Quantum kinetic models of open quantum systems in semiconductor theory

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1 Quantum kinetic models of open quantum systems in semiconductor theory Chiara Manzini Scuola Normale Superiore di Pisa Westfälische Wilhelms-Universität Münster 29 Aprile 2005 Quantum kinetic models of open quantum systems in semiconductor theory p.1/30

2 Outlook 1- Motivation 2- Quantum kinetic formulation 3- Analytical difficulties 4- Open quantum systems: two examples 5- The three well-posedness results 6- New tools and perspectives 7- Final considerations Quantum kinetic models of open quantum systems in semiconductor theory p.2/30

3 1- Motivation Trend in semiconductor technology: miniaturization = Challenge: simulation tools Features of novel devices = Quantum effects Prototype: Resonant Tunneling Diode Quantum kinetic models of open quantum systems in semiconductor theory p.3/30

4 Density Density Doping Quantum kinetic models of open quantum systems in semiconductor theory p.4/30

5 QS Quantum description Quantum System with d degrees of freedom: a pure state of QS ψ L 2 (R d ; C), a physical state of QS ˆρ density matrix on L 2 (R d ; C), i.e. ρ(x,y) L 2 (R 2d ; C) kernel ρ(x,y) = j N λ j ψ j (x)ψ j (y) }{{} pure state {ψ j } j complete orthonormal set, λ j 0, j λ j = 1. The position density:, n(x) = j λ j ψ j 2 (x) L 1 (R d ; R + ) finite mass. Quantum kinetic models of open quantum systems in semiconductor theory p.5/30

6 Quantum Evolution: reversible dynamics { i d dt ψ j = 1 2 xψ j + V ψ j, j N, x R d V (x,t) = n(x,t) = j λ j ψ j 2 (x,t) ρ(x,y,t) = j N λ j ψ j (x,t)ψ j (y,t), {ψ j (t)} j L 2 (R d ; C) E.g. evolution of an electron ensemble with d d.o.f., ballistic regime, mean-field approximation Quantum kinetic models of open quantum systems in semiconductor theory p.6/30

7 Quantum Evolution: reversible dynamics { i d dt ψ j = 1 2 xψ j + V ψ j, j N, x R d V (x,t) = n(x,t) = j λ j ψ j 2 (x,t) w(x,v) = F η v j N λ j ψ j (x + η 2 )ψ j(x η 2 ) Quantum Liouville equation: Wigner-Poisson { wt + v x w Θ[V ]w = 0 x V (x,t) = n[w](x,t) = w(x,v,t)dv F η v {Θ[V ]w}(x,v):=i [V (x + η/2) V (x η/2)]f v η w(x,η) Quantum kinetic models of open quantum systems in semiconductor theory p.6/30

8 2- Quantum Kinetic description A physical state of QS a quasiprobability on the phase space, w : (x,v) R 2d R = ( w(x,v) := F η v ρ x + η 2,x η ) L 2 (R 2d ; R) 2 w is the Wigner function for the physical state of QS The position density n[w](x,t) := w(x,v,t)dv w L 2 (R 2d ; R) n[w] well-defined, non-negative Quantum kinetic models of open quantum systems in semiconductor theory p.7/30

9 3- Analytical difficulties w L 2 (R 2d ; R) NOT n[w] well-defined w(x,v,t) > < 0 n[w](x,t) := w(x,v,t)dv > < 0 Quantum counterpart : {ψ j (t)} j L 2 (R d ; C), λ j 0, j λ j = 1 n(t) = j λ j ψ j 2 (t), n(t) L 1 <, t (mass cons.) ψ j (t) 2 L 2 = const. First a-priori estimate for Schr.-Poisson systems Quantum kinetic models of open quantum systems in semiconductor theory p.8/30

