2 T(x - v τ) i ix T(x + v τ) i ix x T = ((dt/dx),, ) ( q = c T (x i ) v i ( ) ) dt v ix τ v i dx i i ( (dt = cτ ) ) v 2 dx ix,, () i x = const. FIG. 2

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つかさいまいだ
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1 Y. Kondo Department of Physics, Kinki University, Higashi-Osaka, Japan (Dated: September 3, 27) [] PACS numbers: I. m cm 3 24 e =.62 9 As m = kg A. Drude-orentz Drude orentz N. i v i j = N q i v i = i= N ( e) v i () i= v i E Newton m d v i dt = e E (2) t v i e E t () j = ne2 m E t (3) n = N/ ( )τ t = τ (3) j σ j = σ E (4) σ = ne2 τ m (5) 2. FIG. : Drude-orentz UR: Electronic address: kondo@phys.kindai.ac.jp i E i () q = N E i v i (6) i=

i v i j = N q i v i = i= N ( e) v i () i= v i E Newton m d v i dt = e E (2) t v i e E t () j = ne2 m E t (3) n = N/ ( )τ

2 2 T(x - v τ) i ix T(x + v τ) i ix x T = ((dt/dx),, ) ( q = c T (x i ) v i ( ) ) dt v ix τ v i dx i i ( (dt = cτ ) ) v 2 dx ix,, () i x = const. FIG. 2: x = const. E i T E i = ct (7) c q = x (dt/dx) x = x i x = x i τ v ix x = x i + v ix τ E + i E + i = ct (x i + v ix τ) ( = c T (x i ) + ( dt ) dx )v ixτ (8) E i ( E i = c T (x i ) ( dt ) dx )v ixτ (9) x = x i q = N E i v i i= = ct (x i v i τ) v i i = ( c T (x i ) T ) v i τ v i () i N i= i v ix, v iy, v iz v 2 ix = v 2 iy = v 2 iz (2) i v 2 ix = 2cT 3m N (3) E i = ct ( ) dt q x = κ dx κ κ = 2nc2 T τ 3m 3. Wiedelmann-Franz (4) (5) σ κ Wiedelmann Franz σ/(κt ) Wiedelmann-Franz κ/(σt ) 2 8 WK 2 σ κ κ σt = 3 2 ( kb e ) 2 (6) c = 3 2 k B Wiedelmann-Franz κ/(σt ). 8 WK 2 B. 2 k BT x, y, z v x, v y, v z 3 2 mv2 x = 2 mv2 x = 2 mv2 x = 2 k BT (7)

= N E i v i i= = ct (x i v i τ) v i i = ( c T (x i ) T ) v i τ v i () i N i= i v ix, v iy, v iz v 2 ix = v 2 iy = v 2 iz (2) i v 2 ix = 2cT 3m N (3) E i = ct ( ) dt q x = κ dx κ κ = 2nc2 T τ 3m 3.

3 3 k B = J/K x, y, z x, y, z 2 k BT N E = 3 C ( 2 k BT ) N = 3 2 Nk BT (8) C = de dt = 3Nk BT (9) C. Drude-orentz 9 II. A. Schrödinger Schrödinger V ( r, t) Schrödinger ψ( r, t) i = 2 t 2m 2 ψ( r, t) + V ( r, t)ψ( r, t) (2) Ĥ = 2 2m 2 + V ( r, t) (2) (Hamiltonian ) i ψ t = Ĥψ (22) 2 ψ( r, t) 2 t r ψ( r, t) 2 d r = (23) A Â Âψ = aψ, (24) A a a a Â ψ Â ˆp = i B. t ˆp = i x i x ϕ(x) = pϕ k (x) (25) ϕ k (x) e ikx (26) p = k ψ(x) ϕ k (x) ψ(x) = c k e ikx dk (27) c k e ikx ψ(x) Fourier e ikx C. () Hamiltonian Ĥ = 2 2m 2 (28) ϕ() = ϕ() = ϕ k (x) sin kx (29) sin k = (3)

i ψ t = Ĥψ (22) 2 ψ( r, t) 2 t r ψ( r, t) 2 d r = (23) A Â Âψ = aψ, (24) A a a a Â ψ Â ˆp = i B.

