THERMODYNAMIC PRINCIPLES IN MODELING NANO - SCALE TRANSPORT IN SEMICONDUCTORS. NANOLAB, May C. Ringhofer, Arizona State University

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1 THERMODYNAMIC PRINCIPLES IN MODELING NANO - SCALE TRANSPORT IN SEMICONDUCTORS NANOLAB, May 2003 C. Ringhofer, Arizona State University 1

2 INTRODUCTION Transport of electrons and holes in semiconductors are described by a hierarchy of transport equations, ranging from fluid dynamic equations for plasmas to the Schrödinger equation. The more detailed the description, the more involved the numerical solution of the transport picture becomes. In addition, phenomena which drive fluid regimes, such as collisions, are not so well understood in the quantum mechanical framework. Phenomena not present in the macroscopic description, such as tunneling have to be taken into account. On very small spatial scales (5 nm or less) many of the macroscopic mechanisms become irrelevant. However, on an intermediate scale ( nm) all of the above effects play a role. 2

3 gate SiO 2 source drain N + channel N + p-type substrate Oxide thickness = 1.2 nm Channel length = 25 nm Source/Drain length = 50 nm Channel width = 0.5 µm Junction depth = 30 nm Substrate thickness = 64 nm Substrate doping: N A =10 19 cm -3 Doping of the source-drain regions: N D = cm -3 3

4 4

5 The goal is therefore to find amalgamates of macroscopic and microscopic (classical and quantum mechanical) descriptions which, approximately at least, account for all relevant effects. 5

6 CONTENTS S1 FREE TRANSPORT: Describes the motion of one single particle, or an ensemble of independent particles, in a vacuum. Serves to introduce different formulations of quantum mechanical transport and appropriate notations. S2 ENERGY BANDS: Describes the motion of a single particle in a periodic structure (a crystal). Homogenization problem for the Schrödinger equation via the Bloch decomposition. S3 COLLISIONS AND THE FERMI - GOLDEN RULE: Models interactions of electrons with the crystal lattice (phonons). 6

7 Quantum mechanical description results in a many body problem with a non-constant number of bodies. Approximations and non - local extensions of Fermi s Golden Rule S4 MACROSCOPIC APPROXIMATIONS: MOMENT CLOSURES AND QUANTUM HYDRODYNAMICS Moment closures of the Wigner equation using either single state (low temperature) assumptions or thermodynamic (room temperature) assumptions. Yields simple local conservation laws with possibly non - local potentials. S5 MOMENT CLOSURES AND ENTROPY STRUC- TURES: Sets up basic framework for entropy closures in the classical case. 7

8 S6 THE QUANTUM ENTROPY CLOSURE: Translates the maximum entropy principle into the thermodynamic setting. Yields non - local fluid equations which conserve a concave functional. S7 GENERALIZATION OF CLASSICAL COLLISION OPERATORS In analogy to the classical Boltzmann equation, the form of the collision operator and the logarithmic form of the entropy results from simple symmetry considerations. Derive a generalization of collision operators based on a given set of conservation properties. S8 RELATIVE ENTROPIES Derive generalized collision operators in the case that the dominant collision mechanism is not particle - particle collisions but collisions with a background gas. Gives a way to model phenomenologically quantum phonon collisions (assuming Fermi s Golden Rule). 8

9 S9 EFFECTIVE POTENTIALS Rudimentary introduction of quantum mechanical effects and modifications in the classical classical Boltzmann equations and in classical Hamiltonian systems. Introduction of a quantum potential. 9

10 S1 FREE TRANSPORT Formulate transport of a single particle as a Hamiltonian system. Yields in the classical case the Hamilton-Jacobi equations for the density function. Observations are made by integrating the observable quantity against this density function. In the quantum mechanical transport picture observations are made by computing the trace of an observation operator with respect to a basis given by the state of the system. 10

11 It is convenient to use three different formulations at the same time: 1. The Schrödinger equation for the wave function. 2. The Liouville equation for the density matrix of a mixed state (an ensemble of particles). 3. The Wigner equation for the Wigner function. The advantage of the Wigner function is that it relates the quantum mechanical picture to the classical one in the limit 0. More importantly, observations are made by integrating the classical observable quantity against the Wigner function. 11

12 The relation between observation operators and functions is given by the Wigner - Weyl transform. 12

13 S1 A. CLASSICAL FREE TRANSPORT One particle with mass m in a vacuum under the influence of a force F. Newton s second law: F = ma Trajectories: x = v, v = F m, F (x, t) = xv f(x, v, t): particle density for position x, velocity v and time t f(x(t), v(t), t) = 0 d dt t f(x, v, t) + v x f 1 m xv v f = 0 (1) Any function of the energy E := m v(t)2 2 + V (x(t)) is conserved. Φ d dt Φ(mv(t)2 2 + V (x(t))) = 0, 13

14 Expectations: E(κ): expectation of an observable κ(x, v) at time t under the probability f. E(κ) = κ(x, v)f(x, v, t) dxv c.f. : The local mass density at x = y. n(y, t) = E(δ(x y)) = f(y, v, t) dv 14

15 S1 B. FREE QUANTUM TRANSPORT IN THE SCHRÖDINGER PICTURE The Schrödinger equation for the evolution of the wave function ψ(x, t) under the Hamiltonian H. i t ψ(r, t) = Hψ = 2 2m rψ + V (r, t)ψ Observable quantities are replaced by self - adjoint operators. A: Self adjoint operator. E(A) = ψ(r, t) A[ψ](r, t) dr A self adjoint E(A) R c.f. : The local mass density at r = y. A[ψ](r, t) = δ(r y)ψ(r, t) n(y, t) = E(A) = ψ(y, t) 2 15

16 S1 C. DENSITY MATRICES AND INTEGRAL KERNELS OF OPERATORS ρ(r, s, t): density matrix, A d (r, s): integral kernel of the operator A. ρ(r, s, t) = ψ(r, t)ψ(s, t), A[ψ](r, t) = A d (r, s)ψ(s, t) ds Expectation of A expressed via the density matrix: E(A) = T r(a d ρ) = Evolution of the density matrix: i t ψ(r, t) = Hψ(r, t) = A d (r, s)ρ(s, r, t) drs H d (r, s)ψ(s, t) ds, H d (r, s) = 2 δ(r s) + V (r, t)δ(r s) 2m 16

17 The quantum Liouville (Heisenberg) equation for the density matrix: [.,.]: Commutator i t ρ = H d ρ ρ H d =: [H d, ρ] d := [H, ρ] (2) [H, ρ] = 2 2m ( s r )ρ + (V (r) V (s))ρ The density matrix for a mixed state (an ensemble of particles) ρ(r, s, t) = n ψ n (r, t)ψ n (s, t), satisfies the same equation (2) as the single state density matrix. 17

18 S1 D. WIGNER FUNCTIONS The Wigner function f(x, ξ, t) f(x, ξ, t) = (2π) 3 ρ(x 1 2 y, x y, t)eiy ξ dy (3) ξ: wave vector Local density: f(x, ξ, t) dξ = ρ(x, x, t) = n(x, t) The Wigner Equation Most easily written in terms of pseudo differential operators (PDO). A(ξ, y): Symbol of the (PDO) A(ξ, i ξ )f(ξ) = (2π) 3 A(ξ, y)f(ξ ) exp[iy (ξ ξ )] dyξ 18

19 If ρ(r, s, t) satisfies the Heisenberg equation (2) then f(x, ξ, t) satisfies the Wigner equation t f = m ξ xf+θ[v ]f, Θ[V ] = i [V (x+ 1 2i ξ) V (x 1 2i ξ)] (4) momentum) and com- Relation to the classical case: Rename velocities v = m ξ ( ξ: pare to (1): t f c = m ξ xf c + 1 xv ξ f c (5) Remark: (4) coincides with (5) for quadratic V. This allows to define the quantum equivalent (= operator A) of any observable quantity κ(x, ξ) in the classical case by requiring T r(a d ρ) = κ(x, ξ)f(x, ξ, t) dxξ 19

20 S1 E. WIGNER TRANSFORM AND WEYL QUANTIZATION The Wigner transform: W [a](x, ξ) = a(x 1 2 y, x+1 2 y)eiy ξ dy, f(x, ξ, t) = W [ρ](x, ξ, t) Traces of matrix products become L 2 products T r(a b) = a d (r, s)b d (s, r) drs = W [a](x, ξ)w [b](x, ξ) dxξ Expectations of operators: E(a) = T r(a d ρ) = W [a]w [f](x, ξ) dxξ A self adjoint (observable) operator a and a observable function function κ(x, ξ) are related by κ = W [a], a = W 1 [κ], W 1 [κ](r, s) = (2π) 3 κ( r + s 2, ξ)eiξ(r s) dξ 20

21 Real functions κ correspond to self adjoint operators with integral kernels W 1 [κ] Positive observables correspond positive definite operators but NOT to positive functions The Wigner function f = W [ρ] is not positive and therefore not a probability. 21

22 Example: Local moment densities, operators corresponding to powers of ξ m class j (z) = ξ j f c (z, ξ, t) dξ, m quant j (z) = δ(x z)ξ j f(x, ξ, t) dxξ, a(r, s) = ( i) j δ( r + s 2 z) j δ(r s), φ(r)a[ψ](r)dr = ( i) j j y [φ(z 1 2 y)ψ(z y)] y=0 E(A) = ( i) j j y [ψ (z 1 2 y)ψ(z y)] y=0 22

23 S1 F. THE HAMILTONIAN STRUCTURE OF FREE TRANSPORT conservation of an en- Classical Hamiltonian systems: ergy E(x, ξ) t f + v E x f x E v f = 0 Classical Energy: E(x, v) = m 2 v 2 + V (x) = 2 2m ξ 2 + V (x) The classical commutator t f + [E, f] c = 0, [E, f] c := 1 ( ξe x f x E ξ f) The classical commutator conserves any function of the energy and any function of f φ(e)[e, f] c dxξ = φ(f)[e, f] c dxξ = 0, φ 23

24 The Hamiltonian structure of the Heisenberg equation: t ρ + i [H, ρ] = 0, [H, ρ] = H d ρ ρ H d Cyclicity of the commutator: T r(a [b, c]) = T r([a, b] c), φ(b): function of the self adjoint operator (density matrix) b; defined via diagonalization over an orthonormal basis b(r, s) = j ψ j (r)λ j ψ j (s), φ(b)(r, s) = j ψ j (r)φ(λ j )ψ j (s), [φ(b), b] = 0 T r(φ(b) [b, c]) = T r([φ(b), b] c) = 0 The expectation of any operator function of H or ρ is conserved T r(φ(ρ) t ρ) = T r(φ(h) t ρ) = 0 24

25 The commutator in the Wigner picture w = W [ρ], H = W [E] t f+ m ξ xf Θ[V ]f = 0, Θ[V ] = i [V (x+ 1 2i ξ) V (x 1 2i ξ)] t f + [E, f] w = 0, [E, f] w = i W ([W 1 (E), W 1 (f)]) The classical energy E = 2 2m ξ 2 +V (x) and any function of E in the operator sense, i.e. W (Φ(W 1 (E))) is conserved in the Wigner picture. 25

26 So, altogether, there are three equivalent quantum mechanical descriptions of free transport, each with a Hamiltonian structure, i.e. a commutator. Each conserves any function of the energy, where in the q.m. picture the function is an operator function. Classical: Φ(E) t f c dxξ = [Φ(E), E] c f c dxξ = 0 Density matrices: T r(φ(h) t ρ) = i T r([φ(h), H]ρ) = 0 Wigner functions: W (Φ(W 1 (E))) t f dxξ = [W (Φ(W 1 (E))), E] w f dxξ = 0 26

27 The commutator in the Wigner picture can be conveniently expressed in terms of the operator product of two functions f g = W [W 1 [f] W 1 [g]] f g(x, ξ) = f(x+ i 2 ξ, ξ i 2 x)g(x, ξ) = g(x i 2 ξ, ξ+ i 2 x)f(x, ξ) This gives for the Wigner commutator [E, f] w = i (E f f E) [ 2 2m ξ 2 +V (x), w] w = m ξ xw i [V (x+ 1 2i ξ) V (x 1 2i ξ)]w [ 2 2m ξ 2 +V (x), w] w = 1 ξe x w i [E(x+ 1 2i ξ, ξ) E(x 1 2i ξ, ξ)]w 27