10 3- Analytical difficulties w L 2 (R 2d ; R) NOT n[w] well-defined e kin (x,t) := 1 2 v 2 w(x,v,t)dv: w(x,v,t) > < 0 e kin (x,t) > < 0 Quantum cp. : E kin (t) := 1 2 j λ j ψ j (t) L j λ j ψ j (t) 2 L V (t) 2 L 2 }{{} E 2 pot(t) =const. (energy cons.) ψ j (t) 2 L 2 = const. = can NOT apply physical cons. laws Quantum kinetic models of open quantum systems in semiconductor theory p.8/30

11 Advantages w L 2 (R 2d ; R) is necessary for a physical state of QS L 2 -setting is suitable for numerical approximation (cf.[arnold... 96]) pseudo-differential operator Θ[V ]w L 2 x,v = [V (x + η/2) V (x η/2)]f v η w L 2 x,η < Θ[V ]w,w > L 2 v = 0 (skew-simmetry) w t + v x w Θ[V ]w = 0 = w(t) L 2 x,v = const. Quantum kinetic models of open quantum systems in semiconductor theory p.9/30

12 4- Open QS: irreversible dynamics Motivation for quantum kinetic approach Two examples: electron ensemble with d d.o.f. in RTD [Frensley90]: Time-dependent boundary conditions (b.c.) for w model ideal reservoir Boundary-value problem for Wigner-Poisson: inflow time-dependent b.c. for w mixed Dirichlet-Neumann b.c. for V Remark: Reformulation with {ψ j } j impossible: λ j NOT const. Quantum kinetic models of open quantum systems in semiconductor theory p.10/30

13 4- Open QS: irreversible dynamics Motivation for quantum kinetic approach Two examples: electron ensemble with d d.o.f. in RTD electron ensemble with d d.o.f. in GaAs: electron-phonon interaction Semiclassical cp. : Boltzmann for semiconductors Density-matrix form. : NOT suitable for boundaryvalue problems Quantum kinetic models of open quantum systems in semiconductor theory p.10/30

14 Open QS: irreversible dynamics w t + v x w Θ[V ]w = Qw Qw = 1 τ (w M ρ) [Degond,...03] QDD,QET Quantum kinetic models of open quantum systems in semiconductor theory p.11/30

15 Open QS: irreversible dynamics w t + v x w Θ[V ]w = Qw Qw = 1 τ (w M ρ) [Degond,...03] QDD,QET Boltzmann-like collision operator [Demeio,...04] simulation Quantum kinetic models of open quantum systems in semiconductor theory p.11/30

16 Open QS: irreversible dynamics w t + v x w Θ[V ]w = Qw Qw = 1 τ (w M ρ) [Degond,...03] QDD,QET Boltzmann-like collision operator [Demeio,...04] simulation scattering term for electron-phonon interaction [Fromlet,...99] Quantum kinetic models of open quantum systems in semiconductor theory p.11/30

17 Open QS: irreversible dynamics w t + v x w Θ[V ]w = Qw Qw = 1 τ (w M ρ) [Degond,...03] QDD,QET Boltzmann-like collision operator [Demeio,...04] simulation scattering term for electron-phonon interaction [Fromlet,...99] quantum Fokker-Planck term [Castella,...00] [Jüngel,...05] QHD Classical cp. :Vlasov-Fokker-Planck Quantum kinetic models of open quantum systems in semiconductor theory p.11/30

18 Quantum Fokker-Planck term Qw = βdiv v (vw)+σ v w+2γdiv v ( x w) + α x w (QFP) with α,β,γ 0, σ > 0. [Caldeira...83]: QS={electrons+reservoir} β ξ coupling, σ T temp., γ,α 1 T, ǫ ξ T Lindblad condition: α σ γ 2 + β 2 /16 Model Term Accuracy Lindblad Classical β div v (vw) + σ v w O(ǫ 2 ) NOT ok Frictionless σ v w O(ǫ) ok Quantum (QFP) O(ǫ 3 ) ok Quantum kinetic models of open quantum systems in semiconductor theory p.12/30

19 5- Three well-posedness results WP system with inflow, time-dependent b.c. : d = 1, global-in-time well-p. [M,Barletti04] d = 3, local-in-time well-p. [M05] WPFP system, d = 3 [Arnold,Dhamo,M05]: global-in-time well-p., NEW a-priori estimates NEW strategy for well-p. of WP(FP) Common feature: L 2 -setting, ONLY kinetic tools Quantum kinetic models of open quantum systems in semiconductor theory p.13/30

