遍歴電子磁性とスピン揺らぎ理論 - 京都大学大学院理学研究科 集中講義

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1 August 3, 2009

2 Title of Lecture: SCR Spin Fluctuation Theory 2 / 179

3 Part I Introduction Introduction Stoner-Wohlfarth Theory Stoner-Wohlfarth Theory Hatree Fock Approximation Stoner-Wohlfarth Free Energy Predictions of Stoner Wohlfarth Theory Comparison with Experiment 3 / 179

4 l. Introduction Outline of Lectures 8 3 : SCR Stoner-Wohlfarth SCR 8 3 : SCR 8 4 : 8 4 : 8 4 : 4 / 179

5 Main Theme of Lectures 10 ( ,2 ) SCR!! SCR The SCR spin fluctuation theory is not Self-Consistent / 179

6 Theoretical Development 1938 Stoner 1951 Wohlfarth 1972 Murata-Doniach 1973 Moriya-Kawabata (SCR) Unified Theory 1979 Takahashi-Moriya(FeSi ) 1980 Moriya-Usami ( ) 1985 Lonzarich-Taillefer Spin Fluctuations in Itinerant Electron Magnetism (Springer) 1986 T-induced Ferro (Moriya) 1986 Takahashi ( ) 1991 Yamada (metamagnetism) 1990 Takahashi ( ) 1998 FeSi ( ) 2001 Takahashi (, ) 2003 Takahashi ( ) 2006 Takahashi, Nakano ( ) 6 / 179

7 Magnetism in Metals : 2 ( ) : ( ) : ( )? 7 / 179

8 2. Stoner-Wohlfarth Theory 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 n n 8 / 179 N = k k n k n k = N 0 n + n

9 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 9 / 179

10 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 ) 10 / 179

11 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 11 / 179

12 Stoner-Wohlfarth Free Energy Stoner-Wohlfarth F(M,T) = F(0, 0) a(t)m 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 12 / 179

13 Basis of Stoner-Wohlfarth Theory Stoner-Wohlfarth 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) 13 / 179

14 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) 14 / 179 F 1

15 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(0,0), T c M 2 (H,T) = M 2 (0,0)[1 T 2 /Tc 2 ] + 2χ 0 H M2 (0,0) M(H,T) 15 / 179

16 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 16 / 179 H = a(t)m + b(t)m 3

17 Summary: Difficulties of Stoner Wohlfarth Theory (T > Tc ) χ(t) = M(T) H = 2χ 0T 2 c M 2 (0,0) 1 T 2 T 2 c ( ) T C : 17 / 179

18 Part II SCR Spin Fluctuation Theory Effects of Non-linear Mode-Mode Couplings Curie-Weiss Law of Magnetic Susceptibility Moriya-Kabata Theory Magnetic Excitations Effect of non-linearity SCR Spin Fluctuation Theory New Origin of Curie-Weiss Law 18 / 179

19 3. Effects of Non-linear Mode-Mode Coupling N 0 3 (gµb)2 S(S + 1) = k BTχ(T) χ(t) N0(gµB)2 S(S + 1) 3k BT p eff = N0p2 eff 3k BT : = p (S + 1)/S 1 p s (p s = gµ BS: ) CW ( p eff /p s 1 ) 19 / 179

20 Fluctuation-Dissipation Theorem ( ) H = H 0 M qh z q (t) Mq(t) z = dt χ zz (q,ω)e iω(t t ) H q (t ) χ zz (q,ω) = i [M z q (0),Mz q (t)] eiωt dt ( ) 20 / 179 {M z q (0),Mz q (t)} eiωt dt = coth(βω/2)imχ zz (q,ω)

21 General Form of Equal-Time Correlation Functions (T > T c ) S i S i = 3 N0 2 S q S q = 3 ( ) dω βω N0 2 2π coth Imχ(q, ω) 2 q 3k BT N 2 0 q dω π q Imχ(q, ω) ω = 3k BT N 2 0 χ(q, 0) q : coth(βω/2) 2/βω Kramers-Kronig : Reχ(q, 0) = 1 π Reχ(q, ω) = 1 π Z dω Imχ(q, ω ) ω dω Imχ(q, ω ) ω ω, Imχ(q, ω) = 1 Z dω Reχ(q, ω ) π ω ω 21 / 179

22 Origin of Curie-Weiss Law Heisenberg Model Curie-Weiss J k B T c 1/χ(q,0) J : χ(q,0) χ(t) S(S + 1) = S i S i = 3k BT N0 2 χ(q,0) 3k BT χ(t) N 0 q 22 / 179