28 The conservation of any function of energy is a consequence of the cyclicity of the commutator in the classical as well as the quantum mechanical case. f(x, ξ)[g, h] c (x, ξ) dxξ = [f, g] c (x, ξ)h(x, ξ) dxξ = f(x, ξ)[g, h] w (x, ξ) dxξ = [f, g] w (x, ξ)h(x, ξ) dxξ 28

29 S2 ENERGY BANDS Consider transport of one particle (electron) in a periodic medium, i.e. a crystal. This is described by a Schrödinger equation with an additional periodic potential. Under the assumption that the lattice period is much smaller than the length scale under consideration, we obtain a homogenization problem for the Schrödinger equation. The homogenization problem can be solved in terms of the Bloch transform and the solution of a sequence of eigenvalue problems. 29

30 The result is semi- discrete Schrödinger equation where the quadratic term in the kinetic energy is replaced by the eigenvalue, dependent on a parameter vector, the energy band. If the eigenvalue is replaced by a quadratic around its minimum we obtain the so called effective mass approximation. (joint work with F. Poupaud). 30

31 LATTICE POTENTIALS Transport of one electron in a periodic structure (crystal). Structure modeled through periodic interaction potential. Schrödinger equation: t ψ = i 2m xψ i [V + V L]ψ V L : rapid periodic oscillations due to interaction with crystal lattice. V L (x) = V L (x + λlz), z Z, V = 1 λ2v (x/λ) L: Lattice matrix. Consider spatial scale much larger than lattice. Homogenization for λ 0. Decomposition into Bloch waves 31

32 Bloch Wave Decomposition Evolution equation with a linear differential operator with highly oscillatory coefficients t u = Λ(x, x λ, x)u Λ lattice periodic in fast variable x λ Λ(x, y + Lz, w) = Λ(x, y, w), z Z, L : lattice matrix, det(l) = 1 Bloch decomposition: Slicing and Fourier transform Step 1: slicing u 1 (η, z, t) = u(λlz + λξ, t), η L [0, 1] d z : slow variable x 32

33 Periodicity: (u 1 coupled in z) u 1 (η + Lz, z, t) = u 1 (η, z + z, t) Step 2: Fourier transform in z u 2 (η, k, t) = λ d c z u 1 (η, z, t) exp(ik T Lz), k L T B, B = [ π, π] d L T B: Brillouin zone, k Fourier dual of slow variable x, (c = (2π) d/2 ) Periodicity: u 2 (η + z, k, t) = u 2 (η, k, t) exp( ik T Lz ) decoupled in k! 33

34 Step 3: Inverse Fourier transform u 1 (η, z, t) = cλ d L T B u 2(η, k, t) exp( ik T Lz)dk, B = [ π, π] d Norms: L T B u 2(η, k, t) 2 dk = z u 1 (η, z, t) 2 34

35 BLOCH TRANSFORM OF LINEAR DIFFERENTIAL OPERATORS Transformation of Λ: Step 1: Transformation on sliced space : Let derivatives work on the fast variables only. Λ(x, x λ, x)u(x) = Λ 1 (η, z, ξ )u 1 (η, z), Λ 1 (ξ, z, ξ ) = Λ(λLz + λη, η, 1 λ ξ) 35

36 Step 2: Transformation in Fourier transformed space: semi - discrete pseudo differential operators Define (2π) 3 α 3 z A( k,α )f(η, k) = L T B/α A(iαLz)f(η, k ) exp(iα(k k ) T Lz)dk Integral replaced by Riemann sum. Reduces to classical PDO acting on k for α 0: α 0, αlz p A( k,0 )f(η, k) = (2π) 3 A(ip)f(η, k ) exp(i(k k ) T p)dk dp Transform Λ 1 Λ 2 : Λ 1 u 1 (η, z) = λ d L T B (Λ 2u 2 )(η, k) exp( ik z)dk, 36

37 Λ 2 = Λ 2 (η, k,1, ξ ) = Λ 1 (ξ, il 1 k,1, η ) = Λ( iλ k,1 +λη, η, 1 λ ξ) Gives evolution problem with quasiperiodic boundary conditions in η, amenable to asymptotics. t u 2 = Λ 2 u 2 = Λ( iλ k,1 + λη, η, 1 λ ξ)u 2 Boundary conditions in ξ : u 2 (η + z, k, t) = u 2 (η, k, t) exp( ik T Lz ) Boundary conditions independent of λ! 37

38 APPLICATION TO THE SCHRÖDINGER EQUATION Schrödinger: t ψ = i 2m xψ i [V + V L]ψ Scaling: Choose length and time scales such that V L oscillatory and large x x x 0, t t, V (x) = mx2 0V s ( x ), t 0 x 0 qt 2 0 is highly V L (x) = mx2 0 qt λ 2V Ls( x ) λx 0 V Ls lattice periodic x λ, gives t ψ = ih 2 xψ i h [V (x) + 1 λ 2V L( x λ )]ψ 38

39 or Λ(x, η, x ) = ih 2 x 2 i h [V (x) + 1 λ 2V L(η)] h : scaled Planck constant h = t 0 mx 2 0 Transform of Λ : ψ ψ 1 ψ 2, Λ Λ 1 Λ 2, Λ 2 = Λ 2 (η, k,1, ξ ) = Λ( iλ k,1 + λη, η, 1 λ ξ) t ψ 2 (η, k, t) = ih 2λ 2 ξ 2 i h [V ( iλ k,1 + λξ) + 1 λ 2V L(η)] Periodicity: ψ 2 (η + z, k, t) = ψ 2 (η, k, t) exp( ik T Lz ), k L T B 39

40 THE BLOCH TRANSFORMED SCHRÖDINGER EQUATION t ψ 2 (η, k, t) = ih 2λ 2 ξ 2 i h [V ( iλ k,1 + λξ) + 1 λ 2V L(η)] ψ 2 (η + z, k, t) = ψ 2 (η, k, t) exp( ik T Lz ), k L T B boundary conditions independent of λ, only weakly dependent on k, solve in terms of eigenfunctions 40

41 Eigenfunctions t ψ 2 (η, k, t) == ih 2λ 2 ξ 2 i h [V ( iλ k,1 + λξ) + 1 λ 2V L(η)] Take eigenfunctions of the dominant part of the (self adjoint) operator. Eigenvalues ε m and eigenfunctions φ m ; both dependent on k through the boundary conditions. ε m (k)φ m (η, k) = h2 2 η 2 φ + V L (η)φ m, φ m (η + z, k) = φ m (η, k) exp( ik T Lz ), ε m (k): m- th energy band. 41

42 Wannier functions ψ 2 completely in one energy band, i.e. a superposition of eigenfunctions with the same index and different wave vectors. ψ 2 (η, k, t) = σ(k, t)φ 0 (η, k) t ψ 2 (η, k, t) = i hλ 2ε 0(k)ψ 2 i h σ(k, t)v ( iλ k,1 + λξ)φ 0 (η, k) Multiplication with σ does not commute with the PDO V ( iλ k,1 + λξ). However, V only O(λ) dependent on k,1. Approximation: (Poupaud + CR) t ψ 2 (η, k, t) = i hλ 2ε 0(k)ψ 2 i h V ( iλ k,1 + λξ)ψ 2 (η, k, t) 42

43 After reversing the Bloch transform we obtain an effective Schrödinger equation with a modified kinetic energy. t ψ 2 (η, k, t) = i hλ 2ε 0(λk)ψ 2 i h V ( i a k,λa + λξ)ψ 2 t ψ(λ(η + Lz), t) = (2π) 3 z [ i hλ 2ε 0(λk) i h V ( iλ k,1+λξ)]ψ(λ(η+lz ), t) exp[ik T L(z z )] for small λ this is the Riemann sum for t ψ(x, t) = (2π) 3 B/λa [ i hλ 2ε 0( λk) i h V ( 1 ia k + λξ)]ψ(x, t) exp[ik (x x )]dkdx 43

44 So we obtain the homogenized Schrödinger equation t ψ(x, t) = [ i hλ 2ε 0( λ i x) i V (x)]ψ(x, t) h or in unscaled variables i t ψ(x, t) = [ 1 λ 2ε 0(iλ x ) + V (x)]ψ(x, t) Wigner functions: (Markowich and Degond semi- discrete Wigner function) t f(x, ξ, t) + [ 1 λ 2ε 0(λξ) + V (x), f] w = 0 44

45 Effective mass approximation If we assume that ε 0 around ξ = 0 is of the form ε 0 (ξ) = 2 2 ξt Aξ + O( ξ 3 ), A := ε 0 (0) and let λ 0 we obtain the effective mass matrix approximation t f(x, x, t) + [ 2 2 ξt Aξ + V (x), f] w = 0 with A 1 the effective mass matrix. If ε 0 is isotropic then the effective mass is a scalar. 45

46 So the effect of the periodic lattice potential after homogenization is to change the flight path (or the geometry). The form of the energy bands depends only on the lattice potential appearing in the corresponding eigenvalue problem. In the absence of a lattice potential the eigenvalue problem reads ε(k)φ m (η, k) = h2 2 η 2 φ, ε(k) = 2 2 m k 2 and we obtain the vacuum Hamiltonian. 46

47 S3 COLLISIONS AND THE FERMI - GOLDEN RULE Particle trajectories are not only bent by the presence of the crystal. They are changed by collisions with the vibrating crystal lattice. In these collisions energy between the particles and the lattice is exchanged. This is modeled by the creation and destruction of pseudo particles (phonons). In crystals this is by far the most important collision mechanism (more frequent than particle - particle collisions). 47

48 The energy exchange is governed by Fermi s Golden Rule, stating that the amount of particle energy gained / lost in the process is given by the dominant frequency of the lattice. The exact description of phonon collisions is given by an infinite system of many body Schrödinger for a nonconstant number of particles. The Fermi Golden Rule can be derived (not rigorously) from a long time average of this system.

49 The Fermi Golden Rule is local in space and time while the quantum collison picture is non-local in all variables. We derive asymptotic corrections to the Fermi Golden Rule which are mildly non-local through a gradient expansion. (joint work with H. Kosina and M. Nedjalkov) 48

50 LITERATURE I. Levinson, Translational invariance in uniform fields and the equation for the density matrix in the Wigner representation, Soviet Phys.JETP, vol. 30, no. 2, pp , A. Bertoni, P. Bordone, R. Brunetti, and C. Jacoboni, The Wigner function for electron transport in mesoscopic systems, J.Phys.:Condensed Matter, vol. 11, pp , M. Nedjalkov, R. Kosik, H. Kosina, and S. Selberherr, A Wigner Equation for Nanometer and Femtosecond Transport Regime, in Proceedings of the 2001 First IEEE Conference on Nanotechnology, (Maui, Hawaii), pp , IEEE, Oct

51 P. Argyres: Quantum kinetic equations for electrons in high electric and phonon fields, Phys. Lett. A 171 North Holland (1992). N. Ashcroft, M. Mermin: Solid State Physics, Holt - Saunders, New York (1976). J. Barker, D. Ferry: Phys. Rev. Lett. 42 (1997). F. Fromlet,P. Markowich,C. Ringhofer: A Wignerfunction Approach to Phonon Scattering, VLSI Design 9 pp (1999). 50

52 1. Collisions of particles with each other 2. Collisions of particles with a background gas 3. Collisions of particles with a crystal lattice (modeled as pseudo particles phonons ) Q: Collision operator Q[f](ξ, t) = t f(ξ, t) = Q[f](ξ, t) P[f](ξ, ξ )f(ξ, t) P[f](ξ, ξ)f(ξ, t)dξ P : Probability of ξ ξ (scattering cross section). For 2. and 3. Q[f] is linear. 51

53 S3 A THE SEMI-CLASSICAL CASE AND FERMI S GOLDEN RULE In the semi -classical Boltzmann equation the FGR models the interaction of electrons with vibrations of the crystal lattice (pseudo - particles, phonons) FGR states specified amount / loss of energy in scattering event P(ξ, ξ ) = ν=±1 K ν (ξ, ξ )e βε(ξ ) δ(ε(ξ) ε(ξ ) + ν ω) ε(ξ): Energy corresponding to momentum ξ; ω: dominant eigen - frequency of the phonon lattice. 52

54 P(ξ, ξ ) = ν=±1 K ν (ξ, ξ )e βε(ξ ) δ(ε(ξ) ε(ξ ) + ν ω) Symmetry: (Principle of detailed balance) K ν (ξ, ξ ) = K ν (ξ, ξ) Implies the Maxwellian f = exp( βε(ξ)) is in the kernel of Q. Derived from a joint density f ep (ξ, η, t) for electrons with momentum ξ and phonons with momentum q under some equilibrium assumptions on the phonons. 53