20 WP system with inflow b.c. d = 1, 3 { { t + v x Θ[V (t) + V e (t)]}w(t) = 0, t 0, x V (x,t) = n[w](x,t) = w(x,v,t)dv (x,v) Ω R d, Ω R d open, bounded, convex, V e V e (x,t), (x,t) R d R + w(s,v,t) = γ(s,v,t), (s,v) Φ in,t 0, V (x,t) = 0, x Ω,t 0, w(x,v, 0) = w 0 (x,v) Φ in := { (s,v) Ω R d v n(s) > 0 } Easy: non homogeneous Dirichlet/Neumann b.c. Quantum kinetic models of open quantum systems in semiconductor theory p.14/30

21 Weighted space: X k := L 2 (Ω R d ; (1 + v 2k )dxdv), d = 1, 3 k = 1, 2 cf. [Markowich,... 89,Arnold,... 96] Prop. : d = 3, w X 2 n[w] L 2 C w X2 Quantum kinetic models of open quantum systems in semiconductor theory p.15/30

22 Weighted space: X k := L 2 (Ω R d ; (1 + v 2k )dxdv), d = 1, 3 k = 1, 2 cf. [Markowich,... 89,Arnold,... 96] Prop. : d = 3, w X 2 n[w] L 2 C w X2 = V solution of Poisson pb. with n[w] L 2 (Ω): V W 1,2 0 (Ω) W 2,2 (Ω), V W 2,2 C n[w] L 2 Define P : X 2 W 2,2 (R 3 ) P : w X 2 V Pw V a.e. Ω,Pw = 0 outside Σ with Σ open, Ω Σ and Pw W 2,2 (R 3 ) C w X2 Quantum kinetic models of open quantum systems in semiconductor theory p.15/30

23 WP system with inflow b.c. in X 2 w t + v x w Θ[Pw]w = 0, t 0, w(s,v,t) = γ(s,v,t), (s,v) Φ in,t 0, w(x,v, 0) = w 0 (x,v) X 2 for all u X 2, Θ[U]u X2 : contains v 2 i Θ[U]u L 2 F η v { v 2 i Θ[U]u } = 2 η i { [U(x + η/2) U(x η/2)]f v η u} contains ηi, 2 x i U and v i, v 2 i u Quantum kinetic models of open quantum systems in semiconductor theory p.16/30

24 WP system with inflow b.c. in X 2 w t + v x w Θ[Pw]w = 0, t 0, w(s,v,t) = γ(s,v,t), (s,v) Φ in,t 0, w(x,v, 0) = w 0 (x,v) X 2 for all u X 2, Θ[U]u X2 : contains v 2 i Θ[U]u L 2 F η v { v 2 i Θ[U]u } = 2 η i { [U(x + η/2) U(x η/2)]f v η u} Prop. : Θ[U]u X2 C U W 2,2 (R 3 ) u X2 Θ[Pw]w X2 C Pw W 2,2 (R 3 ) w X2 C w 2 X 2 Quantum kinetic models of open quantum systems in semiconductor theory p.16/30

25 WP system with inflow b.c. in X 2 { { t T γ(t) Θ[Pw](t)}w(t) = 0, t 0, w(x,v, 0) = w 0 (x,v) X 2 w(t) D(T γ(t) ) := {u X 2 v x u X 2,u Φin = γ(t)} Remark u 1,u 2 D(T γ(t) ) u 1 u 2 D(T 0 ) p : [0, + ) X 2, s.t. p(t) D(T γ(t) ), t 0 D(T γ(t) ) = p(t) + D(T 0 ) (cf.[barletti00]) Quantum kinetic models of open quantum systems in semiconductor theory p.17/30

