4 213 5 8 4.1.1 () f A exp( E/k B ) f E = A [ k B exp E ] = f k B k B = f (2 E /3n). 1 k B /2 σ = e 2 τ(e)d(e) 2E 3nf 3m 2 E de = ne2 τ E m (4.1) E E τ E = τe E = / τ(e)e 3/2 f de E 3/2 f de (4.2) f (3.2) F = (3.21) f = f + f 1 v f = f 1 /τ, f 1 f 1 = τv f. (4.3) V J J = ( e) τv(v f )dr. V f x v v x v 2 x = v 2 /3 v 2 j x () = e τv 2 f x dr = e x τv 2 3 n x. j = ( e)d n, D = τv 2 /3. (4.4) D (diffusion constant) D = τ 3 v2 = τk B m = µ e k B (4.5) (4.5) (Einstein relation) 4-1
E 1/2 3/2 1.18 () +3/2 +3/2 1.93 () +1/2 +1/2 1.18 1. +1/2 1.1 1 E [1] 4.1.2 B (3.2) F E 4.1 y x E = (E x,, ) j y 4.1 E int = (, E y, ) j y = (Hall effect) R H = E y J x B z (4.6) (Hall coefficient) E y j y = (4.6) (E.11b) j = ˆσE σ xx = ne2 m A l = ne2 τ m 1 + (ω c τ) 2 E y = (A t /A l )E x (4.7) E, σ xy = ne2 ωc τ 2 m 1 + (ω c τ) 2, (4.8) E A t R H = 1 ne ω c (A 2 l + A2 t ) (4.9) ω c τ 1 R H = 1 τ 2 E ne τ 2 E = 1 Γ(2s + 5/2)Γ(5/2) n( e) (Γ(s + 5/2)) 2 = r H n( e) (4.1) s ( e e )r H (Hall factor) 1 (s = ) 1 ( 1 )τ (s = ) (E.9) (4.1) B y z y + x + + + + + + + J y J x 4.1 z x y J y () 4-2
4.2 2 (heat current, thermal flux) ( phonon) (phonon dragg) (Joule heating) 4.2.1 n x (thermal flux density) j qx j qx = nv x (E µ) = v x (E µ)f(e)d(e)de (4.11) (temperature gradient) κ n (thermal conductivity) j q = ˆκ j qx κ n = / x (4.12) 4.2.2, x 1, x1 B (a) B 2 2 B B A A B J V (b) (4.12) (thermoelecric effect) (a) A 1 2 B 1 2 AB V AB (Seebeck effect) ( = 1 2 ) S AB = V AB (4.13) (Seebeck coefficient) (b) AB J Q J (Peltier effect) Q AB J Π AB = Q AB J (4.14) (Peltier coefficient) (a) BAB J (x ) / x (temperature gradient) Q/ x 4-3
(homson effect) (homson coefficient) τ = Π AB = S AB, Q/ x J( / x) τ A τ B = ds AB d (4.15) (4.16) (Kelvin relations or homson relations) ( F) S A ( ) τ A ( ) d (4.17) S AB = S A S B (4.18) S V V (4.18) (thermocouple) 2 4.2.3 (3.2)(3.21) f/ t = v f + F m vf = f f τ(e). (4.19) f f f f = f. f E (E E F )/k B a f = f E a E a = f E ( k B ) E E F k B 2 = f E F E E f = E F E f E. v f = v E f E = mv f E. (4.2a) (4.2b) E (4.19) (4.2) [ f = f τ(e)v ee + E ] F E f E. (4.21) x E x x (3.23) [ j x = e nv x = e v x f(e)d(e)de = e vxτ 2 ee x + E ] F E f x E D(E)dE. 4-4
J n-type heating Cooling heating p-type 4.2 p n j x = S = E x / x = v 2 xτ E F E e / f E D(E)dE = 1 e [ E F vxτ 2 f E D(E)dE τe 2 f E D(E)dE / τe f ] E D(E)dE (4.22) vx 2 2E/3m (4.21) f / E = f /k B τ E s S = 1 [ ] τe E E F = 1 [( ) ] 5 e τ E e 2 + s k B E F. (4.23) S E F s e +e S τ E E F (4.5) 4.2.4 n p PC CPU 4.3 3.2 4-5
1 InSb a B * (nm) 1 1.1 n c 1/3 ab*=.26 Ge:Sb Ge:As Ge:P CdS:Cl CdS:In Si:Sb Si:P Si:As GaP:Zn WSe :a 2 WO :Na 3 5 6 7 1/3 logn c CH OH:Li 3 Ar:Na Xe:Hg 4.3 ( ) n c cm 3 (4.24) P. Edwards and M. Sienko, Phys. Rev. B 17, 2575 (1978) ( ) (metal-insulator transition) ([2] [5]) () () n c a B 4.3 n 1/3 c a B =.26 (4.24) (4.24) Mott (Mott Mott s criterion) 4.3 Mott BEC 5 4-6