55 The semi - classical Boltzmann equation t f c (x, ξ, t) + [V (x) + ε(ξ), f c ] c = Q F GR [f c ] The quantum Boltzmann equation for the Wigner function t f(x, ξ, t) + [V (x) + ε(ξ), f] w = Q[f] Quantum phonon operator Q non-local in x, ξ, t. Only in the effective mass approximation. 54

56 OUTLINE Derivation of Fermi s Golden Rule: Many body Schrödinger equations and the Fröhlich Hamiltonian The phonon trace: Integrating out the phonons using an Ansatz Asymptotics: Weak electron - phonon interactions. The Levinson equation. Highly oscillatory, non - local integral operator A gradient ex- Local - in - time asymptotics: Corrections to the Fermi Golden Rule. pansion. Numerical results. 55

57 S3 B. DERIVATION OF THE FGR The Fock-space Schrödinger equation: ψ m (r, k 1,.., k m, t) ψ m : wave function for one electron with position r and m phonons with wave vectors k 1,.., k m i t Ψ = HΨ, Ψ = (ψ 0, ψ 1,...) i t ψ m = (HΨ) m := H e ψ m + H p ψ m + n H ep mnψ n H: Fröhlich Hamiltonian H e : Free particle. H p : Free phonon. H ep mn Interaction of electron with phonon (generation, destruction). 56

58 Hamiltonians Free electron Hamiltonian, acts only on r; V : mean field potential H e = 2 2m r + V (r), E(r, ξ) = 2 2m ξ 2 + V (r) Free phonon Hamiltonian (acts only on k m = (k 1,.., k m ) H p ψ m (r, k m ) = H p (k m, s m )ψ m (r, s m )ds m H e and H p are diagonal elements in the Fröhlich Hamiltonian H 57

59 Electron phonon interaction Hamiltonian Creation of a phonon: Hmn ep = Hmn ep+ H ep+ mn ψ n (r, k m ) = δ m n 1 k m l = (k 1,.., k l 1, k l+1,.., k m ) + H ep mn m l=1 F (k l )e ir k lψ n (r, k m l ) Destruction of a phonon: H self adjoint m=0 φ m (r, k m ) (HΨ) m (r, k m )drdk m = m=0 ψ m (r, k m ) (HΦ) m (r, k m )drdk m 58

60 H ep mn is the adjoint of H ep+ nm H ep mn ψ n (r, k m ) = δ m n+1 n l=1 k m l+ (s) = (k 1,.., k l 1, s, k l,.., k m ) F (s) e ir s ψ n (r, k m l+ (s))ds 59

61 S3 C. THE PHONON TRACE The density matrix corresponding to Ψ ρ mn (r, k m, r, k n ) = ψ m (r, k m ) ψ n (r, k n ) Find equation for phonon trace: (T r p ρ)(r, r ) = m=0 ρ mm (r, k m, r, k m )dk m using asymptotics quantum Boltzmann equation for single electron density matrix. 60

62 PROBLEM: Quantum mechanics is time reversible S(Ψ) m (r, k m, t) = ψ m (r, k m, t) S(i t Ψ) = i t SΨ, S(HΨ) = HS(Ψ) know Ψ(t = T ); solve same equation backward in time to compute Ψ(t = 0). FGR not time reversible; loss of information by building the phonon trace. 61

63 APPROACHES 1. Start from simplified physical model; phonons as bath of harmonic oscillators (Caldeira, Legget); mathematical justification incomplete (Erdos); derive quantum Fokker - Planck operator (Arnold, Markowich) 2. Start with randomized interaction potential (Papanicolao, Shi Jin et al); derive FGR; weak theory. 3. Direct weak e-p interaction asymptotics (Levinson, Argyres, Barker & Ferry); mathematical justification incomplete (Frommlet, Markowich & Ringhofer); arrive at Levinson (Barker - Ferry) equation FGR. 62

64 4. Direct weak e-p interaction asymptotics theory complete; arrive at Pauli master equation (Castella, Degond)

65 S3 D. WEAK ELECTRON - PHONON INTERACTIONS Notation: i t ρ mn [H e + H p, ρ mn ] = H ep m,m 1 ρ m 1,n + H ep m,m+1 ρ m+1,n ρ m,n 1 H ep n 1,n ρ m,n+1h ep n+1,n t ρ mn Aρ mn = (Bρ) mn A diagonal in m, n, commutes with taking the phonon trace m=0 (Aρ) mm dk m = [H e, (T r p ρ)] free streaming operator acting on T r p ρ 63

66 Assume B small and iterate, S A (t) semigroup operator ρ j+1 = S A (t)(ρ I ) + t 0 S A(t τ)(bρ j )(τ)dτ, j = 0, 1,... 64

67 S3 E. THE LEVINSON (BARKER-FERRY) EQUATION Formulated in terms of the Wigner function (classical equivalent to the density function in the Boltzmann equation) f(x, ξ, t) = T r p ρ(x 1 2 y, x y, t)eiy ξ dy Remark: Since we iterated the semigroup operator the L-B-F equation will be non-local in time. t t f(x, ξ, t) + [ε + V, f] w = 0 dt dξ [S(ξ, ξ, t t )f(x, ξ, t ) S(ξ, ξ, t t )f(x, ξ, t )] Remark: The semigroup operator is expressed in terms of tracing back characteristics in the Wigner picture. (Only true for linear potentials V.) 65

68 The scattering cross section: Write everything in terms of momentum p = ξ S(p, p, t) = ν=±1 a ν cos[ 1 t 0 (ε(q(p, τ)) ε(q(p, τ)) + ν ω)dτ] q(ξ, τ) := p τ x V, ε(p) = p 2 2m ω: eigen - frequency of the lattice Remark: The collisions are never completed! The integral kernel is highly oscillatory for 0. Sano (1995): Numerical solution of the B-F equation The zero field case (V = const) S(p, p, t) = ν=±1 a ν cos[ t (ε(p) ε(p ) + ν ω)] 66

69 THE SPATIALLY HOMOGENEOUS ZERO - FIELD LEVINSON EQUATION Scaling and dimensionless formulation: ε(p 0 ) = KT, t 0 = 1 ωλ, λ = O( ) t f(p, t) = Q λ [f](p, t) = t 0 dt dp 1 λ [S(p, p, t t λ )f(p, t ) S(p, p, t t λ )f(p, t )] dimensionless cross section S(p, p, t) = ν=±1 a ν cos[t(ε(p) ε(p ) + νω)] a ν = O(1) and Ω (scaled emission energy) = O(1). 67

70 S3 F. FERMI S GOLDEN RULE - A HEURISTIC ARGUMENT Q λ [f](p, t) = t 0 dt dp 1 λ [S(p, p, t t λ )f(p, t ) S(p, p, t t λ )f(p, t )] Assume: lim t S(p, p, t) = 0 Q λ [f](p, t) dp [S 0 (p, p )f(p, t) S 0 (p, p)f(p, t)] S 0 (p, p ) = 0 S(p, p, t)dt 68

71 cross section: S(p, p, t) = 0 ν=±1 cos(zt)dt = δ(z) yields FGR. a ν cos[t(ε(p) ε(p ) + νω)] Since lim t S(p, p, t) 0 this becomes a weak limit. 69

72 THEOREM 1: Define the functional Y λ (φ, f) = 0 dt dp{φ(p, t)q λ [f](p, t)} for smooth, compactly supported test functions φ. Then lim Y λ(φ, f) = λ 0 holds for fixed φ with S 0 (p, p ) = 0 dt ν=±1 dp{φ(p, t)q 0 [f](p, t)} a ν δ(ε(p) ε(p ) + νω) 70

73 THEOREM 2: (first order perturbation) Remark: 0 dt Y λ (φ, f) = dp{φ(p, t)(q 0 [f] + λq 1 [ t f])(p, t)} + o(λ) Q 1 acts on t f (non-local in time); Q 1 only defined weakly in p 71

74 Weak formulation of Q 1 [ t f]: φ(p)q 1 [ t f](p)dp = d dε : a ν ν=±1 ln( ε ε + νω ) d d dε dε [(φ φ) t f]dpdp directional derivative perpendicular to surfaces of equal energy. Corresponds (formally) to a first order scattering cross section S 0 (p, p ) = S 1 (p, p, t ) = S = S 0 + λs 1 ν=±1 ν=±1 a ν δ(ε(p) ε(p ) + νω) a ν (ε(p) ε(p ) + νω) 2 t 72

75 REMARK: Parabolic bands ε(p) = p 2 2 : polar coordinates and radial derivatives

76 S3 G. NUMERICAL VERIFICATION THEOREM 3: Let Γ be a time - direction smoothing operator (Γf)(t) = Then 0 γ(t t )f(t )dt. ΓQ λ [f] = Q 0 (Γf) + λq 1 [Γ t f] + o(λ) pointwise in t weakly in p. 73

77 Computations: Parabolic bands ε = p 2 2m a n = const: δ function interaction potential Symmetry: f(p, t) = f(ε(p), t) Compare smoothed energies: εγq λ [f]dp 74

78 FIGURE f time energy 3 75

79 400 FIGURE energy time 76

80 FIGURE f (filtered) time energy 3 77

81 40 35 FIGURE 4 K*Q[f] Q 0 [K*f] Q fgr [K*f] energy time 78

82 80 70 FIGURE 5 K*Q[f] Q 0 [K*f] Q fgr [K*f] energy time 79

83 SOLVING THE LEVINSON EQUATION REMARK: t f λ = Q λ [f λ ] t f 01 = Q 0 [f 01 ] + λq 1 [ t f 01 ] f λ f 01 not covered by the theory. Extremely difficult to solve; implicit; Q 1 singular integral kernel 80

84 Example: t f 01 = Q 0 [f 01 ] + λq 1 [Q 0 [f 01 ]] yields an ill posed equation. Solve for expansion terms directly. f 01 = f 0 + λf 1 t f 0 = Q 0 [f 0 ] t f 1 = Q 0 [f 1 ] + Q 1 [ t f 0 ] = Q 0 [f 1 ] + Q 1 [Q 0 [f 0 ]] 81

85 1.4 FIGURE initial condition energy (mev) 82

86 FIGURE f time (ps) energy (mev)

87 FIGURE f filtered time (ps) energy (mev)

88 FIGURE f 0 +lambda*f time (ps) energy (mev)

89 approximation at t= ps FIGURE10 f 0 f 0 +lam*f energy (mev) f approximation at t=1.7082ps FIGURE11 f 0 f 0 +lam*f energy (mev) f 0.7 FIGURE FIGURE13 approximation at t=2.5623ps f 0 f 0 +lam*f 1 f approximation at t=3.4164ps f 0 f 0 +lam*f 1 f energy (mev) energy (mev) 86

90 CONCLUSIONS The first order correction produces a significant change in the transient behavior. This can explain collisional broadening in actual devices. FUTURE WORK The non-zero field case ( the intra - collisional field effect). Numerics for non - radially symmetric solutions ( MC). 87

91 How to solve the first order approximation using weighted particle methods.

92 S4. MACROSCOPIC APPROXIMATIONS I: MOMENT CLOSURES AND QUANTUM HYDRODYNAMICS Assuming proximity to a form of equilibrium kinetic equations can be replaced by macroscopic equations for a few observable quantities, i.e. mass, momentum and energy. These macroscopic (or hydrodynamic) equations are obtained by taking the moments of the kinetic equation (the classical Boltzmann equation or the Wigner equation in the quantum mechanical case.) This leads to the problem of closure, i.e. expressing higher order moments in terms of lower order moments, using an Ansatz for the kinetic density or the Wigner function. 88

93 The non-local nature of the quantum mechanical equations results in this closure being non-local. In a gradient expansion this non-locality involves higher order derivatives of the potential. For discontinuous potentials (arising from the presence of barriers) the gradient approximation is inappropriate. The nonlocality can be treated by introducing an effective quantum potential from approximations to thermodynamic equilibrium. This leads to smoothed potential approaches. (joint work with: N. Ben-Abdallah, C. Gardner, I. Gasser,G. Kliemeck, P. Markowich) 89

