3．交換相互作用 1．モット絶縁体、ハバード・モデル 2．交 …

1 Covalency).. -1 .. tight binding model). odd number of electrons : metals even number of electrons: insulators or (semi)metals electronic correlation).. w=zt << U . U.. t H = t r cr r c r r r i i + j , j , . + U nir nir zt 1 : Mott transition r i,j n U. nir , = c ir , cir , . -2 .. b b 1 r r . a a H = K ab J ab + 2 s1 s2 . 2 . e2 r r J ab = a (1) b (2) a (1) b (2)dr1dr2. r12. transfer . Pauli .. 2 . f Heff m m Heff i f Heff i = 2t 2 1 r r . Ei Em Heff = 2S1 S 2 . m U 2 . Heff t2. (. = a1 a2 a2 a1 + a2 a1 a1 a2 . U , . ) .. 4t 2. a1 a2 a2 a1 = (a1 a2 + a1 a2 )(a2 a1 + a2 a1 ). J=. U. , . = a1 a2 a2 a1 = a1 a1 a1 a1 a2 a2 = n1 n1 n2 . + a1 a2 a2 a1 a1 a1 a2 a2 S1+ S 2 . + a1 a2 a2 a1 a1 a1 a2 a2 S1 S 2+. + a1 a2 a2 a1 + a1 a1 a1 a1 a2 a2 + n1 n1 n2 . = (n1 + n1 ) (n1 + n1 )(n2 + n2 ) (n1 n1 )(n2 n2 ) S1+ S 2 S1 S 2+. 1 1. 2 2. z z 2 S 2 S. 1 r r 1 2. = 2S1 S 2. 2. -3 covalency).. x2. -bond y2 -bond d x2 y2.

2 -y3 d xy y1. x3 -x1. -y4. -x4. d x2 y2 d xy p =. 1. ( x1 + y2 + x3 y4 ) p =. 1. ( y1 + x2 y3 x4 ). 2 2. = d x 2 y 2 + p = d xy + p . p-d . Ed d x2 y2 = d x 2 y 2 + p . t Ed t . p H = . E p . Ep t (Ed E ) (E p E ) t 2 = 0. t2. Ed + E p (Ed E p ) + 4t 2 Ed Ed E p E=. 2. Ep t2. Ed + anti-bonding AB = d x 2 y 2 p . Ed E p = 2 ( t << Ed E p ) t E t =. p Ed E p bonding B = p + d x 2 y 2 Ed E p .. 2 t2. p f p = =. 4 (. 4 Ed E p ). 2. NMR . 19F NMR in KMnF3, KNiF3, K2 NaCrF6. (R. G. Shulman and S. Sugano, Phys. Rev. 130 (1963) 503.). F . d- . p .. s . Fermi contact field H res KNiF3. KMnF3. =60 MHz .. H0 = , H res (1 + K ) = H 0. N. H 0 H res K=. H res H res = MHz, H 0 = G. K2 NaCrF6. z d- . 1 3 z j 1 . [ ]. 2. Hd = h N 2 Ad S z I z Ad S x I x Ad S y I y , Ad = g B 3.. j 2 rj 5. r j . rd (. H hf = Ad S x , Ad S y , 2 Ad S z ). Ad H 0. = ( sin cos , sin sin , 2 cos ). g B. r r H H A . K d = hf 2 0 = d 3 cos 2 1.