G [1] M. Lundstrom, Fundamentals of Carrier ransport (Cambridge, 2). [2] N. F. Mott, Metal-Insulator transitions (CRC Press, 199); (1996). [3] H. Kamimura and H. Aoki, he Physics of Interacting Electrons in Disordered Systems (Oxford, 199). [4] (, 22). [5] D. Stauffer and A. Aharony, Introduction to Percolation heory (CRC Press, 1994); (, 21). E B F = e(e + v B) (3.2) (3.21) f 1 f f e f (E + v B) k = f 1 τ (p = k) (E.1) f/ k f f 2 de = v dp f / k = ( f / E)v f v B ( v ) f 1 E a ev E f E e (v B) f 1 k = f 1 τ (E.2) f 1 = eτ(v E a ) f E 3.3 (E.4) v E = v E a + eτ m (v B) E a, E = E a eτ m B E a E a = 1 1 + ω 2 c τ 2 [ E + eτ m B E + ( eτ m ) 2 (B E)B ω c = e B m f 1 f 1 = eτe 1 + ω 2 c τ 2 [ v + eτ m v B + ( eτ m ) 2 (B v)b (E.3) (E.4) ], (E.5) ] f E. (E.6) (E.7) B = (,, B z )E = (E x, E y, ) v z = (E.7) f 1 f 1 = e f ( τ [v x E 1 + (ω c τ) 2 E ω c τ 2 ) ( x 1 + (ω c τ) 2 E ωc τ 2 )] y + v y 1 + (ω c τ) 2 E τ x + 1 + (ω c τ) 2 E y (E.8) 4-7
j x = en v x f = f + f 1 v x f v k j x = 2 ( e)v x f(k) dk (2π) 3 = e2 4π 3 τv 2 x 1 + (ω c τ) 2 (E x (ω c τ)e y ) f E dk. (E.9) vx 2 E ξ(e) vxξ(e)dk 2 = 2 3m Eξ(E)dk. (E.9) (E.1) f = A exp( E/k B ) D(E) = A D E 1/2 (E.1) f E = f k B, n = A D (E.9) [ j x = ne2 τ m 1 + (ω c τ) 2 f E 1/2 de = 2A D 3k B E (ne 2 /m )(A l E x A t E y ) (A l A t ) E 3/2 f de ωc τ 2 ] E x 1 + (ω c τ) 2 E y, (E.11a) E (E.11b) E (4.2) j y xy ( ) j = ne2 Al A t m E. (E.12) A t A l F 2 AB BAB m + m B m 2 B A B + V BA = Π BA ( ) Π BA ( + ) + (τ B τ A ) = dv BA d V BA ( ) Π BA( + ) + dπ BA d + τ B τ A =, + τ B τ A = d d ( ΠBA ) = τ B τ A. S AB = Π AB, ds AB d = τ A τ B (F.1) G A() () 2 ab ( A L R) H = H a + H b () H a H b = Hab H aa = H bb = 4-8
A ψ(t) = c a (t)ϕ a e E at/ + c b (t)ϕ b e E bt/ (G.1) Schrödinger H ψ = i ψ/ t c a H a e ie at/ + c b H b e ie bt/ = i[ċ a a e ie at/ + ċ b b e ie bt/ ] a b c a c b dc a dt = i c bh dc b dt = i c ah bae iωt. ω E b E a abe iω t, A A 2 ω 2 a c a () = 1c b () = (G.13) c a c b dc a /dt = dc b /dt = c () a (t) = 1 c () b (t) = (G.2) (G.3) c (2) a (t) = 1 1 2 t c (1) a (t) = 1, c (1) b (t) = i [ t dt H ab(t )e iω t t dt H ba(t )e iω t ] H ba(t )e iωt dt,, c (2) b (t) = i t H ba(t )e iω t dt. (G.4a) (G.4b) H (t) = V cos ωt (G.5) ω a b ω V ab = a V b (G.4b) c b (t) i V ba i V ba t dt cos ωt e iω t = V ba 2 sin[(ω ω)t/2] e i(ω ω)t/2. ω ω [ e i(ω +ω)t 1 ω + ω ] + ei(ω ω)t 1 ω ω (G.6) 4-9
ω ω ω + ω ω ω P b (t) = c b (t) 2 V ba 2 2 sin 2 [(ω ω)t/2] (ω ω) 2 (G.7) 2 sin 2 [(ω ω)π/2] (ω ω) 2 (G.7) t b (ω ω)/(2π) V ba 2 t ω H = ee z cos ωt 1-1 -5 5 1 ω ω (G.8) (G.5) (G.7) P b (t) pe 2 2 sin 2 [(ω ω)t/2] (ω ω) 2, p e a z b (G.9) (G.9) a b 2 2 a b 4.4 (a) (G.9) c a () = 1 c b () = 1 (G.9) 4.4(b) (sitimulated emission, or insused emission) b 4.4(c) (spontaneous emission) LED (G.8) H ωk = ie z[a kλ 2ϵ V e ikr a kλ e ikr ] s (G.1) a k k V s ω k V s ϵ E 2 E cos ω k t 1 n k b, n k a, n k + 1 W = 2π ( ) H ba 2 δ(e b E a ω k ) = 2πe2 ωk 1, n k+1 a kλ 2ϵ V z 2, n kλ 2 δ(e b E a ω k ) s ( ) = 2πe2 ωk a z b 2 (n k + 1)δ(E b E a ω k ) 2ϵ V s (G.11) (a) Œõ zžû (b) U ± úo (c) Ž R úo 4.4 2 4-1
n k 1 (G.6) 1 H ba = (V ba/2)e iωt (G.13) c a H ab = V ab 2 eiωt (G.12) dc a dt = i 2 c bv ab e i(ω ω)t, dc b dt = i 2 c av ba e i(ω ω)t. d 2 c b dt 2 (G.13) + i(ω ω ) dc a dt + V ab 2 (2) 2 = (G.14) λ ± 1 2 (δ ± δ 2 + V ab 2 / 2 ), δ ω ω (G.15) c a () = 1c b () = c b (t) = c + e iλ +t + c e λ t (G.16) ω R (Rabi frequency) c b (t) = i V ab ω R eiδt/2 sin(ω R t/2), ( ) c a (t) = e [cos iδt/2 ωr t i δ ( )] (G.17) ωr t sin 2 ω R 2 ω R (ω ω) 2 + V ab 2 / 2 (G.18) (G.17) δ ab (Rabi oscillation) 4-11