94 LITERATURE D. Ferry, H. Grubin: Modelling of quantum transport in semiconductor devices, Solid State Phys. 49, pp , C. Gardner: The quantum hydrodynamic model for semiconductor devices, SIAM Journal on Applied Mathematics, vol. 54, pp , C. L. Gardner and C. Ringhofer: Smooth quantum potential for the hydrodynamic model Physical Review, vol. E 53, pp , C.Gardner, C.Ringhofer: Approximation of thermal equilibrium for quantum gases with discontinuous potentials and application to semiconductor devices, SIAM Journal on Applied Mathematics, vol. 58, pp ,

95 I. Gasser, A. Jüngel: The quantum hydrodynamic model for semiconductors in thermal equilibrium, Z. Angew. Math. Phys. 48, pp ,1997. I. Gasser and P. A. Markowich: Quantum Hydrodynamics, Wigner Transforms and the Classical Limit, Asympt. Analysis, Vol. 14, No. 2, pp ,

96 The quantum Boltzmann equation t f + [E, f] w = Q[f] (6) Derive equation for certain moments: densities of observable quantities M j (x, t) = κ j (ξ)f(x, ξ, t) dξ Usually: Mass, momentum and energy κ 0 = 1, κ 1 = 1 ξε(ξ), κ 2 = ε(ξ) Physical space Taking the moments of (6) yields system of conservation laws in physical space of the form t M(x, t) + N = R, N(x, t) = κ(ξ)[e, f] w (x, ξ, t) dξ, R(x, t) = κ(ξ)q[f](x, ξ, t) dξ The closure problem: Express N, R in terms of M. 92

97 t M(x, t) + N = R, (7) Remark: The moment equations (7) are exact. The approximation is in the closure expression for N and R. R is usually modeled by simple relaxation terms. 93

98 S4 A. COMPUTING THE MOMENTS If the goal is to derive small perturbations to classical transport (i.e. local moment equations), the effective mass approximation has to be used. Computing N: Weak formulation, using the cyclicity of the commutator φ(x)n j (x, t) dx = φ(x)κ j (ξ)[e, f] w dxξ = [φκ j, E] w f dxξ [f, g] w = i ν=±1 νf(x + iν 2 ξ, ξ iν 2 x)g(x, ξ) [φκ j, V ] w = i φ(x) ν=±1 [φκ j, ε] w = [ε, φκ j ] w = i κ j(ξ) νκ j (ξ iν 2 x)v (x) ν=±1 νε(ξ iν 2 x)φ(x) 94

99 ε(ξ) quadratic [φκ j, ε] w = κ j (ξ) m ξ xφ(x) Same as classical moments. κ j (ξ) quadratic [φκ j, V ] w = 1 φ(x) ξκ j (ξ) x V (x) Under the effective mass approximation the balance equations for mass momentum and energy are the same as in the classical case. N j (x, t) = x κ j (ξ) m ξf(x, ξ, t) dξ+ xv (x) 1 f(x, ξ, t) ξκ j (ξ) dξ 95

100 QUANTUM HYDRODYNAMICS M 0 = κ 0 = 1, κ 1 = ξ, κ 2 = 2 ξ 2 2m f dξ, M 1 = ξf dξ, M 2 2 ξ 2 = 2m f dξ N 0 = M 1, N 1 = x P + M 0 x V, N 2 = x q + V M 1 P = 2 m ξξ T f dξ, q = 3 2m 2 ξ 2 ξf dξ (8) Closure problem: Express N in terms of M 96

101 S4 B. CLOSURES 1. Single state closures: Assume the density matrix is made up of one single wave function. Gives zero temperature hydrodynamics with a quantum correction in the potential. Starting with a nonlinear Schrödinger equation gives a temperature law T = T (n) via an entalpy relation. 2. Thermodynamic closures: Assume Gallilean invariance and the concept of a Maxwellian, i.e. a thermodynamic equilibrium. Gives non-local interaction of particles via a potential. 97

102 S4 C. SINGLE STATE CLOSURE Take one singe wave function (i.e. one single particle) with the effective mass approximation i t ψ = 2 2m xψ + V (x)ψ Write ψ in polar coordinates ψ = e a+ib f(x, ξ) = exp[iy ξ + ν=±1 a(x + ν 2 y) + iνb(x + ν y] dy 2 Two degrees of freedom (a, b) requires only a two moment closure M 0, M 1 (mass, momentum). M 0 = e 2a, M 1 = e 2a x b, P = 2 m e2a (( x b) x a) Change variables: e 2a = n, x b = mu 98

103 The single state QHD equations: M 0 = n, M 1 = mnu, P = n(mu 2 2 4m 2 x ln(n)) (a) t n + x (nu) = 0, (9) (b) t (mnu) + x [n(mu 2 2 4m 2 x ln(n))] + n xv = 0 99

104 THE BOHM POTENTIAL The term 4m 2 2 x ln(n)) in (9)(b) is written as a correction to the potential. x [n 2 x ln(n)] = 2n x ( x n ) n A potential which is locally dependent on n can be written as a temperature via the entalpy relation. d dn (na(n)) = nb (n) x (na(n)) = n x B(n), i t ψ = 2 2m xψ + (V (x) + B( ψ 2 ))ψ 100

105 t (mnu) + x [n(mu 2 2 4m 2 x ln(n))] + n x (V + B) = 0 t (mnu) + x [n(mu 2 + A(n)I)] + n x (V 2 c.f.: B(n) = T ln(n) implies A(n) = T 2m x n n ) = 0 (10) (10) is called the Bohm potential. Many body problems and Bohmian mechanics: Solve many Schrödinger equations in Lagrangian coordinates with particles and the Bohm potential. t (mn j u j ) + x [mn j u 2 j ] + n j x (V 2 x ntot ) = 0 2m ntot 101

106 S4 D. ENTROPIC CLOSURE AND SMOOTHED POTENTIALS Assume that (in a moving frame ) the Wigner function relaxes towards thermal equilibrium. Use the quantum mechanical definition of thermal equilibrium (the entropy minimizer) to close the moment system. Closure problem reduces to finding a usable expression for thermal equilibrium. 102

107 LOCAL MAXWELLIANS Classical hydrodynamics: Close the Boltzmann equation by ansatz of thermal equilibrium f nut (x, ξ, t) = M(n, T, ξ m u) with M the Maxwellian M(n, T, ξ) = n 2πT exp( ξ 2 2mT ). Quantum equivalent: close by quantum Maxwellian M = W (e βh ) Problem: e βh does not depend explicitly on local density n. 103

108 Fix: introduce quasi-fermi level M βφ (x, ξ) = W exp[ β(h + φ)] dependent on β and Fermi - level φ; Relate φ, β to mass and local temperature: 1 ξ 2 2m M βφ (x, ξ, t) dξ = ( n ) 3 2 nt Problem reduced to computing the exponential of a matrix (operator). ρ βφ (r, s) = exp[ β(h + φ)], (H + φ)ψ(r) := 2 2 rψ + (V + φ)ψ 104

109 S4 E. HOW TO COMPUTE THE EXPONENTIAL OF A SELF ADJOINT OPERATOR (MATRIX) Done via the semigroup generated by H + φ, leading to the Bloch equation. β exp( βh) = H exp( βh), exp( βh) β=0 = δ(r s) (= I) Definition: The anti-commutator for density matrices and Wigner functions {a, b} = 1 (a b + b a), 2 {f, g} w = W ({W 1 f, W 1 g}) 105

110 THE BLOCH EQUATION β W 1 M βφ = {H+φ, W 1 M βφ }, W 1 M 0φ (r, s) = δ(r s) β M βφ = { ξ 2 2m + V (x) + φ(x), M βφ} w, M β=0,φ (x, ξ) = 1 (11) Closure: compute higher order moments of M βφ (x, ξ). This can be done approximately by using asymptotics in the Bloch equation giving M βφ = M 0 βφ + λm1 βφ, with λ some expansion parameter, with explicit formulas for M 0 βφ, M1 βφ. Then compute β, φ via ( ) n = 3nT 1 ξ 2 m [M 0 βφ + λm1 βφ ] dξ 106

111 S4 F. ASYMPTOTICS IN THE BLOCH EQUATION AND APPROXIMATE QUANTUM MAXWELLIANS The Bloch equation for the quantum Maxwellain in the Wigner picture: β M βφ = 1 2 ν=±1 β M βφ (x, p) = 2 8m xm p 2 2m M 1 2 [ 2 ξ + iν 2 x 2 + (V + φ)(x + iν 2m 2 ξ)]m βφ, ν=±1 M β=0,φ (x, p) = 1 (12) (V +φ)(x+ iν 2 p)m Two forms of asymptotics: 1. Semiclassical: λ = 2 ; 107

112 Gives same result as the single state closure plus additional temperature. Closure terms in (8): P = n(mu 2 + T I 2 4m 2 x ln(n)), q = 0 Conservative form of the Bohm potential. Inconsistent for discontinuous potentials V. 2. Small external potential and slowly varying densities H = εev (x) + φ(εx) P = nt I n 2 x( 2 ln n nγ[v ]), q = 0 1 Γ = 0 γ2 exp[ 2 (1 γ 2 ) x ]dγ 8T Particles see only a smoothed version Γ[V ] of the potential. Non- local effects in the pressure tensor. Heat flux q from Chapman - Enslog with BGK operator.

113 0.1 U x V

114 Results: Works quite well to produce small corrections to classical regimes (Tunneling into oxides). Ferry et al (2000) Fichtner et al (2001) Qualitatively but not quantitatively correct in pure quantum cases (resonant tunneling diodes). Shortcomings: 1. Gallilean invariance 2. β in e βh is constant

115 Current Voltage 109

116 CONCLUSIONS AND OPEN PROBLEMS Quantum entropy closures natural extension of classical thermodynamic approach. Resulting QHD reproduces structural properties. Derivation does not rely on explicit form of quantum collision operator. Complicated, non - local, needs approximations. Type? Shocks? Numerical methods? Experiments. 110

117 S5 MOMENT CLOSURES AND ENTROPY STRUCTURES This section is a brief review of maximum entropy closures and the structure induced by the entropy. Given a collision operator, we derive equations for the conserved quantities of this operator. This yields an open system which is closed by assuming that the collision operator drives the entropy to its maximum locally. Since we locally maximize the entropy when closing the system we obtain a concave functional of the moments which is conserved by the moment equations. 111

118 LITERATURE M. Junk, Domain of definition of Levermore s fivemoment system, J. Stat. Phys. 93 (1998), pp M. Junk, Maximum entropy for reduced moment problems, Math. Methods and Models in the Applied Sciences 10 (2000), pp C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), pp I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy, volume 37, Second edition

119 S5 A. Moment Equations Consider the classical transport equation with a general (nonlinear) collision operator. t f + [E, f] c = Q[f, f] We do not need to know the details of Q, i.e. of the type of particle interactions considered. We only need to know conserved quantities and the production of a certain entropy, i.e. the increase of a concave functional. Conserved quantities: κ 1 (ξ),.., κ J (ξ) κq[f, f] dξ = 0, f, κ = (κ 0,.., κ J ) T, κ 0 = 1 113

120 The local entropy: G[f](x, t) = g(f(x, ξ, t)) dξ, g (f)q[f] dξ 0, f This implies for the whole system, because of the cyclicity of the classical commutator: t G[f] dx 0 Usually the logarithmic entropy g (f) = ln(f), G[f] = f(1 ln(f)) dξ is used. 114

121 Moments: Derive approximate equations for quantities conserved during collisions. M(x, t) = (M 0,.., M J ) T = (κ 0,.., κ J ) T f(x, ξ, t)dξ t M + N = 0, N(x, t) = κ(ξ)[e, f] c (x, ξ, t)dξ Closure problem: Express N in terms of the primary variables M, N = N(M, x, t). 115

122 S5 B. MAXIMUM ENTROPY CLOSURE G dxconserved during free flight. Q drives G dx towards its maximum (the equilibrium). Compute N by maximizing the entropy S loc locally in x under the constraint of given moments. Constrained minimization problem f(x, ξ, t) = φ M (ξ) : G(φ) := φ(1 ln(φ)) dξ, G(φ) max G(φ M ) = max{g(f) : κ(ξ)fdξ = M} 116

123 G(φ M ) = max{g(f) : κ(ξ)fdξ = M} This extremal problem can be solved by using Lagrange - multipliers. κφ M (ξ) dξ = M DG(φ M )(ψ) = ψg (φ M ) dξ = 0, ψ : κψ dξ = 0 φ M (ξ) = g 1 ( a: Lagrange multipliers J j=0 a j κ j (ξ)), κφ M dξ = M 117