4 4 2 k k = nπ (3) 2 ϕ k (x) = sin kx (32) E k = 2 k 2 2m = 2 π 2 n 2 2m 2 (33) D. (2) ϕ(x) = ϕ(x + ) ϕ k (x) = e ikx (34) k k = 2πn/ E k = 2 k 2 2m = 2 2 π 2 n 2 m 2 (35) E E + E k E + E = 2 (k + k) 2 2m E = 2 k k (36) m lim k E 2 m k k = k = 2mE/ k k = 2m 2m 2m 2 E = k 2 E = 2mE E E k k + k k 2π/ = m E (37) π 2E ±k 2 N = D(E) E (38) ( ) m D(E) = 2 = 2m (39) π 2E π E D(E) III. A.. Coulomb 3 = x, y, z Hamiltonian Ĥ = 2 2m 2 (4) x, y, z ψ(x, y, z) = ψ(x +, y, z) = ψ(x, y +, z) = ψ(x, y, z + ) (4) Hamiltonian 4 ψ k (r) = e ık r (42) k x = 2π n, k y = 2π n 2, k z = 2π n 3 (43) n, n 2, n 3 k E k = 2 2m (k2 x + k 2 y + k 2 z) = 2 2m (2π )2 (n 2 + n n 2 3) (44) k k 2π/ (2π/) 3 = (2π) 3 / (2π) 3 (45) E E + E k 2 2 2m = E (k + k) 2 2 2m = E + E

k 2 E = 2mE E E k k + k k 2π/ = m E (37) π 2E ±k 2 N = D(E) E (38) ( ) m D(E) = 2 = 2m (39) π 2E π E D(E) III. A.

5 5 k, k + k k 4πk 2 k 2π/ (2π) 3 4πk2 k (46) D(E) E D(E) D(E) = 4π 2 (2m 2 )3/2 E (47) D(E) k, k E 2 k k/m = E, k = 2mE/ 2. up,down k k F N e N e = 4π 3 k3 F (2π) 3 2 (48) (2π) 2 3 Fermi Fermi Fermi k F Fermi Fermi Fermi cm 24 Fermi 3 8 cm Fermi v F k F /m 3 8 cm/s Pauli 3. Fermi Fermi f (E) 2k B T T = E F FIG. 3: Fermi T = f(e) E f(e) = { E < EF E > E F (49) 3 T k B T Fermi k B T Pauli Fermi k B T Fermi Pauli Fermi k B T k B T Fermi k B T 3 f(e) = e (E µ)/k BT + E (5) Fermi µ N e D(E) N e = 2 de (5) e (E µ)/k BT + µ τ ϵ τ n τ {n τ } = (n, n 2, ) N = τ n τ E N = τ ϵ τ n τ Z N = {n τ } e β P τ ϵ τ n τ

6 6 Ξ = = e βnµ Z N N= e βnµ N= {n τ } = = τ N= {n τ } n τ e β P τ ϵτ nτ e β P τ (µ ϵτ )nτ e β(µ ϵτ )nτ Fermi n τ =, n τ e β(µ ϵ)nτ = + e β(µ ϵ) Ξ = τ { + e β(µ ϵτ ) } τ n τ P r (n τ ) P r (n τ ) = eβ(µ ϵτ )nτ σ = eβ(µ ϵτ )nτ + e β(µ ϵτ ) {n σ } eβ(µ ϵσ)nσ Ξ σ σ τ n τ n τ = n τ P r (n τ ) = n τ =, e β(ϵτ µ) + Fermi p k B T/2 Fermi k B T 2D(E F )k B T k B T 2D(E F )(k B T ) 2 T C e 4D(E F )k 2 BT Fermi D(E F ) D(E F ) = 4π 2 (2m 2 )3/2 ( 2 kf 2 N e 2m )/2 = 3 (52) 4 E F C e 3N e k B k B T E F (53) k BT E F k B T E F E F k B 4 4 K (54) 3 K IV. A.. NaCl a b 5 P R P R P = ma + nb (55)

7 7 (a) (b) (c) (a') (b') (c') FIG. 4: (a) (b) (c) (a ),(b ) (c ) (a),(b) (c) b a b' a' FIG. 5: B. X X X. X a k = 2π/λ X X Ae ıωt FIG. 6: X Ae ı(ωt 2ka)... Ae ıωt { + e 2ıka + e 4ıka + e 6ıka + } { ıωt e2nıka Ae for e = 2ıka e 2ıka NAe ıωt for e 2ıka = (56) e 2ıka e 2ıka =

