Evolution en temps des défauts dans les cristaux

Transcription

1 Evolution en temps des défauts dans les cristaux Gabriel STOLTZ (CERMICS, Ecole des Ponts & MICMAC team, INRIA Rocquencourt) Travail en collaboration avec E. Cancès Séminaire de Mathématiques Appliquées, Collège de France Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

2 Microscopic origin of macroscopic dielectric properties (1) In a dielectric material, the presence of an electric field causes the nuclear and electronic charges to slightly separate, inducing a local electric dipole This generates an induced response inside the material (reorganization of the electronic density), screening the applied field Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

3 Microscopic origin of macroscopic dielectric properties (2) Dielectric material: can polarize in presence of external fields density electric field external ν D, div D = 4πν polarization δρ P, div P = 4πδρ total ρ E, div E = 4πρ D = E+P Constitutive equation: ε M = 3 3 symmetric real matrix with ε M 1 D = ε M E P = (ε M 1)E = (1 ε 1 M )D Time-dependent fields: the response of the material is not instantaneous, but given by a convolution with some response function. With E(t) = W(t) where W(t) is the macroscopic potential, ) div (ε M (ω) Ŵ(ω) = 4π ν(ω) Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

4 Outline Some background material Description of perfect crystals Crystals with defects: static picture Time evolution of defects in crystals: effective perturbations Response to an effective potential Linear response and definition of the polarization Static polarization in some adiabatic limit Time evolution of defects in crystals: nonlinear dynamics Well-posedness of the nonlinear Hartree dynamics Definition of the macroscopic dielectric permittivity [CS12] E. Cancès and G. Stoltz, A mathematical formulation of the random phase approximation for crystals, accepted in Ann. I. H. Poincare-An. (arxiv ) Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

5 Some background material Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

6 Density operators for a finite system of N electrons in R 3 Bounded, self-adjoint operator on L 2 (R 3 ) such that 0 γ 1 and Tr(γ) = N. In some orthonormal basis of L 2 (R 3 ), γ = + i=1 n i φ i φ i, 0 n i 1, + i=1 n i = N For the Slater determinant ψ(x 1,...,x N ) = (N!) 1/2 det(φ j (x i )) 1 i,j N, N γ ψ = φ i φ i Electronic density ρ γ (x) = + i=1 i=1 Kinetic energy T(γ) = 1 2 Tr( γ ) = 1 2 ˆ n i φ i (x) 2 with ρ γ 0 and ρ γ = N. R 3 + i=1 n i φ i 2 L 2 (R 3 ) Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

7 The Hartree model for finite systems Hartree energy E Hartree ρ nuc (γ) = Tr ( 1 2 γ )+ 1 2 D(ρ γ ρ nuc,ρ γ ρ nuc ) where ˆ D(f,g) = R 3 ˆ f(x)g(x ) R 3 x x ˆ f(k)ĝ(k) dx dx = 4π R 3 k 2 dk is the classical Coulomb interaction, defined for f,g L 6/5 (R 3 ), but which can be extended to { C = f S (R 3 ) f L 1 loc (R3 ), 1 f( ) } L 2 (R 3 ) Variational formulation { } inf (γ), γ S(L2 (R 3 )), 0 γ 1, Tr(γ) = N, Tr( γ) < E Hartree ρ nuc More general models of density functional theory: correction term E xc (γ) [Sol91] J.-P. Solovej, Invent. Math., 1991 Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

8 Euler-Lagrange equations for the Hartree model Nonlinear eigenvalue problem, ε F Lagrange multiplier of Tr(γ) = N γ 0 = + i=1 n i φ i φ i, ρ 0 (x) = H 0 φ i = ε i φ i, φ i,φ j = δ ij, 1 if ε i < ε F n i = [0,1] if ε i = ε F 0 if ε i > ε F H 0 = 1 2 +V0, V 0 = 4π(ρ nuc ρ 0 ). When ε N < ε N+1 (gap): + i=1 + i=1 γ 0 = 1 (,εf ](H 0 ), H 0 = 1 2 +V0, n i φ i (x) 2, n i = N, V 0 = 4π(ρ nuc ρ 0 ), Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

9 The Hartree model for crystals (1) Thermodynamic limit, periodic nuclear density ρ nuc per, lattice R (az) 3 ( ) 2π 3 with unit cell Γ, reciprocal lattice R a Z with unit cell Γ ˆ Bloch-Floquet transform: unitary L 2 (R 3 ) L 2 per(γ)dq Γ f q (x) = f(x +R)e iq (x+r) = (2π)3/2 Γ f(q +K)e ik x R R K R Any operator commuting with the spatial translations τ R (R R) can be decomposed as (Af) q = A q f q, and σ(a) = q Γ σ(a q ) Bloch matrices: A K,K (q) = e K,A q e K L 2 per (Γ), e K (x) = Γ 1/2 e ik x F(Av)(q +K) = K R A K,K (q)fv(q +K ) [CLL01] I. Catto, C. Le Bris, and P.-L. Lions, Ann. I. H. Poincaré-An, 2001 [CDL08] E. Cancès, A. Deleurence and M. Lewin, Commun. Math. Phys., 2008 Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

