email: takahash@sci.u-hyogo.ac.jp April 30, 2009
Outline 1. 2. 3. 4. 5. 6. 2 / 260
Today s Lecture: Itinerant Magnetism 60 / 260
Multiplets of Single Atom System HC HSO : L = i l i, S = i s i, J = L + S H SO : L, S, S 2, L 2 = E(L,S), (2L + 1)(2S + 1) H SO ( ) : J, J 2, S 2, L 2 = E(J,L,S), (2J + 1) 61 / 260
Magnetism of Atoms ( ) Hund S ( l 1 ) L J 62 / 260
Origin of the Hund s Rule 2 φ a, φ b 2 2 Ψ S (r 1,r 2 ) = 1 2 [φ a (r 1 )φ b (r 2 ) + φ b (r 1 )φ a (r 2 )], (S = 0) Ψ A (r 1,r 2 ) = 1 2 [φ a (r 1 )φ b (r 2 ) φ b (r 1 )φ a (r 2 )], (S = 1) Ψ A r 1 r 2 ( S = 1, S = 0) 63 / 260
Magnetic Ions in Crystals 1. : ( ) 2. Hund 3. 64 / 260
Magnitudes of Various Interactions ( 1 cm 1 = 1.4388 K) 3d 4f 5f 10 4 10 5 10 4 10 5 10 4 10 5 10 2 10 3 10 3 3 10 3 10 4 10 10 2 10 2 10 3 d: 4f: 5f: 65 / 260
Physical Units in Magnetism M : µb H : MH χ : M = χh µ B p = M/N A µ B [ ] h = 2µB H [ ] χ = χ/(na µ 2 B ) [ ] 66 / 260
Stoner-Wohlfarth Theory Stoner (1938) Wohlfarth (1951) Band : Stoner (T = 0) 67 / 260
Magnetic Susceptibility of Simple Metals : χ ρ(εf ) 1 10 4 [K 1 ] E F : EF k B T 1 : χ J : J kb T c k B T 68 / 260
Heitler-London Model Chemical Bonding of a Hydrogen Molecule 2 e +e +e e H = H 1 + H 2 + V H 1 = 1 e 2 2m p2 1 r 1 R e 2 a r 1 R b, H 2 = 1 e 2 2m p2 2 r 2 R e 2 a r 2 R b V = e 2 r 1 r 2 69 / 260
Quantum Mechanics of Many Particle Systems ( ) Ψ(r,r ) = Ψ(r,r) Slater 2 : c νσ, c νσ c µσc + c c νσ νσ µσ = δ µνδ σσ Slater 2 ˆΨ(r) = P νσ 70 / 260 0 Ψ(r)Ψ(r )c β c α 0 = φ α(r)φ β (r ) φ β (r)φ α(r ) φν(r)cνσ: ( r )
Reference: Second Quantization 1 φ α A B n α = 1 n β = 1 φ β φ α A φ β B = φ α, φ β n α = 1, n β = 1 ( ): c α: c β : φ α φ β φ α φ β 71 / 260
Second Quantized Hamiltonian 2 1 Ĥ = Ĥ0 + Ĥ1 Ĥ 0 = [ ] 1 d 3 rψ σ (r) 2m p2 + V(r) Ψ σ (r) σ Ĥ 1 = 1 d 3 r d 3 r Ψ σ 2 (r)ψ σ (r )v(r r )Ψ σ (r )Ψ σ (r) σ,σ Heitler-London V (r) = e2 r R a e2 r R b, v(r r ) = e2 r r 72 / 260
Approximate Hamiltonian 2» 1 2m p2 e2 φ i(r) = ε 0φ i(r), (i = a, b) r R i 6 Slater 4, φ i (r)α, φ i (r)β, 2 : ˆΨ σ (r) φ a (r)c aσ + φ b (r)c bσ Ĥ = [ ] ε 0 (ˆn aσ + ˆn bσ ) + (t ba c bσ c aσ + h.c.) + U[ˆn a ˆn a + ˆn b ˆn b ] σ 2 : 3 : 73 / 260
Limit of Strong Correlation 6 2 S = S 1 + S 2 S = 1 3 (E = 2ε 0 ) c a c b 0, 1 2 (c a c b + c a c b ) 0, c a c b 0 S = 0 1 (E = 2ε 0 + U) 1 2 (c a c a c b c b ) 0 2 1 Ψ 1 = 1 2 (c a c b c a c b ) 0, Ψ 2 = 1 2 (c a c a + c b c b ) 0 74 / 260