26 WP system with inflow b.c. in X 2 { { t T γ(t) Θ[Pw](t)}w(t) = 0, t 0, w(x,v, 0) = w 0 (x,v) X 2 w : [0, + ) X 2, s.t. w(t) D(T γ(t) ), t 0 w(t) = p(t) + u(t), with u : [0, + ) D(T 0 ) { { t T 0 L p (t) Θ[Pu](t)}u(t) = Q p (t) u(x,v, 0) = w 0 (x,v) p(x,v, 0) D(T 0 ) L p (t)u(t) := Θ[Pp](t)u(t) + Θ[Pu](t)p(t) Q p (t) := { t T γ(t) Θ[Pp](t)}p(t) Choose p s.t. Q p C[0, + ),Q p L 1 (0,T], T (1) Quantum kinetic models of open quantum systems in semiconductor theory p.17/30

27 WP system with inflow b.c. in X 2 : local-in-t solution { { t T γ(t) Θ[Pw](t)}w(t) = 0, t 0, Theorem w(x,v, 0) = w 0 (x,v) X 2 γ s.t. pas in (1), w 0 D(T γ(0) ), t max s.t.! classical solution w(t), t [0,t max ). If t max <, then lim tրtmax w(t) X2 =. Ex. : γ C 1 ([0, );L 2 ( Ω R d ; (v n(s))(1 + v 4 )dsdv)) A priori estimates are needed for global-in-t result Quantum kinetic models of open quantum systems in semiconductor theory p.18/30

28 WPFP system, d = 3 { t + v x Θ[V (t)]}w(t) = Qw(t) x V (x,t) = n[w](x,t) w(x,v,t = 0) = w 0 (x,v) w w(x,v,t), (x,v,t) R 2d [0, ) Qw = βdiv v (vw) + σ v w + 2γdiv( x w) + α x w Assume α σ > γ 2 Q uniformly elliptic in x and v. Classical cp. : Vlasov-PFP with non-linear term: x V v w, FP term: β div v (vw) + σ v w. Frictionless FP term: σ v w hypoelliptic case. Quantum kinetic models of open quantum systems in semiconductor theory p.19/30

29 Existing results density matrix ˆρ : [Arnold,...03] global-in-t solution, by conservation laws. Quantum Kinetic: L 1 -analysis [Cañizo,...04] global-in-t solution, by ˆρ(t) 0, cons. laws: w(t) L 1 (R 2d ; R) n[w](t) L 1 (R d ) (mass cons.) n[w](t) L 1 = const. e kin [w](x,t)> e kin [w](x,t)< INSTEAD keep to L 2 -setting, purely kinetic analysis Quantum kinetic models of open quantum systems in semiconductor theory p.20/30

30 Weighted space: X 2 := L 2 (R 6 ; (1 + v 4 )dxdv) Prop. : w X 2 n[w] L 2 C w X2 = V (x) := 1 4π x n[w](x), x R3 : V / L p, p, V = x 4π x 3 n[w] L6. F v Θ[V ]z(x,η) = i (V (x + η/2) V (x η/2)) }{{} =:δv (x,η) For η fixed, δv (.,η) L C η 1 2 n[w] L 2 F v z(x,η) Θ[V ]z L 2 C n[w] L 2 η 1 2 Fv z L 2 Quantum kinetic models of open quantum systems in semiconductor theory p.21/30

31 Weighted space: X 2 := L 2 (R 6 ; (1 + v 4 )dxdv) V / L p, p, V = x 4π x 3 n[w] L6. Θ[V ]z L 2 C n[w] L 2 η 1 2 Fv z L 2 Prop. : w,z, v z X 2 Θ[V ]z X2 C{ z X2 + v z X2 } w X2 parabolic reguralization is needed Quantum kinetic models of open quantum systems in semiconductor theory p.21/30

32 The linear equation w t = Aw(t), t > 0, Aw := v x w + Qw w(t = 0) = w 0 X 2, {e ta,t 0} C 0 semigroup on X 2 (cf. [Arnold,...02]) w(t) L 2 e 3 2 βt w 0 L 2, t 0 w(x,v,t) = G(x,v,t) w 0 (cf. [Sparber,...03]) v w(t) X2 B t 1/2 e κt w 0 X2, 0 < t T 0 (cf. classical cp. VPFP [Carpio98]) Quantum kinetic models of open quantum systems in semiconductor theory p.22/30

