes- {111} 34 Magnetic anisotropy

Compound erg cm-3 erg cm-3

Origins of magnetic anisotropy

•Single-ion anisotropy Spin-orbit interaction •Anisotropic exchange •Magnetic-dipolar interactions Interactions between pairs of magnetic ions 35 Magnetic-dipolar contribution to the anisotropy Cubic crystal For a pair of neighboring ions: E()  w ex Dipolar interaction M  2 1   l1    3

cos ϕ=α1  4 6 2 3   q1  1   ... x  7 35  Quadrupolar interaction 2 2 2 2 2 2 Ea  2Nq1  2  23 1 3  const

N – number of atoms per volume unit K1  2Nq – cubic anisotropy constant vcc: fcc: K  16/9Nq K  Nq 1 1 36 Magnetic-dipolar contribution to the anisotropy

 2 1   4 6 2 3  E()  l1    q1  1   ... cos ϕ=α  3   7 35 

Cubic crystal 2 2 2 2 2 2 Ea  K11  2  23 1 3 ...

Hexagonal crystal 2 4 Ea  Ku1 sin   Ku2 sin  ... θ Ku1 ~ l  q

Contributes to the shape anisotropy

This films demonstrate in-plane anisotropy37 Single-ion anisotropy (crystal-field theory)

Spinel CoFe2O4 [001] [111] Co2+ in the octahedral coordination + trigonal distortion along [111]

O2- Splitting of the 3d shell [100] In the ligand field:

x2-y2 ; z2 Co2+ 3d7 xy; xz; yz

38 Single-ion anisotropy (crystal-field theory)

[001] Co2+ in the octahedral coordination [111] + trigonal distortion along [111]

[100]

Splitting of the 3d shell In the ligand field + trigonal distortion

x2-y2 ; z2

7 3d Maxima of electronic density xy; xz; yz – in the plane (111) Maxima of electronic density – along the axis [111] 39 Single-ion anisotropy (crystal-field theory)

[001] Co2+ in the octahedral coordination [111] + trigonal distortion along [111]

Splitting of the 3d shell In the ligand field + trigonal distortion [100]

x2-y2 ; z2

7 3d Orbital moment xy; xz; yz L||[111]

Spin-orbit interaction wSO  L S  LS cos  -angle between S и [111] rd 2+ 7 3 Hunds rule: Co (3d )   0 40 Single-ion anisotropy (crystal-field theory)

[001] Co2+ in the octahedral coordination [111] + trigonal distortion along [111] wSO  L S  LS cos  -angle between S и [111]

[100]

Co2+ (3d7)   0

Averaging over 4 types of Co2+ positions with distortions along different {111} axes:

2 2 2 2 2 2 wa  NLS( x y   y z   z x )

K1  0 41 [Slonczewski, JAP 32, S253 (1961) ] Temperature dependence of magneto-crystalline anisotropy

Classical theory gives: n(n1) n   2 E  K  Kn (T) M S (T) a n    Kn (0)  M S (0)  Cubic anisotropy 10 [Zener, Phys. Rev. 96, 1335 (1954)] K (T)  M (T )  n   S    Exp: Kn (0)  M S (0)  Theory: Uniaxial anisotropy 3 Kn (T )  M S (T )     Fe Kn (0)  M S (0) 

Competition between different anisotropies gives rise to spin-reorinetation transitions 42 Magnetic ordering, magnetic anisotropy and the mean-field theory

Ferro- and antiferromagnets in an external field 43 Ferromagnets in applied magnetic field: domain walls displacement

Cubic anisotropy (K1>0)

[100]

44 TechnicalFerromagnets magnetizationin applied magnetic processes: field: magnetizationrotation of magnetization rotation e.a. M M e.a.

H H

Reversible rotation towards the field Irreversible rotation towards the field

45 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet

Χmax=-1/w2

H M

H =-w M H S 2 S 46 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet

If the anisotropy is weak: H Condition for the spin-flop H M S (spin-reorienetation transition) H

H 47 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet

If the anisotropy is weak: If the anisotropy is strong: H HS H M M Spin-flip C H H

H H

H 48H