Energy Levels in the Limit of Strong Correlation 2 1 Ψ 1, Ψ 2 ĤΨ 1 = 2ε 0 Ψ 1 + 2tΨ 2 ĤΨ 2 = (2ε 0 + U)Ψ 2 + 2tΨ 1 (2ε 0 E)(2ε 0 + U E) 4t 2 = 0 { E = 2ε 0 +U/2± U 2 /4 + 4t 2 2ε0 + U + 4t 2 /U, + 2ε 0 4t 2 /U, 75 / 260
Limit of Weak Correlation U = 0 ( ) c ±σ = 1 (c aσ ± c 2 bσ ) c +σ c +σ = 1 2 (c aσ c aσ + c bσ c bσ + c aσ c bσ + c bσ c aσ) ˆn +σ + ˆn σ = ˆn aσ + ˆn bσ, ˆn +σ ˆn σ = c aσ c bσ + c bσ c aσ Ĥ 0 = σ = σ [ε 0(ˆn +σ + ˆn σ ) + t(ˆn +σ ˆn σ )] [ε +ˆn +σ + ε ˆn σ ], (ε ± = ε 0 ± t) 76 / 260
Energy Schemes in Two Opposite Limits U 2ε 0 + U + E 2ε 0 + U 2t 2(ε 0 t) ε 0 2ε 0 2ε 0 E 2(ε 0 + t) ( E t2 U ) (t < 0 ) 77 / 260
Model of Itinerant Electron Magnetism Hubbard Model: H = kσ = kσ t ij c iσ c jσ + U i ε k c kσ c kσ + U i n i n i M z B, n i n i M z B M z = 2µ B S z, S z = i s z i (2µB ) M = 1 n k n k = N 0 2 2 n n N = k k n k n k = N 0 n + n 78 / 260
Hartree-Fock Approximation U i n i n i = U iσ (n i n + n i n n n ) = U kσ n kσ n σ N 0 U n n H = ( (ε kσ µ)c N 2 ) kσ c kσ I 4 M2, (I = U/N 0 ) kσ ε kσ =ε k + IN/2 σ, = IM + h/2 79 / 260
Free Energy and Thermodynamic Relations F(h,µ,T) = IM 2 + F 0, F 0 = kt kσ ln(1 + e β(ε kσ µ) ) ( ) N(h, µ,t) = F µ = X f (ε kσ ) = X Z dερ(ε)f (ε + σ ) kσ σ M(h, µ,t) = F h = 1 X σf (ε kσ ) 2 kσ = 1 Z dερ(ε)[f (ε + ) f (ε )] 2 ρ(ε) ρ(ε) = X k δ(ε ε k ) 80 / 260
Free Energy as a Function of Magnetization (Legendre ) F(M,N,T) = F(h,µ,T) + hm + µn µ(m,n,t), h(m,n,t) N, M F(M,N, T) N F(M,N, T) M ««F(h, µ,t) µ F(h, µ,t) h = µ + + N µ N + + M h N = µ F(h, µ,t) = h + µ = h ««µ F(h, µ,t) h + N M + + M h M 81 / 260
Free Energy in the Ground State { µ 0 + δµ +, (for Majority Spin) µ 0 = µ 0 + δµ, (for Minority Spin) = IM + h/2 δµ, N = 2M = µ0 +δµ+ µ0 +δµ µ0 ρ(ε)dε + ρ(ε)dε = 2 ρ(ε)dε µ0 +δµ+ µ 0 +δµ dερ(ε) = ρ 0 [(δµ + ) (δµ )] 82 / 260
Free Energy and Equation of State δµ M 2ρ 0 δµ + ρ 0 2 + = 0 ( ρ 2 0 ρ 2 0 = 1 ρ 0 M + 1 2ρ 3 0 ρ 0 3ρ 0 ) M 3 + = IM + h 2 ( ) h 1 2 = I M + 1 ρ 0 F(M,0) = F(0,0) + 2ρ 3 0 ( 1 ρ 0 I ( ρ 2 0 ρ 2 0 ρ 0 3ρ 0 ) M 2 + 1 4ρ 3 0 ) M 3 + ( ρ 2 0 ρ 2 0 ) ρ 0 M 4 + 3ρ 0 83 / 260
Temperature Dependence dερ(ε)f (ε) = µ dερ(ε) + π2 µ0 3 ρ (µ)(kt) 2 + = dερ(ε) ρ 0 δµ(t) + π2 3 ρ 0(kT) 2 +, δµ(t) = π2 ρ 0 (kt) 2 + 3 ρ 0 2M = 2 [ρ(µ 0 + δµ) + π2 = 2 ρ 0 [ 1 π2 3 ( ρ 2 0 ρ 2 0 3 ρ (µ 0 + δµ)(kt) 2 + ) ] (kt) 2 + + ρ 0 ρ 0 ] + O( 3 ) 84 / 260
Reference: Sommerfeld Expansion Sommerfeld f (x) Z dxf (x)g(x) = = Z µ Z µ dxg(x) + X n=1 g n(kt) 2n 2n 1 x 2n 1 G(x) x=µ dxg(x) + π2 6 (kt)2 G (µ) + 7π4 360 (kt)4 G (µ) + g n = (2 2 2(n 1) )ζ(2n) 85 / 260