10 The Hartree model for crystals (2) Nonlinear eigenvalue problem γper 0 = 1 (,εf ](Hper), 0 ρ 0 per = ρ γ 0 per, Hper 0 = 1 2 +V0 per, ˆ ˆ Vper 0 = 4π(ρ nuc per ρ 0 per), ρ 0 per = More explicit expressions using the Bloch decomposition Γ Γ ρ nuc per = N ( H 0 per )q = 1 q 2 + iq V0 per = ε n,q u n,q u n,q n=1 ( γ 0 per )q + = 1 {εn,q ε F } u n,q u n,q n=1 Fermi level obtained from N = 1 Γ + n=1 {q Γ ε n,q ε F } Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

11 The Hartree model for crystals (3) The spectrum of the periodic Hamiltonian is composed of bands σ(h) = n 1[ Σ n,σ + ] n, Σ n = min ε n,q, Σ + n = maxε n,q q Γ q Γ Assume in the sequel that g = Σ N+1 Σ+ N > 0 (insulator) ε F ε F ε 2 (ξ) ε 1 (ξ) Insulator / Conductor semi-conductor N = 2 N = 3 Γ ξ Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

12 Defects in crystals Nuclear charge defect ρ nuc per +ν, expected ground state γ = γ 0 per +Q ν A thermodynamic limit shows that Q ν can be thought of as some defect state embedded in the periodic medium { ( Q ν = argmin Tr 0 H 0 per Q ) ˆ ρ Q (ν 1 )+ 1 } Q Q R 3 2 D(ρ Q,ρ Q ) γper 0 Q 1 γper 0 where, defining Q = γperqγ 0 per 0 and Q ++ = (1 γper)q(1 γ 0 per), 0 } Q = {Q = Q, (1 ) 1/2 Q S 2, (1 ) 1/2 Q ±± (1 ) 1/2 S 1 Generalized trace Tr 0 (Q) = Tr(Q ++ )+Tr(Q ) Density ρ Q L 2 (R 3 ) C [CDL08] E. Cancès, A. Deleurence and M. Lewin, Commun. Math. Phys., 2008 [CL10] E. Cancès and M. Lewin, Arch. Rational Mech. Anal., 2010 Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

13 Time evolution of defects in crystals: effective perturbations Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

14 Defects in a time-dependent setting Formal thermodynamic limit: state γ(t) = γ 0 per +Q(t), Hamiltonian H v γ(t) = H 0 per +v c (ρ Q (t) ν(t)), and dynamics (von Neumann equation) i dγ dt = [Hv γ,γ] Classical formulation: nonlinear dynamics i dq(t) dt v c ( ) = 1 = [ H 0 per +v c (ρ Q(t) ν(t)),γ 0 per +Q(t) ] Denote U 0 (t) = e ith0 per the free evolution. Mild formulation for an effective potential v(t) Q(t) = U 0 (t)q 0 U 0 (t) i ˆ t 0 U 0 (t s)[v(s),γ 0 per +Q(s)]U 0 (t s) ds Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

15 Well-posedness of the mild formulation If initially Q(0) Q, the Banach space allowing to describe local defects in crystals, does Q(t) Q? [CS12, Proposition 1] The integral equation has a unique solution in C 0 (R +,Q) for Q 0 Q and v = v c (ρ) with ρ L 1 loc (R +,L 2 (R 3 ) C). In addition, Tr 0 (Q(t)) = Tr 0 (Q 0 ), and, if γ 0 per Q 0 1 γ 0 per, then γ 0 per Q(t) 1 γ 0 per. This result is based on a series of technical results boundedness of the potential: v L 1 loc (R +,L (R 3 )) 1 stability of time evolution: β Q Q U 0 (t)qu 0 (t) Q β Q Q commutator estimates with γper: 0 i[v,γper] 0 Q C com v C commutator estimates in Q: i[v c ( ),Q] Q C com,q L 2 C Q Q Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

16 Dyson expansion and linear response Response at all orders (formally): Q(t) = U 0 (t)q 0 U 0 (t) + Q 1,v (t)= i Q n,v (t) = i ˆ t ˆ0 t 0 + n=1 Q n,v (t) U 0 (t s) [ v(s),γ 0 per +U 0 (s)q 0 U 0 (s) ] U 0 (t s) ds, U 0 (t s)[v(s),q n 1,v (s)]u 0 (t s) ds for n 2 Obtained by plugging the formal decomposition into the integral equation [CS12, Proposition 5] Under the previous assumptions, Q n,v C 0 (R +,Q) with Tr 0 (Q n,v (t)) = 0, Q n,v (t) Q β 1+ Q0 Q n! ( ˆ t n C ρ(s) L 2 C ds). 0 The formal expansion therefore converges in Q, uniformly on any compact subset of R +, to the unique solution in C 0 (R +,Q) of the integral equation. Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

