¼§À�ÍýÏÀ – Ê×ÎòÅÅ»Ò¼§À�¤È¥¹¥Ô¥ó¤æ¤é¤®

Size: px

Start display at page:

Download "¼§À�ÍýÏÀ – Ê×ÎòÅÅ»Ò¼§À�¤È¥¹¥Ô¥ó¤æ¤é¤®"

あやかこやぎ
4 years ago
Views:

1 Spring semester, 2012

2 Outline 1. 2 / 26

3 Introduction : (d ) : 4f 1970 ZrZn 2, MnSi, Ni 3 Al, Sc 3 In Stoner-Wohlfarth Moriya-Kawabata (1973) 3 / 26

4 Properties of Weak Itinerant Ferromagnets ps Tc Arrott (M 2 vs H/M) Curie-Weiss p s p eff 4 / 26

5 Heisenberg Model H = <i,j> J ij S i S j S = (S x.s y, S z ) S 2 = S(S + 1): S : M = gµb i Sz i 5 / 26

6 A Magnetic Moment in the Magnetic Field (Zeeman) H z = µ H = µ z H, µ = gµ B J Z = J m= J e gµbmh/kbt = e gµbjh/kbt [1 + e gµbh/kbt + + e 2gµBJH/kBT ] = e gµ BJH/k B T [1 e gµ B(2J+1)H/k B T ] 1 e gµ BJH/k B T = sinh[(2j + 1)gµ BH/2k B T ] sinh(gµ B H/2k B T ) F = k B T log Z(H, T ) 6 / 26

7 Curie Law of Magnetic Susceptibility m = gµ B J z = F H = gµ BJB J (x), x = gµ B JH/k B T B J (x) = {(1 + 1/2J) coth[(1 + 1/2J)x] (1/2J) coth(x/2j)}, J + 1 x, (x 1) 3J χ(t ) χ(t ) = m H = (gµ B) 2 J(J + 1) 3k B T = (µ B) 2 p 2 eff 3k B T 7 / 26

8 Molecular Field Approximation : H m = i J J z j H eff = H + J Jj z = H + ζj gµ B N(gµ B ) 2 M, j M = gµ B Ji z i 8 / 26 M N(gµ B) 2 J(J + 1) J(J + 1) H m = 3k B T 3k B T [N(gµ B) 2 H + ζj M]

9 Magnetic Properties of Localized Moment Systems Curie-Weiss χ(t ) = N(gµ B) 2 J(J + 1) J(J + 1)ζJ, T C = 3k B (T T C ) 3k B M(T ) = Ngµ B JB J (x eff ), x eff = gµ B JH eff /k B T = gµ B J(H + ζj M/[N(gµ B ) 2 ]) M(0) Nµ B p s = Ngµ B J, (T 0) p eff J(J + 1) = = p s J J 9 / 26

10 Magnetic Free Energy F (H, T ) F (M, T ) = F (H, T ) + MH, M = H F (M, T ) = F (0, T ) a(t )M b(t )M4 + M H = F M = a(t )M + b(t )M3 + χ(t ) Tc C peff 10 / 26

11 Weak Itinerant Ferromagnets T c (K) p s p eff p eff /p s MnSi Ni 3 Al Sc In ZrZn Zr 0.92 Ti 0.08 Zn Zr 0.8 Hf 0.2 Zn Table: M 0 (0) = Nµ B p s : (T = 0) 11 / 26

12 Rhodes-Wohlfarth Pc/Ps 14 (FeCo)Si 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 Tc(K) Ni Fe 1000 Figure: Rhodes-Wohlfarth 12 / 26

13 Arrott Plot of Magnetization Curve H = a(t )M + b(t )M 3 + = b(t )M[M 2 M 2 0 (T )] + M 2 = a(t ) b(t ) + 1 b(t ) H M, a(t ) : a(t ) (T 2 c T 2 ) b(t ) H M = b(t )[M2 M 2 0 (T )] + 13 / 26

14 Observed Magnetization Curves ZrZn 2 Figure: ZrZn 2 Arrott 14 / 26

15 Observed Magnetization Curves Sc 3 In 14 / 26 Figure: Sc 3 In Arrott

16 Temperature Dependence of Magnetization ZrZn 2 15 / 26 Figure: M 2 vs T 2

17 Temperature Dependence of b(t ) 4 b(t ) 1/F (T ) 60 F(T) (arbitraryunits) T 2 c T 2 (K 2 ) Figure: F (T ) vs T 2 for (ZrTi)Zn 2 16 / 26

18 Temperature Dependence of b(t ) 4 b(t ) 1/F (T ) 55 T 2 c 50 F(T)(arbitraryunits) T 2 (K 2 ) Figure: F (T ) vs T 2 for ZrZn / 26

19 Localized vs Itinerant Magnets M/(N 0 µ B ) 1 Arrott T 3/2 T 2 χ(t ) p eff /p s / 26

20 Stoner-Wohlfarth Theory Stoner-Wohlfarth : N N T 2 ( ) 18 / 26

21 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 N = k k n k n k = N 0 n + n 19 / 26

22 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 20 / 26

23 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 = f (ε kσ ) = dερ(ε)f (ε + σ ) kσ σ M(h, µ, T ) = F h = 1 σf (ε kσ ) 2 kσ = 1 dερ(ε)[f (ε + ) f (ε )] 2 ρ(ε) ρ(ε) = k δ(ε ε k ) 21 / 26

24 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 3. (or M) dερ(ε) + n=1 a n (kt ) 2n ρ (2n 1) (µ) 22 / 26

25 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 23 / 26

26 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 ) = F 1 2ρ 3 R = ρ 2 /ρ 2 ρ /ρ +, F 1 = ρ 2 /ρ 2 ρ /3ρ : ρ ε F 24 / 26

27 Summary of Stoner-Wohlfarth Theory F (M, T ) = F (0, T ) a(t )M b(t )M4 + H = a(t )M + b(t )M 3 + a(t ) = a 0 + a 2 T 2 +, b(t ) = b 0 + b 2 T 2 + H = M[a(T ) + b(t )M 2 + ] = 0, (H = 0) M 2 0 (0) = a 0 b 0, M 2 0 (T ) M 2 0 (0) a 2 b 0 T / 26

28 Difficulties of the Theory Magnetic susceptibility χ(t ) = 1 a(t ) 1 T 2 T 2 c χ(t ) C T T c 26 / 26

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論 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 +