124 The Lagrange multipliers a are functions of the moments M. For simple g this relation can be inverted explicitly. Closure: Write N in terms of a and a in terms of M. N(x, t) = κ(ξ)[e, (g ) 1 (a M (x, t) T κ(ξ))] c dξ, M(x, t) = κ(ξ)(g ) 1 (a M (x, t) T κ(ξ))dξ, Moment equation in entropy variables a: t κ(ξ)(g ) 1 (a T M (x, t)κ(ξ)) dξ + κ[e, (g ) 1 (a T M κ)] cdξ = 0 Physical entropy: Maxwellian. g (z) = ln(z), (g ) 1 = e z, gives the 118

125 S5 C. The Fluid Entropy Kinetic entropy structure translates into a conserved concave functional of the moments. a T M (x, t)κ(ξ) t(g ) 1 (a T M (x, t)κ(ξ)) dxξ+ a T M κ[e, (g ) 1 (a T M κ)] cdxξ = 0 g (φ M ) t φ M dxξ = 0, φ M (x, ξ, t) = (g ) 1 (a T M (x, t)κ(ξ)) t G F (M) = 0, G F (M) = g((g ) 1 (a T Mκ) dxdξ Logarithmic entropy: t G F (M) = 0, G F (M) = M 0 a T M M dxdξ 119

126 S5 D. Concavity of the Fluid Entropy G F G F (M) is a concave functional of the moments since it is obtained through an extremal problem. G F (M) = g(φ M ) dxξ κφ M dξ = M, κφ K dξ = K, κ[αφ M + (1 α)φ K ]dξ = αm + (1 α)k = G(φ(αM + (1 α)k)) G(αφ M + (1 α)φ K ) g(αφ M + (1 α)φ K dξ = αg(φ M ) + (1 α)g(φ K ) αg(φ M ) + (1 α)g(φ K ) dξ 120

127 SUMMARY Given the set of conserved moments κ, close the moment equations by (g ) 1 (a T κ) with g(z) a concave function (bounded from above). This produces a system of conservation laws for the moments M = κf dξ. For this system there is a concave functional G F (M) of the moments which is conserved. This is a nonlinear definition of hyperbolicity, and yields in general stability and well posedness. The function g(z) is chosen such that the collisions drive g towards its local maximum. 121

128 S6 MOMENT CLOSURES II: QUANTUM ENTROPY CLOSURE In the quantum mechanical case moment equations can be defined in analogy to the classical case via the Wigner function. The same sort of closure problem arises. The only information on collision mechanisms needed to derive the moment equations are the conservation properties. We add the information that collisions increase entropy and close the moment equations by maximizing entropy for a given set of moments. The quantum mechanical entropy is defined non-locally as the trace of a density matrix. Therefore, the closure will yield non-local equations. 122

129 These equations are quite complicated, but retain some hyperbolic nature in the sense that there is a conserved concave functional. This is an intermediate step between solving the quantum Boltzmann equation and quantum hydrodynamics. The resulting problem yields an identifiable positive definite density matrix. Local approximations (in the form of derivatives) can be made to give an extension of quantum hydrodynamics. (joint work with P. Degond and F. Mehats) 123

130 LITERATURE Degond P., Ringhofer C., Quantum moment hydrodynamics and the entropy principle, to appear JSP (2003) URL: P. Degond, C. Ringhofer, A note on quantum moment hydrodynamics and the entropy principle, to appear, C. R. Acad. Sci. Paris, Ser1 (2002), URL: Levermore C. D., Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83 (1996) Morozov V. G., Röpke G., Zubarev s method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes, Condensed Matter Physics 1 (1998)

131 Muller I., Ruggeri T., Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy, volume 37, Second edition Zubarev D. N., Morozov V. G., Röpke G., Statistical mechanics of nonequilibrium processes. Vol 1, basic concepts, kinetic theory, Akademie Verlag, Berlin,

132 S6 A. QUANTUM MOMENT EQUATIONS The quantum Boltzmann equation: t f + [E, f] w = Q w [f], (13) Commutator in Wigner representation [E, f] w = i W ([W 1 (E), W 1 (f)]), E(x, ξ) = ε(ξ) + V (x) Basic principle: Assume minimal amount of information on the quantum collision operator Q w. 1. Conservation: Assume conservation local in space. 126

133 κ(ξ): Vector of, locally in space, conserved quantities.conserved quantities: κ j (ξ)q w [w] dξ = 0, j = 0,.., J Yields the moment equations: t M + N = 0, M(x, t) = κf dξ, N(x, t) = Closure problem: Express N in terms of M. κ[e, f] w dξ (14) 127

134 S6 B. THE QUANTUM ENTROPY IN THE DENSITY MATRIX PICTURE Quantum entropy for density matrices: Define a convex functional G d on the space of density matrices via the trace of a convex operator function g. Definition of g(ρ): G d (ρ) = T r(g(ρ)), G d (ρ) : Entropy ρ(r, s) = j ψ j (r)λ j ψ j (s), g(ρ)(r, s) = j ψ j (r)g(λ j )ψ j (s), Entropy production of collision operator in the density matrix picture: Directional derivative of G d (ρ) in direction of Q d (ρ) nonnegative DG d (ρ)q d [ρ] 0, ρ, Q w [w] = W (Q d [W 1 (w)]) 128

135 The classical equivalent: G(f) = g(f) dxξ DG(f)δf = g (f)δf dxξ Theorem: Under the trace the linearization of G d becomes multiplication with g (ρ). G d (ρ) = T r(g(ρ)) DG d (ρ)δρ = T r(g (ρ) δρ) (15) Multiplying the quantum Boltzmann equation (13) by g (ρ) and building the trace: T r(g (ρ) t ρ) = DG d (ρ) t ρ = t G d (ρ) T r(g (ρ) [H, ρ]) = T r([g (r), ρ] H) = 0, T r(g (ρ)q d [ρ]) 0 Therefore: t G d (ρ) 0 holds. 129

136 S6 C. THE QUANTUM ENTROPY IN THE WIGNER PICTURE G w (f) = G d (W 1 f) = W g(w 1 f) dxξ, Q w [f] = W Q d [W 1 f] Linearization: DG w (f)δf = W g (W 1 f)δf dxξ t G w (f) = W g (W 1 f) t f dxξ Integrating the quantum Boltzmann equation (13) against W g (W 1 f) gives t G w (f) 0 130

137 S6 D. MAXIMUM ENTROPY CLOSURE Given the concave functional G w and a given moment vector M, find the flux vector N by maximizing the entropy G w among all Wigner functions with the given moments N(x, t) = κ[e, φ M ] w dξ, G w (φ M ) = max{g w (f) : κf dξ = M, x} Remark: Other than in the classical case, the entropy G w is defined globally in x. This implies that the φ M and therefore N will depend non-locally on M, although conservation properties were assumed local in x. 131

138 S6 E. THE MINIMIZATION PROBLEM G w (φ M ) = max{g w (f) : κ(ξ)f dξ = M(x, t) x} Remark: Existence of minimizer still an open question. Assuming existence, we can compute the minimizer using Lagrange multiplyers. Put constraints in a weak form w.r.t. x: G w (φ M ) = max{g w (f) : λ(x) κ(ξ)f dxξ = λ(x)m(x, t) λ} DG w (φ M )δf = 0, δf(x, ξ) : λ(x)κ(ξ)δf(x, ξ) dxξ = 0, λ 132

139 DG w (φ M )δf = 0, δf(x, ξ) : λ(x)κ(ξ)δf(x, ξ) dxξ = 0, λ From Theorem (15): DG w (φ M )δf = W (g (W 1 (f M )))δf dxξ W (g (W 1 (φ M ))) = λ(x)κ(ξ), φ M = W ((g ) 1 (W 1 (λκ))) for λ = λ M given by the constraints λ M (x) : κ(ξ)φ M (x, ξ)dξ = M(x) x 133

140 Maximum entropy closure t M(x, t) + N(x, t) = 0, N = κ(ξ)[e, φ M ] w dξ = 0 (16) φ M (x, ξ, t) = W ((g ) 1 (W 1 (λ T M κ))), κ(ξ)φ M (x, ξ, t) dxξ = M(x, t) Remark: For the logarithmic entropy g(z) = ln(z), and κ = 1, ξ, ξ 2 2m we obtain φ M (x, ξ, t) = W exp(w 1 λ M (x, t) T κ(ξ)) Compute the matrix exponential of the density matrix corresponding to a quadratic vector polynomial in ξ instead of V (x) + 2 ξ 2 2m Needs some approximation! 134

141 S6 F. THE FLUID ENTROPY N = κ(ξ)[e, φ M ] w dξ = 0, t M(x, t) + N(x, t) = 0, φ M (x, ξ, t) = W ((g ) 1 (W 1 (λ T M κ))) Define the fluid entropy of a moment vector M as G F (M) := G w (φ M ) By construction: λ M t M dx = t G F (M), λ M N dx = 0 Therefore the functional G F (M) is conserved. It remains to show that G F (M) is actually a concave (non-local, non-linear) functional of M. Follows from the extremal properties of φ M : 135

142 Lemma: G F (M) := G w (φ M ) and G w concave, G w (φ M ) = max{g w (f) : implies κ(ξ)f dξ = M(x, t) x} G F (αm + (1 α)m ) αg F (M) + (1 α)g F (M ) 136

143 SUMMARY The maximum entropy closure approach carries through in the quantum mechanical setting if entropy is defined in the usual sense of statistical mechanics. The constraints in the extremal problem have to be defined in a weak form. The relation between entropy variables (Lagrange multipliers) and moments involves an equation for a matrix exponential. Without any additional approximations this is not computationally as easy as in the classical case. 137

144 We require a minimal amount of information on the collision operator. The resulting moment equations automatically have an entropy property. Since the closure is given by W exp(w 1 λ T κ) spatial mass densities will automatically remain nonnegative. 138

145 S6 G. APPROXIMATE CLOSURES From (16) with logarithmic entropy: t M(x, t) + N(x, t) = 0, N = κ(ξ)[e, φ M ] w dξ = 0 (a) φ M (x, ξ, t) = W (exp(w 1 (λ T Mκ))), (17) (b) κ(ξ)φ M (x, ξ, t) dxξ = M(x, t) Given moment vector M, find entropy variables λ M such that (17)(b) holds with f M given by (17)(a). Involves solving an equation for a matrix exponential. Given φ M, compute N. 139

146 THE DIRECT APPROACH Pre-conditioned iteration, using the classical solution as a preconditioner. κ[exp((λ n + λ) T κ) exp(λ T n κ)](x, ξ) dξ = M(x) κw exp(w 1 (λ T n κ))(x, ξ) dξ, λ n+1 = λ n + ω λ ω: Relaxation parameter. Needs only the evaluation of W exp(w 1 (λ T n κ)). LHS can be evaluated exactly. The 140

147 NUMERICAL EVALUATION OF THE EXPONENTIAL 1. Choose set of orthonormal basis functions {ψ m }. 2. For given λ(x), diagonalize A and compute matrix exponential with A(k, m) = ψ k (r)op(λt κ)ψ m (r) dr, A = UΛU H, e A = Ue Λ U H W 1 (λ T κ)(r, s) = km ψ k (r)a(k, m)ψ m (s) exp(w 1 λ T κ)(r, s) = km ψ k (r)e A (k, m)ψ m(s) Possible simplification: Replace the exponential by a monotone rational function via Pade approximation. Replaces matrix diagonalization by inversion. 141

148 THE SEMI-CLASSICAL APPROACH Derive quantum hydrodynamic system, (i.e. O( 2 ) approximation of classical hydrodynamics) which has the right entropy. Write the moment equations (17) as closure of the QBE (13) with a matrix exponential M = t M + N = 0 κw exp(w 1 λ T κ) dξ, N = κ[e, W exp(w 1 λ T κ)] w dξ = 0 By construction we have the identities λ T t M(λ) dx = t W exp(w 1 λ T κ)(λ T κ 1) dxξ, λ T N(λ) dx = 0 142

149 λ T t M(λ) dx = t W exp(w 1 λ T κ)(λ T κ 1) dxξ, λ T N(λ) dx = 0 Expand everything (including the commutator) in terms of 2 : W exp(w 1 λ T κ) = exp(λ T κ) + 2 Ω(λ) +.., [E, f] w = [E, f] c + 2 [E, f] 1 +.., M = M 0 (λ) + 2 M 1 (λ) +.., N = N 0 (λ) + 2 N 1 (λ) +.. Since each of the terms in the expansion are independent of, the entropy conservation identities have to hold term by term. 143

