I ino@hiroshima-u.ac.jp 217 11 14
4 4.1 2 2.4 C el = 3 2 Nk B (2.14) c el = 3k B 2 3 3.15 C el = 3 2 Nk B 3.15 39
2 1925 (Wolfgang Pauli) (Pauli exclusion principle) T E = p2 2m p T N 4
Pauli Sommerfeld 4.2 Ĥ Ĥ = ˆp2 2m = ħ2 ˆk2 2m ( ) ˆp = ħ ˆk ˆk = i = i x, y, z ħ 2 ˆk2 2m ψ(r) = E ψ(r) (4.1) k A ψ k (r) def. = A e ik r = A e ik xx e ik yy e ik zz (4.2) ˆk ψ k (r) = k ψ k (r) ˆk (4.1) E = ħ2 k 2 2m (4.3) E = p2 2m ψ k (r) 4.1 ψ k (r) 2 = A 2 41
y 2 k 2 k x Im Re 4.1 ψ k (r) ( ) ( ) L ψ(x+l, y, z) = ψ(x, y, z) ψ(x, y+l, z) = ψ(x, y, z) (4.4) ψ(x, y, z+l) = ψ(x, y, z) L (4.2) (4.4) e ikxl = 1 e ikyl = 1 e ikzl = 1 k k = 2π ( ) nx, n y, n z n x n y n z (4.5) L n x n y n z L dx L dy L dz ψ(r) ψ(r) = 1 (4.2) A = 1 L 3 ψ(r) = 1 L 3 ei k r (4.6) 42
* 1 (4.5) 4.2 2π L ( ) 2π 3 1 s = +1/2 L s = 1/2 ( ) 2π 3 dk x dk y dk z dn = 2 dk x dk y dk z L V = L 3 dn = 2 V (2π) dk xdk 3 y dk z (4.7) (4.3) E 2mE N(E) (4.3) k = ħ 2mE N(E) (4.7) k < ħ N(E) = 2V dk (2π) 3 x dk y dk z k < 2mE ħ = 2V (2π) 4π ( )3 2mE 3 3 ħ = V ( )3 2m E 3/2 3π 2 ħ (4.8) E E + de dn = d N(E) de de D(E) def. = d N(E) (4.9) de * 1 43
と定義する これに (4.8) 式を代入すると 自由電子気体の状態密度 が得られる V D(E) = 2π2 ( )3 2m E ℏ (4.1) 従って 三次元では D(E) E となる (4.8) 式では 球の体積を用いて状態を数えた が 代わりに円の面積を用いると二次元自由電子気体の状態密度が 線分の長さを用いる と一次元自由電子気体の状態密度が得られる フェルミ準位とフェルミ波数 温度ゼロ T = では 電子は パウリの排他律が許す限りエネルギーの低い状態を占め る 従って 電子の数が増えるにつれて エネルギーの低い状態から順に埋まっていく 温度ゼロにおいて 占有状態と非占有状態を分けるエネルギーを フェルミ準位 (Fermi leel) EF と呼ぶ*2 また エネルギーが EF に等しい状態の波数を フェルミ波数 (Fermi waenumber) kf と定義する 三次元波数空間において フェルミ波数を集めると曲面 になるので これを フェルミ面 (Fermi surface) と呼ぶ 図 4.3 フェルミ球 図 4.2 量子状態の分布 *2 EF は フェルミ エネルギー と呼ばれることもある 44
(4.3) 4.3 k F (Fermi sphere) k < k F k > k F (4.7) N 2V (2π) 3 4π 3 k3 F = N n = N V k F = ( 3π 2 n ) 1/3 (4.11) ψ(r) e i k F r λ F def. = 2π k F (4.12) F ħk F F = ħ k F m (4.13) (4.3) E F E F = ħ2 k 2 F 2m T F (4.14) T F def. = E F k B (4.15) (4.1) E = E F (4.14) D(E F ) = V ( )3 2m EF = V ( ) 3 2m ħ k F 2π 2 ħ 2π 2 ħ 3 2m D(E F ) = mv π 2 ħ 2 k F (4.16) 45
, D(E) D(E F ) 4.4 E F, E (4.11) (4.16) k F = 2π λ F = m ħ F = 2m EF ħ = 2mkB T F ħ = π2 ħ 2 m D(E F ) V = ( 3π 3 n ) 1/3 (4.17) n 2.1 n (4.11) (4.16) 4.1 4.1 Z n k F λ F F E F T F D(E F )/V (/nm 3 ) (/Å) (Å) (km/s) (ev) (1 3 K) (/nm 3 ev) 29Cu 1 84.7 1.36 4.62 1573 7.3 81.6 18.1 47Ag 1 58.6 1.2 5.23 1391 5.5 63.8 16. 79Au 1 59. 1.2 5.22 1394 5.53 64.1 16. 13Al 3 181 1.75 3.59 225 11.7 135 23.3 λ F T F F.5% T (2.17) 2 117 3 F km/s 46
