The microscopic origin of the macroscopic dielectric permittivity of crystals

Transcription

1 The microscopic origin of the macroscopic dielectric permittivity of crystals Gabriel STOLTZ (CERMICS, Ecole des Ponts & MICMAC team, INRIA Rocquencourt) Work in collaboration with E. Cancès and M. Lewin Workshop on Quantum and Atomistic Modeling of Materials Defects Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

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) IPAM, October / 35

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) IPAM, October / 35

4 Outline Some background material Description of perfect crystals Crystals with defects: static picture Static dielectric response of crystals Linear response to an effective perturbation Definition of the macroscopic dielectric permittivity Time evolution of defects in crystals Response to an effective potential Static polarization in some adiabatic limit Well-posedness of the nonlinear Hartree dynamics Frequency dependent macroscopic dielectric permittivity [CS12] E. Cancès and G. Stoltz, to appear in Ann. I. H. Poincare-An. (arxiv ) [CLS11] E. Cancès, M. Lewin and G. Stoltz, in Numerical Analysis of Multiscale Computations, B. Engquist, O. Runborg, Y.-H. R. Tsai. (Eds.), Lect. Notes Comput. Sci. Eng. 82 (2011) [CL10] E. Cancès and M. Lewin, Arch. Rational Mech. Anal 197(1) (2010) Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

5 Some background material Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

6 Some elements on trace-class operators Compact self-adjoint operator A = + i=1 The operator A is called trace-class (A S 1 ) if ρ A (x) = + i=1 λ i φ i (x) 2 belongs to L 1 (R 3 ) and Tr(A) := + λ i = + i=1 i=1 λ i φ i φ i with λ i 0 + i=1 ˆ e i A e i = λ i <. Its density R 3 ρ A A is Hilbert-Schmidt (A S 2 ) if A A S 1, i.e. λ i 2 <. If A is i 1 self-adjoint, its integral kernel is in L 2 (R 3 R 3 ) A(x,y) = λ i φ i (x)φ i (y). i 1 Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

7 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 i (x j )) 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) IPAM, October / 35

8 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) IPAM, October / 35

9 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) IPAM, October / 35

10 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 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) IPAM, October / 35

11 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) IPAM, October / 35

12 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,q ε 1,q Insulator / Conductor semi-conductor N = 2 N = 3 Γ q Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

13 Defects in crystals (1) 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 [HLS05] C. Hainzl, M. Lewin, and E. Séré, Commun. Math. Phys., 2005 (and subsequent works) [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) IPAM, October / 35

14 Defects in crystals (2) Definition of the embedding energy Tr 0 ((H 0 per ε F )Q) := Tr( H 0 per ε F 1/2 (Q ++ Q ) H 0 per ε F 1/2 ) [CL, Theorem 1] Let ν such that (ν 1 ) L 2 (R 3 )+C. Then, there exists at least one minimizer Q ν,εf, and all the minimizers share the same density ρ ν,εf. In addition, Q ν,εf is solution to the self-consistent equation Q ν,εf = 1 (,εf )( H 0 per +(ρ ν,εf ν) 1) 1 (,εf ]( H 0 per ) +δ, where δ is a finite-rank self-adjoint operator on L 2 (R 3 ) such that 0 δ 1 and Ran(δ) Ker ( H 0 per +(ρ ν,εf ν) 1 ε F ). When ν is sufficiently small, δ = 0 and the minimizer is unique. Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

15 Static dielectric reponse of crystals: effective perturbations Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

16 Expansion of the time-independent response Perturbation by a sufficiently small effective potential V L 2 (R 3 )+C : Q V = 1 (,ε 0 F )( H 0 per +V ) ( ) 1 (,ε 0 H F ] 0 per = 1 ( (z H 0 2iπ per V ) 1 ( ) ) z H 0 1 per dz C = Q 1,V + +Q n,v + Q n+1,v The linear response in V reads Q 1,V = 1 ( ) z H 0 1V ( ) 2iπ per z H 0 1 per dz C C ε 0 F σ(h 0 per) Σ + N Σ N+1 Λ Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

17 Expansion of the time-independent response (2) The higher order contributions and the remainder are respectively given by Q k,v = 1 ( ) [ z H 0 1 2iπ per V ( ] z Hper) 0 1 k dz and C Q n+1,v = 1 ( z H 0 2iπ per V ) [ 1 V ( z H 0 ) ] 1 n+1 per dz. C [CL10, Lemma 3] For V sufficiently small in L 2 (R 3 )+C, the operators Q k,v and Q k,v are in Q and Tr 0 (Q k,v ) = 0. For k 6, it holds Q k,v, Q k,v S 1 and Tr(Q k,v ) = 0. Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