150 For the semi-classical expansion we need the effective mass approximation and the moment vector p = ξ f(x, ξ, t) 3 f(x, p, t), W (ρ)(x, p) = ρ(x 2 y, x+ 2 y)eiy p dy E(x, p) = V (x)+ p 2 2m, [E, f] w = i ν=±1 νe(x+ iν 2 p, p iν 2 x)f(x, p) [E, f] c = p E x f x E p f, [E, f] 1 = 1 N 0 = κ[e, e λt κ ] c, N 1 = 24 3 x V : 3 f κ[e, Ω(λ)] c + κ[e, e λt κ ] 1 dp This system satisfies the entropy identities: λ T t (M M 1 )(λ) dx = t [e λt κ + 2 Ω(λ)](λ T κ 1) dxξ, p λ T (N 0 (λ) + 2 N 1 (λ)) dx = 0 144

151 APPROXIMATE COMPUTATION OF THE MATRIX EXPONENTIAL 3 W exp(w 1 λ T κ) = e λt κ + 2 Ω(λ) Asymptotics in the Bloch equation (similar to (12)) in S4). e λt κ + 2 Ω(λ) = u(x, p, β = 1) β u = {λ T κ, u} = 1 2 ν=±1 λ(x+ iν 2 p)κ(p iν 2 x)u(x, p, β), u(x, p, β = 0) = 1 Expand: β u = {λ T κ, u} = λ T κu 2 8 [κ 2 x λ : 2 p u + λ 2 p κ : 2 x u] 145

152 Yields rather complicated set of conservation laws (64 terms in N N 1 ). Satisfies entropy. Possible alternative to other quantum hydrodynamic approaches. Further work. 146

153 S7 GENERALIZATION OF CLASSICAL COLLISION OPERATORS Collision operators can be derived from simple considerations regarding conservation properties and symmetries. Given a certain set of conservation properties and symmetries, these considerations imply the shape of an entropy. In the classical case, this results in a logarithmic entropy and a Maxwellian equilibrium. This leads to a simple generalization in the quantum mechanical picture. Assuming local conservation properties for binary collisions, we derive a quantum collision operator which is local in space and time. 147

154 In analogy to the classical case, this leads to a quantum Maxwellian, which is the operator exponential of the inverse Wigner transform of a quadratic vector polynomial. This will lead in the maximum entropy closure framework to the quantum equivalent of classical hydrodynamics in a thermodynamic setting. (joint work with P. Degond) 148

155 LITERATURE R. Balian, From microphysics to macrophysics, Springer, C. Cercignani, The Boltzmann Equation and Its Applications, vol. 67 of Applied Mathematical Sciences, Springer-Verlag (1988). P. Degond, C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, submitted, J. Stat. Phys.(2002) preprint available at URL: P. Degond, C. Ringhofer, A note on binary quantum collision operators conserving mass momentum and energy, submitted, preprint available at URL: 149

156 P. Degond, C. Ringhofer, A note on quantum moment hydrodynamics and the entropy principle, to appear, Cmpt. Rendu Acad. Sci. Paris (2002), preprint available at URL: H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), pp

157 S7 A: BINARY COLLISIONS We consider collisions of particle pairs in a gas. These are assumed to be instantaneous and local. f(ξ, t): density for wave vector ξ and time t. (x dependence irrelevant here.) Evolution equation: t f = Q[f, f] (18) Q: collision operator, assumed to be bilinear, and locally conserves mass. 151

158 P(ξ, η, ξ, η): Probability (ξ, η) (ξ, η ). ξ, η and ξ, η : Pre- and post collision wave vectors. Formulation as a gain and a loss term: Q[f, f](ξ) = [P(ξ, η, ξ, η)f(ξ )f(η ) P(ξ, η, ξ, η )f(ξ)f(η)] dηη ξ (Implies mass conservation: t f(ξ, t) dξ = 0) Weak formulation of Q: φ(ξ)q[f, f](ξ) dξ = f(ξ)f(η)p(ξ, η, ξ, η )[φ(ξ ) φ(ξ)] dξξ ηη 152

159 S7 B: Additional properties of Q 1. Conservation: Assume that there are an additional set of quantities κ j (ξ), j = 1,.., J which are locally conserved during the collision process. κ j (ξ)q[f, f](ξ) dξ = 0, j = 0,.., J, f (19) We already have mass conservation. (19). So set κ 0 = 1 in Usually conserved quantities are momentum and energy J = 4, κ j = ξj ε(ξ), j = 1, 2, 3, κ 4 = ε(ξ) 153

160 2. Entropy: We will show that there exists a concave functional which is increased by the collision operator. G[f](t) := g(f(ξ, t)) dξ non-decreasing for f the solution of (18) This requires g (f(ξ))q[f, f](ξ) dξ 0, f, t g(f(ξ)) dξ 0 G: concave functional of f, i.e. G[sf 1 + (1 s)f 2 ] sg[f 1 ] + (1 s)g[f 2 ], f 1, f 2, s The form of g will follow from additional symmetry properties of the collision mechanism. 154

161 S7 C: SYMMETRIES IN P We assume two additional properties of the collision mechanism: 1. Pre- and post- collision states are equally likely (Micro-reversibility) 2. The particles are indistinguishable. (Follows from a mean field approximation which allows us to write collisions in the form (18).) Micro - reversibility: P(ξ, η, ξ, η ) = P(ξ, η, ξ, η) ξ and η indistinguishable: P(ξ, η, ξ, η ) = P(η, ξ, η, ξ ) 155

162 Symmetrized weak form of Q Symmetries in P allow us to write the weak version of Q[f, f] in symmetric form φ(ξ)q[f, f](ξ) dξ = f(ξ)f(η)p(ξ, η, ξ, η )[φ(ξ ) φ(ξ)] dξξ ηη = 1 4 [f(ξ)f(η) f(ξ )f(η )]P(..)[φ(ξ )+φ(η ) φ(ξ) φ(η)] dξξ ηη (20) The form (20) is invariant against the permutations of ξ, η, ξ, η. 156

163 S7 D: THE FORM OF THE ENTROPY The symmetrized form (20) implies that a sign can be given to DG[f]Q[f, f] if the entropy production rate g satisfies the algebraic relation g (ab) = g (a) + g (b) This, together with the necessary sign of the entropy production yields g (z) = ln(z). Entropy production: DG[f]Q[f, f] = g (f(ξ))q[f, f](ξ) dξ 0, f Weak symmetrized form 1 4 [f(ξ )f(η ) f(ξ)f(η)]p(..) [g (f(ξ)) + g (f(η)) g (f(ξ )) g (f(η ))] dξξ ηη 0 157

164 g (a) + g (b) = g (ab) 1 4 [f(ξ )f(η ) f(ξ)f(η)]p(..) [g (f(ξ)f(η)) g (f(ξ )f(η ))] dξξ ηη 0 Can be achieved if, in addition g is monotonically decreasing. In summary this gives g (z) = ln(z), G[f](t) = f[1 ln(f)](ξ, t) dξ g(z) = z(1 ln(z)) is a concave function, which implies that G[f] is a concave functional. 158

165 S7 E: EQUILIBRIUM AND THE MAXWELLIAN The concave functional G[f] should increase until reaching its maximum. This maximum is achieved when holds. DG(f)Q[f, f] = 0 Form (19) we know that we achieve a relative maximum for g (f) = a T κ or f = exp(a T κ) Using the conservation properties of Q, we show that Q actually vanishes there. 159

166 THE FORM OF THE COLLISION OPERATOR WITH CONSERVATION PROPERTIES The symmetrized form (20) of Q, together with the conservation of the quantities κ j, j = 1,.., J in (19) implies that the probability P has to be concentrated on surfaces of equal energy, momentum, etc. in R 12. κ j (ξ)q[f, f](ξ) dξ = 1 [f(ξ)f(η) f(ξ )f(η )]P(..)[κ j (ξ )+κ j (η ) κ j (ξ) κ j (η)] dξξ ηη = 0 4 for all f and j = 1,.., J P(ξ, η, ξ, η ) = K(ξ, η, ξ, η ) J j=1 δ(κ j (ξ )+κ j (η ) κ j (ξ) κ j (η)) 160

167 Choose f eq as Conservation ln(f eq (ξ)) = a T κ = J j=0 a j κ j (ξ), P(ξ, η, ξ, η ) = K(ξ, η, ξ, η ) J j=1 δ(κ j (ξ )+κ j (η ) κ j (ξ) κ j (η)) P(ξ, η, ξ, η )[ln(f eq (ξ ) + ln(f eq (η ) ln(f eq (ξ) ln(f eq (η)] = 0 S[f] reaches local maximum at f = f eq and Q[f eq, f eq ] = 0 holds for f eq = the Maxwellian f eq = exp[ J j=0 a j κ j (ξ)] 161

168 S7 F: TRANSPORT f(x, ξ, t): density for position x and wave vector ξ Hamiltonian System: Energy: E(x, ξ) = ε(ξ) + V (x) conserved t f + [E, f] c = 0, [E, f] c = ξ E x f x E ξ f Conservative form: [E, f] c = x (f ξ E) ξ (f x E) = [f, E] c Cyclicity of the commutator: g[e, f] c dxξ = [g, E] c f dxξ Implies conservation of energy and entropy φ(e)[e, f] c dxξ = 0, φ(f)[e, f] c dxξ = 0, 162

169 The Boltzmann equation: t f + [E, f] c = Q[f, f] Entropy: S(f) = f(1 ln(f)) dxdξ t S[f] dx = ln(f) t f dxξ 0 163

170 S 7 E. BINARY QUANTUM COLLISIONS We assume a general collision operator which conserves mass in the sense of traces, i.e. T r(w 1 Q[f]) = 0 = Q[f](x, ξ) dxdξ This yields a binary quantum collision operator in Wigner representation: Q[f](z) = K[f](z, z 1, z, z 1) K[f](z, z 1, z, z 1 ) dz 1z z 1, z = (x, ξ) (21) Here K is some arbitrary (generally nonlinear) operator. From the form (21) and entropy and conservation properties we can deduce the quantum equivalent of the classical binary collision operator. 164

171 The weak and symmetrized formulation As in the classical case, we write Q in symmetric form assuming indistinguishable particles. φ(z)q[f](z) dz = K[f](z, z 1, z, z 1 )(φ(z ) φ(z)) dzz 1 z z 1 indistinguishable: K[f](z, z 1, z, z 1 ) = K[f](z 1, z, z 1, z ) φ(z)q[f](z) dz = 1 4 (K[f](z, z 1, z, z 1 ) K[f](z, z 1, z, z 1))(φ+φ 1 φ φ 1 ) dzz 1z z 1 165

172 Entropy production We assume that the collision operator Q increases the quantum entropy, given by G(f) = T r[ρ (1 ln ρ)], W (ρ) = f The entropy is now globally defined as a trace. However, we still have DG(f)Q[f] = T r[ ln ρ W 1 Q[f]] = W (ln(w 1 f))q[f] dxξ 0 or, 1 4 for (K[f](z, z 1, z, z 1 ) K[f](z, z 1, z, z 1))(φ+φ 1 φ φ 1 ) dzz 1z z 1 0 φ = Ln(f) = W (ln W 1 f) 166

173 The conversion operator Define: Ln(f) = ln(a(f)), A(f) = exp(ln(f)) = exp(w ln W 1 (f)) A maps Wigner functions corresponding to positive density definite matrices into positive functions. 1 4 (K[f](z, z 1, z, z 1 ) K[f](z, z 1, z, z 1)) (ln(a(f)a(f 1 )) ln(a(f )A(f 1 ))) dzz 1z z 1 0 This implies that the entropy is increased whenever K[f](z, z 1, z, z 1 ) a monotone function of A(f)A(f 1) 167

174 Local conservation properties If we conserve a number of quantities κ j, j = 1,.., J during the collision process, this implies, as in the classical case, the shape of the kernel. K[f](z, z 1, z, z 1 ) = A(f)A(f 1)S(z, z 1, z, z 1 ) J j=1 δ(κ j +κ 1j κ j κ 1j ) Kernel given by Ln(f) = J κ j j=0 168