4.3 1926 (Enrico Fermi) (Paul Adrien Maurice Dirac) T µ E (Fermi-Dirac distribution) f FD (E) = 1 e (E µ)/k BT + 1 (4.18) * 3 (4.18) +1 (3.4) f MB (E) (3.8) f BE (E) 1927 (Arnold Johannes Wilhelm Sommerfeld) 1..5 4k B T 1..5 µ = 1 K 3 K 1 K 5 K 1 K K 4.5 µ 2k B T µ µ+2k B T, E 4.6 -.2 -.1.1.2, E (ev) * 3 Ξ i = 1 + e β(ε i µ) ni = 1 β µ ln Ξ 1 i = e β(εi µ) + 1 47
4.5 E E f FD (E) 1 1 E + f FD (E) 1 E µ E = µ d f FD (E) de = E=µ e (E µ)/k BT k B T [ e (E µ)/kbt + 1 ] 2 = 1 4 k B T E=µ 4.5 E = 4k B T T 4.6 (4.1) (4.18) E ( f FD (E) D(E) f 1 e β(e µ) FD 2m + 1 p2) 1 1 e β( 1 2m µ) p2 + 1 f MB (E) D(E) E e βe f MB ( 1 2m p2) 1 e 1 2m βp2 T =.2 T F 4.7 E F 16 K * 4 E 1.5 k B T E F * 4 1358 K k B T =.2 E F 48
Fermi sphere p y ~k F p x E E E F 4k B T Fermi-Dirac E F.6E F 1.5k B T p x p y Maxwell-Boltzman p x 4.7 k B T/E F =.2 49
T T µ E F 3 5 E F * 5, E 4.8 E F 1 2 D(E F) 2k B T 4 3 k BT U(T) U() [ 1 2 2k BT 1 ] 2 D(E F) 4 3 k BT = 2 3 k2 B T2 D(E F ) C el def. = du dt 4k2 B 3 D(E F) T 2k B T E F 4.8 C MB = 3 2 Nk B T D(E F ) 4.8 * 5 µ(t) = E F π2 6 k2 B T2 D (E F ) D(E F ) E F lim µ(t) = E F T T T F µ µ E F 5
E U U(T) = E D(E) f FD (E) de E F N = E F D(E) f FD (E) de U(T) E F N = (E E F ) D(E) f FD (E) de T C el C el def. = du dt = (E E F ) D(E) d f FD dt de d f FD dt E F D(E) D(E F ) C el = D(E F ) (E E F ) d f FD dt de x = β(e E F ) dx dt = d dt C el = D(E F ) βe F ( x dx β dt ) d f FD dx dx β ( ) E EF = x k B T T βe F = E F k B T C el = D(E F) x 2 d f FD dx (4.19) β 2 T dx µ(t) E F 1 f FD (x) = e x + 1 d f FD dx = d ( ) 1 dx e x + 1 x 2 d f FD dx dx = 2 x 2 d f FD dx x dx = 2 2x f FD dx = 4 e x dx = π2 + 1 3 51
* 6 x e x + 1 (4.19) dx = π2 12 C el = γt C el = π2 k 2 B 3 D(E F) T (4.2) γ = π2 k 2 B 3 D(E F) (4.21) D(E F ) γ D(E F ) (4.2) C MB = 3 2 Nk B C el C MB D(E) E D(E F ) E F N = π 2 k 2 B 3 D(E F) T 3 2 N k B = D(E F) E F EF D(E) de = = 2π2 9 D(E F) E F N EF E F = E de EF T T F E 3/2 F [ 2 3 E3/2 ] EF = 3 2 (4.22) C el C MB = π2 3 ( T TF ) 3.3 T T F (4.23) * 6 x e x +1 dx = ( 1) n x e (n+1)x ( 1) n dx = e (n+1)x dx = n+1 n= ζ(2) 2 n= = π2 12 n=1 x e x dx = 1 + e x ( 1) n (n+1) 2 = 1 n 2 2 x e x n= n=1 n=1 1 n 2 52 ( e x ) n dx = n= 1 (2n) 2 = 1 2 n=1 1 n 2 =