3 H0 g B. ( ). 2p . z pz [ ]2. Hp = h N 2 Ap S zp I z Ap S xp I x Ap S yp I y , Ap = g B r 3. 5 2p rp r r (. S = 2 f S for p z H hf ( p z ) = 2 Ap f S x , Ap f S y , 2 Ap f S z ). rp S = 2 f S. r r (. H hf ( p x ) = 2 2 Ap f S x , Ap f S y , Ap f S z ). ( p ) = 2( A f ). px, py r H hf y p S x , 2 Ap f S y , Ap f S z for p x and p x rp (. H hf = 2 Ap ( f f ) S x , Ap ( f f ) S y , 2 Ap ( f f ) S z ). 2 Ap ( f f ) . Kp =. g B. (3 cos 1). 2. z 2s . [ ]. Hs = h N As S xs I x + S ys I y + S zs I z , As =. 8 . 3. g B 2s (0 ). 2. rs r S = 2 fs S. rs (. H hf = 2 As f s S x , Sy , Sz ). 2 As f s . Ks =. g B. K = Kd + K p + Ks =.. g B. [ (. 2 As f s + {Ad + 2 Ap ( f f )} 3 cos 2 1 )]. f , f -f . K2 NaCrF6 KMnF3 KNiF3 . (3d)3 (3d)5 (3d)8. d f 0 f = 0. f 0. fs 0 fs 0. f s = f = 0. f 0 f 0. d . H res KNiF3. KMnF3. Mn2+: fS= %, f -f = %. Ni2+: fS= %, f = %. H res Cr3+: fS= %, f = %. K2 NaCrF6. - superexchange interaction).

4 Molecular orbital Wannier orbital (r r ). r r r kmr (r ). 1. m r Ri =. N. r e ik Ri k r kmr (r ) : Bloch .. H = br cr r c r + U. r r i i + j , j , n r nr i i r i,j n 4b 2 t2. Superexchange ( : J = , b . U Ed E p Goodenough-Kanamori rule .. Hopping . Direct Exchange Hund rule) --- .. Mn Eg S=1/2. Eg T2g : =1/2. H = [. r r { r ~ r } r ~ r ]. S i S j J s + i J s j + L + i J j + L. LaMnO3. ij . d3 x 2 r 2 , d3 y 2 r 2. ab . c . Jahn-Teller . KCuF3 d x2 z 2 , d y2 z2. c . ab .. ( 2 2 2. ). Hanis = 2 xx S x + yy S y + zz S z = DS z + E S x S y 2. ( 2 2. ). = . 0 L n n L 0 2 1 . D = zz xx + yy ( ). 2 . n E n E0. 2 xx yy E = . 2. (. zz > xx > yy ).. Hex = S1 J S 2 . , . Hdip =. (g B ) 2. r r (. r r r r S r S r S1 S 2 3 1 2 2. )( ) . r3 . r .. r r r r r r H = L1 S1 + L2 S 2 + J (n1 g 2 , n1 g 2 ) S1 S 2. r r 3 J ( g1n2 , g1n2 ) S1 S 2.. 2. J anis J (g 2) J. 2. E. Dzyaloshinski-Moriya 2 . r r r r r r r r g1 L1 S1 n1 J (n1 g 2 , g1 g 2 ) S1 S 2 + J ( g1 g 2 , n1 g 2 ) S1 S 2 n1 L1 S1 g1.

5 HDM = + (1 2) . n1 E n1 E g1 . r r r g1 L1 n1 = n1 L1 g1 = - n1 L1 g1 . [r r r ] r r S1 , S1 S 2 = i S1 S 2 [. r r r ] r r S 2 , S1 S 2 = i S 2 S1. HDM.. r [. r r r g L n J (n1 g 2 , g1 g 2 ) S1 , S1 S 2. = 2 1 1 1. ] r r r r [. g L n J ( g1n2 , g1 g 2 ) S 2 , S1 S 2. + 2 2 2. ] . n1 En1 E g1 n2 En 2 E g 2 . r r r [. HDM = D S1 S 2 ]. r r r g L n J (n1 g 2 , g1 g 2 ) g L n J ( g1n2 , g1 g 2 ) . D = 2i 1 1 1 2 2 2 D J g 2J. En1 E g1 E. n1 n2 E n2 E g2 .. r r [. r r r H = JS1 S 2 + D S1 S 2 ]. E = JS 2 cos + DS 2 sin . tan ( ) = . D. J.