Un modèle non linéaire pour la description d un défaut dans un cristal quantique

Un modèle non linéaire pour la description d un défaut dans un cristal quantique Mathieu LEWIN Mathieu.Lewin@math.cnrs.fr (CNRS & University of Cergy-Pontoise) collaboration avec É. Cancès & A. Deleurence (ENPC, Marne-La-Vallée) Collège de France, 8 Janvier 2010 Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 1 / 18

Introduction Describing the electronic state of a crystal in presence of a defect is a major issue in Quantum Chemistry and Physics. electronics and optics: doped semi-conductors, quantum dots,... material science: storage of radioactive waste,... Germanium crystal doped with arsenic or indium. Structural damage created by displacement of zirconium, silicon and oxygen atoms in crystalline zircon (a candidate for storing nuclear waste for over 250 000 years) in the presence of a heavy nucleus. Farnan et al, Nature 445 (2007), 190. Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 2 / 18

Introduction System of interest: infinitely many classical fixed nuclei (perturb. of periodic structure); infinitely many quantum electrons. Difficulties: infinitely many electrons variable = operator of infinite rank; nonlinear model lack of compactness at infinity; long range of Coulomb potential divergences; two scales. References: [CDL1] Cancès, Deleurence & M.L. Comm. Math. Phys. 281 (2008). [CDL2] Cancès, Deleurence & M.L. J. Phys.: Condens. Matter 20 (2008). [CL] Cancès & M.L. Arch. Rat. Mech. Anal. (2010). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 3 / 18

Supercell method Main method at present: supercell method. The supercell model spurious interactions between the defect and its periodic images; inaccuracies for charged defects. Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 4 / 18

Methodology Finite system model for N electrons E per (N) = ēn + o(n) E def (N) = E per (N) + c + o(1) thermodynamic limit N Infinite periodic system ground state γ 0 per Infinite perturbed system ground state γ = γ 0 per + Q [CLL] Catto, Le Bris & Lions. Ann. I. H. Poincaré 18 (2001). [HLSo] Hainzl, M.L. & Solovej. Comm. Pure Applied Math. 76 (2007). [CDL1] Cancès, Deleurence & M.L. Comm. Math. Phys. 281 (2008). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 5 / 18

Reduced Hartree-Fock model for a finite system Electrons (=fermions). Spin neglected for simplicity. Hartree-Fock state: a density matrix γ B(L 2 (R 3 )) s.t. 0 γ 1. tr(γ) = ρ γ = nb of particles in the system. Density of charge: ρ γ (x) = γ(x,x) s.t. tr(γv ) = ρ γ V. N [ ] Example: γ = ϕ i ϕ i Ψ(x 1,...,x N ) = (N!) 1/2 det(ϕ i (x j )). i=1 Reduced Hartree-Fock energy: Ext. elec. field V ext = µ 1 x, where µ = charge distrib. of nuclei. ( ) ˆ E µ rhf (γ) := tr 2 γ + V ext ρ γ + 1 ˆˆ ρ γ (x)ρ γ (y) dx dy R 3 2 R 6 x y Rmk. Not a quantitative model: exchange-correlation missing. Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 6 / 18

Reduced Hartree-Fock model for a finite system ( ) ˆˆ E µ rhf (γ) := tr 2 γ ρ γ (x)µ(y) dx dy + 1 ˆˆ ρ γ (x)ρ γ (y) dx dy R 6 x y 2 R 6 x y Minimizers under constraint tr(γ) = N (, e.g., when N µ) γ = χ (,εf ) (H γ ) + δ, H γ = 2 + V γ, V γ = 4π(ρ γ µ) (mean-field / Fock operator) (Self-Consistent Field) where 0 δ χ {εf }(H γ ) and ε F =Lagrange multiplier=fermi level. Rmk. If δ 0, may be rewritten as ε F 0 2 + [Sol] Solovej. Appendix of Invent. Math. 104 (1991). σ (H γ)!! NX ϕ i 2 µ 1 ϕ i = λ i ϕ i, x i=1 for i = 1...N. Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 7 / 18

Periodic Fermi sea I Discrete subgroup L R 3, compact fundamental domain Γ. Nuclei: µ = µ per = z L χ( z) Γ where χ 0, χ N \ {0} support in Γ. Thermodynamic limit: retain nuclei in a box C L = [ L/2;L/2) 3. Electrons in the whole space [CLL] or in C L, with chosen BC [CDL1]. Impose neutrality: N = C L µ per. As L, γ L CV to the unique L-periodic sol. of γ 0 per = χ (,ε F ) ( H 0 per ), H 0 per = 2 + V 0 per, V 0 per = 4π( ρ γ 0 per µ per ), with ˆ Γ ) (ρ γ µ 0per per = 0. Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 8 / 18