8 8 X λ ke 2ıka = a A a = 2π A b = A c = B b = 2π B a = B c = C c = 2π C b = C a = (62) X Ae ı(ωt k r) e ıωt R i Ae ık R i (57) R A e ık R i e ık (R R i ) (58) A, B, C K = k k Bragg X G k G k G k = k k k = k X k' k G -k' A e ık R i e ık R i (59) K = k k A 59 FIG. 7: Bragg A e ık R l e ı(k a)l m e ı(k b)m n e ı(k c)n (6) 4. K K e ı(k a) = e ı(k b) = e ı(k c) = K a = 2πp, K b = 2πq, K c = 2πr, (6) p, q, r : integer X Bragg A b = A c = A b c A = α(b c) (63) a 2π A a = α(b c) a = 2π (64) α b c A = 2π a (b c) (65) A a (b c) B, C

) (58) A, B, C K = k k Bragg X G k G k G k = k k k = k X k' k G -k' A e ık R i e ık R i (59) K = k k A 59 FIG.

9 9 V. A.. V (x) = n v(x R n ) (69) R n = na Coulomb Coulomb R i v(r R i ) V (r) = i v(r R i ) (66) V (r) = V (r + R) (67) R = la + mb + nc P R P' [ 2 d 2 + V (x)]ϕ(x) = Eϕ(x) (7) 2m dx2 = Na N ϕ(x) = ϕ(x + ) (7) V (x) = ϕ k (x) = e ıkx, k = 2π m (72) v(x) ϕ(x) = c k e ıkx (73) ϕ(x) 73 7 [E k c k + c k V (x)]e ık x = Ec k e ık x k k E k = 2 k 2 2m /2 e ıkx x k E k c k k + c k k = Ec k k e ı(k k)x dx V (x)e ı(k k)x dx e ı(k k)x dx FIG. 8: P P [ 2 2m 2 + V (r)]ϕ(r) = Eϕ(r) (68) 2. a k k e ı(k k)x dx = e ı(2π/)(m m)x dx [ ] = e ı(2π/)(m m)x ı(2π/)(m m) = k = k Dirac δ e ı(k k)x dx = δ(k k) (74)

(x)]ϕ(x) = Eϕ(x) (7) 2m dx2 = Na N ϕ(x) = ϕ(x + ) (7) V (x) = ϕ k (x) = e ıkx, k = 2π m (72) v(x) 72 72 ϕ(x) = c k e ıkx (73) ϕ(x) 73 7 [E k c k + c k

10 ϕ k (x) E k c k + k c k V (x)e ı(k k)x dx = Ec k 2 69 = n V (x)e ı(k k)x dx v(x R n )e ı(k k)x dx = Rn e ı(k k)r n n R n v(x )e ı(k k)x dx x R n x v(x ) x ( R n, R n ) (, ) n n e ı(k k)r n = n N n= e ı(k k)na k k = (2π/a)m m N k k (2π/a)m N n= e ı(k k)na = = = e ı(k k)na e ı(k k)a e ı 2π(n n) e ı(k k)a e ı2π(n n) e ı(k k)a = N 2 k = k + G m G m = (2π/a)m m (E k E)c k + m Na v Gm c k+gm = (75) v G v G = v(x)e ıgx dx (76) a 7 k E = E ϕ k (r) = e ık r, k = 2π (l, m, n) (77) ϕ(r) = c k e ık r (78) e ı(k k) r d 3 r = δ(k k) (79) V (r)e ı(k k) r d 3 r = v G δ(k k G) (8) v G = v(r)e ıg r d 3 r (8) G (E k E)c k + G k v G c k+g = (82) B.. Nearly Free Electron k c k c k 75 c k+gm (E k+gm E)c k+gm + m v Gm c k+gm +G m = (83) G m = G m E E k c k+gm v Gm E k E k+gm c k (84) 75 c k+gm G m = [E k + v + m v Gm 2 E k E k+gm E]c k = (85)