175 This yields a global kernel without any degrees of freedom. The Maxwellian is obtained by requiring local collisions. K[f](z, z 1, z, z 1 ) = A(f)A(f 1)S(z, z 1, z, z 1 ) δ(x x 1 )δ(x x )δ(x x 1 ) J j=1 δ(κ j + κ 1j κ j κ 1j ) Locally conserved quantities: f = Exp[ J j=0 a j (x)κ j (ξ)], Exp(f) := W (exp(w 1 f)) 169

176 SUMMARY The form of binary collisions can be (up to scattering cross sections) determined solely through conservation and entropy principles. This carries over to the quantum mechanical case by just reformulating entropy in a quantum setting via operator traces. This gives a non-local quantum collision operator whose non-locality is defined by the conversion operator A. 170

177 S8 RELATIVE ENTROPIES In semiconductors the dominant collision mechanism is not electron- electron scattering but scattering with the phonon background. This scattering mechanism only conserves mass (charge) but not momentum and energy since they can be exchanged with the lattice. The description of scattering with a background can be almost completely deduced from the conservation properties and the assumption of entropy production. This yields the concept of entropy production relative to a given equilibrium state, adding a linear term to the concave functional and thereby shifting the maximum. 171

178 The quantum version of relative entropy operators can be formulated in the same way as for binary collisions. This gives a natural way to generalize the semiconductor Boltzmann equation to the quantum mechanical setting by assuming a quantized phonon background. The resulting moment equations have to be obtained from an asymptotic analysis in a diffusion scaling, i.e. a version of the Chapman Enskog expansion. The resulting moment closures are diffusion equations (quantum versions of drift - diffusion or energy transport). The entropy property guarantees the well - posedness, i.e. the right sign of the diffusion coefficients. (joint work with J.P. Bourgade, P. Degond, F. Mehats) 172

179 LITERATURE C. Cercignani, The Boltzmann equation and its applications, Applied Math. Sciences, Springer, P. Degond, F. Mehats, C. Ringhofer, On a quantum energy transport model, Manuscript, A. Kersch, W. Morokoff, Transport simulation in microelectronics, Birkhäuser, Basel, C. Ringhofer: Dissipative discretization methods for approximations to the Boltzmann equation, Math. Mod. Meth. in Appl. Sci. 11, pp (2001). C. Ringhofer: An Entropy - Based Finite Difference Method for the Energy Transport System,Math. Mod. Meth. in Appl. Sci. 11, pp (2001). 173

180 S8 A. Collisions with a Background Gas Consider binary collisions between two different species. Assume an equilibrium distribution for one and compute the trace (integral) with respect to the background variables. Start with a two particle density function F (ξ, η, t) ξ, η momentum vectors of species 1 and 2. The spatially homogeneous case t F = Q[F ] The two species collision term written as gain - loss term (automatically conserves mass). Q[F ](ξ, η) = P(ξ, η, ξ, η)f (ξ, η ) P(ξ, η, ξ, η )F (ξ, η) dξξ ηη 174

181 Q[F ](ξ, η) = P(ξ, η, ξ, η)f (ξ, η ) P(ξ, η, ξ, η )F (ξ, η) dξξ ηη Weak formulation: U(ξ, η)q[f ](ξ, η) dξη = P(ξ, η, ξ, η )F (ξ, η)[u(ξ, η ) U(ξ, η)] dξηξ η 175

182 S8 B. Symmetries We still assume micro-reversibility but (obviously) not indistinguishability. P(ξ, η, ξ, η ) = P(ξ, η, ξ, η) Implies in weak formulation U(ξ, η)q[f ](ξ, η) dξη = (22) 1 2 P(ξ, η, ξ, η )[F (ξ, η) F (ξ, η )][U(ξ, η) U(ξ, η )] dξξ ηη 176

183 CONSERVED QUANTITIES Assume that the sum of a given set of quantities (i.e. the total energy) is conserved in the collision process. κ, σ properties of species 1 and 2 whose sum is conserved during collisions. P(ξ, η, ξ, η ) = K(ξ, η, ξ, η ) J j=1 δ(κ j (ξ)+σ j (η) κ j (ξ ) σ j (η )) Spatially homogeneous Boltzmann equation: t F = Q[F ] (23) 177

184 S8 C. Kernel and Entropy of the two Species Operator Because the two particle collision operator is linear any concave functional of F can serve as entropy. In the same way, any function of the sum of conserved quantities is in the kernel of Q. The form (22) implies g (F )Q[F ](ξ, η)dξη 0, g 0 Kernel of Q: Any function of the total conserved quantities (from (23)). F (ξ, η) = F 0 (κ(ξ) + σ(η)) Q[F 0 ] = 0 178

185 THE REDUCED OPERATOR Molecular chaos Ansatz: Assume ξ and η approximately statistically independent. (This implies that there are many particles and many collisions.) F (ξ, η, t) = f(ξ, t)w(η, t), f dξ = 1, w dη = 1 Yields two separate, coupled Boltzmann equations for f and w. t f(ξ, t) = Q w [f](ξ) = Q[f(ξ)w(η)] dη t w(η, t) = Q f [w](η) = Q[f(ξ)w(η)] dξ 179

186 In the case of binary collisions between same species f = w would hold and the two equations would be the same. Instead we make an Ansatz for background gas w(η, t) = w 0 (η). φ(ξ)q w [f](ξ) dξ = 1 2 P(ξ, η, ξ, η )[f(ξ)w(η) f(ξ )w(η )][φ(ξ) φ(ξ )] dξξ ηη 180

187 S8 D. THE ENTROPY AND THE KERNEL OF THE REDUCED OPERATOR The kernel and entropy of the reduced operator φ(ξ)q w [f](ξ) dξ = (24) is determined by choosing one particular kernel element of the two particle operator. This element has to be a Maxwellian (i.e. an exponential function of conserved quantities.) Write Q in terms of an element of the kernel U(ξ, η)q[f ](ξ, η) dξη = 1 2 F 0 (ξ, η)p(ξ, η, ξ, η )[ F F 0 (ξ, η) F F 0 (ξ, η )][U(ξ, η) U(ξ, η )] dξξ ηη 181

188 Choose one kernel element. Set F 0 (ξ, η) = exp(b T (κ(ξ) + σ(η))) = f 0 (ξ)w 0 (η) Reduced operator: 1 2 Q w0 [f] = f 0 (ξ)w 0 (η)p(ξ, η, ξ, η )[ f f 0 (ξ) f f 0 (ξ )][φ(ξ) φ(ξ )] dξξ ηη The kernel of Q w0 [f] is now given by any multiple of f

189 Any concave function of f f can be used as entropy for 0 Q w0. g ( f f 0 )Q w [f] 0 for all g 0 with f 0 = exp[ b T κ] Remark: Q w [f] only conserves mass but none of the other quantities. 183

190 S8 E. Transport We now consider the spatially inhomogeneous equation with the relative entropy collision operator. The entropy principle, in the classical case, implies that w 0 can be a function of energy only. t f(x, ξ, t) + [E, f] c = Q w [f], E(x, ξ) = V (x) + ε(ξ) [E, f] c should conserve the relative entropy g ( f f 0 )[E, f] c dxξ = 0 This implies f 0 = exp[ βe] 184

191 S8 F. A QUANTUM VERSION OF THE RELATIVE ENTROPY COLLISION OPERATOR The above derivation translates immediately to an operator conserving a given set of quantities. In the quantum case the choice of entropy functional is restricted to the logarithm. U(z, z 1 )Q[F ](z, z 1 ) dzz 1 = 1 2 P(z, z 1, z, z 1 )[A(F )(z, z 1) A(F )(z, z 1 )][U(z, z 1) U(z, z 1 )] dzz z 1 z 1 A(F ) = exp(ln(f )) = exp[w (ln W 1 F )] z := (x, ξ), z 1 := (x 1, η) 185

192 Reduced operator: φ(z)q w [f](z) dz = 1 2 P(z, z 1, z, z 1 )[A(f)(z)A(w)(z 1) A(f)(z )A(w)(z 1 )][φ(z) φ(z )] dzz From the conservation properties we obtain the necessary shape of w for the kernel element F 0 : A(f 0 )A(w 0 ) = exp[ln(f 0 )(z) + Ln(w 0 )(z 1 )] = ψ(κ(ξ) + σ(η)) f 0 = Exp(b T (x)κ(ξ)), w 0 = Exp(b T (x)σ(η)) 186

193 This implies any classical concave function of A(f) A(f 0 ) produces classical entropy, i.e. g ( A(f) A(f 0 ) )Q w 0 [f] 0, g 0 We need however an operator trace. For the logarithmic entropy g (z) = ln(z) we obtain [Ln(f 0 ) Ln(f)]Q w0 [f] 0, T r[(w 1 (b T κ) + ln(ρ)) W 1 Q[f]] 0, f = W (ρ) DG(ρ)W 1 Q[f] 0, G(ρ) = T r[ρ (1 W 1 (b T κ) ln(ρ))] 187

194 In order for the quantum commutator in the transport picture to conserve the quantum entropy we need T r((w 1 (b T κ) + ln(ρ)) [E, ρ] w ) = 0, E = ε + V which implies b T κ = βε + βv, β = const Therefore the equilibrium has to be given by a Maxwellian with a constant temperature! 188

195 SUMMARY Using the conversion operator A, which maps positive operators into positive functions, we define a quantum version of collisions with a background gas which dissipates entropy relative to an equilibrium of the background. Because of the trace property, the form of the relative entropy is restricted to the logarithm. If the entropy is conserved by the commutator the background equilibrium has to be a Maxwellian with a constant temperature. 189

196 S8 G. A PHENOMENOLOGICAL QUANTUM PHONON OPERATOR We assume the phonon - background to be in equilibrium. The phonons are quantized. The phonon background can only gain or lose a fixed amount ω of energy in the collision. This corresponds to creation or destruction of a phonon. The conservation properties are local in space. 190

197 The equilibrium: w 0 (η) = Exp( βσ), f 0 (ξ) = Exp( βε), A(f 0 ) = exp( βε) The scattering cross - section: φ(z)q[f](z) dz = (25) (A(f)e βε A(f )e βε )e βε βσ P(z, z p, z, z p )(φ(z) φ(z ))dzz p z z p with z = (x, ξ), z p = (x p, ξ p ) P(z, z p, z, z p) = δ(x x p )δ(x x )δ(x x p)k(z, z p, z, z p) The kernel: 4 j=1 δ(κ j + σ j κ j σ j ) A(f) = e a(x) βε, f = Exp(a(x) βε) 191

198 S8 H. RELATIVE ENTROPY CLOSURES AND DIFFUSION EQUATIONS Collision operators which dissipate an entropy relative to a steady state without a flux lead to diffusion equations. These are obtained through the Chapman - Enskog, or Hilbert expansion. The relative entropy guarantees the welposedness, i.e. the sign of the diffusion coefficients. Boltzmann: t f + [E, f] = Q[f], Conservation properties, entropy and kernel: κq[f] dξ = 0, DG[f]Q(f) 0 f, Q[(g ) 1 (a T κ)] = 0 a (26) 192

199 We consider the function as well as the operator framework, i.e. g can be a function or an operator function. and [.,.] can be the classical or the quantum commutator. Usual moment closure: t M + κ[e, f M ] dξ = 0, Q[f M ] = 0, M := κf dξ We consider the case when κ[e, f M ] dξ = 0 holds. (For instance when f M is an even function of ξ. Diffusion scaling: α t f + [E, f] = 1 Q[f], α << 1 α 193

200 CHAPMAN ENSKOG WITH RELATIVE ENTROPY Define nonlinear projection onto the kernel of Q P (f) = (g ) 1 (a T κ), κp (f) dξ = κf dξ Consider small deviation from equilibrium: f = f 0 + αf 1, f 0 = P (f), κf 1 = 0 Rescaled equation: α t (f 0 + αf 1 ) + [E, f 0 + αf 1 ] = 1 α Q[f 0 + αf 1 ], 194

201 Limiting problem: κ( t f 0 + [E, f 1 ]) dξ = 0, [E, f 0 ] = DQ[f 0 ]f 1 (27) f 0 = (g ) 1 (a T κ) This is the same set of equations as obtained from Chapman-Enskog in the case κ[e, f M ] dξ = 0. (27) is a diffusion equation. Well-posed sign of diffusion coefficients. To use (27) Q has to be chosen sufficiently simple to be able to invert DQ[f 0 ]f 1. If the entropy relation (26) holds then (27) produces entropy. 195