23 Moriya-Kawabata Theory Curie-Weiss 1970 Curie-Weiss : SCR ( ) 23 / 179

24 Moriya-Kawabata Theory Curie-Weiss 1970 Curie-Weiss : SCR ( SW ) 2 4 ( ) 23 / 179

25 Band Splitting and Magnetic Ordering vs : =? order T c magnetic moment T c = T m? T T m SW T Stoner ( ) T c?? ( No) 24 / 179

26 Effect of Collective Excitations ( ) C = γt + bt 3 + γ ( ) (Landau ) 25 / 179

27 Spin Waves in Itinerant Magnets MnSi : T = 5 K 26 / 179

28 Effect of Fluctuations H(x,p) = p2 2m mω2 x 2 H(x,p) exp( ω/k B T) k B T log(k B T/ ω), (Classical) F(T) = ω 2 + k BT log(1 e ω/kbt ), (Quantum) x 2 = 1 [ ] 1 H = mω2 mω e ω/kbt 1 27 / 179 k BT mω 2 ( ω k B T)

29 Free Energy with Spatial Fluctuations Stoner-Wohlfarth (q = 0) Mq (q 0) Ψ[{M q },M,T] = F SW (M,T) + q 0 1 2χ 0 (q) M q M q M q : 1/χ 0 (q) ω 2 Curie-Weiss γ 28 / 179

30 Non-Linear Effect of Phonons ( ) ( ) F(T) = [ ] 1 2 ω qs + k B T log(1 e ω qs/k BT ), ω qs = v qs q qs ( ) ω(v ) : 1 2 mω2 (V )x 2 29 / 179

31 Free Energy of SCR Theory Stoner Wohlfarth Ψ[{M q },M,T] = F SW (M,T) + Φ({M q }) Φ({M q }) = 1 2χ q 0 (q) M q M q b {q i }M q1 M q2 M q3 M q4 + 1/χ(q) 30 / 179 ω 2 q χ 1 (q),

32 A Simple Example of the Effect of Nonlinearity F(x) F(x) = x^2 F(x) = x^ *x^4 F(x) = c + 2*x^ x : F(x) = a 0 x 2 + b 0 x 4 a eff x 2 a { a 0, x 2 0 a eff = a 0 + a, x 2 a x 2 = a 0 31 / 179

33 Thermodynamics of SCR Model : exp[ F(M,T)/k B T] = exp[ Ψ({M q })/k B T] {M q} = e F SW (M,T)/k B T exp[ Φ({M q })/k B T] {M q} H = F(M,T) M = 1 χ(t) M + b(t)m / 179

34 Variational Approach ( ) Φ({M q }) Φ 0 ({M q }) = q (Ω q M q 2 + Ω q M q 2 ) F FSW e βφ0({mq}) exp( β[φ Φ 0 ]) exp[ βφ({m q })] = {M q} {M q} = e 1 βf0 e βφ({mq}) exp(β[φ Φ 0 ]) Z {M q} = e βf0 exp[ β(φ Φ 0 )], Z = e βf0 = {M q} e βφ0({mq}) 33 / 179

35 Upper Bound of Free Energy F(Ω,Ω,M 0,T) = F SW (M,T) + F 0 + F exp[ F/k B T] = exp[ (Φ Φ 0 )/k B T] F Φ Φ 0 ( Ω q, Ω q, M ): F = F SW + F 0 + Φ Φ 0 e X = e x = 1 x x2 + = exp[ x ( x2 x 2 ) + ], x X 1 2 ( x2 x 2 ) 0 34 / 179

36 A Simple Example of Non-linear Model φ(x) = 1 2 ax bx4, e βf = dxe βφ(x) (SW : φ(x) ) F φ(x 0 ) = 1 2 ax bx4 0, ax 0 + bx 3 0 = 0 x = x0 + δx ( : ) φ(x) 1 2 ax bx aδx2, F = φ(x 0 ) + F 0 (a + 3bx0 2 ) F 0 (a) = 1 2 k BT log(2πk B T/a), e βf 0 = dxe βax2 /2 = 2π/βa 35 / 179

37 A Simple Example (2) (x = x 0 + δx) φ(x) φ(x0 ) + φ 0(x), φ 0 (x) = 1 2 a δx 2 e β[f φ(x 0 )] = e βf 0 e βf 0 dxe βφ0(x) e β[φ(x) φ(x 0 ) φ 0(x)] = e βf 0 e β[φ(x) φ(x 0 ) φ 0(x)] = e β(f 0+ F) = e βf0 dxe βφ0(x), e βf0 = ( dxe βa δx 2 /2 2πkB T = a ) 1/2 F = φ(x 0 ) + F 0(a ) + F, F = φ(x) φ(x 0 ) φ 0(x) 36 / 179