Periodic Fermi sea II Bloch-Floquet transform: H 0 per commutes with all translations {τ z} z L. Hper 0 Hper(ξ), 0 Hper(ξ) 0 = 1 Γ 2 ( i + ξ)2 + Vper 0 on L 2 per(γ) Brillouin zone: Γ = {ξ R 3 : ξ x π, x Γ}. ε F ε F ε 2 (ξ) ε 1 (ξ) Insulator / semi-conductor Conductor N = 2 N = 3 Assumption: insulator in the following. Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 9 / 18 Γ ξ

Introducing a defect Take µ = µ per + ν where ν= local defect. Goal: construct solution of γ = χ (,εf ) (H γ ) + δ, H γ = /2 + V γ, V γ = 4π ( ρ γ µ per ν ) with ε F gap. Meaning of ρ γ? Correct functional setting? Idea [CIL,HLSS]: write everything relatively to γ 0 per. ( ) Q = γ γ 0 per = χ (,εf ) (H γ ) χ (,εf ) ( H 0 per ) + δ, H γ = H 0 per + W Q, W Q = 4π ( ρ Q ν ). Fermi sea γ ε F σ(h γ) electrons [CIL] Chaix, Iracane + Chaix, Iracane & Lions. J. Phys. B 22 (1989). [HLSS] Hainzl, M.L., Séré & Solovej. J. Phys. A 76 (2007). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 10 / 18

Functional analysis setting Schatten classes: Let A= self-adjoint compact operator, A = i λ i ϕ i ϕ i (λ i 0). A Hilbert-Schmidt (A S 2 ) when i λ i 2 <. A trace-class (A S 1 ) when i λ i <. Then tr(a) = i λ i = k e k,ae k for any orth. basis {e k }. Density ρ A (x) = i λ i ϕ i (x) 2 L 1 (R 3 ). The sum k e k,ae k can CV for one basis and not for other ones! Generalization [HLSé]: A is γper 0 -trace-class (A S0 1 ) when A S 2 and A = γper 0 Aγ0 per, A++ = (γper 0 ) A(γper 0 ) S 1. Define tr 0 (A) = tr(a + A ++ ). ( A A = A + ) A + A ++ f (x)g(y) Notation: D(f,g) = R 6 x y dx dy = 4π C := {f : D(f,f ) < }. R 3 ˆf (k)ĝ(k) k 2 dk and [HLSé] Hainzl, M.L. & Séré. Commun. Math. Phys. 257 (2005). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 11 / 18

Existence of bound states Theorem (Existence of bound states [CDL1]) For all ε F in the gap and ν such that ν 1 L 2 (R 3 ) + C, there exists at least one solution Q to Eq. ( ), such that (1 ) 1/2 Q S 2, (1 ) 1/2 Q ±± (1 ) 1/2 S 1 and ρ Q L 2 (R 3 ) C. The associated density ρ Q is unique, hence so is the mean-field op. H γ. Only δ can vary among solutions. These states are the ones obtained in the thermo. limit. If ε F is fixed in the gap and ν is small enough, then ε F / σ(h Q ), hence δ 0 and Q is unique. Furthermore one has tr 0 (Q) = 0. Variational proof [CIL,HLSS]: formally E µper+ν rhf (γ) E µper+ν rhf (γper) 0 = tr ( HperQ 0 ) ˆˆ ρ Q (x)ν(y) dx dy R 6 x y + 1 ˆˆ ρ Q (x)ρ Q (y) dx dy := F ν (Q). 2 R 6 x y Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 12 / 18

Variational interpretation F ν (Q) ε F tr 0 Q = tr 0 (H 0 per ε F)Q D(ν,ρ Q ) + 1 2 D(ρ Q,ρ Q ). Note 0 γ 1 γ 0 per Q (γ 0 per) Q 2 Q ++ Q tr 0 (H 0 per ε F )Q = tr H 0 per ε F (Q ++ Q ) tr H 0 per ε F Q 2 0. Theorem (Variational interpretation [CDL1]) For all ε F in the gap, F ε F tr 0 is strongly continuous and convex on K := {Q = Q : γ 0 per Q (γ 0 per), (1 ) 1/2 Q S 2, (1 ) 1/2 Q ±± (1 ) 1/2 S 1 }. Minimizers Q K are exactly the solutions of ( ). Any such Q also minimizes the pb ( ) E ν (q) = inf R K, tr 0 (R)=q Fν (R) where q = tr 0 (Q). One has E ν (q) < E ν (q q ) + E 0 (q ) for all q q and any minimizing sequence for ( ) is precompact in Q. Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 13 / 18