33 The non-linear equation T (0, ) fixed w t = Aw(t)+(Θ[V ]w)(t), t [0,T], w(t = 0) = w 0, Y T := {z C([0,T];X 2 ) v z C((0,T];X 2 ), t 1/2 v z(t) X2 C T t (0,T)}. T < T 0, the map w Y T w s. t. w(t) = e ta w 0 + t 0 e (t s)a (Θ[Ṽ ]w)(s)ds, t [0,T] is a strict contraction on some ball of Y τ with τ( w 0 X2 ). Quantum kinetic models of open quantum systems in semiconductor theory p.23/30

34 The non-linear equation T (0, ) fixed w t = Aw(t)+(Θ[V ]w)(t), t [0,T], w(t = 0) = w 0, Y T := {z C([0,T];X 2 ) v z C((0,T];X 2 ), t 1/2 v z(t) X2 C T t (0,T)}. Existence and Uniqueness Theorem w 0 X 2, t max s.t.! mild solution w Y T, T < t max. If t max <, then lim tրtmax w(t) X2 =. Quantum kinetic models of open quantum systems in semiconductor theory p.23/30

35 Global-in-t well-posedness A priori estimates w(t) 2 e 3 2 βt w 0 2, t 0, for v 2 w(t) 2 < 0 t T, some L p -bounds for x V (t) are needed Idea: get bounds for x V (t) depending on w(t) 2 Strategy: exploit dispersive effects of the free-str. (cf. [Perthame96]) = estimate(define) x V via integral equation definition of n[w] is by-passed Anticipation: global-in-t well-posedness WITHOUT WEIGHTS Quantum kinetic models of open quantum systems in semiconductor theory p.24/30

36 A priori estimate for electric field: WP case R 3 v E(x,t)= V (x,t)= 1 x 4π x n(x,t) 3 w(x,v,t) dv= w 0 (x vt,v) dv+ R 3 v t 0 R 3 v Correspondingly E into E 0 (x,t) = 1 x 4π x 3 x E 1 (x,t) = 1 4π x x 3 x (Θ[V ]w)(x vs,v,t s)dsdv R 3 v R 3 v w 0 (x vt,v)dv t 0 (Θ[V ]w)(x vs,v,t s)ds Quantum kinetic models of open quantum systems in semiconductor theory p.25/30

37 A priori estimate for electric field: WP, E 1 n 1 (x,t) = t 0 (Θ[V ]w)(x vs,v,t s)dsdv Classical cp. : Vlasov-Poisson [Perthame96] t n 1 (x,t) = E v w (x vs,v,t s)dsdv R 3 v 0 = div x R 3 v t 0 s(e w)(x vs,v,t s)dsdv Θ[V ]w = F 1 η (δv ŵ) = F 1 η ( W[E] v w ) since δv (x,η) = V (x + η 2 ) V (x η 2 ) = η E(x rη)dr =: W[E](x,η) η Quantum kinetic models of open quantum systems in semiconductor theory p.26/30

38 A priori estimate for electric field: WP, E 1 n 1 (x,t) = t 0 (Θ[V ]w)(x vs,v,t s)dsdv Θ[V ]w = Fη 1 1 (δv ŵ) = Fη ( W[E] v w ) n 1 (x,t) = since δv (x,η) = V (x + η 2 ) V (x η 2 ) = η R 3 v = div x E(x rη)dr =: W[E](x,η) η t 0 R 3 v F 1 η ( W[E] v w )(x vs,v,t s)ds t 0 s F 1 η (W[E]ŵ)(x vs,v,t s)ds Quantum kinetic models of open quantum systems in semiconductor theory p.26/30