11 v G = vg ) c k [ E E k + v + m ] v Gm 2 E k E k+gm (86) E φ k (x) = e ıkx c k [ + v Gm e ıgmx ] (87) E k E k+gm m G m Nearly Free Electron Approximation NFE E k = E k+gm c k+gm c k k 2 = (k + G m ) 2, (k = G m /2) NFE k G m 75 (E k + v E)c k v Gm c k+gm + v Gm c k+gm + v Gm+ c k+gm+ + = c k c k+gm (E k+gm + v E)c k+gm v G m c k+g + v G m c k + v G m+ c k+g + = c k c k+gm (E k + v E)c k + v Gm c k+gm = (88) (E k+gm + v E)c k+gm + v Gm c k = E k + v E v Gm E k+gm + v E = (89) v Gm E = 2 (E k + E k+gm ) + v ± 2 (Ek E k+gm ) v Gm 2 (9) k = G m /2 E k = G m /2 E = E Gm /2 + v ± v Gm (9) E 9 G m = ±(2π/a)m m k = ± π a, ±2π a, (92) 2 v Gm -2p/a -p/a Energy Gap Energy Gap p/a 2p/a FIG. 9: Nearly Free Electron Approximation NFE 3. k = G m /2(m 2 v Gm k = G m /2 Bragg G m = 2π/a φ + (x) cos( πx a ), φ (x) sin( πx a ) (93) a/2 k potential FIG. : (a) (b) (a) (b) 4. Bragg (a) (b)

.. +v G m c k+g + v G m c k + v G m+ c k+g + = c k c k+gm (E k + v E)c k + v Gm c k+gm = (88) (E k+gm + v E)c k+gm + v Gm c k = E k + v E v Gm E k+gm + v E = (89) v Gm E = 2 (E k + E k+gm ) + v ± 2

12 2 2 2m k 2 = 2 2m k + G 2 k ( G G ) = 2 G (94) G G k G C. Brillouin zone Brillouin zone Brillouin zone Brillouin zone FIG. : 2 Brillouin zone Brillouin zone 2Brillouin zone 2 Brillouin zone 3Brillouin zone 3 2Brillouin zone 4 5 Brillouin zone Brillouin zone (2π) n / unit cell 2π/a simple cubic (2π) 3 /a 3 Brillouin zone dim : 3 dim : 2π 2π a = a = N (95) (2π) 3 (2π)3 = = N N Brillouin zone Brillouin zone 2 D. Bloch k p = k k p = (k + G m ) k. k k φ k (x) = c k+gm e ı(k+gm)x, (96) m G m = ± 2π a m m k 2π/a k Brillouin zone 2π/a Brillouin zone 96 φ k (x) = e ıkx u k (x) (97) u k (x) = c k+gm e ıgmx (98) m u k (x) a u k (x + a) = c k+gm e ıgm(x+a) m = c k+gm e ıg mx m = u k (x) (99) e ıg ma = φ k (x) φ k (x + R n ) = e ıkr n φ k (x), R n = na () Bloch Bloch k 2

dim : 3 dim : 2π 2π a = a = N (95) (2π) 3 (2π)3 = = N 2 4 4 3 4 N Brillouin zone Brillouin zone 2 D. Bloch k p = k k p = (k + G m ) k.

13 3 u k (r) exp(ıkr) ) FIG. 2: Bloch E. Tight Binding. NFE NFE 3s 3s NFE NFE NFE 2. Bloch k Brillouin zone k u k (x) unit cell 3 2. Tight Binding NFE N NFE ψ(r) = i c i ϕ(r R i ) () -p/a E Energy Gap p/a Energy Gap k ϕ(r) R i i c i ± Bloch c i e ık R i ψ k (r) = e ık Ri ϕ(r R i ) (2) N i Bloch ψ k (r + R) = e ık R i ϕ(r R i + R) N i = e ık (Rj+R) ϕ(r R j ) = e ık R ψ(3) k (r) N j R i R R j 3. FIG. 3: Brillouin zone NFE 3. Hamiltonian ψ k (x) = e ıkrn ϕ(x R n ), (4) N Hamiltonian n d 2 Ĥ = 2 + V (x) (5) 2m dx2

Tight Binding NFE N NFE ψ(r) = i c i ϕ(r R i ) () -p/a E Energy Gap p/a Energy Gap k ϕ(r) R i i c i ± Bloch c i e ık R i ψ k