17 Definition of the polarization (1) Aim: Justify the Adler-Wiser formula for the polarization matrix Damped linear response: standard linear response as η 0 Q η 1,v (t) = iˆ t U 0 (t s) [ v(s),γ 0 per] U0 (t s) e η(t s) ds { L polarization operator χ η 1 0 : (R,C ) Cb 0(R,L2 (R 3 ) C) v ρ Q η 1,v linear response operator E η = vc 1/2 χ 0 vc 1/2 acting on L 1 (R,L 2 (R 3 )) ˆ f 2, E η f 1 L 2 (L 2 ) = F t f 2 (ω), E η (ω)f t f 1 (ω) L 2 (R 3 ) dω R Bloch decomposition: for a.e. (ω,q) R Γ and any K R, F t,x (E η f)(ω,q +K) = K R E η K,K (ω,q)f t,x f(ω,q +K ) [Adler62] S. L. Adler, Phys. Rev., 1962 [Wiser63] N. Wiser, Phys. Rev., 1963 Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

18 Definition of the polarization (2) [CS12, Proposition 7] The Bloch matrices of the damped linear response operator E η read E η K,K (ω,q) = 1 Γ (q) Γ q +K q +K Tη K,K (ω,q), where the continuous functions T η K,K are uniformly bounded: T η K,K (ω,q) = Γ u m,q,e ik x u n,q+q L 2 per u n,q+q,e ik x u m,q L 2 per ε n,q+q ε m,q ω iη dq (the sum is over 1 n N < m and 1 m N < n) The Bloch matrices of the standard linear response are recovered as η 0, the convergence being in S (R R 3 ) Static polarizability (ω = 0) recovered in some adiabatic limit Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

19 Time evolution of defects in crystals: nonlinear dynamics Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

20 Time-dependent Hartree dynamics for defects Well-posedness of the mild formulation For ν L 1 loc (R +,L 2 (R 3 )) W 1,1 loc (R +,C), and γper 0 Q 0 1 γper 0 with Q 0 Q, the dynamics ˆ t [ ] Q(t)=U 0 (t)q 0 U 0 (t) i U 0 (t s) v c (ρ Q(s) ν(s)),γper+q(s) 0 U 0 (t s) ds 0 has a unique solution in C 0 (R +,Q). For all t 0, Tr 0 (Q(t)) = Tr 0 (Q 0 ) and γ 0 per Q(t) 1 γ 0 per. Idea of the proof: (i) short time existence and uniqueness by a fixed-point argument; (ii) extension to all times by controlling the energy E(t,Q) = Tr 0 (H 0 perq) D(ρ Q,ν(t))+ 1 2 D(ρ Q,ρ Q ) Classical solution well posed under stronger assumptions on Q 0,ν Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

21 Macroscopic dielectric permittivity (1) Starting from Q 0 = 0, the nonlinear dynamics can be rewritten as Q(t) = Q 1,vc(ρ Q ν)(t)+ Q 2,vc(ρ Q ν)(t) In terms of electronic densities: [(1+L)(ν ρ Q )](t) = ν(t) r 2 (t) Properties of the operator L = χ 0 v c : ρ Q1,vc( ) For any 0 < Ω < g, the operator L is a non-negative, bounded, self-adjoint operator on the Hilbert space { H Ω = L 2 (R,C) supp(f t,x ) [ Ω,Ω] R 3}, endowed with the scalar product ˆ Ω ˆ F t,x 2(ω,k)F t,x 1(ω,k) 2, 1 L 2 (C) = 4π Ω R 3 k 2 dωdk. Hence, 1+L, considered as an operator on H Ω, is invertible. Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

22 Macroscopic dielectric permittivity (2) Linearization: given ν H Ω, find ρ ν such that (1+L)(ν ρ ν ) = ν Homogenization limit: spread the charge as ν η (t,x) = η 3 ν(t,ηx) and consider the rescaled potential W η ν(t,x) = η 1 v c (ν η ρ νη ) ( t,η 1 x ) When L = 0, the potential is W η ν = v c (ν) [CS12, Proposition 14] The rescaled potential Wν η converges weakly in H Ω to the unique solution W ν in H Ω to the equation ( ) div ε M (ω) [F t W ν ](ω, ) = 4π[F t ν](ω, ) where ε M (ω) (for ω ( g,g)) is a smooth mapping with values in the space of symmetric 3 3 matrices, and satisfying ε M (ω) 1. The matrix ε M (ω) can be expressed using the Bloch decomposition Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

23 Perspectives Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24

24 Perspectives and open issues Metallic systems (no gap: many estimates break down) Longtime behavior of the defect Influence of electric and magnetic fields (rather than a local perturbation as was the case here) Interaction of electronic defects with phonons (lattice vibrations) GW methods (the polarization matrix enters the definition of the self-energy) Gabriel Stoltz (ENPC/INRIA) Collège de France, Juin / 24