1 1,,,.,,,,,.,.,,,.,, Computer-Aided Design).,,, Boltzmann,, [1]., Anderson, []., Anderson, Schrödinger [3],[4]., nm,,.,,,.,, Schrödinger.,, [5],[6].,,.,,, 1
.,, -. -, -., -,,,. Wigner-Boltzmann Schrödinger, de Broglie, Madelung, Bohm [7],[8], Schrödinger Madelung Wigner-Wyle [9]., R d,d=1,, 3, Single-state Schrödinger i t ψ i(x, t) = m ψ i (x, t)+v (x, t)ψ i (x, t) (1), Wigner-Wyle. V (x, t) i E i ψ i. ψ ψ, ρ(x, x )= i ψ i (x)ψ i (x )α i (). α i i., k, T, β =1/kT., ψ i Schrödinger, Heisenberg : i ρ t = m (Δ x Δ x ) ρ +(V (x) V (x )) ρ. (3), Wigner rotated Fourier : 1 f w (r, p) = (π ) d ρ(r + 1 r,r 1 r )e ipr dr. (4) Heisenberg x = r + r / x = r r / Fourier, Q(f w ), Wigner Wigner-Boltzmann : θ[v ] θ[v ]f w = i t f w(r, p)+ p m rf w (r, p) θ[v ]f w = Q(f w ). (5) 1 (π ) d (V (r + r r ) V (r ) ) f w (r, p)e i(p p )r dp dr (6)., Tayler,
3 t f w(r, p)+ p m rf w r V p f w 0 α=1 α ( 1) α 4 α (α + 1)! ( rv p f w ) α+1 = Q(f w ) (7) θ[v ]f w r V p f w (8) Wigner-Boltzmann Boltzmann 3 Wigner A(p) A = A(p)f w (r, p)dp (9). u macroscopic fluid velocity p /m p = mu + p (10),,,, n def = < 1 >= f w (r, p)dp, (11) def P ij = p i p j p m = i p j m f w(r, p)dp, (1) W def = p m = 1 mnu 1 Tr(P ij) (13). Wigner-Boltzmann Chapman-Enskog [5],[6]. Wigner-Boltzmann A =1 p p /m 0 1, α +1 3 (7) 4., τ p τ w, : n t + 1 Π i =0, i =1,, 3, (14) m Π j t + (u i Π j P ij )= n V x j mnu j τ p, (15) W t + (u i W u j P ij + q i )= Π i V (W W 0). (16) m τ w, W 0. n, u, Π i = mnu i,i=1,, 3 (17) 3
4. q i., Wigner,,,., Fermi-Dirac Boltzmann ρ(x, x )= i ψ i (x)ψ i (x )ce βe i (18) β =1/kT,, β ρ(x, x )= ( 4m x ρ(x, x )+x ρ(x, x ) ) 1 (V (x)+v (x )) ρ(x, x ) (19), Bloch. f w0 Wigner x = r + r / x = r r / (19) Wigner-Weyle ( ) β f w 0 (r, p) = 8m r p f w0 (r, p) m V β (r, p p )f w0 (r, p )dp (0). V β (r, p) = ) 1 1 (V (π ) d (r + r r )+V(r ) e ipr dr (1). (0) Taylor ( ) β f w 0 (r, p) = 8m r p f w0 (r, p) m. ε = f w0 f w0 (r, p) = α=0 α ( 1) α (α)!4 α α r V (r)p α f w0 () ε k φ k (r, p) (3) k=0, Wigner., (3) () ε, Wigner, f w0 φ 0 + εφ 1, (4), p β( φ 0 = Ae m +V ), (5) )) p β( φ 1 = Ae m +V ) 1 ( β r V + (( β3 r V ) + p 8m 3 m r V (6)., Wigner, ( p β( f w0 (r, p) Ae m +V ) 1 β ( r V β )) ) (( r V ) + p 8m 3 m r V + O( 4 ) (7) 4
, f w0 (r, p) ( ( p β( = Ae m +V ) 1+ β V 8m x k 5 + β3 4m ( V x k ) + β3 4m p kp V l x k x l ) ) + O( 4 ) (8)., k =1,, 3, l =1,, 3.,, ( ( n = Ce βv 1+ β V 1m x k ( ) + β3 V ) ) + O( 4 ), (9) 4m x k P ij = n β δ ij β 1m n V x j + O( 4 ), (30) W = 1 mnu + 3. n β + β V 4m n x k + O( 4 ) (31) 4 - P ij W (15) (16).,, (14) (15) n t + x (nu i)=0, (3) t (mnu i)+ ( ) mnu i u j + kt n + β x j 1m n V = n V mnu i (33) x j τ p. β (9),. V = 1 log n + O( ). (34) x j β x j Ancona (33) ( ) β n V 1m x j x j 1m x j = n 6m ) ln(n) x j ( 1 ) n n x j (n (35) [5]., (33) t (mnu i)+ (mnu i u j + kt n) x j 6m n ( 1 ) n n x = n V mnu i (36) j τ p. J j = qnu j q = e. -., (36) 5