T F 1/1 3.3 T/T F (4.17) n (3.21) C ph T 3 C el T T 4.9 Cu T < 5 K C/T T 2 C(T) T = γ + AT2 C/T T 2 γ exp =.688 mj/mol K 2 4.1 D(E F )/V (4.21) γ th =.53 mj/mol K 2 C el T C ph T 3 25 C/T (mj/mol K 2 ) Cu T 2 (K 2 ) 4.9 Cu [1] γ, c (J/mol K) 2 15 1 5 1 Cu 2 3 4, T (K) 4.1 Cu [2] (T < 5 K) C el =.688 T mj/mol K 2.48 T 3 mj/mol K 2 53
4.2 γ exp [3] (4.16) γ th γ exp γ th Z (mj/mol K 2 ) (mj/mol K 2 ) γ exp /γ th Li 1 1.65.75 2.2 Na 1 1.38 1.12 1.2 K 1 2.8 1.73 1.2 Rb 1 2.63 1.99 1.3 Cu 1.69.5 1.4 Ag 1.64.64 1. Au 1.69.64 1.1 Be 2.171.49.35 Mg 2 1.26.99 1.3 Ca 2 2.73 1.5 1.8 Zn 2.64.75.85 Cd 2.69.95.73 Al 3 1.35.91 1.5 In 3 1.66 1.23 1.3 Tl 3 1.47 1.31 1.1 C a 4.49 Si 4 1.14 Sn 4 1.78 1.38 1.3 Pb 4 2.99 1.5 2. As 5.191 1.29.15 Sb 5.119 1.61.7 Bi 5.85 1.79.5 a 54
4.1 (4.21) (4.2) D(E) (4.1) (4.3) γ D(E F ) D(E F ) γ exp γ th 4.2 C Si γ exp As Sb Bi γ exp 4.11 γ = π2 k 2 B D(E F ) 3 V D(E F )/V * 7 4.4 1933 * 7 D(E F )/V d f d f m m = γ exp γ th 55
2.6 4.12 E ee d p(t) dt = ee 4.12 τ eeτ d = e τ m E (2.9) p( ) τ 1.5 1..5 4.11 C 1 V 2 Mn Ni Cu Pd Ag 3 4 5 6 7 8 Pt Au Pb γ (J/cm 3 K 2 ) 56
E p y p p x p E E F p ~k F p x 1 p p x p p p x 4.12 57
d F (2.7) d τ d j (2.11) σ = e 2 n τ m τ F (2.12) 2 (2.13) κ = c 2 3 nτ (2.13) c C el /N F ( ) 2 (4.2) 2 F = ħkf = 2E F m m (4.22) 2D(E F) E F 3N κ = Cel 2 F 3N nτ κ = 1 3N π2 k 2 B 3 D(E F) T 2E F m nτ = 1 κ = π2 k 2 B T 3 nτ m (4.24) (2.15) 58
(2.12) σ = e 2 nτ m nτ m κ σt = 1 ( ) 2 πkb 2.44 1 8 WΩ/K 2 3 e 1.11 2 2.3 2.3 2.9 ρ τ l = τ 2 T (2.17) 2 = 117 km/s 3 F l = F τ (4.25) 4.1 F (4.25) 2.12 τ l 4.13 F l(t) τ(t) l(t) τ(t) σ(t) 1 l(t) 1 τ(t) ρ(t) l(t) 4.13 T 1 K l l 59
, 1 6 1 5 1 4 Ag Cu Au 1 3 Al 1 2 1 1 1 1, T (K) 4.13 l l 2.12 4.13 l 1 1 K l 5 l Å Å Å Å Å 6
4.5 T T F 3 n C el = π2 k 2 B 3 D(E F) T σ = e 2 nτ τ m T 2 = 117 km/s Cu = 157 km/s 3 F κ = π2 k 2 B T nτ 3 m l = F τ ( ) Å Å Å [1] W. S. Corak, M. P. Garfunkel, C. B. Satterthwaite, and A. Wexler, Atomic Heats of Copper, Siler, and Gold from 1 K to 5 K, Phys. Re. 98, 1699 (1955). [2] G. K. White and S. J. Collocott, Heat Capacity of Reference Materials: Cu and W, J. Phys. Chem. Ref. Data 13, 1251 (1984). [3] G. R. Stewart, Measurement of low-temperature specific heat, Re. Sci. Instrum. 54, 1 (1983). 61