38 A Simple Example (3) F φ(x) δx φ(x) = a 2 (x 0 + δx)2 + b 4 (x 0 + δx)4 = φ(x 0 ) + a 2 δx2 + b 4 (6x 0 2 δx 2 + δx 4 ) + x 0 [a + b(x δx 2 )]δx δx F = φ(x) φ(x0 ) φ 0(x) x0, a F = a 2 δx2 + b 4 (6x 0 2 δx 2 + δx 4 ) a 2 δx2 δx 2 = k ( ) 2 BT a, kb T δx4 = 3 a 37 / 179

39 A Simple Example (4) F (φ(x 0 ), F 0 (a ), F ) F = 1 2 ax bx ( ) 2 k 2πkB T BT log a ( kb T a ) + 14 b [ 6x 0 2 ( kb T a a ) ( kb T + 3 a ) 2 ] a ( a = a + 3b x0 2 + k ) BT k B T a, a = δx 2 x0 [ ( x0 a + 3b x0 2 + k )] BT a = 0 38 / k BT

40 4. SCR Spin Fluctuation Theory Approximate Free Energy of SCR Model F = F SW + F 0 + Φ Φ 0 Ω q, Ω q F0 2 e βf 0 = X Z e βφ 0({M q}) = Π q dm qe βφ 0({M q}) = Π q {M q} F 0 = k B T X q "! 1 2 log πk B T Ω + log q!# πk B T Ω q 4 πk BT Ω q Φ 0 Φ 0 = ( ) Ω z q M q 2 + Ω q Mq 2 = 3 2 k BT q q 39 / 179 M q 2 = k BT 2Ω z, Mq 2 = k BT q Ω q! 1/2 1 = 3 2 N 0k B T! 3 πk B T 5 Ω q

41 Approximate Free Energy (2) Φ Φ = q 1 2χ 0 (q) M q M q b M q1 M q2 M q3 M q4 + {q i } M q1 M q2 M q3 M q4 = M0 4 + [ M2 0 2 Mq M q + 4 Mq z Mz q ] {q i } + q,q [ M q M q M q M q + 2 α M q 2 = k BT 2Ω z, Mq 2 = k BT q Ω q q M α q M α q M α q Mα q ] 40 / 179

42 Variational Minimum Conditions Ω 0 : F(Ω q,ω q,m 0,T) Ω q = 1 2χ q bm bt q Ω q 1 Ω q + 2 Ω q = Ω 0 + (Ω q Ω 0 ) = Ω Aq2 (T > Tc ): Ω q = Ω q = 0 Ω q Ω q = 1 2χ q bt q 1 Ω q bt q 1 Ω q Ω q : 41 / 179

43 Origin of Curie-Weiss Law in SCR Theory 1 2χ(0) = 1 2χ 0 (0) b p M p M p ( 2 ) χ(0) ( )λ(t) χ(q) = χ(0) 1 + q 2 /κ 2, κ2 = 1/λ 2 χ 1 (0) SCR (Self-Consistent Renormalization) : χ(0) 2 χ(0) 42 / 179

44 Time-Dependence of Order Parameter = W k B T (W : ) : W = kb Θ : W J kb T c : kb T W 43 / 179

45 Self-Consistent Equation T = T c χ 1 (0) = 0: 0 = 1 2χ 0 (0) b p M p M p (T c ) M p M p (T) = p p M p M p (T c ) b χ 1 (0) M p M p 0 dωn(ω)imχ(q, ω) ωγ q Imχ(q,ω) = χ(0) κ 2 + q 2 ω 2 + Γ 2, Γ q = Γ 0 q(κ 2 + q 2 ), (κ = 1/λ) q 44 / 179 κ 2

46 Numerical Examples 45 / 179

47 Comparison: SCR vs SW Theories SCR Stoner-Wohlfarth SCR Theory SW Theory (Boson) (Fermion) (T T c) (T 2 Tc 2 ) T/T c 1 M 2 Ms 2 T 2 M 2 = Ms 2 (1 T 2 /Tc 2 ) T/T c 1 M 2 (Tc 4/3 T 4/3 ) H = am + bm 3 (T > T c) SCR 46 / 179

48 Summary SCR 4 ( ) (2 4 ) ( ) ( ( ) ( ) : MnSi, Ni 3 Al, ZrZn 2, Sc 3 In 47 / 179

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