14 4 V (x) 69 k Ẽk Ẽ k = = N ψ k (x)ĥψ k(x)dx ψ k (x)ψ k(x)dx ψ k(x)ψ k (x)dx n n exp( ık(r n R n )) ϕ(x R n )ϕ(x R n )dx (6) ϕ(x) n = n ϕ(x) ϕ(x) ϕ(x R n ) 2 dx = ϕ(x ) 2 dx = ϕ(x) a (, ) (, ) n n ϕ(x) x = R n R n ϕ(x R n ) ϕ(x R n ) n = n ± = ϕ(x R n± )ϕ(x R n )dx ϕ(x a)ϕ(x )dx S (7) S ψ k(x)ψ k (x)dx = { N }{{} +S( }{{} e ıka + } e ıka {{ } )} n n =n n =n n =n+ = + 2S cos ka (8) 6 = N ψ k(x)ĥψ k(x)dx n n exp( ık(r n R n )) ϕ(x R n )Ĥϕ(x R n)dx n = n, n = n ± E = = E = = ϕ(x R n )Ĥϕ(x R n)dx ϕ(x)ĥϕ(x)dx (9) ϕ(x R n± )Ĥϕ(x R n)dx ϕ(x )Ĥϕ(x)dx () ψ k(x)ĥψ k(x)dx = { E +E N }{{} ( }{{} e ıka + e } ıka {{ } )} n n =n n =n n =n+ = E + 2E cos ka () 8 6 S S E Ẽ k = E + 2(E E S) cos ka (2) simple cubic ϕ(r) s Ẽ k = E + 2(E E S)(cos k x a + cos k y a + cos k x a) (3) Tight Binding 4. NFE Tight Binding k 2 NFE k k 2 Brillouin zone NFE ϵ k k 2 ka 2 Ẽ k = const. (E E S)a 2 k 2 (4) m 2 m = 2(E E S)a 2 (5) Ẽ k const + 2 k 2 2m (6)

k(x)dx n n exp( ık(r n R n )) ϕ(x R n )Ĥϕ(x R n)dx n = n, n = n ± E = = E = = ϕ(x R n )Ĥϕ(x R n)dx ϕ(x)ĥϕ(x)dx (9) ϕ(x R n± )Ĥϕ(x R n)dx ϕ(x )Ĥϕ(x)dx () ψ k(x)ĥψ k(x)dx = { E +E N }{{} ( }{{} e ıka +

15 5 simple cubic Ẽ k const + 2 k 2 2m (7) NFE p p 4 E E S 2 E E S E S 5. TB TB NFE e ık r u k (r) Bloch ϕ k (r) 8 Bloch ṽ kk = ϕ k (r)v(r) ϕ k (r)d 3 r (9) v(r) ϕ k (r) ϕ k (r) Bloch NFE G.. v ev Bloch ψ(x) = ϕ(k)e ık(x x) dk (2) F. Coulomb /r Coulomb Coulomb NFE TB Bloch k v kk = }{{} matrix element ϕ k (r) } {{ } v(r) }{{} ϕ k (r) } {{ } creation interaction annihilation d 3 r (8) k v(r) k v kk Bloch V D ϕ(k) k = k δk δk k φ(x) x δx /δk 2 t = t = t ψ(x, t) = ϕ(k)e ı[k(x x) Ekt/ ] dk (2) E k k E k E k + ( de k dk ) k (k k ) (22) t = t ψ(x, t) e ı[e k ( de k dk ) k k ]t/ ϕ(k)e ık[(x x ) ( de k dk ) k t/ ] dk (23) e t = x t = t x + (de k dk ) k t (24) v = (de k dk ) k (25) E k / = ω k e ı(kx ω kt) ω k /k dω k /dk

Coulomb /r Coulomb Coulomb NFE TB Bloch k v kk = }{{} matrix element ϕ k (r) } {{ } v(r) }{{} ϕ k (r) } {{ } creation interaction annihilation d 3 r (8) k v(r) k v kk Bloch V D

16 6 k v k = de k k = dk }{{} m E k = 2 k 2 /2m (26) E empty states k v = p/m filled states 2. Bloch Bloch m v k = k m (27) (a) (b) (c) FIG. 4: (a) (b) (c) 3. δt E δe = ee }{{} v k δt }{{} force distance (28) k k + δk δe = de k δk (29) dk δk = e d( k) Eδt = ee (3) dt ṗ = ee Bloch ection H. Bloch 2N 4 J = e k v k δt (e/ )Eδt 5(b) E G ω ω I. Fermi Brillouin zone Fermi 4 Fermi Fermi Fermi i, Na, K, Rb, Cs Brillouin zone Fermi NFE