Properties of bound states Theorem (Properties of bound states [CL]) Fix ε F gap. Assume ν L 1 (R 3 ) L 2 (R 3 ) is small enough with R 3 ν 0. Then Q is not trace-class. If L is anisotropic, then ρ Q is not in L 1 (R 3 ). Reason: ρ Q = L(ρ Q ν) + o(ρ Q ν). L can be explicitely computed. f L 1 (R 3 ) lim k 0 L(f )(σ k ) = f (0)σ T Lσ for σ S 2. L 0 is given by the Adler-Wiser formula [Adl,Wis] + 2 (k x )u n,q,u n,q L 2per (Γ) k T Lk = 8π Γ N n=1 n =N+1 Γ ( ǫn,q ǫ n,q ) 3 dq, where ǫ n,q,u n,q = eigenvalues/vectors of Bloch Hamiltonian H 0 per (q). Always L 0; for anisotropic materials, L ci 3. [CL] Cancès & M.L. Arch. Rat. Mech. Anal. (2010). [Adl] Adler, Phys. Rev. 126 (1962). [Wis] Wiser, Phys. Rev. 129 (1963). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 14 / 18

Dielectric permittivity Theorem (Homogenization limit [CL]) Fix ε F gap and take ν η = η 3 ν( η). Let V η = (ρ Qη ν η ) 1. Then W η (x) = η 1 V η (x/η) CV weakly in C as η 0 to the unique sol. of div(ε M W ) = 4πν where ε M is a 3 3 symmetric matrix I 3, the (electronic contribution to the) macroscopic dielectric tensor of the perfect crystal. Explanation: ρ Q = L(ρ Q ν) + o(ρ Q ν) ν ρ Q = (1 + L) 1 ν +. ε M obtained from strong limit of U η (1 + L) 1 U η, U η = dilation unitary operator. Calculation: ε := vc 1/2 (1 + L)vc 1/2 where v c (ν) := ν 1. Bloch matrix { ε K,K (q)} K,K (2πZ) 3, q [ π, π)3. lim η 0 ε 0,0 (ση) = 1 + σ T Lσ, lim η 0 ε 0,K (ση) = β K σ, lim η 0 ε K,K (ση) = C K,K. Then [BarRes86]: ε M = 1 + L K,K (2πZ) 3 \{0} β K [C 1 ] K,K β K [BarRes86] Baroni and Resta, Phys. Rev. B 33 (1986). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 15 / 18

Numerics I Two different scales: use of Bloch-Floquet transform to discretize periodic pb; use of localized Wannier basis for the locally perturbed pb. Ex: Maximally Localized Wannier functions [MV]. 1D simulation [CDL2]: Yukawa potential, Z = 2. 1.5 1 0.5 Modulus of the Maximally Localized Wannier Orbitals [MV] for the first two bands (red and green), the 3rd and 4th bands (blue and orange). 0 0 5 10 15 20 [MV] Marzari, Vanderbilt. Phys. Rev. B 56 (1997). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 16 / 18

Numerics II Comp. of Q: relaxed constraint algorithms [Can]. ν = δ 0.3 2δ 0 3.0 0.4 2.8 0.2 2.6 0.0 2.4-0.2 2.2-0.4 2.0-0.6 1.8 1.6-0.8-6 -4-2 0 2 4-1.0-10 -8-6 -4-2 0 2 4 6 8 10 Polarization of the Fermi sea in the presence of defect, calculated with 28 MLWFs. As good as supercell calculation in a basis set of size 1000. Left: ρ γ 0 per and ρ γ. Right: ρ γ ρ γ 0 per. [Can] Cancès, Le Bris Int. J. Quantum Chem. 79 (2000). Cancès, J. Chem. Phys. 114 (2001). Kudin, Scuseria, Cancès, J. Chem. Phys. 116 (2002). Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 17 / 18

Conclusion Possible to describe an infinite quantum system perturbed locally: use perfect system as reference; variational (as soon as the perturbation is localized enough); divergences; well-behaved associated numerical methods. Perspectives: simulations on real systems; theory of realistic models; time-dependent setting; relaxation of nuclei; correlation? metals? Mathieu LEWIN (CNRS / Cergy) Cristaux quantiques avec défauts Collège de France, 08/01/10 18 / 18