Magnetism. Andreas Wacker Mathematical Physics Lund University

Transcription

1 Magnetism Andreas Wacker Mathematical Physics Lund University

2 Overview B=μ0(H+M) B: Magnetic field (T), satisfies div B=0 M: Magnetization (density of magnetic moments) H: H-field (A/m), satisfies curl H=jfree (+Ḋ) μ0 =4π x 0-7 Vs/Am vacuum permeability Relate M to material properties Often M = B 0 χ>0: paramagnetic χ<0: diamagnetic M for vanishing B: ferromagnetic Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

3 Magnetic moment μ= d 3 r r j (r ) μ For loop μ = IA I Energy in a magnetic field E = B q 3 = d r n r r v Object with mass m and charge q with identical distribution n(r) ( p q A) m Quantum mechanics: L = l ħ v= Bohr magneton μ= q 3 L d r q n(r)r A(r ) ] [ m eℏ ev B= =57.88 me T Electron spin ħs: = g e B s Landé factor ge= Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

4 Susceptibility from independent electrons p e A ² L S Hamiltonian H = V r g e B s B with l =, s = me ℏ ℏ constant B= B e z use A= B r= B x e y B y e x = p V r B B l z g e s z e x y B H me 8 me H e μ z= = μ B l z + ge s z x + y B Bz 4m e System with known paramagnetic diamagnetic eigenstates Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

5 Larmor diamagnetism Expectation value of magnetic moment e μ z= μ B l z +g e s z x + y B 4m e Ground state with zero angular momentum: Density n of electrons gives magnetization: e x y M =n = n B 4 me = / 0 negative Order of magnitude: ~ -0-5 Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

6 Susceptibility from independent electrons p e A ² L S Hamiltonian H = V r g e B s B with l =, s = me ℏ ℏ constant B= B e z use A= B r= B x e y B y e x = p V r B B l z g e s z e x y B H me 8 me H e μ z= = μ B l z + ge s z x + y B Bz 4m e System with known paramagnetic diamagnetic eigenstates Atom/Ion with angular momentum j (ls) Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

7 Paramagnetism due to thermal orientation of spins p = H + V (r )+ μ B B ( l z + g e s z ) me Atom with spin, eigenvalues m s=±/ μ = μ l +g s z B z e Energy E 0 ±g e B B / z ( ) gμ B B gμ B B canonical distribution s z = tanh kb T 4 kb T Density n of atoms gives magnetization: n g e μ B M =n( ) g e μ B s z = B 4 kbt = / 0 Curie law: ~ T Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

8 Susceptibility from independent electrons p e A ² L S Hamiltonian H = V r g e B s B with l =, s = me ℏ ℏ constant B= B e z use A= B r= B x e y B y e x = p V r B B l z g e s z e x y B H me 8 me H e μ z= = μ B l z + ge s z x + y B Bz 4m e System with known paramagnetic diamagnetic eigenstates Crystal with Bloch waves k Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

9 Pauli paramagnetism for spins in metals Equilibrium for B=0 Levels are spin-degenerate Conduction band filled until Fermi level Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

10 Pauli paramagnetism for spins in metals Equilibrium for finite B Levels E / =E0±gμBB/ Conduction band filled until Fermi level g B B n n = D E F g B B n n = D E F g B g B M= n n = D E F B 4 = / 0 independent on temperature Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

11 Summary Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

12 Main trends Any material has a diamagnetic component of the order χ~-0-5 Atoms/compounds with finite magnetic moment (e.g. O, transition elements with partially filled d,f shells) are paramagnetic with χ~+/t, at room temperature up to χ~+0-3 Metals exhibit a less pronounced paramagnetism (independent on temperature) or are diamagnetic χ~±0-5 (record Bi χ=-6.6*0-5) Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

13 The fun part or on the way to a Nobel price Energy in magnetic field E= μ B= VMB=V χ B /μ 0 Levitating frog at 6 T A. Geim 997 IG Nobel price Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

14 Interacting spins Interaction between magnetic moments E=μ 0 μ μ 4 π d3 μ 0=4 π 0 7 Vs/Am, μ B=58 μ ev/t and T=N/Am Provides only 50 μev for d=0.nm The electromagnetic interaction is too small on atomic scale The interaction is resulting from the exchange term of Coulomb interaction Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