39 A priori estimate for electric field: WP, E 1 div x ( 1 4π E 1 (x,t) = 1 4π x x 3 n 1 (x,t) = ) x x t 3 0 s Fη 1 (W[E]ŵ)(x vs,v,t s)dvds Lemma : F 1 η (W[E]ŵ)(x vs,v,t s)dv L 2 x s 3/2 E(t s) L 2 x w(t s) L 2 x,v Quantum case: just in L 2. Classical case: L p -version. Quantum kinetic models of open quantum systems in semiconductor theory p.27/30

40 A priori estimate for electric field: WP, E 1 div x ( 1 4π E 1 (x,t) = 1 4π x x 3 n 1 (x,t) = ) x x t 3 0 s Fη 1 (W[E]ŵ)(x vs,v,t s)dvds Lemma : F 1 η (W[E]ŵ)(x vs,v,t s)dv L 2 x s 3/2 ( E 0 (t s) L 2 + E 1 (t s) L 2) w(t s) L 2 Prop. : E 0 (t) L 2 C T t ω with ω [0, 1) E 1 (t) L 2 C ( ) T, sup s [0,T] w(s) L 2 t 1/2 ω. NEW : w C([0,T];L 2 x,v) E 1 [w] L 1 t((0,t];l 2 x) Quantum kinetic models of open quantum systems in semiconductor theory p.27/30

41 A priori estimate for electric field: WP, E 0 E 0 (x,t) = 1 4π x x 3 x w 0 (x vt,v)dv Ex. Strichartz for free-str. [Castella-Perthame 96], w 0 (x vt,v)dv t 1 2 w0 L 1 x (L 6/5 v ), t > 0. L 6/5 x Let t (0,T] w 0 (x vt,v)dv E 0 (t) 2 C L 6/5 x C T t ω, ω [0, 1) w 0 (x vt,v)dv L 6/5 x (HP1) CC T t ω Quantum kinetic models of open quantum systems in semiconductor theory p.28/30

42 WPFP: global-in-t, smooth solution w(x,v,t) = G(t,x,v) w 0 + t 0 G(s, x, v) (Θ[V ]w)(t s)ds E 1 L 1 t ((0,T];L[2,6] x ) by parabolic reguralization. Classical cp. : VPFP [Castella98] Remark: E(t) L 3 is needed for bootstraping vw(t) 2, v 2 w(t) 2 C, 0 t T, T. Theorem[Arnold,Dhamo,M05] w 0 X 2 s.t. (HP1) holds,!w global mild solution, w(t) C (R 6 ), t > 0, n(t),e(t) C (R 3 ), t > 0. Quantum kinetic models of open quantum systems in semiconductor theory p.29/30

43 WPFP: global-in-t, smooth solution w(x,v,t) = G(t,x,v) w 0 + t 0 G(s, x, v) (Θ[V ]w)(t s)ds E 1 L 1 t ((0,T];L[2,6] x ) by parabolic reguralization. Remarks NOT used pseudo-conformal law (cf. classical cp.) w C([0,T];L 2 x,v) E 1 [w] L 1 t((0,t];l [2,6] x ) V 1 [w](x,t) := 1 x 4π x E 3 1 [w](x,t) V 1 [w] L 1 t ((0,T];L[6, ] x ) fixed-point-map for w WPFP: global-in-t, smooth solution, without weights Quantum kinetic models of open quantum systems in semiconductor theory p.29/30

44 7- Final considerations kinetic analysis, L 2 -framework: physically consistent, suitable for real device simulation, admissable parallelism with classical cp. by-pass the definition of the particle density Perspectives WPFP: hypoelliptic case, WP d = 3, all-space case a priori estimates in the bounded domain case long-time-behaviour: decay t of E(t), n(t) (cf. [Sparber,...04],[Perthame96]) Quantum kinetic models of open quantum systems in semiconductor theory p.30/30

Quantum Fokker-Planck models: kinetic & operator theory approaches

TECHNISCHE UNIVERSITÄT WIEN Quantum Fokker-Planck models: kinetic & operator theory approaches Anton ARNOLD with F. Fagnola (Milan), L. Neumann (Vienna) TU Vienna Institute for Analysis and Scientific