Stoner-Wohlfarth Free Energy Stoner-Wohlfarth(SW) F(M,T) = F(0, 0) + 1 2 a(t)m2 + 1 4 b(t)m4 + a(t) = 1 ρ I + π2 R 6ρ (kt)2 +, b(t) = F1 2ρ 3 R = ρ 2 /ρ 2 ρ /ρ +, F 1 = ρ 2 /ρ 2 ρ /3ρ H = F M = a(t)m + b(t)m3 + : ρ ε F 86 / 260
Basis of Stoner-Wohlfarth Theory SW E band + E Coulomb 1. ε kσ = ε k σ, = µ B H + IM, (I = U/N) 2. : Fermi Sommerfeld dερ(ε)f (ε) = µ dερ(ε) + n=1 3. (or M) a n (kt) 2n ρ (2n 1) (µ) 87 / 260
Predictions by SW theory Stoner-Wohlfarth (T < T c ) : H = F M = a(t)m + b(t)m3 + : Iρ(εF ) > 1 (Stoner ) T = 0 a(0) < 0 Tc : a(t c ) = 0 [ ] 6(Iρ 1) 1/2 kt c = π 2, a(t) = a(0)(1 T 2 /Tc 2 ) R M0 (T = 0): H = 0 a(0)m + b(0)m 3 = 0 [ ] a(0) 1/2 [ ] 2(Iρ 1) 1/2 M 0 = = ρ T c b(0) 88 / 260 F 1
Origin of Band Splitting 1. E band = + Nm2 ρ(ε F ) : m = 0 m > 0 2. E Coulomb : 89 / 260 E Coulomb = Un n = U(n 2 /4 m 2 ), E Coulomb = Um 2
Band Splitting: Schematic Example : U > ρ(ε F )/N (Stoner ) 90 / 260
Magnetic Isotherm : H = 0 [ ] a(t) 1/2 M(T) = = M 0 [1 T 2 /Tc 2 b(t) ]1/2 M 2 (H,T) = a(t) b(t) + 1 b(t) H M(H,T) M 2 (H,T) = M 2 (0,0)[1 T 2 /Tc 2 ] + M 2 2χ 0 H (0,0) M(H,T) 91 / 260
Characteristic Properties of Itinerant Magnets M/(N 0 µ B ) 1 Arrott T 3/2 T 2 χ(t) CW CW p eff /p s 1 1 Arrott : M 2 H/M 92 / 260 H = a(t)m + b(t)m 3
Experimental - Magnetic Isotherm Arrott Plot: M 2 vs H/M Sc 3 In: Takeuchi, Masuda (1979) ZrZn 2 : Ogawa (1968) 93 / 260
Experimental - Magnetic Moment : T 2 ZrZn 2 : Ogawa (1972) 94 / 260
Rhodes-Wohlfarth Plot T c Pc/Ps 14 (FeCo)Si 12 10 8 6 4 2 Pd-Fe Pd-Co (FeCo)Si Pd-Rh-Fe Sc-In Pd-Ni (FeCo)Si Pd-Ni Pd-Co Pd-Fe Pd-Fe Pd-Fe Pd-Co Ni-Cu Ni-Pd Pd-Cu CoB CrBr 3 EuO Gd MnB MnSb FeB 0 0 200 400 600 800 Tc(K) Ni Fe 1000 p C (p C + 2) = p 2 eff, χ(t) = N 0(gµ B) 2 p 2 eff/3k B(T T c) 95 / 260
Magnetovolume Effect (ω = δv /V ) F(M,T,ω) = V 2κ ω2 + F(0,T,ω) + 1 2 a(t,ω)m2 + 1 4 b(t,ω)m4 a(t,ω) = a(t,0) Cω +, C = 1 a 2 ω ( ) ω = κ V CM2 = κ V CM2 0 (T) + κ V C[M2 M 2 0 (T)] T > T c ω = 0 ( ) 96 / 260
Volume Dependence of Magnetism T c : a(t c,ω) = 0 a T δt c + a ω ω = a (T c,0)δt c 2Cω = 0 δt c = (2Cω)/a (T c,0) (T = 0) : a(0,ω) + b(0,ω)m 2 s = 0 2Cω + 2b(0,ω)M s δm s = 0, M s = M 0 (0) δm s = (Cω)/b(0,ω)M s 97 / 260
Summary: Success of SW Theory Stoner-Wohlfarth Arrott M 2 = a(t) + b H/M M(T) M(0) T 2 4 b(t) b(0) T 2 : M 2 98 / 260
Summary: Difficulties of the Theory (T > Tc ) χ(t) = M(T) H = 2χ 0T 2 c M 2 (0,0) 1 T 2 T 2 c T C 99 / 260