15 One-Particle Quantum Mechanics State described by wave function Ψ(r,t) Spin function a b a r,, t Combine r, t = b r,, t Ψ(r,s,t) d3r: probability to find particle with spin s at time t in volume d3r around r Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

16 Two-Particle Quantum Mechanics Two-particle state wave function Ψ(r,s;r,s;t) Ψ(r,s,r,s,t) d3rd3r: probability to find at time t particle with spin s in volume d3r around r and particle with spin s in volume d3r around r p p = H V A r, S V B r, S V AB r, r ma mb i ℏ r, s ; r, s ; t = H t Simple states: Product states product r, s ; r, s = a r s b r s Note: Only few states are product states! Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

17 Symmetry for identical particles p p (,)= H +V A ( r, S )+ +V B ( r, S )+V AB ( r, r ) ma mb Identical particles: m A =m B, V A =V B, V AB=V BA Symmetry for permutation of particles: H, = H, Wavefunction can be chosen symmetric or antisymmetric Symmetry postulate for electrons (and other fermions) States are antisymmetric in particle indices r, s ; r, s ;t = r, s ; r, s ; t Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

18 Product states become Slater determinants a r, s Eigenstates of single-particle Hamiltonian Slater r, s ; r, s = a r s b r s a r s b r s ] [ a r s b r s =! a r s b r s Note: Only few states are Slater states! But form a ON basis for the space of antisymmetric many-particle states No state with a=b and α=β Pauli principle Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

19 Symmetry of wave function modifies Coulomb repulsion: Exchange interaction Pair correlation: P (r, r )= ψ(r, s ; r, s ) s s States: ϕ a (r )= i k r i k ' r e with spin α, ϕ b (r)= e with spin β V V P (r, r )= cos(k k ' ) (r r ) χ α (s)χ*β (s) V s ( Coulomb energy Ψ ab e Ψ ab 4 π ϵ0 r r ) = P (r, r )e UV c IV c 3 3 = d r d r = δ α,β 4 π ϵ0 r r V V Direct interaction Exchange Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

20 Ferromagnetism of some metals Reduced Coulomb repulsion for parallel spins but requires occupation of Bloch states with higher energy EF for Cu Parallel spins are energetically favorable if I D(E F )V c > Large density of states in d-bands provides ferromagnetism in Fe,Co, Ni From NIST Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

21 Singlet and Triplet Symmetrize spatial and spin part separately Normalization positive if exchange dominates Effective spin-spin interaction Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

22 The Heisenberg Model H Heisenberg= (n, m) J S n S m + g μ B B S n n (n,m): only neighboring sites are included, each pair twice Effective Hamiltonian to describe excitations of spins for fixed spatial states good for transition elements with partially filled f-shells From NIST Spectroscopic Notation S+LJ Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

23 Mean (or Molecular)-field Approximation H Heisenberg= (n, m) J S n S m + g μ B B S n n Rewrite S n S m= S n S n S m S m S n S m S n S m S n S m small correlation neglect J J MF H Heisenberg = gμ B Sn B Sn+δ + Sn S m δ g μb n (n, m) ( ) = B eff n Single spins in an effective magnetic field Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se / constant

24 Solution for spin / J eff eff MF H = gμ S B with B =B g μ Sn+ δ Heisenberg B n n n δ n B eff g μ B B n Assume B eff e S = tanh n z nz kbt ( ) Same result for all sites n, ν neighbors (n+δ) z J S ν J S z eff Self consistency: B n = B e z = B ez g μb g μb δ ( ) ( g μ B B ν J S z g μ B B ν J S z S z + kbt 6 kbt ( ) ( ) 3 ) Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

25 Linear term provides susceptibility g μ B B ν J S z g μ B B ν J S z S z + kbt 6 kbt ( ) ( 3 ) g μ B S z g μ B J M= B with T c= 4kB Vc 4Vc k B (T T c ) = diverges at Tc 0 Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

26 Experimental data Iron has Tc=043 K ~ T T c.33 from Landoldt Börnstein Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se /

27 Finite magnetization at vanishing B field for T<Tc g μ B B ν J S z g μ B B ν J S z S z + kbt 6 kbt ( ) ( Include cubic term, B=0: g μ B S z M= Vc g μb T c T / ± 3 Vc Tc ( ) Lund university / Science Faculty / Mathematical Physics / Andreas.Wacker@fysik.lu.se / )