δt E δe = ee }{{} v k δt }{{} force distance (28) k k + δk δe = de k δk (29) dk δk = e d( k) Eδt = ee (3)

17 7 E Energy Gap E G (a) (b) (c) k -p/a (a) p/a filled states empty states E Energy Gap E G (b') (c') k -p/a (a2) E p/a Energy Gap FIG. 6: (a) Brillouin zone Fermi Brillouin zone Brillouin zone 2π/a (b) Brillouin zone (c) Brillouin zone (b ) Fermi (b) (c ) Fermi (b ) E G J. -p/a (b) p/a 2p/a FIG. 5: (a) (a2) (b) k k + 2π/a Be, Mg, Ca, Sr, Ba 6 Chapter 9 in Solid State Physics, Saunders College, by Ashcroft and Mermin, ISBN # Fermi Al Cu, Ag, Au d k. K Ge E G = 4.3 ev Si E G = 4. ev GaAs InAs CdS 3k B.7 ev n e n e e E G/k B T (3)

18 8 σ σ e E G/k B T (32) E G 2. Ge As As As + Ge Ge Ge As As + Coulomb m Coulomb As + e 2 /ϵr a B = ϵ 2 m e 2 = m m ϵa B (33) Eg = m e 4 2ϵ 2 2 = m ϵ 2 m E g hydrogen (34) m.25m ϵ = 6 a B 6a B / E g. ev.7 ev Ge As donner Ga Ga acceptor K. VI. A.. N n R n u n U(R) R n n + U((R n+ + u n+ ) (R n + u n )) = U(a + u n+ u n ) U(a) + 2 K(u n+ u n ) 2 (35) u n+ u n a K R = a U(x) u n+ u n U T = NU(a) + N n= u N+ = u 2 K(u n+ u n ) 2 (36) FIG. 7: n M 2 t 2 u n N n = U T u n = K(u n u n ) + K(u n+ u n ) = K( u n + 2u n u n+ ) (37)

19 9 37 M 2 t 2 Q q u n (t) = Q q (t)e ıqr n (38) = K( e ıqa + 2 e ıqa )Q q = 4K sin 2 ( qa 2 )Q q (39) u n+n = u n q = 2π n (4) = Na 37 Q q 39 4K ω = ± M sin(qa ) (4) 2 q u n (t) = Q q e ı(qr n ωt) (42) q 8 Brillouin zone π/a < q < π/a q /a K ω ±c q, c = M a (43) c w p/a FIG. 8: q 2. q = q = 9 q = q = (a) (b) FIG. 9: q = (a) (b) 3. 3N (n + /2) ω ω/2 n ω phonon photon Brillouin zone ω D k B T ω D k B T

q 8 Brillouin zone π/a < q < π/a q /a K ω ±c q, c = M a (43) c w p/a FIG. 8: q 2.

20 2 3N C C = 3Nk B Dulong-Petit Debye θ D = ω D k B (44) T θ D θ D K T θ D k B T ω ω > k B T ω < k B T q ω = cq q < k B T/ c ( ) 3 4π kb T (2π) 3 3 (45) 3 c 3 () (2) k B /2 C k B ( kb T c ) 3 (46) T 3 Debye B.. n R n u n V (x) = N v(x R n ) (47) n= Ṽ (x) = N v(x R n u n ) (48) n= V (x) δv (x) N δv (x) = Ṽ (x) V (x) = v (x R n )u n (49) n= v (x) v(x) u n = q 5 49 δv (x) = q N Q q n= Q q e ıqrn (5) v (x R n )e ıqr n (5) k Bloch k Bloch [δv ] k k = φ k (x)δv (x)φ k(x)dx (52) [δv ] k k = q = q N Q q n= e ıqr n N Q q e ı(q k +k)r n n= } {{ } Nδ(q k +k G m ) v k k = a v (x R n )e ı(k k)x dx v (x )e ı(k k)x dx } {{ } v k k /N (53) v (x)e ı(k k )x dx (54) n G m (2π/a)m [δv ] k k = v k kq q δ(q k + k G m ) (55) 55 k k q = k k + G m ω q q ω q q 55 k q = k k + G m k k q = k k+g m k 2 q = k k + G m G m