6 τ p t J i + kt qτ p m. n qτ p m 6m n ( 1 ) n n x = qτ p j m n V J i (37) μ = eτ p m (38) Einstein D = μ kt e (39) V = eϕ (40), (37) J i = ed n eμn (ϕ + 6em 1 ) n n. (3), μ 1 Ω R d (d 1) : x j (41) λ Δϕ = n f, (4) ( t n + div n (ϕ ln(n)+b Δρ ) ρ ) =0. (43) ρ = n, b = /6em. λ., γ n = bδρ/ρ - (DD), DD. ) v = ϕ ln(n)+b Δρ ρ (44) 4 : λ Δϕ = n f, (45) t n + div(n v) =0, (46) b Δρ ln(n)+ϕ = v. ρ (47),,, (ϕ, v, ρ), (47) ρ [11].,, -. 6
7 5 - - (DD),,, [10]. - (QDD),, [11], [1], QDD [13]. : (A.1) Ω R d d =1,, 3 (A.) Ω Ω D Dirichlet Ω N Neumann Ω \ (Ω D Ω N ) (A.3) H 1 (Ω) L p (Ω) H (Ω) W 1,q (Ω) 1/p +1/q =1/ p, q (, ]., θ (0, 1), C>0, a W 1,q (Ω), θ a 1/θ, g L (Ω) ψ D W 1,q (Ω) div(a ψ) =g, ψ ψ D H0(Ω 1 Ω N ) ψ W 1,q (Ω), ψ W 1,q (Ω) C( ψ D W 1,q (Ω) + g L (Ω)). (A.4) (ϕ D,v D,u D ) (H 1 (Ω) L (Ω)) 3. (A.5) f L (Ω). QDD, ρ = n = e u (47), λ Δϕ = e u f, (48) b (ρ u)+ρu = ρ (ϕ v), in Ω (49) div(n v) =0, (50). ϕ = ϕ D, u = u D, v = v D,on Ω D, (51) ϕ ν = u ν = v ν =0,on Ω N (5), (ϕ, v, u) QDD. (49) ρ = e u u, ρ [14] (48)-(50), (ϕ, v, u) [13] w L (Ω) (P1) ϕ 7
8 λ Δϕ = e w f. (53) (P) v div(e w v) =0. (54) (P3) u b (e w u)+e w u = e w (ϕ v). (55) (P1) (P3) Lax-Milgram X = {w L (Ω) : U w U} T : X X, T (w) =u Stampacchia, T, Schauder [13]: 1 (ϕ, v, u) (H 1 (Ω) L (Ω)) 3 (48)-(5) Ω ϕ ϕ ϕ, v v v, U u U ϕ, ϕ, v, v, U, U 1( (51)-(5) (48)-(50) H 1 (Ω) L (Ω)., A(ρ) = Δρ/ρ [1],, (A.3) T [13]. v D W 1,q T L p 1/p +1/q =1/ QDD., T, QDD ρ. 6,. {t k } k N τ k = t k t k 1 Euler (ϕ, n, u) QDD : n k n k 1 div( n k n k (ϕ k + γ τ n)) k = 0, k (56) b (ρ k u k )+ρ k u k = ρk (ϕk v k ), (57) λ Δϕ k = n k f. (58) n(x, 0) = n 0 (x), ϕ k =0, u k = ϕ b /, n k = n D, on Ω (59) 8
9 QDD, ϕ b. Lyapunov : ( ) ). W k = Ω (b ρ k +(n k (ln n k 1) + 1) + λ ϕk )dx (60) W k+1 W k (61) [15]. QDD, - [14]. Ω i Ω Ω = i Ω i,ω i QDD (49)-(50) G = ϕ + b ρ/ρ, J = e G η, η = e v (6) F = ρ u, ρ = e u (63) Tikhonov-Samarskii [16] (49) (50) (49), (50) Green n t dx e G η ds =0, (64) Ω i Ω i ν b ρ u Ω i ν ds + ρudx + 1 ρ(ϕ v)dx (65) Ω i Ω i xi+1 n t dx = J i+1/ J i 1/, (66) xi+1/ b(f i+1/ F i 1/ ) u i ρdx = 1 xi+1/ 1/ (ϕ i v i ) ρdx (67) 1/ [,+1 ] F J F i+1/ or J i+1/ = η i+1 η i xi+1 e θ, θ = G or u (68) dx, (66), (67),. xi+1 e θ dx θ 9