202 Entropy conservation of the free streaming operator: g (f 0 + αf 1 )[E, f 0 + αf 1 ] dξ = 0, Dg (f 0 )f 1 [E, f 0 ] + g (f 0 )[E, f 1 ] dξ = 0 This implies in (27) g (f 0 ) t f 0 dξ = Dg (f 0 )f 1 DQ[f 0 ]f 1 dξ To show: Dg (f 0 )f 1 DQ[f 0 ]f 1 dξ 0 g (f 0 +αf 1 )Q[f 0 +αf 1 ] dξ 0 Dg (f 0 )f 1 DQ[f 0 ]f 1 dξ 0 196

203 BGK OPERATORS Simplest operator which allows inversion of DQ[f 0 ]. Q[f] = P (f) f, P (f) = (g ) 1 (a T κ), κp (f) dξ = κf dξ Fredholm alternative: DP (f 0 )f 1 f 1 = g κg dξ = 0, κf 1 dξ = 0 Entropy production of the BGK operator: P (f) can be written as solution of the extremal problem G(P (f)) = max{g(φ) : κ(φ f) dξ = 0} This (and the concavity of G) implies g (f)q[f] dξ 0 197

204 EXAMPLES Quantum Drift - Diffusion t M 0 + x M 1 = 0, M 1 = x V M 0 x p 2 Exp(a(x) βε) 198

205 S9 EFFECTIVE POTENTIALS Introduce quantum mechanics in complicated realistic transport pictures by modifying the role of the potential. The quantum transport equation is replaced by a classical transport equation with an effective potential. What is the effective force acting on particles, and how does it act? Has to produce a non-local effect of potential barriers on particles, i.e. barriers seem smaller and their effect is felt from a distance. 199

206 Quantum potentials can be introduced either through a many body formulation or a thermodynamic formulation. Both versions allow for the introduction of many body effects via a particle - particle - particle - mesh approach. The entropy principle suggests a natural way to define effective potentials as the simplest approximation to the phenomenological phonon operator. (joint work with D. Ferry, D. Vasileska) 200

207 LITERATURE R. Akis, S. Milicic, D. K. Ferry,D. Vasileska: An effective potential method for including quantum effects into the simulation of ultra-short and ultra-narrow channel MOSFETs, Proceedings of the 4th International Conference on Modeling and Simulation of Microsystems, Hilton Head Island, SC, March 19-21, pp , C. Herring, E. Vogt: Transport and Deformation-Potential Theory for Many-Valley Semiconductors with Anisotropic Scattering, Phys. Rev., Vol. 101, 944, S. E. Laux: On particle-mesh coupling in Monte Carlo semiconductor device simultion, IEEE Trans. CAD Integr. Circ. Syst., Vol. 15, 1266,

208 C. Ringhofer, C. Gardner, D. Vasileska, Effective Potentials and Quantum Fluid Models: A Thermodynamic Approach, to appear Inter. J. on High Speed Electronics and Systems (2003). L. Shifren, C. Ringhofer, D.Ferry: A Wigner function based quantum ensemble Monte Carlo study of a resonant tunneling diode, to appear, IEEE Electron Device Letters, URL: chris, R. E. Wyatt: Quantum Wavepacket Dynamics with Trajectories: Wavefunction Synthesis along Quantum Paths, Chem. Phys. Lett. 313, ,

209 S9 A. CORRECTIONS TO THE BOLTZMANN EQUATION The classical transport picture, i.e. a classical Hamiltonian system together with an integral operator with positive kernel, can be simulated efficiently via particle and Monte Carlo methods. t f + [E, f] c = Q[f], [E, f] c = 1 ( ξe x f ξ f x E) (28) f n = f(x, ξ, t n ) given Free streaming step: Operator splitting t f + [E, f] c = 0, f(t = t n ) = f n, f n+1/2 := f(t n+1 ) Collision step: t f = Q[f], f(t = t n ) = f n+1/2, f n+1 = f(t n+1 203

210 This can be done very efficiently by approximating f(x, p, t) by a superposition of δ functions. f(x, ξ, t) = n δ(x x n (t))δ(ξ ξ n (t)) Free streaming step: Move particles along characteristics. This step is exact. x n = 1 ξe(x, ξ), ξ n = 1 xe(x, ξ) Collision step: Q[f] = P(ξ, ξ)f(x, ξ, t) P(ξ, ξ )f(x, ξ, t) dξ Probabilistic interpretation: dp[ξ ξ in (t, t + t)] = tp(ξ, ξ)dt Convergent of order N 1/2. 204

211 The quantum transport picture: t f + [E, f] w = Q[f] (29) Problem: δ(x x n (t))δ(ξ ξ n (t)) not a solution of the Wigner equation. t f + [E, f] w = 0 does not possess characteristics. Basic principle of effective potential approaches: Replace the quantum transport equation (29) by the classical transport picture (28) with a modified potential V Q, the so called quantum potential, and solve t f + [ 2 ξ 2 2m + V Q, f] c = Q[f] 205

212 S9 B. EFFECTIVE POTENTIAL APPROACHES Effective potentials in macroscopic descriptions (from Section S4): The Bohm potential: V Q = V 2 x n 2m n In the context of particle discretizations the derivatives of the density n have to be computed from an ensemble of particles. The smoothed potential: V Q = ΓV, Γ = 1 0 γ2 exp[ 2 (1 γ 2 ) 8T x ]dγ 206

213 S9 C. GAUSSIAN SMOOTHING Feynman: derived from equilibrium consideration. Alternative: particles in a many body framework The many body problem an the BBGKY hierarchy The many body Wigner function for N particles F (x 1,.., x N, ξ 1,.., ξ N ). t F +[ 2 ξ 2 2m +V, F ] w = 0, ξ := (ξ 1,.., ξ N ), x := (x 1,.., x N ) The Coulomb potential: binary interaction between particles. V(x) = 1 N 1 N j 1 j=2 k=1 V C (x j x k ), V C (x) = δ(x) 207

214 MEAN FIELD THEORY Integrate out all but one variable, and define f j = F dx j ξ j, x j := x 1,.., x j 1, x j+1,.., x N, ξ j := ξ 1,.., ξ j 1, ξ j+1,.., ξ N t f j + [ 2 ξ 1 2 2m, f j] c + [V, F ] w dx j ξ j = 0 Close the hierarchy by expressing [V, F ] w dx j ξ j in terms of f j. Statistical independence (molecular chaos): Assume the particles are independently distributed (N >> 1) F (x, ξ) = f 1 (x 1, ξ 1 ),.., f N (x N, ξ N ) 208

215 Two choices: 1. The particles are identical (f 1 =.., f N ) yields the Poisson equation [V, F ] c dx j ξ j = [V, f j ] c (x j, ξ j ), V = f j dξ All equations are identical! 2. The particles are not identical but given by δ functions f j (x j, ξ j ) = δ(x j y j )δ(ξ j η j ) [V, F ] c dx j 1 ξ j = [V j, f j ] c (x j, ξ j ), V j (x j ) = k j V C (x j y k ) The difference is given by either solving the Poisson equation as a differential equation or by convoluting the Green s function with the δ functions. 209

216 V = f j dξ, V j (x j ) = k j V C (x j y k ) The Green s function approach requires O(N 2 ) operations and is not feasible. P 3 M Methods (Particle,Particle,Particle,Mesh) The two can be combined by evaluating the Green s function only for close particles and solving the Poisson equation for far away particles. 210

217 Assume the other particles are not given by δ functions, but by Wigner functions corresponding to Gaussian wave packets. f j (x, ξ) = W [ψ j (r)ψ j (s) ], ψ j (r) = exp[ c k η j 2 +ik (r y j )] dk 2 f j (x, ξ) = exp[ c ξ η j 2 2 x y j 2 ] 2c V j (x) = k j Γ G V C (x y k ) (30) Γ G : Convolution of V C with the Gaussian x 2 2c. Replace the P 3 M potential with a Gaussian smoothed potential and replace [V, f j ] w by [Γ G V, f j ] c ; i.e. move the particles along classical trajectories. 211

218 S9 D QUANTUM POTENTIALS IN THE ENTROPY FRAMEWORK Assume the modified quantum operator from Section S8. Replace by a spatially local operator which has the same kernel but the classical entropy relative to the quantum equilibrium. Obtain a classical Boltzmann equation with a modified general Hamiltonian. Generalization of existing quantum potential approaches to the entropy framework. 212

219 The collision operator for background scattering (from (25)) φ(ξ)q[f](x, ξ, t) dξ = (31) ( A(f) e βε βv A(f ) e βε βv )S(x, ξ, ξ )(φ(ξ) φ(ξ ))dξ, ξ ) with A(f) the conversion operator A(f) = exp[ln(f)]. Equilibrium and entropy: G[f] = A(f eq ) = e βε βv, V, f = Exp( βε βv ), Replace (31) by f[1 Ln(f) βε βv ] dxξ, DG[f]Q[f] 0, f φ(ξ)q[f](x, ξ, t) dξ = f ( A 1 (e βε βv ) f A 1 (e βε βv ) )S(x, ξ, ξ )(φ(ξ) φ(ξ ))dξ, ξ ) 213

220 Has the same equilibrium but dissipates the relative entropy G[f] = f[1 ln(f)+ln(exp( βε βv ))] dxξ, DG[f]Q[f] 0, f This relative entropy has to be conserved by the modified classical Hamiltonian. Replace the Hamiltonian 2 ξ 2 2m + V (x) by a modified Hamiltonian E q(x, ξ). (ln(f) ln(exp( βε βv )))[E q, f] c dxξ = 0 Therefore the modified energy E q has to be of the form E q (x, ξ) = ln(exp( βε βv )) If we approximate ln(exp( βε βv )) by exp( βε βv Q (x)), we obtain the classical Boltzmann equation with a quantum corrected potential appearing only in the free - streaming operator. 214

221 S9 E THERMAL EQUILIBRIUM AND THE BLOCH EQUATION (REVISITED) The effective Hamiltonian E q is given solely through the expression of the thermal equilibrium Exp( βε βv ) in some suitable setting. This is done in the same way as in the case of thermodynamic quantumhydrodynamics via the semigroup produced by ε + V in the Wigner function framework. The resulting effective Hamiltonian E q corresponds to a quantum potential which is dependent on the Wave vector as well. Exp( βe), E = ε(ξ) + V (x) 215

222 The Bloch equation for Exp( βe) Exp( βe) = v(x, ξ, γ = 1), γ v = β{e, v} w, v(x, ξ, γ = 0) = 1 The symbol {E, v} w denotes the quantum mechanical anti-commutator in the Wigner picture, i.e. {E, v} w = 1 2 W [W 1 E W 1 v + W 1 v W 1 E] {E, v} w = 1 2 ν=±1 E(x + iν 2 ξ, ξ iν 2 x)v(x, ξ, γ) To obtain E q we take logarithms in the Bloch equation. Setting Exp( βε) = exp( βe q ) and v = e βu gives E q (x, ξ) = u(x, ξ, γ = 1), γ u = e βu {E, e βu } w, u(x, ξ, γ = 0) = 0 γ u = 2 2m ξ m (β xu β 2 x u 2 ) + e βu {V, e βu } w 216

223 Two forms of asymptotics: Semiclassical ( 0) and the Born approximation (inverting the Laplacian). Semiclassical approximation: E q = E + 2 8m [β xv β2 3 V 2 2 β 2 3m 2 x V : ξ 2 ] Born Approximation: E q (x, ξ) = exp[ β 2 x 8m ]sinh(z) V (x), z z = iβ 2 ξ x 2m E q (x, ξ) [1 β2 4 (ξ x ) 2 24m 2 ] exp[ β 2 x ]V (x) (32) 8m The last approximation can be performed (for small) even for the discontinuous barrier potential. 217

224 Compare (32) to the Gaussian smoothing (30). size c of the electron now depends on its energy. The The effective transport equation t f + 1 x (f ξ E q ) 1 ξ (f x E q ) = Q[f] φ(x, ξ)q g [f](x, ξ) dξ = 1 2 K(x, ξ, x, ξ )e βeq [fe βeq f e βe q][φ φ ]dxξx ξ This form increases the semi-classical entropy G(f) = f(1 ln(f) βe q ) dxξ 218

225 gate SiO 2 source drain N + channel N + p-type substrate Oxide thickness = 1.2 nm Channel length = 25 nm Source/Drain length = 50 nm Channel width = 0.5 µm Junction depth = 30 nm Substrate thickness = 64 nm Substrate doping: N A =10 19 cm -3 Doping of the source-drain regions: N D = cm

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