. n R n u n V (x) = N v(x R n ) (47) n= Ṽ (x) = N v(x R n u n ) (48) n= V (x) δv (x) N δv (x) = Ṽ (x) V (x) = v (x R n )u n (49) n= v (x) v(x) u n = q 5 49 δv (x) = q N Q q n= Q q e ıqrn (5) v (x R n

21 2 (a) k' k q (b) k' k -q FIG. 2: D. Coulomb ξ aξ ξ 2 k, k, q Brillouin zone k, q Brillouin zone k Brillouin zone k 2. C. Tight Binding Model Coulomb polaron ϵ = aξ + 2 bξ2 (56) ϵ min ϵ min = a2 2b at ξ = a b (57) 2ϵ min ϵ = 2aξ + 2 bξ2 (58) ϵ ϵ min = 2a2 b at ξ = 2a b (59) ϵ min < 2ϵ min 2 k k' k - q q k' + q FIG. 2: E.

22 22 k k W k k W k k v k 2 k Q q 2 (6) E k k = q + G (6) dk E k E k = ± ω q (62) Pauli k τ τ = k W k k (63) x δt x k x k x + δk, δk = e Eδt (64) δk k x v kx v kx + v kx k x δk (65) v kx = k x /m v kx v kx ee δt (66) m m n J = e k (v kx ee m δt) = ne2 E δt (67) m 22 τ 67 δt τ J = σe (68) σ = ne2 τ m (69) FIG. 22: τ Q q 2 T τ T σ T (7) VII. BOSE-EINSTEIN CONDENSATION A. Bose-Einstein Condensation Bose-Einstein condensation Bose

23 µ 23 2. Bose BEC BEC Bose p de Broglie λ = h/p T < p > mk B T de Broglie λ db h mkb T (72) de Broglie BEC FIG. 23: Bose T > T c T T c T < T c) Taken from http://amo.phy.gasou.edu/bec.html.

23 23 µ Bose BEC BEC Bose p de Broglie λ = h/p T < p > mk B T de Broglie λ db h mkb T (72) de Broglie BEC FIG. 23: Bose T > T c T T c T < T c) Taken from n λ 3 db (73) Bose BEC Bose Bose Bose p Bose-Einstein < n p >= exp[(ϵ p µ)/k B T ] (7) µ µ k B T Bose Boltzmann µ k B T condensation 4 K 2 K Bose 4 2 BEC B. 9 eiden Kamerlingh Onnes n n n. e p e p e p FIG. 24: Bose T µ ) T µ )T = K Bose p = ϵ p = Fermi Fermi K Meissner effect H c

24 24 Hc(T), in gauss In Hg Sn Pb Tl Temperature, in K Cooper Fermi Bose Bose BEC K Fermi sea Fermi sea Pauli ψ (r, r 2 ) = k g k e ık r e ık r 2 (74) FIG. 25: 26 siglet 74 ψ (r r 2 ) = (α β 2 β α 2 ) k g k cos k (r r 2 ) (75) Hc - 4 p M Hc Applied Magnetic Field Ba Hc - 4 p M Hc Hc Applied Magnetic Field Ba FIG. 26: H c Mesissner type I type II H c Meissner type I H c2 vortex type II H c type I Hc2 α up spin β down spin triplet singlet singlet r r 2 Schrödinder E Eg k = 2ϵ k g k + k >k F V kk g k (76) 2 2 2ϵ k V kk = V (r)e ı(k k) r dr (77) r 76 g k E < 2E F V (r) { V ϵk E V kk = F < ω c (78) otherwise 2. Cooper 956 Coper Cooper Cooper Fermi sea gk g k = V 2ϵ k E (79) 79 k g k V = k>k F (2ϵ k E) (8)

25 25 V EF + ω c = N() dϵ E F 2ϵ E = 2 N() ln(2e F E + 2 ω c ) (8) 2E F E N()V E 2E F 2 ω c e 2/N()V < 2E F (82) Fermi sea 2E F 2E F V = ψ (r) = (α β 2 β α 2 ) k cos k r 2ξ k + E (83) ξ k = ϵ k E F and E = 2E F E > (84) 83 k ξ k < E Fermi ξ k N()V E = 2 ω c e 2/N()V ω c ω c 78 E p v F Cooper x h/ p hv F /E T c E E k B T c Cooper ξ hv F /k B T c ae F /k B T C a 3. Coulomb 2 K C. BCS Cooper Fermi sea Fermi sea BCS [].

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345