10 xi+1 e θ dx = h i+1 e θ i+1 +θ i (69) (69) (68),., θ,,, xi+1 e θ dx = h i+1e θi+1 B(θ i+1 θ i ) (70)., B( ) Bernoulli J i+1/ = 1 h i+1 (B(G i+1 G i )n i+1 B(G i G i+1 )n i ), (71) F i+1/ = 1 h i+1 e u i+1 B(u i+1 u i )(u i+1 u i ) (7), QDD : n k n k 1 τ k = B(Gk i+1 Gk i )nk i+1 (B(Gk i Gk i+1 )+B(Gk i Gk i 1 ))nk i + B(Gk i 1 Gk i )nk i 1 h,(73) b(ρk i+1b(u k i+1 u k i )(u k i+1 u k i ) ρ k i B(u k i u k i 1)(u k i u k i 1)) h +Λ k i u k i = Λk i (ϕk i v k i ). (74), Λ i = +1 ρdx, Tikhonov-Samarskii,., Boltzmann., DD G = ϕ Bell Scharfetter Gummel Scharfetter-Gummel [17],, CAD. 7,,., Schrödinger. -,,.,, 10
11 [18].,,. [ 1 ] T.Grasser, T-W.Tang, H.Kosina, S.Selberherr, A review of hydrodynamic and energy-transport models for semiconductor device simulation, IEEE Proc., 91(003), 51-74. [ ] S.Kotani, Ljapunov indices determine absolutely continuous spectra of sationary random one-dimensional Schrödinger operators, Proc. of Taniguchi Symp. SA. Katata(198),5-47. [3],,,38(1986),53-61. [4], II,,40(1986),193-01. [ 5 ] M.G.Ancona and G.J.Iafrate, Quantum correction to the equation of state of an electron gass in a semiconductor, Phys.Rev.B, 39(1989), 9536-9540. [ 6 ] C.L.Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl.Math., 54(1994), 409-47. [ 7 ] D.Bohm, A suggested interpretation of the quantum theory in terms of hidden variables I, Phys.Rev., 85(195),166-179. [ 8 ] D.Bohm, A suggested interpretation of the quantum theory in terms of hidden variables II, Phys.Rev., 85(195),180-193. [ 9 ] R.E.Wyatt, Quantum dynamics with trajectories: Introduction to quantum hydrodynamics, Springer,(005). [10] J.W.Jerome, Analysis of charge transport, Springer-Verlag Berlin Heidelberg, (1996). [11] R.Pinnau and A.Unterriter, The stationary current-voltage characteristics of the quantum drift-diffusion model, SIAM J.Numer.Anal., 37(1999), 11-45. [1] A.Unterreiter, The thermal equibrium solution of a generic bipolar quantum hydrodynamic model, Comm. Math. Phys., 188(1997), 69-88. [13] S.Odanaka, A numerical scheme for quantum hydrodynamics in a semiconductor, RIMS Kokyuroku, Kyoto University,1495(006), 51-59. [14] S.Odanaka, Multidimensional discretization of the stationary quantum drift-diffusion model for ultrasmall MOSFET structures, IEEE Trans. CAD of ICAS, 3(004), 837-84. [15] S.Gallego and F.Méhats, Entropic discretization of a quantum drift-diffusion model, SIAM J. Numer.Anal., 43(005), 188-1849. [16] G.I.Maruchuk, Methods of numerical mathematics, Springer-Verlag, (198). [17] D.L.Scharfetter and H.K.Gummel, Large signal analysis of a silicon Read diode oscillator, IEEE Trans. Elec. Dev., 16(1969), 64-77. [18] S.Odanaka, A high-resolution method for quantum confinement transport simulations in MOSFETs, IEEE Trans. CAD of ICAS, 6(007), 80-85. ( 007 5 9 ) ( ) 11