18 Independent particle polarizability [CL10, Proposition 1] If V L 2 (R 3 )+C, the operator Q 1,V is in Q and Tr 0 (Q 1,V ) = 0. If V L 1 (R 3 ), then Q 1,V is trace-class and Tr(Q 1,V ) = 0. The independent particle polarizability operator χ 0 defined as χ 0 V = ρ Q1,V is continuous L 1 (R 3 ) L 1 (R 3 ) and L 2 (R 3 )+C L 2 (R 3 ) C Potential generated by a charge defect: V = v c ( ) = 1 Linear reponse at the density level: L = χ 0 v c ( ) = ρ Q1,vc( ) This linear response is a fundamental tool to prove that Q ν S 1 and ρ ν := ρ Qν L 1 (R 3 ) in general. Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

19 Static dielectric reponse of crystals: macroscopic dielectric permittivity Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

20 Linear response in the nonlinear Hartree model (1) Screening of the bare defect charge by the response of the Fermi sea Effective perturbation v c (ν ρ ν ) ρ ν = L(ν ρ ν )+r 2,ν, r 2,ν = ρ Q2,vc(ν ρν) so that ν ρ ν = (1+L) 1 ν (1+L) 1 r 2,ν [CL10, Proposition 2] The operator L is a bounded, self-adjoint and nonnegative operator on C; hence 1+L is invertible. Homogenization limit: The nonlinear terms disappear in some homogenized limit where the charge is spread out in space ν η (x) = η 3 ν(ηx) Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

21 Linear response in the nonlinear Hartree model (2) Consider the rescaled potential generated by the screened defect W η ν(x) = η 1 v c (ν η ρ νη ) ( η 1 x ) When L = 0, the potential is W η ν = v c (ν) [CL10, Theorem 3] There exists a 3 3 symmetric matrix ε M 1 such that, for all ν L 1 (R 3 ) L 2 (R 3 ), the rescaled potential Wν η weakly converges in C as η 0 to the unique solution W ν of the equation ) div (ε M W ν = 4πν. The matrix ε M (ω) can be expressed using the Bloch decomposition It gives the electronic contribution to the dielectric permittivity Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

22 Time evolution of defects in crystals: effective perturbations Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

23 The time-dependent Hartree dynamics Finite system described by the density matrix γ(t), von Neumann equation i dγ(t) ] = [Hγ(t) 0 dt,γ(t), Hγ 0 = 1 2 +V nuc +v c (ρ γ ) When a perturbation v(t) is added, the dynamics is modified as i dγ(t) ] = [Hγ(t) 0 dt +v(t),γ(t), Formal thermodynamic limit: state γ(t) = γ 0 per +Q(t) and dynamics i dγ dt = [ H v γ,γ ], H v γ(t) = H 0 per +v c (ρ Q (t) ν(t)) [Chadam76] J. M. Chadam, The time-dependent Hartree-Fock equations with Coulomb two-body interaction, Commun. Math. Phys. 46 (1976) [Arnold96] A. Arnold, Self-consistent relaxation-time models in quantum mechanics, Commun. Part. Diff. Eq. 21(3-4) (1996) Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

24 Defects in a time-dependent setting: the dynamics Classical formulation: nonlinear dynamics i dq(t) dt = [ 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 Mild formulation for the nonlinear dynamics 0 U 0 (t s)[v(s),γ 0 per +Q(s)]U 0 (t s) ds Replace v(s) by v c (ρ Q(s) ν(s)) in the above formula Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

25 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) IPAM, October / 35

26 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) IPAM, October / 35

27 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) IPAM, October / 35

28 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 ) Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

29 Recovering the static polarizability in some adiabatic limit The static polarizability corresponds to formally setting ω = 0 ) Ẽ static (h) = vc 1/2 ( ρ Q static 1,v 1/2 c (h) on L 2 (R 3 ), with Q1,V static = 1 (z H 2iπ per) 0 1 V(z Hper) 0 1 dz C Adiabatic limit: long times t/α, slowly evolving perturbation v(αt) Q α 1,v(t) = i ˆ t/α [CS12, Proposition 10] U 0 ( t α s ) [v(αs),γ 0 per ] U0 ( t α s ) ds. Define (Ẽ 0 f)(t) = Ẽ static (f(t)). Then, for any function f S(R R 3 ), lim Ẽ α f = Ẽ 0 f in S (R R 3 ). α 0 Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

30 Time evolution of defects in crystals: nonlinear dynamics Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

31 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) IPAM, October / 35

32 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 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) IPAM, October / 35

33 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) IPAM, October / 35

34 Perspectives Gabriel Stoltz (ENPC/INRIA) IPAM, October / 35

35 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) IPAM, October / 35