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 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: 4.11021J H Potential energy of a magnetic moment U H cos in the field H: H In a field of 1MA/m: 1.2 1023 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 Ncoth kT Langevine function L 5 Paramagnetism: independent magnetic moments in external field Full magnetization along the field: 1 MH M Ncoth kT H At RT in the field of 1МА/м: 0.003 N 2 M NL() 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 gB J z B kT J z J, J 1.... J H Gd3+ Resulting magnetization along the field: Fe3+ M NgJB BJ Brillouin function Cr3+ J 1 At small : M NgJB ... 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 gB 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 gB NJ S g S H z B g B Self-consistent solution Sz SBS kT 8 Back to ferromagnets: Weiss molecular field H H=0 S J ) >( z < S Sz SBS () g S(H H ) B m kT 2 non-zero solutions: Sz kT gB Sz H 2 opposite orientations NJS NJ =0 of the order parameter 9 Ferromagnets (S=1/2): Curie temperature H T1 J H=0 S TC>T1 T2>TC ) ( S B Variations of these curves with T: Sz SBS () kT gB Sz H NJS NJ =0 JN Ordering temperature:T C k Sz changes continuously with T At T>TC Sz 0 nd 2 order phase transition 10 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 gB 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 (S=1/2) in applied field H H 0 TC S J ) >( z < S Sz SBS () g S(H H ) B m kT Non-zero magnetization above TC Susceptibility H 1 Above T S B Curie-Weiss C z T TC k(T TC ) k(T TC ) law in small fields 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)|<<M0 Small deviation of M from M0 Characteristic period 2 2 of magnetization variation λ (M(r)) =M0 =const Length of M is conserved λ>> 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: M0 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 TN [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 cosn ... 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 l1 3 cos ϕ=α1 4 6 2 3 q1 1 ..